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abstract: 'We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to the secular determinant. In this way the problem reduces to finding the roots of a polynomial function of just one variable, the parameter in the Hamiltonian operator. As illustrative examples we consider a particle in a one-dimensional box with a polynomial potential, the periodic Mathieu equation, the Stark effect in a polar rigid rotor and in a polar symmetric top.'
address:
- ' Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, Mexico'
- ' INIFTA, División Química Teórica, Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina'
author:
- Paolo Amore and Francisco M Fernández
title: 'Exceptional points of the eigenvalues of parameter-dependent Hamiltonian operators'
---
Introduction
============
In many applications of quantum mechanics to physical problems the Hamiltonian operator $H(\lambda )$ depends on a parameter $\lambda $. For example, in the case of atoms and molecules in electric or magnetic fields the parameter $\lambda $ is related to the intensity of the external field so that $H(0)$ is the Hamiltonian operator for the isolated atom or molecule. One of the approximate methods for calculating the energies $E(\lambda )$ of such problems is perturbation theory that is based on the Taylor expansion of the energies $E(\lambda )$ and eigenfunctions $\psi
(\lambda )$ about $\lambda =0$. The resulting series may be divergent or they may have finite convergent radii[@RS78; @S82; @F01] (and references therein).
When $H(\lambda )$ is an analytic function of $\lambda $ (the typical case being $H(\lambda )=H_{0}+\lambda H^{\prime }$) the convergence of the perturbation series radii are determined by exceptional points (EPS) in the complex $\lambda $ plane where two or more eigenvalues coalesce. This coalescence is different from degeneracy in that the corresponding eigenvectors become linear dependent at an EP. For this reason there has been great interest in the accurate calculation of EPS. Among the models analyzed are the Mathieu equation[@BC69; @H81; @HG81; @FAC87; @V98; @SX99; @ZRKBS12; @FG14], a polar rigid rotor in an electric field[@FC85; @FG14], a polar symmetric top in an electric field[FAC87,MMFC87]{} and a particle in a box with a linear potential[@FG14]. In all those cases only a pair of eigenvalues coalesce at each EP that is also called a branch point, double point or critical point. The EPS on the imaginary axis proved to be relevant to the study of $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians[@FG14] (and references therein).
The EPS can be estimated by a suitable analysis of the perturbation series[@FAC87; @F01] but there are more accurate techniques[@F01; @BC69; @H81; @HG81; @V98; @SX99; @ZRKBS12; @FG14], most of which are based on the secular equation for a truncated matrix representation of the Hamiltonian operator in a suitable basis set of eigenvectors. In most of these cases, one obtains the eigenvalues $E(\lambda )$ from the roots of a nonlinear equation $Q(E,\lambda )=0$ and it is well known that the branch points are simultaneous solution of this equation and $\partial $ $Q(E,\lambda
)/\partial E=0$. This approach and its variants[@F01; @BC69; @H81; @HG81; @V98; @SX99; @ZRKBS12; @FG14] have proved suitable for the calculation of reasonably accurate EPS. However, finding the roots of two nonlinear equations requires a judicious application of efficient algorithms (see, for example, the remarkable calculation of double points carried out many years ago for the characteristic values of the Mathieu equation[@BC69]).
The purpose of this paper is to point out that the calculation of the EPS is considerably facilitated by the application of the discriminant[S1851,G81,KB16,BPR03]{} to the polynomial function that determines the approximate energies of the quantum-mechanical problem. The advantage is that the two nonlinear equations in two variables reduce to one nonlinear equation in one variable. As a result, the estimation of the EPS reduces to the calculation of the roots of one nonlinear equation in the parameter $\lambda $. Another advantage of this approach is that most computer-algebra software have built-in algorithms for the calculation of the discriminant of a polynomial. The resultant of two polynomials and the discriminant of a polynomial are known since long ago in the mathematical literature[S1851,G81,KB16,BPR03]{} and have already been applied to the analysis of physical problems. Some of the examples are the determination of singularities in the eigenvalues of parameter-dependent matrix eigenvalue problems[@HS91], the analysis of the properties of two-dimensional magnetic traps for laser-cooled atoms[@D02], the description of optical polarization singularities[@F04], the EPS for the eigenvalues of a modified Lipkin model[@HSG05], the location of level crossings between eigenvalues of parameter-dependent symmetric matrices[BR06,B07,BR07a,BR07b]{}, the solution of two equations with two unknowns that appear in the study of gravitational lenses[@KB16]. In all these applications the resultant and discriminant have been applied to polynomials of finite degree coming from matrices of finite dimension where the technique yields exact results. In this paper, on the other hand, we focus on quantum-mechanical problems defined on infinite Hilbert spaces so that the matrix representations of the Hamiltonian operators and their characteristic polynomials are approximate due to necessary truncation.
In section \[sec:Par-dep\_Ham\] we briefly discuss some properties of parameter-dependent Hamiltonians, in section \[sec:PB\] we apply the approach to a particle in a box with two different potentials, in section \[sec:Mathieu\] we consider the periodic solutions to the Mathieu function, sections \[sec:RR\] and \[sec:sym\_top\] are devoted to the Stark effect in a polar rigid rotor and a symmetric top, respectively. Finally, in section \[sec:conclusions\] we summarize the main results and draw conclusions. In order to make this paper sufficiently self-contained we add two appendices with a discussion of three-term recurrence relations for the expansion coefficients of the wavefunctions and a slight introduction to the resultant of two polynomials and the discriminant of a polynomial.
Parameter-dependent Hamiltonians {#sec:Par-dep_Ham}
================================
The purpose of this paper is the analysis of Hamiltonian operators $H(\lambda )$ that depend on a parameter $\lambda $. Their eigenvalues $E_{n}(\lambda )$ and eigenvectors (or eigenfunctions) $\psi _{n}$ depend on this parameter. This kind of problems may exhibit EPS $\lambda _{EP}$ in the complex $\lambda $-plane at which two (or more) eigenvalues coalesce: $E_{m}\left( \lambda _{EP}\right) =E_{n}\left( \lambda _{EP}\right) $. This phenomenon is different from ordinary degeneracy in that the corresponding eigenvectors $\psi _{m}$ and $\psi _{n}$ become linearly dependent at an exceptional point $\lambda _{EP}$. If we manage to obtain a power-series expansion$$E_{n}(\lambda )=\sum_{j=0}E_{n,j}\lambda ^{j},
\label{eq:PTseriesE}$$for example by means of perturbation theory, its radius of convergence will be determined by the EP closest to the origin of the complex $\lambda $-plane[@RS78; @S82; @F01].
One can obtain EPS from a nonlinear equation of the form $Q(E,\lambda )=0$, from which one commonly obtains the eigenvalues $E_{n}(\lambda )$. It is well known that the branch points of the eigenvalues as functions of the complex parameter $\lambda $ are common roots of the nonlinear equations $\left\{ Q(E,\lambda )=0,\partial Q(E,\lambda )/\partial E=0\right\} $. In this paper we consider a simple and straightforward way of obtaining the EPS that enables one to reduce the two nonlinear equations of two variables to just one nonlinear function of $\lambda $.
One of the simplest ways of obtaining the equation $Q(E,\lambda )=0$ is based on the matrix representation $\mathbf{H}$ of the Hamiltonian operator $H$ in a given orthonormal basis set $\left\{ \left\vert i\right\rangle
,i=0,1,\ldots \right\} $. If the basis set is infinite we resort to an approximate truncated matrix representation $\mathbf{H}_{N}$ with elements $H_{ij}=\left\langle i\right\vert H\left\vert j\right\rangle $, $i,j=0,1,\ldots ,N-1$. The approximate energies are roots of the characteristic polynomial $$p_{N}(E,\lambda )=\left\vert
\mathbf{H}_{N}-E\mathbf{I}_{N}\right\vert =0,
\label{eq:p_N(E,lambda)}$$where $\mathbf{I}_{N}$ is the $N\times N$ identity matrix. The roots of $p_{N}(E,\lambda )=0$ give us approximate eigenvalues $E_{n}(\lambda )$ and, consequently, we expect to obtain the EPS with increasing accuracy by increasing $N$.
Since the EPS are complex values of $\lambda $ for which two eigenvalues coalesce, and taking into account that $p_{N}(E,\lambda )$ is a polynomial function of $E$, it is clear from the discussion of the discriminant in \[app:Discriminant\] that the EPS can be obtained from the roots $\lambda _{i}^{[N]}$ of the one-variable function$$F_{N}(\lambda )=Disc_{E}\left( p_{N}(E,\lambda )\right).
\label{eqF_N=Disc(p_N)}$$Since the characteristic polynomial comes from a truncation of the matrix representation of the Hamiltonian, one expects some of the roots of $F_{N}(\lambda )$ to be spurious. However, the sequences of roots that converge when $N$ increases are expected to yield the actual EPS. Another advantage of this approach is that most computer-algebra software have built-in algorithms for the calculation of the discriminant of a polynomial. It is worth noticing that this approach applies even in the case that more than two eigenvalues coalesce at an EP (see \[app:Discriminant\]). The approach just outlined is particularly simple and practical when $H=H_{0}+\lambda H^{\prime
}$ because in this case $F_{N}(\lambda )$ is a polynomial function of $\lambda $.
Suppose that $E(0)$ is an isolated simple eigenvalue of $H_{0}$ and that there are two real numbers $a$ and $b$ such that$$\left\Vert H^{\prime }\Phi \right\Vert \leq a\left\Vert H_{0}\Phi
\right\Vert +b\left\Vert \Phi \right\Vert,
\label{eq:Kato_inequality}$$where $\left\Vert f\right\Vert =\sqrt{\left\langle f\right\vert \left.
f\right\rangle }$, for all $\Phi $ in the state space. Under such conditions there is a unique eigenvalue $E(\lambda )$ of $H$ near $E(0)$ and $E(\lambda
)$ is analytic in a neighbourhood of $\lambda =0$ in the complex $\lambda $ plane[@RS78; @S82; @F01] (and references therein). All the examples discussed in this paper satisfy this condition because $\left\Vert H^{\prime
}\Phi \right\Vert \leq b\left\Vert \Phi \right\Vert $ as we will see below.
If $H(\lambda )$ is Hermitian for $\lambda $ real, then the EPS cannot be real. In all the examples discussed here $H(\lambda )^{\ast }=H(\lambda
^{\ast })$ for $\lambda $ complex so that $H(\lambda ^{\ast })\psi
_{n}^{\ast }=E_{n}(\lambda )^{\ast }\psi _{n}^{\ast }$ and, consequently, $E_{n}(\lambda )^{\ast }=E_{m}(\lambda ^{\ast })$. Suppose that $E_{n}(\lambda )$ and $E_{k}(\lambda )$ coalesce at $\lambda _{EP}$, therefore $E_{n}\left( \lambda _{EP}\right) ^{\ast }=E_{k}\left( \lambda
_{EP}\right) ^{\ast }\Rightarrow E_{m}\left( \lambda _{EP}^{\ast }\right)
=E_{j}\left( \lambda _{EP}^{\ast }\right) $ and, by virtue of the preceding result, $\lambda _{EP}^{\ast }$ is also an EP. In other words, the EPS appear in pairs of complex-conjugate numbers.
Suppose that there exists a unitary operator $U$ such that $U^{\dagger
}H(\lambda )U=H(-\lambda )$; therefore $U^{\dagger }H(\lambda )\psi
_{n}=H(-\lambda )U^{\dagger }\psi _{n}=E_{n}(\lambda )U^{\dagger }\psi _{n}$, from which it follows that $E_{n}(\lambda )=E_{m}(-\lambda )$. We conclude that if $\lambda _{EP}$ is an EP, then $-\lambda _{EP}$ is also an EP. It follows from the two preceding results that there may be quadruplets of EPS: $\lambda _{EP}$, $\lambda _{EP}^{\ast }$, $-\lambda _{EP}$ and $-\lambda
_{EP}^{\ast }$ when $\Re \lambda _{EP}\neq 0$.
Particle in a one-dimensional box with a perturbation {#sec:PB}
=====================================================
Our first example is given by a particle in a one-dimensional box with an interaction potential. The Schrödinger equation for such a problem can be written in dimensionless form as$$\begin{aligned}
&&-\psi ^{\prime \prime }(x)+\lambda V(x)\psi (x)=E\psi (x), \nonumber \\
&&\psi (-1) =\psi(1)=0. \label{eq:PB_V}\end{aligned}$$If $V(x)=H^{\prime }$ is a continuous function of $x$ then $\left\vert
V(x)\right\vert \leq C$ when $x\in (-1,1)$ and $\left\Vert H^{\prime }\Phi
\right\Vert \leq C^{2}\left\Vert \Phi \right\Vert $. Consequently, every eigenvalue $E_{n}(\lambda )$ is an analytical function of $\lambda $ for all $\lambda <\left\vert \lambda _{EP}\right\vert $, where $\lambda _{EP}$ is the EP closest to the origin of the complex $\lambda $ plane. In order to obtain a suitable matrix representation for this Hamiltonian we resort to the basis set of eigenfunctions of $H_{0}=-d^{2}/dx^{2}$:$$\left\langle x\right. \left\vert n\right\rangle =\sin \left(
\frac{(n+1)\pi (x+1)}{2}\right) ,\;n=0,1,\ldots .
\label{eq:PB_basis}$$
If $V(x)$ is a polynomial function of $x$ the analytical calculation of its matrix elements $\left\langle m\right\vert V\left\vert n\right\rangle $ is straightforward. The simplest example is $V(x)=x$. The canonical transformation $U^{\dagger }xU=-x$, $U^{\dagger }d/dxU=-d/dx$ yields $U^{\dagger }H(\lambda )U=H(-\lambda )$ and, therefore, we expect the EPS to appear as quadruplets ($\Re \lambda _{EP}\neq 0$) and doublets ($\Re \lambda
_{EP}=0$).
In order to discard any spurious root $\lambda _{i}^{[N]}$ of the polynomial $F_{N}(\lambda )$ of degree $N(N-1)$ we keep only those that satisfy $\left\vert \lambda _{EP}^{[N]}-\lambda _{EP}^{[N-1]}\right\vert <10^{-3}$. Figure \[fig:PBX1EPS\] shows some of the EPS for this problem; they clearly exhibit the symmetry just mentioned with respect to the real ($\lambda _{EP}$,$\lambda _{EP}^{\ast }$) and imaginary ($\lambda _{EP}$,$-\lambda _{EP}$) axes of the complex $\lambda $ plane. A larger number of EPS on the imaginary axis was obtained recently in a study of $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians[@FG14].
If $V(x)=x^{2}$ the canonical transformation discussed above leaves $H(\lambda )$ invariant and we only expect doublets (that is to say: symmetry with respect to the real axis). Figure \[fig:PBX2EPS\] confirms this conclusion. In this case it is convenient to treat the even ($n=0,2,\ldots $) and odd states ($n=1,3,\ldots $) separately.
Mathieu equation {#sec:Mathieu}
================
One of the most widely studied periodic problems is the Mathieu equation that we write here in the following form$$\psi ^{\prime \prime }(x)+\left[ E-2\lambda \cos (2x)\right] \psi
(x)=0, \label{eq:Mathieu}$$so that we can relate it to the linear operator $H=-d^{2}/dx^{2}+2\lambda \cos (2x)$. We consider the two cases of periodic solutions, those of period $\pi $ ($\psi (x+\pi )=\psi
(x)$) and those of period $2\pi $ ($\psi (x+2\pi )=\psi (x)$) and each class can be separated into even and odd. The four cases can be reduced to tridiagonal matrix representations or three-term recurrence relations; in what follows we show the main parameters (see \[app:Rec-Rel\]) for each of them.
Period $\pi $ even: $$\begin{aligned}
\left\vert j\right\rangle &=&\frac{\sqrt{2}+\left( 1-\sqrt{2}\right) \delta
_{j0}}{\sqrt{\pi }}\cos (2jx),\;j=0,1,\ldots, \nonumber \\
A_{j+1} &=&\left[ 1+\left( \sqrt{2}-1\right) \delta _{j0}\right]
\lambda ,\;B_{j}=4j^{2}-E. \label{eq:Mathieu_pi_even}\end{aligned}$$
Period $\pi $ odd$$\begin{aligned}
\left\vert j\right\rangle &=&\sqrt{\frac{2}{\pi }}\sin
[(2j+2)x],\;j=0,1,\ldots, \nonumber \\
A_{j+1} &=&\lambda ,\;B_{j}=4(j+1)^{2}-E.
\label{eq:Mathieu_pi_odd}\end{aligned}$$
Period $2\pi $ even$$\begin{aligned}
\left\vert j\right\rangle &=&\sqrt{\frac{2}{\pi }}\cos \left[ (2j+1)x\right]
,\;j=0,1,\ldots, \nonumber \\
A_{j+1} &=&\lambda ,\;B_{j}=(2j+1)^{2}+\lambda \delta _{j0}-E.
\label{eq:Mathieu_2pi_even}\end{aligned}$$
Period $2\pi $ odd$$\begin{aligned}
\left\vert j\right\rangle &=&\sqrt{\frac{2}{\pi }}\sin \left[ (2j+1)x\right]
,\;j=0,1,\ldots, \nonumber \\
A_{j+1} &=&\lambda ,\;B_{j}=(2j+1)^{2}-\lambda \delta _{j0}-E.
\label{eq:Mathieu_2pi_odd}\end{aligned}$$
In all these cases we find that $\left\Vert H^{\prime }\Phi \right\Vert \leq
\left\Vert H^{\prime }\Phi \right\Vert $ because $\left\vert H^{\prime
}\right\vert =\left\vert \cos (2x)\right\vert \leq 1$. Besides, the canonical transformation $U^{\dagger }xU=x+\pi /2$, $U^{\dagger }d/dxU=d/dx$ leads to $U^{\dagger }H(\lambda )U=H(-\lambda )$ and again we expect the distribution of EPS to exhibit symmetry with respect to both axes.
Since in the four cases we have tridiagonal matrices, we resort to the recurrence relation for the determinants $D_{N}$ discussed in \[app:Rec-Rel\]. Thus, the problem reduces to obtaining the roots of a polynomial $F_N(\lambda )=Disc_{E}(D_{N}(E,\lambda ))$ of degree $N(N+1)$. Present results are shown in figures \[fig:Mathieupi\] and \[fig:Mathieu2pi\]. In the case of period $\pi $ the distribution of EPS for each parity symmetry (even and odd) exhibit the characteristic symmetry with respect to both axes. However, in the case of those of period $2\pi $ the even functions exhibit EPS $\lambda _{EP}$ and $\lambda _{EP}^{\ast }$ while the odd functions exhibit the remaining ones $-\lambda _{EP}$ and $-\lambda _{EP}^{\ast }$. The reason is that for period $\pi $ $D_{N}^{e,o}(E,-\lambda )=D_{N}^{e,o}(E,\lambda )$ while for period $2\pi $ $D_{N}^{e}(E,-\lambda )=D_{N}^{o}(E,\lambda )$, where the superscripts $e$ and $o$ stand for even and odd parity, respectively.
Present results agree with those of Blanch and Clemm[@BC69] in the whole $\lambda $ plane and the ones obtained by Fernández and Garcia[@FG14] on the imaginary axis. Both appear to be the most accurate EPS available in the literature.
Polar rigid rotor in a uniform electric field {#sec:RR}
=============================================
In this section we consider a rigid rotor with dipole moment $\mu $ in a uniform electric field of intensity $F$. The kinetic energy of the Hamiltonian operator is $\mathcal{L}^{2}/(2I)$, where, $\mathcal{L}^{2}$ is the square of the angular momentum and $I$ is the moment of inertia. The interaction with the field is $-\mu F\cos \theta $, where $\theta $ is the angle between the dipole moment and the field direction. This model has proved useful for the analysis of the rotational Stark effect in linear polar molecules[@TS55]. The Schrödinger equation can be written in dimensionless form as $H(\lambda )\psi =\epsilon (\lambda )\psi $, where$$H(\lambda )=H_{0}+\lambda H^{\prime }=L^{2}-\lambda \cos \theta,
\label{eq:Rigid_Rotor}$$$L^{2}=\mathcal{L}^{2}/\hbar ^{2}$, $\lambda =2I\mu F/\hbar ^{2}$ and $\epsilon =2IE/\hbar ^{2}$. As in the preceding example we have $\left\Vert
H^{\prime }\Phi \right\Vert \leq \left\Vert \Phi \right\Vert $ and because of the transformation $U^{\dagger }\theta U=\theta +\pi \Rightarrow
U^{\dagger }H(\lambda )U=H(-\lambda )$ we expect a distribution of EPS that is symmetric with respect to both axes in the complex $\lambda $ plane.
In order to apply present approach we resort to a matrix representation of the Hamiltonian operator in the basis set of eigenfunctions of $L^{2}$ and $L_{z}$$$\begin{aligned}
L^{2}\left\vert l,m\right\rangle &=&l(l+1)\left\vert l,m\right\rangle
,\;l=0,1,\ldots , \nonumber \\
L_{z}\left\vert l,m\right\rangle &=&m\left\vert l,m\right\rangle
,\;m=0,\pm 1,\ldots ,\pm l. \label{eq:Angular_momentum_basis}\end{aligned}$$It is well known that the coefficients $c_{i}$ of the expansion$$\psi =\sum_{i=0}^{\infty }c_{i}\left\vert M+i,m\right\rangle
,\;M=|m|, \label{eq:psi_RR_expansion}$$satisfy a three-term recurrence relation like those discussed in the \[app:Rec-Rel\] with[@TS55; @F01]$$A_{i}=-\lambda \left[ \frac{i(i+2M)}{4(i+M)^{2}-1}\right] ^{1/2},\;B_{i}=(i+M)(i+M+1)-\epsilon. \label{eq:A_i,B_i_RR}$$
The calculation of the EPS $\lambda _{EP}$ from the determinants $D_{N}(\epsilon ,\lambda )=0$ (see \[app:Rec-Rel\]) is straightforward and some results are shown in figure \[fig:RREPS\] for $M=0,1,2,3$. It is worth noticing that the EPS for $M=0$ are close to those for $M=2$ while the EPS for $M=1$ are close to those for $M=3$. Notice that the distribution of EPS exhibits the symmetry with respect to both axes mentioned above.
Polar symmetric top in a uniform electric field {#sec:sym_top}
===============================================
The rotational Stark effect in a symmetric-top molecule is commonly studied by means of the model Hamiltonian[@TS55; @S63; @HO91]$$\begin{aligned}
H &=&-\frac{\hbar ^{2}}{2I_{B}}\left[ \frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\sin \theta \frac{\partial }{\partial \theta }+\frac{1}{\sin ^{2}\theta }\frac{\partial ^{2}}{\partial \phi ^{2}}+\left( \frac{\cos
^{2}\theta }{\sin ^{2}\theta }+\frac{I_{B}}{I_{C}}\right) \frac{\partial ^{2}}{\partial \chi ^{2}}-\frac{2\cos \theta }{\sin ^{2}\theta }\frac{\partial
^{2}}{\partial \phi \partial \chi }\right] \nonumber \\
&&-\mu F\cos \theta, \label{eq:symmetric_top_1}\end{aligned}$$where $I_{C}$ is the moment of inertia about the symmetry axis, $I_{B}$ is the other moment of inertia, $\phi $, $\theta $ and $\chi $ are the Euler angles, $\mu $ is the dipole moment of the molecule and $F$ is the intensity of the uniform electric field. Clearly, $\theta $ is the angle between the dipole moment and the field direction.
The Schrödinger equation $H\psi =E\psi $ is separable if we write $\psi
(\theta ,\phi ,\chi )=\Theta (\theta )e^{iM\phi }e^{iK\chi }$, where $M,K=0,\pm 1,\pm 2,\ldots $ are the two rotational quantum numbers that remain when the field is applied. The resulting eigenvalue equation in dimensionless form is$$\begin{aligned}
&&\left( \frac{1}{\sin \theta }\frac{d}{d\theta }\sin \theta \frac{d}{d\theta }-\frac{M^{2}}{\sin ^{2}\theta }-\frac{\cos ^{2}\theta }{\sin
^{2}\theta }K^{2}+\frac{2\cos \theta }{\sin ^{2}\theta }KM-\lambda \cos
\theta +\epsilon \right) \Theta =0, \nonumber \\
&&\epsilon =\frac{2I_{B}}{\hbar ^{2}}E-\frac{I_{B}}{I_{C}}K^{2},\;\lambda =\frac{2I_{B}}{\hbar ^{2}}\mu F. \label{eq:symmetric_top_2}\end{aligned}$$
If we write $H(\lambda )=H_{0}+\lambda H^{\prime }$ then we realize that $\left\Vert H^{\prime }\Phi \right\Vert \leq \left\Vert \Phi \right\Vert $ as in the two preceding examples. Besides, the transformation used in the case of the rigid rotor leads to $U^{\dagger }H(\lambda )U=H(-\lambda )$ and once again we expect a distribution of EPS that is symmetric with respect to both axes.
In this case we obtain a suitable matrix representation of the Hamiltonian in the basis set of eigenvectors $\left\vert J,M,K\right\rangle $ of the free symmetric top ($\lambda =0$). The coefficients $c_{i}$ of the expansion $$\psi =\sum_{i=0}^{\infty }c_{i}\left\vert J_{0}+i,M,K\right\rangle
,\;J_{0}=\max (|M|,|K|), \label{eq:psi_symmetric_top_expansion}$$satisfy a three-term recurrence relation with[@TS55; @S63; @HO91]$$\begin{aligned}
B_{i} &=&\left( J_{0}+i\right) \left( J_{0}+i+1\right) -\lambda \frac{MK}{\left( J_{0}+i\right) \left( J_{0}+i+1\right) }-\epsilon, \nonumber \\
A_{i} &=&-\lambda \frac{\sqrt{\left[ \left( J_{0}+i\right) ^{2}-K^{2}\right] \left[ \left( J_{0}+i\right) ^{2}-M^{2}\right] }}{\left(
J_{0}+i\right) \sqrt{4\left( J_{0}+i\right) ^{2}-1}}.
\label{eq:A_i,B_i_symmetric_top}\end{aligned}$$Therefore, we can obtain the EPS from the secular determinants $D_{N}(\epsilon ,\lambda )=0$ for a sufficiently large dimension $N$ (see \[app:Rec-Rel\]).
The distribution of the EPS can be predicted from the set of equalities$$\begin{aligned}
&&D_{N}(\epsilon ,M,K,\lambda )=D_{N}(\epsilon ,K,M,\lambda
)=D_{N}(\epsilon ,-M,-K,\lambda )=D_{N}(\epsilon ,-K,-M,\lambda )=
\nonumber
\\
&&D_{N}(\epsilon ,-M,K,-\lambda ) =D_{N}(\epsilon ,K,-M,-\lambda
)=D_{N}(\epsilon ,M,-K,-\lambda )=\nonumber \\ &&D_{N}(\epsilon
,-K,M,-\lambda ). \label{eq:D_N_symmetric_top_symmetry}\end{aligned}$$Figure \[fig:SYMTOPM0K0\] shows that the distribution of EPS for $M=K=0$ is symmetric with respect to both axes. On the other hand, Figure [fig:SYMTOPM1K1M1KM1]{} shows that the distribution of EPS for either $MK=1$ or $MK=-1$ is symmetric with respect to the real axis but the union of both sets is symmetric with respect to both axes.
Present results are considerably more accurate than those obtained earlier for this model[@FAC87; @MMFC87].
Conclusions {#sec:conclusions}
===========
In this paper we have shown that the discriminant is an extremely useful tool for the location of EPS in the eigenvalues of parameter-dependent Hamiltonian operators. In all the previous studies that we are aware of, the approach was applied to operators on finite Hilbert spaces with matrix representations of finite dimension that lead to characteristic polynomials of finite degree[@SM98; @HS91; @D02; @F04; @HSG05; @BR06; @B07; @BR07a; @BR07b; @KB16]. Here, on the other hand, the Hilbert spaces have infinite dimension so that the truncated $N$-dimensional matrix representations, as well as the corresponding characteristic polynomials are approximate. However, the location approach based on the discriminant applies successfully producing sequences of roots that converge towards the actual EPS as $N$ increases. Any spurious root is easily identified because it does not form part of a convergent sequence.
All the examples chosen in the present study are of mathematical or physical interest. Our results for the Mathieu equation agree with the most extended and accurate ones available in the literature[@BC69] and those for the Stark effects in the rigid rotor and symmetric top are either more extended or more accurate than the ones published previously[F01,FAC87,FG14,FC85,MMFC87]{}.
Resultant and discriminant {#app:Discriminant}
==========================
In this section we summarize those properties of the discriminant of a polynomial that are relevant for present paper. The resultant of two polynomials$$\begin{aligned}
f(x) &=&\sum_{j=0}^{m}a_{m-j}x^{j}, \nonumber \\
g(x) &=&\sum_{j=0}^{n}b_{m-j}x^{j}, \label{eq:polynomials}\end{aligned}$$is given by the determinant$$Res_{x}(f,g)=\left\vert
\begin{array}{cccccccc}
a_{0} & 0 & \cdots & 0 & b_{0} & 0 & \cdots & 0 \\
a_{1} & a_{0} & \cdots & 0 & b_{1} & b_{0} & \cdots & 0 \\
a_{2} & a_{1} & \ddots & 0 & b_{2} & b_{1} & \ddots & \vdots \\
\vdots & \vdots & \ddots & a_{0} & \vdots & \vdots & \ddots & b_{0} \\
a_{m} & a_{m-1} & \cdots & \vdots & b_{n} & b_{n-1} & \cdots & \vdots \\
0 & a_{m} & \ddots & \vdots & 0 & b_{n} & \ddots & \vdots \\
\vdots & \vdots & \ddots & a_{m-1} & \vdots & \vdots & \ddots & b_{n-1}
\\
0 & 0 & \cdots & a_{m} & 0 & 0 & \cdots & b_{n}\end{array}\right\vert . \label{eq:Res(f,g)_1}$$It can be proved that$$Res_{x}(f,g)=a_{0}^{m}b_{0}^{n}\prod_{i=1}^{m}\prod_{j=1}^{n}\left(
\xi _{i}-\mu _{j}\right), \label{eq:Res(f,g)_2}$$where $\xi _{i}$ and $\mu _{j}$ are the roots of the polynomials $f$ and $g$, respectively. The discriminant of $f(x)$ is defined as$$Disc_{x}(f)=\frac{(-1)^{m(m-1)/2}}{a_{m}}Res_{x}(f,f^{\prime }),
\label{eq:Disc(f)_1}$$and in this case we have$$Disc_{x}(f)=a^{2m-2}\prod_{i<j}\left( \xi _{i}-\xi _{j}\right)
^{2}. \label{eq:Disc(f)_2}$$
Suppose that the nonlinear equation $Q(E,\lambda )=0$ gives us the eigenvalues $E(\lambda )$ of a quantum-mechanical system. If this equation is a polynomial function of $E$, then the roots of $F(\lambda
)=Disc_{E}(Q(E,\lambda ))$ are the exceptional points $\lambda _{EP}$ in the complex $\lambda $ plane where at least two eigenvalues coalesce. We appreciate that the advantage of resorting to the discriminant is that we only have to search for the roots of a nonlinear function of just one variable. In all the examples studied here the nonlinear equation $Q(E,\lambda )$ is a polynomial function of both $E$ and $\lambda $ so that $F(\lambda )$ is a polynomial function of $\lambda $ (see \[app:Rec-Rel\], and the examples). Consequently, the calculation is particularly simple because there are efficient algorithms for finding the roots of polynomials. Besides, most computer-algebra software enable one to obtain analytical expressions for $F(\lambda )$ because the discriminant is given by a determinant. Thus, the only numerical step of the calculation reduces to finding the roots of the polynomial $F(\lambda )$.
As an illustrative example we consider a trivial toy problem that we deem to be quite interesting: the Hamiltonian operator in matrix form$$\mathbf{H}(\beta ,\lambda )=\left(
\begin{array}{ccc}
3-\lambda & \beta & 0 \\
\beta & 2 & \beta \\
0 & \beta & 1+\lambda\end{array}\right). \label{eq:H_3x3}$$When $\beta =0$ the three eigenvalues cross at $\lambda =1$ and the three eigenvectors are degenerate (they are obviously linearly independent). However, when $\beta \neq 0$ the eigenvalues do not cross for real values of $\lambda $ and exhibit avoided crossings as shown in Figure \[fig:AC\] for $\beta =0.1$.
The characteristic polynomial is $$Q(E,\lambda )=\frac{\left( 2-E\right) \left(
50E^{2}-200E-50\lambda ^{2}+100\lambda +149\right) }{50},
\label{eq:Q_3x3}$$
so that$$Disc_{E}(Q(E,\lambda ))=\frac{\left( 50\lambda ^{2}-100\lambda
+51\right) ^{3}}{31250}. \label{eq:Disc(Q)_3x3}$$We appreciate that the three eigenvalues coalesce at any of the two EPS $\lambda _{EP}=1+\sqrt{2}i/10$ and $\lambda _{EP}^{\ast }$ that are branch points of order two. The structure of the avoided crossings in this toy model is similar to that of the modified Lipkin model for $N=3$[@HSG05].
At $\lambda _{EP}$ (a similar analysis can be carried out at $\lambda
_{EP}^{\ast }$) the matrix $\mathbf{H}$ has only one eigenvalue $E\left(
\lambda _{EP}\right) =2$ and only one eigenvector$$\mathbf{v}_{1}=\frac{1}{2}\left(
\begin{array}{c}
-i \\
\sqrt{2} \\
i\end{array}\right), \label{eq:v_1}$$so that $\mathbf{H}$ is defective. By means of the Jordan chain$$\begin{aligned}
\left( \mathbf{H}-2\mathbf{I}_{3}\right) \mathbf{v}_{2}
&=&\mathbf{v}_{1},
\nonumber \\
\left( \mathbf{H}-2\mathbf{I}_{3}\right) \mathbf{v}_{3}
&=&\mathbf{v}_{2}, \label{eq:Jordan_chain}\end{aligned}$$where $\mathbf{I}_{3}$ is the $3\times 3$ identity matrix, we obtain two additional vectors $\mathbf{v}_{2}$ and $\mathbf{v}_{3}$ and the matrix$$\mathbf{U}=\left(
\begin{array}{ccc}
\mathbf{v}_{1} & \mathbf{v}_{2} & \mathbf{v}_{3}\end{array}\right) =\left(
\begin{array}{ccc}
-i/2 & 5\sqrt{2} & 0 \\
1/\sqrt{2} & 5i & 50\sqrt{2} \\
i/2 & 0 & 50i\end{array}\right), \label{eq:U_3x3}$$that converts $\mathbf{H}$ into a Jordan canonical form$$\mathbf{U}^{-1}\mathbf{HU}=\left(
\begin{array}{ccc}
2 & 1 & 0 \\
0 & 2 & 1 \\
0 & 0 & 2\end{array}\right). \label{eq:Jordan_form}$$In this simple case we can easily obtain the EPS directly from the eigenvalues but in most nontrivial problems the use of the discriminant leads to far simpler expressions.
Three-term recurrence relations {#app:Rec-Rel}
===============================
Suppose that there is an orthonormal basis set $\left\{ \left\vert
i\right\rangle ,i=0,1,\ldots \right\} $ such that$$H\left\vert i\right\rangle =H_{i,i-1}\left\vert i-1\right\rangle
+H_{i,i}\left\vert i\right\rangle +H_{i,i+1}\left\vert
i+1\right\rangle, \label{eq:three_term_rec_rel_H}$$where $H_{i,j}=H_{i,j}^{\ast }=H_{j,i}$. Therefore, if we expand$$\psi =\sum_{i}c_{i}\left\vert i\right\rangle,$$the Schrödinger equation $H\psi =E\psi $ becomes a three-term recurrence relation for the coefficients $c_{j}$:$$\begin{aligned}
A_{i}c_{i-1}+B_{i}c_{i}+A_{i+1}c_{i+1}
&=&0,\;A_{i}=H_{i-1,i},\;B_{i}=H_{ii}-E, \nonumber \\
&&i=0,1,2\ldots ,\;c_{-1}=0. \label{eq:three_term_rec_rel_c_i}\end{aligned}$$One commonly obtains approximate energies by means of the truncation condition $c_{j}=0$, $j>N$, so that the roots of the characteristic polynomial given by the secular determinant$$D_{N}=\left\vert
\begin{array}{cccccc}
B_{0} & A_{1} & 0 & \cdots & \cdots & 0 \\
A_{1} & B_{1} & A_{2} & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \cdots & \vdots \\
0 & 0 & \cdots & A_{N-1} & B_{N-1} & A_{N} \\
0 & 0 & \cdots & 0 & A_{N} & B_{N}\end{array}\right\vert, \label{eq:D_N}$$converge from above toward the actual energies of the physical problem when $N\rightarrow \infty $. These determinants can be efficiently generated by means of the three-term recurrence relation[@S63; @HO91; @F01]$$D_{N}=B_{N}D_{N-1}-A_{N}^{2}D_{N-2},\;N=0,1,\ldots,
\label{eq:rec_rel_D_N}$$with the initial conditions $D_{-1}=1$, $D_{j}=0$ for $j<-1$. Notice that the dimension of the determinant $D_{N}$ is $N+1$.
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![EPS for the particle in a box with potential $\protect\lambda x$[]{data-label="fig:PBX1EPS"}](PBX1EPS.EPS){width="9cm"}
![EPS for the particle in a box with potential $\protect\lambda x^{2}$ for even (blue circles) and odd (red circles) states[]{data-label="fig:PBX2EPS"}](PBX2EPS.EPS){width="9cm"}
![EPS for the Mathieu equation with period $\protect\pi$ for even (blue circles) and odd (red circles) functions[]{data-label="fig:Mathieupi"}](MATHIEUPI.EPS){width="9cm"}
![EPS for the Mathieu equation with period $2\protect\pi$ for even (blue circles) and odd (red circles) functions[]{data-label="fig:Mathieu2pi"}](MATHIEU2PI.EPS){width="9cm"}
![EPS for the rigid rotor in an electric field for the states with $M=0$ (blue circles) $M=1$ (red diamonds) $M=2$ (green triangles) and $M=3$ (orange stars)[]{data-label="fig:RREPS"}](RREPS.EPS){width="9cm"}
![EPS for the polar symmetric top in a uniform electric field for $M=0 $ and $K=0$[]{data-label="fig:SYMTOPM0K0"}](SYMTOPM0K0.EPS){width="9cm"}
![EPS for the polar symmetric top in a uniform electric field for $(M=1,K=1)$ (blue circles) and $(M=1,K=-1)$ (red circles)[]{data-label="fig:SYMTOPM1K1M1KM1"}](SYMTOPM1K1M1KM1.EPS){width="9cm"}
![Avoided crossings for the three-level model[]{data-label="fig:AC"}](AC.EPS){width="9cm"}
|
---
author:
- 'Yingjie Hu$^1$[^1] and Krzysztof Janowicz$^2$'
bibliography:
- 'references.bib'
title: |
An empirical study on the names of points of interest and\
their changes with geographic distance
---
[^1]: Corresponding author; email: [email protected]\
To cite this paper: Hu, Y. and Janowicz, K., 2018. An empirical study on the names of points of interest and their changes with geographic distance. In *Proceedings of the 10th International Conference on Geographic Information Science (GIScience 2018)*. Aug. 29-31, Melbourne, Australia.
|
---
abstract: 'We study the use of exciton-polaritons in semiconductor microcavities to generate radiation spanning the infrared to terahertz regions of the spectrum by exploiting transitions between upper and lower polariton branches. The process, which is analogous to difference-frequency generation (DFG), relies on the use of semiconductors with a nonvanishing second-order susceptibility. For an organic microcavity composed of a nonlinear optical polymer, we predict a DFG irradiance enhancement of $2.8\cdot10^2$, as compared to a bare nonlinear polymer film, when triple resonance with the fundamental cavity mode is satisfied. In the case of an inorganic microcavity composed of (111) GaAs, an enhancement of $8.8\cdot10^3$ is found, as compared to a bare GaAs slab. Both structures show high wavelength tunability and relaxed design constraints due to the high modal overlap of polariton modes.'
author:
- Fábio Barachati
- Simone De Liberato
- 'Stéphane Kéna-Cohen'
bibliography:
- 'references.bib'
title: 'Generation of Rabi frequency radiation using exciton-polaritons'
---
Introduction
============
Half-light, half-matter quasiparticles called polaritons arise in systems where the light-matter interaction strength is so strong that it exceeds the damping due to each bare constituent. In semiconductor microcavities, polaritons have attracted significant attention due to their ability to exhibit strong resonant nonlinearities and to condense into their energetic ground state at relatively low densities. Such polaritons result from the mixing between an exciton transition ($E_X$) and a Fabry-Perot cavity photon ($E_C$). They exhibit a peculiar dispersion, which is shown in Fig. \[figTMM\]. Around the degeneracy point of both bare constituents, the lower and upper polariton (LP and UP) branches anticross and their minimum energetic separation is called the vacuum Rabi splitting ($\hbar\Omega_R$). It can range from a few meV in inorganic semiconductors to $\sim1$ eV in organic ones[@kasprzak2006bose; @PhysRevLett.98.126405; @ADOM:ADOM201300256; @ph500266d; @PhysRevLett.106.196405]. Radiative transitions from the upper to the lower polariton branch can therefore provide a simple route towards tunable infrared (IR) and terahertz (THz) generation.
![\[figTMM\] Dispersion relation of exciton-polaritons as a function of in-plane wavevector. The interaction between an exciton transition ($E_X$) and a Fabry-Perot cavity mode ($E_C$), both represented by dashed lines, leads to the appearance of lower and upper polariton (LP and UP) branches (solid blue). A radiative transition at the Rabi energy ($\hbar\Omega_R$) occurs between two incident pumps at frequencies $\omega_1$ and $\omega_2$ through difference-frequency generation in a second-order nonlinear semiconductor ($\chi^{(2)}\neq0$). Inset: microcavity showing the two pump beams ($\vec{E}_{1,2}$), incident at angle $\theta_i$, and the Rabi radiation ($\vec{E}_3$), reflected at angle $\theta_3$. The solid blue lines in the $\chi^{(2)}$ layer illustrate the high modal overlap of polariton fields.](Figure11c){width="3.4in"}
Such transitions can be understood as resulting from a strongly coupled $\chi^{(2)}$ nonlinear interaction in which two photons, dressed by the resonant interaction with excitons, interact emitting a third photon. As a consequence of the usual $\chi^{(2)}$ selection rule, such polariton-polariton transitions are forbidden in centrosymmetric systems. To overcome this issue several solutions have been proposed, including the use of asymmetric quantum wells,[@PhysRevB.87.241304; @PhysRevB.89.235309] the mixing of polariton and exciton states with different parity,[@PhysRevLett.107.027401; @1.3519978] and the use of transitions other than UP to LP.[@PhysRevLett.108.197401; @PhysRevLett.110.047402]
Here, we study the use of non-centrosymmetric semiconductors, possessing an intrinsic second-order susceptibility $\chi^{(2)}$, to allow for the generation of Rabi-frequency radiation. The irradiance of the resulting UP to LP transitions, which are analogous to classical difference-frequency generation (DFG), have been calculated using a semiclassical model, yielding DFG irradiance enhancements up to almost four orders of magnitude compared to the ones due to the bare $\chi^{(2)}$ nonlinearity. These enhancements can also be related to those expected for parametric fluorescence. Finally, we highlight the use of a triply-resonant scheme to obtain polariton optical parametric oscillation (OPO).
Semiconductor microcavities are advantageous for nonlinear optical mixing due to their ability to spatially and temporally confine the interacting fields. For small interaction lengths, the efficiency of the nonlinear process does not depend on phase-matching, but instead on maximizing the field overlap.[@Rodriguez:07] To overcome mode orthogonality, while simultaneously satisfying the symmetry requirements of the $\chi^{(2)}$ tensor, a number of strategies have been proposed such as mode coupling between crossed beam photonic crystal cavities with independently tunable resonances[@Burgess:09; @1.3607281; @Rivoire:11] and the use of single cavities supporting both TE and TM modes.[@Zhang:09; @1.3568897] Exciton-polaritons provide a simple solution to this problem because they arise from coupling to a single cavity mode and thus naturally display good modal overlap. Many of the fascinating effects observed in strongly-coupled semiconductor microcavities exploit this property, but these have been principally limited to the resonant $\chi^{(3)}$ nonlinearity inherited from the exciton.
Note that in this paper we define the vacuum Rabi frequency as being equal to the resonant splitting due to light-matter coupling. Although this definition is commonly used in the study of quantum light-matter interactions,[@Loudon-TheQuantumTheoryOfLight-92] it differs from that often employed in the field of microcavity polaritons,[@RevModPhys.85.299] where the vacuum Rabi frequency is defined as being equal to half of the resonant splitting.
This paper is organized as follows. Section \[Theory\] reviews the nonlinear transfer matrix scheme used to calculate frequency mixing in the small-signal regime. In Sec. \[Results\], we calculate the enhancement in irradiance at the Rabi frequency over a bare nonlinear slab for organic and inorganic microcavities and in Sec. \[Discussion\] we discuss the results and highlight some of the peculiarities of both material sets. Conclusions are presented in Sec. \[Conclusion\].
\[Theory\]Theory
================
To calculate the propagation of the incident pump fields and the difference-frequency contribution due to nonlinear layers, we use the nonlinear transfer matrix method introduced by @Bethune:89.[@Bethune:89] This method is applicable to structures with an arbitrary number of parallel nonlinear layers,[@Liu:13] but is restricted to the undepleted pump approximation, where the three fields are essentially independent. First, we propagate the incident pump fields using the standard transfer matrix method. Within each nonlinear layer, these behave as source terms in the inhomogeneous wave equation. Then, we solve for the particular solution and determine the corresponding source field vectors. Finally, we use the boundary conditions and propagate the free fields using the transfer matrix method to obtain the total field in each layer.
\[TMM\]Propagation of the pump fields
-------------------------------------
We begin by calculating the field distribution of the two incident pumps as shown in Fig. \[figTMM\] by using the standard transfer matrix method.[@yeh1988optical; @born1999principles; @1.370757] To simplify the discussion, we consider the pumps to be TE (ŷ) polarized. In our notation, the electric field in each layer $i$ is given by sum of two counter-propagating plane waves $$\mathcal{E}_i^\pm(z,x,t)=\operatorname{Re}\left\{E_i^\pm\thinspace\exp[i(\pm k_{iz}z+k_xx-\omega t)]\right\},
\label{eqE}$$ where the $k_{iz}$ and $k_x$ components of the $\vec{k_i}$ wavevector satisfy the relationship $k_{iz}^2+k_x^2=n_i^2(\omega)\,\omega^2/c^2$, with $n_i$ the refractive index of layer $i$. The forward and backward complex amplitudes of the electric field are represented in vector form as $\mathbf{E}_i=\begin{bmatrix}E_i^+&E_i^-\end{bmatrix}^T$.
For a given incident field $\mathbf{E}_1$, the field in layer $i$ is calculated by $\mathbf{E}_i=T_i\mathbf{E}_1$, where $T_i$ is the partial transfer matrix $$T_i=M_{i(i-1)}\phi_{i-1}\cdots M_{21}.$$
The interface matrix $M_{ij}$, that relates fields in adjacent interfaces $i$ and $j$, and the propagation matrix $\phi_i$, that relates fields on opposite sides of layer $i$ with thickness $d_i$, are given by $$M_{ij}=\frac{1}{2k_{iz}}
\begin{bmatrix}
k_{iz}+k_{jz}&k_{iz}-k_{jz}\\k_{iz}-k_{jz}&k_{iz}+k_{jz}
\end{bmatrix}
\label{eqtm}$$ and $$\phi_i=
\begin{bmatrix}
\exp(ik_{iz}d_i)&0\\0&\exp(-ik_{iz}d_i)
\end{bmatrix}.
\label{eqphi}$$
Inclusion of nonlinear polarizations
------------------------------------
To obtain the difference-frequency contribution within a nonlinear layer, we must solve the inhomogeneous wave equation for the electric field $$\nabla^2\mathcal{E}-\mu\epsilon\frac{\partial^2\mathcal{E}}{\partial t^2}=\mu\frac{\partial^2\mathcal{P}^{NL}}{\partial t^2},
\label{eqWave}$$ where the source term $$\mathcal{P}^{NL}(z,x,t)=\epsilon_0\chi^{(2)}\mathcal{E}^2(z,x,t)
\label{eqPol}$$ is the second-order nonlinear polarization, $\mu$ is the magnetic permeability and $\epsilon$ the permittivity. By using a polarization term of the same form as Eq. (\[eqE\]), Eq. (\[eqWave\]) can be written in the frequency domain as $$\left[-(k^{NL})^2+\omega_{NL}^2n^2(\omega_{NL})\mu_0\epsilon_0\right]\mathbf{E}=-\omega_{NL}^2\mu_0\mathbf{P}^{NL},$$ with wavevector $k^{NL}$, $\mu(\omega_{NL})=\mu_0$ and $\epsilon(\omega_{NL})=n^2(\omega_{NL})\epsilon_0$. The nonlinear polarization thus generates a bound source field at the same frequency given by $$\mathbf{E}_s=\frac{\mathbf{P}^{NL}}{\frac{(k^{NL})^2}{\omega_{NL}^2\mu_0}-n^2(\omega_{NL})\epsilon_0}.
\label{eqES}$$
If we consider the presence of two pump fields $\mathbf{E}_1(\omega_1)$ and $\mathbf{E}_2(\omega_2)$, with $\omega_1>\omega_2$, the $\mathcal{E}^2(z,x,t)$ term in Eq. (\[eqPol\]) can be written as $$\begin{split}
\mathcal{E}^2(z,x,t)&=\operatorname{Re}\left\{E_1^+\thinspace\exp\left[i\left(k_z^1z+k_x^1x-\omega_1t\right)\right]\right.\\
&\quad\left.+E_1^-\thinspace\exp\left[i\left(-k_z^1z+k_x^1x-\omega_1t\right)\right]\right.\\
&\qquad\left.+E_2^+\thinspace\exp\left[i\left(k_z^2z+k_x^2x-\omega_2t\right)\right]\right.\\
&\qquad\quad\left.+E_2^-\thinspace\exp\left[i\left(-k_z^2z+k_x^2x-\omega_2t\right)\right]\right\}^2.
\end{split}$$
Expanding $\mathcal{E}^2(z,x,t)$ leads to terms related to frequency doubling ($\omega_{NL}=2\omega_1$ or $2\omega_2$) and rectification ($\omega_{NL}=0$), sum-frequency ($\omega_{NL}=\omega_1+\omega_2$) and difference-frequency generation ($\omega_{NL}=\omega_1-\omega_2$). The terms contributing to the latter ($\equiv \omega_3$) are given by $$\begin{split}
\mathcal{P}^3(z,x,t)&=\epsilon_0\chi^{(2)}\operatorname{Re}\left\{\left(E_1^+E_2^{+*}\thinspace\exp\left[i\left(k_z^1-k_z^2\right)z\right]\right.\right.\\
&\quad\left.\left.+E_1^+E_2^{-*}\thinspace\exp\left[i\left(k_z^1+k_z^2\right)z\right]\right.\right.\\
&\qquad\left.\left.+E_1^-E_2^{+*}\thinspace\exp\left[-i\left(k_z^1+k_z^2\right)z\right]\right.\right.\\
&\qquad\quad\left.\left.+E_1^-E_2^{-*}\thinspace\exp\left[-i\left(k_z^1-k_z^2\right)z\right]\right)\right.\\
&\qquad\qquad\left.\times\thinspace\exp\left(i\left[\left(k_x^1-k_x^2\right)x-\omega_3t\right]\right)\right\}.
\end{split}$$
Co-propagating waves ($\pm,\pm$) generate terms with perpendicular wavevector $k^{3-}_{z}=k_z^1-k_z^2$, whereas counter-propagating waves ($\pm,\mp$) generate terms with $k^{3+}_{z}=k_z^1+k_z^2$. Their contributions can be handled separately when pump depletion is ignored, so we divide the polarization term into two components
$$\begin{aligned}
\mathbf{P}^{3-}&=\epsilon_0\chi^{(2)}
\begin{bmatrix}
E_1^+E_2^{+*}\\E_1^-E_2^{-*}
\end{bmatrix}\\
\mathbf{P}^{3+}&=\epsilon_0\chi^{(2)}
\begin{bmatrix}
E_1^+E_2^{-*}\\E_1^-E_2^{+*}
\end{bmatrix},\end{aligned}$$
with their source fields given by Eq. (\[eqES\]) and the perpendicular component of $k^{NL}$ taking the values of $k_z^{3-}$ or $k_z^{3+}$, respectively.
In addition to the bound fields, there are also free fields with frequency $\omega_3$ that are solutions to the homogeneous wave equation. The free field in a nonlinear layer $j$ is obtained from the bound field amplitudes $\mathbf{E}_{js}$ and the boundary conditions at the interfaces. By imposing continuity of the total tangential electric and magnetic fields across interfaces i–j and j–k, an effective free field source vector can be defined as $$\mathbf{S}_j=\left(\phi_j^{-1}M_{js}\phi_{js}-M_{js}\right)\mathbf{E}_{js},
\label{eqS}$$ where the source matrices with the subscript $s$, $M_{js}$ and $\phi_{js}$, are identical to the ones given by Eqs. (\[eqtm\]) and (\[eqphi\]), with $k_{iz}$ and $k_{jz}$ taking the values of $k^3_{jz}$ and $k^{3\pm}_{jz}$, respectively.
The total nonlinear field is then given by the sum of independent source field vectors $\mathbf{S}_j$ propagated using the transfer matrix method reviewed in Sec. \[TMM\]. In particular, for the case where only layer $j$ is nonlinear, we obtain $$\begin{split}
\begin{bmatrix}
E_{3T}\\0
\end{bmatrix}
&=M_{N(N-1)}\cdots M_{21}
\begin{bmatrix}
0\\E_{3R}
\end{bmatrix}\\
&\quad+M_{N(N-1)}\cdots M_{(j+1)j}\mathbf{S}_j\\
&=T_N
\begin{bmatrix}
0\\E_{3R}
\end{bmatrix}
+
\begin{bmatrix}
R_j^+\\R_j^-
\end{bmatrix},
\label{eqEk1}
\end{split}$$
with $$\mathbf{R}_j=T_N{T_j}^{-1}\mathbf{S}_j.$$
Therefore, the reflected and transmitted components of the $\mathbf{E}_3$ field can be calculated by
$$\begin{aligned}
\label{eqE3R}
E_{3R}&=-\frac{R_j^-}{T_{22}}\\
\label{eqE3T}
E_{3T}&=R_j^+-\frac{T_{12}}{T_{22}}R_j^-.\end{aligned}$$
The angle dependence of the reflected difference-frequency field can be expressed as $$|k_3|\thinspace\sin\theta_3^{\pm}=|k_1|\thinspace\sin\theta_1\pm|k_2|\thinspace\sin\theta_2,$$
where the $\pm$ sign must match the wavevector component $k^{3\pm}_z$ when both pumps are incident on the same side of the normal.[@PhysRev.128.606] Because the first layer is taken to be air with $n(\omega)=1$, if we consider both pumps to be incident with the same angle $\theta_1=\theta_2=\theta_i$, we obtain for the cases of $k^{3-}_z$ and $k^{3+}_z$
$$\begin{aligned}
\label{eqAng3-}
\sin\theta_3^{-}&=\frac{\omega_1\thinspace\sin\theta_i-\omega_2\thinspace\sin\theta_i}{\omega_1-\omega_2}=\sin\theta_i\\
\label{eqAng3+}
\sin\theta_3^{+}&=\left(\frac{\omega_1+\omega_2}{\omega_1-\omega_2}\right)\thinspace\sin\theta_i.\end{aligned}$$
Equation (\[eqAng3-\]) shows that the DFG component due to co-propagating waves exits the structure at the same angle as the incident pumps, resembling the law of reflection. Conversely, according to Eq. (\[eqAng3+\]), the component due to counter-propagating pump waves is very sensitive to any angle mismatch between the pumps and easily becomes evanescent for small DFG frequencies.
\[Results\]Results
==================
Organic polymer cavity
----------------------
In this section, we investigate the use of organic microcavities for Rabi frequency generation. Due to the large binding energy of Frenkel excitons, organic microcavities can readily reach the strong coupling regime at room temperature and have shown Rabi splittings of up to 1 eV.[@ADOM:ADOM201300256; @ph500266d] Demonstrations of optical nonlinearities have been more limited than in their inorganic counterparts, but a variety of resonant[@PhysRevB.69.235330; @PhysRevB.74.113312] and non-resonant nonlinearities[@kena2010room; @daskalakis2014nonlinear; @plumhof2014room] have nevertheless been observed in these systems.
Although most organic materials possess a negligible second-order susceptibility, a number of poled nonlinear optical (NLO) chromophores have been shown to exhibit high electro-optic coefficients that exceed those of conventional nonlinear crystals such as LiNbO$_3$ by over an order of magnitude.[@Boyd; @doi:10.1021/cr9000429; @doi:10.1021/cr9000429] In addition, the metallic electrodes needed for polling can also be used as mirrors, providing high mode confinement and a means for electrical injection.
We will consider a thin NLO polymer film enclosed by a pair of metallic (Ag) mirrors of thicknesses 10 nm (front) and 100 nm (back). The model polymer is taken to possess a dielectric constant described by a single Lorentz oscillator $$\epsilon(\omega)=\epsilon_B+\frac{f{\omega_0}^2}{{\omega_0}^2-\omega^2-i\Gamma\omega},
\label{eqDL}$$
where $\epsilon_B$ is the background dielectric constant, $f$ is the oscillator strength, $\omega_0$ is the frequency of the optical transition and $\Gamma$ its full width at half maximum (FWHM). The parameters are chosen to be $\epsilon_B=4.62$, $f=0.91$, $\hbar\omega_0=1.55$ eV and $\hbar\Gamma=0.12$ eV. Experimental values are used for the refractive index of Ag.[@Rakic:98] For simplicity, we ignore the dispersive nature of the second-order nonlinear susceptibility and take $\chi^{(2)}=300$ pm/V. In principle, the Lorentz model could readily be extended to account for the dispersive resonant behavior.[@Boyd]
Figure \[figRmap\] shows the linear reflectance, calculated at normal incidence, as a function of polymer film thickness. The reflectance for film thicknesses below 200 nm shows only the fundamental cavity mode (M1), which is split into UP and LP branches. For these branches, the Rabi energy falls below the LP branch, where there are no further modes available for difference-frequency generation.
![\[figRmap\] Reflectance as a function of pump energy and thickness of the polymer film. Front and back Ag mirrors have thicknesses of 10 nm and 100 nm, respectively. Dielectric parameters: $\epsilon_B=4.62$, $f=0.91$, $\hbar\omega_0=1.55$ eV, $\hbar\Gamma=0.12$ eV and $\chi^{(2)}=300$ pm/V. Dashed horizontal line indicates the exciton energy. At the thickness of 300 nm, indicated by a vertical dashed line, the M1 cavity mode is resonant with the difference-frequency generation of pumps 1 and 2 such that $E_{UP}-E_{LP}=\hbar\Omega_R=E_{M1}$.](Figure2b){width="3.4in"}
By increasing the thickness of the film, low-order modes shift to lower energies and provide a pathway for the DFG radiation to escape. For example, at 300 nm, a triple-resonance condition occurs where the Rabi splitting of the M2 cavity mode matches the M1 energy ($E_{UP}-E_{LP}=\hbar\Omega_R=E_{M1}=0.68$ eV). A second resonance occurs between M3 and LP because $E_{M3}-E_{LP}=E_{LP}=1.25$ eV, but with reduced modal overlap.
The enhancement in DFG irradiance from the microcavity, as compared to a bare nonlinear slab, is shown in Fig. \[figAgDFGmapF\] as a function of the pump energies. The two peaks correspond to the triple-resonance conditions mentioned above, where the left peak corresponds to an enhancement of $2.8\cdot10^2$ at the Rabi energy ($\lambda_3=1.82$ $\mu$m) and the right peak to an enhancement of $3.3\cdot10^2$ at the LP energy ($\lambda_{LP}=996$ nm).
![\[figAgDFGmapF\] DFG irradiance enhancement of the poled NLO polymer model structure with respect to a bare film of equal thickness. Due to the thickness of the second mirror, only reflected fields are considered. The tilted dashed lines correspond to pairs of pump energies that generate the same DFG energy and that match the M1 (left, $\hbar\omega_3=\hbar\omega_{M1}=0.68$ eV) and LP (right, $\hbar\omega_3=\hbar\omega_{LP}=1.25$ eV) energies in the triple-resonance condition. Inset: normalized electric field profiles of the relevant modes, illustrating the excellent modal overlap of the LP and UP branches.](Figure3b){width="3.4in"}
The inset shows the normalized electric field profiles of the relevant modes, which highlight the good modal overlap of the two pump fields in the strong-coupling regime. The small thickness of the front metallic mirror lowers the mutual orthogonality of different modes and accounts for the lack of symmetry of the fields with respect to the center of the film. This loss of orthogonality allows the overlap integral between M3 and LP to be non-zero and the enhanced DFG extraction due to the triple-resonance condition leads to the appearance of the second peak at $\hbar\omega_3=1.25$ eV in Fig. \[figAgDFGmapF\].
Additionally, oblique incidence of the pump beams can be used to tune the DFG energy. As indicated by Eq. (\[eqAng3+\]), the $k^{3+}_z$ component of the DFG signal rapidly becomes evanescent and therefore we shall consider only the $k^{3-}_z$ component. Figure \[figAngles1\] shows the dependence of DFG energy and irradiance on the angle of incidence when $\theta_1=\theta_2=\theta_i$. In the lower panel, as the interacting modes move to higher energies, the triple-resonance condition at the Rabi ($E_{UP}-E_{LP}$) energy is maintained for incidence angles up to 79$^o$. The maximum irradiance is obtained at 57$^o$ for $\hbar\omega_{NL}=0.72$ eV ($\lambda_{NL}=1.72$ $\mu$m). This enhancement is reduced by 3 dB at $\hbar\omega_3=0.74$ eV ($\lambda_3=1.68$ $\mu$m) for 79$^o$. The upper panel shows that the peak at $\hbar\omega_3=1.25$ eV falls out of the triple resonance condition faster with a 3 dB roll-off at 40$^o$.
![\[figAngles1\] Angle dependence of DFG energy and irradiance (kW/m$^2$) for TE polarized pumps incident on the structure with NLO polymer and Ag mirrors when $\theta_1=\theta_2=\theta_i$. Only waves with $k^{3-}_{z}=k_z^1-k_z^2$ are considered. Lower and upper panels show the DFG at the the Rabi and LP energies, respectively. Solid black lines illustrate the energies of the M1 (bottom) and LP (top) modes where DFG radiation can be extracted in triple-resonance. Dashed black lines illustrate a typical linewidth of 100 meV for the LP branch and 50 meV for the M1 mode. Solid white lines indicate the angle dependence of the DFG energy. For the upper panel, as the white line moves out of resonance with the black LP line, the DFG peak is suppressed. For the lower one, a slight increase is observed around 57$^o$ and corresponds to an enhancement of the triple-resonance condition, after which the irradiance rolls off.](Figure8){width="3.4in"}
(111) GaAs cavity
-----------------
The vast majority of resonant nonlinearities observed in inorganic semiconductor microcavities are due to a $\chi^{(3)}$ nonlinearity inherited from the exciton.[@PhysRevB.62.R4825] In the typical $\chi^{(3)}$ four-wave mixing process, two pump (p) polaritons interact to produce signal (s) and idler (i) components such that their wave-vectors satisfy $2\mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i$. Second-order susceptibilities tend to be much larger than their $\chi^{(3)}$ counterparts, but conventionally used (001)-microcavities only allow for nonlinear optical mixing between three orthogonally polarized field components.
A number of commonly used inorganic semiconductors are known to be non-centrosymmetric and to possess high second-order susceptibility tensor elements. Examples include III-V semiconductors, such as gallium arsenide (GaAs) and gallium phosphide (GaP), and II-VI semiconductors, such as cadmium sulfide (CdS) and cadmium selenide (CdSe).[@Boyd; @shen1984principles] To allow for the nonlinear optical mixing of co-polarized waves to occur, we will consider (111) GaAs as the microcavity material,[@1.4833545; @Buckley:14] in contrast to the typical (001)-oriented material.
We consider a $\lambda/2$ (111) bulk GaAs microcavity sandwiched between 20 (25) pairs of AlAs/Al$_{0.2}$Ga$_{0.8}$As distributed Bragg reflectors (DBRs) on top (bottom). The structure is followed by a bulk GaAs substrate with the same dielectric constant as the cavity material, modeled by Eq. (\[eqDL\]) with experimental values $\epsilon_B=12.53$, $f=1.325\cdot10^{-3}$, $\hbar\omega_0=1.515$ eV and $\hbar\Gamma=0.1$ meV.[@PhysRevB.52.1800] Experimental values are also used for the refractive index of Al$_x$Ga$_{1-x}$As.[@1.336070] The nonlinear susceptibility was kept the same as for the NLO polymer ($\chi^{(2)}=300$ pm/V) to allow for a direct comparison of the irradiances. The absolute value chosen has no effect on the enhancement factor. In practice, the largest contribution to the background $\chi^{(2)}$ in GaAs is due to interband transitions and for simplicity we ignore the resonant contribution to $\chi^{(2)}$.
The enhancement in DFG irradiance as compared to a bare GaAs slab of equal thickness is shown in Fig. \[figDBRDFGmapF\]. Due to the much smaller oscillator strength in GaAs, as compared to the NLO polymer, the Rabi splitting of $\hbar\omega_3=5.52$ meV falls in the THz range ($\nu_3=1.33$ THz) with an enhancement of $8.8\cdot10^3$.
![\[figDBRDFGmapF\] DFG enhancement of a $\lambda/2$ (111) GaAs cavity structure with respect to a bare slab. GaAs parameters: $\epsilon_B=12.53$, $f=1.325\cdot10^{-3}$, $\hbar\omega_0=1.515$ eV and $\hbar\Gamma=0.1$ meV.[@PhysRevB.52.1800] The same value of $\chi^{(2)}=300$ pm/V was used as for the NLO polymer. Due to the presence of the substrate, only reflected fields are considered. The tilted dashed line corresponds to pairs of pump energies that generate the same DFG energy. Inset: normalized electric field profiles inside the GaAs layer illustrating the excellent modal overlap of the LP and UP branches.](Figure4b){width="3.4in"}
Figure \[figAngles2\] shows the angle dependence of the DFG energy and irradiance when $\theta_1=\theta_2=\theta_i$. The dashed black line in the upper panel traces the DFG energy, where a logarithmic scale for the irradiance was used due to its rapid decrease with angle of incidence. The lower panel shows a segment of the same data on a linear scale. Tunability down to 3 dB can be obtained up to $\hbar\omega_3=7.21$ meV ($\nu_3=1.74$ THz) at 17$^o$.
![\[figAngles2\] Angle dependence of DFG energy and irradiance (W/m$^2$) for TE polarized pumps incident on the $\lambda/2$ (111) GaAs structure with DBR mirrors when $\theta_1=\theta_2=\theta_i$. Only waves with $k^{3-}_{z}=k_z^1-k_z^2$ are considered. The upper panel shows the angle dependence of DFG irradiance in logarithmic scale, with the dashed black line tracing the DFG energy. The lower panel shows a smaller angular range of the same data in linear scale where a fast decrease of DFG irradiance can be observed as the angle of incidence increases.](Figure9){width="3.4in"}
\[Discussion\]Discussion
========================
In Sec. \[Results\] we showed that the use of polaritonic modes for Rabi frequency generation can lead to irradiance enhancements of almost four orders or magnitude with respect to bare nonlinear slabs. Quantitative estimates can be obtained by considering equal pump irradiances $I_1=I_2=10$ GW/m$^2$. Figure \[figIcomp\] shows the maximum DFG irradiances for the two structures and the reference slabs. For the NLO film with Ag mirrors, the calculated peak DFG irradiances are $I_{DFG}=7.69$ kW/m$^2$ at $\hbar\omega_3=0.68$ eV and $I_{DFG}=4.05$ kW/m$^2$ at $\hbar\omega_3=1.25$ eV. For the $\lambda/2$ (111) GaAs microcavity with DBRs, we find $I_{DFG}=45$ W/m$^2$ at $\hbar\omega_3=5.52$ meV.
![\[figIcomp\] Comparison of the calculated DFG irradiances for the two structures studied. Solid blue (dash-dot red) line represents the NLO polymer (GaAs) cavity with Ag (DBR) mirrors and dotted lines directly below represent the corresponding bare slabs. Top blue (bottom red) energy scale relates to the NLO polymer (GaAs) cavity. The curves have been extracted from the maps shown in Fig. \[figAgDFGmapF\] and Fig. \[figDBRDFGmapF\] by picking out the maximum values among all pairs of pump energies that generate the same DFG energy. Pump irradiances are $I_1=I_2=10$ GW/m$^2$. ](Figure7d){width="3.4in"}
For the organic microcavity, Fig. \[figIcomp\] shows that the irradiance due to DFG at the Rabi energy exceeds the one at the LP energy, as expected due to the higher mode overlap. In Fig. \[figAgDFGmapF\], however, a higher enhancement was found at the LP energy. This apparent contradiction arises from normalizing each point by the corresponding DFG irradiances of the bare polymer slab.
There is also a substantial difference in cavity field enhancement for both material sets. Metal losses in the polymer cavity prevent a significant enhancement of the UP and LP electric fields with $|E_{peak}/E_{in}|=1.2$, where $E_{peak}$ and $E_{in}$ are the peak and incident fields, respectively. In contrast, for the GaAs microcavity an enhancement of 15 is obtained. Despite this field enhancement, the irradiance shown in Fig. \[figIcomp\] is 170 times lower at the Rabi energy for the inorganic microcavity than for the organic one. This is a consequence of the $\omega_{NL}^2$ factor in the source field given by Eq. (\[eqES\]), making DFG at smaller energies increasingly difficult.
Finally, we can use Fig. \[figIcomp\] to evaluate the tunability of the structures at normal incidence. For the first structure, the FWHM of the $\hbar\omega_3=0.68$ eV DFG peak is 0.045 eV, indicating that the same structure can be used for DFG generation from 1.76 $\mu$m to 1.88 $\mu$m by adjustment of the pumps only. For the GaAs structure, the FWHM of the $\hbar\omega_3=5.52$ meV DFG peak is 0.12 meV, indicating a tunability from $\nu_3=1.32$ THz to $\nu_3=1.35$ THz.
We should note that although in our calculation two pumps were used, similar enhancements are anticipated for (spontaneous) parametric fluorescence ($I_2=0$). In addition, the triply-resonant scheme introduced for the organic microcavity where the signal is resonant has further consequences. First, coupled-mode theory analysis of triply-resonant systems has shown the existence of critical input powers to maximize nonlinear conversion efficiency.[@Burgess:09; @Burgess:09OE] These are found to be inversely proportional to the product of the Q-factors. Lower Q-factors are thus advantageous for high power applications. Second, the scheme is also well-suited for realizing a more conventional $\chi^{(2)}$ polariton OPO. In this case, the oscillation threshold can be shown to depend inversely on the product of Q-factors.
Since in general, any $\chi^{(2)}$ medium will also have a non-zero $\chi^{(3)}$, these structures will display a change in refractive index proportional to the square of the applied electric field, an effect known as self/cross-phase modulation. The power dependance of the refractive index can lead to rich dynamics such as multistability and limit-cycle solutions.[@PhysRevA.83.033834; @PhysRevA.89.053839]
\[Conclusion\]Conclusion
========================
We studied the potential for generating Rabi-frequency radiation in microcavities possessing a non-vanishing second-order susceptibility. Using a semiclassical model based on nonlinear transfer matrices in the undepleted pump regime, we calculated the Rabi splitting and the DFG irradiance enhancement for an organic microcavity, composed of a poled nonlinear optical polymer, and for an inorganic one, composed of GaAs. In the first case, we obtained a Rabi splitting of $\hbar\omega_3=0.68$ eV ($\lambda_3=1.82$ $\mu$m) and an enhancement of two orders of magnitude, as compared to a bare polymer film. In the second case, we found a Rabi splitting of $\hbar\omega_3=5.52$ meV ($\nu_3=1.33$ THz) and an enhancement of almost four orders of magnitude, as compared to a bare GaAs slab. These results show the potential of the use of polaritonic modes for IR and THz generation. Both model structures display a high degree of frequency tunability by changing the wavelength and angle of incidence of the incoming pump beams. Similar enhancements are anticipated for parametric fluorescence and the triply-resonant scheme introduced for the optical microcavity can be exploited to realize monolithic $\chi^{(2)}$ OPOs.\
FB and SKC acknowledge support from the Natural Sciences and Engineering Research Council of Canada. SDL acknowledges support from the Engineering and Physical Sciences Research Council (EPSRC), research grant EP/L020335/1. SDL is Royal Society Research Fellow.
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abstract: 'We apply noncommutative geometry to a system of N parallel D-branes, which is interpreted as a quantum space. The Dirac operator defining the quantum differential calculus is identified to be the zero-momentum mode of the supercharge for strings connecting D-branes. As a result of the calculus, Connes’ Yang-Mills action functional on the quantum space reproduces the dimensionally reduced U(N) super Yang-Mills action as the low energy effective action for D-brane dynamics. Several features that may look [*ad hoc*]{} in a noncommutative geometric construction are shown to have very natural physical or geometric origin in the D-brane picture in superstring theory.'
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ł Ł ø Ø
UU-HEP/96-07\
NI9018\
.5in
[**Noncommutative Geometry and D-Branes** ]{}
.5in Pei-Ming Ho and Yong-Shi Wu\
.3in [*Department of Physics, University of Utah\
Salt Lake City, Utah 84112*]{}
.5in
Introduction
============
D-branes [@DLP] are extended dynamical objects in string theory, on which string endpoints can live (having the Dirichlet boundary condition). In recent developments, recognition [@Pol; @PCJ] of these nonperturbative degrees of freedom has played a central role in understanding string-dualities, M-theory unification, and small distance structure of space (or space-time) [@Reviews; @DKPS]. One remarkable feature of D-branes is that when there are $N$ parallel D-branes, their coordinates are lifted to $N \times N$ matrices [@Wit], and their low-energy dynamics is described by dimensional reduction of ten-dimensional $U(N)$ super Yang-Mills gauge theory. This reminds us of noncommutative geometry [@Con], in which coordinates as local functions are allowed to be noncommuting. Indeed, there are striking similarities between the D-brane dynamics and the non-commutative geometric construction of the standard model [@CL]: the parallel D-branes versus the multi-sheet space-time, the inter-brane connections versus the Higgs fields, and so on. Moreover, noncommutative geometric features also appear in a recently conjectured light-cone formulation for eleven-dimensional M-theory [@BFSS]. We feel, as a warm-up for exploring the possible uses of noncommutative geometry in string theory including M-theory, it is instructive to first examine more closely the connection between noncommutative geometry and D-brane dynamics.
In string theory, one is used to start from bosonic degrees of freedom. For a D$p$-brane in bosonic string theory, at low energies the dynamics is described by two kinds of fields living on the brane. Let us denote the coordinates on the D-brane by $y^i$, where $i=0,1,2,\cdots,p$. There is the $U(1)$ gauge field $A^i(y)$, $i=0,1,\cdots,p$, coupled to the motion of string endpoints in the tangential directions on the brane. There is also the Higgs field $\Phi^a(y)$, $a=p+1,p+2,\cdots, 25$, corresponding to vibrations of the D-brane (or the motion of string endpoints) in the normal directions. The effective action of a D-brane is the Dirac-Born-Infeld action [@Lei], whose leading term in the gradient expansion is the usual Maxwell action. In superstring theory, the content of the fields living on a D-brane is enlarged to include $\Psi$, the fermionic super-partners of $A^i$ and $\Phi^a$.
When there are $N$ parallel D-branes (with microscopic separations), all fields $A^i$, $\Phi^a$ and $\Psi$ become anti-Hermitian $N\times N$ matrices [@Wit], describing the effects of short open strings ending on different D-branes. At low energies and to the leading order in the gradient expansion, the effective action for such a system in superstring theory should be the dimensionally-reduced $U(N)$ super Yang-Mills action [@Wit] \[SUSY-YM\] S= d\^[p+1]{}y Tr(-F\_F\^+), where $\m,\n=0,1,\cdots,9$, and $\Psi$ is a Majorana-Weyl spinor in 10 dimensions. Both $A_{\m}$ and $\Psi$ are in the adjoint representation of $U(N)$. We will use $ i,j,k, \cdots $ for indices of values $0,1,\cdots,p$, and $a,b,c, \cdots $ for indices of values $p+1,p+2,\cdots,9$. In this convention, $F_{\m\n}$ splits into &F\_[ij]{}=\_i A\_j-\_j A\_i+\[A\_i,A\_j\],\
&F\_[ia]{}=\_i \_a+\[A\_i,\_a\]\_i \_a,\
&F\_[ab]{}=\[\_a,\_b\], due to dimensional reduction from 10 to $p+1$. Similarly, =\^i\_i+\^a\[\_a,\]. Explicitly, the low energy effective action for $N$ D-branes is \[eff-act\] S&=&d\^[p+1]{}y Tr(-F\_[ij]{}F\^[ij]{} -\_i\_a\^i\^a -\[\_a,\_b\]\[\^a,\^b\]\
&&+(\^i\_i+\^a\[\_a,\])).
On the other side, Connes’ noncommutative geometry [@Con] generalizes differential calculus and geometry to spaces, called “quantum spaces”, on which the algebra of functions (including coordinates) is noncommutative. In this generalization, ordinary smooth manifolds may allow new noncommutative differential calculi. Noncommutative geometric ideas have been used to reformulate the action for the standard model [@CL] and the $SU(5)$ grand unified theory [@CFF1]. This is done by starting from a certain noncommutative algebra acting on the fermion fields and then introducing appropriate Dirac operator to formulate Connes’ action functional (including both Yang-Mills and fermion parts) on a multi-sheet space-time, with the inter-sheet distances directly related to the vacuum expectation values of the Higgs fields. In ref. [@Cha], some supersymmetric Yang-Mills actions are reformulated as Connes’ action functional on certiain quantum spaces. In this note, we will show that the D-brane action (\[eff-act\]) can be rewritten as Yang-Mills-Connes action functional on a quantum space representing D-branes.
In Sec.\[QC-YM\] we will first review basics of quantum differential calculus and Yang-Mills gauge theory on a quantum space, and then in Sec.\[QC-D\] we will consider a certain class of quantum spaces in detail, which is used in this paper to model a system of $N$ D-branes. Subsequently we deduce the Dirac operator that defines the desired quantum calculus from the string supercharge in Sec.\[D-S\] and find in Sec.\[D-Q\] that the corresponding Yang-Mills-Connes functional for this calculus is equivalent to the action (\[eff-act\]). In Sec.\[D-T\] we comment on the relation of T-duality with the choice of the Dirac operator. In the concluding section, we summarize in retrospect several features, which look [*ad hoc*]{} in a generic noncommutative geometric construction but become very natural when put in the context of D-branes in string theory.
Yang-Mills-Connes Functional {#QC-YM}
============================
A quantum space is described by a $*$-algebra[^1] of functions $\cA$ on the quantum space. A differential calculus on a quantum space is an extension of $\cA$ to a graded $*$-algebra $\O^*(\cA)=\oplus_{n=0}\O^{(n)}(\cA)$, where $\O^{(0)}(\cA)=\cA$ and $\O^{(n)}(\cA)$ are right $\cA$-modules.[^2] An element in $\O^{(n)}(\cA)$ is called a differential form of degree $n$ or an $n$-form. The differential algebra $\O^*(\cA)$ also needs to be equipped with the exterior derivative $d$. The exterior derivative is a map $d:\O^{(n)}(\cA)\rightarrow\O^{(n+1)}(\cA)$ satisfying the graded Leibniz rule $d(\o_1\o_2)=(d\o_1)\o_2+(-1)^{n_1}\o_1(d\o_2)$ for $\o_1\in\O^{(n_1)}(\cA)$ and $\o_2\in\O^*(\cA)$ and the nilpotency condition $d^2=0$. Typically an element in $\O^{(n)}(\cA)$ can be written as $\xi^{\m_1}\cdots \xi^{\m_n}a_{\m_1\cdots\m_n}$ for some $a_{\m_1\cdots\m_n}\in\cA$, where $\{\xi^{\m}\}$ is a basis of one-forms in $\O^{(1)}(\cA)$.
In Connes’ formulation of noncommutative geometry [@Con], all information about a quantum space is encoded in the spectral triple $(\cA,D,\cH)$, where $D$ is an anti-self-adjoint operator (called the Dirac operator) acting on $\cH$, which is a Hilbert space with a $*$-representation $\pi$ of $\cA$, namely, the $*$-anti-involution is realized as the Hermitian conjugation: $\pi(a^*)=\pi(a)^{\dagger}$, $\forall a\in\cA$, where the Hermitian conjugation is denoted by $\dagger$. Using $D$ one can define a noncommutative differential calculus $\O^{*}(\cA)$ and extend $\pi$ to a representation $\pih$ of $\O^*(\cA)$ [@CFF0]. This procedure is shown by an example in Sec.\[QC-D\].
Other essential ingredients of a quantum space are the inner product and integration over $\O^*(\cA)$. Let $\la\cO\ra_{\cH}$ denote the regularized average of an operator $\cO$ on the Hilbert space $\cH$, e.g. \[inner\] \_=\_ , where $\Lambda$ is the cutoff for the spectrum of the Dirac operator. The inner product on $\O^{(n)}(\cA)$ is defined by [@CFF0] [^3] ø\_1|ø\_2\_[Ø]{}=(ø\_1)\^(ø\_2)\_ for $\o_1,\o_2\in\O^{(n)}(\cA)$. The integration of $\o\in\O^*(\cA)$ is defined by [@CFF0] ø=(ø)\_, where $\pih$ is a representation of $\O^*(\cA)$ on $\cH$.
The distance between two states in the Hilbert space is defined by [@Con] \[dist\] dist(\_1,\_2) =sup{ |\_1|(a)\_1-\_2|(a)\_2|: a, |(da)|\^2 1}, where $\|\cdot\|$ is the operator norm and $\la\cdot|\cdot\ra$ is the inner product on the Hilbert space.
A gauge field theory fits into this framework easily. The group of unitary elements $\cU=\{u: uu^* =u^* u=1, u\in\cA\}$ in $\cA$ acts on $\cH$ as a group of transformations, which is identified with the gauge group. For example, if $\cA$ is the algebra of $N\times N$ matrices of complex functions on a manifold, the group $\cU$ is the gauge group $U(N)$ on the manifold. The gauge field is a one-form $A=\xi^{\m}A_{\m}\in\O^{(1)}(\cA)$ for some $A_{\m}\in\cA$. It transforms under $u\in\cU$ as $A\rightarrow A^u=uAu^*+u(du^*)$, which implies that $\pih(A)\rightarrow \pih(A^u)=U\pih(A)U^{\dagger}+U[D,U^{\dagger}]$, where $U=\pih(u)$ (and $U^{\dagger}=\pih(u^*)$). The modified Dirac operator $\Dt=D+\pih(A)$ is covariant. The gauge-covariant field strength is defined as usual by $F=dA+A^2$. The only new ingredient so far in this straightforward generalization is the quantum differential calculus behind each expression above. The Yang-Mills-Connes action functional defined by [@Con] \[YM-M\] S=F|F \_[Ø]{} + |, where $\la\cdot|\cdot\ra$ is the inner product on $\cH$, is another natural but non-trivial ingredient of the noncommutative generalization. For a non-Abelian gauge field $A=dx^{\m}A_{\m}$ on a classical manifold, the distance defined by (\[dist\]) with $\Dt=\dels+\As$ between two vectors in the fibers located at two points on the manifold is the length of the shortest path on the manifold which connects the two vectors by parallel transport [@Con1].
A Class of Quantum Calculi {#QC-D}
==========================
A basic idea in applying noncommutative geometry to field theory is that the geometry of the relevant quantum space is determined by matter or, more precisely, by fermion fields. Thus one is led to consider a special class of quantum calculi, where the $*$-algebra $\cA$ is a noncommutative algebra acting on fermion fields, and the Dirac operator acting on the fermions is of the form \[gD\] D=\^D\_, where the $\g^{\m}$’s are usual $\g$-matrices. The Hilbert space $\cH$ is one for fermions, of the form $\cH=S\otimes\cH_0$, where $S$ is a representation of the Clifford algebra (e.g. $S=\C^{32}$ for a Dirac spinor in 10 dimensions), and $\cH_0$ is the Hilbert space in which the algebra $\cA$ acts with a representation $\pi_0$. The representation of $a\in\cA$ on $\cH$ is $\pi(a)=1\otimes\pi_0(a)$. From now on we will suppress the symbol of tensor product $\otimes$.
In the universal differential calculus $\O^*\cA$ [@Con], a differential one-form $\r\in\O^{(1)}\cA$ is a formal expression $\r=\sum_{\a}a_{\a}db_{\a}$, where $a_{\a}$, $b_{\a}$ are elements in $\cA$. To simplify the notation, we will omit the index $\a$ in the following.
With the help of the Dirac operator, the representation $\pi$ of $\cA$ on $\cH$ is extended to $\O^*\cA$ [@Con] by defining, for $\r\in\O^{(1)}\cA$, [^4] ()=a\[D,b\]. By (\[gD\]), it is $
\pi(\r)=\g^{\m}\sum a[D_{\m},b]=\g^{\m}\r_{\m}.
$ The representation of a two-form $\o=\sum adbdc$ is thus (ø)=a\[D,b\]\[D,c\], and similarly for forms of higher degrees.
In particular, the representation of $d\r=\sum dadb$ is \[dr\] (d) &=&\^ (\[D\_,\_\]-a\[\[D\_,D\_\],b\])\
&&+g\^(\[D\_,\_\]-a\[D\_,\[D\_,b\]\]), where $\g^{\m\n}=\frac{1}{2}[\g^{\m},\g^{\n}]$ and $g^{\m\n}=\frac{1}{2}\{\g^{\m},\g^{\n}\}$ is the metric.
The general differential calculus $\O^*(\cA)$ [@Con] is defined by the quotient Ø\^\*()=Ø\^\*/J, where $J=ker\pi + d(ker\pi)$. This means that two differential forms $\o_1$ and $\o_2$ of the same degree will be considered the same if $(\o_1-\o_2)\in J$.
To find $\O^{(2)}(\cA)$, the differential calculus of degree two, we consider a one-form $\r\in\ker\pi$. According to (\[dr\]), if the zero-curvature condition for the unperturbed Dirac operator \[DD\] \[D\_,D\_\]=0 is satisfied, then \[dr1\] (d)=-a\[g\^D\_D\_,b\](). For a non-Abelian gauge theory on a classical manifold, $D_{\m}$ can be $\del_{\m}$ plus a pure gauge to satisfy (\[DD\]).
Denote the degree-two component of $J$ by $J^{(2)}$, then $\pi(J^{(2)})$ is composed of all elements (\[dr1\]) for all $\r=\sum adb\in ker\pi$. We will focus on the cases for which $D_{\m}$ satisfies (\[DD\]) and \[JA\] (J\^[(2)]{})=().
The representation $\pi$ defined above is, in general, not a good representation of $\O^*(\cA)$ because one and the same differential form may admit many equivalent expressions of the form $\sum adbdc\cdots$, so that the representation is not unique.
A good representation is given by $\pih=P_J\circ\pi$ [@CFF0] where $P_J$ is the projection perpendicular to $\pi(J)$. By (\[dr\]) and (\[JA\]), it follows that for a two-form $\o$, (ø)=\^ø\_ for some $\o_{\m\n}\in\cA$. In particular, by (\[DD\]), \[dr2\] (d)= \^(\[D\_,\_\]-\[D\_,\_\]).
It can be shown that the conditions (\[DD\]) and (\[JA\]) also imply that for a three-form $\o$, (ø)=\^ø\_ for some $\o_{\m\n\k}\in\cA$, and similarly for higher degrees.
Let $\xi^{\m}$ denote the basis of one-forms which is represented by $\g$-matrices: $\pi(\xi^{\m})=\g^{\m}$. Then it follows that the calculus $\O^*(\cA)$ is generated by elements in $\cA$ and one-forms $\xi^{\m}$, where the $\xi^{\m}$’s anticommute with each other and commute with elements in $\cA$ as in the classical case. The only possible source of noncommutativity is $\cA$.
Dirac Operator and Supercharge {#D-S}
==============================
The low-energy dynamics of $N$ parallel D-branes is described by a field theory on the D-brane world volume. So we may try to reformulate the D-brane action (\[eff-act\]) in terms of the quantum calculus discussed in last section. The key is to find an appropriate Dirac operator for the fermion fields, which are massless spinor states of strings connecting the D-branes. The Dirac operator should be motivated from string theory: Indeed we find a natural candidate to be the supercharge operator for strings connecting D-branes, truncated in the subspace of massless spinor states.
It was shown by Witten [@Wit82] that the quantized zero-momentum modes in a supersymmetric non-linear $\sigma$-model in $1+1$ dimensions can be identified with the de Rham complex of the target space. The bosonic fields $X^{\m}$ are the coordinates on the target space. The fermionic fields $\psi^{\m}$ are Majorana spinors on the world sheet, which splits into two Majorana-Weyl spinors $\psi^{\m}_+, \psi^{\m}_-$ in $1+1$ dimensions. By canonical quantization, $\psi^{\m}_+$ and $\psi^{\m}_-$ satisfy two anticommuting sets of Clifford algebra: $\{\psi^{\m}_A,\psi^{\n}_B\}=g^{\m\n}\d_{AB}$, where $A, B=+, -$ and $g^{\m\n}$ is the metric of the target space. The supercharge $Q$ on the world sheet for zero-momentum modes is also a Majorana spinor and has two Weyl components $Q_+$ and $Q_-$: Q\_=\^\_P\_, where the momentum $P^{\m}$ acts on functions of $X^{\m}$ as a derivative. Let $Q=\frac{1}{2}(Q_+ +iQ_-)$ and $Q^*=\frac{1}{2}(Q_+ -iQ_-)$. It is remarkable that the supercharges $Q$ and $Q^*$ realize the exterior derivative $d$ and its adjoint $d^*$ [@Wit82]. $\psi^{\m}=\psi^{\m}_+ +i\psi^{\m}_-$ and $\psi^{\m *}=\psi^{\m}_+ -i\psi^{\m}_-$ correspond to differential one-forms and inner derivatives, respectively. Hermitian conjugation realizes Poincaré duality.
All these are also true for closed strings. For an open string with Neumann boundary conditions, however, certain modification is necessary, because the zero-momentum modes of the right-moving and left-moving sectors of $\psi^{\m}$ are identified. Then there is only one set of Clifford algebra, and the supercharge of zero-momentum modes has only one independent component \[Q-2\] Q\_0=\^\_0 P\_. After canonical quantization, $\psi^{\m}_0$’s become $\g$-matrices acting on massless spinor states, which are (after GSO projection) Majorana-Weyl spinors in the supermultiplet of a Yang-Mills theory in 10 dimensions [@GSW]. Being the Dirac operator for these spinors, the supercharge $Q_0$ realizes the exterior derivative in the target space in the sense of Connes.
Later it has also been argued by Witten [@Wit85] that the generalized Dirac operator in the full superstring theory (including nonzero-momentum sector), the so-called Dirac-Ramond operator, is the supercharge on the world sheet for the following three reasons: (1) Its zero-momentum limit is the usual Dirac operator. (2) It annihilates physical states. (3) It anticommutes with an analogue of the chirality operator: $(-1)^F$.
Motivated by these observations, we try to deduce in the following the Dirac operator for the quantum space representing D-branes from the supercharge on the world sheet of strings ending on D-branes.
It is well known [@DLP] that for open strings ending on a D$p$-brane, we have for $X^a$, $a=p+1,\cdots,9$, Dirichlet boundary conditions, which originate from $T$-duality for Neumann boundary conditions. Since $T$-duality simply reverses the relative sign on the left-moving and right-moving modes for both $X^{\m}$ and $\psi^{\m}$, the appropriate boundary conditions for $\psi^{\m}$ on a string with Dirichlet boundary conditions are still [@PCJ] \^\_+(0,)=\^\_-(0,),\^\_+(,)=\^\_-(,), (except changes in sign for dualized directions) of the same type as for open strings with Neumann boundary conditions. As mentioned above, the supercharge (\[Q-2\]) acts on the states of massless spinors. Upon identifying these massless string states with the field $\Psi$ in the super Yang-Mills theory on a 9-brane, the string supercharge reduces (or truncates) to the Dirac operator in 10 dimensional spacetime. For a D-brane of lower dimensions, the momentum operators in directions normal to the brane vanish due to the Dirichlet boundary conditions, hence the supercharge becomes a Dirac operator on the $(p+1)$ dimensional world volume of the D$p$-brane. However, to fully describe the dynamics of massless fields on a D-brane, one has to include the effects of the tadpole diagram for closed strings created from the D-brane. Although it was mentioned before that the supercharge has two components $Q,Q^*$ on a closed string, the boundary conditions on the brane identify $\psi^{\m}$ and $\psi^{\m*}$ up to a sign [@Pol] (after all, a closed string tadpole diagram can be viewed as an open string disk diagram), so half of the supersymmetry is broken. Hence the D-brane is a BPS state [@Pol] and there is only one independent component of the supercharge on a closed string created from a D-brane. The momentum of the closed string is shifted by the gauge field $\phi^a$ normal to the brane, which originates from a pure gauge transformation $\L^a=\phi^a x^a$ in the dual picture. Including contributions from both open and closed strings, the Dirac operator obtained by truncating the supercharge on strings ending on a D-brane is \[old-Dirac\] D=\^i\_i+\^a\_a.
D-branes as Quantum Space {#D-Q}
=========================
Now we are able to formulate precisely how to interpret the system of $N$ parallel D-branes (with microscopic separations) as a quantum space. We take a $p+1$ dimensional coordinate system on one of the branes as the world volume coordinates, and treat the structure arising from the strings connecting D-branes as the “internal” structure that defines a quantum space. (Closed string tubes connecting two branes can be viewed as loops of open strings ending on different branes.) Each D-brane has a label $r$ ($r=1,2,\cdots, N$), and an open (oriented) string from D-brane $r$ to D-brane $s$, and the states on such string may be labeled by an ordered pair, $(r,s)$, of indices. (These indices are also called Chan-Paton labels, since such an open string is dual to an open string with usual Chan-Paton labels $(r,s)$ [@PCJ].) In particular, the massless spinor states of the open string connecting D-brane $r$ and D-brane $s$ result in fermion fields living on the D-brane world volume, which therefore also carry the Chan-Paton labels $(r,s)$. Thus the fermion field $\Psi$ is an $N\times N$ matrix with entries being Majorana-Weyl spinors in 10 dimensions, which are naturally anti-Hermitian, belonging to the adjoint representation of $U(N)$: exchanging the labels $r$ and $s$ leads to inverting the orientation of the string. The Hilbert space on which the Dirac operator acts is taken to be the space of $N\times N$ matrices of Dirac spinors, which is larger than the configuration space of $\Psi$, since the Dirac operator always reverses the chirality.
To define a quantum space representing the D-branes, in addition to $\cH$ we need to specify the other two elements in the spectral triple $(\cA,D,\cH)$. Recall that the algebra $\cA$ defines the gauge group as the group $\cU$ of unitary elements of $\cA$. Hence we take $\cA$ to be $M_N(\C)\otimes
L^2(\R^{p+1})$, the algebra of $N\times N$ matrices of square-integrable functions on the $p+1$ dimensional D-brane world volume, [^5] so that $\cU$ is the $U(N)$ gauge group. The representation $\pi$ of $\cA$ on the fermion Hilbert space $\cH$ is simply the matrix multiplication. The Dirac operator is chosen to be a natural generalization of the operator (\[old-Dirac\]) to the multi-brane case. By introducing a Wilson line $A^a=\phi_a\equiv diag(\phi^a_1,\cdots,\phi^a_N)$ in the T-dual picture, the Dirac operator resulting from the (truncated) supercharge operator is found to be \[old-Dirac2\] D=\^i\_i+\^a\_a, where the $\phi_a$’s are $N\times N$ matrices, automatically satisfying \[phiphi\] \[\_a,\_b\]=0, a,b=p+1,,9. The first term in eq. (\[old-Dirac2\]) is the classical Dirac operator on the $(p+1)$ dimensional D-brane world volume. Viewing a $p$-brane as dimensional reduction of a $9$-brane, one can think of the second term as the remnant of the dimensionally reduced $(9-p)$ directions along which the partial derivatives $\del_a$ vanish but the pure gauge terms survive. (Recall the statement following (\[dr1\]).)
The quantum calculus considered in Sec.\[QC-D\] is applicable to the present case. Now let us show that the Yang-Mills-Connes functional (\[YM-M\]) with the Dirac operator (\[old-Dirac2\]) reproduces the super Yang-Mills action (\[eff-act\]) describing the dynamics of $N$ D-branes.
The Dirac operator (\[old-Dirac2\]) satisfies (\[DD\]) because of (\[phiphi\]), and for generic $\phi^a$ it also satisfies (\[JA\]), thus according to the discussions in Sec.\[QC-D\], for generic $\phi^a$ the calculus $\O^*(\cA)$ on D-branes is generated by $\cA$ and $dx^{\m}$. [^6] The only noncommutativity resides in $\cA$, the algebra of matrices $M_N(\C)\otimes L^2(\R^{p+1})$. The one-forms $dx^{\m}$ are represented by $\g$-matrices: $\pih(dx^{\m_1}\cdots dx^{\m_k})=\g^{\m_1\cdots\m_k}$ and so they anticommute with each other and commute with elements in $\cA$.
The gauge field is a one-form A=dx\^i A\_i+dx\^a A\_a, where $A_i$ and $A_a$ are required to be anti-Hermitian. It modifies the Dirac operator to \[new-Dirac\] =\^i\_i+\^a\_a, where $\nabla_i=\del_i+A_i$ and \_a=\_a+A\_a.
The field strength, F=dx\^dx\^F\_=dA+A\^2, is given by &F\_[ij]{}=\_i A\_j-\_j A\_i+\[A\_i,A\_j\],\
&F\_[ia]{}=\_i \_a+\[A\_i,\_a\]\_i\_a,\
&F\_[ab]{}=\[\_a,\_b\]-\[\_a,\_b\]=\[\_a,\_b\], where we have used (\[phiphi\]).
The first term in the Yang-Mills functional (\[YM-M\]) involes the trace of the Hilbert space in (\[inner\]), which is composed of three kinds of traces. The first is the trace of $\g$-matrices, which gives rise to the contraction of the components of two $F_{\m\n}$. The second is the trace over square-integrable functions on $\R^{p+1}$, which turns into the integration over the world volume of the D$p$-brane. The trace of $N\times N$ matrices remains explicit as in (\[eff-act\]). It is then straightforward to see that by constraining physical states to be anti-Hermitian $N\times N$ Majorana-Weyl spinors $\Psi$ after Wick rotation, the Yang-Mills functional (\[YM-M\]) for this quantum space is equivalent to the effective action (\[eff-act\]) for $N$ D-branes. Obviously the same formulation can be applied to other cases, for example, the $N=2$ super Yang-Mills theory in four dimensions as dimensionally reduced from $N=1$ super Yang-Mills theory in six dimensions.
T-duality and Dirac operator {#D-T}
============================
Using T-duality, we can define another Dirac operator by taking the T-dual of the supercharge on an open string with Neumann boundary conditions. Introducing a Wilson line in the dual picture: $A^a=diag(\phi^a_1,\cdots,\phi^a_N)$ in some compactified dimensions of radius $R^a$, the open string with Chan-Paton labels $(r,s)$ is T-dual to an open string stretching between two D-branes at positions $x^a_r=\phi^a_r$ and $x^a_s=\phi^a_s$ (or a simultaneous translation of them) [@PCJ]. The momentum $p^a$ in the compactified direction is shifted by the gauge field: p\^a=+(\^a\_s-\^a\_r), where $n$ is the quantum number for momentum $p^a$ in the dual picture and becomes the winding number in a compactified dimension of radius $R'^a=\a'/R^a$. As we are focusing on low energy modes, we set $n^a=0$.
To describe all string states at the same time, it is natural to put the $p^a$’s for all possible string configurations into an antisymmetric matrix P\^a\_[rs]{}=\^a\_s-\^a\_r, which is in the Lie algebra of $SO(N)\subset U(N)$. The Dirac operator as the “total” supercharge (\[Q-2\]) therefore becomes D\_[dual]{}=\^i\_i+\^[a]{}P\_a, where $i=0,\cdots,p$, and $a=p+1,\cdots,9$.
Consider the case of two D$8$-branes ($N=2$). The matrix $P^a$ ($a=9$) is (
[cc]{} 0 & (x\^a\_1-x\^a\_2)\
(x\^a\_2-x\^a\_1) & 0
). When the distance $|x^a_1-x^a_2|$ between two branes is large, the gauge group is $U(1)^2$. Hence the algebra of functions $\cA$ is taken to be diagonal $2\times 2$ matrices for this case. This is precisely the two-sheet model Connes considered [@Con] and the distance (\[dist\]) for this case is $|(x^a_1-x^a_2)|^{-1}$, the inverse of the actual distance. This is not surprising because we are using the Dirac operator obtained from the supercharge in the dual picture and T-duality inverses the length. It is interesting to note that if we take the inverse of every element of $P^a$ to “correct” this inversion in length: i.e. use P’\^a\_[rs]{}=(x\^a\_s-x\^a\_r)\^[-1]{} to replace $P^a$ in the Dirac operator for $N$ D$8$-branes, then the new Dirac operator will define the geometry of $N$-sheets separated by the actual distances $|(x^a_r-x^a_s)|$. The algebra $\cA$ in this case is the algebra of diagonal $N\times N$ matrices, as appropriate for the $U(1)^N$ gauge symmetry.
Discussions
===========
In this paper, we have interpreted the system of N parallel D-branes as a quantum space in the sense of noncommutative geometry. The associated Yang-Mills-Connes action functional on this quantum space is shown to reproduce the dimensionally reduced U(N) super Yang-Mills action as the low energy effective action for D-brane dynamics. To conclude, in this section we note in retrospect that several features that would look [*ad hoc*]{} in a noncommutative geometric construction actually have very natural physical or geometric interpretation in the D-brane picture in string theory.
First, the source of noncommutativity resides in the matrix algebra $\cA$, which arises naturally due to the Chan-Paton labels of the fermion fields, which in turn originate from the strings ending on different D-branes. In other words, parallel D-branes provide a physical realization of “multi-sheet space-time” and a geometric origin for the gauge group $U(N)$. One may wonder whether our universe could really be such a system of D-branes or, equivalently, have spacetime of a discrete Kaluza-Klein structure.
Second, the choice of the Dirac operator (\[old-Dirac2\]) is dictated by the D-brane picture, where the addition of the second term is due to the fluctuations in the position of the D-branes. In particular, the commutativity (\[phiphi\]) that makes the condition (\[DD\]) satisfied is not an ansatz as in usual noncommutative geometric reformulation of super Yang-Mills action [@Cha]; it is deduced here from T-duality of the D-branes: the inter-brane separations is dual to a Wilson line for pure gauge configuration [@PCJ; @Wit].
Third, in the Yang-Mills-Connes action functional (\[eff-act\]), the $\phi_a$ that is introduced in the unperturbed Dirac operator (\[old-Dirac2\]) appears only in the combination $\Phi_a
= A_a + \phi_a$. In the D-brane picture this is a reflection of the fact that $\phi_a$ stands for classical inter-brane separation, while $A_a$ its quantum fluctuations, as is consistent with $\phi_a$ being diagonal and constant and with the commutativity constraint (\[phiphi\]). In accordance to T-dulaity, in string theory it is the total $\Phi_a$ (together with $A_i$) that stands for the D-brane “coordinates” (divided by $\alpha'$, the string tension) lifted to a matrix [@Wit]. We note that such interpretation is not available in usual noncommutative geometric construction.
Finally, in general the Yang-Mills-Connes action functional is not necessarily supersymmetric. However, in the present case, our Yang-Mills-Connes action functional (\[eff-act\]) happens to be supersymmetric. This is closely related to the fact that we start with a very special fermion field content in a special dimensionality (dimensional reduction of a Majorana-Weyl spinor in ten dimensions), which is inherited from superstring theory.
From the above discussions, we see that there is a close relationship and deep internal consistency between noncommutative geometry (at least on discrete Kaluza-Klein space-time) and D-brane dynamics at low energies. An interesting question arises: whether or not this close relationship of D-brane dynamics with noncommutative geometry can be extended to a deeper level? (Either to the full D-brane dynamics which should be described by a supersymmetric and non-abelian generalization of the Dirac-Born-Infeld action, or to superstring theory or even M-theory.) This seems to call for a generalization of noncommutative geometry to superstrings or M-theory that incorporates D-branes.
Acknowledgement
===============
P.M.H. is grateful to Bruno Zumino for encouragements, Zheng Yin for discussions and the hospitality of the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge. This work is supported in part by the Rosenbaum fellowship (P.M.H.) and by U.S. NSF grant PHY-9601277.
.8cm
[*Note Added*]{}: When we are completing the paper, we learn that in a recent preprint of M. Douglas, hep-th/9610041, a comparison between the D-brane action and noncommutative geometric construction of the standard model action is briefly discussed (without much detail).
[10]{}
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[^1]: An algebra with a map $*$: $a\rightarrow a^*\in\cA$ which is an anti-involution, i.e., $(a^*)^*=a$ and $(ab)^*=b^* a^*$ for $a,b\in\cA$, is called a $*$-algebra.
[^2]: That is, if $\o\in\O^{(n)}(\cA)$ then $\o a\in\O^{(n)}(\cA)$ for all $a\in\cA$.
[^3]: The inner product and integration can also be defined in terms of the Dixmier trace [@Con]; what we use here is more familiar to physicists.
[^4]: We will simply write $a$ to stand for $\pi(a)$ for $a\in\cA$ in the following.
[^5]: It is also possible to define $\cA$ to be the algebra of the $U(N)$ gauge group represented in its adjoint representation on $\cH$.
[^6]: Strictly speaking, our notation for $dx^a$ ($a=p+1,\cdots,9$) is inappropriate because while $dx^i$ ($i=0,\cdots,p$) is exact, $dx^a$ is not exact but a closed one-form $\sum_{\a}a_{\a}db_{\a}$ for $a_{\a}$, $b_{\a}$ being some matrices in $\cA$.
|
---
abstract: 'We prove the non-vanishing of special $L$-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special $L$-values and considering Shintani cone decomposition.'
address:
- |
Ulsan National Institute of Science and Technology\
Ulsan\
Korea
- |
Ulsan National Institute of Science and Technology\
Ulsan\
Korea
author:
- JAESUNG KWON
- 'HAE-SANG SUN'
title: 'Non-vanishing of special $L$-values of cusp forms on GL(2) with totally split prime power twists'
---
Introduction
============
Studying the non-vanishing of special $L$-values is interesting as fascinating mathematical techniques and ingredients are involved. There are plenty of non-vanishing results from which important arithmetic consequences arise.
Rohrlich [@rohrlich1984onl] shows the non-vanishing of the special values of cyclotomic modular $L$-values by estimating the Galois averages of fast convergent series expressions of the special $L$-values. As a consequence, Rohrlich [@rohrlich1984onl] deduces that the Mordell-Weil groups of CM elliptic curves over the cyclotomic $\Z_p$-extension of $\Q$ are finitely generated.
One can consider more general settings. Friedberg and Hoffstein [@friedberg1995nonvanishing] shows the non-vanishing of automorphic forms of GL(2) for infinitely many quadratic twists by studying the infinite sum of twisted $L$-values, which is obtained by Rankin-Selberg convolution with metaplectic Eisenstein series on GSp(4). Rohrlich [@rohrlich1989nonvanishing] proves that the $L$-values of automorphic forms on ${\operatorname{GL}}(2)$ are non-vanishing for infinitely many GL(1) twists by modifying the idea of Shintani cone decomposition [@shintani1979aremark] to count the number of ideals with bounded norm in an arithmetic progression. Namikawa [@namikawa2017p] presents the non-vanishing result for cyclotomic $L$-values of automorphic forms of GL(2) when the weight is greater than two by constructing the Mazur-Tate-Teitelbaum $p$-adic $L$-function and using the non-vanishing result of Jacquet and Shalika [@jacquet1976anon]. Order [@order2019rankin] claims the non-vanishing of cyclotomic $L$-values of non-dihedral automorphic forms on GL(2) over totally real fields.
The main result of the present paper is to show the non-vanishing of the special $L$-values of automorphic forms on ${\operatorname{GL}}(2)$ over general number fields twisted by Hecke characters of $p$-power order and totally split prime power conductors, which is a generalization of Rohrlich [@rohrlich1984onl Theorem] or Luo and Ramakrishnan [@luo1997determination Proposition 2.2].
Main Theorems {#Main:Theorems}
-------------
Let us give some notations and settings. Let $F$ be a number field, $\fN$ an integral ideal of $F$, $S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$ the space of cuspidal automorphic forms over $F$ of a cohomological weight $(\mathbf{k},\mathbf{m})$ (see Namikawa [@namikawa2017p]), type $J$ and level $\fN$ with a central character $\chi$. Let $\fp$ be a prime ideal of $F$ lying above an odd prime number $p$ and $\Xi_\fp$ the set of Hecke characters of $p$-power order with $\fp$-power conductors. Let $f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$, $\varphi\in\Xi_\fp$, $L(s,f\otimes\varphi)$ the $L$-function attached to $f$ and $\varphi$, $K_f$ the Hecke field of $f$ over $\Q$, which is the field adjoining all the Fourier-Whittaker coefficients of $f$ to $\Q$, and $n_0:=\textrm{max}\{m\in\Z|\ \mu_{p^m}\subset K_f\}$.
To obtain fast convergent series expressions of twisted modular $L$-values, namely the approximate functional equation of the $L$-function, we have to find the Fourier-Whittaker expansions of automorphic forms and its Mellin transforms. Also we need Atkin-Lehner theory for automorphic forms of GL(2) over $F$. These ingredients will be developed in Section \[cuspform\] and \[sp:Lvalue\] together with a brief introduction to automorphic forms of GL(2) over $F$. Define the Galois averages of special $L$-values by $$L_{\textrm{av}}(f\otimes\varphi):=\frac{1}{[K_f(\varphi):K_f]}\sum_{\s\in\text{Gal}(K_f(\varphi)/K_f)} L\Big(\frac{k}{2},f\otimes\varphi^\s\Big) .$$
In Section \[esti:arith:progress\], we estimate the number of ideals of bounded norm and the lower bound of absolute norms of elements in an arithmetic progression, which make it possible to calculate the limit of the Galois averages for our case. These are achieved by using the coherent cone decomposition (see Rohrlich [@rohrlich1989nonvanishing]). In Section \[hecke:char\] and Section \[estimation\], we will discuss the Galois averages of twisted $L$-values and obtain their estimations, which play the key role in the proofs of main theorems. We would like to remark that for a prime $\fp$ of $F$ which does not split totally, the corresponding estimations of the Galois averages over the characters are worse than ones in the present paper due to the existence of the corank one group of $p$-adic units.
We set $\theta\in[0,1]$ as a bound of an exponent of the eigenvalues of a Hecke eigenform, in other words, $\theta$ is a number in $[0,1]$ such that for any $\e>0$ and integral ideals $\fa$ of $F$, one has $$\label{rama:peter:bdd:intro}
|a_f(\fa)|\ll_\e N(\fa)^{\frac{k-1}{2}+\theta+\e}$$ where $a_f(\fa)$ is the $\fa$-th Hecke eigenvalue of $f$ and $N$ is the norm map of $F/\Q$.
Let us assume that $\fp$ is a totally split prime and coprime to $h_F\fd_F\fN$. Then we obtain an estimation of the Galois averages of special $L$-values which depends on $\theta$:
\[main:thm:0:intro\] Suppose some parity condition (see (\[parity:cond\]) and Remark \[parity:cond:rem\]) on $F$, $(\mathbf{k},\mathbf{m})$ and $J$. For $a>1/|W|$, $\e>0$ and $\theta<\frac{1}{4}$, an estimation of $L_{\operatorname{av}}(f\otimes\psi_n)$ is given by $$\begin{aligned}
\begin{split}\label{main:thm:1:eq}
L_{\operatorname{av}}(f\otimes\psi_n)=a_f(\O_F)+o(1)
\end{split}\end{aligned}$$ as $n$ tends to the infinity.
Note that the more optimal $\theta$ we have, the better estimation on the error of the Galois averages we have. By applying Theorem \[main:thm:0:intro\] together with the bound $\theta=7/64$ obtained by Blomer-Brumley [@blomer2011ramanujan] and Nakasuji [@nakasuji2012generalized], we obtain the following non-vanishing result:
\[main:thm:1:intro\] Let $\fp$ be a totally split prime ideal of $F$ lying above $p$ and coprime to $h_F\fd_F\fN$. Suppose the parity condition on $F$, $(\mathbf{k},\mathbf{m})$ and $J$. For a non-zero eigenform $f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$ of Hecke operators $T_\fa$ for all integral ideals $\fa$ of $F$, we have $$L\Big(\frac{k}{2},f\otimes\varphi\Big)\neq 0$$ for almost all Hecke characters $\varphi$ over $F$ of $p$-power order and $\fp$-power conductor.
Kim-Sun [@kim2017modular] obtain partial results toward the mod $p$ non-vanishing of cyclotomic modular $L$-values by studying the homological nature of modular symbols. Sun [@sun2018generation] proves that the Hecke field of a newform can be generated by a single special value of the modular $L$-function, by studying an additive variant of the Galois average, which supersedes the non-vanishing results. The first named author plans to generalize the results in Kim-Sun [@kim2017modular] and Sun [@sun2018generation] to the current setting.
Notations
---------
Let us provide some notations which will be used globally in this paper. Let $F$ be a number field, $\A_F$ the adele ring of $F$, $\A_F^{(\infty)}$ the finite adele of $F$, $F_\infty:=F\otimes_\Q \R$ the infinite adele of $F$, $\O_F$ the integer ring of $F$, $\fd_F$ the different ideal of $F/\Q$, $d_F$ a finite idele of $\fd_F$, $D_F$ the discriminant of $F$, $h_F$ the class number of $F$, and $|\cdot|_{\A_F}=|\cdot|_{F_\infty}|\cdot|_{\A_F^{(\infty)}}$ the idelic norm. For any place $v$ of $F$, denote by $\O_{F,v}$ the completion of $\O_F$ with respect to $v$. Let $\widehat{A}:=A\otimes_\Z\widehat{\Z}$ for a $\Z$-algebra $A$. Let $p$ be an odd prime number coprime to $D_F$, $\zeta_m$ a primitive $p^m$-th root of unity and $\mu_m$ the set of $p^m$-th roots of unity. Let us set by $N$ the norm map of $F/\Q$ (or $\prod_{\fp|p}N_{F_\fp/\Q_p}$) and by ${\operatorname{Tr}}$ the trace map of $F/\Q$ (or $\sum_{\fp|p}{\operatorname{Tr}}_{F_\fp/\Q_p}$). For $z\in\C$, denote $\textbf{e}(z):=\textrm{exp}(2\pi i z)$.
Cusp forms {#cuspform}
==========
Cusp forms on GL(2)
-------------------
In this section, we will briefly give the definition of cuspidal automorphic forms on ${\operatorname{GL}}_2(\A_F)$ of cohomological weight. From now on, these are called cusp forms for simplicity. All the settings in this section come from Hida [@hida1994critical] or Namikawa [@namikawa2017p].
Let us give some notations. Let $I_F:=\text{Gal}(F/\Q)$ and $\Z[I_F]$ the free $\Z$-module generated by $I_F$. Let ${\operatorname{id}}$ and ${\operatorname{c}}\in I_F$ be the identity map and complex conjugation on $F$ (or on $\C$), respectively. Let $\Sigma(\R)$ be the set of real places of $F$, $\Sigma(\C)$ the set of complex places of $F$, and $J$ a subset of $\Sigma(\R)$. Note that $\Sigma(\R)$ and $\Sigma(\C)$ can be considered as subsets in $I_F$. Let $\textbf{k}=\sum_{\s\in I_F}k_\s \s$ and $\textbf{m}=\sum_{\s\in I_F}m_\s \s$ be elements in $\Z[I_F]$ satisfying following conditions:
1. $k_\s\geq 2$,
2. $k_\s+2m_\s=k_\tau+2m_\tau \text{ for any }\s,\tau\in I_F$,
3. $k_\s=k_{\s{\operatorname{c}}} \text{ for any }\s\in\Sigma(\C).$
Let $\textbf{t}=\sum_{\s\in I_F}\s$, $\textbf{n}=\textbf{k}-2\textbf{t}$, and $\textbf{n}^*=\sum_{\s\in I_F}n_\s^*\s\in\Z[I_F]$, where $n_\s^*=n_\s+n_{\s{\operatorname{c}}}+2$ for $\s\in\Sigma(\C)$ and $n_\s^*=0$ for $\s\in\Sigma(\R)\cup\Sigma(\C){\operatorname{c}}$. Let $\fN$ be a non-zero integral ideal of $F$ and define subgroups of ${\operatorname{GL}}_2(\widehat{\O}_F)$ by $$\begin{aligned}
U_0(\fN)&:=\bigg\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in {\operatorname{GL}}_2(\widehat{\O}_F) : c\in \widehat{\fN} \bigg\} , \\
U_1(\fN)&:=\bigg\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in U_0(\fN) : d\in 1+\widehat{\fN} \bigg\} .\end{aligned}$$ Let us denote $\mathbf{x}_\s=\left(\begin{smallmatrix} X_\s \\ Y_\s \end{smallmatrix}\right)$, $\mathbf{x}=\otimes_{\s\in\Sigma(\C)}\mathbf{x}_\s$ where $X_\s$, $Y_\s$’s are indeterminates. For a commutative ring $R$ with unity and $\mathbf{d}=\sum_{I_F}d_\s \s\in\Z[I_F]$, let $L(\mathbf{d},R):=\otimes_{\s\in I_F}L(d_{\s},R)$ where $L(d_{\s},R)$ is the space of homogeneous polynomials of variable $\mathbf{x}_\s$ of degree $d_\s$ with the coefficients in $R$. Note that there is a usual action of GL(2) on $L(\mathbf{d},R)$ (Namikawa [@namikawa2017p Section 3]). Let $C_{F,\infty}^+$ be the maximal compact subgroup of SL$_2(F_\infty)$, which is given by $$C_{F,\infty}^+=\prod_{\s\in\Sigma(\R)}\operatorname{SO}_2(\R)\times\prod_{\s\in\Sigma(\C)}\operatorname{SU}_2(\C) .$$
\[adel:cuspform:defn\] Let $\chi:F^\times\backslash\A_F^\times\rightarrow\C^\times$ be a Hecke character of modulus $\fN$ satisfying $\chi_\infty(z_\infty)=z_\infty^{-(\textbf{n}+\textbf{2m})}$ for $z_\infty\in F_\infty^\times$. A cusp form on ${\operatorname{GL}}_2(\A_F)$ of weight $(\textbf{k},\textbf{m})$, a type $J$, a level $U_0(\fN)$, and a central character $\chi$, is a $C^\infty$-function $f:{\operatorname{GL}}_2(\A_F)\rightarrow L(\mathbf{n}^*,\C)$ such that
$D_\s f=\big(\frac{n_\s^2}{2}+n_\s\big)f$ for $\s\in I_F$, where $D_\s$ is the Casimir operator for ${\operatorname{GL}}_2$ corresponding to $\s$.
$f(\g z g u)=\chi(z)\chi_\fN(u)f(g)$ for $\g\in {\operatorname{GL}}_2(F)$, $z\in \A_F^\times$, $g\in{\operatorname{GL}}_2(\A_F)$ and $u\in U_0(\fN)$ where $\chi_\fN\big(\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)\big):=\prod_{v|\fN}\chi(d_v)$.
$f(g u_{\infty})(\mathbf{x})=\textbf{e}\big(\sum_{\s\in J}k_\s\theta_\s-\sum_{\s\in\Sigma(\R)\backslash J}k_\s\theta_\s\big)f(g)\big(\otimes_{\s\in\Sigma(\C)}u_\s\mathbf{x}_\s\big)$ for $g\in {\operatorname{GL}}_2(\A_F)$ and $$u_\infty=\bigg(\bigg(\begin{pmatrix} cos(2\pi\theta_\s) & sin(2\pi\theta_\s) \\ -sin(2\pi\theta_\s) & cos(2\pi\theta_\s) \end{pmatrix}\bigg)_{\s\in\Sigma(\R)},(u_\s)_{\s\in\Sigma(\C)}\bigg)\in C_{F,\infty}^+ .$$
$\int_{F\backslash \A_F}f(\left(\begin{smallmatrix} u & 0 \\ 0 & 1 \end{smallmatrix}\right)g)(\textbf{s})du=0$ for $g\in {\operatorname{GL}}_2(\A_F)$ where $du$ is a Haar measure on $F\backslash\A_F$. From now on, we denote by $S_{(\textbf{k},\textbf{m}),J}(\fN,\chi)$ the space of the aforementioned cusp forms on ${\operatorname{GL}}_2(\A_F)$.
Let us denote by $K_j$ the $j$-th modified Bessel function of the second kind, and write $y_\infty=(y_\s)_\s\in F_\infty^\times$. Let $W_{\textbf{k},\textbf{m}}:F_{\infty}^\times\rightarrow L(\mathbf{n}^*,\C)$ be the Whittaker function defined by $$W_{\textbf{k},\textbf{m}}(y_\infty)(\mathbf{x}):=\prod_{\s\in\Sigma(\R)}W_{\textbf{k},\textbf{m},\s}(y_\s)\cdot\bigotimes_{\s\in\Sigma(\C)}W_{\textbf{k},\textbf{m},\s}(y_\s)(\mathbf{x}_\s)$$ where $W_{\textbf{k},\textbf{m},\s}(y_\s):=|y_\s|^{-m_\s}\textbf{e}(i|y_\s|)$ for $\s\in\Sigma(\R)$ and $$\begin{aligned}
W_{\textbf{k},\textbf{m},\s}(y_\s)(\textbf{x}_\s):=&\sum_{j_\s=0}^{n_{\s}^*}\left(\begin{smallmatrix} n_{\s}^* \\ j_\s \end{smallmatrix}\right) y_\s^{-m_\s}{\overline}{y}_\s^{-m_{\s{\operatorname{c}}}}\bigg(\frac{y_\s}{i|y_\s|}\bigg)^{n_{\s{\operatorname{c}}}+1-j_\s} \\
&\times K_{j_\s-1-n_{\s{\operatorname{c}}}}(4\pi|y_\s|)X_\s^{n_{\s}^*-j_\s}Y_\s^{j_\s}\end{aligned}$$ for $\s\in\Sigma(\C)$. Then we have the Fourier-Whittaker expansion of a cusp form on ${\operatorname{GL}}_2(\A_F)$:
\[four:whit:exp:hida\] Let $\mathscr{F}$ be the group of fractional ideals of $F$. For $f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$, there exists a function $a_f:\mathscr{F}\rightarrow\C$ satisfying following properties:
$a_f(\fa)=0$ if $\fa\in\mathscr{F}$ is not integral.
We have the Fourier-Whittaker expansion of $f$ by $$f\bigg(\begpmat y & x \\ 0 & 1 \endpmat\bigg)(\mathbf{x})=|y|_{\A_F}\sum_{\xi\in F^\times,[\xi]=J}a_f(\xi y\fd_F)W_{\mathbf{k},\mathbf{m}}(\xi y_\infty)(\mathbf{x})\mathbf{e}_F(\xi x)$$ for $x,y\in\A_F$, where $[\xi]:=\{\s\in\Sigma(\R):\xi^\s>0\}$, $y_\infty\in F_\infty$ is the infinite part of $y$, and $\mathbf{e}_F:\A_F/F\rightarrow\C^\times$ is the additive character such that $\mathbf{e}_F(z_\infty):=\prod_{\s\in\Sigma(\R)\cup\Sigma(\C)}\mathbf{e}(\operatorname{Tr}_{F_\s/\R}(z_\s))$ for $z_\infty=(z_\s)_\s\in F_\infty$.
Approximate functional equations {#sp:Lvalue}
================================
Let $\fN$ be a integral ideal of $F$, $\chi$ a Hecke character of modulus $\fN$ satisfying $\chi_\infty(z_\infty)=z_\infty^{-(\textbf{n}+\textbf{2m})}$ for $z_\infty\in F_\infty^\times$, and $f\in S_{(\textbf{k},\textbf{m}),J}(\fN,\chi)$. In this section, we obtain an approximate functional equation of the $L$-function $L(s,f)$ of a cusp form $f$ in a spirit of Luo-Ramakrishnan [@luo1997determination].
Let $U_F:=F_\infty^\times\times\widehat{\O}_F^\times$ be the maximal compact subgroup of $\A_F^\times$. Let ${\operatorname{GL}}_2^+(F_\infty)=\prod_{\s\in\Sigma(\R)}{\operatorname{GL}}^+_2(\R)\times\prod_{\s\in\Sigma(\C)}{\operatorname{GL}}_2(\C)$, $F_{\infty,+}^\times:=\prod_{\s\in\Sigma(\R)}\R^\times_{>0} \times \prod_{\s\in\Sigma(\C)}\C^\times$ be identity connected components of $F_\infty^\times$ and ${\operatorname{GL}}_2(F_\infty)$, respectively. We can fix a representative $\{a_i\}_{i=1}^{h_F}\subset\A_F^{(\infty),\times}$ of the class group $\text{Cl}(F)\cong F^\times\backslash\A_F^\times/U_F$ of $F$ such that the corresponding integral ideals $\{\fa_i\}_{i=1}^{h_F}$ of $F$ are coprime to $\fN$. Let us set $t_i:=\left(\begin{smallmatrix} a_i & 0 \\ 0 & 1 \end{smallmatrix}\right)$, then by the strong approximation theorem, we have $$\begin{aligned}
\label{strong:approx}
F^\times\backslash\A_F^\times\cong \coprod_{i=1}^{h_F} a_i\cdot\big(\O_F^\times\backslash U_F\big) ,\
\text{GL}_2(\A_F)=\coprod_{i=1}^{h_F}\text{GL}_2(F) t_i\text{GL}_2^+(F_\infty)U_1(\fN) .\end{aligned}$$
Define a number $[\textbf{d}]\in\Z$ by $|a|^{2[\textbf{d}]}:=a^{\textbf{d}}a^{\textbf{d}{\operatorname{c}}}$ for $\textbf{d}\in\Z[I_F]$. In our case, $[\textbf{n}+2\textbf{m}]=n_\s+2m_\s$, which is independent on $\s\in I_F$ due to our assumption on the weight $(\mathbf{k},\mathbf{m})$. So from now on, we will write $k=[\textbf{n}+2\textbf{m}]+2$.
Recall that the $L$-function $L(s,f)$ of $f$ is given by the analytic continuation of the following Dirichlet series $$\begin{aligned}
\sum_{0\neq\fa<\O_F}\frac{a_f(\fa)}{N(\fa)^s} \text{ for }\mathfrak{R}(s)>\frac{k+2}{2}\end{aligned}$$ where $\fa$ runs over the set of nonzero integral ideals of $F$.
Integral representation of special $L$-values
---------------------------------------------
To obtain an approximate functional equation of the special $L$-value $L\big(\frac{k}{2},f\big)$, we need to compute the Mellin transform of $f$.
For $\mathbf{j}=\sum_{\s\in\Sigma(\C)}j_\s \s\in\Z[I_F]$, we define the $\mathbf{j}$-th component $f_\mathbf{j}$ of a $L(\textbf{n}^*,\C)$-valued function $f$ by $f_\mathbf{j}(g):=\prod_{\s\in \Sigma(\R)}f_\s(g)\prod_{\s\in \Sigma(\C)}f_{\s,j_\s}(g)$ where $g\in{\operatorname{GL}}_2(\A_F)$ and $$f(g)(\mathbf{x})=\prod_{\s\in\Sigma(\R)}f_\s(g)\cdot\bigotimes_{\s\in\Sigma(\C)}\sum_{j_\s=0}^{n_\s^*} f_{\s,j_\s}(g) X_\s^{n_\s^*-j_\s}Y_\s^{j_\s} .$$ By Proposition \[four:whit:exp:hida\], we obtain $$\begin{aligned}
\label{jth:comp:f}\begin{split}
f_\mathbf{j}\bigg(\begpmat y & x \\ 0 & 1 \endpmat\bigg)=&|y|_{\A_F}\sum_{\xi\in F^\times,[\xi]=J}a_f(\xi y\fd_F)W_{\mathbf{k},\mathbf{m}}^\mathbf{j}(\xi y_\infty)\textbf{e}_F(\xi x)
\end{split}\end{aligned}$$ where $$W_{\mathbf{k},\mathbf{m}}^\mathbf{j}(y_\infty):=\prod_{\s\in\Sigma(\R)}W_{\mathbf{k},\mathbf{m}}^{\s}(y_\s)\cdot\prod_{\s\in\Sigma(\C)}W_{\mathbf{k},\mathbf{m}}^{\s,j_\s}(y_\s)$$ and $W_{\mathbf{k},\mathbf{m}}^\s(y_\s)(\mathbf{x}_\s)=\sum_{j_\s=0}^{n_\s^*} W_{\mathbf{k},\mathbf{m}}^{\s,j_\s}(y_\s)X_\s^{n_\s^*-j_\s}Y_\s^{j_\s}$ for $\s\in\Sigma(\C)$. Let us set the Haar measure $d^\times y$ on $F^\times\backslash\A_F^\times$ which satisfies the following conditions: $$d^\times y_\sigma = \left\{ \begin{array}{ll}
\frac{dy_\sigma}{y_\sigma} & \textrm{if $\s\in\Sigma(\R)$}\\
\frac{drd\phi}{2\pi r} & \textrm{if $\s\in\Sigma(\C)$ and $y_\sigma=re^{i\phi}\in F_\s^\times$}\\
\end{array} \right.
\text{, } \int_{\O^\times_{F,v}}d^\times y_v=1\text{ for }v\nmid\infty .$$ Note that we can easily obtain the following equality: $$\{\xi\in F^\times: [\xi]=J\}=\{\xi\epsilon: \xi\in P_{F,J},\ \epsilon\in\O_{F,+}^\times\}$$ where $\O_{F,+}^\times$ is the group of totally positive units of $F$ and $P_{F,J}$ is a representative set $\{\xi\}$ of $\O_F^\times\backslash F^\times$ such that $[\xi]=J$. Thus by using (\[strong:approx\]), (\[jth:comp:f\]), and the above equation, the Mellin transform of $f_{\mathbf{n}^*/2}$ is given by
$$\begin{aligned}
\begin{split}\label{mellin:transform1}
&\int_{F^\times\backslash\A_F^\times}f_{\textbf{n}^*/2}\bigg(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix}\bigg)|y|^{s-1}_{\A_F} d^\times y \\
=&\sum_{i=1}^{h_F}\sum_{\epsilon\in\O_{F,+}^\times}\sum_{\xi\in P_{F,J}}\int_{\O_F^\times\backslash U_F}a_f(a_i\xi\epsilon y\fd_F)|a_i\xi\epsilon y|_{\A_F}^{s}W_{\mathbf{k},\mathbf{m}}^{\textbf{n}^*/2}(\xi\epsilon y_\infty)d^\times y \\
\end{split}\end{aligned}$$
for $\mathfrak{R}(s)>1+\max_{\s\in I_F}m_\s$. As the integral in the above equation is invariant under the change of variable $y\mapsto uy$ for $u\in\O_F^\times$, (\[mellin:transform1\]) becomes $$\begin{aligned}
\begin{split}\label{mellin:transform1'}
&\sum{}^*\int_{\O_F^\times\backslash U_F}a_f(a_i\xi\epsilon uy\fd_F)|a_i\epsilon uy|_{\A_F}^{s}W_{\mathbf{k},\mathbf{m}}^{\textbf{n}^*/2}(\xi\epsilon uy_\infty)d^\times y \\
=&\frac{1}{[\O_F^\times:\O_{F,+}^\times]}\sum_{i=1}^{h_F}\sum_{\xi\in P_{F,J}}\int_{\widehat{\O}_F^\times}a_f(a_i\xi y^{(\infty)}\fd_F)|a_i y^{(\infty)}|_{\A_F^{(\infty)}}^{s}d^\times y^{(\infty)} \\
&\phantom{blaaaaaaaaaaaaaaaank}\times\int_{F_\infty^\times}W_{\mathbf{k},\mathbf{m}}^{\textbf{n}^*/2}(\xi y_\infty)|y_\infty|_{F_\infty}d^\times y_\infty
\end{split}\end{aligned}$$ where $$\sum{}^*=\frac{1}{[\O_F^\times:\O_{F,+}^\times]}\sum_{u\in\O_F^\times/\O_{F,+}^\times}\sum_{\epsilon\in\O_{F,+}^\times}\sum_{i=1}^{h_F}\sum_{\xi\in P_{F,J}}.$$ By the following integration formula (Hida [@hida1994critical Section 7]) $$\int_0^\infty y^s{K_{j}(ay)}\frac{dy}{y}=2^{s-2}a^{-s}\Gamma\Big(\frac{s+j}{2}\Big)\Gamma\Big(\frac{s-j}{2}\Big)\text{ if }\mathfrak{R}(s\pm j)>0$$ and our assumption on the weight $(\mathbf{k},\mathbf{m})$, we have
$$\begin{aligned}
\begin{split}\label{mellin:transform2}
&\int_{F_{\infty}^\times} W_{\mathbf{k},\mathbf{m}}^{\textbf{n}^*/2} (\xi y_\infty)|y_\infty|^{s}_{F_\infty}d^\times y_\infty \\
=&\prod_{\s\in\Sigma(\R)}\int_{\R^\times} W_{\mathbf{k},\mathbf{m}}^\s(\xi^\s y_\s)|y_\s|^s d^\times y_\s \cdot\prod_{\s\in\Sigma(\C)}\int_{\C^\times} W_{\mathbf{k},\mathbf{m}}^{\s,n_{\s{\operatorname{c}}}+1}(\xi^\s y_\s)|y_\s|^{2s}d^\times y_\s \\
=&\prod_{\s\in\Sigma(\R)}\frac{2}{|\xi^\s|^{s}}\int_0^\infty e^{-2\pi y}y^{s-m_\s}\frac{dy}{y}\cdot\prod_{\s\in\Sigma(\C)}\frac{1}{|\xi^\s|^{2s}}\int_0^\infty \left(\begin{smallmatrix} n_\s^* \\ n_{\s{\operatorname{c}}}+1 \end{smallmatrix}\right) r^{2(s-m_\s)}K_0(4\pi r)\frac{dr}{r} \\
=&\frac{2^{|\Sigma(\R)|}}{|N(\xi)|^s}\prod_{\s\in\Sigma(\C)}\frac{1}{4}\left(\begin{smallmatrix} n_\s^* \\ n_{\s{\operatorname{c}}}+1 \end{smallmatrix}\right)\cdot\prod_{\s\in I_F}(2\pi)^{-(s-m_\s)}\Gamma(s-m_\s)
\end{split}\end{aligned}$$
for $\xi\in P_{F,J}$ and $\mathfrak{R}(s)>\max_{\s\in I_F}m_\s$. Combining the equations (\[mellin:transform1’\]) and (\[mellin:transform2\]), we obtain
$$\begin{aligned}
\begin{split}\label{mellin:transform}
&\int_{F^\times\backslash\A_F^\times}f_{\textbf{n}^*/2}\bigg(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix}\bigg)|y|^{s-1}_{\A_F} d^\times y \\
=&\Gamma_{F,\mathbf{k},\mathbf{m}}(s)\sum_{i=1}^{h_F}\sum_{\xi\in P_{F,J}}\frac{1}{N(\xi\fd_F)^s}\int_{\widehat{\O}_F^\times} a_f(a_i\xi y\delta_F)|a_i y|^{s}_{\A_F^{(\infty)}}d^\times y \\
=&\Gamma_{F,\mathbf{k},\mathbf{m}}(s)\sum_{i=1}^{h_F}\sum_{\fa}\frac{a_f(\fa_i\fa\fd_F)}{N(\fa_i\fa\fd_F)^s}=\Gamma_{F,\mathbf{k},\mathbf{m}}(s) L(s,f)
\end{split}\end{aligned}$$
for $\mathfrak{R}(s)>\max_{\s\in I_F}(1,m_\s)$, where $\fa$ runs over the set of non-zero principal fractional ideals of $F$ and $$\Gamma_{F,\mathbf{k},\mathbf{m}}(s):=\frac{2^{|\Sigma(\R)|}|D_F|^s}{[\O_F^\times:\O_{F,+}^\times]}\prod_{\s\in\Sigma(\C)}\frac{1}{4}\left(\begin{smallmatrix} n_\s^* \\ n_{\s{\operatorname{c}}}+1 \end{smallmatrix}\right)\cdot\prod_{\s\in I_F}(2\pi)^{-(s-m_\s)}\Gamma(s-m_\s) .$$ Note that the last equality holds due to the Proposition \[four:whit:exp:hida\] (1).
Fricke involution
-----------------
To find an integral representation of the $L$-function which converges uniformly on the entire complex plane, we need to cut the integral representation (\[mellin:transform\]) of completed $L$-function of $f$ into two parts by using the Fricke involution, which is similar to the classical case.
Let us write $\chi=\chi_\infty\chi^{(\infty)}$ where $\chi_\infty$ and $\chi^{(\infty)}$ are the infinite part and the finite part of $\chi$, respectively. For any finite place $v$ of $F$, let $\varpi_v$ be a uniformizer of $\O_{F,v}$. Let us define the Fricke involution $W_\fN$ for $f\in S_{\mathbf{k},\mathbf{m},J}(\fN,\chi)$ by taking the right translation using an element $\left(\begin{smallmatrix} 0 & -1 \\ \varpi_\fN & 0 \end{smallmatrix}\right)$ of ${\operatorname{GL}}_2(\A_F^{(\infty)})$: $$W_{\fN}f(g)(\mathbf{x}):=N(\fN)^{1-\frac{k}{2}}(\chi|\cdot|_{\A_F}^{k-2})^{-1}(\text{det}(g))f\bigg(g\begpmat 0 & -1 \\ \varpi_\fN & 0 \endpmat \bigg)(\mathbf{x})$$ where $\chi|\cdot|_{\A_F}^{k-2}$ is a normalization of $\chi$ and $\varpi_\fN:=\prod_{v|\fN}\varpi_v^{\text{ord}_v(\fN)}$. Then we have the following fact:
\[fricke:hecke\] If $f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$ is a Hecke eigenform, then $$W_{\fN}f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,{\overline}{\chi})$$ and it is also a Hecke eigenform.
See Hida [@hida1994critical Section 8] or Namikawa [@namikawa2017p Section 5.1].
Using the decomposition ${\operatorname{GL}}_2(\A_F)={\operatorname{GL}}_2(F_\infty)\times{\operatorname{GL}}_2(\A_F^{(\infty)})$, we have $$\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ \varpi_\fN & 0 \end{pmatrix}=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\cdot\bigg(\begin{pmatrix} 1 & 0 \\ 0 & y_\infty \end{pmatrix}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},\begin{pmatrix} \varpi_\fN & 0 \\ 0 & y^{(\infty)} \end{pmatrix}\bigg)$$ where ${\operatorname{GL}}_2(F)$ acts on ${\operatorname{GL}}_2(\A_F)$ diagonally. From this matrix identity and the definition of cusp forms, it is easy to obtain that $$\begin{aligned}
\begin{split}\label{fricke:formula}
W_{\fN}f\bigg(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix}\bigg)(\textbf{x})=&N(\fN)^{1-\frac{k}{2}}\textbf{e}\bigg(\sum_{\s\in J}\frac{k_\s}{4}-\sum_{\s\in\Sigma(\R)\backslash J}\frac{k_\s}{4}\bigg)|y|_{\A_F}^{2-k} \\
&\times f\bigg(\begin{pmatrix} \varpi_\fN y^{-1} & 0 \\ 0 & 1 \end{pmatrix}\bigg)\bigg(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\textbf{x}\bigg) .
\end{split}\end{aligned}$$ Also from the definition of $W_\fN$ and Proposition \[fricke:hecke\], we can check that $$W_\fN^2 f=\chi^{(\infty)}(-1)f=(-1)^{[F:\Q](k-2)}f .$$ From these formulas, we can obtain the analytic continuation and the functional equation of a completed $L$-function $\Lambda(s,f)$ attached to $f$, which is defined by $$\Lambda(s,f):=\Gamma_{F,\mathbf{k},\mathbf{m}}(s)N(\fN)^{\frac{s}{2}}L(s,f).$$
\[Lftn:analytic:conti\] If we have $$\label{parity:cond}
(-1)^{|\Sigma(\R)|(k-2)}C_{F,J,\mathbf{k}}^2=1 ,$$ then the completed $L$-function $\Lambda(s,f)$ of $f$ has the analytic continuation to $\C$ and the following functional equation: $$\label{functional:eq}
\Lambda(s,f)=C_{F,J,\mathbf{k}}\Lambda(k-s,W_\fN f)$$ where $$C_{F,J,\mathbf{k}}=(-1)^{|\Sigma(\R)|+\sum_{\s\in\Sigma(\C)}(n_\s+1)}\mathbf{e}\bigg(\sum_{\s\in\Sigma(\R)\backslash J}\frac{k_\s}{4}-\sum_{\s\in J}\frac{k_\s}{4}\bigg).$$
\[parity:cond:rem\] The condition (\[parity:cond\]) is satisfied, for example, when the weights $k_\s$ are all even.
By splitting integral (\[mellin:transform\]) into two pieces, applying a change of variable $y\mapsto\varpi_\fN y^{-1}$ and putting (\[fricke:formula\]) into one of the integrals, one can obtain the following integral representation: $$\begin{aligned}
\Lambda(s,f)=&N(\fN)^{\frac{s}{2}}\int_{|y|_{\A_F}\geq |\varpi_\fN|_{\A_F}^{1/2}} f_{\mathbf{n}^*/2}\bigg(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix}\bigg)|y|_{\A_F}^{s-1}d^\times y \\
&+C_{F,J,\mathbf{k}}N(\fN)^{\frac{k-s}{2}}\int_{|y|_{\A_F}\geq |\varpi_\fN|_{\A_F}^{1/2}} W_{\fN}f_{\mathbf{n}^*/2}\bigg(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix}\bigg)|y|_{\A_F}^{k-1-s}d^\times y .
$$ From the above formula, we can easily obtain our functional equation.
We follow Luo-Ramakrishnan [@luo1997determination]. Let $\Phi$ be an infinitely differentiable function on $\R_{>0}^\times$ with compact support and $\int_0^\infty \Phi(y)\frac{dy}{y}=1$. Define $$\begin{aligned}
V_{1,s}(x)&:=\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}\kappa(t)\Gamma_{F,\mathbf{k},\mathbf{m}}(s+t) x^{-t}\frac{dt}{t}\\
V_{2,s}(x)&:=\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}\kappa(-t)\Gamma_{F,\mathbf{k},\mathbf{m}}(s+t) x^{-t}\frac{dt}{t}\end{aligned}$$ where $\kappa(t):=\int_0^\infty\Phi(y)y^t\frac{dy}{y}$. By shifting the contour, one can show that $V_{1,s}$ and $V_{2,s}$ satisfy following: $$\begin{aligned}
\begin{split}\label{aux.func.esti}
V_{1,s}(x)&=O(\Gamma_{F,\mathbf{k},\mathbf{m}}(\mathfrak{R}(s)+j))x^{-j})\text{ for all }j\geq 1 \text{ as }x\rightarrow\infty \\
V_{2,s}(x)&=\Gamma_{F,\mathbf{k},\mathbf{m}}(s)+O\bigg(\Gamma_{F,\mathbf{k},\mathbf{m}}\Big(\mathfrak{R}(s)-\frac{1}{2}\Big)x^{\frac{1}{2}}\bigg) \text{ as }x\rightarrow 0
\end{split}\end{aligned}$$ where the implicit constants depend only on $j$ and $\Phi$. For $\mathfrak{R}(s)>\frac{k+2}{2},\ \max_{\s\in I_F}m_\s-2$ and $y>0$, we have $$\begin{aligned}
\begin{split}\label{series:exp}
\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}\kappa(t)\Gamma_{F,\mathbf{k},\mathbf{m}}(s+t) L(s+t,f) y^t\frac{dt}{t}=\sum_{0\neq\fa<\O_F}\frac{a_f(\fa)}{N(\fa)^s}V_{1,s}\bigg(\frac{N(\fa)}{y}\bigg) .
\end{split}\end{aligned}$$ By the Residue theorem, we have $$\begin{aligned}
\Gamma_{F,\mathbf{k},\mathbf{m}}(s) L(s,f)=&\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}\kappa(t)\Gamma_{F,\mathbf{k},\mathbf{m}}(s+t) L(s+t,f)y^t\frac{dt}{t} \\
&+\frac{1}{2\pi i}\int_{-2+i\infty}^{-2-i\infty}\kappa(t)\Gamma_{F,\mathbf{k},\mathbf{m}}(s+t) L(s+t,f)y^t\frac{dt}{t} .\end{aligned}$$ Putting equations (\[functional:eq\]) and (\[series:exp\]) in the above equation, we obtain $$\begin{aligned}
\begin{split}\label{approx:func:eq}
\Lambda(s,f)=&N(\fN)^{\frac{s}{2}}\sum_{0\neq\fa<\O_F}\frac{a_f(\fa)}{N(\fa)^s}V_{1,s}\bigg(\frac{N(\fa)}{y}\bigg) \\
&+C_{F,J,\mathbf{k}}N(\fN)^{\frac{k-s}{2}}\sum_{0\neq\fa<\O_F}\frac{a_{W_\fN f}(\fa)}{N(\fa)^{k-s}}V_{2,k-s}\bigg(\frac{N(\fa)y}{N(\fN)}\bigg)
\end{split}\end{aligned}$$ when $(-1)^{|\Sigma(\R)|(k-2)}C_{F,J,\mathbf{k}}^2=1$.
Twisted cusp forms {#twisted:cuspforms}
------------------
We will define and discuss a cusp form twisted by a Hecke character. In this subsection, we follow Hida [@hida1994critical Section 6].
Let $\fc$ be an integral ideal of $F$ coprime to $\fd_F$ and $\varphi:F^\times\backslash\A_F^\times\rightarrow\C^\times$ a Hecke character of finite order with conductor $\fc$. Denote by $(\fc^{-1}/\O_F)^\times$ the set of elements of $\prod_{v|\fc}F_{v}$ corresponing to the elements of $\fc^{-1}/\O_F$ whose annihilator ideal is same as $\fc$. For $a\in\A_F^{(\infty)}$, define the Gauss sum $G(\varphi,a)$ of $\varphi$ (Hida [@hida1994critical Section 6]) by $$G(\varphi,a):=\varphi^{-1}(d_F)\sum_{u\in (\fc^{-1}/\O_F)^\times}\varphi(\varpi_\fc u)\textbf{e}_F(d_F^{-1}au)$$ which is clearly independent on a choice of $d_F$. Especially, we put $G(\varphi,1^{(\infty)})=G(\varphi)$. Then we have the following lemma for the Gauss sums:
\[gauss:sums\]
1. For any $a\in\A_F^{(\infty)}$, we have $G(\varphi){\overline}{\varphi}(a)=G(\varphi,a)$.
2. $|G(\varphi)|^2=N(\fc)$.
3. ${\overline}{G(\varphi)}=\varphi^{(\infty)}(-1)G({\overline}{\varphi})$.
4. $G({\overline}{\varphi})G(\varphi)=\varphi^{(\infty)}(-1)N(\fc)$.
Those are immediate consequences of Neukirch [@neukirch2013algebraic Chapter VII, Proposition 7.5] by using a bijection $(\fc^{-1}/\O_F)^\times\cong(\widehat{\O}_F/\widehat{\fc})^\times$ defined by $u\mapsto \varpi_{\fc}u$.
Define a function $f\otimes\varphi:\text{GL}_2(\A_F)\rightarrow L(\mathbf{n}^*,\C)$ by $$\label{twisted:cuspform:decomp:eq}
f\otimes\varphi(g)(\mathbf{x}):=G(\varphi)^{-1}\varphi(\det(g))\sum_{u\in(\fc^{-1}/\O_F)^\times}\varphi(\varpi_\fc u)f\bigg(g\begpmat 1 & u \\ 0 & 1 \endpmat\bigg)(\mathbf{x}).$$ Then we have the following proposition:
\[twisted:cuspform:decomp\] We have $f\otimes\varphi\in S_{k}(\fN\cap\fc^2,\chi\varphi^2)$. Also we have $a_{f\otimes\varphi}(\fa)=a_f(\fa)\varphi(\fa)$.
A Hecke character $\varphi$ can be considered as a ray class character. Then our proof is immediate by Hida [@hida1994critical Section 6].
Also we have the following relation between the Fricke involution and the twisting by Hecke characters:
\[fricke:twist:commute\] If $\fc$ and $\fN$ are coprime, then we have $$W_{\fN\fc^2}(f\otimes\varphi)=\iota(\varphi)(W_{\fN}f)\otimes{\overline}{\varphi}$$ where $\iota(\varphi):=N(\fc)^{1-k}\chi(\varpi_\fc){\overline}{\chi}_\fN\big(\left(\begin{smallmatrix} d & -v\varpi_{\fc} \\ -\varpi_{\fN\fc}u & \varpi_{\fc} \end{smallmatrix}\right)\big)\varphi(\varpi_{\fN\fc^2})G({\overline}{\varphi})^2$, a complex number of absolute value $1$.
In this proof, we keep using the bijection in the proof of Lemma \[gauss:sums\]. By our assumption, $\varpi_{\fc}$ and $\varpi_{\fN}\varpi_{\fc} u$ are coprime for any $u\in(\fc^{-1}/\O_F)^\times$, thus there exist $d$ and $\varpi_{\fc}v\in\widehat{\O}_F$ such that $$\varpi_{\fc}d-\varpi_{\fN}\varpi_{\fc}u\varpi_{\fc}v=1.$$ From this, we can easily check that the map $\varpi_{\fc}u\mapsto\varpi_{\fc}v$ is well-defined automorphism on $(\widehat{\O}_F/\widehat{\fc})^\times$. By the definition of the Fricke involution, the equation (\[twisted:cuspform:decomp:eq\]), and Proposition \[twisted:cuspform:decomp\], we have $$\begin{aligned}
W_{\fN\fc^2}(f\otimes\varphi)(g)(\mathbf{x})=&N(\fN\fc^2)^{1-\frac{k}{2}}(\chi\varphi^2|\cdot|^{k-2})^{-1}(\det(g))G(\varphi)^{-1}\varphi(\det(g))\varphi(\varpi_{\fN\fc^2}) \\
&\times\sum_{u\in(\fc^{-1}/\O_F)^\times}\varphi(\varpi_\fc u)f\bigg(g\begpmat 0 & -1 \\ \varpi_{\fN\fc^2} & 0 \endpmat\begpmat 1 & u \\ 0 & 1 \endpmat\bigg)(\mathbf{x}) .
$$ Put the identities $$\begpmat 0 & -1 \\ \varpi_{\fN\fc^2} & 0 \endpmat\begpmat 1 & u \\ 0 & 1 \endpmat=\begpmat \varpi_{\fc} & 0 \\ 0 & \varpi_{\fc} \endpmat\begpmat 1 & v \\ 0 & 1 \endpmat\begpmat d & -v\varpi_{\fc} \\ -\varpi_{\fN\fc}u & \varpi_{\fc} \endpmat\begpmat 0 & -1 \\ \varpi_{\fN} & 0 \endpmat$$ and $\varpi_{\fN\fc^2}uv\equiv -1\mod\widehat{\fc}$ into the above equation, then by the Definition \[adel:cuspform:defn\] and Proposition \[fricke:hecke\], we obtain $$\begin{aligned}
W_{\fN\fc^2}(f\otimes\varphi)(g)(\mathbf{x})=&N(\fc)^{2-k}\chi(\varpi_\fc){\overline}{\chi}_\fN\big(\left(\begin{smallmatrix} d & -v\varpi_{\fc} \\ -\varpi_{\fN\fc}u & \varpi_{\fc} \end{smallmatrix}\right)\big) G(\varphi)^{-1}{\overline}{\varphi}(\det(g))\varphi(\varpi_{\fN\fc^2}) \\
&\times\sum_{u\in(\fc^{-1}/\O_F)^\times}{\overline}{\varphi}(\varpi_\fc v)W_\fN f\bigg(g\begpmat 1 & v \\ 0 & 1 \endpmat\bigg)(\mathbf{x}) .
$$ Using Lemma \[gauss:sums\], we can rewrite the above equation as $$\begin{aligned}
W_{\fN\fc^2}(f\otimes\varphi)=N(\fc)^{1-k}\chi(\varpi_\fc){\overline}{\chi}_\fN\big(\left(\begin{smallmatrix} d & -v\varpi_{\fc} \\ -\varpi_{\fN\fc}u & \varpi_{\fc} \end{smallmatrix}\right)\big)G({\overline}{\varphi})^2\varphi(\varpi_{\fN\fc^2})(W_\fN f)\otimes{\overline}{\varphi} ,\end{aligned}$$ hence we are done.
\[prime:power:conductor\] Note that if $\fc$ is a prime power, then one can easily observe that $\varphi(\varpi_\fc)=1$, thus $\iota(\varphi)=N(\fc)^{1-k}\chi(\varpi_\fc){\overline}{\chi}_\fN\big(\left(\begin{smallmatrix} d & -v\varpi_{\fc} \\ -\varpi_{\fN\fc}u & \varpi_{\fc} \end{smallmatrix}\right)\big)\varphi(\varpi_{\fN})G({\overline}{\varphi})^2$ if $\fc$ is a prime power.
Finally, we can find a fast convergent series expression of the twisted special $L$-values: Suppose that $\fc$ and $\fd_F\fN$ are coprime, and $(-1)^{|\Sigma(\R)|(k-2)}C_{F,J,\mathbf{k}}^2=1$. Then by the equation (\[approx:func:eq\]) and Proposition \[twisted:cuspform:decomp\], \[fricke:twist:commute\], we have $$\begin{aligned}
\label{fast:int:exp}
\begin{split}
\Gamma_{F,\mathbf{k},\mathbf{m}}\Big(\frac{k}{2}\Big) L\Big(\frac{k}{2},f\otimes\varphi\Big)
=&\sum_{0\neq\fa<\O_F}\frac{a_f(\fa)\varphi(\fa)}{N(\fa)^{k/2}}V_{1,\frac{k}{2}}\bigg(\frac{N(\fa)}{y}\bigg) \\
&+C_{F,J,\mathbf{k}}\sum_{0\neq\fa<\O_F}\frac{a_{W_\fN f}(\fa){\overline}{\varphi}(\fa)\iota(\varphi)}{N(\fa)^{k/2}}V_{2,\frac{k}{2}}\bigg(\frac{N(\fa)y}{N(\fN\fc^2)}\bigg) .
\end{split}\end{aligned}$$
Galois Averages of Hecke characters {#hecke:char}
===================================
In this section, we are going to discuss the Galois averages of Hecke characters, which play a crucial role to show the non-vanishing of the special $L$-values.
Let $K/\Q$ be a finite extension and set $n_0:=\textrm{max}\{m\in\Z|\ \mu_{m}\subset K\}$. For a Hecke character $\varphi:F^\times\backslash\A_F^\times\rightarrow\C^{\times}$ of finite order, or a ray class character, we define the Galois averages of $\varphi$ over $K$ by $$\begin{aligned}
&\varphi_{\textrm{av}}:=\frac{1}{[K(\varphi):K]}\sum_{\sigma\in\textrm{Gal}(K(\varphi)/K)}\varphi^\sigma
\\
&\varphi^\iota_{\textrm{av}}:=\frac{1}{[K(\varphi):K]}\sum_{\sigma\in\textrm{Gal}(K(\varphi)/K)}\iota(\varphi^\sigma){\overline}{\varphi}^\sigma\end{aligned}$$ where $K(\varphi)$ is the field determined by adjoining all the values of $\varphi$ to $K$.
Let $\fp$ be a prime ideal of $F$ lying above $p$. We assume that $\fp$ and $\fd_F\fN$ are coprime, which allows us to use the discussion in Subsection \[twisted:cuspforms\]. We assume that $p$ is coprime to $h_F$, and $\fa_i$ is coprime to $\fp$ for all $i=1,\cdots,h_F$. Let us fix an embedding $F\hookrightarrow F_\fp$. Let $\text{Cl}(F,\mathfrak{m})$ be the ray class group of a modulus $\mathfrak{m}$ of $F$. Let us denote $$\O_{F,\fp}:=\varprojlim_m \O_F/\fp^m,\ {\overline}{\O_F^\times}:=\varprojlim_m \O_F^\times\text{ mod }\fp^n,\text{ and }\text{Cl}(F,\fp^\infty):=\varprojlim_m \text{Cl}(F,\fp^m).$$ Then we have the following exact sequences: $$\label{ray:class:group:seq}
\begin{tikzcd} 1 \arrow{r} & \overline{\O_{F}^\times} \arrow{r} & \O_{F,\fp}^\times \arrow{r} & \text{Cl}(F,\fp^\infty) \arrow{r} & \text{Cl}(F) \arrow{r} & 1 \end{tikzcd} ,$$ $$\begin{tikzcd} 1 \arrow{r} & 1+\fp\O_{F,\fp} \arrow{r} & \O_{F,\fp}^\times \arrow{r}{\text{mod }\fp} & (\O_{F,\fp}/\fp)^\times \arrow{r} & 1 \end{tikzcd} .$$ Let $\Delta$ be the torsion part of $\text{Cl}(F,\fp^\infty)$, $W$ be a split image of $(\O_{F,\fp}/\fp)^\times$ in $\O_{F,\fp}^\times$, and $\Gamma^\p:=1+\fp\O_{F,\fp}$. By decomposing (\[ray:class:group:seq\]) into the tortion part and the pro $p$-part, then we obtain the following exact sequences $$\begin{tikzcd} W \arrow{r} & \Delta \arrow{r} & \text{Cl}(F) \arrow{r} & 1 \end{tikzcd} ,
\begin{tikzcd} \Gamma^\p \arrow{r} & \text{Cl}(F,\fp^\infty)_p \arrow{r} & 1 \end{tikzcd} .$$
Let us denote $\mu_{\infty}:=\varinjlim_{n}\mu_n$ and $\Xi_\fp:={{\rm{Hom}}}_{\text{cont}}(\text{Cl}(F,\fp^\infty),\mu_{\infty})$. Then there is a unique element $(\psi_n)_n\in\varinjlim_{n}{{\rm{Hom}}}_{\text{cont}}(\text{Cl}(F,\fp^n),\mu_{\infty})\cong\Xi_\fp$ corresponds to $\psi$. From now on, let us say that $\psi\in\Xi_\fp$ is primitive if and the conductor of $\psi_n$ is $\fp^{n+n_0}$ for all $n$. Let $\psi=(\psi_n)_n$ be a primitive element of $\Xi_\fp$. Let $\mathcal{E}:=\ker(\psi)\cap\Gamma^\p$, then we have the following split exact sequence $$\begin{tikzcd} 1 \arrow{r} & \mathcal{E} \arrow{r}{\subset} & \Gamma^\p \arrow{r}{\psi} & \mu_{\infty} \arrow{r} & 1\end{tikzcd} .$$
Let $\Gamma$ be a split image of $\mu_{\infty}$ in $\Gamma^\p$, whose $\Z_p$-rank is one. From the above exact sequences, we obtain the surjection $$\label{rayclassgroup:surjection}
\begin{tikzcd} \Delta\times\Gamma^\p\cong\Delta\times\mathcal{E}\times\Gamma \arrow[r, two heads] & \text{Cl}(F,\fp^\infty) \end{tikzcd} ,$$ hence $\psi_n$ can be considered as an element of ${{\rm{Hom}}}_{\text{cont}}(\Delta\times\mathcal{E}\times\Gamma,\mu_{\infty})$ for each $n$.
Let us define a filtration on $\Gamma^\p$ by $\Gamma^\p_n:=1+\fp^n\O_F$. Let us denote $\mathcal{E}_n:=\mathcal{E}\cap\Gamma^\p_n$ and $\Gamma_n:=\Gamma\cap\Gamma^\p_n$. As $\psi_n$ has a $p$-power order and $(p,h_F)=1$, we have $\psi_n(\Delta)=\{1\}$. So we have $\ker(\psi_n)\supset \Delta\times\mathcal{E}\times\Gamma_{n+n_0}$. The following proposition tells us that nonzero integral elements which make the Galois averages non-zero, are distributed sparsely.
\[gal:av:nonzero\] Let $(h_F,p)=1$. Let $\fa$ be an element of $\operatorname{Cl}(F,\fp^\infty)$ and $\psi=(\psi_n)_n\in\Xi_\fp$. Then for $n\geq 1$, we have that $\psi_{n,\operatorname{av}}(\fa)\neq 0$ if and only if $$\fa\in\bigcup_{(\fb,\epsilon,\gamma)\in\Delta\times\mathcal{E}\times\Gamma_n} \{\fb\epsilon\gamma\}=\bigcup_{(\fb,\epsilon,\gamma)\in\Delta\times(\mathcal{E}/\mathcal{E}_n)\times\Gamma_n^\p}\{\fb\epsilon\gamma\}.$$
Assume that $\psi_{n,\textrm{av}}(\fa)\neq 0$, then we have $\psi_n(\fa)\in\mu_{\infty}$ by our assumption. As the conductor of $\psi_n$ is $\fp^{n+n_0}$, $\psi$ factors through $\Gamma/\Gamma_{n}\cong\mu_{{n-1}}$, hence $\psi_n(\fa)=\zeta_{{n-1}}^r$ for some $r\in\Z_{\geq0}$. Our assumption $\psi_{n,\textrm{av}}(\fa)\neq 0$ also says that $${\operatorname{Tr}}_{K(\zeta_{{n-1}}^r)/K}(\zeta_{{n-1}}^r)\neq 0 ,$$ thus we have $n-1-v_p(r)\leq n_0$. Note that we can write $\fa=\fb\epsilon\gamma$ for some $(\fb,\epsilon,\gamma)\in\Delta\times\mathcal{E}\times\Gamma$, thus we have $$\psi_n(\fa)=\psi_n(\gamma)=\zeta_{{n-1}}^r\in\mu_{{n_0}},$$ which implies that $\gamma\in\Gamma_{n}/\Gamma_{n+n_0}\cong\mu_{n_0}$.
Conversely, if $\fa=\fb\epsilon\gamma$ for some $(\fb,\epsilon,\gamma)\in\Delta\times\mathcal{E}\times\Gamma_n$, then $\psi_n(\fa)=\psi_n(\gamma)\in\mu_{{n_0}}$. Thus, $\psi_n(\fa)=\zeta_{{n_0}}^r$ for some $r\in\Z_{\geq0}$. Hence the Galois average of $\psi_n(\fa)$ is given by $$\psi_{n,\text{av}}(\fa)=\frac{1}{[K(\psi_n):K]}\sum_{\s\in\textrm{Gal}(K(\psi_n)/K)}(\zeta_{{n_0}}^r)^{\s}=\zeta_{{n_0}}^r\neq 0 .$$
Assume that $\fa=\fb\epsilon\g$ for some $\g\in \Gamma^\p$. Then by (\[rayclassgroup:surjection\]), we have $\fb\epsilon\g=\fb^\p\epsilon^\p\g^\p$ for some $\fb^\p\epsilon^\p\g^\p\in \Delta\times\mathcal{E}\times\Gamma$. Then $\psi_n(\gamma)=\psi_n(\gamma^\p)$, which implies that $\g\in\gamma^\p\Gamma=\Gamma$. The reverse direction is clear as $\Gamma\subset\Gamma^\p$. In conclude, $\g\in\Gamma_{n}$.
For $\epsilon_1,\epsilon_2\in\mathcal{E}$, we can easily check that $\epsilon_1\Gamma^\p_n=\epsilon_2\Gamma^\p_n$ if and only if $\epsilon_1\epsilon_2^{-1}\in\mathcal{E}_n$.
The following lemma allow us to estimate the size of $\psi_{n,\text{av}}^\iota$.
\[root:num:gal:av\] For any integral ideal $\fa$ of $F$, We have $$|\psi^\iota_{n,\operatorname{av}}(\fa)|\ll_{F,n_0,\fp} N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}$$ where $\operatorname{f}(F,\fp)$ is the residue degree of $F$ at $\fp$.
If $\fa$ is not coprime to $\fp$, then clearly the above inequality holds. Thus $[\fa]_n\in\text{Cl}(F,\fp^n)$. Then $\fa=\fa_i\a$ for some $i$ and $\a\in\O_F$ which is coprime to $\fp$. By the definition of $\iota(\varphi)$ (see Proposition \[fricke:twist:commute\] and Note \[prime:power:conductor\]), we have $$\begin{aligned}
\begin{split}\label{rootnum:char}
|\psi^\iota_{n,\operatorname{av}}(\fa)|
=\frac{1}{N(\fp)^{n+n_0}}\bigg|\sum_{\beta}{\overline}{\psi}_{n,\operatorname{av}}(\fa_i\a\beta d_F^{-2})\sum_{uv=\beta}\textbf{e}_F\bigg(\frac{u+v}{d_F\varpi_\fp^{n+n_0}}\bigg)\bigg|
\end{split}\end{aligned}$$ where $\beta,u$ and $v$ runs over $(\O_F/\fp^{n+n_0})^\times$. As $\a d_F^{-2}$ is coprime to $\fp$, we can abbreviate $\a\beta d_F^{-2}$ by $\beta$. By Proposition \[gal:av:nonzero\], $\psi_{n,\textrm{av}}(\beta)\neq 0$ if and only if $\beta\in W\cdot\mathcal{E}/\mathcal{E}_{n+n_0}\cdot\Gamma_n/\Gamma_{n+n_0}$. So (\[rootnum:char\]) is bounded by $$\begin{aligned}
\begin{split}\label{rootnum:char:2}
\frac{1}{N(\fp)^{n+n_0}}\bigg|\sum_{\k\in W}\sum_{\epsilon\in\mathcal{E}/\mathcal{E}_{n+n_0}}\sum_{\gamma\in\Gamma_n/\Gamma_{n+n_0}}{\overline}{\psi}_{n,\text{av}}(\gamma)\sum_{u}\textbf{e}_F\bigg(\frac{u+\k\epsilon\gamma u^{-1}}{d_F\varpi_\fp^{n+n_0}}\bigg)\bigg| .
\end{split}\end{aligned}$$ By Bruggeman-Miatello[@bruggeman1995estimates Proposition 9], which is about the estimation on the Kloosterman sums, (\[rootnum:char:2\]) is less than $$\begin{aligned}
&\frac{1}{N(\fp)^{n+n_0}}\sum_{\k\in W}\sum_{\epsilon\in\mathcal{E}/\mathcal{E}_{n+n_0}}\sum_{\gamma\in\Gamma_n/\Gamma_{n+n_0}}\bigg|\sum_{u}\textbf{e}_F\bigg(\frac{u+\k\epsilon\gamma u^{-1}}{d_F\varpi_\fp^{n+n_0}}\bigg)\bigg| \\
\leq&\frac{2}{N(\fp)^{\frac{n+n_0}{2}}}\sum_{\k\in W}\sum_{\epsilon\in\mathcal{E}/\mathcal{E}_{n+n_0}}\sum_{\gamma\in\Gamma_n/\Gamma_{n+n_0}} 1 = \frac{2|W|p^{n_0\cdot\operatorname{rk}_{\Z_p}(\Gamma)}p^{(n+n_0-1)\cdot\operatorname{rk}_{\Z_p}(\mathcal{E})}}{N(\fp)^{\frac{n+n_0}{2}}} .\end{aligned}$$ Note that we can observe that $\text{rk}_{\Z_p}(\mathcal{E})=\text{rk}_{\Z_p}(\Gamma^\p)-\text{rk}_{\Z_p}(\mu_{\infty})=\text{f}(F,\fp)-1$ and $\text{rk}_{\Z_p}(\Gamma)=\text{rk}_{\Z_p}(\mu_{\infty})=1$, where $\text{rk}_{\Z_p}(M)$ is the $\Z_p$-rank of $M$. Thus we can conclude the proof.
Number of elements in arithmetic progressions {#esti:arith:progress}
=============================================
In this section, we obtain an estimation of the number of elements and the absolute norms of elements in arithmetic progressions by using the idea of Rohrlich [@rohrlich1989nonvanishing], which plays a key role to estimate our twisted special $L$-values.
For each $\fa=([\fa_n]_n)_n\in\text{Cl}(F,\fp^\infty)$, let $\fa_n$ be an integral ideal of $F$ which is a representative of a ray class $[\fa_n]_n\in\text{Cl}(F,\fp^n)$ of $\fa_n$ modulo $\fp^n$. Similarly, for each $\a=(\a_n)_n\in\gamma^\p$, define $\langle\a\rangle_n\in\O_F\backslash\{0\}$ by a representative of $\a_n\in\Gamma^\p/\Gamma^\p_n$. Then by Proposition \[gal:av:nonzero\], we have the following lemma:
\[gal:av:nonzero:integral\] Let $(h_F,p)=1$. Let $\fa$ be an integral ideal of $F$ and $\psi=(\psi_n)_n\in\Xi_\fp$ a primitive element. Then for $n\geq 1$, we have that $\psi_{n,\operatorname{av}}(\fa)\neq 0$ if and only if $$\fa\in\bigcup_{(\fb,\epsilon)\in\Delta\times(\mathcal{E}/\mathcal{E}_n)}\{\fb_n\langle\epsilon\rangle_n\g:\g\in(1+\fp^n)\}.$$
It is an immediate consequence of Proposition \[gal:av:nonzero\].
Let us denote by $F_\R:=F\otimes_\Q \R$ the real Minkowski space of $F$, and $C_F^0$ a fundamental domain of $F_\R/\O_F^\times$, where $\O_F^\times$ acts on $F_\R$ via the natural embedding $F\hookrightarrow F_\R$. For $n\in\Z_{>0}$, $x>0$ and $\a\in\O_{F}$ coprime to $\fp$, let us denote $U_{\a,n}(x):=\#\{\beta\in\a(1+\fp^n)\cap C_F^0: |N(\beta)|\leq x\}$.
Let us recall the following fact about coherent cone decomposition. By Rohrlich [@rohrlich1989nonvanishing Proposition 5], there is a finite collection $\mathscr{B}$ of coherent $\Z$-cones in $F$ such that $$\label{coherent:cone:decomp}
\O_F\backslash\{0\}\subset\bigcup_{u\in\O_F^\times}\bigcup_{B\in\mathscr{B}} uB .$$ Then we have the following estimations:
\[lattice:esti:1\] For $n\in\Z_{>0}$, $x>0$ and $\a\in\O_F$ coprime to $\fp$, we have $$U_{\a,n}(x)\ll_{F} \operatorname{max}\Big(\frac{x}{N(\fp)^n},1\Big).$$
The inequality holds by the equation (21) in Rohrlich [@rohrlich1989nonvanishing Proposition 5] since $$\a(1+\fp^n)\cap C_F^0\subset(\O_F\backslash\{0\})/\O_F^\times\subset\bigg(\bigcup_{u\in\O_F^\times}\bigcup_{B\in\mathscr{B}}uB\bigg) /\O_F^\times=\bigcup_{B\in\mathscr{B}} B$$ by the equation (\[coherent:cone:decomp\]). Also we can observe that the implicit constant in the equation (21) in Rohrlich [@rohrlich1989nonvanishing Proposition 5] depends only on the coherent cone decomposition $\mathscr{B}$ of $\O_F$, which is a finite collection of cones and depends only on $F$.
\[lattice:esti:2\] Let $\fp$ be a totally split prime over $F/\Q$. If $\a\in (1+\fp^{n})\backslash\{1\}$, then we have $|N(\a)|\gg_F N(\fp)^{n}$.
Let $\a=1+\beta\in (1+\fp^{n})\backslash\{1\}\cap C_F^0$. Then by the equation (\[coherent:cone:decomp\]), we have $$N(\fp)^n\leq |N(\beta)|=|N(\a-1)|=|N(\a)|\prod_{\s\in I_F}|1-\s(\a)^{-1}| .$$ Note that for $B\in\mathscr{B}$, there is a basis $\{z_j\}_{j=1}^{d}$ of $F$ over $\Q$ such that $B=\Z_{>0}\langle\{z_j\}_{j=1}^{d}\rangle$ where $d=[F:\Q]$. Let $\s\in I_F$ and $z=\sum_{j=1}^d n_j z_j\in B$, then we have $\max(\{n_j\}_{j=1}^d)\ll_B |\s(z)|$ as $B$ is a coherent $\Z$-cone (Rohrlich [@rohrlich1989nonvanishing]). Thus the above equation becomes $$N(\fp)^n\leq |N(\a)|\prod_{\s\in I_F} (1+|\s(\a)|^{-1})\ll_F |N(\a)|.$$ as $\a\in (1+\beta\in (1+\fp^{n})\backslash\{1\}\cap C_F^0\subset\bigcup_{B\in\mathscr{B}}B$ by the equation (\[coherent:cone:decomp\]). For $\a\langle\kappa\rangle_n+\beta\in (1+\fp^{n})\backslash\{1\}$, there exists $\a^\p\in(1+\fp^{n})\backslash\{1\}\cap C_F^0$ and $u\in\O_F^\times$ such that $\a=\a^\p u$. Thus we have $N(\a)=N(\a^\p)\gg_F N(\fp)^{n}$ by the above inequality. Hence we are done.
\[lattice:esti:3\] Let $\fp$ be a totally split prime over $F/\Q$ above $p$, and $(h_F,p)=1$. Define $\mathfrak{i}$ be the identity element of $\Delta$. For $\fb\in\Delta\backslash\{\mathfrak{i}\}$, we have $N(\fb_n)\gg_F N(\fp)^{n/|\Delta|}$.
Let $\fb\in\Delta\backslash\{\mathfrak{i}\}$. If $\fb_n^{|\Delta|}=\O_F$, then $\fb_n=\O_F$, which is a contradiction. So a generator of $\fb_n^{|\Delta|}$ is an element of $(1+\fp^n)\backslash\{1\}$. By Lemma \[lattice:esti:2\], we have $N(\fb_n)\gg_F N(\fp)^{n/|\Delta|}$
Galois averages of the special $L$-values {#estimation}
=========================================
In this section, we obtain an estimation on the twisted special $L$-values which allows us to verify the non-vanishing of $L$-values under the assumption that $\fp$ splits completely over $\Q$.
Let us recall that $\fp$ is a prime ideal of $F$ lying above $p$ and coprime to $h_F\d_F\fp$ and $\psi=(\psi_n)_n$ is a primitive element of $\Xi_\fp$ where $\Xi_{\fp}={{\rm{Hom}}}_{\text{cont}}(\text{Cl}(F,\fp^\infty),\mu_{\infty})$, thus the conductor of $\psi_n$ is $\fp^{n+n_0}$ for each $n$. For $x>0$ and $\a\in\O_F$ which is coprime to $\fp$, we set $U_{\a,n}(x)=\#\{\beta\in\a(1+\fp^n)\cap C_F^0:|N(\beta)|\leq x\}$ and $C_F^0=F_\R/\O_F^\times$. Let $f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$ be a Hecke eigenform and $K_f$ the Hecke field of $f$ over $\Q$ (cf. subsection \[Main:Theorems\]). Let $(-1)^{|\Sigma(\R)|(k-2)}C_{F,J,\mathbf{k}}^2=1$. For a Hecke character $\varphi:F^\times\backslash\A_F^\times\rightarrow\C^\times$ of finite order with conductor $\fc$, define the Galois average of the twisted special $L$-value by $$L_{\textrm{av}}(f\otimes\varphi):=\frac{1}{[K_f(\varphi):K_f]}\sum_{\s\in\text{Gal}(K_f(\varphi)/K_f)} L\Big(\frac{k}{2},f\otimes\varphi^\s\Big) .$$ Then by (\[fast:int:exp\]), the Galois average $L_{\textrm{av}}(f\otimes\psi_n)$ is given by
$$\begin{aligned}
&\frac{1}{\Gamma_{F,\mathbf{k},\mathbf{m}}\big(\frac{k}{2}\big)}\sum_{0\neq\fa<\O_F}\frac{a_f(\fa)\psi_{n,\text{av}}(\fa)}{N(\fa)^{k/2}}V_{1,\frac{k}{2}}\bigg(\frac{N(\fa)}{y}\bigg) \label{1st:Lvalue} \\
+&\frac{C_{F,J,\mathbf{k}}}{\Gamma_{F,\mathbf{k},\mathbf{m}}\big(\frac{k}{2}\big)}\sum_{0\neq\fa<\O_F}\frac{a_{W_\fN f}(\fa)\psi^\iota_{n,\text{av}}(\fa)}{N(\fa)^{k/2}}V_{2,\frac{k}{2}}\bigg(\frac{N(\fa)y}{N(\fN\fp^{2n+2n_0})}\bigg) . \label{2nd:Lvalue}\end{aligned}$$
as $f\otimes\varphi\in S_{(\mathbf{k},\mathbf{m}),J}(\fN\cap\fc^2,\chi\varphi^2)$ by Proposition \[twisted:cuspform:decomp\]. Now we will estimate the quantity $L_{\textrm{av}}(f\otimes\psi_n)$ for each $n$. To do this, we need a bound for the Hecke eigenvalues of $f$. Let us assume that the coefficients satisfy $$|a_f(\fp)|\leq 2N(\fp)^{\frac{k-1}{2}+\theta}$$ for any prime ideals $\fp$ of $\O_F$ and a number $\theta\in [0,\frac{1}{2}]$ to be specified in Section \[det\]. Hence, for $\e>0$ and each integral ideal $\fa$ of $\O_F$, we have $$\label{rama:peter:bdd}
|a_f(\fa)|\leq 2d(\fa)N(\fa)^{\frac{k-1}{2}+\theta}\ll_\e N(\fa)^{\frac{k-1}{2}+\theta+\e}$$ where $d(\fa)$ is the number of the integral ideals of $F$ dividing $\fa$.
From now on, we do not consider the variables related to $f$ and $F$ in the implicit constants of our estimations. First, we estimate the last term (\[2nd:Lvalue\]) of the averaged $L$-value by following:
\[1st:gal:av:Lvalue:error:2\] For $\e>0$ and $y>0$, we have that $$\begin{aligned}
(\ref{2nd:Lvalue})
\ll_{\e,\fp} & \frac{N(\fp)^{n(2\theta+2\e+\frac{3}{2}-\frac{1}{\operatorname{f}(F,\fp)})}}{y^{\theta+\e+\frac{1}{2}}}\end{aligned}$$ where $\operatorname{f}(F,\fp)$ is the residue degree of $F/\Q$ at $\fp$.
Note that $W_\fN f$ is also an Hecke eigenform by Proposition \[fricke:hecke\]. By the Ramanujan-Petersson bound (\[rama:peter:bdd\]) and Proposition \[root:num:gal:av\], we obtain $$\begin{aligned}
\begin{split}\label{1st:gal:av:Lvalue:error:2:esti}
&\sum_{0\neq\fa<\O_F}\frac{a_{W_\fN f}(\fa)\psi^\iota_{n,\operatorname{av}}(\fa)}{N(\fa)^{k/2}}V_{2,\frac{k}{2}}\bigg(\frac{N(\fa)y}{N(\fN\fp^{2n+2n_0})}\bigg) \\
\ll_{\e,\fp} &N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\sum_{0\neq\fa<\O_F} N(\fa)^{\theta+\e-\frac{1}{2}}V_{2,\frac{k}{2}}\bigg(\frac{N(\fa)y}{N(\fN\fp^{2n+2n_0})}\bigg) \\
=&N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\sum_{m\in\Z_{>0}}\sum_{N(\fa)=m}m^{\theta+\e-\frac{1}{2}}V_{2,\frac{k}{2}}\bigg(\frac{my}{N(\fN\fp^{2n+2n_0})}\bigg) \\
\ll_\e &N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\sum_{m>0} m^{\theta+\e-\frac{1}{2}}V_{2,\frac{k}{2}}\bigg(\frac{my}{N(\fN\fp^{2n+2n_0})}\bigg)
\end{split}\end{aligned}$$ where in the last inequality, we use the fact that $\sum_{N(\fa)=m}1\leq d(m)\ll_\e m^\e$. We can split the last term of the above equation into two parts: $$\begin{aligned}
(\ref{1st:gal:av:Lvalue:error:2:esti})=I+II,\text{ where }I=\sum_{m>\frac{N(\fN\fp^{2n+2n_0})}{y}},\ II=\sum_{0<m<\frac{N(\fN\fp^{2n+2n_0})}{y}}\end{aligned}$$ Using (\[aux.func.esti\]), for $j\geq \frac{1}{2}+\theta$, we have $$\begin{aligned}
\begin{split}
I&\ll N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\sum_{m>\frac{N(\fN\fp^{2n+2n_0})}{y}} m^{\theta+\e-\frac{1}{2}}\bigg(\frac{m y}{N(\fN\fp^{2n+2n_0})}\bigg)^{-j} \\
&\ll N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\frac{N(\fN\fp^{2n+2n_0})^j}{y^j}\int_{\frac{N(\fN\fp^{2n+2n_0})}{y}}^\infty x^{\theta+\e-\frac{1}{2}-j} dx \\
&\ll_\e N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\frac{N(\fN\fp^{2n+2n_0})^{\theta+\e+\frac{1}{2}}}{y^{\theta+\e+\frac{1}{2}}} \ll \frac{N(\fp)^{n(2\theta+2\e+\frac{3}{2}-\frac{1}{\operatorname{f}(F,\fp)})}}{y^{\theta+\e+\frac{1}{2}}} .
\end{split}\end{aligned}$$ Similarly, we have $$\begin{aligned}
\begin{split}
II&\ll N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\sum_{0<m<\frac{N(\fN\fp^{2n+2n_0})}{y}} m^{\theta+\e-\frac{1}{2}} \\
&\ll N(\fp)^{n(\frac{1}{2}-\frac{1}{\operatorname{f}(F,\fp)})}\int_0^{\frac{N(\fN\fp^{2n+2n_0})}{y}} x^{\theta+\e-\frac{1}{2}}dx \ll_\e\frac{N(\fp)^{n(2\theta+2\e+\frac{3}{2}-\frac{1}{\operatorname{f}(F,\fp)})}}{y^{\theta+\e+\frac{1}{2}}} .
\end{split}\end{aligned}$$ Hence we can conclude our proof.
Let us assume that $\fp$ is totally split prime of $F$ lying above $p$. Then we have $\operatorname{f}(F,\fp)=1$, which implies that $\mathcal{E}=\{1\}$. By Lemma (\[gal:av:nonzero:integral\]), Lemma \[lattice:esti:2\] and Lemma \[lattice:esti:3\], we can rewrite (\[1st:Lvalue\]) as $$\begin{aligned}
&a_f(\O_F)\frac{V_{1,\frac{k}{2}}\big(\frac{1}{y}\big)}{\Gamma_{F,\mathbf{k},\mathbf{m}}\big(\frac{k}{2}\big)}+\sum_{\substack{\fb\in\Delta \\ N(\fb_n)>c_F N(\fp)^{n/|\Delta|}}}\frac{a_f(\fb_n)}{N(\fb_n)^{k/2}}\frac{V_{1,\frac{k}{2}}\big(\frac{N(\fb_n)}{y}\big)}{\Gamma_{F,\mathbf{k},\mathbf{m}}\big(\frac{k}{2}\big)} \label{1st:Lvalue:main} \\
+&\sum_{\fb\in\Delta}\sum_{\substack{\a\in (1+\fp^n)\cap C_F^0 \\ |N(\a)|>c_F N(\fp)^{n}}}\frac{a_f(\fb_n\a)\psi_{n,\text{av}}(\fb_n\a)}{N(\fb_n\a)^{k/2}}\frac{V_{1,\frac{k}{2}}\big(\frac{N(\fb_n\a)}{y}\big)}{\Gamma_{F,\mathbf{k},\mathbf{m}}\big(\frac{k}{2}\big)}\label{1st:Lvalue:error}\end{aligned}$$ for some constant $c_F$ which depends only on $F$, where (\[1st:Lvalue:main\]) and (\[1st:Lvalue:error\]) turn out to be the main term and the error term of (\[1st:Lvalue\]), respectively. An estimation of (\[1st:Lvalue:error\]) for the case of $\text{f}(F,\fp)=1$, can be obtained by following:
\[1st:gal:av:Lvalue:error:1\] Assume that $\operatorname{f}(F,\fp)=1$. For $\e>0$ and $y>c_F N(\fp^n)$, we have that $$\begin{aligned}
(\ref{1st:Lvalue:error})
\ll_{\e,\fp} N(\fp)^{n(1+\frac{1}{|\Delta|})(\theta+\e-\frac{1}{2})}+\frac{y^{\frac{1}{2}+\theta+\e}}{N(\fp)^{n(1+\frac{1}{|\Delta|})}}.\end{aligned}$$
By the bound (\[rama:peter:bdd\]), (\[1st:Lvalue:error\]) is bounded by $$\begin{aligned}
(\ref{1st:Lvalue:error})&\ll_{\e} \sum_{\fb\in\Delta}\sum_{\substack{\a\in(1+\fp^n)\cap C_F^0 \\ |N(\a)|>c_F N(\fp)^{n} }} \frac{V_{1,\frac{k}{2}}\big(\frac{N(\fb_n\a)}{y}\big)}{N(\fb_n\a)^{\frac{1}{2}-\theta-\e}} .\end{aligned}$$ Let us split the above sum by $\sum_{\fb\in\Delta}(*)+\sum_{\fb\in\Delta}(**)$ where $$(*)=\sum_{\substack{\a\in(1+\fp^n)\cap C_F^0 \\ c_F N(\fp)^{n}<|N(\a)|\leq y/N(\fb_n) }},\ (**)=\sum_{\substack{\a\in(1+\fp^n)\cap C_F^0 \\ |N(\a)|>y/N(\fb_n) }}$$ For $x>0$ and $\a\in\O_{F}$ coprime to $\fp$, define $u_{\a,n}(x):=\#\{\beta\in\a(1+\fp^n)\cap C_F^0:|N(\beta)|=x\}$. Then $U_{\a,n}(x)=\sum_{m\leq x} u_{\a,n}(m)$. By the estimate (\[aux.func.esti\]) and the Abel summation formula, $(*)$ is bounded by $$\begin{aligned}
(*)=&\frac{1}{N(\fb_n)^{\frac{1}{2}-\theta-\e}}\sum_{c_F N(\fp)^{n}<m\leq y/N(\fb_n)}\frac{u_{1,n}(m)}{m^{\frac{1}{2}-\theta-\e}} \\
\ll&_\e \frac{1}{N(\fb_n)^{\frac{1}{2}-\theta-\e}}\bigg(\frac{U_{1,n}(N(\fp)^n)}{N(\fp)^{n(\frac{1}{2}-\theta-\e)}}+\frac{U_{1,n}(y/N(\fb_n))}{(y/N(\fb_n))^{\frac{1}{2}-\theta-\e}}+\int_{c_FN(\fp)^n}^{y/N(\fb_n)}\frac{U_{1,n}(x)}{x^{\frac{3}{2}-\theta-\e}}dx\bigg).\end{aligned}$$ Set $j\geq 1$. Then similarly, $(**)$ is bounded by $$\begin{aligned}
(**)=&\frac{y^j}{N(\fb_n)^{\frac{1}{2}-\theta-\e+j}}\sum_{m>y/N(\fb_n)}\frac{u_{1,n}(m)}{m^{\frac{1}{2}-\theta-\e+j}} \\
\ll&_\e \frac{y^j}{N(\fb_n)^{\frac{1}{2}-\theta-\e+j}}\bigg(\frac{U_{1,n}(y/N(\fb_n))}{(y/N(\fb_n))^{\frac{1}{2}-\theta-\e+j}}+\int_{y/N(\fb_n)}^{\infty}\frac{U_{1,n}(x)}{x^{\frac{3}{2}-\theta-\e+j}}dx\bigg).\end{aligned}$$ By the Lemma \[lattice:esti:1\] and Lemma \[lattice:esti:3\] on the above equations, then we are done.
We have the following estimation on the averaged special $L$-values:
\[main:thm:1\] Assume that $\operatorname{f}(F,\fp)=1$. For $a>1$ and $\e>0$, an estimation of $L_{\operatorname{av}}(f\otimes\psi_n)$ is given by $$\begin{aligned}
\begin{split}\label{main:thm:1:eq}
&L_{\operatorname{av}}(f\otimes\psi_n)-a_f(\O_F)\frac{V_{1,\frac{k}{2}}\big(\frac{1}{y}\big)}{\Gamma_{F,\mathbf{k},\mathbf{m}}\big(\frac{k}{2}\big)} \ll_{\e,\fp} N(\fp)^{n(\theta+\e-\frac{1}{2})/|\Delta|} \\
&+N(\fp)^{n(a(\frac{1}{2}+\theta+\e)-(1+\frac{1}{|\Delta|}))}+N(\fp)^{n((2\theta+2\e+\frac{1}{2})-a(\theta+\e+\frac{1}{2}))}
\end{split}\end{aligned}$$ as $n$ tends to the infinity.
For $y>0$, by the bound (\[rama:peter:bdd\]), the second term of (\[1st:Lvalue:main\]) is bounded by $$\sum_{\substack{\fb\in\Delta \\ N(\fb_n)>c_F N(\fp)^{n/|\Delta|}}}\frac{a_f(\fb_n)}{N(\fb_n)^{k/2}}\frac{V_{1,\frac{k}{2}}\big(\frac{N(\fb_n)}{y}\big)}{\Gamma_{F,\mathbf{k},\mathbf{m}}\big(\frac{k}{2}\big)} \ll_{\e,\fp} N(\fp)^{n(\theta+\e-\frac{1}{2})/|\Delta|}$$ since the function $V_{1,\frac{k}{2}}$ is bounded on the positive real line.
Set $y=y_n=N(\fp)^{an}$ for $a>1$, then by the above equation, the main term (\[1st:Lvalue:main\]), Lemma \[1st:gal:av:Lvalue:error:2\] and Lemma \[1st:gal:av:Lvalue:error:1\], we can prove the theorem.
Non-vanishing of $L$-values {#det}
===========================
In this section, we use the Ramanujan-Petersson bound, which is obtained by Blomer-Brumley [@blomer2011ramanujan], and Nakasuji [@nakasuji2012generalized] to show the non-vanishing result.
Let $f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$ be a non-zero eigenform of Hecke operators $T_\fa$ for all integral ideals $\fa$ of $F$. Then by Miyake [@miyake1971onautomorphic Lemma 2], we have $a_f(O_F)\neq 0$ if $f \neq 0$. We have the non-vanishing of the special $L$-values as a corollary of Theorem \[main:thm:1\]:
Let $\fp$ be a totally split prime ideal of $F$ lying above $p$ and coprime to $h_F\fd_F\fN$. Let $(-1)^{|\Sigma(\R)|(k-2)}C_{F,J,\mathbf{k}}^2=1$. For a non-zero eigenform $f\in S_{(\mathbf{k},\mathbf{m}),J}(\fN,\chi)$ of Hecke operators $T_\fa$ for all integral ideals $\fa$ of $F$, we have $$L\Big(\frac{k}{2},f\otimes\varphi\Big)\neq 0$$ for almost all Hecke characters $\varphi$ over $F$ of $p$-power orders and $\fp$-power conductors.
To make the error terms of (\[main:thm:1:eq\]) converges to $0$ when $n$ goes to $\infty$, the numbers $a$ and $\theta$ must satisfy the following inequalities: $$\begin{aligned}
\theta<\frac{1}{2},\
1<a,\
a\Big(\theta+\frac{1}{2}\Big)<1+\frac{1}{|\Delta|},\
a\Big(\theta+\frac{1}{2}\Big)>2\theta+\frac{1}{2}
.\end{aligned}$$ As we have set $\theta\in[0,\frac{1}{2}]$, those are clearly equivalent to, $$0\leq\theta<\frac{1}{2},\ \frac{2\theta+\frac{1}{2}}{\theta+\frac{1}{2}}<a<\frac{1+\frac{1}{|\Delta|}}{\theta+\frac{1}{2}} .$$ Note that $|\Delta|\geq 2$, where the equality holds when $p=3$ and $h_F=1$. So we can find $a$ satisfying the above inequalities if $$\theta<\frac{1}{4}+\frac{1}{2|\Delta|}$$ By Blomer-Brumley [@blomer2011ramanujan Theorem 1] and Nakasuji [@nakasuji2012generalized Corollary 1.2], one has $\theta=7/64<\frac{1}{4}+\frac{1}{2|\Delta|}$, hence we can choose such $a$.
Let $\varphi$ be a Hecke character of $p$-power order and conductor $\fp^n$, or a ray class character of $p$-power order and conductor $\fp^n$. Then by Theorem \[main:thm:1\] and the estimation (\[aux.func.esti\]), we have $$\label{gal:av:limit}
\lim_{n\rightarrow\infty}L_{\textrm{av}}(f\otimes\varphi)=a_f(\O_F)\neq 0$$ Hence we have $$L_{\textrm{av}}(f\otimes\varphi)=\frac{1}{[K_f(\varphi):K_f]}\sum_{\s\in\text{Gal}(K_f(\varphi)/K_f)} L\Big(\frac{k}{2},f\otimes\varphi^\s\Big)\neq 0$$ for sufficiently large $n$. On the other hand, by Hida [@hida1994critical Theorem 11.1], we have $$\label{shimura:rec}
L\Big(\frac{k}{2},f\otimes\varphi\Big)^\s=0 \text{ if and only if } L\Big(\frac{k}{2},f^\s\otimes\varphi^\s\Big)=0$$ for any $\s\in\text{Aut}_\Q(\C)$. Assume that $L\big(\frac{k}{2},f\otimes\varphi\big)=0$ for infinitely many Hecke character $\varphi$ of $p$-power order and $\fp$-power conductor. Then by (\[shimura:rec\]), we have $L_{\textrm{av}}(f\otimes\varphi)=0$ for infinitely many such $\varphi$. This contradicts to the equation (\[gal:av:limit\]).
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF- 2017R1A2B4012408).
[99]{}
V. Blomer and F. Brumley, On the Ramanujan conjecture over number fields, Ann. of Math. 174 (2011), 581-605. R. Bruggeman and R. Miatello, Estimates of Kloosterman sums for groups of real rank one, Duke Math. J. 80 (1995), No. 1, 105-137. S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic $L$-functions on GL(2), Ann. of Math. 142 (1995) 385-423. H. Hida, On the critical values of $L$-functions of ${\operatorname{GL}}(2)$ and ${\operatorname{GL}}(2)$ $\times$ ${\operatorname{GL}}(2)$, Duke Math. J. 74 (1994), no. 2, 431-529. H. Jacquet and J. Shalika, A non-vanishing theorem for zeta functions of GL$_n$, Invent. Math. 38 (1976/77), no. 1, 1-16. M. Kim and H. Sun, Modular symbols and modular $L$-values with cyclotomic twists, Submitted W. Luo and D. Ramakrishnan, Determination of modular forms by twists of critical $L$-values, Invent. Math. 130 (1997), no. 2, 371-398. T. Miyake, On automorphic forms on GL$_2$ and Hecke operators, Ann. of Math. 94 (1971), no. 1 , 174-189 M. Nakasuji, Generalized Ramanujan conjecture over general imaginary quadratic fields, Forum Math. 24 (2012), no. 1, 85-98 K. Namikawa, On $p$-adic $L$-function associated with cusp forms on ${\operatorname{GL}}_2$, Manuscripta Math. 153 (2017), no. 3-4, 563-622 J. Neukirch, Algebraic number theory, Springer-Verlag (1992). J. Order, Rankin-Selberg $L$-functions in cyclotomic fields, III, Rreprint D. E. Rohrlich, Nonvanishing of $L$-functions for ${\operatorname{GL}}(2)$, Invent. Math. 97 (1989), 391-403. D. E. Rohrlich, On $L$-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 409-423. T. Shintani, A remark on zeta functions of algebraic number fields. In: Automorphic Forms, Representation theory and Arithmetic. (Proc. Bombay Colloquium 1979). Berlin-Heidelberg-New York: Springer 1981 H. Sun, Generation of cyclotomic Hecke fields by modular $L$-values with cyclotomic twists, to appear in Amer. J. Math.
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abstract: 'In this paper it is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey space $\mathcal{M}_{L(\log L),\lambda}$, but $M$ is still bounded on $\mathcal{M}_{L(\log L),\lambda}$ for radially decreasing functions. The boundedness of the iterated maximal operator $M^2$ from $\mathcal{M}_{L(\log L),\lambda}$ to weak Zygmund-Morrey space $\WM_{L(\log L),\lambda}$ is proved. The class of functions for which the maximal commutator $C_b$ is bounded from $\mathcal{M}_{L(\log L),\lambda}$ to $\WM_{L(\log L),\lambda}$ are characterized. It is proved that the commutator of the Hardy-Littlewood maximal operator $M$ with function $b \in \B(\rn)$ such that $b^- \in L_{\infty}(\rn)$ is bounded from $\mathcal{M}_{L(\log L),\la}$ to $\WM_{L(\log L),\lambda}$. New pointwise characterizations of $M_{\alpha} M$ by means of norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.'
address:
- |
Institute of Mathematics\
Academy of Sciences of the Czech Republic\
Žitná 25\
115 67 Praha 1, Czech Republic
- |
Department of Mathematics\
Faculty of Science and Arts\
Kirikkale University\
71450 Yahsihan, Kirikkale, Turkey
- |
Department of Mathematics\
Faculty of Science and Arts\
Kirikkale University\
71450 Yahsihan, Kirikkale, Turkey
author:
- Amiran Gogatishvili
- Rza Mustafayev
- Müjdat Aǧcayazi
title: 'Weak-type estimates in Morrey spaces for maximal commutator and commutator of maximal function'
---
[^1]
Introduction
=============
Given a locally integrable function $f$ on $\rn$ and $0\leq \a <n$, the fractional maximal function $M_{\a}f$ of $f$ is defined by $$M_{\a}f(x):=\sup_{Q \ni x}|Q|^{\frac{\a-n}{n}}\int_Q |f(y)|\,dy,
\qquad (x\in\rn),$$ where the supremum is taken over all cubes $Q$ containing $x$. The operator $M_{\a}:~f \rightarrow M_{\a}f$ is called the fractional maximal operator. $M: = M_{0}$ is the classical Hardy-Littlewood maximal operator.
The study of maximal operators is one of the most important topics in harmonic analysis. These significant non-linear operators, whose behavior are very informative in particular in differentiation theory, provided the understanding and the inspiration for the development of the general class of singular and potential operators (see, for instance, [@stein1970; @guz1975; @GR; @tor1986; @stein1993; @graf2008; @graf]).
Let $f\in\Lloc$. Then $f$ is said to be in $\B (\rn)$ if the seminorm given by $$\|f\|_{*}:= \sup_Q\frac{1}{|Q|}\int_Q |f(y)-f_Q|dy$$ is finite.
Given a measurable function $b$ the maximal commutator is defined by $$C_b(f)(x) : = \sup_{Q \ni x}\frac{1}{|Q|}\int_{Q}
|b(x)-b(y)||f(y)|dy$$ for all $x\in\rn$.
This operator plays an important role in the study of commutators of singular integral operators with $\B$ symbols (see, for instance, [@GHST; @LiHuShi; @ST1; @ST2]). The maximal operator $C_b$ has been studied intensively and there exist plenty of results about it. Garcia-Cuerva et al. [@GHST] proved the following statement.
\[Cb\] Let $1 < p < \infty$. $C_b$ is bounded on $L_p(\rn)$ if and only if $b\in\B(\rn)$.
Given a measurable function $b$ the commutator of the Hardy-Littlewood maximal operator $M$ and $b$ is defined by $$[M,b]f (x): = M(bf)(x) - b(x)Mf(x)$$ for all $x\in\rn$.
The operator $[M,b]$ was studied by Milman et al. in [@MilSchon] and [@BasMilRu]. This operator arises, for example, when one tries to give a meaning to the product of a function in $H^1$ and a function in $\B$ (which may not be a locally integrable function, see, for instance, [@bijz]). Using real interpolation techniques, in [@MilSchon], Milman and Schonbek proved the $L_p$-boundedness of the operator $[M,b]$. Bastero, Milman and Ruiz [@BasMilRu] proved the next theorem.
\[BMR\] Let $1 < p < \infty$. Then the following assertions are equivalent:
[(i)]{} $[M,b]$ is bounded on $L_p(\rn)$.
[(ii)]{} $b \in \B(\rn)$ and $b^- \in L_{\infty}(\rn)$. [^2]
The opertors $C_b$ and $[M,b]$ enjoy weak-type $L(1 + \log^+ L)$ estimate.
\[HuYang\] The following assertions are equivalent:
[(i)]{} There exists a positive constant $c$ such that for each $\la >0$, inequality $$\label{eq0003.7}
|\{x \in\rn : C_b(f)(x)> \la \}| \le c
\int_{\rn}\frac{|f(x)|}{\la}\left(1+\log^+
\left(\frac{|f(x)|}{\la}\right)\right)dx.$$ holds for all $f\in L(1 + \log^+ L)(\rn)$.
[(ii)]{} $b\in \B (\rn)$.
[@AGKM Theorem 1.6] \[thm3495195187t\] Let $b \in \B(\rn)$ such that $b^-\in L_{\infty}(\rn)$. Then there exists a positive constant $c$ such that $$\begin{aligned}
|\{x \in\rn : |[M,b] f(x)| > \la \}| \leq c c_0\left(1+\log^+
c_0\right)\int_{\rn}\frac{|f(x)|}{\la}\left(1+\log^+\left(\frac{|f(x)|}{\la}\right)\right)dx, \label{weak}
\end{aligned}$$ for all $f\in L\left(1 + \log^+ L\right)$ and $\la >0$, where $c_0=\|b^+\|_{*}+\|b^-\|_{\infty}$.
Operators $C_b$ and $[M,b]$ essentially differ from each other. For example, $C_b$ is a positive and sublinear operator, but $[M,b]$ is neither positive nor sublinear. However, if $b$ satisfies some additional conditions, then operator $C_b$ controls $[M,b]$.
[@AGKM Lemma 3.1 and 3.2]\[pointwise1\] Let $b$ be any non-negative locally integrable function. Then $$\label{eqPointwise}
|[M,b]f(x)|\leq C_b(f)(x) \quad (x \in \rn)$$ holds for all $f \in L_1^{\loc}(\rn)$.
If $b$ is any locally integrable function on $\rn$, then $$\label{eq.0001}
|[M,b]f|(x)\leq C_b(f)(x)+ 2b^-(x) Mf(x) \quad (x \in \rn)$$ holds for all $f \in L_1^{\loc}(\rn)$.
We recall the following statement from [@AGKM].
[@AGKM Theorem 1.13]\[thm1\] Let $b \in \B(\rn)$. Suppose that $X$ is a Banach space of measurable functions defined on $\rn$. Moreover, assume that $X$ satisfies the lattice property, that is, $$0 \le g \le f \quad \Rightarrow \quad \|g\|_X \ls \|f\|_X.$$ Assume that $M$ is bounded on $X$. Then the operator $C_b$ is bounded on $X$, and the inequality $$\|C_b f\|_X \le c \|b\|_* \|f\|_X$$ holds with constant $c$ independent of $f$.
Moreover, if $b^- \in L_{\infty}(\rn)$, then the operator $[M,b]$ is bounded on $X$, and the inequality $$\|[M,b] f\|_X \le c (\|b^+\|_* + \|b^-\|_{\infty})\|f\|_X$$ holds with constant $c$ independent of $f$.
The proof of previous theorem is based on the following inequalities.
[@AGKM Corollary 1.11 and 1.12]\[lem1111111\] Let $b\in \B(\rn)$. Then, there exists a positive constant $c$ such that $$\label{eq.0002}
C_b(f)(x)\leq c \|b\|_{*} M^2f(x) \qquad (x\in\rn)$$ for all $f \in L_1^{\loc}(\rn)$.
Moreover, if $b^- \in L_{\infty}(\rn)$, then, there exists a positive constant $c$ such that $$\label{eqPointwise3}
|[M,b]f(x)|\leq c \left(\|b^+\|_{*}+\|b^-\|_{\infty}\right)M^2f(x)$$ for all $f \in L_1^{\loc}(\rn)$.
The classical Morrey spaces $\mathcal{M}_{p, \lambda} \equiv \mathcal{M}_{p, \lambda} (\rn)$, were introduced by C. Morrey in [@M1938] in order to study regularity questions which appear in the Calculus of Variations, and defined as follows: for $0 \le \lambda \le n$ and $1\le p \le \infty$, $$\mathcal{M}_{p,\lambda} : = \left\{ f \in \Lploc:\,\left\| f\right\|_{\mathcal{M}_{p,\lambda }} : =
\sup_{x\in \rn, \; r>0 }
r^{\frac{\lambda-n}{p}} \|f\|_{L_{p}(B(x,r))} <\infty\right\},$$ where $\Bxr$ is the open ball centered at $x$ of radius $r$.
Note that $\mathcal{M}_{p,0}(\rn) = L_{\infty}(\rn)$ and ${\mathcal M}_{p,n}(\rn) = L_{p}(\rn)$.
These spaces describe local regularity more precisely than Lebesgue spaces and appeared to be quite useful in the study of the local behavior of solutions to partial differential equations, a priori estimates and other topics in PDE (cf. [@giltrud]).
The boundedness of the Hardy-Littlewood maximal operator $M$ in Morrey spaces $\mathcal{M}_{p,\lambda}$ was proved by F. Chiarenza and M. Frasca in [@ChiFra1987]: It was shown that $Mf$ is a.e. finite if $f \in \mathcal{M}_{p,\lambda}$ and an estimate $$\label{ChiFr}
\|Mf\|_{\mathcal{M}_{p,\lambda}} \le c \|f\|_{\mathcal{M}_{p,\lambda}}$$ holds if $1 < p < \infty$ and $0 < \lambda < n$, and a weak type estimate replaces for $p = 1$, that is, the inequality $$\label{ChiFrWeakType}
t |\{Mf > t\} \cap B(x,r)| \le cr^{n - \lambda}
\|f\|_{\mathcal{M}_{1,\lambda}}$$ holds with constant $c$ independent of $x,\,r,\,t$ and $f$.
In [@gogmus], it is proved that the Hardy-Littlewood maximal operator $M$ is bounded on $\M_{1,\la}$, $0 \le \la < n$, for radially decreasing functions, that is, the inequality $$\label{gogmus}
\|Mf\|_{\M_{1,\la}} \ls \|f\|_{\M_{1,\la}},~f \in \mf^{\rad,\dn}$$ holds with constant independent of $f$, and an example which shows that $M$ is not bounded on $\M_{1,\la}$, $0 < \la < n$ is given.
Combining Theorem \[lem1111111\] with inequalities and , it is easy to generalize Theorems \[Cb\] and \[BMR\] to Morrey spaces (see Theorems \[thm3.1\] and \[thm4.5\]).
In this paper the Zygmund-Morrey and the weak Zygmund-Morrey spaces are defined. In order to investigate the boundedness of the maximal commutator $C_b$ and the commutator of maximal function $[M,b]$ on Zygmund-Morrey spaces we start to study the boundedness properties of the Hardy-Littlewood maximal operator on these spaces. It is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey spaces $\mathcal{M}_{L(\log L),\la}$, but $M$ is still bounded on $\mathcal{M}_{L(\log L),\la}$ for radially decreasing functions. The boundedness of the iterated maximal operator $M^2$ from Zygmund-Morrey spaces $\mathcal{M}_{L(\log L),\la}$ to weak Zygmund-Morrey spaces $\WM_{L(\log L),\la}$ is proved. The class of functions for which the maximal commutator $C_b$ is bounded from $\mathcal{M}_{L(\log L),\la}$ to $\WM_{L(\log L),\la}$ are characterized. It is proved that the commutator $[M,b]$ is bounded from $\mathcal{M}_{L(\log L),\la}$ to $\WM_{L(\log L),\la}$, when $b \in \B(\rn)$ such that $b^- \in L_{\infty}(\rn)$. New pointwise characterizations of $M_{\alpha} M$ by means of norm of Hardy-Littlewood maximal function in Morrey space are given.
The paper is organized as follows. In Section \[sect2\] notations and preliminary results are given. Boundedness of maximal commutator and commutator of maximal function in Morrey spaces are investigated in Section \[sect6\]. New characterizations of $M_{\a}M$ are obtained in section \[sect7\]. In Section \[sect7.3\] it is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey spaces $\mathcal{M}_{L(\log L),\la}$, but $M$ is still bounded on $\mathcal{M}_{L(\log L),\la}$ for radially decreasing functions. The boundedness of the iterated maximal operator from $\mathcal{M}_{L(\log L),\la}$ to $\WM_{L(\log L),\la}$ is proved in Section \[sect7.5\]. In Section \[sect8\] weak-type estimates for maximal commutator and commutator of maximal function in Zygmund-Morrey spaces are proved.
Notations and Preliminaries {#sect2}
===========================
Now we make some conventions. Throughout the paper, we always denote by $c$ a positive constant, which is independent of main parameters, but it may vary from line to line. However a constant with subscript such as $c_1$ does not change in different occurrences. By $a\ls b$ we mean that $a\le c b$ with some positive constant $c$ independent of appropriate quantities. If $a\ls b$ and $b\ls a$, we write $a\approx b$ and say that $a$ and $b$ are equivalent. For a measurable set $E$, $\chi_E$ denotes the characteristic function of $E$. Throughout this paper cubes will be assumed to have their sides parallel to the coordinate axes. Given $\la > 0$ and a cube $Q$, $\la Q$ denotes the cube with the same center as $Q$ and whose side is $\la$ times that of $Q$. For a fixed $p$ with $p\in [1,\infty)$, $p'$ denotes the dual exponent of $p$, namely, $p'=p/(p-1)$. For any measurable set $E$ and any integrable function $f$ on $E$, we denote by $f_E$ the mean value of $f$ over $E$, that is, $f_E=(1/|E|)\int_E f(x)dx$. Unless a special remark is made, the differential element $dx$ is omitted when the integrals under consideration are the Lebesgue integrals.
For the sake of completeness we recall the definitions and some properties of the spaces we are going to use.
Let $\Omega$ be any measurable subset of $\rn$, $n\geq 1$. Let $\mf(\Omega)$ denote the set of all measurable functions on $\Omega$ and $\mf_0 (\Omega)$ the class of functions in $\mf (\Omega)$ that are finite a.e.
For $p\in (0,\infty]$, we define the functional $\|\cdot\|_{p,\Omega}$ on $\mf(\Omega)$ by
$$\|f\|_{p,\Omega}:=
\begin{cases}
(\int_{\Omega} |f(x)|^p \,dx)^{1/p} &\text{if } \ \ p<\infty,\\
\esup_{\Omega} |f(x)| &\text{if } \ \ p=\infty.
\end{cases}$$
The Lebesgue space $L_p(\Omega)$ is given by $$L_p(\Omega):= \{f\in \mf(\Omega): \|f\|_{p,\Omega}<\infty\}$$ and it is equipped with the quasi-norm $\|\cdot\|_{p,\Omega}$.
Denote by $\mf^{\rad,\dn} = \mf^{\rad,\dn}(\rn)$ the set of all measurable, radially decreasing functions on $\rn$, that is, $$\mf^{\rad,\dn} : = \{f \in \mf(\rn):\, f(x) = \vp(|x|),\,x \in \rn
\,\mbox{with}\,\vp \in \mf^{\dn}(0,\infty)\}.$$
Recall that $Mf \ap Hf$, $f \in \mf^{\rad,\dn}$, where $$Hf (x) : = \frac{1}{|B(0,|x|)|} \int_{B(0,|x|)} |f(y)|\,dy$$ is $n$-dimensional Hardy operator. Obviously, $Hf \in \mf^{\rad,\dn}$, when $f \in \mf^{\rad,\dn}$.
The non-increasing rearrangement (see, e.g., [@BS p. 39]) of a function $f \in \mf_0 (\rn)$ is defined by $$f^*(t) : =\inf\left\{\la >0 : |\{x\in\rn: |f(x)|>\la \}| \leq t
\right\}\quad (0<t<\infty).$$ Then $f^{**}$ will denote the maximal function of $f^*$ defined by $$f^{**} (t) : = \frac{1}{t} \int_0^t f^* (s)\,ds, ~ (t > 0).$$
The Zygmund class $L(\log^+ L)(\Omega)$ is the set of all $f \in
\mf(\Omega)$ such that $$\int_{\Omega} |f(x)| (\log^+ |f(x)|)\,dx < \infty,$$ where $\log^+ t = \max \{\log t,0\}$, $t>0$. Generally, this is not a linear set. Nevertheless, considering the class $$L(1 + \log^+ L)(\Omega) = \left\{ f \in \mf (\Omega):\, \|f\|_{L(1 +
\log^+ L)(\Omega)} : = \int_{\Omega} |f(x)|\, (1 + \log^+
|f(x)|)\,dx < \infty \right\},$$ we obtain a linear set, the Zygmund space.
The size of $M^2$ is given by the following inequality.
[@CPer Lemma 1.6]\[lem002.8\] There exists a positive constant $c$ such that for any function $f$ and for all $\la >0$, $$|\{x \in\rn : M^2f(x)> \la \}| \leq c
\int_{\rn}\frac{|f(x)|}{\la}\left(1+\log^+
\left(\frac{|f(x)|}{\la}\right)\right)dx.$$
The following important result regarding $\B$ is true.
\[lem2.4.\] For $p\in (0,\infty)$, $\B(p)(\rn)=\B(\rn)$, with equivalent norms, where $$\|f\|_{\B(p)(\rn)}:=\sup_{Q}\left(
\frac{1}{|Q|}\int_{Q}|f(y)-f_{Q}|^p dy\right)^{\frac{1}{p}}.$$
A continuously increasing function on $[0,\infty]$, say $\Psi: [0,\infty]\rightarrow [0,\infty]$ such that $\Psi (0)=0$, $\Psi(1)=1$ and $\Psi(\infty)=\infty$, will be referred to as an Orlicz function. If $\Psi$ is an Orlicz function, then $$\Phi (t)=\sup \{ts-\Psi(s); s\in [0,\infty]\}$$ is the complementary Orlicz function to $\Psi$.
The Orlicz space denoted by $L^{\Psi} = L^{\Psi}(\rn)$ consists of all measurable functions $g: \rn \rightarrow \R$ such that $$\int_{\rn} \Psi \left(\frac{|g(x)|}{\a}\right)dx <\infty$$ for some $\a >0$.
Let us define the $\Psi$-average of $g$ over a cube $Q$ of $\rn$ by $$\|g\|_{\Psi,Q}=\inf \left\{\a>0: \frac{1}{|Q|}\int_{Q}
\Psi\left(\frac{|g(x)|}{\a}\right)dx\leq 1\right\}.$$
When $\Psi$ is a Young function, i.e. a convex Orlicz function, the quantity $$\|f\|_{\Psi}=\inf \left\{\a>0: \int_{\rn}
\Psi\left(\frac{|f(y)|}{\a}\right)dy\leq 1\right\}$$ is well known Luxemburg norm in the space $L^{\Psi}$ (see [@RR]).
A Young function $\Psi$ is said to satisfy the $\nabla_2$-condition, denoted $\Psi \in \nabla_2$, if for some $K > 1$ $$\Psi (t) \le \frac{1}{2K} \Psi (Kt) ~\mbox{for all} ~ t > 0.$$ It should be noted that $\Psi (t) \equiv t$ fails the $\nabla_2$-condition.
[@k]\[k\] The Hardy-Littlewood maximal operator is bounded on $L^{\Psi}$, provided that $\Psi \in \nabla_2$.
Combining Theorem \[k\] and \[thm1\], we obtain the following statement.
Let $b \in \B(\rn)$ and $\Psi \in \nabla_2$.
Then the operator $C_b$ is bounded on $L^{\Psi}$, and the inequality $$\|C_b f\|_{L^{\Psi}} \le c \|b\|_* \|f\|_{L^{\Psi}}$$ holds with constant $c$ independent of $f$.
Moreover, if $b^- \in L_{\infty}(\rn)$, then the operator $[M,b]$ is bounded on $L^{\Psi}$, and the inequality $$\|[M,b] f\|_{L^{\Psi}} \le c (\|b^+\|_* + \|b^-\|_{\infty})\|f\|_{L^{\Psi}}$$ holds with constant $c$ independent of $f$.
If $f\in L^{\Psi}(\rn)$, the Orlicz maximal function of $f$ with respect to $\Psi$ is defined by setting $$M_{\Psi}f(x)=\sup_{x\in Q} \|f\|_{\Psi,Q},$$ where the supremum is taken over all cubes $Q$ of $\rn$ containing $x$.
The generalized Hölder’s inequality $$\label{genHolder}
\frac{1}{|Q|}\int_{Q} |f(y)g(y)|dy \leq \|f\|_{\Phi,Q}
\|g\|_{\Psi,Q},$$ where $\Psi$ is the complementary Young function associated to $\Phi$, holds.
The main example that we are going to be using is $\Phi(t)=t(1+\log^+ t)$ with maximal function defined by $M_{L(1 +
\log^+ L)}$. The complementary Young function is given by $\Psi(t)\thickapprox e^t$ with the corresponding maximal function denoted by $M_{\exp L}$.
We define the weak $L(1 + \log^+ L)$-average of $g$ over a cube $Q$ of $\rn$ analogously by $$\|g\|_{WL(1 + \log^+ L),Q}=\inf \left\{\a>0:
\sup_{t>0}\frac{1}{|Q|}\frac{|\{x\in Q: |g(x)|>\a
t\}|}{\frac{1}{t}\left(1+\log^+ \frac{1}{t}\right)}\leq 1\right\}.$$
Let $0<\la <n$. The Zygmund-Morrey spaces $\mathcal{M}_{L(\log L),\la}(\rn) \equiv \mathcal{M}_{L(1 + \log^+ L),\la}(\rn)$ and the weak Zygmund-Morrey spaces $\WM_{L(\log L),\la}(\rn) \equiv \WM_{L(1 + \log^+ L),\la}(\rn)$ are defined as follows: $$\begin{aligned}
\mathcal{M}_{L(1 + \log^+ L),\la}(\rn): = & \{ f \in \mf
(\rn):\,\|f\|_{\mathcal{M}_{L(1+\log^+
L),\la}}:=\sup_{Q}|Q|^{\frac{\la}{n}}\|f\|_{L(1 + \log^+ L),Q}<\infty \}, \\
\WM_{L(1 + \log^+ L),\la}(\rn) : = & \{f \in \mf
(\rn):\,\|f\|_{\WM_{L(1+\log^+
L),\la}}:=\sup_{Q}|Q|^{\frac{\la}{n}}\|f\|_{WL(1 + \log^+ L),Q}<\infty \},
\end{aligned}$$ respectively. Note that $\mathcal{M}_{L(1+\log^+
L),\la}$ is a special case of Orlicz-Morrey spaces ${\mathcal L}^{\Phi,\phi}$ (with $\Phi (t) = t (1 + \log^+ t)$ and $\phi (t) = t^{\lambda}$, $t > 0$) defined in [@sst Definitions 2.3]. As we know, a weak version has not been defined yet in such form.
Boundedness of maximal commutator and commutator of maximal function in Morrey spaces {#sect6}
=====================================================================================
In this section we investigate boundedness of maximal commutator and commutator of maximal function in Morrey spaces.
The following theorem is true.
\[thm3.1\] Let $1<p<\infty$, $0\leq\la\leq n$. The following assertions are equivalent:
[(i)]{} $b\in \B(\rn)$.
[(ii)]{} The operator $C_b$ is bounded on $\mathcal{M}_{p,\lambda
}$.
[${\rm (i)}\Rightarrow {\rm (ii)}$]{}. Suppose that $b\in\B(\rn)$. By Theorem \[lem1111111\] and inequality it follows that $C_b$ is bounded in Morrey space $\mathcal{M}_{p,\lambda }$ and the following inequality holds: $$\|C_b(f)\|_{\mathcal{M}_{p,\lambda }}\lesssim \|b\|_{*}\,
\|f\|_{\mathcal{M}_{p,\lambda }}.$$
[${\rm (ii)}\Rightarrow {\rm (i)}$]{}. Assume that there exists $c > 0$ such that $$\|C_b(f)\|_{\mathcal{M}_{p,\lambda}} \le c \|f\|_{\mathcal{M}_{p,\lambda}}$$ for all $f \in \mathcal{M}_{p,\lambda}$. Obviously, $$\|f\|_{\mathcal{M}_{p,\lambda }}\thickapprox \sup_{Q'}
\left(|Q'|^{\frac{\la-n}{n}}\int_{Q'}|f(y)|^pdy\right)^{\frac{1}{p}}.$$ Let $Q$ be a fixed cube. We consider $f=\chi_Q$. It is easy to compute that $$\label{eq0000005.1}
\begin{split}
\|\chi_Q\|_{\mathcal{M}_{p,\lambda }}&\thickapprox \sup_{Q'}
\left(|Q'|^{\frac{\la-n}{n}}\int_{Q'}\chi_Q(y)dy\right)^{\frac{1}{p}}=\sup_{Q'}\left(|Q'\cap
Q||Q'|^{\frac{\la-n}{n}} \right)^{\frac{1}{p}} \\
& =\sup_{Q'\subseteq Q}\left(|Q'||Q'|^{\frac{\la-n}{n}}
\right)^{\frac{1}{p}}=|Q|^{\frac{\la}{np}}.
\end{split}$$
On the other hand, since $$C_b(\chi_Q)(x)\gtrsim \frac{1}{|Q|}\int_{Q}|b(y)-b_Q|dy \quad
\mbox{for all} \quad x\in Q.$$ then $$\label{eq0000005.2}
\begin{split}
\|C_b(\chi_Q)\|_{\mathcal{M}_{p,\lambda }} & \thickapprox \sup_{Q'}
\left(|Q'|^{\frac{\la-n}{n}}\int_{Q'}|C_b(\chi_Q)(y)|^pdy\right)^{\frac{1}{p}}
\\
&\gtrsim |Q|^{\frac{\la}{np}}\frac{1}{|Q|}\int_{Q}|b(y)-b_Q|dy.
\end{split}$$ Since by assumption $$\|C_b(\chi_Q)\|_{\mathcal{M}_{p,\lambda }}\lesssim
\|\chi_Q\|_{\mathcal{M}_{p,\lambda }},$$ by and , we get that $$\frac{1}{|Q|}\int_{Q}|b(y)-b_Q|dy \lesssim c.$$
Combining Theorem \[lem1111111\] with inequality , we get the following statement.
\[thm3.11\] Let $0 < \la < n$. Assume that $b\in \B(\rn)$. Then the operator $C_b$ is bounded on $\mathcal{M}_{1,\lambda}$ for radially decreasing functions.
The following theorem was proved in [@x].
\[thm4.5\] Let $1<p<\infty$, $0\le\la \le n$. Suppose that $b$ be a real valued, locally integrable function in $\rn$. The following assertions are equivalent:
[(i)]{} $b$ is in $\B(\rn)$ such that $b^-\in L_{\infty}(\rn)$.
[(ii)]{} The commutator $[M,b]$ is bounded in $\mathcal{M}_{p,\lambda }$.
[${\rm (i)}\Rightarrow {\rm (ii)}$]{}. Assume that $b$ is in $\B(\rn)$ such that $b^-\in L_{\infty}(\rn)$. By Theorem \[lem1111111\] and inequality it follows that $[M,b]$ is bounded in Morrey space $\mathcal{M}_{p,\lambda }$ and the following inequality holds: $$\|[M,b]f\|_{\mathcal{M}_{p,\lambda }}\lesssim
\left(\|b^+\|_{*}+\|b^-\|_{\infty}\right)\,
\|f\|_{\mathcal{M}_{p,\lambda }}.$$
Combining Theorem \[lem1111111\] with inequality , we obtain the following statement.
\[thm4.51\] Let $0 < \la < n$. Suppose that $b$ is in $\B(\rn)$ such that $b^-\in L_{\infty}(\rn)$. Then $[M,b]$ is bounded on $\mathcal{M}_{1,\lambda}$ for radially decreasing functions.
Some auxillary results {#sect7}
======================
To prove the theorems in the next sections we need the following results.
\[thm6.3\] Let $0\leq \a < n$. Then $$\begin{split}
M_{\a}(Mf)(x) = \sup_{Q \ni x}|Q|^{\frac{\a-n}{n}}\int_{Q}Mf &
\thickapprox
\sup_{Q \ni x}|Q|^{\frac{\a}{n}}\|f\|_{L(1 + \log^+ L),Q} \\
&\thickapprox \sup_{Q \ni x}|Q|^{\frac{\a-n}{n}}\int_{Q}|f|\left(1+\log^+
\frac{|f|}{|f|_{Q}}\right)
\end{split}$$ holds for all $f\in\Lloc$.
The statement of Theorem \[thm6.3\] follows by the following lemmas.
\[lem6.1\] The inequality $$\frac{1}{|Q|}\int_{Q}Mf(y)dy \ls \sup_{Q\subset Q'}\|f\|_{L(1 +
\log^+ L),Q'}$$ holds for all $f\in \Lloc$ with positive constant independent of $f$ and $Q$.
Let $Q$ be a cube in $\rn$ and $f=f_1+f_2$, where $f_1=f\chi_{3Q}$. Then $$\label{eq434565}
\frac{1}{|Q|}\int_{Q}Mf(y)dy \leq
\frac{1}{|Q|}\int_{Q}Mf_1(y)dy+\frac{1}{|Q|}\int_{Q}Mf_2(y)dy.$$ We recall simple geometric observation: for a fixed point $x \in Q$, if a cube $Q'$ satisfies $Q' \ni x$ and $Q' \cap (3Q)^c \neq \emptyset$, then $Q \subset 3 Q'$. Hence $$Mf_2 (x) = \sup_{Q' \ni x} \frac{1}{|Q'|}\int_{Q'}|f_2(y)|dy \le \sup_{Q\subset 3Q'}\frac{1}{|Q'|}\int_{Q'}|f(y)|dy.$$ Consequently, we have that $$\label{eq434596}
\frac{1}{|Q|}\int_{Q}Mf_2(y)dy \lesssim \sup_{Q\subset
Q'}\frac{1}{|Q'|}\int_{Q'}|f(y)|dy.$$ Since for any cube $Q'$ $$\frac{1}{|Q'|}\int_{Q'}|f(y)|dy\leq \|f\|_{L(1 + \log^+ L),Q'},$$ we get $$\label{eq996677}
\frac{1}{|Q|}\int_{Q}Mf_2(y)dy \lesssim \sup_{Q\subset Q'}\|f\|_{L(1
+ \log^+ L),Q'}.$$
On the other hand $$\frac{1}{|Q|}\int_{Q}Mf(y)dy \ls \|f\|_{L(1 + \log^+ L),Q}$$ for all $f$ such that $\supp f \subset Q$ (see [@CPer p. 174]). Thus $$\label{eq996633}
\frac{1}{|Q|}\int_{Q}Mf_1(y)dy\lesssim
\frac{1}{|3Q|}\int_{3Q}Mf_1(y)dy \lesssim \|f\|_{L(1 + \log^+
L),3Q}.$$
From , and , it follows that $$\label{eq44005500}
\begin{split}
\!\!\frac{1}{|Q|}\int_{Q}Mf(y)dy \lesssim \sup_{Q\subset
Q'}\|f\|_{L(1 + \log^+ L),Q'} + \|f\|_{L(1 + \log^+ L),3Q} \lesssim \sup_{Q\subset Q'}\|f\|_{L(1 + \log^+ L),Q'}.
\end{split}$$
We recall the following statement (see, for instance, [@IwMar p. 175]). For the completeness we give the proof.
\[stein\] Note that the estimation $$\int_Q M(f \chi_Q) \ap \int_Q |f| \,\bigg( 1 + \log^+ \frac{|f|}{|f|_Q}\bigg),$$ holds for all $f \in L_1^{\loc}(\rn)$ with positive constants independent of $f$ and $Q$.
Let $Q$ be a cube in $\rn$. We are going to use weak type estimates (see [@stein1969], for instance): there exist positive constants $c_1 <1$ and $c_2 > 1$ such that for every $f\in \Lloc$ and for every $t>1/|Q|\int_{Q}|f|$ we have $$c_1 \int_{\{x\in Q: |f(x)|>t\}}\frac{|f(x)|}{t}\,dx \le |\{x\in Q:
M(f\chi_Q)(x)>t\}| \le c_2 \int_{\{x\in Q:
|f(x)|>t/2\}}\frac{|f(x)|}{t}\,dx.$$ We have that $$\begin{aligned}
\int_{Q}M(f\chi_Q) & = \int_0^{\infty}|\{x\in Q: M(f\chi_Q)(x)>\la\}|d\la \\
& = \int_0^{|f|_Q}|\{x\in Q: M(f\chi_Q)(x)>\la\}|d\la \\
&+ \int_{|f|_Q}^{\infty}|\{x\in Q: M(f\chi_Q)(x)>\la\}|d\la\\
&=|Q| |f|_Q + \int_{|f|_Q}^{\infty}|\{x\in Q: M(f\chi_Q)(x)>\la\}|d\la \\
& \geq |Q| |f|_Q + c_1\int_{|f|_Q}^{\infty} \left(\int_{\{x\in Q:
|f(x)|>\la\}}|f(y)|\,dy\right)\,\frac{d\la}{\la}\\
& = |Q| |f|_Q + c_1 \int_{\{x\in Q:
|f(x)|>|f|_Q\}} \left( \int_{|f|_Q}^{|f(x)|} \frac{d\la}{\la}\right)\,|f(x)|\,dx \\
& = |Q| |f|_Q + c_1 \int_{\{x\in Q:
|f(x)|>|f|_Q\}} |f(x)|\,\log \left( \frac{|f(x)|}{|f|_Q}\right)\,dx \\
&\geq c_1 \int_{Q}|f|\left(1+\log^+ \frac{|f|}{|f|_{Q}}\right).
\end{aligned}$$ On the other hand, $$\begin{aligned}
\int_{Q}M(f\chi_Q) & = \int_0^{\infty}|\{x\in Q: M(f\chi_Q)(x)>\la\}|d\la \\
& \ap \int_0^{\infty}|\{x\in Q: M(f\chi_Q)(x)>2\la\}|d\la \\
& = \int_0^{|f|_Q}|\{x\in Q: M(f\chi_Q)(x)>2\la\}|d\la \\
&+ \int_{|f|_Q}^{\infty}|\{x\in Q: M(f\chi_Q)(x)>2\la\}|d\la\\
& \le |Q| |f|_Q + c_2\int_{|f|_Q}^{\infty} \left(\int_{\{x\in Q:
|f(x)|>\la\}}|f(y)|\,dy\right)\,\frac{d\la}{\la}\\
& = |Q| |f|_Q + c_2 \int_{\{x\in Q:
|f(x)|>|f|_Q\}} |f(x)|\,\log \left( \frac{|f(x)|}{|f|_Q}\right)\,dx \\
&\le c_2 \int_{Q}|f|\left(1+\log^+ \frac{|f|}{|f|_{Q}}\right).
\end{aligned}$$
\[lem90099009\] Inequalities $$\begin{split}
\frac{1}{|Q|}\int_{Q}|f|\left(1+\log^+ \frac{|f|}{|f|_{Q}}\right)
\ap \|f\|_{L(1 + \log^+ L),Q}
\end{split}$$ hold for all $f\in \Lloc$ with positive constants independent of $f$ and $Q$.
Since $$1\leq \frac{1}{|Q|}\int_{Q}\frac{|f|}{|f|_{Q}}\left(1+\log^+
\frac{|f|}{|f|_{Q}}\right),$$ then $$|f|_{Q} \leq \|f\|_{L(1 + \log^+ L),Q}.$$ Using the inequality $\log^+ (ab)\leq \log^+ a +\log^+ b$, $a,b
\in\R^+$, we get $$\begin{aligned}
\frac{1}{|Q|}\int_{Q}|f|\left(1+\log^+
\frac{|f|}{|f|_{Q}}\right) &\\
&\hspace{-2cm}=\frac{1}{|Q|}\int_{Q}|f|\left(1+\log^+\left(
\frac{|f|}{\|f\|_{L(1 + \log^+ L),Q}}\frac{\|f\|_{L(1 + \log^+ L),Q}}{|f|_{Q}}\right)\right) \\
&\hspace{-2cm}\leq \frac{1}{|Q|}\int_{Q}|f|\left(1+\log^+
\frac{|f|}{\|f\|_{L(1 + \log^+ L),Q}}\right)
\\
&\hspace{-2cm}+\frac{1}{|Q|}\int_{Q}|f|\log^+\frac{\|f\|_{L(1 + \log^+ L),Q}}{|f|_{Q}}\\
&\hspace{-2cm}\leq \|f\|_{L(1 + \log^+
L),Q}+|f|_{Q}\log^+\frac{\|f\|_{L(1 + \log^+ L),Q}}{|f|_{Q}}.
\end{aligned}$$ Since $\frac{\|f\|_{L(1 + \log^+ L),Q}}{|f|_{Q}}\geq 1$ and $\log
t\leq t$ when $t\geq 1$, we get $$\begin{split}
\frac{1}{|Q|}\int_{Q}|f|\left(1+\log^+ \frac{|f|}{|f|_{Q}}\right)
\leq 2\|f\|_{L(1 + \log^+ L),Q}.
\end{split}$$ On the other hand, since $$\|f\|_{L(1 + \log^+ L),Q} = \frac{1}{|Q|}\int_{Q} |f|\left(1+\log^+ \frac{|f|}{\|f\|_{L(1 + \log^+ L),Q}}\right),$$ on using $|f|_{Q} \leq \|f\|_{L(1 + \log^+ L),Q}$, we get that $$\|f\|_{L(1 + \log^+ L),Q} \le \frac{1}{|Q|}\int_{Q}|f|\left(1+\log^+
\frac{|f|}{|f|_{Q}}\right).$$
[***Proof of Theorem \[thm6.3\].***]{} By Lemma \[lem6.1\], we get that $$\sup_{Q \ni x}|Q|^{\frac{\la-n}{n}}\int_{Q}Mf(y)dy\lesssim
\sup_{Q\ni x}|Q|^{\frac{\la}{n}} \sup_{Q \subset Q'}\|f\|_{L(1+\log^+ L),Q'} \le \sup_{Q \ni x}|Q|^{\frac{\la}{n}} \|f\|_{L(1+\log^+ L),Q}.$$ The equivalence $$\sup_{Q \ni x}|Q|^{\frac{\la}{n}}\|f\|_{L(1+\log^+ L),Q} \thickapprox
\sup_{Q \ni x}|Q|^{\frac{\la-n}{n}}\int_{Q}|f|\left(1+\log^+
\frac{|f|}{|f|_{Q}}\right)$$ is obvious in view of Lemma \[lem90099009\].
By Lemma \[stein\], we have that $$\sup_{Q \ni x}|Q|^{\frac{\la-n}{n}}\int_{Q}|f|\left(1+\log^+
\frac{|f|}{|f|_{Q}}\right)\lesssim
\sup_{Q \ni x}|Q|^{\frac{\la-n}{n}}\int_{Q}Mf.$$
$$\hspace{17.5cm}\square$$
The following corollaries follow from Theorem \[thm6.3\].
Inequalities $$\label{eq2340958671897252}
M^2f(x) \thickapprox M_{L(1 + \log^+ L)}f(x) \thickapprox \sup_{x\in
Q}\frac{1}{|Q|}\int_{Q}|f|\left(1+\log^+ \frac{|f|}{|f|_{Q}}\right)$$ holds for all $x\in\rn$ and $f\in \Lloc$ with positive constants independent of $x$ and $f$.
\[thm134321r19\] Let $0<\la <n$. The equivalency $$\|Mf\|_{\mathcal{M}_{1,\la}} \thickapprox
\|f\|_{\mathcal{M}_{L(1 + \log^+ L),\la}} \thickapprox \sup_{Q}|Q|^{\frac{\la-n}{n}}\int_{Q}|f|\left(1+\log^+ \frac{|f|}{|f|_{Q}}\right)$$ holds with positive constants independent of $f$.
Note that $M^2f \thickapprox M_{L(1 + \log^+ L)}f$ was proved in [@CPer] (see, also [@graf p. 159]). For the second part of see [@CarPas], [@Leck], [@LeckN] and [@CPer]. The equivalence $\|Mf\|_{\mathcal{M}_{1,\la}} \thickapprox \|f\|_{\mathcal{M}_{L(1 + \log^+ L),\la}}$ is a special case of [@sst Lemma 3.5].
Note on the boundedness of the maximal function on Zygmund-Morrey spaces {#sect7.3}
========================================================================
In this section we prove that the Hardy-Littlewood maximal operator $M$ is bounded on $\M_{L(1 + \log^+ L),\la}$, $0 < \la < n$, for radially decreasing functions, and we give an example which shows that $M$ is not bounded on $\M_{L(1 + \log^+ L),\la}$, $0 < \la < n$.
In order to prove the main result of this section we need the following auxiliary lemmas.
\[lem999.1\] Assume that $0 < \la < n$. Let $f \in \mf^{\rad,\dn}(\rn)$ with $f(x) = \vp (|x|)$. The equivalency $$\|f\|_{\M_{L(1 + \log^+ L),\la}} \ap \sup_{x>0} x^{\la - n} \int_0^x
\frac{1}{t} \int_0^t
|\vp (\rho)|\rho^{n-1}\,d\rho\,dt$$ holds with positive constants independent of $f$.
Recall that $$\|f\|_{\M_{L(1 + \log^+ L),\la}} \ap
\sup_{B}|B|^{\frac{\la-n}{n}}\int_{B}M f = \| M_{\la}(Mf)\|_{\infty}, ~ f \in \mf(\rn).$$ Since $M_{\la}(f)(y) \gs \frac{1}{|B(0,|y|)|^{1 - \la /n}}\int_{B(0,|y|)} |f(z)|\,dz$, in view of $Mf \ap Hf$, $f \in
\mf^{\rad,\dn}$, switching to polar coordinates, we have that $$\begin{aligned}
M_{\la}(Mf)(y) & \gs \frac{1}{|B(0,|y|)|^{1 - \la /n}}\int_{B(0,|y|)} |Mf(z)|\,dz \\
& \ap \frac{1}{|B(0,|y|)|^{1 - \la /n}}\int_{B(0,|y|)} |Hf(z)|\,dz
\\
& = \frac{1}{|B(0,|y|)|^{1 - \la /n}}\int_{B(0,|y|)}
\frac{1}{|B(0,|z|)} \int_{B(0,|z|)} |f(w)|\,dw\,dz \\
& \ap \frac{1}{|B(0,|y|)|^{1 -\la /n}}\int_{B(0,|y|)}
|z|^{-n} \int_0^{|z|} |\vp(\rho)|\rho^{n-1}\,d\rho\,dz \\
& \ap |y|^{\la - n}\int_0^{|y|} \frac{1}{t} \int_0^t
|\vp(\rho)|\rho^{n-1}\,d\rho\,dt.
\end{aligned}$$ Consequently, $$\begin{aligned}
\|f\|_{\M_{L(1 + \log^+ L),\la}} & \gs \esup_{y \in \rn} |y|^{\la - n}\int_0^{|y|} \frac{1}{t} \int_0^t
|\vp(\rho)|\rho^{n-1}\,d\rho\,dt \\
& = \sup_{x > 0} x^{\la - n}\int_0^x \frac{1}{t} \int_0^t
|\vp(\rho)|\rho^{n-1}\,d\rho\,dt,
\end{aligned}$$ where $f (\cdot) = \vp (|\cdot|)$.
On the other hand, $$\begin{aligned}
\|f\|_{\M_{L(1 + \log^+ L),\la}} & \ls \sup_{B}|B|^{\frac{\la-n}{n}}\int_0^{|B|}(M f)^* (t)\,dt \\
& \ap \sup_{B}|B|^{\frac{\la-n}{n}}\int_0^{|B|}f^{**}(t)\,dt \\
& = \sup_{B}|B|^{\frac{\la-n}{n}}\int_0^{|B|} \frac{1}{t}\int_0^t f^{*}(s)\,ds\,dt \\
& = \sup_{B}|B|^{\frac{\la-n}{n}}\int_0^{|B|} \frac{1}{t}\int_0^t |\vp(s^{\frac{1}{n}})|\,ds\,dt \\
& \ap \sup_{B}|B|^{\frac{\la-n}{n}}\int_0^{|B|} \frac{1}{t}\int_0^{t^{\frac{1}{n}}} |\vp(\rho)|\rho^{n-1}\,d\rho\,dt \\
& \ap \sup_{B}|B|^{\frac{\la-n}{n}}\int_0^{|B|^{\frac{1}{n}}} \frac{1}{x}\int_0^{x} |\vp(\rho)|\rho^{n-1}\,d\rho\,dx \\
& = \sup_{x > 0} x^{\la - n}\int_0^x \frac{1}{t} \int_0^t
|\vp(\rho)|\rho^{n-1}\,d\rho\,dt,
\end{aligned}$$ where $f (\cdot) = \vp (|\cdot|)$.
\[cor999.1\] Assume that $0 < \la < n$. Let $f \in \mf^{\rad,\dn}(\rn)$ with $f(x) = \vp (|x|)$. The equivalency $$\|Mf\|_{\M_{L(1 + \log^+ L),\la}} \ap \sup_{x>0} x^{\la - n} \int_0^x \frac{1}{y} \int_0^y
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt\,dy$$ holds with positive constants independent of $f$.
Let $f \in \mf^{\rad,\dn}$ with $f(x) = \vp (|x|)$. Since $Mf \ap Hf$ and $Hf \in \mf^{\rad,\dn}$, by Lemma \[lem999.1\], switching to polar coordinates, we have that $$\begin{aligned}
\|Mf\|_{\M_{L(1 + \log^+ L),\la}} & \ap \sup_{x>0} x^{\la - n} \int_0^x
\frac{1}{y} \int_0^y
\left(\frac{1}{|B(0,t)|} \int_{B(0,t)} |f(y)|\,dy \right)t^{n-1}\,dt\,dy \\
& \ap \sup_{x>0} x^{\la - n} \int_0^x \frac{1}{y} \int_0^y
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt\,dy.
\end{aligned}$$
\[lem999.2\] Assume that $0 < \la < n$. Let $f \in \mf^{\rad,\dn}$ with $f(x) = \vp (|x|)$. The inequality $$\|Mf\|_{\M_{L(1 + \log^+ L),\la}} \ls \|f\|_{\M_{L(1 + \log^+
L),\la}},~ f \in \mf^{\rad,\dn}$$ holds if and only if the inequality $$\begin{aligned}
\sup_{x>0} x^{\la - n} \int_0^x \frac{1}{y} \int_0^y
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt\,dy & \\
& \hspace{-3cm} \ls \sup_{x>0} x^{\la - n} \int_0^x
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt,~ \vp \in \mf^{+,\dn}(\R_+)
\end{aligned}$$ holds true.
The statement immediately follows from Lemma \[lem999.1\] and Corollary \[cor999.1\].
\[lem999.5\] Let $0 <\la < n$. Then inequality $$\label{eq.999}
\sup_{x>0} x^{\la - n} \int_0^x \frac{1}{y} \int_0^y
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt\,dy \ls \sup_{x>0} x^{\la - n} \int_0^x
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt$$ holds for all $\vp \in \mf^{+,\dn}(\R_+)$.
Indeed: $$\begin{aligned}
\sup_{x>0} x^{\la - n} \int_0^x \frac{1}{y} \int_0^y
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt\,dy & \\
& \hspace{-5cm} =
\sup_{x>0} x^{\la - n} \int_0^x y^{n - \la - 1} y^{\la - n} \int_0^y
\frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt\,dy \\
& \hspace{-5cm} \le \sup_{y > 0} y^{\la - n} \int_0^y \frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt \cdot \left( \sup_{x > 0} x^{\la - n} \int_0^x y^{n - \la - 1} dy \right) \\
& \hspace{-5cm} \ap \sup_{y > 0} y^{\la - n} \int_0^y \frac{1}{t} \int_0^t
\vp (\rho)\rho^{n-1}\,d\rho\,dt.
\end{aligned}$$
\[main11\] Assume that $0 < \la < n$. The inequality $$\|Mf\|_{\M_{L(1 + \log^+ L),\la}} \ls \|f\|_{\M_{L(1 + \log^+ L),\la}}$$ holds for all $f \in \mf^{\rad,\dn}$ with constant independent of $f$.
The statement follows by Lemmas \[lem999.2\] and \[lem999.5\].
We give an example which shows that $M$ is not bounded on $\M_{L(1 +
\log^+ L),\la}$, $0 < \la < n$. For simplicity let $n = 1$ and $\la
= 1 / 2$. Consider even function $f$ defined as follows: $$f(x) = \sum_{k=0}^{\infty} \chi_{[k^2 \ln^2 (k+e),k^2 \ln^2 (k+e) +
1]}(x), \qquad x \ge 0.$$ It is easy to see that $Mf$ and $M^2 f$ are even functions. Obviously, $$\begin{aligned}
Mf(x) \ap & \sum_{k = 0}^{\infty} \chi_{[k^2 \ln^2 (k + e),k^2 \ln^2 (k + e) + 1]}(x) \\
& + \sum_{k = 0}^{\infty} \frac{1}{x - k^2 \ln^2 (k + e)}\chi_{[k^2 \ln^2 (k + e) + 1, k^2 \ln^2 (k + e) + 1 + m_k]}(x) \\
& + \sum_{k = 0}^{\infty} \frac{1}{(k+1)^2\ln^2 (k + 1 + e) + 1 -
x}\chi_{[k^2 \ln^2 (k + e) + 1 + m_k, (k+1)^2 \ln^2 (k + 1 +
e)]}(x), ~ x \ge 0,
\end{aligned}$$ where $$m_k = \frac{(k+1)^2 \ln^2 (k + 1 + e) - k^2 \ln^2 (k + e) - 1}{2}, ~ k = 0,1,2,\ldots.$$ Then $$\begin{aligned}
\|f\|_{\M_{L(1 + \log^+ L),1 / 2}(\R)} & \ap \|Mf\|_{{\mathcal M}_{1,1/2}(\R)} = \sup_{I}|I|^{-1/2} \int_I Mf \\
& \le \sup_{I:\,|I| \le 1}|I|^{-1/2} \int_I Mf + \sup_{I:\,|I| > 1}|I|^{-1/2} \int_I Mf.
\end{aligned}$$ It is easy to see that $$\sup_{I:\,|I| \le 1}|I|^{-1/2} \int_I Mf \le \sup_{I:\,|I| \le 1}|I|^{1/2} \le 1.$$ Since $$\int_{j^2 \ln^2 (j+e)}^{(j+1)^2 \ln^2 (j+e+1)} Mf (x)\,dx \ap (1 + 2 \ln (1
+ m_j)), ~ j = 0,1,2,\ldots,$$ we have that $$\begin{aligned}
\sup_{I:\,|I| > 1}|I|^{-1/2} \int_I Mf (x)\,dx & = \sup_{m \ge 2} \sup_{I:\,m - 1 < |I| \le m}|I|^{-1/2} \int_I Mf (x)\,dx \\
& \ls \sup_{m \ge 2} m^{-1/2} \int_0^m Mf (x)\,dx \\
& \le \sup_{m \ge 2} m^{-1/2} \sum_{i^2 \ln^2 (j+e) < m}\int_{j^2
\ln^2 (j+e)}^{(j+1)^2 \ln^2 (j+e+1)} Mf (x)\,dx \\
& \ap \sup_{m \ge 2} m^{-1/2} \sum_{i^2 \ln^2 (j+e) < m} (1 + 2 \ln (1
+ m_j)) \\
& \ls \sup_{m \ge 2} m^{-1/2} \sum_{i^2 \ln^2 (j+e) < m} \ln (j + e)
\\
& \ls \sup_{m \ge 2} m^{-1/2} m^{1/2} = 1,
\end{aligned}$$ we have that $$\|f\|_{\M_{L(1 + \log^+ L),1 / 2}(\R)} \ls 1 + 1 = 2.$$
On the other hand, it is easy to see that $$\begin{aligned}
M^2 f(x) & \ge \frac{1}{x - (k^2 \ln^2 (k+e) + 1)} \int_{k^2 \ln^2 (k + e) + 1}^x \frac{dt}{t - k^2 \ln^2 (k + e)} \\
& = \frac{\ln (x - k^2 \ln^2 (k+e))}{x - (k^2 \ln^2 (k+e) + 1)} \\
& \ge \frac{\ln (x - k^2 \ln^2 (k+e))}{x - k^2 \ln^2 (k+e)}
\end{aligned}$$ for any $x \in [k^2 \ln^2 (k + e) + e,k^2 \ln^2 (k + e) + m_k]$.
Thus $$\begin{aligned}
M^2 f(x) & \ge \sum_{k = 0}^{\infty} \frac{\ln (x - k^2 \ln^2 (k+e))}{x - k^2 \ln^2 (k+e)}\chi_{[k^2 \ln^2 (k + e) + e,k^2 \ln^2 (k +
e) + m_k]}(x).
\end{aligned}$$ Finally, $$\begin{aligned}
\|M f\|_{\M_{L(1 + \log^+ L),1 / 2}(\R)} & \ap \|M^2 f\|_{{\mathcal M}_{1,1/2}(\R)} \\
& \gs \sup_k (k \ln (k + e))^{-1} \int_0^{k^2 \ln^2 (k + e)} M^2 f (x)\,dx \\
& \ge \sup_k (k \ln (k + e))^{-1} \sum_{j=1}^{k-1}\int_{j^2 \ln^2 (j + e) + e}^{j^2 \ln^2 (j + e) + m_j} M^2 f (x)\,dx\\
& \ge \sup_k (k \ln (k + e))^{-1} \sum_{j=1}^{k-1}\int_{j^2 \ln^2 (j + e) + e}^{j^2 \ln^2 (j + e) + m_j} \frac{\ln (x - k^2 \ln^2 (k+e))}{x - k^2 \ln^2 (k+e)} \,dx \\
& = \sup_k (k \ln (k + e))^{-1} \sum_{j=1}^{k-1}\int_e^{m_j} \frac{\ln x}{x}\,dx \\
& \gs \sup_k (k\ln(k + e))^{-1} \sum_{j=1}^{k-1} \ln^2 m_j \\
& \gs \sup_k (k\ln(k + e))^{-1} \sum_{j=1}^{k-1} \ln^2 (j + e) \\
& \gs \sup_k (k\ln(k + e))^{-1} k \ln^2 (k + e) \\
& = \sup_k \ln (k + e) = \infty.
\end{aligned}$$
Weak-type estimates in Morrey spaces for the iterated maximal function {#sect7.5}
======================================================================
In this section the boundedness of the iterated maximal operator $M^2$ from Zygmund-Morrey spaces $\mathcal{M}_{L(1 + \log^+ L),\la}$ to weak Zygmund-Morrey spaces $\WM_{L(1 + \log^+ L),\la}$ is proved.
\[thm53432\] Let $0<\la <n$. Then the operator $M^2$ is bounded from $\mathcal{M}_{L(1 + \log^+ L),\la}$ to $\WM_{L(1 + \log^+ L),\la}$ and the following inequality holds $$\label{eq88008802}
\begin{split}
\|M^2f\|_{\WM_{L(1 + \log^+ L),\la}}\leq c \|f\|_{\mathcal{M}_{L(1 +
\log^+ L),\la}}
\end{split}$$ with positive constant $c$ independent of $f$.
Let $Q$ be any cube in $\rn$ and let $f=f_1+f_2$, where $f_1=f\chi_{4Q}$. By subadditivity of $M^2$ we get $$M^2f \leq M^2f_1 +M^2f_2.$$ Since for any cube $Q'$ conditions $z\in 2Q\cap Q'$ and $Q'\cap\{\rn\backslash 4Q\}\neq\emptyset$ imply $Q\subset 4Q'$, we have $$\label{eq998855}
Mf_2(z)=M(f\chi_{\rn\backslash 4Q})(z)\leq \sup_{Q\subset
4Q'}\frac{1}{|Q'|}\int_{Q'}|f|
%\lesssim \sup_{Q\subset
%Q'}\frac{1}{|Q'|}\int_{Q'}|f|$$ for any $z\in 2Q$. Thus for any $z\in \rn$ $$\label{eq1122223333}
Mf_2(z)\leq \chi_{2Q}(z)\sup_{Q\subset
4Q'}\frac{1}{|Q'|}\int_{Q'}|f| +\chi_{\rn\backslash 2Q}(z)Mf(z).$$ Applying to both sides of the inequality by operator $M$ for any $y\in Q$ we get $$M^2f_2(y)\leq M(\chi_{2Q})(y)\sup_{Q\subset
4Q'}\frac{1}{|Q'|}\int_{Q'}|f| +M(\chi_{\rn\backslash 2Q}Mf)(y).$$ Since $M(\chi_{2Q})(y)=1$, $y\in Q$, by the inequality we arrive at $$M^2f_2(y)\leq \sup_{Q\subset 4Q'}\frac{1}{|Q'|}\int_{Q'}|f|
+\sup_{Q\subset 2Q'}\frac{1}{|Q'|}\int_{Q'}Mf \lesssim
\sup_{Q\subset Q'}\frac{1}{|Q'|}\int_{Q'}Mf.$$ Consequently, for $y\in Q$ $$\label{eq5.15}
M^2f(y)\lesssim M^2(f\chi_{4Q})(y) + \sup_{Q\subset
Q'}\frac{1}{|Q'|}\int_{Q'}Mf.$$ In view of inequality $$\label{log}
1+\log^+ (ab)\leq (1+\log^+ a)(1+\log^+ b),$$ by Lemma \[lem002.8\], for any $\a
>0$ and $t>0$ we have that $$\begin{split}
\left|\left\{x\in Q: M^2(f\chi_{4Q})(x)>\a t
\right\}\right| \\
& \hspace{-2cm}\leq \left|\left\{x\in \rn:
M^2(f\chi_{4Q})(x)>\a t \right\}\right| \\
& \hspace{-2cm} \leq c \int_{\rn}\frac{|(f\chi_{4Q})(x)|}{\a
t}\left(1+\log^+
\left(\frac{|(f\chi_{4Q})(x)|}{\a t}\right)\right)dx \\
& \hspace{-2cm} \leq c \frac{1}{\a}\left(1+\log^+
\frac{1}{\a}\right) \int_{4Q}\frac{|f(x)|}{t}\left(1+\log^+
\left(\frac{|f(x)|}{t}\right)\right)dx .
\end{split}$$ We get that $$\begin{split}
\frac{\left|\left\{x\in Q: M^2(f\chi_{4Q})(x)>\a t
\right\}\right|}{\frac{1}{\a}\left(1+\log^+ \frac{1}{\a}\right)}
\leq c \int_{4Q}\frac{|f(x)|}{t}\left(1+\log^+
\left(\frac{|f(x)|}{t}\right)\right)dx .
\end{split}$$ Consequently, $$\begin{split}
\sup_{\a >0}\frac{1}{|Q|}\frac{\left|\left\{x\in Q:
M^2(f\chi_{4Q})(x)>\a t \right\}\right|}{\frac{1}{\a}\left(1+\log^+
\frac{1}{\a}\right)} \leq c \, \frac{1}{|4Q|}
\int_{4Q}\frac{|f(x)|}{t}\left(1+\log^+
\left(\frac{|f(x)|}{t}\right)\right)dx .
\end{split}$$ Thus $$\begin{split}
\inf\left\{t>0 : \sup_{\a
>0}\frac{1}{|Q|}\frac{\left|\left\{x\in Q: M^2(f\chi_{4Q})(x)>\a
t \right\}\right|}{\frac{1}{\a}\left(1+\log^+ \frac{1}{\a}\right)}
\leq 1 \right\} &
\\
&\hspace{-5cm}\leq \inf\left\{t>0 : \frac{1}{|4Q|}
\int_{4Q}\frac{c|f(x)|}{t}\left(1+\log^+
\left(\frac{|f(x)|}{t}\right)\right)dx\leq 1 \right\}\\
\\
&\hspace{-5cm}\leq \inf\left\{t>0 : \frac{1}{|4Q|}
\int_{4Q}\frac{c|f(x)|}{t}\left(1+\log^+
\left(\frac{c|f(x)|}{t}\right)\right)dx\leq 1 \right\}
\end{split}$$ that is, $$\label{eq880088}
\begin{split}
\|M^2(f\chi_{4Q})\|_{WL(1 + \log^+ L),Q}\leq \|c f\|_{L(1 + \log^+
L),4Q}=c\|f\|_{L(1 + \log^+ L),4Q}.
\end{split}$$ For the second summand in right hand side of the inequality applying the inequality we obtain $$\label{eq5784756}
\begin{split}
\left\|\sup_{Q\subset Q'}\frac{1}{|Q'|}\int_{Q'}Mf
\right\|_{WL(1 + \log^+ L),Q} \lesssim \sup_{Q\subset Q'}\frac{1}{|Q'|}\int_{Q'}Mf
\lesssim \sup_{Q\subset Q'}\|f\|_{L(1 + \log^+ L),Q'}.
\end{split}$$ By inequalities , and we get $$\label{eq88008801}
\begin{split}
\|M^2f\|_{WL(1 + \log^+ L),Q}\leq c \sup_{Q\subset 4Q'}\|f\|_{L(1 +
\log^+ L),Q'}.
\end{split}$$ Thus $$\begin{aligned}
\sup_{Q}|Q|^{\frac{\la}{n}}\|M^2f\|_{WL(1 + \log^+ L),Q} & \leq c \, \sup_{Q}|Q|^{\frac{\la}{n}}\sup_{Q\subset
4Q'}\|f\|_{L(1 + \log^+ L),Q'}\\
&\leq c \,
\left(\sup_{Q}|Q|^{\frac{\la}{n}} \sup_{Q\subset
4Q'}|Q'|^{-\frac{\la}{n}}\right) \,\sup_{Q}|Q|^{\frac{\la}{n}}\|f\|_{L(1 + \log^+ L),Q}\\
&\thickapprox \sup_{Q}|Q|^{\frac{\la}{n}}\|f\|_{L(1 +
\log^+ L),Q},
\end{aligned}$$ that is, $$\begin{split}
\|M^2f\|_{\WM_{L(1 + \log^+ L),\la}}\leq c \|f\|_{\mathcal{M}_{L(1 +
\log^+ L),\la}}.
\end{split}$$
Weak-type estimates in Morrey spaces for maximal commutator and commutator of maximal function {#sect8}
==============================================================================================
In this section the class of functions for which the maximal commutator $C_b$ is bounded from $\mathcal{M}_{L(1 + \log^+
L),\la}$ to $\WM_{L(1 + \log^+ L),\la}$ are characterized. It is proved that the commutator of the Hardy-Littlewood maximal operator $M$ with function $b \in \B(\rn)$ such that $b^- \in
L_{\infty}(\rn)$ is bounded from $\mathcal{M}_{L(1 + \log^+ L),\la}$ to $\WM_{L(1 + \log^+ L),\la}$.
\[main10\] Let $0<\la <n$. The following assertions are equivalent:
[(i)]{} $b\in\B(\rn)$.
[(ii)]{} The operator $C_b$ is bounded from $\mathcal{M}_{L(1 +
\log^+ L),\la}$ to $\WM_{L(1 + \log^+ L),\la}$.
[${\rm (i)}\Rightarrow {\rm (ii)}$]{}. Assume that $b\in\B(\rn)$. By Theorem \[lem1111111\] and Theorem \[thm53432\] operator $C_b$ is bounded from $\mathcal{M}_{L(1 + \log^+ L),\la}$ to $\WM_{L(1 + \log^+ L),\la}$ and the following inequality holds $$\label{eq8800887645}
\begin{split}
\|C_b(f)\|_{\WM_{L(1 + \log^+ L),\la}}\leq c \|b\|_{*}
\|f\|_{\mathcal{M}_{L(1 + \log^+ L),\la}}
\end{split}$$ with positive constant $c$ independent of $f$.
[${\rm (ii)}\Rightarrow {\rm (i)}$]{}. Assume that the inequality $$\label{eq34544724r2346657u}
\begin{split}
\|C_b(f)\|_{\WM_{L(1 + \log^+ L),\la}}\leq c \|f\|_{\mathcal{M}_{L(1
+ \log^+ L),\la}}.
\end{split}$$ holds with positive constant $c$ independent of $f$. Let $Q_0$ be any cube in $\rn$ and let $f=\chi_{Q_0}$.
By Theorem \[thm6.3\], $$\begin{split}
\|\chi_{Q_0}\|_{\mathcal{M}_{L(1 + \log^+ L),\la}} & \thickapprox
\sup_{Q}|Q|^{\frac{\la-n}{n}}\int_{Q}\chi_{Q_0}\left(1+\log^+
\frac{\chi_{Q_0}}{(\chi_{Q_0})_{Q}}\right)\\
&=\sup_{Q:\,Q\cap Q_0 \neq \emptyset}|Q|^{\frac{\la}{n}}\frac{|Q\cap Q_0|}{|Q|}\left(1+\log
\frac{|Q|}{|Q\cap Q_0|}\right).
\end{split}$$ Obviously, $$\|\chi_{Q_0}\|_{\mathcal{M}_{L(1 + \log^+ L),\la}} \gs
|Q_0|^{\frac{\la}{n}}.$$ Let $\ve \in (0,1-\la / n)$. Since the function $(1+\log t)/ t^{\ve}$ is bounded on the interval $[1,\infty)$, we get $$\begin{aligned}
\|\chi_{Q_0}\|_{\mathcal{M}_{L(1 + \log^+ L),\la}} & \ls
\sup_{Q:\,Q\cap Q_0 \neq \emptyset}|Q|^{\frac{\la}{n}}\frac{|Q\cap
Q_0|}{|Q|}\left(
\frac{|Q|}{|Q\cap Q_0|}\right)^{\ve} \\
& = \sup_{Q:\,Q\cap Q_0 \neq \emptyset}|Q|^{\frac{\la}{n}+\ve-1}|Q\cap Q_0|^{1-\ve} \\
& = \sup_{Q \subseteq Q_0}|Q|^{\frac{\la}{n}+\ve-1}|Q\cap Q_0|^{1-\ve} =|Q_0|^{\frac{\la}{n}}.
\end{aligned}$$ Thus $$\label{eq56789345678}
\|\chi_{Q_0}\|_{\mathcal{M}_{L(1 + \log^+ L),\la}} \thickapprox
|Q_0|^{\frac{\la}{n}}.$$ On the other hand $$\begin{split}
\|C_b(\chi_{Q_0})\|_{W\mathcal{M}_{L(1 + \log^+ L),\la}}&=\sup_{Q}|Q|^{\frac{\la}{n}}\|C_b(\chi_{Q_0})\|_{WL(1 + \log^+ L),Q}\\
&\geq |Q_0|^{\frac{\la}{n}}\|C_b(\chi_{Q_0})\|_{WL(1 + \log^+
L),Q_0}.
\end{split}$$ Note that $$\begin{split}
\|C_b(\chi_{Q_0})\|_{WL(1 + \log^+ L),Q_0} &\\
&\hspace{-2cm}=\inf \left\{\la>0:
\sup_{t>0}\frac{1}{|Q_0|}\frac{|\{x\in Q_0: |C_b(\chi_{Q_0})(x)|>\la
t\}|}{\frac{1}{t}\left(1+\log^+ \frac{1}{t}\right)}\leq 1\right\} \\
&\hspace{-2cm}\geq \inf \left\{\la>0: \frac{2}{|Q_0|}|\{x\in Q_0:
|C_b(\chi_{Q_0})(x)|> 2\la\}|\leq 1\right\}.
\end{split}$$ Since for any $x\in Q_0$ $$C_b(\chi_{Q_0})(x)\geq \frac{1}{|Q_0|}\int_{Q_0}|b(x)-b(y)|dy \geq
\frac{1}{2|Q_0|}\int_{Q_0}|b(y)-b_{Q_0}|dy,$$ then $$\begin{split}
\frac{2}{|Q_0|}\left|\left\{x\in Q_0: |C_b(\chi_{Q_0})(x)|> 2\,
\frac{1}{4|Q_0|}\int_{Q_0}|b(y)-b_{Q_0}|dy\right\}\right|=2.
\end{split}$$ Thus $$\|C_b(\chi_{Q_0})\|_{WL(1 + \log^+ L),Q_0} \geq
\frac{1}{4|Q_0|}\int_{Q_0}|b(y)-b_{Q_0}|dy.$$ Consequently, $$\label{eq87678656566}
\begin{split}
\|C_b(\chi_{Q_0})\|_{W\mathcal{M}_{L(1 + \log^+ L),\la}}\gtrsim
|Q_0|^{\frac{\la}{n}}\frac{1}{|Q_0|}\int_{Q_0}|b(y)-b_{Q_0}|dy.
\end{split}$$ By , and we arrive at $$\frac{1}{|Q_0|}\int_{Q_0}|b(y)-b_{Q_0}|dy \ls c.$$
Combining Theorems \[lem1111111\] and \[main11\], we get the following statement.
Let $0<\la <n$. Assume that $b\in\B(\rn)$. Then the operator $C_b$ is bounded on $\mathcal{M}_{L(1 + \log^+ L),\la}$ for radially decreasing functions.
The following theorems hold true.
\[thm4567895432\] Let $0<\la <n$ and $b$ is in $\B(\rn)$ such that $b^-\in
L_{\infty}(\rn)$. Then the operator $[M,b]$ is bounded from $\mathcal{M}_{L(1 + \log^+ L),\la}$ to $\WM_{L(1 + \log^+ L),\la}$ and the following inequality holds $$\|[M,b]f\|_{\WM_{L(1 + \log^+ L),\la}}\leq
c\left(\|b^+\|_{*}+\|b^-\|_{L_{\infty}}\right)
\|f\|_{\mathcal{M}_{L(1 + \log^+ L),\la}}$$ with positive constant $c$ independent of $f$.
The statement follows by Theorem \[lem1111111\] and Theorem \[thm53432\].
\[thm456789543223\] Let $0<\la <n$ and $b$ is in $\B(\rn)$ such that $b^-\in
L_{\infty}(\rn)$. Then the operator $[M,b]$ is bounded on $\mathcal{M}_{L(1 + \log^+ L),\la}$ for radially decreasing functions, and the following inequality holds $$\|[M,b]f\|_{\mathcal{M}_{L(1 + \log^+ L),\la}}\leq
c\left(\|b^+\|_{*}+\|b^-\|_{L_{\infty}}\right)
\|f\|_{\mathcal{M}_{L(1 + \log^+ L),\la}},~ f \in \mf^{\rad,\dn},$$ with positive constant $c$ independent of $f$.
The statement follows by Theorems \[lem1111111\] and \[main11\].
[^1]: The research of A. Gogatishvili was partly supported by the grants P201-13-14743S of the Grant Agency of the Czech Republic and RVO: 67985840, by Shota Rustaveli National Science Foundation grants no. 31/48 (Operators in some function spaces and their applications in Fourier Analysis) and no. DI/9/5-100/13 (Function spaces, weighted inequalities for integral operators and problems of summability of Fourier series). The research of the first and second authors was partly supported by the joint project between Academy of Sciences of Czech Republic and The Scientific and Technological Research Council of Turkey
[^2]: Denote by $b^+(x)=\max\{b(x),0\}$ and $b^-(x)=-\min\{b(x),0\}$, consequently $b=b^+-b^-$ and $|b|=b^++b^-$.
|
---
abstract: 'Type-Iax supernovae (SN Iax) are stellar explosions that are spectroscopically similar to some type-Ia supernovae (SN Ia) at maximum light, except with lower ejecta velocities[@Foley13; @Li03]. They are also distinguished by lower luminosities. At late times, their spectroscopic properties diverge from other SN[@Jha06; @Phillips07; @Sahu08; @McCully14], but their composition (dominated by iron-group and intermediate-mass elements[@Foley13; @Stritzinger14]) suggests a physical connection to normal SN Ia. These are not rare; SN Iax occur at a rate between 5 and 30% of the normal SN Ia rate[@Foley13]. The leading models for SN Iax are thermonuclear explosions of accreting carbon-oxygen white dwarfs (C/O WD) that do not completely unbind the star[@Jordan12; @Kromer13; @Fink14], implying they are “less successful” cousins of normal SN Ia, where complete disruption is observed. Here we report the detection of the luminous, blue progenitor system of the type-Iax SN 2012Z in deep pre-explosion imaging. Its luminosity, colors, environment, and similarity to the progenitor of the Galactic helium nova V445 Puppis[@Kato08; @Woudt09; @Goranskij10], suggest that SN 2012Z was the explosion of a WD accreting from a helium-star companion. Observations in the next few years, after SN 2012Z has faded, could test this hypothesis, or alternatively show that this supernova was actually the explosive death of a massive star[@Valenti09; @Moriya10].'
author:
- |
Curtis McCully$^1$, Saurabh W. Jha$^1$, Ryan J. Foley$^{2,3}$, Lars Bildsten$^{4,5}$,\
Wen-fai Fong$^6$, Robert P. Kirshner$^6$, G. H. Marion$^{7,6}$, Adam G. Riess$^{8,9}$,\
& Maximilian D. Stritzinger$^{10}$
bibliography:
- 'progenitor-12Z.bib'
date: 'August 7, 2014\'
title: 'A luminous, blue progenitor system for a type-Iax supernova'
---
Department of Physics and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway, New Jersey 08854, USA.
Astronomy Department, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, Illinois 61801, USA.
Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801, USA.
Department of Physics, University of California, Santa Barbara, California 93106, USA.
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA.
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138, USA.
Department of Astronomy, University of Texas at Austin, Austin, Texas 78712, USA.
Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA.
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, Maryland 21218, USA.
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark.
SN 2012Z was discovered[@Cenko12] in the Lick Observatory Supernova Search on UT 2012-Jan-29.15. It had an optical spectrum similar to the type-Iax (previously called SN 2002cx-like) SN 2005hk[@Jha06; @Phillips07; @Sahu08] (see Extended Data Fig. \[fig:spec\]). The similarities between SN Iax and normal SN Ia make understanding SN Iax progenitors important, especially because no normal SN Ia progenitor has been identified. Like core-collapse SN (but also slowly-declining, luminous SN Ia), SN Iax are found preferentially in young, star-forming galaxies[@Foley09; @Lyman13]. A single SN Iax, SN 2008ge, was in a relatively old (S0) galaxy with no indication of current star formation to deep limits[@Foley10_ge]. Non-detection of the progenitor of SN 2008ge in *Hubble Space Telescope* (*HST*) pre-explosion imaging restricts its initial mass $\lesssim$12 M$_{{\odot}}$, and combined with the lack of hydrogen or helium in the SN 2008ge spectrum, favours a white dwarf progenitor[@Foley10_ge].
Deep observations of NGC 1309, the host galaxy of SN 2012Z, were obtained with *HST* in 2005–2006 and 2010, serendipitously including the location of the supernova before its explosion. To pinpoint the position of SN 2012Z with high precision, we obtained follow-up *HST* data in 2013. Colour-composite images made from these observations before and after the supernova are shown in Fig. \[fig:image\], and photometry of stellar sources in the pre-explosion images near the supernova location is reported in Extended Data Table \[tab:stars\]. We detect a source, called S1, coincident with the supernova at a formal separation of 00082 $\pm$ 00103 (equal to 1.3 $\pm$ 1.6 pc at 33 Mpc, the distance to NGC 1309[@Riess09b; @Riess11]). The pre-explosion data reach a 3$\sigma$ limiting magnitude of M$_{\rm V} \approx -$3.5, quite deep for typical extragalactic SN progenitor searches[@Smartt09], but certainly the possibility exists that the progenitor system of SN 2012Z was of lower luminosity and would be undetected in our data (as has been the case for all normal SN Ia progenitor searches to date[@Li11]). However, the locations of SN 2012Z and S1 are identical to within 0.8$\sigma$, and we estimate only a 0.24% (2.1%) probability that a random position near SN 2012Z would be within 1$\sigma$ (3$\sigma$) of any detected star, making a chance alignment unlikely (see Methods, and Extended Data Fig. \[fig:chance\]). We also observe evidence for variability in S1 (plausible for a pre-supernova system; Extended Data Table \[tab:variability\]), at a level exhibited by only 4% of objects of similar brightness. We thus conclude there is a high likelihood that S1 is the progenitor system of SN 2012Z.
The color-magnitude diagram (CMD) presented in Fig. \[fig:cmd\] shows S1 to be luminous and blue, yet in an odd place for a star about to explode. If its light is dominated by a single star, S1 is moderately consistent with a [$\approx\!$ ]{}18.5 M$_{{\odot}}$ main-sequence star[@Bertelli09], an [$\approx\!$ ]{}11 M$_{\odot}$ blue supergiant early in its evolution off the main sequence, or perhaps a [$\approx\!$ ]{}7.5 M$_{{\odot}}$ (initial mass) blue supergiant later in its evolution (with core helium-burning in a blue loop, where models are quite sensitive to metallicity and rotation[@Georgy13]). None of these stars are expected to explode in standard stellar evolution theory, particularly without any signature of hydrogen in the supernova[@Smartt09].
The SN 2012Z progenitor system S1 is in a similar region in the CMD to some Wolf-Rayet stars[@Shara13], highly evolved, massive stars, that are expected to undergo core collapse and may produce a supernova. If S1 were a single Wolf-Rayet star, its photometry is most consistent with the WN subtype and an initial mass [$\approx\!$ ]{}30–40 M$_{\odot}$, thought perhaps to explode with a helium-dominated outer layer as a SN Ib[@Groh13], and unlikely to produce the structure and composition of ejecta seen in SN Iax[@Jha06; @Foley13; @McCully14; @Stritzinger14]. Moreover, isochrones[@Bertelli09] fit to the neighbouring stars (Extended Data Fig. \[fig:isochrone\]) yield an age range of [$\approx\!$ ]{}10–42 Myr, longer than the 5–8 Myr lifetime of such a massive Wolf-Rayet star.
S1 may be dominated by accretion luminosity; its brightness in B and V is not far from the predicted thermal emission of an Eddington-luminosity Chandrasekhar mass WD (a super-soft source; SSS; Fig. \[fig:cmd\]). However, its V-I and V-H colours are too red for a SSS model. A composite scenario, with accretion power dominating the blue flux, and another source providing the redder light (perhaps a fainter, red donor star) may be plausible.
The leading models of SN Iax[@Jordan12; @Kromer13; @Fink14] are based on C/O WD explosions, so S1 may be the companion star to an accreting WD. Although there are a variety of potential progenitor systems (including main-sequence and red giant donors, which are inconsistent with S1 if they dominate the system’s luminosity), in standard scenarios no companion star can have an initial mass greater than [$\approx\!$ ]{}7 M$_{{\odot}}$; otherwise, there would not be enough time to form the primary C/O WD that explodes. Thus, the photometry of S1 suggests that if it is the companion to a C/O WD, recent binary mass transfer must have played a role in its evolution. One model for a luminous, blue companion star is a relatively massive ([$\approx\!$ ]{}2 M$_{{\odot}}$ when observed) helium star[@Iben91; @Kato08; @Liu10], formed after binary mass transfer and a common envelope phase (e.g., a close binary with initial masses [$\approx\!$ ]{}7 and 4 M$_{\odot}$). Although the model parameter space has not been fully explored, the predicted region for helium star donors in a binary system with a 1.2 M$_{{\odot}}$ initial-mass accreting C/O WD[@Liu10] in the CMD is shown in Fig. \[fig:cmd\], and S1 is consistent with being in this region. The evolutionary timescale for such a model is also well-matched to the ages of nearby stars (Extended Data Fig. \[fig:isochrone\]).
SN 2012Z and the star S1 have an interesting analogue in our own Milky Way Galaxy: the helium nova V445 Puppis[@Kato08; @Woudt09; @Goranskij10], thought to be the surface explosion of a near-Chandrasekhar mass helium-accreting WD. Though S1 is somewhat brighter than the pre-explosion observations of V445 Pup, their consistent colours, similar variability amplitude[@Goranskij10], and the physical connection between V445 Pup and likely SN Iax progenitors[@Jordan12; @Kromer13; @Fink14] is highly suggestive. Indeed, two SN Iax (though not SN 2012Z itself) have shown evidence for helium in the system[@Foley09; @Foley13]. In this model, a low helium accretion rate could lead to a helium nova (like V445 Pup), whereas a higher mass-transfer rate could result in stable helium burning on the C/O WD, allowing it to grow in mass before the supernova. The accretion is expected to begin as the helium star starts to evolve and grow in radius; indeed, the S1 photometry is consistent with the evolutionary track of a helium star with a mass (after losing its hydrogen envelope) of [$\approx\!$ ]{}2 M$_{{\odot}}$, on its way to becoming a red giant[@Kato08].
Though the scenario of a helium-star donor to an exploding carbon-oxygen white dwarf is a promising model for the progenitor and supernova observations, we cannot yet rule out the possibility that S1 is a single star that itself exploded. Fortunately, by late 2015, SN 2012Z will have faded below the brightness of S1, and *HST* imaging will allow us to distinguish these models. Our favoured interpretation of S1 as the companion star predicts that it will still be detected (though perhaps modified by the impact of its exploding neighbour, a reduction in accretion luminosity, or a cessation of variability). On the other hand, if S1 has completely disappeared, it will be a strong challenge to models of SN Iax, and perhaps significantly blur the line between thermonuclear white-dwarf supernovae and massive-star core-collapse supernovae, with important impacts to our understanding of stellar evolution and chemical enrichment.
Supernova 2012Z was discovered just weeks after the passing of our dear friend and colleague, Weidong Li, whose work on the Lick Observatory Supernova Search, SN 2002cx-like supernovae, and Hubble Space Telescope observations of SN progenitors, continues to inspire us. That those three foci of Weidong’s research converge here in this paper makes our hearts glad, and we dedicate this paper to his memory.
We thank the SH$_0$ES team for assistance with data from *HST* program GO-12880, E. Bertin for the development of the STIFF software to produce color images, and A. Dolphin for software and guidance in photometry. This research at Rutgers University was supported through NASA/[*HST*]{} grant GO-12913.01, and National Science Foundation (NSF) CAREER award AST-0847157 to S.W.J. NASA/*HST* grant GO-12999.01 to R.J.F. supported this work at the University of Illinois. At UC Santa Barbara, this work was supported by NSF grants PHY 11-25915 and AST 11-09174 to L.B. The Danish Agency for Science, Technology, and Innovation supported M.D.S. through a Sapere Aude Level 2 grant.
Support for *HST* programs GO-12913 and GO-12999 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
C.M., S.W.J., and R.J.F. performed the data analysis and were chiefly responsible for preparing the manuscript and figures. W.F., R.P.K., G.H.M., and A.G.R. assisted in developing the proposal to obtain *HST* observations, including acquiring supporting ground-based photometry and spectroscopy. L.B. provided insight into models for progenitor systems. M.D.S. analyzed ground-based photometry and spectroscopy of the supernova, used as input for this paper. All authors contributed to extensive discussions about, and edits to, the paper draft.
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to S.W.J. ([email protected]).
Methods {#methods .unnumbered}
=======
0.05in **Observations and reduction.** SN 2012Z provides a unique opportunity to search for a SN Iax progenitor, because of deep, pre-explosion *HST* observations. Its face-on spiral host galaxy NGC 1309 was also the site of the nearby, normal type-Ia SN 2002fk, and as such was targeted in 2005 and 2006 with the *HST* Advanced Camera for Surveys (ACS) and in 2010 with the *HST* Wide-Field Camera 3 (WFC3) optical (UVIS) and infrared (IR) channels, to observe Cepheid variable stars in order to anchor the SN Ia distance scale (*HST* programs GO-10497, GO-10802, and GO-11570, PI: A. Riess; GO/DD-10711, PI: K. Noll). To measure the location of SN 2012Z with high precision, we used *HST* WFC3/UVIS images of NGC 1309 (fortuitously including SN 2012Z) taken on UT 2013-Jan-04 (program GO-12880; PI: A. Riess), as well as targeted *HST* WFC3/UVIS images of SN 2012Z taken on UT 2013-Jun-30 (GO-12913; PI: S. Jha).
The 2005–2006 ACS images include 2 visits totaling 9600s of exposure time in the F435W filter (similar to Johnson $B$), 14 visits for a total exposure of 61760 sec in F555W (close to Johnson $V$) and 5 visits for 24000s in F814W (analogous to Cousins $I$). These are showing the blue, green, and red channels, respectively of panels a–c in Fig. \[fig:image\]. We re-reduced the archived data, combining the multiple exposures (including sub-sampling and cosmic ray rejection) using the AstroDrizzle software from the DrizzlePac package, with the results shown in panels b and c of Fig. \[fig:image\]. The bottom right panels d and e of Fig. \[fig:image\] show our combined *HST* WFC3/UVIS F555W (blue; 1836 sec) + F625W (green; 562 sec) + F814W (red; 1836 sec) images from January and June 2013, with SN 2012Z visible.
We used the DrizzlePac TweakReg routine to register all of the individual flatfielded (“[flt]{}”) frames to the WFC3/UVIS F555W image taken on UT 2013-Jan-04. The typical root-mean-square (rms) residual of individual stars from the relative astrometric solution was 0009, corresponding to 0.18 pixels in ACS and 0.23 pixels in WFC3/UVIS. We drizzle the ACS images to the native scale of UVIS, 004 per pixel (20% smaller than the native 005 ACS pixels) and subsample the ACS point-spread function (psf) correspondingly with a pixel fraction parameter of 0.8.
**Photometry and astrometry.** In Extended Data Table \[tab:stars\], we present photometry of sources in the region based on the *HST* ACS images from 2005–2006 (F435W, F555W, and F814W), as well as WFC3/IR F160W data (6991 sec of total exposure time) from 2010. The stars are sorted by their proximity to the SN position, and their astrometry is referenced to SDSS images of the field, with an absolute astrometric uncertainty of 0080 (but this is irrelevant for the much more precise relative astrometry of SN 2012Z and S1). We photometered the *HST* images using the psf-fitting software DolPhot, an extension of HSTPhot. We combined individual flt frames taken during the same *HST* visit at the same position, and then used DolPhot to measure photometry using recommended parameters for ACS and WFC3.
The WFC3/UVIS images of SN 2012Z from January and June 2013 provide a precise position for the supernova of R.A. = 3$^{\textrm{h}}$22$^{\textrm{m}}$0539641, Decl. = $-$1523149390 (J2000) with a registration uncertainty of 00090 (plus an absolute astrometric uncertainty of 008, irrelevant to the relative astrometry). As shown in Fig. \[fig:image\] we detect a stellar source (called S1) in the pre-explosion images coincident with the position of the SN with a formal separation, including centroid uncertainties, of 00082 $\pm$ 00103, indicating an excellent match and strong evidence for S1 being the progenitor system of SN 2012Z.
**Chance alignment probability.** To estimate the probability of a chance alignment, we use the observed density of sources detected with signal-to-noise ratio S/N $>$ 3.0 (in any filter) and S/N $>$ 3.5 (in all bands combined, via DolPhot) in a 200 $\times$ 200 pixel (8${\mbox{$^{\prime\prime}$}}$ $\times$ 8${\mbox{$^{\prime\prime}$}}$) box centered on SN 2012Z (452 sources), and find only a 0.24% (2.1%) chance that a random position would be consistent with a detected star at 1$\sigma$ (3$\sigma$). Moreover, only 171 of these stars are as bright as S1, so *a posteriori* there was only a 0.09% (0.80%) chance of a 1$\sigma$ (3$\sigma$) alignment with such a bright object. As shown in Extended Data Fig. \[fig:chance\], these results are not especially sensitive to the size of the region used to estimate the density of sources, at least down to 50 $\times$ 50 pixels (2${\mbox{$^{\prime\prime}$}}$ $\times$ 2${\mbox{$^{\prime\prime}$}}$) around SN 2012Z. Nearer than this, the density of sources increases by a factor of a few, though with substantially larger uncertainty given the low number of sources (including S1 itself). We base our fiducial chance alignment probability on the larger region where the density of sources stabilizes with good statistics, but our qualitative results do not depend on this choice.
Given the surface brightness of NGC 1309, we crudely estimate [$\approx\!$ ]{}160 M$_{\odot}$ in stars projected within an area corresponding to our 1$\sigma$ error circle ([$\approx\!$ ]{}8 pc$^2$). These should be roughly uniformly distributed throughout this small region, so our chance alignment probability should accurately quantify the probability that SN 2012Z originated from an undetected progenitor that was only coincidentally near a detected source like S1.
The pre-explosion data for SN 2012Z are the deepest ever for a SN Iax, reaching M$_{\rm V} \approx -$3.5; the next best limits come from SN 2008ge[@Foley10_ge], which yielded no progenitor detection down to M$_{\rm V} \approx -$7. The star S1, at M$_{\rm V} \approx -$5.3, would not have been detected in any previous search for SN Iax progenitors. This implies that our chance alignment probability calculation can be taken at face value; there have not been previous, unsuccessful “trials” that would reduce the unlikelihood of a chance coincidence. Viewed in the context of progenitor searches for normal SN Ia, in only two cases: SN 2011fe[@Li11] and SN 2014J[@Kelly14], would a star of the luminosity of S1 have been clearly detected in pre-explosion data. Other normal SN Ia, like SN 2006dd[@Maoz08], have progenitor detection limits in pre-explosion observations just near, or above, the luminosity of S1.
**Variability of S1.** NGC 1309 was imaged over 14 epochs in F555W with ACS before SN 2012Z exploded. Examining these individually, we find some evidence for variability in S1; the photometry is presented in Extended Data Table \[tab:variability\]. Formally, these data rule out the null hypothesis of no variability at 99.95% (3.5$\sigma$), with $\chi^2 =$ 36.658 in 13 degrees of freedom. However, most of the signal is driven by one data point (MJD 53600.0; a 4.2$\sigma$ outlier); excluding this data point (though we find no independent reason to do so) reduces the significance of the variability to just 91.1% (1.7$\sigma$). To empirically assess the likelihood of variability, we looked to see how often stars with the same brightness as S1 (within 0.5 mag) doubled in brightness relative to their median like S1 in one or more of the 14 epochs; we find that just 4% do so (4 of the nearest 100, 9 of the nearest 200, and 20 of nearest 500 such stars). As variability might be expected in a SN progenitor before explosion (from non-uniform accretion, for example), combining this with the chance alignment probability strengthens the identification of S1 as the progenitor system of SN 2012Z. It also disfavours the possibility that S1 is a compact, unresolved star cluster.
**Properties of S1 and nearby stars.** In Fig. \[fig:cmd\] we present colour-magnitude diagrams of S1 and other objects for comparison. In the figure, S1 has been corrected for Milky Way reddening (E(B-V)$_{\rm{MW}}$ = 0.035 mag, corresponding to A$_{\rm{F435W}}$ = 0.14 mag, A$_{\rm{F555W}}$ = 0.11 mag, A$_{\rm{F814W}}$ = 0.06 mag), and host reddening (E(B-V)$_{\rm{host}}$ = 0.07 mag; A$_{\rm{F435W}}$ = 0.28 mag, A$_{\rm{F555W}}$ = 0.22 mag, A$_{\rm{F814W}}$ = 0.12 mag) based on narrow, interstellar absorption lines in high-resolution spectroscopy of SN 2012Z itself[@Stritzinger14b]. This low extinction is consistent with the photometry and spectroscopy of SN 2012Z, as well as its location in the outskirts of a face-on spiral host. For the potential Galactic analogue V445 Puppis progenitor, we correct its photometry for Galactic and circumstellar reddening[@Woudt09].
Fig. \[fig:cmd\] also shows stellar evolution tracks[@Bertelli09] for stars with initial masses of 7, 8, 9, 10, and 11 M$_{{\odot}}$, adopting a metallicity of 0.87 solar, based on the H II region metallicity gradient[@Riess09b] for NGC 1309 interpolated to the SN radial location. The Eddington-luminosity accreting Chandrasekhar-mass white dwarfs are shown as the large purple dots, with each subsequent dot representing a change in temperature of 1000 K. These “super-soft” sources are fainter in F555W for higher temperatures (and bluer F435W$-$F555W colours) as the fixed (Eddington-limited) bolometric luminosity emerges in the ultraviolet for hotter systems. The shaded blue region represents the range of helium-star donors for C/O WD SN progenitor models starting with a 1.2 M$_{{\odot}}$ white dwarf[@Liu10]. We converted the model temperatures and luminosities to our observed bands assuming a blackbody spectrum. The expected temperature and luminosity for this class of models is expected to vary with white dwarf mass, and therefore, we regard this region as approximate; its shape, size, and location is subject to change.
S1 is inconsistent with all confirmed progenitors of core-collapse SN (exclusively SN II), which are mostly red supergiants[@Smartt09]. The blue supergiant progenitor of, e.g., SN 1987A, was significantly more luminous and likely more massive than S1. However, we caution that our theoretical expectations for massive stars could be modified if S1 is in a close binary system where mass transfer has occurred.
The stars detected in the vicinity of S1 provide clues to the nature of recent star formation in the region. They include red supergiants (like S2 and S3 from Table \[tab:stars\]) as well as objects bluer and more luminous than S1 (like S5). We show CMDs including these stars in Extended Data Fig. \[fig:isochrone\]; the stars plotted have a signal-to-noise ratio S/N $>$ 3.5 in the displayed filters, and were required to be no closer than 3 pixels to a brighter source to avoid photometric uncertainties from crowding. Model isochrones[@Bertelli09] imply that these stars span an age range of [$\approx\!$ ]{}10–42 Myr. These tracks favour an initial mass for S1 of [$\approx\!$ ]{}7–8 M$_{\odot}$ (neglecting mass transfer) if it is roughly coeval with its neighbours. In other words, if S1 were a 30–40 M$_{\odot}$ initial-mass Wolf-Rayet star with a predicted lifetime of only 5–8 Myr[@Groh13], it would be the youngest star in the region.
-0.35in {width="112.00000%"} -0.1in
{width="\textwidth"}
-0.2in
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-0.2in
{width="5.8in"}
-0.2in
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-0.2in
Star R.A. (J2000) Decl. (J2000) F435W (mag) F555W (mag) F814W (mag) F160W (mag)
------ -------------------------------- --------------- ---------------- ---------------- ---------------- ----------------
S1 3$^{\sf h}$22$^{\sf m}$0539591 $-$1523149350 27.589 (0.122) 27.622 (0.060) 27.532 (0.135) 26.443 (0.321)
S2 3$^{\sf h}$22$^{\sf m}$0540280 $-$1523149402 $>$29.221 28.551 (0.116) 27.463 (0.093) 26.032 (0.238)
S3 3$^{\sf h}$22$^{\sf m}$0539483 $-$1523148102 28.466 (0.258) 27.221 (0.041) 25.435 (0.022) 23.699 (0.027)
S4 3$^{\sf h}$22$^{\sf m}$0538460 $-$1523150314 28.258 (0.211) 27.308 (0.046) 25.717 (0.028) 23.887 (0.033)
S5 3$^{\sf h}$22$^{\sf m}$0541013 $-$1523149362 25.778 (0.029) 25.918 (0.014) 25.888 (0.030) 25.435 (0.136)
S6 3$^{\sf h}$22$^{\sf m}$0537564 $-$1523150702 $>$28.807 28.386 (0.116) 26.553 (0.055) 24.986 (0.085)
S7 3$^{\sf h}$22$^{\sf m}$0541970 $-$1523148454 28.539 (0.286) 27.942 (0.078) 27.703 (0.146) 26.436 (0.322)
S8 3$^{\sf h}$22$^{\sf m}$0537049 $-$1523146690 $>$28.847 28.763 (0.164) 27.279 (0.101) 25.032 (0.092)
S9 3$^{\sf h}$22$^{\sf m}$0536477 $-$1523150634 27.683 (0.127) 27.470 (0.050) 26.266 (0.042) 24.753 (0.069)
S10 3$^{\sf h}$22$^{\sf m}$0539605 $-$1523154062 $>$28.912 29.305 (0.273) 26.969 (0.077) 25.214 (0.102)
S11 3$^{\sf h}$22$^{\sf m}$0537904 $-$1523145218 27.196 (0.083) 27.329 (0.045) 26.949 (0.076) $>$26.614
S12 3$^{\sf h}$22$^{\sf m}$0537556 $-$1523144202 $>$28.897 28.302 (0.109) 27.095 (0.085) 25.381 (0.131)
S13 3$^{\sf h}$22$^{\sf m}$0536527 $-$1523145054 28.085 (0.181) 27.560 (0.055) 26.852 (0.069) 25.740 (0.171)
S14 3$^{\sf h}$22$^{\sf m}$0540432 $-$1523155778 $>$29.360 28.651 (0.113) 26.078 (0.026) 24.180 (0.042)
S15 3$^{\sf h}$22$^{\sf m}$0543040 $-$1523144706 27.937 (0.164) 28.465 (0.122) 28.078 (0.208) $>$26.512
S16 3$^{\sf h}$22$^{\sf m}$0542376 $-$1523143726 $>$28.765 28.163 (0.092) 26.905 (0.072) 25.126 (0.094)
S17 3$^{\sf h}$22$^{\sf m}$0538194 $-$1523142494 $>$28.926 29.007 (0.207) 26.957 (0.077) 26.230 (0.260)
S18 3$^{\sf h}$22$^{\sf m}$0535086 $-$1523152402 27.831 (0.143) 27.819 (0.069) 26.969 (0.077) 25.581 (0.145)
S19 3$^{\sf h}$22$^{\sf m}$0539387 $-$1523142018 28.670 (0.302) 28.417 (0.118) 26.616 (0.056) 25.072 (0.094)
S20 3$^{\sf h}$22$^{\sf m}$0537179 $-$1523156186 $>$28.814 28.643 (0.147) 26.660 (0.059) 24.854 (0.075)
S21 3$^{\sf h}$22$^{\sf m}$0534566 $-$1523154618 $>$28.813 $>$29.632 27.291 (0.102) 25.572 (0.138)
S22 3$^{\sf h}$22$^{\sf m}$0539179 $-$1523140298 $>$28.809 29.135 (0.235) 27.219 (0.100) 25.685 (0.161)
S23 3$^{\sf h}$22$^{\sf m}$0543718 $-$1523156194 28.340 (0.241) 29.474 (0.331) $>$28.665 $>$26.633
S24 3$^{\sf h}$22$^{\sf m}$0535072 $-$1523156022 $>$28.835 28.097 (0.089) 26.975 (0.078) 25.936 (0.202)
S25 3$^{\sf h}$22$^{\sf m}$0538396 $-$1523139958 28.367 (0.230) 28.382 (0.119) 26.046 (0.035) 24.549 (0.057)
S26 3$^{\sf h}$22$^{\sf m}$0533025 $-$1523150438 27.518 (0.109) 27.466 (0.055) 26.284 (0.048) 25.134 (0.094)
S27 3$^{\sf h}$22$^{\sf m}$0546232 $-$1523149126 $>$28.841 28.581 (0.137) 27.093 (0.084) 25.445 (0.126)
S28 3$^{\sf h}$22$^{\sf m}$0539871 $-$1523139614 $>$28.808 28.522 (0.132) 27.052 (0.084) 25.443 (0.137)
S29 3$^{\sf h}$22$^{\sf m}$0539210 $-$1523139558 $>$28.908 28.386 (0.117) 28.395 (0.288) 26.075 (0.241)
S30 3$^{\sf h}$22$^{\sf m}$0540792 $-$1523138970 $>$28.952 28.692 (0.154) 27.315 (0.102) 25.459 (0.135)
S31 3$^{\sf h}$22$^{\sf m}$0532555 $-$1523146310 $>$28.961 $>$29.657 28.039 (0.201) 25.475 (0.127)
S32 3$^{\sf h}$22$^{\sf m}$0532419 $-$1523152534 27.612 (0.122) 27.707 (0.062) 27.701 (0.150) $>$26.591
S33 3$^{\sf h}$22$^{\sf m}$0537008 $-$1523159726 28.700 (0.320) 28.055 (0.088) 28.035 (0.206) $>$26.599
S34 3$^{\sf h}$22$^{\sf m}$0542642 $-$1523159482 27.164 (0.086) 27.302 (0.045) 26.856 (0.072) $>$26.520
S35 3$^{\sf h}$22$^{\sf m}$0538581 $-$1523160310 28.710 (0.329) 27.696 (0.062) 28.152 (0.227) $>$26.624
S36 3$^{\sf h}$22$^{\sf m}$0534751 $-$1523158186 $>$28.843 $>$29.672 27.447 (0.120) 25.444 (0.127)
S37 3$^{\sf h}$22$^{\sf m}$0547012 $-$1523152990 26.530 (0.051) 26.607 (0.024) 26.564 (0.053) 25.154 (0.095)
S38 3$^{\sf h}$22$^{\sf m}$0535077 $-$1523139774 $>$28.916 $>$29.672 27.501 (0.123) 25.776 (0.173)
S39 3$^{\sf h}$22$^{\sf m}$0538399 $-$1523137882 $>$28.831 27.847 (0.069) 27.555 (0.138) 25.455 (0.123)
S40 3$^{\sf h}$22$^{\sf m}$0542445 $-$1523160202 $>$28.775 27.545 (0.057) 26.175 (0.040) 24.407 (0.050)
S41 3$^{\sf h}$22$^{\sf m}$0533343 $-$1523141682 26.522 (0.049) 26.491 (0.022) 26.209 (0.041) 25.470 (0.128)
S42 3$^{\sf h}$22$^{\sf m}$0531449 $-$1523150958 28.411 (0.246) 29.178 (0.236) $>$28.665 $>$26.583
Date (UT) MJD Exposure (sec) Counts ($e^-$) F555W (mag)
------------ --------- ---------------- ---------------- --------------
2005-08-06 53588.7 2400 171 (71) 28.36 (0.45)
2005-08-17 53600.0 2400 656 (75) 26.94 (0.13)
2005-08-24 53606.8 2400 282 (67) 27.79 (0.26)
2005-08-27 53610.0 2400 392 (64) 27.49 (0.18)
2005-09-02 53615.6 2400 389 (69) 27.48 (0.19)
2005-09-03 53617.0 2400 308 (66) 27.77 (0.23)
2005-09-05 53618.6 2400 392 (71) 27.49 (0.20)
2005-09-07 53621.0 2400 329 (66) 27.66 (0.22)
2005-09-11 53624.0 2400 429 (68) 27.37 (0.17)
2005-09-16 53629.8 2400 242 (67) 27.98 (0.30)
2005-09-20 53633.4 2400 327 (64) 27.70 (0.21)
2005-09-27 53640.5 2400 193 (66) 28.25 (0.37)
2006-10-24 54032.6 2080 388 (76) 27.26 (0.21)
2006-10-07 54015.3 2080 264 (75) 27.62 (0.31)
|
---
author:
- |
Peter Christian Bruns\
Universität Bonn, Helmholtz–Institut für Strahlen– und Kernphysik (Theorie),\
D-53115 Bonn, Germany\
E-mail:
- |
Ulf-G. Meißner\
Universität Bonn, Helmholtz–Institut für Strahlen– und Kernphysik (Theorie)\
and Bethe Center for Theoretical Physics, D-53115 Bonn, Germany\
and\
Forschungszentrum Jülich, Institut für Kernphysik (Theorie)\
and Jülich Center for Hadron Physics, D-52425 Jülich, Germany\
E-mail:
title: Infrared regularization with vector mesons and baryons
---
Introduction
============
In a recent work [@BM04], the scheme of infrared regularization (developed in its original form by Becher and Leutwyler [@BL]) was extended to the case of explicit meson resonances interacting with soft pions, as e.g. the first step towards a systematic inclusion of vector mesons in the meson sector of Chiral Perturbation Theory (ChPT). It is the aim of this article to provide a further extension of infrared regularization to the situation where baryons as well as vector mesons are present. This can be considered as a synthesis of the results in [@BM04] and [@BL]. Such an extension is not only of interest in itself, but can be applied to a plethora of observables, where vector and axial-vector mesons are known to play an important role, such as the electroweak form factors of the nucleon. As one example we mention the contribution of the $\pi\rho$ loop to the strangeness form factors of the nucleon [@Meissner:1997qt; @Hammer:1997vt].
Infrared regularization (IR) is a solution to the following problem. The presence of mass scales which must be considered as ’heavy’ compared to the masses and momenta of the soft pions will in general mess up the usual power counting rules of ChPT by which the perturbation series of the effective theory is ordered [@Wein; @GL84; @GL85]. This observation was first made when baryons were incorporated in the framework of ChPT [@GSS] (for a recent review on baryon ChPT we refer to [@Bernard:2007zu]). The procedure of infrared regularization separates the (dimensionally regularized) one-loop graphs of baryon ChPT into a part which stems from the soft pion contribution and a part generated from loop momenta close to the ’heavy’ scale. The latter portion of the loop graph, called the ’regular’ part, will usually not be in accord with the low-energy power counting, but can always be absorbed in local terms derived from the effective Lagrangian. It is therefore dropped from the loop graph, and only the first part, called the infrared singular part of the loop integral, is kept. Though both the vector mesons and the baryons interact as ’heavy’ particles with the pions, the power counting is different for the two species, at least for the kinds of Feynman graphs we consider in this work. There, the vector mesons appear only as internal lines with small momenta far from their mass shell, so that the resonance propagator is counted as $O(q^{0})$ (where $q$ indicates some small momentum scale or Goldstone boson mass), while the baryon propagator is counted as $O(q^{-1})$, since the baryon is pushed from its mass shell only by a small amount due to its interaction with the soft pions and vector mesons. In the graphs we treat here, only one single baryon is present, with the baryon line running through the diagram undergoing only soft interactions. The number of the virtual vector mesons, however, is not fixed. The pion propagator is counted as $O(q^{-2})$, as usual, both the pion momentum and the pion mass being of $O(q)$. Appropriate powers of $q$ are also assigned to vertices from the effective Lagrangian. Finally, the measure of every $d$-dimensional loop integration is booked as $O(q^{d})$. This counting scheme applies, of course, to tree graphs, but also to the infrared singular, or soft, parts of the loop graphs. The regular parts of the loop graphs are not guaranteed to obey the power counting rules. These general remarks will be exemplified in the following sections.
Before working out the case where both baryons and vector mesons appear in a Feynman diagram, we will briefly review the scheme of infrared regularization for loop integrals where only one heavy scale shows up. This will not only serve to give a unified presentation of the method, but also provide some results needed for an application of the general scheme to the axial form factor of the nucleon. Before starting with the presentation of the formalism, let us mention that the loop integrals studied in this work have also been treated, using a different regularization scheme which is in some respect complementary to the one used here, in Ref. [@Mainz03].
IR regularization in the pion-nucleon system {#sec:PiN}
============================================
When only pions and nucleons are treated as explicit fields of the effective theory, the fundamental loop integral one has to consider is $$\label{eq:IMB}
I_{MB}(p^2) = \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{((p-l)^2-m^2)(l^2-M^2)} .$$ Here, $M$ is the pion mass (being of chiral order $O(q)$) and $m$ is the nucleon mass. Applying the low-energy power counting scheme outlined in the introduction, one would assign a chiral order of $q^{d-3}$ to this integral. We will see in a moment that only an appropriately extracted low-energy part of $I_{MB}$ will obey this power counting requirement.
All the other pion-nucleon loop integrals are either only trivially modified by the infrared regularization scheme, or they can be derived from eq. (\[eq:IMB\]) (see sec. 6 of [@BL]). For example, the scalar tadpole integral containing only the pion propagator is not modified at all, as there is no ’hard momentum’ structure present that could lead to a nonvanishing regular part of this integral. Thus we have $$I_{M}^{IR} = I_{M} = \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-M^{2}} ,$$ and a direct calculation gives $$\label{eq:IM}
I_{M}^{IR} = \frac{\Gamma(1-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}M^{d-2} .$$ Note the typical structure of the $d$-dependent power of the pion mass. For arbitrary values of the dimension parameter $d$, $I_{M}^{IR}$ is in general proportional to fractional powers of $M^{2}$, and will even diverge in the so-called chiral limit where the quark masses $m_{u},m_{d}$ go to zero (so that also $M\rightarrow 0$) for small enough $d$ (we mostly consider the two–flavor case here but the method can be trivially extended to include also strange quarks). Such terms will never occur in the regular parts of the loop integrals which stem from the high-momentum region of the integration: those parts are always expandable in the pion mass. The baryon tadpole integral, e.g., is $$I_{B} = \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-m^{2}},$$ and is trivially expandable in $M^{2}$, since it has no pion mass dependence at all. Therefore one has $I_{B}^{IR}=0$. These remarks may serve to explain the terminology ’infrared singular’ vs. ’regular’.
Returning to eq. (\[eq:IMB\]), we must extract the part of this integral that is proportional to some $d$-dependent power of $M$, like in eq. (\[eq:IM\]). One can think of this extraction prescription as an operational definition of infrared regularization. The method is explained in full generality in [@BL]. Here, we concentrate on the case with on-shell momentum $p$, i.e. $p^{2}=m^{2}$. This shows all the features we need for the demonstration, and also yields the result we will use in our application of the scheme in sec. \[sec:PiNV\]. We introduce a Feynman parameter integration in the usual way: $$\label{eq:intlz}
I_{MB} =
\int\frac{d^{d}l}{(2\pi)^{d}}\int_{0}^{1}\frac{dz}{[((p-l)^{2}-m^{2})z
+ (l^{2}-M^{2})(1-z)]^{2}}.$$ Performing the standard steps, we find for $p^{2}=m^{2}$: $$I_{MB} = -m^{d-4}\frac{\Gamma(2-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{0}^{1}\frac{dz}{(z^{2}-\alpha z + \alpha)^{2-\frac{d}{2}}} ,$$ where we have defined $\alpha = M^{2}/m^{2}$ (note the difference to ref.[@BL], where this letter is reserved for $\alpha_{\mathrm{BL}}= M/m$). Fractional powers of the small variable $\alpha$ will be produced near $z=0$: there, the integrand is approximately $(\alpha)^{\frac{d}{2}-2}$. For small enough $d$, there would be an infrared singularity for $M\rightarrow 0$ located in parameter space at $z=0$. It can already be seen from eq. (\[eq:intlz\]) that the parameter region near $z=0$ is associated with the low-energy portion of the integral: In this region, only the ’soft’ pion propagator is weighted in the loop integration, while the hard momentum structure of the nucleon propagator dominates near $z=1$. The extraction of the part of the integral proportional to $d$-dependent powers of $\alpha$ now proceeds as follows: the parameter integration is split into two parts like $$\label{eq:split}
\int_{0}^{1} = \int_{0}^{\infty} - \int_{1}^{\infty}.$$ We will first show that the second integral on the r.h.s. is regular, i.e. expandable, in the variable $\alpha$. This is easy to see, because for $z\geq 1$, the integrand can be expanded like $$\label{eq:expintR}
(z^{2}-\alpha z + \alpha)^{\frac{d}{2}-2}
=
z^{d-4}\sum_{k=0}^{\infty}\frac{\Gamma(2-\frac{d}{2}+k)}{\Gamma(2-\frac{d}{2})}
\frac{\alpha^{k}}{k!}\biggl(\frac{z-1}{z^{2}}\biggr)^{k}.$$ Interchanging integration and summation (which is a valid operation at least for some range of $d$), one gets for the regular part $$\label{eq:R}
R \equiv m^{d-4}\frac{\Gamma(2-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{1}^{\infty}\frac{dz}{(z^{2}-\alpha z + \alpha)^{2-\frac{d}{2}}}
= m^{d-4}\frac{\Gamma(2-\frac{d}{2})}{(3-d)(4\pi)^{\frac{d}{2}}} + O(\alpha).$$ At this point we should make the remark that the extension of the parameter integration to infinity will lead to divergences as $d$ increases. The infrared singular or regular parts are then defined as follows: The parameter integrals are computed for the range of $d$ where they are well-defined, and the result will be continued analytically to arbitrary values of $d$. This amounts to the suppression of power divergences of the parameter integrals, which have nothing to do with the infrared singularity at $z=0$, and will be cancelled anyway on the r.h.s. of eq. (\[eq:split\]).
Next we must show that the first term on the r.h.s. of eq. (\[eq:split\]) is proportional to a $d$-dependent power of $\alpha$. To see this, we substitute $z=\sqrt{\alpha}y$ to get $$\int_{0}^{\infty} \frac{dz}{(z^{2}-\alpha z + \alpha)^{2-\frac{d}{2}}}
= \sqrt{\alpha}^{d-3}\int_{0}^{\infty}\frac{dy}{(y^{2}-\sqrt{\alpha} y + 1)^{2-\frac{d}{2}}}.$$ The remaining integral on the r.h.s. will not produce $d$-dependent powers of $\alpha$, since the integrand can be expanded in $\sqrt{\alpha}$ similar to eq. (\[eq:expintR\]). Thus we have found that the parameter integral from zero to infinity is proportional to a $d$-dependent power of $\alpha$. The infrared singular part $I_{MB}^{IR}$ of the loop integral therefore equals $$\begin{aligned}
\label{eq:IMBIR}
I_{MB}^{IR} &=& -m^{d-4}\frac{\Gamma(2-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{0}^{\infty}\frac{dz}{(z^{2}-\alpha z + \alpha)^{2-\frac{d}{2}}} \nonumber \\
&=& -\frac{m^{d-4}\sqrt{\alpha}^{d-3}}{2(4\pi)^{\frac{d}{2}}}
\sum_{k=0}^{\infty}\frac{\sqrt{\alpha}^{k}}{k!}\Gamma
\biggl(\frac{k+1}{2}\biggr)\Gamma\biggl(\frac{3+k-d}{2}\biggr).\end{aligned}$$ As $d\rightarrow 4$, this has the well-known leading term $$I_{MB}^{IR}(d\rightarrow 4) = \frac{1}{16\pi}\biggl(\frac{M}{m}\biggr) + \ldots,$$ where the dots indicate terms of higher order in $M/m$. This is clearly in accord with the low-energy power counting. In contrast to that, the first term of the expansion of the regular part $R$ obviously violates this counting when $d\rightarrow 4$. The infrared regularization now prescribes to drop $R$ from the loop contribution and substitute $I_{MB}^{IR}$ for $I_{MB}$. Moreover, the poles of $I_{MB}^{IR}$ in $d-4$ are also absorbed in a renormalization of the masses and coupling constants of the effective Lagrangian. Again, for a more general and comprehensive treatment of the IR scheme in the pion-nucleon system, the reader should consult the original article of Becher and Leutwyler [@BL].
IR regularization for vector mesons and pions {#sec:PiV}
=============================================
In this section, we consider another case of infrared regularization, first examined in [@BM04]. The internal baryon lines from the preceding section are now replaced by vector meson lines, however, the vector mesons do not show up as external particles in the graphs we consider here (an example for the treatment of such a graph can be found in sec. 11 of [@BM04]). The only external particles here are pions (or, in some cases, soft photons etc.), so that there are only small external momenta of order $O(q)$ flowing into the loop. The fundamental scalar loop integral is in this case $$\label{eq:IMV}
I_{MV}(q^2) = \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{((q-l)^2-M_{V}^2)(l^2-M^2)} ,$$ where $M_{V}$ is the mass of the heavy meson resonance and $q$ is some small external momentum (small with respect to the resonance mass $M_{V}$). This is, in principle, the same function as in eq. (\[eq:IMB\]), so the reader might ask why we devote this section to the examination of this case. The point is that the extraction of the infrared singular part of $I_{MV}$ must proceed along different lines here. As shown in [@BM04], a splitting like that of eq. (\[eq:split\]) does not amount to a separation into infrared singular and regular parts as in the preceding section. This can be traced back to the fact that the extension of the parameter integrals to infinity leads to a singular behaviour of the loop function near $q^{2}= 0$. Of course, also $I_{MB}^{IR}$ has such a singularity as the external momentum squared goes to zero (see sec. 5.4 of [@BL]), but the point $p^{2}=0$ lies far outside the low-energy region in that case. Here, however, the point $q^{2}=0$ lies at the center of the low-energy region, so the infrared singular and regular parts can not be expressed as parameter integrals from zero or one to infinity, respectively. For example, a ’regular’ part defined as being proportional to the parameter integral from one to infinity would not be expandable around $q^{2}=0$, as it should be in order to be able to absorb the corresponding terms in a renormalization of the local operators in the effective Lagrangian.
To circumvent this difficulty, we will take up the following simple idea from [@BM04]. We observe that $I_{MV}$ is analytic at $q^{2}=0$, the only singularity being the threshold branch point at $q^{2}=(M_{V}+M)^{2}$, which is far outside the range of small $q^{2}$-values. Expanding $I_{MV}$ in $q^{2}$, the analyticity properties in that variable are obvious. Each coefficient in this expansion can be split into an infrared singular and a regular part almost like in eq. (\[eq:split\]). Extracting the infrared singular part proportional to $d$-dependent powers of the pion mass of each coefficient, and resumming the series, one arrives at a well-defined expression for the infrared singular part of the loop integral $I_{MV}$ that is (by construction) expandable in the small variable $q^{2}/M_{V}^{2}$.
To start with the analysis, let us first consider the special case where the external momentum vanishes: $q=0$. Then we have $$\begin{aligned}
I_{MV}(0) &=& \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{(l^2-M_{V}^2)(l^2-M^2)} \\
&=& \frac{1}{M^{2}-M_{V}^{2}}\biggl(\int\frac{d^{d}l}{(2\pi)^{d}}
\frac{i}{l^{2}-M^{2}}-\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-M_{V}^{2}}\biggr).\end{aligned}$$ In the last step of this equation, we have already achieved the splitting into an infrared singular and a regular part, which is quite trivial here. The first term is proportional to a $d$-dependent power of $M$ (compare eq. (\[eq:IM\])), while the second part is clearly expandable in $M^{2}$ (because $M_{V}^{2}\gg M^{2}$). Moreover, the pion mass expansion of the first term starts with $M^{d-2}$, in agreement with the low-energy power counting for $I_{MV}$: In contrast to the baryon propagator in sec. \[sec:PiN\], the resonance propagator is counted as $O(q^{0})$ since, in the low-energy region, the off-shell momentum of the internal resonance line is far below its mass shell. The counting for the pion propagator and the loop measure is the same as before, and one is lead to the power counting result $q^{d-2}$ for $I_{MV}$. Therefore we can write $$\label{eq:IMVIR0}
I_{MV}^{IR}(0) = \frac{1}{M^{2}-M_{V}^{2}}\int\frac{d^{d}l}{(2\pi)^{d}}
\frac{i}{l^{2}-M^{2}}.$$ It should be clear that there will only be corrections of $O(q^{2})$ to this result when we compute the regularized integral for nonvanishing $q$. We introduce the following variables, $$\label{eq:ab}
\tilde\alpha = \frac{M^{2}}{M_{V}^{2}}\qquad,\qquad \tilde\beta = \frac{q^{2}}{M_{V}^{2}},$$ which we assume to be small in the sense that $\tilde\alpha,\tilde\beta\ll
1$. Just like in sec. \[sec:PiN\], we use the Feynman parameter trick to write $$I_{MV}(q^{2}) = -M_{V}^{d-4}\frac{\Gamma(2-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{0}^{1}\frac{dz}{(\tilde\beta z^{2}+z(1
-\tilde\alpha-\tilde\beta)+\tilde\alpha)^{2-\frac{d}{2}}}.$$ We rewrite this expression as follows: $$I_{MV}(q^{2}) =
-M_{V}^{d-4}\frac{\Gamma(2-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{0}^{1}\frac{dz}{(z(1-\tilde\alpha-\tilde\beta)
+\tilde\alpha)^{2-\frac{d}{2}}}\biggl(1+\frac{\tilde\beta z^{2}}{z(1
-\tilde\alpha-\tilde\beta)+\tilde\alpha}\biggr)^{\frac{d}{2}-2} .$$ This can be expanded according to $$I_{MV}(q^{2}) = -\frac{M_{V}^{d-4}}{(4\pi)^{\frac{d}{2}}}
\sum_{k=0}^{\infty}\frac{\Gamma(\frac{d}{2}-1)}{k!\Gamma(\frac{d}{2}-1-k)}
\int_{0}^{1}\frac{\Gamma(2-\frac{d}{2})dz}{(z(1-\tilde\alpha-\tilde\beta)
+\tilde\alpha)^{2-\frac{d}{2}}}
\biggl(\frac{\tilde\beta z^{2}}{z(1-\tilde\alpha-\tilde\beta)+\tilde\alpha}\biggr)^{k}.$$ In the next step, we extract the infrared singular part of each term in the sum. In each parameter integral, we substitute $z=\tilde\alpha y$ and extend the integration range to infinity: $$\label{eq:Ik}
I_{k} \equiv \int_{0}^{1}dz \frac{z^{2k}}{(z(1-\tilde\alpha-\tilde\beta)
+\tilde\alpha)^{2+k-\frac{d}{2}}} \rightarrow
\int_{0}^{\infty}dy\frac{\tilde\alpha^{\frac{d}{2}-1+k}y^{2k}}{(1+y(1
-\tilde\alpha-\tilde\beta))^{2+k-\frac{d}{2}}}.$$ Divergences of the parameter integral due to the extension of the upper limit to infinity are again handled by analytic continuation in $d$. It can be expressed in terms of Gamma functions: $$\int_{0}^{\infty}dy\frac{y^{2k}}{(1+y(1-\tilde\alpha-\tilde\beta))^{2+k-\frac{d}{2}}}
= \frac{\Gamma(2k+1)\Gamma(1-\frac{d}{2}-k)}{(1
-\tilde\alpha-\tilde\beta)^{2k+1}\Gamma(2-\frac{d}{2}+k)}.$$ This is clearly expandable in the small variables $\tilde\alpha$ and $\tilde\beta$, so that the r.h.s. of eq. (\[eq:Ik\]) in fact has the proper form of an infrared singular contribution, being proportional to $d$-dependent powers of $\tilde\alpha$. Moreover, it is not difficult to see that the parameter integrals from $1$ to infinity are completely regular in the small variables. So, putting pieces together, and using the following identity for Gamma functions: $$\label{eq:GammaId}
\frac{\Gamma(\frac{d}{2}-1)}{\Gamma(\frac{d}{2}-1-k)}
= (-1)^{k}\frac{\Gamma(2-\frac{d}{2}+k)}{\Gamma(2-\frac{d}{2})}, \qquad k\in\mathbf{N},$$ we can sum the series of infrared singular terms and write $$\label{eq:IMVIR}
I_{MV}^{IR}(q^{2}) = -\frac{M_{V}^{d-4}}{(4\pi)^{\frac{d}{2}}}
(\tilde\alpha)^{\frac{d}{2}-1}\sum_{k=0}^{\infty}\frac{(-\tilde\alpha\tilde\beta)^{k}}{(1
-\tilde\alpha-\tilde\beta)^{2k+1}}\frac{\Gamma(2k+1)\Gamma(1-\frac{d}{2}-k)}{\Gamma(k+1)}.$$ Note that the leading term in this result is of chiral order $O(q^{d-2})$, as predicted by the power counting scheme. All the terms we have separated off from $I_{MV}$ are regular in both small parameters, and only the infrared singular terms remain. For $\tilde\beta=0$, the result for $I_{MV}^{IR}(0)$, which was already established in eq. (\[eq:IMVIR0\]), is reproduced.
It is perhaps worth noting that the result of eq. (\[eq:IMVIR\]) can be obtained in a different way, which goes back to Ellis and Tang [@Ellis; @Tang]. Though they use their method in the pion-nucleon sector, a variant of it is also applicable here. Their prescription to obtain the soft momentum contribution of a loop integral is the following: Expand the propagators of the heavy particles [*as if*]{} the loop momentum were small, and then interchange summation and loop integration. It is claimed that this prescription eliminates the ’hard momentum’ contributions present in the full loop graph. If this is true, and the concept of hard vs. soft momentum effects is a well-defined one, the result of the procedure should reproduce the infrared singular part of the loop integral in question. Let us see how this works out for our example. Following the prescription just described step by step, we make the following set of transformations: $$\begin{aligned}
I_{MV}(q^{2}) &\rightarrow& \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-M^{2}}
\sum_{k=0}^{\infty}\frac{(2q\cdot l)^{k}}{(q^{2}+l^{2}-M_{V}^{2})^{k+1}} \\
&\rightarrow& \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-M^{2}}
\sum_{k=0}^{\infty}\frac{(2q\cdot l)^{k}}{(q^{2}+M^{2}-M_{V}^{2})^{k+1}} \\
&\rightarrow& \sum_{k=0}^{\infty}\int\frac{d^{d}l}{(2\pi)^{d}}
\frac{i(2q\cdot l)^{k}}{(l^{2}-M^{2})(q^{2}+M^{2}-M_{V}^{2})^{k+1}} .\\\end{aligned}$$ While the first step is just the expansion of the vector meson propagator pole imposed by the prescription, the second step deserves a comment: There, we have used the same trick of partial fractions to split off some hard momentum contributions as in the treatment of $I_{MV}(0)$, but now this was performed $k+1$ times. In the last step, summation and integration were interchanged. Using the formula $$\label{eq:supertadpole}
\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i(q\cdot l)^{2n}}{l^{2}-M^{2}}
= (-q^{2}M^{2})^{n}M^{d-2}\frac{\Gamma(n+\frac{1}{2})}{
\Gamma(\frac{1}{2})}\frac{\Gamma(1-\frac{d}{2}-n)}{(4\pi)^{\frac{d}{2}}}$$ and the fact that the loop integrals in the series vanish if $k$ is odd, the result of the transformation is $$\label{eq:IVMsoft}
I_{MV}^{\mathrm{soft}}(q^{2}) = \frac{M^{d-2}}{(4\pi)^{\frac{d}{2}}}
\sum_{n=0}^{\infty}\frac{(-4q^{2}M^{2})^{n}}{(q^{2}+M^{2}-M_{V}^{2})^{2n+1}}
\frac{\Gamma(n+\frac{1}{2})\Gamma(1-\frac{d}{2}-n)}{\Gamma(\frac{1}{2})}.$$ Extracting a factor of $M_{V}^{d-4}$, and using the identity $$\label{eq:GammaId2}
\frac{\Gamma(2n+1)}{\Gamma(n+1)}
= 4^{n}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(\frac{1}{2})},$$ we see by comparing to eq. (\[eq:IMVIR\]) that $I_{MV}^{\mathrm{soft}} =
I_{MV}^{IR}$.
In [@BM04], the result for $I_{MV}^{IR}$ was given in a different form (see eqs. (7.10) and (8.4) of that reference). Of course, it is equivalent to the result derived above: it is shown in app. \[app:closedform\] that the two different forms just amount to a reordering of the corresponding expansions. We will find that the form of eq. (\[eq:IMVIR\]) is most practical for our purposes, in particular, the chiral expansion can almost immediately be read off from that formula.
In closing this section, we note that the spin or the parity of the resonance obviously do not play a major role in the above considerations. Though we will concentrate on the case of vector mesons, most of what we have said would also apply for other meson resonances, like e.g. scalar or axial-vector mesons.
Pion-nucleon system with explicit meson resonances {#sec:PiNV}
==================================================
Now that we have collected the results for the fundamental one-loop integrals in the pion-nucleon and the vector meson-pion sector, we are prepared to extend the framework of infrared regularization once more, and apply it to Feynman graphs where nucleons, pions as well as vector mesons take part in the same loop. The simplest example where this situation occurs is the triangle graph consisting of one pion, one nucleon and one vector meson line. Due to baryon number conservation, the nucleon line must run through the complete diagram. We shall assume that only a small momentum $k$ (small in the usual sense) is transferred at the vector meson-pion vertex. Such a graph will typically contribute to some nucleon form factor in the region of small momentum transfer. With this application in mind, and with the excuse that it will simplify the presentation a bit, we will further specify to on-shell nucleons. Let $p$ and $\bar p$ be the four-momenta of the incoming and the outgoing nucleon. Then we have $$\label{eq:kinppbar}
p^{2} = m^{2}=\bar p^{2}
=(p+k)^{2} \Rightarrow k^{2}=2\bar p\cdot k=-2p\cdot k.$$ Using these kinematic relations, we can rewrite the fundamental scalar loop integral $$\label{eq:IMBV}
I_{MBV}(k^{2}) \equiv \int\frac{d^{d}l}{(2\pi)^{d}}
\frac{i}{((p-l)^{2}-m^{2})((k+l)^{2}-M_{V}^{2})(l^{2}-M^{2})}$$ with the help of the usual Feynman parameter trick, as $$\label{eq:intlxy}
\int\frac{d^{d}l}{(2\pi)^{d}}\int_{0}^{1}\int_{0}^{1-y}
\frac{2idxdy}{(y((p-l)^{2}-m^{2})+x((k+l)^{2}-M_{V}^{2})+(1-x-y)(l^{2}-M^{2}))^{3}}.$$ Doing the loop integration in the usual manner, we get $$\label{eq:intxy}
I_{MBV}(k^{2})
= m^{d-6}\frac{\Gamma(3-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{0}^{1}\int_{0}^{1-y}\frac{dxdy}{(y^{2}-\alpha y+\alpha
+ \beta(x^{2}+xy)+x(\gamma-\alpha-\beta))^{3-\frac{d}{2}}},$$ where the definitions $$\alpha = \frac{M^{2}}{m^{2}} \ll 1, \qquad \beta
= \frac{k^{2}}{m^{2}} \ll 1, \qquad
\gamma = \frac{M_{V}^{2}}{m^{2}} \sim O(1).$$ were used. In particular, we have assumed here that the nucleon and the meson resonance are roughly of the same order of magnitude (in the real world, we have $\gamma \sim 2/3$ for the rho resonance, which is good enough for our purposes).\
We should remark here that there is, of course, a second graph with the same topology, where the pion and the resonance line are interchanged (see fig. \[fig:triangles\]). The expression for the corresponding scalar loop integral is $$\label{eq:IMBV2}
\tilde I_{MBV}(k^{2}) \equiv
\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{((\bar p-l)^{2}-m^{2})((l-k)^{2}-M_{V}^{2})(l^{2}-M^{2})}.$$ However, the reader can convince himself that, due to the on-shell kinematics specified in eq. (\[eq:kinppbar\]), this will give exactly the same expression as in eq. (\[eq:intxy\]). Thus, we can focus on the integral $I_{MBV}$.
In analogy to sec. \[sec:PiV\], it will be instructive to begin with the special case where $k=0$. Since we must split off the terms where only propagators of heavy particles occur, we can obviously apply the same partial fraction method that led to eq. (\[eq:IMVIR0\]). The remaining pion-nucleon integral can be dealt with as in sec. \[sec:PiN\]. This gives $$\label{eq:IMBVIR0}
I_{MBV}^{IR}(0) = \frac{I_{MB}^{IR}(m^{2})}{M^{2}-M_{V}^{2}}.$$ Having the standard method described of sec. \[sec:PiN\] in mind, we note that this equals $$\begin{aligned}
I_{MBV}^{IR}(0) &=& m^{d-6}\frac{\Gamma(3-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{0}^{\infty}\int_{0}^{\infty}\frac{dxdy}{(y^2-\alpha y+\alpha
+x(\gamma-\alpha))^{3-\frac{d}{2}}} \\
&=& \frac{m^{d-6}}{(\gamma-\alpha)}
\frac{\Gamma(2-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}}
\int_{0}^{\infty}\frac{dy}{(y^{2}-\alpha y+\alpha)^{2-\frac{d}{2}}} \end{aligned}$$ (compare the last line with the l.h.s of eq. (\[eq:IMBIR\])). Once again, the possible divergence for large $d$ at $x\rightarrow\infty$ was regularized by analytic continuation from small $d$ as before. It is reassuring to see that eq. (\[eq:IMBVIR0\]) is reproduced in this way.
The extension of both parameter integrations to infinity is the natural generalization of the prescription used in sec. \[sec:PiN\]. This can be seen as follows. In eq. (\[eq:intlxy\]), we chose our Feynman parameters such that the pion propagator contributes with the weight one at $x=y=0$, while the vector meson and the nucleon propagator have their maximum weight at $(x,y) = (1,0)$ and $(0,1)$, respectively. Consequently, the infrared singularity is located in parameter space at the point $(x,y) =
(0,0)$. Indeed, a look at eq. (\[eq:intxy\]) confirms that the integrand in that expression is approximately $\alpha^{\frac{d}{2}-3}$ in the region where $x\sim y\sim 0$, thus producing infrared singular terms there. Far from this region, the integrand is expandable in $\alpha$, as we will show later. To be more specific, imagine the positive quarter of the parameter plane, i.e. $Q_{+}\equiv \lbrace(x,y):x\geq 0, y\geq 0\rbrace$, split in two parts, the first being the triangle one integrates over in the full loop integral (see eq. (\[eq:intxy\])) and the second part its complement in $Q_{+}$, named $R_{+}$ (see fig. \[fig:rays\] for an illustration). In the latter parameter region, no infrared singularities are located, hence, the integration over this region should yield a proper regular part (there is a qualification here, see below). Geometrically, one might imagine that all lines from the ’soft point’ $(0,0)$ (the location of the infrared singularity in parameter space) to the ’hard line’ from $(0,1)$ to $(1,0)$ are extended to infinity to get the infrared singular part of the loop integral. In a more general setting, there may be soft or hard points, lines, surfaces etc., depending on the number of soft and hard pole structures in the integral under consideration. Extending all connecting lines from the soft point, line etc. to the hard point (line$\ldots$) to infinity, one always achieves a splitting in the original parameter region and a region where no infrared singularities in parameter space are present, analogous to $R_{+}$. It is possible to show that this geometric picture leads to the same results as the prescription given in sec. 6 of [@BL]. This picture might serve as a guide when trying to split arbitrary one-loop graphs into infrared singular and regular parts, however, one should always convince oneself that both those parts have the correct properties, and that their sum equals the original integral. One has to be a bit careful here, as the following qualification shows. As explained in sec. \[sec:PiV\], extending the integration range to infinity might lead to unphysical singularities in variables in which the original integral is analytic (at least in the low-energy region). Those singularities will be harmless if they are located far from the low-energy region, as e.g. the singularity at $s=0$ of the infrared singular parts in the pion-nucleon sector, but in other cases, they can be disturbing. Following the method of [@BM04], we avoided this problem in sec. \[sec:PiV\] by expanding the original loop integral in the small variable $\tilde\beta$ beforehand. Only then could the integration range be extended to infinity, in each coefficient of the expansion. We must expect that a similar phenomenon will occur in the present case.
To exclude this from the start, we expand $I_{MBV}$ in analogy to eq. (\[eq:IMVIR\]): $$\begin{aligned}
I_{MBV}(k^{2}) &=& \frac{m^{d-6}}{(4\pi)^{\frac{d}{2}}}
\sum_{j=0}^{\infty}\frac{\Gamma(\frac{d}{2}-2)}{\Gamma(\frac{d}{2}-2-j)}
\int_{0}^{1}\int_{0}^{1-y}\frac{\Gamma(3-\frac{d}{2})dxdy(\beta x(x+y))^{j}}{j!(y^2
-\alpha y+\alpha+x(\gamma-\alpha-\beta))^{3-\frac{d}{2}+j}} \\
&=& \frac{m^{d-6}}{(4\pi)^{\frac{d}{2}}}
\sum_{j=0}^{\infty}\frac{\Gamma(\frac{d}{2}-2)}{\Gamma(\frac{d}{2}-2-j)}
\int_{0}^{1}\int_{0}^{1-y}\frac{\Gamma(3-\frac{d}{2})dxdy(\sum_{l=0}^{j}{j\choose l}
\beta^{j}x^{l+j}y^{j-l})}{j!(y^2-\alpha y
+\alpha+x(\gamma-\alpha-\beta))^{3-\frac{d}{2}+j}}. \end{aligned}$$ There is still some $\beta$-dependence in the denominator, but in that combination, it will turn out to be harmless. We extend the parameter integrations to $Q_{+}$ and define $$I_{MBV}^{IR}(k^{2}) =
\frac{m^{d-6}}{(4\pi)^{\frac{d}{2}}}
\sum_{j=0}^{\infty}\frac{\Gamma(\frac{d}{2}-2)}{
\Gamma(\frac{d}{2}-2-j)}\int_{0}^{\infty}\int_{0}^{\infty}
\frac{\Gamma(3-\frac{d}{2})dxdy(\sum_{l=0}^{j}{j\choose l}
\beta^{j}x^{l+j}y^{j-l})}{j!(y^2-\alpha y+\alpha+x(\gamma
-\alpha-\beta))^{3-\frac{d}{2}+j}}.$$ Since the denominator is always positive for $0<\alpha\ll 1, |\beta|\ll 1$, the $x$-integration can readily be done using the formula $$\label{eq:xparint}
\int_{0}^{\infty}dx\frac{x^{n}}{(a+bx)^{D}}
= \frac{a^{n+1-D}}{b^{n+1}}\frac{\Gamma(n+1)\Gamma(D-(n+1))}{\Gamma(D)},$$ together with eq. (\[eq:GammaId\]). This gives $$\begin{aligned}
I_{MBV}^{IR}(k^{2})
&=& \frac{m^{d-6}}{(4\pi)^{\frac{d}{2}}}\sum_{j=0}^{\infty}\sum_{l=0}^{j}
\frac{\Gamma(j+l+1)\Gamma(2-\frac{d}{2}-l)}{\Gamma(j-l+1)
\Gamma(l+1)}\frac{(-\beta)^{j}}{(\gamma-\alpha-\beta)^{j+l+1}}\times \\
& & \times\int_{0}^{\infty}\frac{y^{j-l}dy}{(y^{2}-\alpha y+\alpha)^{2-\frac{d}{2}-l}}.\end{aligned}$$ In the next step, we can write down the chiral expansion of the $y$-integral, using the same method as in sec. \[sec:PiN\]. The generalized formula is $$\label{eq:yparint}
\int_{0}^{\infty}dy\frac{y^{n}}{(y^{2}-\alpha y+\alpha)^{D}}
= \sqrt{\alpha}^{n+1-2D}\sum_{k=0}^{\infty}\frac{\sqrt{\alpha}^{k}}{k!}
\frac{\Gamma(\frac{n+k+1}{2})\Gamma(\frac{2D+k-(n+1)}{2})}{2\Gamma(D)}.$$ Obviously, the chiral expansion of this integral can straightforwardly be read off from the series on the r.h.s. Inserting this result, we get $$\label{eq:IMBVIR}
I_{MBV}^{IR}(k^{2}) = \frac{m^{d-6}}{(4\pi)^{\frac{d}{2}}}
\sum_{j,k=0}^{\infty}\sum_{l=0}^{j}\frac{(-\beta)^{j}
\sqrt{\alpha}^{d-3+j+l+k}}{(\gamma-\alpha-\beta)^{j+l+1}}
\frac{\Gamma(j+l+1)\Gamma(\frac{j+k-l+1}{2})
\Gamma(\frac{3-d-l-j+k}{2})}{2\Gamma(j-l+1)\Gamma(k+1)\Gamma(l+1)}.$$ The only expressions we have not expanded in the small variables are the factors of $(\gamma-\alpha-\beta)$ in the denominator, but this can of course be done: the corresponding geometric series is absolutely convergent due to the assumption that $\gamma\sim O(1)$.
To complete the proof that eq. (\[eq:IMBVIR\]) is the correct infrared singular part of $I_{MBV}$, we have to show that all the terms we dropped in the extraction procedure described above are regular in $\alpha$. Those terms are proportional to parameter integrals over the region $R_{+}$, of the general type $$\begin{aligned}
R_{j} &=& \int_{R_{+}}\frac{dxdy(x(x+y))^{j}}{(y^{2}
-\alpha y+\alpha+x(\gamma-\alpha-\beta))^{3-\frac{d}{2}+j}} \\
&=& \int_{z=1}^{\infty}\int_{x=0}^{z}\frac{dxdz(xz)^{j}}{((z-x)^2
-\alpha(z-1)+x(\gamma-\beta))^{3-\frac{d}{2}+j}}.\end{aligned}$$ Here we have traded the variable $y$ for $z\equiv x+y$. One finds that the function $$f(z,x) = \frac{(z-x)^{2}+x(\gamma-\beta)}{z-1}$$ has the property $$f(z,x) \geq \mathrm{min}\lbrace 4,(\gamma-\beta)\rbrace$$ in $R_{+}$, provided that the parameters $\beta$ and $\gamma$ are in their typical low-energy ranges. This is already sufficient to ensure that the integrand of $R_{j}$ can safely be expanded in $(z-1)\alpha$, and that integration and summation of the corresponding series can be interchanged. This proves the regularity of the integrals $R_{j}$ in $\alpha$.
It is also possible to show that the result for $I_{MBV}^{IR}$ can be obtained in a way that is closely analogous to the prescription of Ellis and Tang (see the end of sec. \[sec:PiV\]). The proof can be found in app. \[app:TangIR\].
Application: Axial form factor of the nucleon {#sec:AxFF}
=============================================
A typical example for an application where the triangle graph treated in sec. \[sec:PiNV\] shows up is a contribution to some form factor of the nucleon at low momentum transfer. As a specific example, we consider the nucleon form factor of the isovector axial-vector current in a theory with an explicit rho resonance field. A representation for this form factor, using the infrared regularization scheme in the pion-nucleon sector, has been given by Schweizer [@Schweizer]. In that framework, the contributions due to the various baryon or meson resonances are contained in the low-energy coefficients (LECs) of the effective pion-nucleon Lagrangian, or, more correctly: The contributions from tree-level resonance exchange can be described as an infinite sum of contact terms derived from the pion-nucleon Lagrangian. The inclusion of explicit resonance fields therefore amounts to a resummation of higher order terms, which is often advantageous (as discussed in detail in ref. [@Kubis:2000zd]). Moreover, a theory with explicit resonance fields can serve to achieve an understanding of the numerical values of the LECs, relying on the assumption that the lowest-lying resonances give the dominant contributions to those coefficients. This is usually called the principle of resonance saturation, and has been very successfull in ChPT, see e.g. [@RoleofRes; @AspectsPiN]. Assuming this principle to be valid, the pion-nucleon LECs can be expressed through the masses of the resonances and the couplings of the resonances to the nucleons and pions. Such relations are most useful, of course, if the resonance masses and couplings are sufficiently well known. In this section, we will compute the contribution to the axial form factor of the nucleon that is described by the two triangle graphs of fig. (\[fig:triangles\]), with one nucleon, one pion and one resonance line, in the framework of the extended infrared regularization scheme developed in the preceding sections. First, we must set up the necessary formalism and collect the various terms from the effective Lagrangian we need for the computation.
By Lorentz invariance, the matrix element of the axial current $$A_{\mu}^{i}(x) \equiv \bar q(x)\gamma_{\mu}\gamma_{5}\frac{\tau^{i}}{2}q(x)$$ between one-nucleon states can be parametrized as $$\label{eq:AxMatEl}
\langle N'(\bar p) | A_{\mu}^{i}(x) | N(p)\rangle
= \bar u'(\bar p)\biggl(G_{A}(t)\gamma_{\mu}
+G_{P}(t)\frac{k_{\mu}}{2m}+G_{T}(t)
\frac{\sigma_{\mu\nu}k^{\nu}}{2m}\biggr)\gamma_{5}\frac{\tau^{i}}{2}u(p)e^{ikx}.$$ In the above expressions, $q$ is the quark field spinor, $q^{T}=(u,d)$, $N'$ is the outgoing nucleon with momentum $\bar p$, $N$ labels the incoming nucleon with momentum $p$, and $k=\bar p-p$, $t\equiv k^{2}$. The symbols $\tau^{i}$ denote the usual Pauli matrices. Finally, $\bar u'$ and $u$ are the Dirac spinors associated with the outgoing and incoming nucleon, respectively. Assuming perfect isospin symmetry and charge conjugation invariance, as we will do here, leads to $G_{T}\equiv 0$. The relation of $G_{A}$ and $G_{P}$ to the quantities $F_{1,2}$ used in [@Schweizer] is $F_{1}(t) = G_{A}(t), 2mF_{2}(t) = -G_{P}(t)$. For later reference, we give the representation of $G_{A}$ up to order $q^{3}$ that can be found in [@Schweizer]: $$\begin{aligned}
\label{eq:GaSchweizer}
G_{A}(t) &=& g_{A}+4\bar d_{16}M^{2}+d_{22}t-
\frac{g_{A}^{3}M^{2}}{16\pi^{2}F^{2}} \nonumber \\
&+& \frac{g_{A}M^{3}}{24\pi mF^{2}}\biggl(3+3g_{A}^{2}-4c_{3}m+8c_{4}m\biggr) + O(q^{4}). \end{aligned}$$ Here $F$ and $g_{A}$ denote the pion decay constant and the nucleon axial charge in the chiral limit, respectively, while the coefficients $c_{i},d_{i}$ are LECs showing up in the pion-nucleon effective Lagrangian at order two and three, respectively. For a precise definition of the underlying Lagrangian see ref. [@Fettes:1998ud].
We now turn back to the calculation of the axial form factor in the presence of vector mesons. We write down the relevant terms in the effective Lagrangian and give the necessary rules for the vertices and propagators required for the calculation. First, the lowest order chiral Lagrangian for the pion-nucleon interaction reads $$\label{eq:L1N}
\mathcal{L}^{(1)}_{N}
= \bar\psi(i\slashed{D}-m)\psi + \frac{g_{A}}{2}\bar\psi \slashed{u}\gamma_{5}\psi.$$ Here, $\psi$ is the nucleon spinor, $\psi^{T}=(p,n)$, and the matrix $u_{\mu}$ collects the pion fields $\pi^{a}$ via $$\begin{aligned}
u &=& \exp{\biggl(\frac{i\vec\tau\cdot\vec\pi}{2F}\biggr)}, \\
u_{\mu} &=& i\lbrace u^{\dagger},\partial_{\mu}u\rbrace
+ u^{\dagger}r_{\mu}u-ul_{\mu}u^{\dagger}, \\
r_{\mu} &=& v_{\mu}+a_{\mu},\quad l_{\mu} = v_{\mu}-a_{\mu}.\end{aligned}$$ In the last line, we have introduced external isovector vector and axial-vector sources, $v_{\mu}$ and $a_{\mu}$, $$v_{\mu} = v_{\mu}^{i}\frac{\tau^{i}}{2}, \quad a_{\mu} = a_{\mu}^{i}\frac{\tau^{i}}{2}.$$ The covariant derivative $D_{\mu}$ in eq. (\[eq:L1N\]) is defined as $$\begin{aligned}
D_{\mu} &=& \partial_{\mu}+\Gamma_{\mu}, \\
\Gamma_{\mu} &=& \frac{1}{2}[u^{\dagger},\partial_{\mu}u]
-\frac{i}{2}u^{\dagger}r_{\mu}u-\frac{i}{2}ul_{\mu}u^{\dagger}.\end{aligned}$$ >From eq. (\[eq:L1N\]), one derives the $\bar N N\pi$ vertex rule $$\frac{g_{A}}{2F}\slashed{q}\gamma_{5}\tau^{a}$$ for an outgoing pion of momentum $q$.
Now we turn to the effective Lagrangians involving the vector meson fields. We choose a representation in terms of an antisymmetric tensor field $W_{\mu\nu}$ [@RoleofRes; @Ecker; @BM04]. The corresponding free Lagrangian is $$\mathcal{L}_{W}^{\mathrm{kin}}
= -\frac{1}{2}\langle D^{\mu}W_{\mu\nu}D_{\rho}W^{\rho\nu}\rangle
+ \frac{1}{4}M_{V}^{2}\langle W_{\mu\nu}W^{\mu\nu}\rangle,$$ where $$W_{\mu\nu} = \frac{1}{\sqrt{2}}W_{\mu\nu}^{i}\tau^{i}
= \left( \begin{array}{cc} \frac{\rho^{0}}{\sqrt{2}}\quad & \rho^{+} \\
\rho^{-}\quad & -\frac{\rho^{0}}{\sqrt{2}} \\ \end{array} \right)_{\mu\nu}.$$ The brackets $\langle\ldots\rangle$ denote the trace in isospin space. From $\mathcal{L}_{W}^{\mathrm{kin}}$, one derives the tensor field propagator in momentum space, $$T_{\mu\nu,\rho\sigma}^{ij}(k) = \frac{i\delta^{ij}}{M_{V}^{2}}
\frac{g_{\mu\rho}g_{\nu\sigma}(M_{V}^{2}-k^{2})+g_{\mu\rho}
k_{\nu}k_{\sigma}-g_{\mu\sigma}k_{\nu}k_{\rho}-(\mu
\leftrightarrow \nu)}{M_{V}^{2}-k^{2}}.$$ At lowest chiral order, the interaction of the rho meson with the pions is given by [@RoleofRes] $$\label{eq:LWInt}
\mathcal{L}_{W}^{\mathrm{int}}
= \frac{F_{V}}{2\sqrt{2}}\langle F^{+}_{\mu\nu}W^{\mu\nu}\rangle
+ \frac{iG_{V}}{2\sqrt{2}}\langle[u_{\mu},u_{\nu}]W^{\mu\nu}\rangle.$$ In the first term we have used the definitions $$\begin{aligned}
F^{\pm}_{\mu\nu} &=& uF_{\mu\nu}^{L}u^{\dagger}\pm u^{\dagger}F_{\mu\nu}^{R}u, \\
F^{L}_{\mu\nu} &=& \partial_{\mu}l_{\nu}-\partial_{\nu}l_{\mu}-i[l_{\mu},l_{\nu}], \\
F^{R}_{\mu\nu} &=& \partial_{\mu}r_{\nu}-\partial_{\nu}r_{\mu}-i[r_{\mu},r_{\nu}].\end{aligned}$$ The external sources $r_{\mu},l_{\mu}$ are counted as $O(q)$, so that $F^{\pm}_{\mu\nu}$ is of chiral order $O(q^{2})$. Also, $u_{\mu}$ is of $O(q)$. Therefore, $\mathcal{L}_{W}^{\mathrm{int}}$ leads to vertices of chiral order $O(q^{2})$. Using the method of external sources to derive Greens functions from the generating functional, we must extract the amplitudes linear in the source $a_{\mu}$ to compute the matrix element of eq. (\[eq:AxMatEl\]). For the triangle graphs considered here, we need the vertex that connects the vector meson and the pion with the external axial source. From eq. (\[eq:LWInt\]), we find the corresponding vertex rule $$\epsilon^{iac}\biggl(\frac{G_{V}}{F}(q_{\mu}g_{\nu\tau}-q_{\nu}g_{\mu\tau})-\frac{F_{V}}{2F}(k_{\mu}g_{\nu\tau}-k_{\nu}g_{\mu\tau})\biggr).$$ Here $i,a,c$ are the isospin indices associated with the axial source $a_{\tau}$, the pion and the vector meson field $W_{\mu\nu}$, respectively, $q$ is the four-momentum of the outgoing pion. Since $k$ and $q$ are counted as small momenta, the chiral order of this vertex rule is in accord with the power counting for the interaction Lagrangian.
The restrictions of chiral symmetry are not that strong for the interaction of the vector mesons with the nucleons: Here, there are terms of chiral order $O(q^{0})$. The leading terms of the interaction Lagrangian have been given in ref. [@BM95]. For the $SU(2)$ case we consider here, the relevant terms are $$\begin{aligned}
\label{eq:LWN}
\mathcal{L}_{NW} &=& R_{V}\bar\psi\sigma_{\mu\nu}W^{\mu\nu}\psi
+ S_{V}\bar\psi\gamma_{\mu}D_{\nu}W^{\mu\nu}\psi \nonumber \\
& & +T_{V}\bar\psi\gamma_{\mu}D_{\lambda}W^{\mu\nu}D^{\lambda}D_{\nu}\psi
+ U_{V}\bar\psi\sigma_{\lambda\nu}W^{\mu\nu}D^{\lambda}D_{\mu}\psi.\end{aligned}$$ In the notation of [@BM95], we have $R_{V}=R_{D}+R_{F},S_{V}=S_{D}+S_{F}$, etc. The definition of $\sigma_{\mu\nu}$ is standard, $\sigma_{\mu\nu} = i[\gamma_{\mu},\gamma_{\nu}]/2.$ It turns out that only the piece proportional to $G_{V}$ from eq. (\[eq:LWInt\]) and the piece proportional to $R_{V}$ from eq. (\[eq:LWN\]) contribute at lowest order to the diagrams computed here, the chiral expansion of which starts at $O(q^{3})$ (given that a scheme like infrared regularization is used that preserves the power counting rules). To keep the presentation short, we show only the contribution from those terms and therefore neglect some higher order contributions. However, this will be sufficient to compare our results to the representation up to $O(q^{3})$ given by Schweizer [@Schweizer].
The evaluation of the first graph gives (see fig. (\[fig:triangles\])) $$I_{1} = \int\frac{d^{d}l}{(2\pi)^{d}}\biggl(\frac{G_{V}}{F}(l_{\mu}g_{\nu\tau}
-l_{\nu}g_{\mu\tau})\epsilon^{iac}\biggr)\frac{i}{l^{2}-M^{2}}
\biggl(-\frac{g_{A}}{2F}\slashed{l}\gamma_{5}\tau^{a}\biggr)\frac{iT^{\mu\nu,\rho\sigma}_{cd}(l-k)}
{\slashed{\bar p}-\slashed{l}-m}i\sigma_{\rho\sigma}
\tau^{d}\frac{R_{V}}{\sqrt{2}},$$ and the second one gives $$I_{2} = \int\frac{d^{d}l}{(2\pi)^{d}}\frac{R_{V}}{\sqrt{2}}
i\sigma_{\mu\nu}\tau^{c}
\frac{iT^{\mu\nu,\rho\sigma}_{cd}(l+k)}{\slashed{p}-\slashed{l}-m}\biggl(\frac{g_{A}}{2F}\slashed{l}
\gamma_{5}\tau^{a}\biggr)\frac{i}{l^{2}-M^{2}}\biggl(\frac{G_{V}}{F}(l_{\rho}
g_{\sigma\tau}-l_{\sigma}g_{\rho\tau})\epsilon^{ida}\biggr).$$ We have left out the Dirac spinors $\bar u,u$ here. Since we consider on-shell nucleons, we can use the Dirac equation to simplify the numerators of the integrals, $\bar u(\bar p)\slashed{\bar p}=\bar u(\bar p)m$, $\slashed{p}u(p) = mu(p)$. We will make some remarks on the computation of $I_{1}$ (the computation of $I_{2}$ can be done analogously). In a first step, we reduce the full loop integral to a linear combination of scalar loop integrals, which have been treated in detail in the preceding sections. Scalar loop integrals without a pion propagator denominator are dropped using infrared regularization, since they are pure regular parts. Therefore we can replace $l^{2}\rightarrow M^{2}$ everywhere in the numerator.
With the abbreviation $$g_{1} = \frac{2\sqrt{2}g_{A}G_{V}R_{V}}{M_{V}^{2}F^{2}}$$ we get $$\begin{aligned}
I_{1} &=& g_{1}\tau^{i}\biggl(\int\frac{d^{d}l}{(2\pi)^{d}}
\frac{2im\slashed{l}((M^{2}-k\cdot l)(l^{\rho}-k^{\rho})\sigma_{\rho\tau}
+\frac{i}{2}(k_{\tau}-l_{\tau})(\slashed{l}\slashed{k}-\slashed{k}\slashed{l}))
\gamma_{5}}{((\bar p-l)^{2}-m^{2})((l-k)^{2}-M_{V}^{2})(l^{2}-M^{2})} \\
&+& \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i((M^{2}-k\cdot l)(l^{\rho}-k^{\rho})
\sigma_{\rho\tau}+\frac{i}{2}(k_{\tau}-l_{\tau})(\slashed{l}\slashed{k}
-\slashed{k}\slashed{l}))\gamma_{5}}{((l-k)^{2}-M_{V}^{2})(l^{2}-M^{2})} \\
&-& \int\frac{d^{d}l}{(2\pi)^{d}}\frac{2im\slashed{l}l^{\rho}
\sigma_{\rho\tau}\gamma_{5}}{((\bar p-l)^{2}-m^{2})(l^{2}-M^{2})}\biggr).\end{aligned}$$ For completeness, we shall give the relevant loop integrals with tensor structures in app. \[app:LoopInt\]. Using the coefficient functions defined there, the result for the first integral $I_{1}$ can be written as $$\label{eq:I1decomp}
I_{1} = -ig_{1}\tau^{i}(\gamma_{\tau}I_{1}^{(\gamma)}+k_{\tau}I_{1}^{(k)}
+\bar p_{\tau}I_{1}^{(p)})\gamma_{5},$$ where the coefficients read $$\begin{aligned}
I_{1}^{(\gamma)} &=& 2m(M_{V}^{2}-k^{2})C_{1}+mk^{2}(I_{MBV}^{A}
+I_{MBV}^{B})(M^{2}+M_{V}^{2}-k^{2})\\
& & -mM^{2}(M^{2}+M_{V}^{2}-k^{2})I_{MBV} + mM^{2}I_{MB}, \\
I_{1}^{(k)}
&=&
2m^{2}(M_{V}^{2}-k^{2})(3C_{2}+C_{3}+4C_{4})+4m^{2}M^{2}(I_{MBV}^{A}+I_{MBV}^{B})
\\
& & -2m^{2}(M_{V}^{2}-k^{2})(3I_{MBV}^{A}+I_{MBV}^{B})
+2m^{2}(I_{MBV}^{A}-I_{MBV}^{B})(M^{2}+M_{V}^{2}-k^{2}) \\
& & -4m^{2}M^{2}I_{MBV} + t_{MV}^{(1)}+(M_{V}^{2}-2k^{2})I_{MV}^{(1)}+(k^{2}-M_{V}^{2})I_{MV} \\
& & +(M^{2}+M_{V}^{2}-k^{2})(I_{MV}-I^{(1)}_{MV}),\\
I_{1}^{(p)} &=&
2m^{2}(M_{V}^{2}-k^{2})(3C_{2}-C_{3}-2C_{4})+4m^{2}M^{2}(I_{MBV}^{A}
-I_{MBV}^{B}) \\ & & -2m^{2}(I_{MBV}^{A}-I_{MBV}^{B})(M^{2}+M_{V}^{2}-k^{2})
+2t^{(0)}_{MV}-2m^{2}I_{MB}^{(1)} \\
& & +(M^{2}+M_{V}^{2}-k^{2})(I_{MV}^{(1)}-I_{MV})-I_{M}.\end{aligned}$$ What concerns the evaluation of $I_{2}$, we note that it is given by $$\label{eq:I2decomp}
I_{2} = -ig_{1}\tau^{i}(\gamma_{\tau}I_{2}^{(\gamma)}
+k_{\tau}I_{2}^{(k)}+p_{\tau}I_{2}^{(p)})\gamma_{5},$$ with $$I_{2}^{(\gamma)}=I_{1}^{(\gamma)}, \quad I_{2}^{(k)} = I_{1}^{(k)},
\quad I_{2}^{(p)}=-I_{1}^{(p)}.$$ The sum of both graphs therefore gives $$I_{1+2} = I_{1}+I_{2} = -ig_{1}\tau^{i}(\gamma_{\tau}I_{1+2}^{(\gamma)}
+k_{\tau}I_{1+2}^{(k)})\gamma_{5},$$ with $$I_{1+2}^{(\gamma)} = 2I_{1}^{(\gamma)}, \quad I_{1+2}^{(k)} = 2I_{1}^{(k)}+I_{1}^{(p)}.$$ In the sum of the two graphs, the contribution proportional to $(\bar
p+p)_{\tau}$ cancels, as was to be expected on general grounds (see the remarks following eq. (\[eq:AxMatEl\])). We are now in a position to display the decomposition of the graphs as a linear combination of the scalar loop integrals worked out in the preceding sections: $$\begin{aligned}
I_{1+2}^{(\gamma)}
&=& c_{MBV}^{(\gamma)}I_{MBV}^{IR}+c_{MB}^{(\gamma)}I_{MB}^{IR}
+c_{MV}^{(\gamma)}I_{MV}^{IR}, \label{eq:I12gdecomp} \\
I_{1+2}^{(k)}
&=& c_{MBV}^{(k)}I_{MBV}^{IR}+c_{MB}^{(k)}I_{MB}^{IR}+c_{MV}^{(k)}I_{MV}^{IR}
+c_{M}^{(k)}I_{M}. \label{eq:I12kdecomp}\end{aligned}$$ The expressions for the coefficients $c^{(\gamma)}$ read $$\begin{aligned}
c_{MBV}^{(\gamma)}
&=& \frac{4m}{(d-2)k^{2}(k^{2}-4m^{2})}\biggl[m^{2}M_{V}^{6}
+(k^{2}((d-5)m^{2}+M^{2})-2m^{2}M^{2})M_{V}^{4} \\
&-& ((M^{2}+m^{2}(2d-7))k^{4}-2(d-2)m^{2}M^{2}k^{2}-m^{2}M^{4})M_{V}^{2}\\
&+& (d-3)k^{2}m^{2}(k^{2}-M^{2})^{2}\biggr], \\
c_{MB}^{(\gamma)} &=&
-\frac{2m}{(d-2)k^{2}(k^{2}-4m^{2})}\biggl[(M_{V}^{2}+(d-3)k^{2})((2m^{2}
-M^{2})k^{2}+2m^{2}(M^{2}-M_{V}^{2}))\biggr], \\
c_{MV}^{(\gamma)} &=& \frac{2m}{(d-2)(k^{2}-4m^{2})}
\biggl[(M_{V}^{2}+(d-3)k^{2})(k^{2}-M^{2}-M_{V}^{2})\biggr],\end{aligned}$$ and the coefficients $c^{(k)}$ are given by $$\begin{aligned}
c_{MBV}^{(k)}
&=& \frac{2m^{2}}{(d-2)k^{4}(k^{2}-4m^{2})}\biggl[((d-2)k^{2}
-4(d-1)m^{2})M_{V}^{6} \\
&-& 2((d-2)k^{4}+(dM^{2}-2(d+1)m^{2})k^{2}-4(d-1)m^{2}M^{2})M_{V}^{4}\\
&+& ((d-2)k^{6}+4(d-5)m^{2}k^{4}-2(d-4)M^{2}k^{4}+((d-2)k^{2}
-4(d-1)m^{2})M^{4})M_{V}^{2}\\ &-& 4(d-3)k^{2}m^{2}(k^{2}-M^{2})^{2}\biggr], \\
c_{MB}^{(k)} &=&
\frac{1}{(d-2)k^{4}(k^{2}-4m^{2})}\biggl[(4(d-3)m^{2}(2m^{2}-M^{2})
-(d-2)(2m^{2}+M^{2})M_{V}^{2})k^{4} \\
&+&
2m^{2}((d-2)M_{V}^{4}+(d-4)M^{2}M_{V}^{2}+4m^{2}((d-3)M^{2}+2M_{V}^{2}))k^{2}
\\
&+& 8(d-1)m^{4}(M^{2}-M_{V}^{2})M_{V}^{2}\biggr],\\
c_{MV}^{(k)}
&=& -\frac{4m^{2}}{(d-2)k^{2}(k^{2}-4m^{2})}\biggl[(M_{V}^{2}
+(d-3)k^{2})(k^{2}-M^{2}-M_{V}^{2})\biggr], \\
c_{M}^{(k)} &=& \frac{M_{V}^{2}}{k^{2}}.\end{aligned}$$ Here we used the abbreviations $k^{4}\equiv (k^{2})^{2}$ and $k^{6}\equiv (k^{2})^{3}$. In view of the denominators of the coefficients $c^{(\gamma,k)}$, which contain powers of $k^{2}$, it is advantageous to expand the scalar loop integrals in the small variable $\beta
= \gamma\tilde\beta$ first. From eqs. (\[eq:IMVIR\],\[eq:IMBVIR\]), we find $$I_{MV}^{IR} = \frac{I_{M}}{m^{2}(\alpha-\gamma)}
+\beta\biggl(\frac{(d\gamma-(d-4)\alpha)I_{M}}{m^{2}d(\alpha-\gamma)^{3}}\biggr)+\ldots$$ and $$\begin{aligned}
I_{MBV}^{IR}
&=& \frac{I_{MB}^{IR}}{m^{2}(\alpha-\gamma)}\\
&+& \beta\biggl(\frac{((d-1)(\gamma-\alpha)(\alpha-2)
-2(4\alpha-\alpha^{2}))m^{2}I_{MB}^{IR}+((d-3)\alpha
-(d-1)\gamma)I_{M}}{2(d-1)m^{4}(\gamma-\alpha)^{3}}\biggr)\\
&+& O(\beta^{2}).\end{aligned}$$ Inserting the $\beta$-expansions of the scalar loop integrals in the expressions for $I_{1+2}^{(\gamma,k)}$ from eqs. (\[eq:I12gdecomp\],\[eq:I12kdecomp\]), one observes that the poles in the variable $\beta\sim k^{2}$ cancel. In the final step, we must insert the expansion of the scalar loop integral $I_{MB}^{IR}$ in the second small variable $\alpha\sim M^{2}$, which can directly be read off from eq. (\[eq:IMBIR\]). The expression for $I_{M}$ is given in eq. (\[eq:IM\]). Doing this, taking the limit $d\rightarrow 4$ and comparing to the decomposition of the matrix element in eq. (\[eq:AxMatEl\]), one finds the following $O(q^{3})$-contribution of $I_{1}$ and $I_{2}$ to the axial form factor $G_{A}$: $$\label{eq:resultGA}
G_{A}^{1+2} = -\frac{g_{1}}{3\pi}M^{3} + O(q^{4})
= -\frac{2\sqrt{2}g_{A}G_{V}R_{V}}{3\pi M_{V}^{2}F^{2}}M^{3} + O(q^{4}).$$ There are no terms of lower order. This is in accord with the power counting for the two graphs, which predicts a chiral order of $q^{3}$ for this contribution to $G_{A}$. In order to compare this with the $q^{3}$-terms in $G_{A}$ worked out in [@Schweizer], one can proceed as follows. Looking at fig. \[fig:triangles\], and imagining the vector meson lines shrinking to a point vertex (corresponding to a limit where the mass $M_{V}$ tends to infinity, with $G_{V}R_{V}/M_{V}$ fixed), it is intuitively clear that the result corresponds to a pion-nucleon loop graph with an $O(q^{2})$ contact term replacing the vector meson line. Such contributions are parametrized by the two LECs $c_{3}$ and $c_{4}$ in [@Schweizer], see eq. (\[eq:GaSchweizer\]). In fact, identifying $$\label{eq:c4}
\frac{2\sqrt{2}G_{V}R_{V}}{M_{V}^{2}} = -c_{4},$$ one reproduces exactly the corresponding terms in the representation based on the pure pion-nucleon theory, cf. eq. (\[eq:GaSchweizer\]). That eq. (\[eq:c4\]) is a good guess can be seen like that: Comparing the $\rho N$-coupling used here, namely, the term proportional to $R_{V}$ in eq. (\[eq:LWN\]), to a more conventional one using a vector field representation for the rho field, $$\mathcal{L}_{NV} = \frac{1}{2}g_{\rho NN}\bar\psi\biggl(\gamma^{\mu}
\mathbf{\rho_{\mu}\cdot\tau}-\frac{\kappa_{\rho}}{2m}
\sigma^{\mu\nu}\partial_{\nu}\mathbf{\rho_{\mu}\cdot\tau}\biggr)\psi,$$ one deduces $$g_{\rho NN}\kappa_{\rho} = -\frac{4\sqrt{2}mR_{V}}{M_{V}}.$$ Using this in eq. (\[eq:c4\]), we get $$c_{4} = \frac{g_{\rho NN}\kappa_{\rho}G_{V}}{2mM_{V}} = \frac{\kappa_{\rho}}{4m}.$$ In the last step, we have assumed a universal rho coupling, $M_{V}G_{V} \equiv F^{2}g_{\rho\pi\pi} = F^{2}g_{\rho NN}$ as well as the KSFR relation $M_{V}^{2}=2F^{2}g_{\rho NN}^{2}$ [@KSFR66] (see also the recent discussion in the framework of effective field theory in ref. [@Djukanovic:2004mm]). This agrees with the rho-contribution to $c_{4}$ found in [@AspectsPiN]. Furthermore, there is no rho contribution to the LEC $c_{3}$ according to this work.
The result of eq. (\[eq:c4\]) is not surprising for itself, but the agreement of our findings with previous resonance saturation analyses demonstrates one very important thing, namely, that the variants of the infrared regularization scheme derived in the previous sections are consistent with the standard case of infrared regularization in the pion-nucleon sector used in [@Schweizer].
As a side remark, we note that the leading order result from the triangle graphs shows no $t-$dependence and therefore gives no contribution to the axial radius. However, the leading contribution to the axial radius can also be related to meson resonances, namely, to a tree-level exchange of an axial-vector meson. The pertinent calculation can be found in [@Schindler], where the axial vector meson-couplings to the pions and nucleons are fitted to experimental data for $G_{A}(t)$ (for an earlier study based on chiral Lagrangians, see [@Gari:1984qs]). Equivalently, it can be parametrized by a certain LEC, named $d_{22}$ in [@Schweizer] (see eq. (\[eq:GaSchweizer\])). As already mentioned at the beginning of sec. \[sec:AxFF\], the difference between the two approaches just amounts to a resummation of higher order terms. Compared to the leading order term, the $t$-dependent part derived from the triangle graphs is suppressed by factors of the small variable $\alpha$, which is a reflection of the fact that the infrared regularized loop integrals preserve the chiral power counting.
Summary {#sec:summary}
=======
In this paper we have presented an extension of the infrared regularization scheme that allows for an inclusion of explicit (vector and axial-vector) meson resonances in the single-nucleon sector of ChPT. For the processes we have considered here, the meson resonances do not appear as external particles, and the corresponding power counting rules for the internal resonance lines are set up such that the resonance four-momentum is considered to be small compared to its mass. The infrared regularization scheme extracts the part of the one-loop graphs to which this power counting scheme applies (for any value of the dimension parameter $d$ used in dimensional regularization), while the remaining parts of the loop graphs will in general violate the power counting requirements, but can be absorbed in a renormalization of the local terms of the effective Lagrangian.
After a short review of the infrared regularization procedure used for the pion-nucleon and the vector meson-pion system in sec. \[sec:PiN\] and \[sec:PiV\], respectively, we have combined the analyses of these sections in sec. \[sec:PiNV\]. There, we consider the simplest example of a Feynman graph where nucleons, pions as well as (vector) meson resonances show up. It is shown how to extract the infrared singular part of such a graph, and the power counting requirements are verified. It should be clear from this example how the infrared singular parts of more complicated one-loop graphs (with more nucleon, pion and resonance lines) can be worked out (some remarks on the general case can be found at the beginning of sec. \[sec:PiNV\], and in sec. 6 of [@BL]). Finally, in sec. \[sec:AxFF\], we have applied the extended scheme to compute a vector meson induced loop contribution to the axial form factor of the nucleon, and demonstrate that the result agrees with the result for $G_{A} (t)$ given by Schweizer [@Schweizer] in combination with the resonance saturation analysis for the pion-nucleon LECs in [@AspectsPiN]. There are, of course, many other possibilities for applications of the scheme developed here. Finally, we add the remark that a suggestion for an extension of infrared regularization to the multiloop case was made in ref. [@LP].
Explicit expression for [$I_{MV}^{IR}$]{} {#app:closedform}
=========================================
In eq. (8.4) of ref. [@BM04], a closed expression for $I_{MV}^{IR}$ for $d\rightarrow 4$ was given that looks much simpler than our result, eq. (\[eq:IMVIR\]), where there is still an infinite sum to be performed. On the other hand, it is quite tedious to work out the chiral expansion of the result of [@BM04], due to the rather complicated expressions $$\label{eq:x1x2}
x_{1,2} = \frac{1}{2\tilde\beta}\biggl(\tilde\beta+\tilde\alpha
-1 \pm \sqrt{(\tilde\beta+\tilde\alpha -1)^{2}-4\tilde\alpha\tilde\beta}\biggr)$$ used there. These two expressions are nothing but the zeroes of the Feynman parameter integral encountered in the computation of the loop integral. The corresponding chiral expansions start with $$\begin{aligned}
x_{1} &=& -\tilde\alpha + \ldots , \\
x_{2}^{-1} &=& -\tilde\beta + \ldots ,\end{aligned}$$ showing that $x_{1}$ is small and negative, while $x_{2}$ tends to minus infinity for $\tilde\beta\rightarrow 0_{+}$. In order to show the equivalence of the two results for $I_{MV}^{IR}$, we employ the following relations, $$\begin{aligned}
\tilde\beta(x_{1}+x_{2}) &=& \tilde\beta+\tilde\alpha-1, \label{eq:sumx1x2} \\
\tilde\beta x_{1}x_{2} &=& \tilde\alpha , \label{eq:prodx1x2}\end{aligned}$$ to write $$\frac{(\tilde\alpha\tilde\beta)^{k}}{(1-\tilde\alpha-\tilde\beta)^{2k+1}}
= -\frac{1}{\tilde\beta
x_{2}}\frac{(\frac{x_{1}}{x_{2}})^{k}}{(1+\frac{x_{1}}{x_{2}})^{2k+1}}=
-\frac{1}{\tilde\beta x_{2}}\sum_{m=0}^{\infty}\frac{(-1)^{m}
\Gamma(2k+m+1)}{m!\Gamma(2k+1)}\biggl(\frac{x_{1}}{x_{2}}\biggr)^{k+m}.$$ Inserting this in eq. (\[eq:IMVIR\]) yields $$\begin{aligned}
\label{eq:interstep}
I_{MV}^{IR}(q^{2})
&=& \frac{M_{V}^{d-4}(\tilde\alpha)^{\frac{d}{2}-1}}{(4\pi)^{\frac{d}{2}}
\tilde\beta x_{2}}\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\frac{\Gamma(2k+m+1)
\Gamma(1-\frac{d}{2}-k)\Gamma(2k+1)}{\Gamma(2k+1)\Gamma(k+1)\Gamma(m+1)}
\biggl(-\frac{x_{1}}{x_{2}}\biggr)^{k+m} \nonumber \\
&=&
\frac{M_{V}^{d-4}(\tilde\alpha)^{\frac{d}{2}-2}x_{1}}{(4\pi)^{\frac{d}{2}}}
\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\frac{\Gamma(2k+m+1)\Gamma(1
-\frac{d}{2}-k)}{\Gamma(k+1)\Gamma(m+1)}\biggl(-\frac{x_{1}}{x_{2}}\biggr)^{k+m}. \end{aligned}$$ In the second line we made use of eq. (\[eq:prodx1x2\]). Now we change the summation indices according to $$j = k+m, \qquad l = k = j-m,$$ and use the following sum formula for Gamma functions, $$\label{eq:GammaSum}
\sum_{l=0}^{j} \frac{\Gamma(j+l+1)\Gamma(x-l)}{\Gamma(j-l+1)\Gamma(l+1)}
= (-1)^{j}\frac{\Gamma(x-j)\Gamma(-x)}{\Gamma(-x-j)}, \qquad j\in\mathbf{N},$$ for $x=1-\frac{d}{2}$. We shall give a short outline of a proof for eq. (\[eq:GammaSum\]): Dividing this equation by $\Gamma(x-j)$, both sides are just polynomials in $x$ of degree $j$, with coefficient $1$ in front of $x^{j}$. Consequently, one only has to prove that both polynomials have the same set of zeroes, namely $\lbrace -1,-2,\ldots,-j\rbrace $. This is not difficult, making use of $$\begin{aligned}
\sum_{l=i}^{j} \frac{(-1)^{j-l}\Gamma(j-i+1)}{\Gamma(j-l+1)
\Gamma(l+1)}\prod_{p=0}^{i-1}(l-p) &=&
\sum_{l=i}^{j}\frac{(-1)^{j-l}\Gamma(j-i+1)}{\Gamma(j-l+1)\Gamma(l-i+1)} \\
= \sum_{n=0}^{j-i}{j-i \choose n}(-1)^{j-i-n} &=& (1-1)^{j-i} = 0 \end{aligned}$$ for $0 < i < j$. These remarks should be sufficient to complete the proof of eq. (\[eq:GammaSum\]).
Returning to eq. (\[eq:interstep\]), we employ eq. (\[eq:GammaSum\]) to write $$\begin{aligned}
\label{eq:resultIMVsoft}
I_{MV}^{IR}(q^{2})
&=&
\frac{M_{V}^{d-4}(\tilde\alpha)^{\frac{d}{2}-2}x_{1}}{(4\pi)^{\frac{d}{2}}}
\sum_{j=0}^{\infty}\sum_{l=0}^{j}\frac{\Gamma(j+l+1)\Gamma(1
-\frac{d}{2}-l)}{\Gamma(j-l+1)\Gamma(l+1)}\biggl(-\frac{x_{1}}{x_{2}}\biggr)^{j}
\nonumber \\
&=&
\frac{M_{V}^{d-4}(\tilde\alpha)^{\frac{d}{2}-2}x_{1}}{(4\pi)^{\frac{d}{2}}}
\sum_{j=0}^{\infty}\frac{\Gamma(\frac{d}{2}-1)\Gamma(1
-\frac{d}{2}-j)}{\Gamma(\frac{d}{2}-1-j)}\biggl(\frac{x_{1}}{x_{2}}\biggr)^{j}. \end{aligned}$$ The last line of eq. (\[eq:resultIMVsoft\]) is exactly the result that was derived in sections 7 and 8 of [@BM04]. This can be seen by substituting $$a\rightarrow -x_{1},\quad b\rightarrow -x_{2}^{-1},\quad
d\rightarrow \frac{d}{2}-2$$ in eq. (7.7) of that reference, and multiplying the result with $$-\frac{\Gamma(2-\frac{d}{2})M_{V}^{d-4}}{(4\pi)^{\frac{d}{2}}}(-\tilde\beta x_{2})^{\frac{d}{2}-2}\quad,$$ as explained at the beginning of sec. 8 of [@BM04]. In the limit $d\rightarrow 4$, the series of eq. (\[eq:resultIMVsoft\]) can be summed up to give $$\label{eq:IMVd4}
I_{MV}^{IR}(q^{2},d\rightarrow 4) = 2x_{1}\lambda-\frac{1}{16\pi^{2}}
\biggl(x_{1}(1-\ln\tilde\alpha)-(x_{1}-x_{2})\ln\biggl(1-\frac{x_{1}}{x_{2}}
\biggr)\biggr),$$ where $$\lambda = \frac{M_{V}^{d-4}}{16\pi^{2}}\biggl(\frac{1}{d-4}-
\frac{1}{2}(\ln(4\pi)-\gamma+1)\biggr).$$ Eq. (\[eq:IMVd4\]) is identical to eq. (8.4) of [@BM04].
Alternative derivation of [$I_{MBV}^{IR}$]{} {#app:TangIR}
=============================================
In this appendix, we present an alternative derivation of the infrared singular part of the loop integral $I_{MBV}(k^{2})$ (see eqs. (\[eq:IMBV\],\[eq:IMBVIR\])) using the prescription of Ellis and Tang we have already explained at the end of sec. \[sec:PiV\]. However, at some places we will also use Feynman parameter integrals, so the derivation outlined here is to some extent a mixture of the standard infrared regularization procedure and the method of Ellis and Tang. In complete analogy to the steps performed at the end of sec. \[sec:PiV\], we start with $$\begin{aligned}
\label{eq:Tangsteps}
I_{MBV}(k^{2}) &\rightarrow&
\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{((p-l)^{2}-m^{2})(l^{2}-M^{2})}
\sum_{j=0}^{\infty}\frac{(-2k\cdot l)^{j}}{(k^{2}+l^{2}-M_{V}^{2})^{j+1}} \nonumber\\
&\rightarrow&
\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{((p-l)^{2}-m^{2})(l^{2}-M^{2})}
\sum_{j=0}^{\infty}\frac{(-2k\cdot l)^{j}}{(k^{2}+M^{2}-M_{V}^{2})^{j+1}} \nonumber \\
&\rightarrow& \sum_{j=0}^{\infty}\int\frac{d^{d}l}{(2\pi)^{d}}
\frac{i(-2k\cdot l)^{j}}{(k^{2}+M^{2}-M_{V}^{2})^{j+1}((p-l)^{2}-m^{2})(l^{2}-M^{2})} .\end{aligned}$$ Now we could use the procedure outlined in [@Tang] to expand the nucleon propagator, together with an interchange of summation and integration. In the present case, however, it is easy to see that it is equivalent to use the common Feynman parameter trick for the remaining loop integrals [*and*]{} extend the parameter integration to infinity like in sec. \[sec:PiN\]. For $I_{MB}$ (see eqs. (\[eq:IMB\]) and (\[eq:intlz\])), the splitting of eq. (\[eq:split\]) corresponds to $$\label{eq:Tangsplit}
\frac{1}{((p-l)^{2}-m^{2})(l^{2}-M^{2})}
= \frac{1}{p^{2}-m^{2}-2l\cdot p+M^{2}}\biggl(\frac{1}{l^{2}-M^{2}}
-\frac{1}{(p-l)^{2}-m^{2}}\biggr).$$ This has also been noted in [@BL], see eqs. (22,23) of that reference. On the other hand, the first term on the r.h.s of eq. (\[eq:Tangsplit\]) is exactly the integrand that gives the soft momentum contribution in the sense of Ellis and Tang, see e.g. eq. (7) in [@Tang]. Therefore, it is consistent to continue the series of steps in eq. (\[eq:Tangsteps\]) with $$\begin{aligned}
\label{eq:nextTangstep}
\ldots &\rightarrow&
\sum_{j=0}^{\infty}\int_{0}^{\infty}\frac{dz}{(k^{2}+M^{2}-M_{V}^{2})^{j+1}}
\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i(-2k\cdot l)^{j}}{[((p-l)^{2}
-m^{2})z+(l^{2}-M^{2})(1-z)]^{2}} \nonumber \\
&\equiv& I_{MBV}^{\mathrm{soft}}(k^{2}).\end{aligned}$$ The loop-integration can be done using the following generalization of eq. (\[eq:supertadpole\]): $$\label{eq:megatadpole}
\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i(k\cdot l)^{2n}}{(l^{2}-M^{2})^{r}}
= (-1)^{r-1}(-k^{2}M^{2})^{n}M^{d-2r}\frac{\Gamma(n+\frac{1}{2})}{
\Gamma(\frac{1}{2})}\frac{\Gamma(r-\frac{d}{2}-n)}{(4\pi)^{\frac{d}{2}}\Gamma(r)}$$ for $r,n \in \mathbf{N}$. This gives $$I_{MVB}^{\mathrm{soft}}(k^{2})
= \sum_{j=0}^{\infty}\sum_{i=0,i\in 2\mathbf{N}}^{j}
\frac{m^{d-6}2^{i}{j\choose
i}(-\beta)^{j-\frac{i}{2}}}{(\gamma-\alpha-\beta)^{j+1}}
\frac{\Gamma(\frac{i+1}{2})\Gamma(2-\frac{d}{2}-\frac{i}{2})}{
\Gamma(\frac{1}{2})(4\pi)^{\frac{d}{2}}}\int_{0}^{\infty}\frac{dzz^{j-i}}{
(z^{2}-\alpha z+\alpha)^{2-\frac{d}{2}-\frac{i}{2}}}.$$ Here we have also used the on-shell kinematics specified in eq. (\[eq:kinppbar\]). The sum over the even integers $i$ extends only to $j-1$ if $j$ is odd. Defining new indices, $$J = j-\frac{i}{2}, \qquad l=\frac{i}{2},$$ and reordering the series correspondingly gives the following expression for $I_{MBV}^{\mathrm{soft}}(k^{2})$: $$\sum_{J=0}^{\infty}\sum_{l=0}^{J}\frac{m^{d-6}(-\beta)^{J}4^{l}
\Gamma(J+l+1)\Gamma(l+\frac{1}{2})\Gamma(2-\frac{d}{2}-l)}{(4\pi)^{\frac{d}{2}}
(\gamma-\alpha-\beta)^{J+l+1}\Gamma(J-l+1)
\Gamma(2l+1)\Gamma(\frac{1}{2})}\int_{0}^{\infty}
\frac{z^{J-l}dz}{(z^{2}-\alpha z+\alpha)^{2-\frac{d}{2}-l}}.$$ Using eq. (\[eq:GammaId2\]), it is straightforward to see that this equals $I_{MBV}^{IR}$ of eq. (\[eq:IMBVIR\]), as expected (the remaining parameter integral can be done with the help of eq. (\[eq:yparint\])).
Loop Integrals {#app:LoopInt}
==============
Here we list the decomposition of the loop integrals with tensor structures in the numerator, which we need in sec. \[sec:AxFF\]. All loop integrals in this appendix are understood as the infrared singular parts of the full loop integrals, but we will suppress the superscript $IR$ for brevity. As a consequence, all loop integrals that do not contain a pion propagator are already dropped here, since they have no infrared singular part. Also, we will use the mass shell condition $p^{2}=m^{2}$ for the nucleon momentum $p$. We start with $$\int\frac{d^{d}l}{(2\pi)^{d}}\frac{il^{\mu}}{((p-l)^{2}-m^{2})(l^{2}-M^{2})}
= p^{\mu}I^{(1)}_{MB},$$ where $$I^{(1)}_{MB} = \frac{1}{2m^{2}}\biggl(M^{2}I_{MB}-I_{M}\biggr).$$ In complete analogy, $$\int\frac{d^{d}l}{(2\pi)^{d}}\frac{il^{\mu}}{((l-k)^{2}-M_{V}^{2})(l^{2}-M^{2})}
= k^{\mu}I^{(1)}_{MV},$$ with $$I^{(1)}_{MV} = \frac{1}{2k^{2}}\biggl((k^{2}+M^{2}-M_{V}^{2})I_{MV}-I_{M}\biggr).$$ Integrals of type $MB$ and $MV$ are also needed with a tensor structure $l^{\mu}l^{\nu}$ in the numerator. They are decomposed as $$\int\frac{d^{d}l}{(2\pi)^{d}}\frac{il^{\mu}l^{\nu}}{((l-k)^{2}-M_{V}^{2})(l^{2}-M^{2})}
= g^{\mu\nu}t^{(0)}_{MV}(k)+\frac{k^{\mu}k^{\nu}}{k^{2}}t^{(1)}_{MV}(k),$$ where the coefficients of the tensor structures are given by $$\begin{aligned}
(d-1)t^{(0)}_{MV}(k) &=&
\frac{4k^{2}M^{2}-(k^{2}+M^{2}-M_{V}^{2})^{2}}{4k^{2}}I_{MV}
+\frac{k^{2}+M^{2}-M_{V}^{2}}{4k^{2}}I_{M}, \\
(d-1)t^{(1)}_{MV}(k) &=&
\frac{d(k^{2}+M^{2}-M_{V}^{2})^{2}-4k^{2}M^{2}}{4k^{2}}I_{MV}-
\frac{d(k^{2}+M^{2}-M_{V}^{2})}{4k^{2}}I_{M}.\end{aligned}$$ The corresponding coefficients in the meson-baryon case, $t_{MB}^{(0,1)}(p)$, can be derived from these results by substituting $k\rightarrow p,
M_{V}\rightarrow m$.
We turn now to loop integrals with three propagators. First the vector integral: $$\int\frac{d^{d}l}{(2\pi)^{d}}\frac{il^{\mu}}{((\bar p-l)^{2}
-m^{2})((l-k)^{2}-M_{V}^{2})(l^{2}-M^{2})}
= (k+\bar p)^{\mu}I_{MBV}^{A}+(k-\bar p)^{\mu}I_{MBV}^{B},$$ with $$\begin{aligned}
I_{MBV}^{A} &=& \frac{1}{2k^{2}(4m^{2}-k^{2})}\biggl[(2m^{2}M^{2}
+(2m^{2}-k^{2})(k^{2}-M_{V}^{2}))I_{MBV} \\ &-& k^{2}I_{MV}-(2m^{2}-k^{2})I_{MB}\biggr], \\
I_{MBV}^{B} &=&
\frac{1}{2k^{2}(4m^{2}-k^{2})}\biggl[(2M^{2}(m^{2}-k^{2})+(k^{2}
-M_{V}^{2})(k^{2}+2m^{2}))I_{MBV} \\ &+&3k^{2}I_{MV}-(k^{2}+2m^{2})I_{MB}\biggr].\end{aligned}$$ We remind the reader that we use $\bar p^{2}=m^{2}=(\bar p-k)^{2}$ here. The scalar loop integral with three propagators occuring in this decomposition was named $\tilde I_{MBV}$ in sec. \[sec:PiNV\], eq. (\[eq:IMBV2\]). However, we noted there that it is equal to $I_{MBV}$ for on-shell nucleon momenta.
The tensor integral $$I_{MBV}^{\mu\nu}
= \int\frac{d^{d}l}{(2\pi)^{d}}\frac{il^{\mu}l^{\nu}}{((\bar p-l)^{2}
-m^{2})((l-k)^{2}-M_{V}^{2})(l^{2}-M^{2})}$$ can be decomposed as $$I_{MBV}^{\mu\nu}= g^{\mu\nu}C_{1} + (\bar p+k)^{\mu}(\bar p+k)^{\nu}C_{2}
+(\bar p-k)^{\mu}(\bar p-k)^{\nu}C_{3}
+((\bar p+k)^{\mu}(k-\bar p)^{\nu}+(\bar p+k)^{\nu}(k-\bar p)^{\mu})C_{4}.$$ Here the coefficients $C_{i}$ are given by $$\begin{aligned}
C_{1} &=&
\frac{1}{d-2}\biggl[M^{2}I_{MBV}-\frac{1}{2}(k^{2}-M_{V}^{2}+2M^{2})I_{MBV}^{A}
+\frac{M_{V}^{2}-k^{2}}{2}I_{MBV}^{B}\biggr], \\
C_{2} &=& \frac{1}{k^{2}(4m^{2}-k^{2})}
\biggl[m^{2}M^{2}I_{MBV}+\frac{M_{V}^{2}-k^{2}}{2}\biggl(m^{2}I_{MBV}^{B}
+(k^{2}-m^{2})I_{MBV}^{A}\biggr) \\
&-& \frac{2m^{2}-k^{2}}{4}I_{MB}^{(1)} - \frac{k^{2}}{4}I_{MV}^{(1)}
- m^{2}(d-1)C_{1}\biggr],\\
C_{3} &=& \frac{1}{k^{2}(4m^{2}-k^{2})}\biggl[M^{2}(m^{2}+2k^{2})I_{MBV}
-\frac{1}{2}(k^{2}-M_{V}^{2}+2M^{2})((m^{2}+2k^{2})I_{MBV}^{A} \\
&+& (k^{2}-m^{2})I_{MBV}^{B})+\frac{3k^{2}}{4}I_{MV}^{(1)}+
\frac{2m^{2}+k^{2}}{4}I_{MB}^{(1)}-(m^{2}+2k^{2})(d-1)C_{1}\biggr], \\
C_{4} &=&
\frac{1}{k^{2}(4m^{2}-k^{2})}\biggl[M^{2}(m^{2}-k^{2})I_{MBV}
+\frac{k^{2}-M_{V}^{2}}{2}\biggl((2k^{2}+m^{2})I_{MBV}^{A}\\
&+& (k^{2}-m^{2})I_{MBV}^{B}\biggr)
+ \frac{3k^{2}}{4}I_{MV}^{(1)}-\frac{2m^{2}+k^{2}}{4}I_{MB}^{(1)}\\
&-& (m^{2}-k^{2})(d-1)C_{1}\biggr].\end{aligned}$$
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---
abstract: 'This paper presents theoretical star formation and chemical enrichment histories for the stellar halo of the Milky Way based on new chemodynamical modeling. The goal of this study is to assess the extent to which metal-poor stars in the halo reflect the star formation conditions that occurred in halo progenitor galaxies at high redshift, before and during the epoch of reionization. Simple prescriptions that translate dark-matter halo mass into baryonic gas budgets and star formation histories yield models that resemble the observed Milky Way halo in its total stellar mass, metallicity distribution, and the luminosity function and chemical enrichment of dwarf satellite galaxies. These model halos in turn allow an exploration of how the populations of interest for probing the epoch of reionization are distributed in physical and phase space, and of how they are related to lower-redshift populations of the same metallicity. The fraction of stars dating from before a particular time or redshift depends strongly on radius within the galaxy, reflecting the “inside-out" growth of cold-dark-matter halos, and on metallicity, reflecting the general trend toward higher metallicity at later times. These results suggest that efforts to discover stars from $z > 6 - 10$ should select for stars with \[Fe/H\] $\lesssim -3$ and favor stars on more tightly bound orbits in the stellar halo, where the majority are from $z > 10$ and $15-40$% are from $z > 15$. The oldest, most metal-poor stars – those most likely to reveal the chemical abundances of the first stars – are most common in the very center of the Galaxy’s halo: they are in the bulge, but not of the bulge. These models have several implications for the larger project of constraining the properties of the first stars and galaxies using data from the local Universe.'
author:
- Jason Tumlinson
bibliography:
- '/Users/tumlinso/astronat/apj/apj-jour.bib'
- 'ms.bib'
title: 'Chemical Evolution in Hierarchical Models of Cosmic Structure II: The Formation of the Milky Way Stellar Halo and the Distribution of the Oldest Stars'
---
Introduction and Motivation
===========================
How and when the first stars and galaxies formed is a central and abiding question in modern astrophysics. The high-redshift frontier has steadily advanced into the end of reionization, $z \sim 6 - 8$, but the truly first stars lie still beyond our view. It is now recognized, on the one hand, that if the first stars are massive and isolated in their own small dark matter halos [@Abel:02:93; @Bromm:02:23] they will be very faint and difficult to detect directly at high redshift. On the other hand, chemical abundances in the most metal-poor stars may preserve a record of early chemical evolution dating all the way back to the first stars [@Freeman:02:487a; @Beers:05:531]. This effort, sometimes termed “Galactic Archaeology”, has the strategic goal of using the most metal-poor stars in the Milky Way and Local Group to address open questions about the first stars and galaxies as a complement to direct study at high redshift.
The goal of this series of papers is to advance the theoretical basis on which these tests can be performed. A large portion of this theoretical foundation consists in demonstrating that the oldest stars in the Milky Way can be used in such a fashion. Stated another way, it would be fatal to the whole enterprise if it turned out that the Milky Way contained no stars from $z > 6$, or from the Epoch of Reionization. As yet we have no purely empirical basis for claiming that it does; all such statements are still necessarily model-dependent. Dark-matter only numerical simulations [@White:00:327; @Scannapieco:06:285; @Diemand:07:859] have already demonstrated that Milky-Way sized DM halos should contain significant structures that originated before $z > 6$ (roughly the current observational frontier). This is well-established and seems necessarily true in the CDM paradigm. Yet while the dark matter holds some interest in its own right, it is still only the dark scaffolding on which the Galaxy’s stellar populations are built. When we consider the stars, a host of other questions arise: How many low-metallicity stars should we expect in the Milky Way? How many of these should date from $z > 6$, if any? What are the most advantageous places to look for these true survivors, and what should we expect to find there? To what extent are these early extremely metal-poor (EMP) stars obscured by later populations of similar metallicity but of less relevance to the first stars? All these questions must be answered to accomplish the overarching goal of “Galactic Archaeology”, but they have no easy answers.
That metal-poor stars[^1] address the first stars has been taken for granted as a working hypothesis by many observational programs [@Cayrel:04:1117; @Christlieb:02:904; @Frebel:05:871]. Yet this claim rests on no firm empirical proof: though some metal-poor stars are known to have ages consistent with 13 Gyr ago, these ages cannot yet be determined precisely or for a large sample. The observational approach rests rather on the somewhat different claim that these stars are related to the first stars by way of the “chemical clock”; they are considered related to the first star formation in their region of space by virtue of their low metallicity, not by their age. Yet not all \[Fe/H\] $\sim -3$ stars are necessarily created equal, and some may arise much later and be chronologically unrelated to the epoch of first star and galaxy formation in which we are interested. The relationship between the two clocks is likely to be a complex one. Further, even if we take the relationship between EMP stars and the first stars at face value, this assumed relationship does not tell us directly how much and what kind of information EMPs contain about the first stars, or how many stars are needed to answer the open questions.
These issues of interpretation reduce to a core problem that must be solved for the larger enterprise to advance. That is: the readily observable quantities in the present-day Galaxy – the luminosity, temperature, chemical abundances, and orbits of individual stars and their trends in stellar populations – are only indirectly related to the properties of stars and gas in early galaxies that preceded the Milky Way, and to the physical processes that governed their formation. For example, nucleosynthetic yields from supernovae enter newly formed stars only after some degree of dispersal and mixing in interstellar gas. The degree of mixing may vary with time and place. For another example, long-lived stars formed in the shallow potential wells of early galaxies – the exact populations we now study in the MW halo – have been stripped from their parent halo and had their present orbits in the Galaxy set by a long and stochastic series of mergers and interactions that cannot be reconstructed exactly, if at all. These two examples are likely essential ingredients in the chain of events leading from the stars we want to understand to the stars we actually observe. There appears to be little alternative to an attempt at a single model that includes these processes in building quantitative links between theoretical ideas about early galaxies and observations in the present day.
The observational frontier is being advanced by large surveys that detect and measure galactic structure and substructure, and by mining these surveys for metal-poor stars and then measuring their detailed abundances in intensive spectroscopic campaigns. As examples of the former, there is now a proliferating number of known stellar streams [@Yanny:00:825] and faint dwarf galaxies (Belokurov et al. 2007) discovered in the Sloan Digital Sky Survey. Using SDSS, [@Carollo:07:1020] confirmed two distinct chemical and kinematic components of the stellar halo near the Sun, using a much larger sample than available before SDSS. As examples of the latter, the HK [@Beers:92:1987] and HES [@Cohen:04:1107] surveys have been thoroughly mined for metal-poor stars, leading to discovery of the most metal-poor stars known [@Christlieb:02:904; @Frebel:05:871], to the discovery of a class of carbon-enhanced metal-poor stars (Lucatello et al. 2005), to increasingly precise measurements of the stellar metallicity distribution [@Schoerck:08:1172], and to intensive studies of the abundances and distribution of the heaviest chemical elements [@Barklem:05:129]. These discoveries are important motivations for this theoretical work. They provide crucial information about the structure of the Milky Way’s halo for which any full synthetic model must ultimately account. Each of these individual studies provides data on a key piece of the overall puzzle; collectively, they suggest that a very large quantity of information may lie in stellar halo populations. Yet observational studies have a strong incentive to carefully select their target populations, so it is difficult to draw general inferences about the behavior of the full Milky Way system without a model that calculates its predictions at a level of detail commensurate with the observations. Given the complexity of the datasets and the large dynamic range between the extremes of the Milky Way on the one hand and abundances in individual stars on the other, building such model presents a significant challenge.
Paper I in this series [@Tumlinson:06:1] made an attempt at such a model in the form of a new stochastic, hierarchical model of Galactic chemical evolution, based purely on the “Extended Press-Schecter” merger-tree formalism [@Somerville:99:1]. That model favored close tracking of chemical abundances over a realistic dynamical model of halo formation, with the goal of constraining the IMF of the first stars by way of their contributions to chemical abundances at low metallicity. That technique worked adequately for the tests performed in that paper, which tracked gas and chemical enrichment budgets with a high level of detail but which could not calculate the spatial or kinematic distributions of model stellar populations. By itself, it yields only a part of a model. Using a similar semi-analytic merger-tree technique, [@Salvadori:07:647] studied the chemical evolution of the halo in response to the first stars, but with the similar limitation that no detailed treatment of the dynamics was included. At the same time, @Bullock:05:931 and their collaborators built dynamical models of halo formation based in non-cosmological N-body simulations and applied simple chemical evolution prescriptions to assess the relationships between kinematics and chemical abundances [@Robertson:05:872; @Font:06:585; @Font:06:886]. Their work has been compared extensively to surveys of the Milky Way halo and has been shown to provide a realistic description of the halo’s basic properties - mass, degree of substructure, and trends in chemical abundances. While each was successful in their own way, none of these studies excelled at understanding the formation of the first stars and galaxies by locating their descendants in the Milky Way. Paper I lacked any treatment of dynamics, and the @Bullock:05:931 halo models did not simulate the Milky Way in a cosmological “live halo” and did not follow the merger histories of the dwarf galaxies or – most importantly – the formation history of the Milky Way host. Because of these limitations neither the Paper I nor the @Bullock:05:931 models were well-suited for studying the present-day properties and distribution of stellar populations formed at very high redshift.
The need for a strong synthetic and flexible Milky Way model and the limitations of the previous efforts inspire the present study, which attempts to retain the strengths of the previous work while rectifying some of its major weaknesses. With respect to Paper I, the present paper takes a step backwards to reassess a number of key assumptions that underlie the basic approach to probing the early Universe with local signatures; the spirit of the new approach is to develop a workable model with a minimum degree of complexity. These reassessments are possible because the present paper deals with a much more sophisticated realization of the Galaxy’s dark matter halo, and they are necessary because the earlier study made a number of simplifying assumptions that are now testable in detail. The goals of the present study are:
- To assess the fundamental basis for the “Galactic Archaeology” effort [@Beers:05:531] by demonstrating that the Milky Way should contain stellar populations that date from the Dark Ages and the Epoch of Reionization, by extending previous DM-only modeling efforts to account for how the Galaxy’s progenitors acquire and process their gas and build up their chemical abundances.
- To identify the most important global and local physical influences on the chemodynamical assembly of the Milky Way’s stellar halo, to assess the sensitivity of the resulting stellar populations to these influences, and to help guide future in-depth studies.
- To lay out a framework for the calculation of key observational quantities related to Galactic chemical evolution, in absolute physical units where possible, and their sensitivity to the key influences; for example, how does cosmic reionization affect the radial density profile of stellar populations in the halo?
- To suggest methods by which stellar populations that originate at high redshift can be isolated from later populations with which they may be easily confused in random samples; for example, when looking at any random star with \[Fe/H\] $< -2$, what is the chance that it reflects conditions at $z > 6$?
It is with all these ideas in mind that this paper is organized as follows: Section 2 describes the N-body simulations of Milky-Way-like DM halos and § \[section-results1\] describes the mass assembly history of the inner portions of the MW halo, considering dark matter only. Section \[section-methods2\] describes the chemical evolution models that run within the dark-matter halo merger trees. Section \[section-results2\] shows how the models provide a realistic description for the gross properties of the real Milky Way halo, as an observational check on the model construction and parameter choices. Section \[section-results3\] shows the metal abundances of stars from the first MW progenitors and maps out broad trends for their variation within the MW halo at the present time. Section \[section-results4\] discusses the general results of the models and draws some implications for observational studies of metal-poor stars and star formation in the early Universe.
Methods I: N-body Halo Simulations
==================================
Simulation Setup and Initial Conditions
---------------------------------------
The first step in constructing a full chemodynamical model is to simulate the formation of a dark-matter halo that resembles the Milky Way in its important properties. This section describes the construction of the Milky-Way-like halo runs, and their analysis up through the construction of merger trees.
The N-body simulations were run using the Gadget2 code (version 2.0; [@Springel:05:1105]) in parallel mode on a local computing cluster. The initial conditions were created using the GRAFIC1 initial conditions package [@Bertschinger:01:1]. These initial conditions are specified for a WMAP3 cosmology (3-year mean; see Tables 2 and 4 of [@Spergel:07:377]), with matter density $\Omega _m = 0.238$, baryon density $\Omega _b = 0.0416$, vacuum energy density $\Omega _{\Lambda} = 0.762$, Hubble constant $H_0 = 73.2$ km s$^{-1}$ Mpc$^{-1}$, power spectrum slope $n_s = 0.958$, power spectrum normalization $\sigma _8 = 0.761$, and primordial helium abundance $Y_{\rm He} = 0.249$. GRAFIC was fed a WMAP3 power spectrum calculated for these parameters by CMBFAST [@Seljak:96:437]. The output of GRAFIC1 is a set of positions and velocities for cold dark-matter particles that represent a random realization of the density field in a periodic cubic box of comoving size 7.320 $h^{-1}$ Mpc in one dimension (10000 kpc in proper coordinates at $z = 0$). The runs analyzed here began with initial conditions calculated on a grid of 1024 particles in each dimension, which were then binned to obtain initial conditions of size $512^3$ or $256^3$, as desired. At the default $512^3$ resolution, the CDM particles have mass $M_p = 2.64 \times 10^5$ [$M_{\odot}$]{}.
Milky Way Analogue Selection and Refinement
-------------------------------------------
Initial conditions that lead to the formation of a MW-like halo at $z = 0$ were generated as follows. First, 10 random realizations of CDM structure were run with the raw $1024^3$ initial conditions binned to $128^3$ (1 particle for every 512 particles in the raw initial conditions). These runs evolve to $z = 0$ very quickly and are easily searched for MW-like halos for resimulation at the higher resolution. Interesting halos are chosen to have virial masses $M_{200} \approx 1.5 \times 10^{12}$ [$M_{\odot}$]{} and are subject to the additional constraint that they must not experience a major (3:1) merger after approximately $z = 1.5 - 2$. Because there is only one Milky Way to study observationally, the focus of these simulations has been on studying multiple halos at around the same mass, to look for stochastic fluctuation in merger history, rather than to examine the dependence of properties on host halo mass.
From the $128^3$ runs four MW-like halos were obtained to run at $512^3$; their properties are listed in Table \[halo-catalog\]. Snapshots of the particle positions and velocities were generated at 20 Myr intervals before $z = 4$ and 75 Myr intervals from $z = 0 - 4$, for a total of 236 snapshots for each run. Only a portion of each box centered on the halo of interest and typically $4-6$ Mpc on a side, was resimulated at the higher resolution, while the remainder of the box was run at effective $256^3$ to save time. The gravitational smoothing length for all simulations was 100 pc in comoving coordinates.
Halo finding and characterization
---------------------------------
The particles in the simulation snapshots have only six physically independent quantities to be studied: three components of position and three components of velocity. Each particle is assigned additional derived physical quantities: a local gravitational potential and matter density, based on smoothing the discrete particles over a 32-particle kernel using SKID [@Stadel:01], as well as kinetic energy and angular momentum.
With completed N-body runs in hand, the next step is to locate and characterize bound halos and their evolution. This “halo-finding” step is crucial, since the resulting merger trees are only as good as the underlying halo catalogs on which they are based. A combination of halo finding and characterization techniques provides a good compromise between accuracy and flexibility. It is essential to capture the small halos embedded within the larger potential well of the host halo as it forms. The 6-dimensional friends-of-friends (6DFOF) halo finding technique presented by [@Diemand:06:1 Appendix A2] meets both requirements effectively. This algorithm was implemented by the author within the original Washington N-body shop FOF software[^2] by adding a velocity term, $b_v$, to the phase-space “linking length”, so that particles are linked together if their relative positions and velocities satisfy the condition: $$\frac{| \vec{x}_1 - \vec{x}_2 |}{b dx} + \frac{| \vec{v}_2 - \vec{v}_2 |}{b_v} < 1$$ where $b$ is the traditional friends-of-friends spatial linking length, $dx$ is the mean particle spacing, and $b_v$ is the velocity-space linking length added for 6DFOF.
No single pair of values for $b$ and $b_v$ finds all identifiable halos and accurately characterizes their properties at all redshifts. At high redshift where the typical bound halo is small and highly clustered near like halos but substructure is minimal at the resolution of these calculations, the 6DFOF parameters should converge to the plain FOF parameters, $b = 0.2$, so $b_v \rightarrow \infty$. At low redshift, when most of the halos of interest are subhalos of the host, smaller spatial and velocity-space linking lengths are needed to avoid overlinking of substructure with the host and with other substructure. As $z \rightarrow 0$, it turns out that $b = 0.07$ and $b_v = 40$ km s$^{-1}$ lead to good results when compared with manual fits to the velocity profiles of obvious and isolated substructure. This value of $b$ is used until $z = 5$, and then linearly increases to $b = 0.2$ at $z = 12$, while $b_v$ linearly increases as $160(1+z)/3$ from $b_v = 40$ at $z= 0$. These values are pragmatic compromises that are considered valid only because they work well; the halo catalogs are tested against pure friends-of-friends at high $z$ and found to be complete and accurate above the resolution limit, while at low $z$ the maximum circular velocities of subhalos are well characterized (compared with values calculated for isolated subhalos directly from the particle data) even if the outer regions of small halos are not linked with the cores owing to the small linking lengths. The outcome of this process is a catalog for each snapshot output that contains halo position, maximum circular velocity, and a group identification for each particle. Some calculations use halo maximum circular velocity $v_{c}$, while others use halo masses estimated by direct fits to the velocity profiles or, occasionally, an analytic relation between $v_{c}$ and $M_{halo}$. Alternate halo catalogs and merger trees were derived for all snapshots using a pure FOF algorithm (that is, $b=0.2$, $b_v \rightarrow \infty$ for all $z$) for use in tests where accurate masses for large halos are desired and accurate measures of substructure are not.
Merger Tree Construction and Particle Tagging
---------------------------------------------
Merger trees are constructed by linking together halos in successive snapshot outputs according to rules that determine whether an earlier halo in fact enters a later halo. In merger tree parlance, each halo is a “node” in the tree; the earlier halo is known as the “child” halo, while the later halo in time is known as the “parent”. Working from high redshift, the algorithm links together pairs where the candidate child has more than 50% of its particles in the parent. Some fraction of halos becomes unbound by, e.g. gravitational interaction with the much larger host halo. These branches of the tree terminate in a “child” halo with no parent that is marked with a special flag as a disrupted subhalo. These particles that become unbound are typically found in a larger tree node at a later snapshot. This merger tree mechanism makes it easy to study the mass assembly history of the host halo and its substructure. This process also generates a large two-dimensional particle-halo cross reference table that specifies the node in the tree where each particle was located (if it was bound) at each redshift snapshot. This cross-reference table makes it easy to correlate particle properties at $z = 0$ with their history of accretion and merging throughout the merger tree.
Results I: Mass Assembly Histories of the Milky Way Halo and Surviving Subhalos {#section-results1}
===============================================================================
Basics {#tree-explain-section}
------
Dark-matter only simulation of cosmic structure is by now a mature technique that has generated many useful insights into the dynamical assembly of the Galaxy’s dark-matter halo [@Helmi:03:834] and its satellites [@Zentner:03:49; @Kravtsov:04:609]. The reader is referred particularly to the series of papers on the [*Via Lactea*]{} simulations [@Diemand:07:859] and their successors for deep insight into the development of these simulations and their implications for dark matter substructure. The goal of these paper not to advance the frontier of the simulations themselves but to obtain measures of halo assembly that will illuminate the expected distribution of halo stellar populations, particularly those arising before reionization. The first step in such a study is to analyze how particles enter and move through the merger tree. The goal of this effort is to assess how and when the dark halo acquires its mass, and how this behavior might influence the resulting stellar populations. This analysis is illustrated with a merger-tree schematic in Figure \[tree\_example\_plot\].
First, each DM particle is assigned the redshift when it first entered any bound halo according to the group-finding algorithm. This “redshift of first entry”, $z_{entry}$, is used as an approximate measure of when the particle could first be associated with stars. In the example tree, particles $1-6$, $9-14$, and $20-23$ enter new bound halos from the unbound state, and $z_{entry}$ for these groups is 4, 4, and 2, respectively. Particles 7, 8, $15-19$, and $24-25$ enter bound halos that already existed at an earlier redshift. Their $z_{entry}$ is, e.g. 3 for particle 15 and 1 for particles 18 and 19. A useful modification of this is to assign $z_{entry}$ only when the particle has entered the densest or most bound portion of a halo - particles that are loosely bound are easily stripped, and perhaps should not be associated with stars since they do not trace the inner, tightly bound portions of a DM halo where stars form. This modified version of $z_{entry}$ is used here, with the condition that the particle must be among the 10% most bound particles in its halo. This quantity can also be considered proxies for the age of the oldest (metal-poor) stars that could be carried by the particle.
Each particle is also assigned $z_{host}$, the redshift when the particle enters the host halo. The identity of the host halo is determined by working backwards recursively from $z = 0$ and at each redshift choosing the most massive progenitor. The “redshift of host entry”, or $z_{host}$, is always later, and typically much later, than $z_{entry}$ for the same particle. This distinction between $z_{entry}$ and $z_{host}$ reflects the basic hierarchical nature of structure formation, in which small subhalos form and then accrete into larger pieces over time.
A third type of particle behavior in the tree is important for characterizing the phase-space structure of the central regions of the host halo. At late times, $z \lesssim 2$, when the host halo has attained a total mass of $\sim 10^{11}$ [$M_{\odot}$]{}, it rapidly accretes matter in particles that have never previously been in a bound halo. Generally the host halo (as marked in Figure \[tree\_example\_plot\]) is tracked out to the redshift of the last major merger ($z_{LMM}$), and particles that accrete directly into the host at this redshift or later are given a special tag as “accreted particles” that enter through smooth accretion into the host. In the example, particles $16-19$ and $24-25$ are late-accreted particles and are assigned this flag. In many comparisons below, these particles are excluded from phase-space density and structure comparisons because they should not be associated with stars. Of course, any particle that accreted from an unbound state into the host, but was previously bound to and then stripped from a bound halo at an earlier redshift, will not receive this special tag and will be counted as eligible to carry stars.
Mass Assembly History as Expressed by Particles {#mass-section}
-----------------------------------------------
These simple tools can now be used to analyze the mass assembly history of the MW analogue halos. Figure \[zplot-figure\] shows a probability density plot for selected subsets of particles from MW1, using the “pure FOF” (§ 2.3) trees and assigning $z_{entry}$ only when particles that have entered the inner 10% of a bound halo. The color code expresses the relative number of stars in each square bin with black representing the most populated bin in each panel. The upper left panel shows all particles within the virial radius, including those in identifiable subhalos. When all particles are considered, the construction of the halo as expressed by $z_{host}$ extends all the way to the present and many particles that entered bound objects very early enter the host only relatively late, virtually always as the inner, tightly bound portions of an identifiable subhalo. Considered as a whole, the Galactic DM halo grows continuously at all times, up through the present.
When only the inner regions of the halo, within R = 20 kpc of the center of mass, are considered, the picture changes substantially. Now the most populated bins are centered around $z_{entry} \sim 5$, $z_{host} \sim 2-4$. In this panel the effect of large subhalos is plain: they appear as dark horizontal stripes since they build up individually over a range of $z_{entry}$ and enter the host together at a single $z_{host}$. Comparing this panel with the first, it appears that the inner portions of the halo form early with respect to the whole halo.
With a cut on particle binding energy (lower left), the early assembly of the inner halo is even more evident. Now the low-redshift tail of $z_{entry}$ disappears. These are particles that lie within $R \leq 20$ kpc but whose orbital energies place them on trajectories that are either unbound (in a few cases) or which extend far out into the outer regions of the halo and which happen to lie in the 20 kpc sphere at $z = 0$. The overall “center of gravity” in the figure is however, little changed, since the inner 20 kpc sphere is composed mostly of tightly bound particles. This is clearly demonstrated by the lower right panel, which imposes both cuts but is nearly indistinguishable from the panel with the $E_{tot}$ cut alone. This exercise reveals the twofold essence of the hierarchical picture of galaxy formation; first, that large subcomponents of galaxies form independently and merge together later, and second, that the inner, more tightly bound material formed and accreted earlier than outer, more loosely bound material. Simply put, halos form hierarchically and from the inside out.
Figure \[mhist-plot\] examines these results in more depth by showing how $z_{entry}$ and $z_{host}$ relate to binding energy in detail. The right panel is a Lindblad plot of MW1. Three regions of $L_z$-$E_{tot}$ space are extracted and the distribution of $z_{entry}$ and $z_{host}$ are plotted at right. Particles that accrete directly to the host after $z_{LMM}$ are excluded, as are particles that are in identifiable subhalos at $z = 0$. With $z_{entry}$ as the proxy (blue regions), the typical particle that ends up in the lower two regions enters a bound halo at $z \gtrsim 5$, while the less bound material does not enter until significantly later. The open circles with range bars mark the mass-weighted mean and 1$\sigma$ variance for $z_{entry}$ for these three regions. The trend is the same here; more bound particles entered earlier, with half them in bound objects before $z = 6$. The most bound subregion also possesses a longer tail to $z > 10$, which indicates that the very first DM halo antecedents of the Milky Way formed very early indeed.
The distribution of $z_{host}$ shows an even stronger trend with binding energy; it is clear from comparing the black lines in the two lower panels that the most bound portions of the halo formed early, with peaks corresponding to major mergers culminating in the last one at $z = 2.1$, after which there is a steep decline in mass being added on these tightly bound orbits (particles acquired by “smooth accretion" are excluded here). While the middle region receives material from these same high-redshift mergers, it also continues to receive significant material from a series of minor mergers up until $z=0$, including some subhalos that may have their own histories extending back to $z > 5$. Much of this material is in identifiable streams.
From Figures \[zplot-figure\] and \[mhist-plot\] it is clear that although the particles that build the inner regions of the halo join the host at around $z_{LMM}$, they have been in bound subhalos for much longer, $z = 4 - 20$. For region 1, the least bound, the mean redshift of first entry is $\langle z_{entry} \rangle = 3$, while for the most tightly bound subregion 3, $\langle z_{entry} \rangle = 6$ (see range bars in the figure). At high redshift, this generally means formation into a newly formed halo or accretion into an existing one, but at $z = 6$ the host halo (the most massive halo that ends up in the final $z = 0$ halo) is only one of dozens of subhalos that will eventually enter the host. Thus the inner dark-matter halo of the Milky Way is assembled from many pieces that begin their individual histories very early, and likely contain stars that formed over a wide range of redshift back to the earliest epochs. The important implications of these results for stellar populations will be taken up in later sections of the paper (§ \[section-results2\]).
There is one important proviso to these results: because these simulations have finite mass resolution, the redshift of first entry for each particle is strictly a lower limit. That is, a higher-resolution simulation might find that a given parcel of mass entered a bound object at an earlier redshift but that the bound object in question is too small to be resolved by the simulations (which counts only halos with $\geq 32$ particles). However, since $M_p = 2.6 \times 10^5$ [$M_{\odot}$]{} and $32M_p = 8 \times 10^6$ [$M_{\odot}$]{}, all but the smallest progenitor halos at the highest redshifts are already resolved by these simulations. Thus resolution effects should not severely bias these results, and if they do the mean redshifts and oldest stellar ages are pushed to earlier times, and all subregions in $E_{tot}$ will systematically shift together.
One of the chief goals of this work is to assess the extent to which stellar populations in the Milky Way can be used to explore the conditions for galaxy and star formation during the epoch of the first galaxies. The chief conclusion of this investigation into the merger and assembly histories of MW-like dark matter halos is that there is good reason to expect that the MW itself contains stars formed at $z > 6 - 10$, and that [*this conclusion holds independently of how or whether baryonic gas physics are modeled*]{}, provided small halos at high redshift are able to form any stars at all. In summary, this study of MW-like dark-matter halos and their merger tree histories leads to the following conclusions:
- Though the inner region of the halo near the Sun as a structure forms “late” (around the time of the LMM), approximately half of the particles were in place at $z > 6$.
- Halo formation is intrinsically hierarchical, in the sense that pieces form and are then assembled into larger structures. If any aspect of chemical evolution depends on the mass of the halo hosting star formation (i.e. depth of local potential well), a model must account for the full hierarchy of mergers and its development over time.
- Evidently the properties of stellar populations in the inner regions of the halo will depend on details of how baryons behave in small dark-matter halos at $z > 4$. This is of course a topic of interest in itself, with as yet little observational leverage. Since it appears that Milky Way halo stellar populations will be sensitive to dwarf galaxies at high redshift, these populations can be used to answer open questions about star formation in the early Universe more generally. This will be taken up in § 6.
Methods II: Modeling of Gas Processes and Stellar Populations {#section-methods2}
=============================================================
Introduction to the General Approach
------------------------------------
The previous two sections studied the mass assembly history of MW-like halos using dark-matter only simulations. For these models to compare to real observations of Galactic structure it is necessary to translate them somehow into model stellar populations. This is what the Paper I method was designed to do: to implement Galactic chemical evolution calculations in a generic, hierarchical merger tree representation of the Galaxy. With merger trees drawn from N-body simulations, this technique becomes even more powerful, since it is now possible to track the phase-space motions of individual particles and the stellar populations associated with them. The general approach will be to assume that each DM halo found in the simulations, and followed through the merger tree, possesses some gas that has been accreted from the IGM, that this gas forms stars, that these stars return metals and energy to the host halo and to the larger environment, and that future generations of stars form with the now metal-enriched gas. These are all processes that can and have been followed with simple analytic relations as in many past models of chemical evolution. The difference here is that the gas budgets, star formation rates, and feedback processes can be written in terms of, or related back to, the underlying dark-matter scaffolding; while this is technique has a history of application to problems of galaxy formation at large, in cosmological volumes, its application to single galaxies, with full attention paid to detailed chemical evolution, is relatively new [BJ05; Paper I; @Robertson:05:872; @Font:06:585].
When attempting this kind of modeling it is desirable to control and understand the influence of the parameters describing the behavior of the baryons, and to avoid excessively complicated parameterizations. While it is true that the differential equations that control “star formation” and “chemical evolution” in these models describe the underlying physical processes only indirectly, it is not true that the parameterization of our ignorance by establishing these relations then permits any desired solution to arise. Proceeding from the simplest possible relations and testing against observations at each step can identify those quantities, such as gas budget and luminosity, that have close relationships, and those that do not, such as interstellar mixing and reionization. As shown below, even simple descriptions of gas budgets, star formation efficiency, and mass return from small halos provide a workable and informative model of Galactic chemical evolution in the hierarchical context.
Paper I presented a new chemical evolution code that works both within the hierarchical context of galaxy formation and in the stochastic limit of low-metallicity Galactic evolution. In that paper the hierarchical galaxy assembly history was specified by the common technique of halo merger trees [@Somerville:99:1] to decompose the Galaxy into its precursor halos working backward in time. It then calculates the history of star formation and chemical enrichment in these objects working forward in time, keeping track of all individual metal-producing supernovae and assigning metallicity to new star formation stochastically from all prior generations. The second generation of the chemical evolution code offers several improvements over the version used in Paper I. These improvements are described here in detail. The parameter choices adopted for fiducial models appear in Table 2.
Ingredients of Gas and Star Modeling
------------------------------------
### Baryon Assignment {#baryonassignmentsubsection}
Knowing how much gas a galaxy contains is the first step to knowing how many stars it can form, so the assignment of baryonic mass budgets to dark-matter halos is the crucial first step in modeling their chemical evolution. The term “assignment” is used purposely to connote a rule for baryonic accretion, not a rigorous physical model. The most straightforward such rule simply assigns each halo a gas budget $M_{gas}$ such that $M_{gas} / M_{halo} = \Omega _{b} / \Omega _{m}$, which takes the value $0.175$ in the adopted cosmology. Because it does not take into account the effects of internal or external feedback influences on the gas budget, particularly reionization, this prescription is not a very useful one.
Instead, these models adopt a prescription based on that of [@Bullock:05:931] that takes into account heuristically the influence of a photoionizing background from the aggregate star formation in all galaxies. This very simple model assigns a fixed mass fraction of baryons, $f_{bary}$ to all DM halos before reionization, $z_{r}$. After $z_r$, gas accretion and therefore star formation in small halos are suppressed completely below $v_c = 30$ [km s$^{-1}$]{}. Between 30 [km s$^{-1}$]{} and 50 [km s$^{-1}$]{}, the assigned baryon fraction varies linearly from 0 to $f_{bary}$.
This baryon assignment prescription is the minimal one that can be effective at suppressing gas accretion and star formation by low mass halos. It is intended to capture the IGM “filtering mass” (Gnedin 2000) below which halos are too small to retain baryons that have been heated to $T \gtrsim 10^4$ K by global reionization. More recent determinations of this filtering mass (Okamoto et al. 2008) have reduced it somewhat from Gnedin’s original formulation, but the prescription adopted here already permits gas accretion into halos at $M_h \sim 10^9$ [$M_{\odot}$]{} after reionization. The exact minimum mass for baryon accretion should be regarded as uncertain; testing of adopted values for the minimum $v_c$ that can accrete baryons after reionization, down to 15 [km s$^{-1}$]{}, indicates that the results for the assembly of the halo are not too sensitive to this adjustment (see § 5.1).
### Star Formation Efficiency
Stars are formed in discrete “parcels” with a constant efficiency, $\epsilon_*$, such that the mass formed into stars $M_* = \epsilon_* M_{gas} \Delta t$ in time interval $\Delta t$. The star formation efficiency is equivalent to a timescale, $\epsilon _* = 1 / t_*$, on which baryons are converted into stars. The fiducial choice for this parameter is $t_* = 10$ Gyr, or $\epsilon _* = 10^{-10}$ yr$^{-1}$.
### Stellar Initial Mass Function
Paper I devoted significant effort to exploring how the initial mass function of the first stars could be constrained using chemical abundances in EMP stars. IMF variations in the EMP populations themselves are indicated by the relative fraction of carbon-enhanced metal-poor stars (CEMPs) in the halo [@Lucatello:05:833; @Tumlinson:07:1361; @Tumlinson:07:L63]. Refining and strengthening this technique, and applying it to future surveys of EMP stars, is an important goal of this theoretical effort. However, variations in the IMF of the first stars have relatively little effect on the bulk properties of the halo, and IMF variations more generally deserve their own intensive study. Here the IMF is assumed to be invariant at all times and at all metallicities, to simplify the parameter space. IMF variations will be studied carefully once other influences such as gas accretion, star formation, and feedback are better understood in the context of a global model. The invariant IMF adopted here is that of @Kroupa:01:231 [eq. 1 and 2], $dn/dM \propto (m/M_{\odot})^\alpha$, with slope $\alpha = -2.3$ from $0.5 - 140$ [$M_{\odot}$]{} and slope $\alpha = -1.3$ from $0.1 - 0.5$ [$M_{\odot}$]{}.
### Type Ia SNe
To model the chemical evolution of heavy elements for longer than $\sim 100$ Myr it is necessary to include the yields of Type Ia SNe. Type Ia SNe are assumed to arise from thermonuclear explosions triggered by the collapse of a C/O white dwarf precursor that has slowly accreted mass from a binary companion until it exceeds the 1.4 [$M_{\odot}$]{} Chandrasekhar limit. For stars that evolve into white dwarfs as binaries, the SN occurs after a time delay from formation that is roughly equal to the lifetime of the least massive companion, which much evolve off the main sequence before the larger star, now a white dwarf, can accrete its outer envelope, exceed the critical mass, and explode.
This process is captured stochastically as follows. Stars with initial mass $M = 1.5 - 8$ [$M_{\odot}$]{} are considered eligible to eventually yield a Type Ia SN. When stars in this mass range are formed, some fraction of them, $f_{Ia}$, are assigned status as a Type Ia and given a binary companion with mass ratio compared to the pair mass $\mu$. To account for the tendency of the mass ratio to approach 1:1, the binary companion mass is chosen from the probability distribution, $$P(\mu) = 2^{1+\gamma} (1+\gamma) \mu^{\gamma}$$ where $0 \le \mu \leq \frac{1}{2}$ and $\gamma = 2$ [@Greggio:83:217]. The prospective SN Ia is then assigned an explosion time corresponding to the lifetime of the less massive companion.
The chemical evolution results are not sensitive to the choice of $\gamma$, but they do depend on the SN Ia probability normalization, $f_{Ia}$. This parameter is fixed by normalizing to the observed relative rates of Type II and Type Ia SNe for spiral galaxies in the local universe [@Tammann:94:487], such that a fraction $f_{Ia} = 0.015$ of all stars formed with $1.5 - 8$ [$M_{\odot}$]{} eventually experience a Type I SN. This normalization gives a ratio of SN II to Ia of 6 to 1. This parameter also varies with the IMF slope and with the mass limits assumed to lead to a Type II SNe. The adopted choice of $f_{Ia}$ was calculated for a long, steady star formation history going back 10 Gyr, to resemble the SFH of galactic disks, and for the [@Kroupa:01:231] IMF and $10 - 40$ [$M_{\odot}$]{} for Type II SNe.
### Chemical Yields
Typical yields for Type II and Type Ia supernovae are drawn from literature sources. Type II supernovae are assumed to arise from stars of $10 - 40$ [$M_{\odot}$]{}, with mass yields provided by Nomoto (2006, private communication). They represent the bulk yields of core-collapse supernovae with uniform explosion energy $E = 10^{51}$ erg. These models have $M = 0.07 - 0.15$ [$M_{\odot}$]{} Fe per event. For Type Ia supernovae with $1.5 - 8$ [$M_{\odot}$]{} the models adopt the W7 yields of [@Nomoto:97:467] for Fe, with 0.5 [$M_{\odot}$]{} of Fe from each Type Ia SN. Since it is the goal of this study to model the bulk formation of the Galactic halo and to explore kinematic trends, the present models track only bulk metallicity with Fe as the proxy reference element. Relative abundances of other elements and their trends will be studied in a later paper.
### Interstellar gas mixing
Paper I implemented a scheme for stochastic mixing of supernova yields into the interstellar and intergalactic medium. These factors are critical for determining the abundance patterns of individual stars, since the observed abundances are a mixing-weighted average of all prior generations. Since the aims of this paper are focused on studying the global formation and kinematics of the MW halo, and not detailed abundances, these models assume instantaneous mixing within the available gas reservoir. Numerically, the “dilution mass” $M_{dil} = 10^{11}$ [$M_{\odot}$]{}, but in practice is limited to the gas budget of the host halo. The mixing timescale is set to $t_{dil} = 1000$ yr, which is very short compared with the average timestep used in the chemical evolution calculation. These parameters recover the instantaneous mixing limit and are not varied here (they are therefore left out of Table 2).
### Chemical and Kinematic Feedback {#feedbacksubsection}
One possible cause of the observed luminosity-metallicity (L-Z) relation for Local Group dwarf galaxies is supernova-driven mass loss from small DM halos [@Dekel:03:1131]. This process is physically complex and must be included in the models in terms of a highly simplified prescription controlled by one or two adjustable parameters. The present models use a prescription that parallels the scheme developed by [@Robertson:05:872] and used subsequently with good success by [@Font:06:585; @Font:06:886; @Font:08:215] to model the Local Group L-Z relation. Their work specified the mass loss rate in terms of the instantaneous supernova rate, which is directly proportional to the star formation rate for a given IMF. Since the present model tracks star formation events and therefore supernovae individually, it should track mass loss in terms of the number of supernovae per timestep in a way that takes into account the intrinsic time variability in the star formation rate and rate of supernovae from a stochastically sampled IMF. Intuitively, supernova-driven winds should respond to the depth of the host halo’s gravitational potential well, and to the energy imparted to the ejecta by the supernovae. The prescription should also allow for selective loss of gas and metals, following indications that superwinds flowing from nearby galaxies are metal-rich [@Low:99:142]. The total SN energy available to drive the wind is given by $$E_{SNe} = \epsilon _{SN} \sum_{i} N_{SN}^i E_{SN}^i = 10^{51} \epsilon _{SN}\sum_{i} N_{SN}^i E_{51}^i$$ where the sum over index $i$ sums over all supernovae from past timesteps that are just undergoing an explosion in the current timestep, and $\epsilon _{SN}$ is the fraction of total supernova energy that is converted to kinetic energy of the remnant. The kinetic energy in the wind is then given by $$E_{wind} = \frac{1}{2} M_{lost} v_{wind}^2$$ where $M_{lost}$ is the total gas mass lost to unbound winds and $v_{wind}$ the wind velocity. Energy balance requires that $E_{SNe} = E_{wind}$, so that $$10^{51} \epsilon _{SN}\sum_{i} N_{SN}^i E_{51}^i = \frac{1}{2} M_{lost} v_{wind}^2
\label{wind-eq1}$$
For the wind to escape, $v_{wind}$ must exceed the escape velocity of the parent DM halo, where $v_{escape} = 2 v_{c}$ for a halo with maximum circular velocity $v_{c}$. Substituting this condition into Equation \[wind-eq1\] and rearranging: $$M_{lost} = \frac{ 10^{51} \epsilon _{SN} \sum_i N_{SN}^i E_{51}^i }{2 v_{virc}^2 }$$ Normalizing to $v_{circ} = 50$ km s$^{-1}$ and performing additional substitutions: $$M_{lost} \simeq 10^4 \epsilon _{SN}\sum_i N_{SN}^i E_{51}^i \left( \frac {v_{circ}}{50 \,{\rm km\, s}^{-1}} \right)^{-2}
\label{wind-eq2}$$ At each timestep, this mass of gas is becomes unbound and is removed permanently from the gas reservoir. Equation \[wind-eq2\] contains only one free parameter, $\epsilon _{SN}$, which expresses the fraction of the supernova energy that is converted to kinetic energy retained by the wind as it escapes. For instance, if 5% of the total SN energy is converted to kinetic energy, and 3% of this is imparted to the ejected material, then $\epsilon _{SN} = 0.0015$. This approach is well-suited to the stochastic framework of Paper I, since it counts individual supernovae and allows for variations in the number and energy of SNe from timestep to timestep. As desired, this expression includes the total energy available from SNe to drive winds and the depth of the DM gravitational potential that the winds must escape.
The selective loss of metals that should arise when supernovae drive their own ejecta out of the host galaxy is captured by a new parameter $f_{esc}^Z$, which expresses the increased metallicity of the ejected winds with respect to the ambient interstellar medium. At each timestep, a total mass in iron $M_{lost}^{Fe}$ is removed from the gas reservoir of the halo: $$M_{lost}^{Fe} = f_{esc}^Z M_{lost} \frac{M_{ISM}^{Fe}}{M_{gas}}$$ where $M_{ISM}^{Fe}$ is the total mass of iron in the ambient interstellar medium, $M_{gas} \times 10^{\rm [Fe/H]}$. This prescription ensures that, on average, the ejected winds are $f_{esc}^Z$ times more metal-enriched than the ambient interstellar medium. Alternatively, the fraction of metal mass lost from the halo is $f_{esc}^Z$ times higher than the total fraction of gas mass lost. This behavior, and the scaling with halo circular velocity, are consistent with the treatment by [@Robertson:05:872], with two key differences in formulation. First, this prescription is generalized to write the mass loss in terms of the instantaneous number of supernovae per timestep rather than the instantaneous star formation rate. Second, the typical values of $f_{esc}^Z$ are somewhat lower here; for Robertson’s choice of ejection parameters $f_{esc}^Z$ as defined here takes on the value 112 (see their equations $19-21$). As will be shown below, the best fitting value of $f_{esc}^Z$ for these models is somewhat lower, $\sim 50$, because they take into account the full merger tree history of each dark matter halo rather than an average mass assembly history specified by the mass at accretion.
The adopted prescription is certainly not the only prescription of mass and metal loss driven by supernovae that could be adopted. This one is preferred because it takes into account energy balance and its scaling with potential well depth, because it can work with a stochastic supernova rate, and because it is easily written to include selective loss of metals. Its most important feature is that it works; it gives reasonable results with reasonable physical inputs.
### Isochrones and Synthetic Stellar Populations
To compare these model halos to observational data on the real Milky Way and its dwarf satellites, it is necessary to calculate the luminosities and colors of model stellar populations using precalculated isochrones and population synthesis models. Each star formation parcel possesses a metallicity, age (with respect to a fiducial time, usually $z = 0$), and a total initial mass distributed according to the assumed IMF. These three quantities together uniquely specify an isochrone and how it is populated. These models adopt the isochrones of [@Girardi:02:195] and [@Girardi:04:205] for the UBVRIJHK and SDSS $\it ugriz$ systems, respectively, as published on the Padova group website[^3]. Their machine-readable tables provide the absolute magnitudes for each bandpass as a function of initial stellar mass, which are then integrated over the assumed IMF. In this way a total luminosity in each passband is obtained. There is a small adjustment ($\lesssim 0.1$ mag) to correct for the fact that the Padova isochrones terminate at 0.15 [$M_{\odot}$]{} instead of the assumed lower mass limit of 0.1 [$M_{\odot}$]{}. The lowest available metallicity in these isochrones is $Z = 0.0001$, or \[Fe/H\] $= -2.3$. Isochrones with $Z = 0$ were calculated by [@Tumlinson:03:608] and [@Marigo:01:152]; the former did not include ages beyond $10^8$ yr and the latter did not include masses below $0.7$ [$M_{\odot}$]{}, so neither provides the complete coverage of age and metallicity that is required by the present models. Instead, the models adopt the lowest metallicity from the uniform [@Girardi:02:195] set to represent all stellar populations with lower metallicities. This approximation is suitable for the tests performed below, where the luminosity of stellar populations below \[Fe/H\] $=-2.3$ is small compared with those at higher metallicity. With these isochrones it is also possible to derive colors and luminosities for only parts of a stellar population by, e.g. integrating only down the giant branch, which can facilitate comparisons with specific observational results.
### Parcels, Batches, and Particle Tagging
The chemical evolution code tracks gas, stellar mass, and metal abundance budgets through the merger tree. The first step in the calculation is to specify an endpoint or “root halo” for which the complete star formation and chemical enrichment history will be determined. In Figure 3 this is the 25-particle z = 0 halo, which is the host in the example. The endpoint may also be a smaller halo that is disrupted by interaction with the host, in which case its tree terminates at $z > 0$. Next, the highest redshift progenitor halo of the endpoint is determined from the merger tree, and the chemical evolution calculation starts there. The star formation history is calculated using a small number of timesteps, usually 10 $-$ 50 between each redshift snapshot. At each timestep, a star formation “parcel” is created with a single metallically and IMF. The metallicity for the parcel is derived from the present gas metallicity, as described above. Each parcel is a single-age simple stellar population that will be used later to integrate up the star formation history of the complete halo. For example, if the 17-node tree in Figure 3 were computed with $N_{timesteps} = 10$, the resulting number of parcels would be 170. For each node in the tree, there are $N_{timesteps}$ parcels, and when halos merge their lists of parcels (from which supernovae and yields will be drawn to assign metallicity to later parcels) are concatenated. This process continues until the “root” endpoint is reached and all progenitor nodes have been calculated.
Since the final Milky Way halo contains multiple subhalos that have experienced (more or less) independent star formation histories, multiple endpoints or “root halos” must be calculated. More than one endpoint is calculated in “batches” that have a single parameter set, and which collectively express the full star formation and chemical evolution history of the full MW halo.
To explore the spatial, kinematic, and dynamical properties of stellar populations in the Galactic halo it is necessary to assign “stars” to DM particles in the N-body simulation. This “tagging” step proceeds as follows. For each particle in the simulation, a cross-reference table is constructed that contains an index to the node of the tree in which each particle was located at each snapshot output. For the example tree in Figure \[tree\_example\_plot\], this table is an array consisting of 25 rows (particles) and 5 columns (redshifts). In the example particle 1 would have five non-zero entries in its column, while particle 20 would have 3 and 24 only one. To obtain a list of stellar populations associated with each particle, these indices are used to obtain the list of star formation parcels that occurred within the nodes in the tree where that particle resided over time. Each particle is assigned a fraction of that parcel in proportion to its fraction of the mass in each halo node in which it was present. In most cases the particles are screened such that the mass of each star formation parcel is associated with only the inner (densest or most bound) 10% of the particles in each halo node, to approximate the effect of stars forming deep in the potential well of the galaxy and to avoid assigning stars to particles that are only loosely bound to their host. Stars are assigned individually for each parcel, particle, and timestep combination, so complex histories for individual particles are handled correctly. This might occur, for instance, when a particle associated with stars is stripped from a small halo during accretion into a larger halo into an unbound state, only to later accrete into the larger halo. This scheme also makes it easy to study subsets of the stellar populations, say all stars formed before $z = 6$, or with \[Fe/H\] $<-3$, simply by excluding those regions of the cross-reference table. All the later measures of stellar populations are obtained in this way.
Results II: Observational Checks on the Fiducial Model {#section-results2}
======================================================
Stellar Halo Mass and Density Profile
-------------------------------------
The merger-tree-based chemical evolution models, with even simple prescriptions to describe the star formation histories of halo progenitors, give model halos that reproduce the observable properties of the real Milky Way stellar halo and its dwarf satellites to a reasonable degree. Figures \[hmassfig\] to \[mzfig\] show the results of these tests in terms of the total stellar mass, mass density profile, and dwarf satellite properties.
First, the total stellar masses of the model halos agree well with observations and with other theoretical studies. Using star counts for millions of main sequence turnoff (MSTO) stars from SDSS, [@Bell:08:295] found a halo mass of $M_h = (3.7 \pm 1.2) \times 10^8$ [$M_{\odot}$]{} from $1 - 40$ kpc. @Bell:08:295 calculated the halo mass using star counts and an observationally-derived ratio of 4.7 [$M_{\odot}$]{} per MSTO star. This ratio corrects for the fact that only part of the stellar mass function is being observed by estimating the total present-day mass of a stellar population that contains one MSTO star (in this case, with color $0.2 < g-r <0.4$ measured from SDSS photometry). This ratio was determined empirically from observations of the globular cluster Pal 5. This observed ratio, which @Bell:08:295 note may be a lower limit, differs from the ratio 9.5 [$M_{\odot}$]{} per MSTO star that is correct for the IMF assumed here. Therefore the model halos are compared to the @Bell:08:295 result corrected upwards by a ratio 9.5 / 4.7 and find excellent agreement between the model and the data. The total mass of the stellar halo is sensitive to the baryon accretion parameter $f_{bary}$ and to the star formation efficiency parameter $\epsilon _*$. It is satisfying that the fiducial values chosen yield reasonable agreement with the data without any fine-tuning.
Figure \[hmassfig\] also plots the total masses of the model halos ($R \leq 40$ kpc) for three cuts on metallicity. The total mass budgets of stars with low metallicity are reduced more by reionization than stars at higher metallicity. This effect is caused by the suppression of low-metallicity star formation in halos that are too small to accrete gas after reionization. This same effect can be seen as a steepening in the slope of the density profile as $z_r$ increases (Figure \[densfig\]). The significance of this effect for the origin of the inner stellar halo will be discussed further below.
The stellar mass density profile is another useful check from observations. This comparison appears in Figure \[densfig\] for MW1 (top) and MW6 (bottom). The four panels in each row show the stellar mass density profile for three metallicity cuts, and within each panel models for three different redshifts of reionization are shown. For comparison, power laws with slope $-2$, $-3$, and $-4$ are marked with dashed lines. The stellar mass density profiles including stars of all metallicity clearly fall in between slopes of $-2$ and $-3$ with a break apparent at $40-50$ kpc, in agreement with recent observations [@Morrison:00:2254; @Yanny:00:825; @Bell:08:295]. Reasonable stellar masses and similar density profiles were also obtained by [@Bullock:05:931], even though their modeling did not include a “live" halo. Though observed density profiles are not available for the lowest metallicities owning to small numbers of known stars at \[Fe/H\] $\lesssim -3$, their overall normalization is consistent with the global metallicity distribution function (see § 5.3). Alternative density profiles were calculated assuming lower values for the minimum halo circular velocity, $v_c^{min}$, that can accrete baryons after reionization, down to 15 [km s$^{-1}$]{}. These tests indicate that the results for the density / metallicity structure of the halo inside $R \sim 40$ kpc changes very little with lower thresholds; the main progenitor galaxies of the stellar halo are already above $v_c = 20 - 30$ [km s$^{-1}$]{} at reionization. There is a modest gain of order 50% in the number of low-metallcity stars beyond $30-40$ kpc, for models with $v_c^{min} = 15 - 20$ [km s$^{-1}$]{}, which describes low-mass subhalos that form stars predominantly after reionization, if allowed, and are disrupted to form the outer halo. It appears that later tests for the halo within $40$ kpc do not depend on uncertainties in the baryon assignment prescription.
The Milky Way Satellite Luminosity and Metallicity Distributions
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The Milky Way’s dwarf satellite galaxies have emerged as key indicators of the baryonic gas physics that govern star formation within small dark-matter halos. Their importance to the present model lies in their ability to serve as observational checks on the assignment of baryons to small halos and on the star formation histories of small halos under the influence of reionization. Recent discoveries of many faint dwarf galaxies by the Sloan Digital Sky Survey have now provided a measure of the dwarf galaxy luminosity function down to $M_V \sim -3$ [@Tollerud:08:277; @Koposov:08:279]. Their properties are crucial to constructing a realistic model of the Milky Way, since the dwarfs provide information about galaxy formation on mass scales similar to those of the early Milky Way, but of course the true Milky Way progenitors are disrupted and mixed while the dwarf galaxies retain some of their stellar populations. Some of the modeling results in this section are already known to the literature on the subject; they are repeated here to check that the basic model is sound by comparing it to observations and previous theoretical results before moving on to perform novel tests of the origin and distribution of the Galaxy’s oldest, most metal-poor populations.
The central problem posed by the Milky Way’s dwarf population is that it is much smaller than the expected number of subhalos in the host halo of the Milky Way’s total mass; this is the well-known “missing satellites” problem [@Bullock:00:539]. This effect is shown in Figure \[vc\_hist\_plot\], which shows the cumulative distribution of subhalo $v_c$ each in simulated host halo compared to the real Milky Way distribution drawn from [@Simon:07:313]. Note that the halo MW3, with slightly lower $M_{200}$ than the others, appears to better fit the real Milky Way system at $v_c \geq 30$ [km s$^{-1}$]{}. The relations and parameters chosen to model the hierarchical chemical evolution of the host halo should be able to explain the observed properties of the MW dwarf population in terms of their stellar populations; for verification of this model we investigate two relevant properties of the Milky Way dwarf population; the luminosity function (LF) and the luminosity-metallicity (LZ) relation. The LF is sensitive to how baryons are assigned to dark matter halos (as described in § \[baryonassignmentsubsection\]), to how efficiently they form stars, and to the effects to reionization on the assignment and/or removal of baryons from small halos. The LZ relation is sensitive to all these, and also to the prescription for chemical and kinematic feedback (§ \[feedbacksubsection\]).
Models for the Milky Way dwarf galaxy luminosity function appear in Figure \[lumfuncfig\] for three different redshifts of reionization. The observed luminosity function, marked in open circles, was constructed from the observed V-band absolute magnitudes and distances as compiled in Table 1 of [@Tollerud:08:277]. Since the comparison here is to all the dwarf galaxies within the present-day virial radius of the halo, Leo T at 417 kpc is excluded from this comparison. The SDSS-discovered dwarf galaxies are counted 5 times each to account for the fact that SDSS has covered only one-fifth of the sky. Together with the 11 classical dwarfs this weighted total of 50 gives 61 dwarf galaxies brighter than $M_V = -2.7$, with the cumulative total marked with $1\sigma$ errors from counting statistics marked in the figure. The model LFs are corrected for the difficulty of detecting faint dwarfs at large distance by excluding from the model count any dwarf subhalo that lies outside the SDSS completeness radius [@Tollerud:08:277 Equation 2] at its magnitude. Though this simple correction is too sharp to be perfectly realistic [cf. @Koposov:08:279], it adequately captures the basic effect of finite detection limits in the observational surveys.
The fiducial values of $f_{bary} = 0.05$ and $\epsilon _* = 10^{-10}$ provide very good descriptions of the halo mass and density profile, so they are not varied here. To first order, the luminosity of a given dwarf depends linearly on their product, and so the need to have a few dwarf galaxies with $M_v \lesssim -15$ in the model halo to match the real MW means this product cannot be much lower than the fiducial value.
The next most important effect controlling the LFs of dwarf galaxies is the suppression of star formation in small halos by reionization, as illustrated in Figure \[lumfuncfig\] for $z_r = 6.5$, $10.5$, and $14.5$. It is apparent from the figure that the count of detectable dwarf satellite galaxies fainter than $M_V \sim -8$ declines rapidly as reionization moves to earlier redshifts, while brighter dwarfs are only little affected by earlier reionization. The reason for this behavior can be found in an analysis of the dark-matter assembly histories of the subhalos that host dwarf satellites with respect to the thresholds in halo circular velocity that enter the baryon assignment prescription. Recall from § \[baryonassignmentsubsection\] that all halos are permitted to accrete their $f_{bary}$ share of baryons and so form stars prior to $z_{r}$, while only halos with $v_{c} \geq 30$ [km s$^{-1}$]{} can continue to accrete baryons after reionization. These latter halos will be relatively unaffected by the suppression of baryon accretion that reionization imposes, but those halos with $v_{c} \lesssim 30$ [km s$^{-1}$]{} will be limited in the total amount of star formation they can ever have by the time available to accrete gas prior to $z_{reion}$. It is these smaller halos whose total gas budgets are limited by $z_r$ that drop out of the visibility corrected LF for models with earlier reionization. Since the four model halos have different amounts of DM substructure, the best-fitting redshift of reionization varies but typically has $z_r \simeq 8-11$, with $z_r = 10.5$ adopted as the fiducial value for later comparisons. These basic results are in agreement with other recent attempts to explain the luminosity function of MW satellites in semi-analytic models (Koposov et al. 2009; Maccio et al. 2009).
![Luminosity functions for the model dwarf satellite populations of the MW halos for three different redshifts of reionzation, $z_r = 7.5, 10.5$, and 14.5, top to bottom. \[lumfuncfig\]](f8_color){width="3.6in"}
The luminosity-metallicity (LZ) relation for the Milky Way dwarf population is shown in Figure \[mzfig\], compared to the four model halos. The model metallicities are derived by obtaining a model MDF from particle-tagged stellar populations and taking the mean value. A model for MW1 with no SN-driven kinematic feedback (§ \[feedbacksubsection\]) is shown in open circles; all other models have $f_{esc}^Z = 50$, which is used as the fiducial value. As shown in similar modeling efforts [@Dekel:03:1131; @Robertson:05:872], models including selective feedback from small halos provide a good match to the overall trend in the real Milky Way. More information about the formation of the satellites and their role in building the stellar halo is available from theoretical modeling of chemical abundances; see in particular the series of papers by [@Font:06:585; @Font:06:886; @Font:08:215] and [@Johnston:08:936].
The increasing scatter at low \[Fe/H\] is a real effect in the models arising from both the fully hierarchical treatment of structure formation and the stochastic treatment of chemical evolution. First, some of this scatter results from diversity of their merger tree histories, since subhalos that spend more time as small pieces prior to mergers will have lower \[Fe/H\] and $v_c$ at $z = 0$. The baryon accretion prescription also increases the scatter, since some halos have histories that extend longer than others before reionization, and some pop just above the gas accretion threshold after reionization and so can form stars later than those that do not. More scatter is added by the time variation of supernova rates from a stochastically sampled IMF, which inject metals into the gas reservoir and eject metals from the halo with a prescription (§ \[feedbacksubsection\]) that depends on the instantaneous supernova rate and the instantaneous halo mass (through the halo binding energy). All these effects contribute to increasing the scatter in the LZ relation at low luminosity, which appears roughly consistent with scatter in the observed metallicities of the known satellites. However, the data as they stand now are not sufficient to say whether all of these mechanisms acted in the real galaxies or which, if any were dominant.
The Milky Way Halo Metallicity Distribution
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![Model MDFs compared to the HES MDF from [@Schoerck:08:1172]. At top, the observed HES MDF compared with the $R < 20$ kpc MW1 MDF normalized to a total of $N=3439$ stars at all metallicity. In the center panel, the model MDF has been corrected for the average HES visibility function and normalized as before. In the lower panel, the visibility correction is applied as before, but here the model MDFs are normalized to have 1507 stars at \[Fe/H\] $< -2$. In the two lower panels the uncorrected MDFs are marked with dotted lines. Except for the normalization, the dashed (uncorrected) and solid (corrected) curves are the same in all panels. It is apparent that normalizing the model and observed MDFs where the visibility correction is smaller than $\sim 10$ or less leads to reasonable agreement. \[mdffig\] ](f10_color){width="3.5in"}
The Galactic halo metallicity distribution function (MDF) can be used to test the model halos, but it turns out to be a problematic comparison because of biases in the observed MDFs and uncertainties in the model. The MDF comparison is shown in Figure \[mdffig\], where the MDF measured by the Hamburg/ESO survey (HES) of metal-poor stars is indicated in filled squares. Here the model is the particle-tagged MDF for all particles in the host halo within $R = 20$ kpc of the center. In the top panel, the HES MDF is shown with the exact number counts listed in Table 3 of [@Schoerck:08:1172] and error bars from Poisson statistics. Here the model MDF is renormalized to contain the same total number of stars ($N=3439$) as in the HES MDF. With no correction to either data or model, the agreement is very poor; the model MDF is too sharply peaked at \[Fe/H\] $\simeq -1.3$ and has too steep a tail to low metallicity.
The HES was designed to select for EMP stars (\[Fe/H\] $<-3$), so [@Schoerck:08:1172] derive a correction function that accounts for the high degree of incompleteness at \[Fe/H\] $\gtrsim -2$ in the as-observed HES sample. This correction factor must be applied to the model MDF before the model and data can be compared directly. This comparison is performed in the middle panel of Figure \[mdffig\], where the corrected model MDF now appears in the solid line and the uncorrected model appears in the dotted line (this is the same as in the top panel). Here the agreement between model and data is improved by the correction for the bias against stars with \[Fe/H\] $>-3$, but the fit is still unacceptably poor when the model and data are normalized to contain the same total number of stars. Given the very large magnitude of the visibility correction (a factor of 100 at \[Fe/H\] $=-1.7$, increasing at higher metallicity), it is perhaps wise to normalize the data to model in a region where the correction is minimized; that is, where the HES MDF is reliable [*as observed*]{}. This comparison appears in the final panel of the figure, where the agreement in the region where the comparison is meaningful is much improved.
There is still some disagreement remaining at \[Fe/H\] $>-2$, but it is difficult to assess both the theoretical MDF and the observational MDF at these high metallicities. First, the visibility correction for the HES MDF becomes larger than a factor of 100 here, with presumably a large uncertainty in this correction. Second, above \[Fe/H\] $\sim -1.5$ the theoretical MDF becomes sensitive to the presence of star formation in the host halo (the main “trunk” of the tree) at redshifts of $z \sim 3$ and below. The exact star formation history here is uncertain since it involves the formation of the Galactic thick and thin disks, and it is unclear where and how the transition between disk and halo should be managed in the model. Star formation parcels that occur in the host halo after the last major merger are considered to be formed in the disk are are not included in any of the stellar population modeling. If in fact “halo” stars continue to form in the host after this milestone, then the halo MDF can be more populated with high \[Fe/H\] stars. Conversely, a visibility correction that is too large can produce disagreement in the sense seen here. Without exact knowledge of how HES stars are distributed kinematically into the halo or disk, and without a more detailed model of disk formation in the theoretical MDF, no more exact results can be obtained. The agreement seen between the two MDFs at \[Fe/H\] $\lesssim -2$ is, however, encouraging.
Results III: Spatial, Kinematic, and Chemical Properties of Early Halo Stellar Populations {#section-results3}
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Halo Chemical Evolution Histories
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A full halo model includes a self-consistent chemical enrichment history for the subhalos that end up in the host, and the particles that make up that halo. Example chemical enrichment histories appear in Figure \[chemevfig\], which displays two-dimensional histograms of the chemical enrichment histories for six variations on MW1, with three redshifts of reionization with and without feedback. Several features of these distributions deserve comment.
First, the hierarchical nature of halo assembly entails that chemical enrichment is inhomogeneous at a single time; halos of different mass form stars of different metallicity. This basic fact is observed in the Milky Way system today, where the disk forms stars at $\sim Z_{\odot}$ and the SMC forms stars at $0.1Z_{\odot}$. These models are of course constrained to achieve the current dwarf-satellite LZ relation. At some times, the range in metallicities is up to 3 orders of magnitude.
Second, and perhaps more important for the goal of using low-metallicity stars to probe the high-redshift Universe, stars at a given metallicity form over a wide range of redshift. For instance, in the upper right panel where $z_r = 10.5$ and chemical feedback is included, stars with \[Fe/H\] $\sim -2.5$ form from $z = 13 - 5$. In a model where the halo at any early time consists of many smaller pieces, and the typical metallicity evolution is influenced by the mass scale, there must be such a spread in the formation times for stars at a given metallicity. These two trends – a spread in metallicity at a single time and a range of times at a given metallicity – are features that emerge from a fully hierarchical, non-homogeneous treatment of chemical evolution. Taken to their extremes, they would completely eliminate any correlation between metallicity and age and/or redshift; instead, in realistic models these two trends perturb the homogeneous picture where metallicity rises monotonically with time, but there is still a generally upward trend in metallicity over time.
Third, the suppression of baryon accretion and star formation in small halos after reionization affects the formation history of the most metal-poor stars. At any redshift, the relationship imposed between halo mass and chemical feedback (§ 4.2.7) ensures that the lowest metallicity stars are forming in the smallest halos, and these are the halos most affected by reionization. In the models displayed in the lower panels of Figure \[chemevfig\], where $z_r = 6.5$, many redshift-metallicity bins are populated where they are not in models with an earlier reionization. This behavior, if it holds true in the real Galaxy, will have important effects on the use of these low-metallicity stars to probe the first stars. Of course, these star formation and metallicity histories are not readily compared directly to observation, and direct measures of stellar age generally are not effective for more than 10 Gyr before the present. We must therefore seek effective proxies for the age of the earliest populations, which is taken up next.
Isolating and Sampling High-redshift Populations
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For Galactic Archaeology to succeed as a probe of star formation at early times, the metal-poor stars under study must in fact have formed at this early time. Of course, according to the “chemical clock” metal-poor stars are by definition more primitive. But the extended epoch over which, e.g. \[Fe/H\] $\sim -2$ stars form in the fiducial model suggests that stars of low metallicity may form well after reionization. Indeed, this same concern is raised by the discrepancies between the relative chemical abundances of the dwarf galaxies and the MW halo at the same metallicity [@Venn:04:1177], indicating that the two Galactic components had different star formation histories leading to the same metallicity [@Font:06:585]. Thus, metallicity itself is at best an imperfect measure of chronological age in metal-poor stars, and additional evidence that the stars under study in fact probe the earliest times should be sought.
Because the models are based on realistic dynamical simulations, they also express the distribution of formation time for stars of different metallicity from which we can estimate the fraction of stars at each metallicity that date from some high redshift. The interpretation of observational surveys involves two properties of a population: first, the fraction of stars at a given metallicity and within some volume that formed before some redshift $z_{cut}$, and second, how representative is that sample of all the stars in the halo that formed at that redshift. The first quantity, called “purity”, is relevant because we would like to obtain a perfectly “distilled” sample of stars from high redshift, and this quantity measures how well this goal can be achieved (these two terms – “pure” and “distilled” – are used interchangeably here). The second is relevant because once a distilled sample is obtained it is desirable to know if it is a fair representation of all the stars that formed at that metallicity and at that redshift. Generically, one might expect that selecting for a pure sample will tend to reduce its use as a fair sample of its epoch, so that “purity” will correlate inversely with “representativeness”.
Results from these tests are presented in Figure \[fracplot\]. The figure shows purity and representativeness for three different models that differ only in their redshift of reionzation $z_r = 6.5$ (first row), $z_r = 10.5$ (second row), and $z_r = 12.5$ (third row). The first column shows dark matter only; the latter three columns show the results for three different cuts on metallicity, \[Fe/H\] $\leq -2.5, -3$, and $-4$ from left to right. Within each panel, there are three different redshift cuts displayed, $z_{cut} = 6, 10$, and 15 from top to bottom. The five points on each curve represent radial cuts of $R = 5, 10, 30, 50$, and 100 kpc. Several key results of interest emerge from this test. First, it is generically true that the model halos have a significant population of stars from before reionization, and that the fraction from before reionization increases at lower metallicity (that is, the curves shift up from one column to the next). Another notable feature is that the measure of purity increases at small radii - this is a consequence of the “inside out” construction of CDM halos, with inner regions formed from earlier subhalos. This behavior in the dark matter, at left, is preserved in the stars. Next, note that at a given radial cut the representativeness increases for higher $z_{cut}$; this is another effect of the central concentration of stars from the earliest halo progenitors. For example, \[Fe/H\] $< -3$ stars within 5 kpc of the halo center are approximately 50% of all \[Fe/H\] $<-3$ stars from before $z = 15$, while they are only $\sim 15-20$% of all \[Fe/H\] $< -3$ stars from before $z = 10$ (third column). In other words, the chemical clock is gets going earlier in the center of the halo than in the outer regions. Finally, it is notable that these results depend only a little on the redshift of reionization (that is, the curves shift only a little moving down the column as $z_r$ increases, moving more only when $z_{cut} \sim z_r$ ). Thus uncertainty in the redshift of reionization does not substantially undermine the goal of obtaining “distilled” samples of high-redshift stars.
If the earliest, most metal-poor stars in the Milky Way halo lie closer to the center of the halo than the typical metal-poor star, there may be kinematic signatures that could be exploited to further isolate them from confusing populations. Figures \[phasecutfig\] and \[mhistfig\] show how the star formation histories of different halo components correlate with the total binding energy of the particles. Figure \[phasecutfig\] shows that, just as for the dark matter, stellar populations at low metallicity form and assemble from the “inside-out”. This figure shows the distribution of binding energy for halo stellar populations cut at four different metallicities. The cumulative histograms in each panel show the distribution of stellar populations formed before five different redshifts: $z = 3$, 6, 10, 12, and 15. Note that in each panel, the highest-redshift cuts provide a larger fraction of stars on tightly bound orbits (low $E_{tot}$), and that the more loosely bound regions of the halo are filled in later. This is a generic consequence of the “inside-out” growth of dark matter halos, and the tendency of the earliest bound material to lie in the center of large halos at late times. Note also that the lowest-metallicity stars, \[Fe/H\] $<-3$, in the lower two panels, have all formed by $z = 10$, so that later curves lie on top of one another. Only stars at \[Fe/H\] $\gtrsim -2.5$ continue to form after $z \sim 10$.
Figure \[mhistfig\] shows how the star formation histories associated with particles vary with the binding energy of those particles, as a function of redshift and metallicity. The star formation histories for the least bound particles ($E_{tot} \sim -4\times 10^{4}$) begin in small halos that form late compared with the earliest progenitors of the host, so their most metal-poor stars form at $z \sim 10-13$ and their histories end at reionization. Stars on more bound orbits ($E_{tot} \sim -8\times 10^{4}$) extend back to $z \sim 20$ and continue to form in significant numbers at \[Fe/H\] $\sim -2$ after $z = 5$. The most bound particles ($E_{tot} \sim -1.3\times 10^{5}$) carry stars formed at $z > 20$ and have very few \[Fe/H\] $< -2$ stars formed at $z < 5$, when the host is forming stars of much higher metallicity (see \[chemevfig\]). This is caused by the fact that most of these particles have been in the host or its closest merger partners since the very beginning, they formed their metal-poor stars very early, and have not had a merger that brought in metal-poor stars on such tightly bound orbits. Note that the lowest selected region in Figure \[mhistfig\] contains almost ten times as many stars from $z \sim 20$ as the middle region. From these tests it is evident that simple model descriptions of the MW halo imply that significant populations of stars remain from the epoch of reionization, and that observational samples designed to select stars within the inner few kpc of the Galaxy and at low metallicity will have the best chance of finding the true survivors of reionization.
This test suggests that searches for metal-poor stars on the most tightly bound orbits that can be accessed with a given observing strategy will be able to find preferentially older metal-poor stars. Modern observational surveys can already kinematically select more bound populations using radial velocities, proper motions, and a model of the Galactic potential [@Carollo:07:450; @Morrison:09:694]. A simple application of these results would use such a kinematically selected sample to study relative chemical abundances such as \[$\alpha$/Fe\] and $r$ and $s$-process abundances as a function of orbital binding energy for each star. To obtain useful leverage, it would be helpful to study populations within all three binding energy regions marked in Figure \[mhistfig\], where the least bound region contains no stars from $z > 14$ and the most bound region includes no stars from $z < 3$. Of course, since all three regions contain large, metal-poor populations from $z \simeq 4 - 10$, any variation will likely occur in the tails of the distribution of abundances and it may be necessary to obtain significant samples of stars in all three regions to detect any differences in relative chemical abundances with binding energy.
Results IV: Implications and Applications for the Study of Metal-poor Stars {#section-results4}
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The foregoing sections have described the detailed results of the halo-building models. This section summarizes the key conclusions of this study, and then draws implications for the further study of metal-poor halo stars and their use as probes of star formation in the early universe.
- A simple model that takes into account only (a) baryon accretion, (b) star formation, (c) selective loss of metals from small halos, and (d) the effects of reionization on the IGM, can give an effective description of the gross properties of the Milky Way stellar halo: the stellar mass, density profile, metallicity distribution, and the luminosity function and luminosity-metallicity relation of the dwarf satellites (Figures 5 to 10).
- When the formation histories of Milky Way halo progenitor galaxies are summed together, two important trends emerge. First, stars at a given metallicity form over a long period, and second, at a given time there is a wide range of metallicity (Figure \[chemevfig\]). Thus the “time clock” and “chemical clock” are decoupled even over relatively small regions in the Milky Way halo, and detailed modeling is necessary to relate one to the other at any point in time or metallicity.
- Because of the inside-out construction of dark matter halos, stars at a given metallicity are older near the center of the halo when compared with stars at larger radii or on more loosely bound orbits. These inner regions are therefore more pure distillations of high-redshift populations, but are conversely less representative of all populations at that metallicity and redshift (Figure \[fracplot\]). Because reionization suppresses small, presumably low-metallicity halos at late times, large fractions of all \[Fe/H\] $\lesssim -2$ stars are from $ z> 6$, while $20-40\%$ of all stars of \[Fe/H\] $=-4$ to $-3$ formed at $z \gtrsim 10-15$.
- These last two effects combine for the trend seen in Figures \[phasecutfig\] and \[mhistfig\]: as binding energy with respect to the Galaxy’s potential well increases (as stars become more tightly bound), the mean redshift of star formation and the earliest redshift of star formation both shift to earlier times. For example, the fraction of $\lesssim -2$ stars from $z > 15$ increases. Weakly bound halo populations ($R > 40$ kpc) may have very few if any stars from $z \gtrsim 12$ (except in subhalos). Because of general chemical evolution and suppression of late, low-metallicity star formation by reionization this preference for high-redshift stars on tightly bound orbits is stronger in the stars than in the dark-matter alone.
- Because old, metal-poor stars prefer tightly bound orbits but exist throughout the halo (that is, populate a large region of phase space even if they prefer one part of it), there is a natural tension between obtaining a [*distilled*]{} sample of high-$z$ stars and a [*representative*]{} (however defined) sample of high-$z$ stars.
With the knowledge that old stellar populations are preferentially concentrated in the inner regions of the Galactic halo (Figures \[fracplot\], \[phasecutfig\], and \[mhistfig\]), it is now possible to calculate how these populations should be distributed on the sky as a first step to evaluating the ability of present or future observational campaigns to detect the chemical abundance signatures of the first stars. Figure \[skymapfig\] shows the variation of the “purity” of a sample of stars at low metallicity on the sky, \[Fe/H\] $< -3$ and $z \geq 15$. Because these high-redshift, low-metallicity stars are represented in the simulation by a relatively small number of particles, it is necessary to coadd the four model halos and average the probabilities for the resulting composite over relatively large areas of the sky to reduce numerical noise and artificial clustering. Nevertheless, it is clear that the halo populations with the highest fraction of high-redshift stars – those most likely to carry the chemical abundance signatures of the first stars – cluster near the Galactic center. These low-metallicity stars reside “in the bulge”, though they are not “of the bulge”. Near the Galactic center, EMP stars from $z > 15$ account for $10 - 15$% of the population, while far away from the center toward the anticenter and poles they are typically less than $2-5$% of the population.
This test suggests that observational surveys may be able to separate the oldest EMPs from later populations, if they can efficiently select and study EMP stars in the crowded regions within $10-30^{\circ}$ of the Galactic center (5 kpc subtends an angular region 30$^{\circ}$ at 8.5 kpc). While the the HK, Hamburg-ESO, and SDSS surveys have shown the EMPs exist all over the sky, not all EMPs are created equal - those near the Galactic center are generically older than those in the outer regions of the galaxy. And while “low metallicity” is equivalent to “first” on the chemical abundance clock, it is not necessarily so on the chronological clock. Since EMP stars do not otherwise reveal their time of formation, it is important to find some proxy for formation time that will enable correlations of chemical abundance signatures with time. With sufficiently large samples, it should be possible to use this variation on the sky as leverage to isolate the chemical abundances of the first stars in a differential comparison - any differences in the abundances of EMPs stars in the two regions could be caused by earlier time of formation for stars in the innermost Galactic halo.
Stellar abundance surveys to date have tended to avoid the innermost Galaxy, where observations are hindered by the density of stars on the sky and high visual extinction. The HK, HES, and SDSS surveys all focused on high Galactic latitude for these practical reasons. Yet the stellar halo of the Milky Way extends all the way to the center of the potential well, and as shown above the oldest stars concentrate near the very center. While even surveys that avoid the bulge entirely should contain some small percentage of EMP stars from very high redshift, using the leverage provided by the “inside-out” growth of the halo will require obtaining an EMP sample that attempts to maximize the fraction of the very oldest EMP stars. The near-infrared APOGEE survey planned as part of SDSS-III[^4] will be the first major effort to obtain a large set of stellar abundances near the center of the Galaxy, and as such it has a chance of detecting a difference in the abundance patterns of the innermost halo stars and those seen in SDSS/SEGUE, which avoids the bulge.
What sorts of questions might be addressed when observational surveys can probe the innermost, oldest stars in the Galaxy? One such question is motivated by recent studies of the carbon-enhanced metal-poor stars. The frequency of these CEMP stars has been linked to the IMF in the binary mass-transfer model for their abundances [@Lucatello:05:833; @Tumlinson:07:1361]. If the early IMF is limited by the cosmic microwave background, as suggested by @Tumlinson:07:L63, it should have a strict time-dependence driven by the 1+$z$ dependence of the CMB temperature. Thus the CEMP frequency should increase in regions of the Galaxy containing the oldest stars and decrease where the mean stellar age at any metallicity is lower. Given the results above, CEMP frequency should increase for stars with lower binding energy and for stars near the Galactic center. Such a change for stars with the same metallicity could provide key evidence that the CEMP phenomenon is linked to the CMB and the IMF. Similar arguments can be made for the chemical clock provided by the abundances of $r$-process elements in metal-poor stars [@Barklem:05:129]. As [\[Fe/H\]]{} increases, the scatter in r-process abundances appears to decline as the star forming material in the early galaxy became more enriched in r-process elements, and more homogeneously mixed. If the more tightly bound inner halo stars formed earlier, they might be expected to show larger scatter in the r-process abundances than less tightly bound, and so later forming, stars at the same [\[Fe/H\]]{}.
These considerations suggest that a new frontier is opening in the study of metal-poor stars and their use as probes of star formation near the end of the cosmological Dark Ages. Very large samples of EMP stars can now be obtained by surveying large areas of sky. Metallicities can be estimated for large fractions of these samples, but high-precision abundances can be measured for only some smaller subset. To answer the questions of interest, including those about the first stars, these samples must be selected carefully, with knowledge of how the selection relates to the underlying question and how it introduces biases. For judicious selection, it is important to have a full synthetic model such as the one attempted here.
This study has advanced but not fully met all the enumerated goals laid out in Section 1. It is clear that simple halo building models can successfully describe the bulk properties of the Milky Way stellar halo without excessively complex parameterizations (Goal 1). These “vanilla” halo models possess significant populations of stars dating from prior to reionization and allow the chemical abundances and spatial distribution of these stars to be quantified (Goal 3). Reionization and chemical feedback have been identified as key physical influences on the resulting ancient stellar populations (Goal 2), but other physical influences such as internal mixing within galaxies and local ionization effects have been left for future work. These models show that stellar populations can be “distilled” to prefer stars dating from reionization by selecting for low metallicity (\[Fe/H\] $\lesssim -2.5$) and for stars on tightly bound orbits in the center of the halo; these selection criteria have also been quantified (Goal 4). Yet much work remains to be done to refine and extend the framework on which observational tests in “Galactic Archaeology” will be performed. Many important physical ingredients can be studied with departures from the vanilla model; two important examples are non-local metal transport between neighboring halos and departures from a normal IMF. These tests will be performed in future studies, now that the behavior of simpler models is known. Finally, increased numerical and time resolution will allow the very smallest, highest redshift halos to be modeled self-consistently in the proper Galactic context. With a fully realized “Virtual Galaxy”, and observational surveys that can advance into unexplored regions of the galaxy with ever-increasing fidelity, we can hope to find the residues of the first stars in the Milky Way itself.
I gratefully acknowledge the support of the Gilbert and Jaylee Mead Fellowship in Astrophysics in the Yale Center for Astronomy and Astrophysics, where a portion of this work was completed. Paolo Coppi donated computing time for the simulations, and the Yale ITS/HPC staff provided the necessary hardware and software support. I am also grateful to Romain Teyssier for the parallelized version of GRAFIC1. Thanks are also due for helpful comments from an anonymous referee. Support for a portion of this work has come from the Director’s Discretionary Research Fund at STScI.
[^1]: This paper adopts the [@Beers:05:531] terminology for labeling metal-poor stars, e.g. EMPs, or extremely metal-poor stars, have \[Fe/H\] $<-3$, and so on.
[^2]: http://www-hpcc.astro.washington.edu/tools/fof.html
[^3]: http://pleiadi.pd.astro.it/
[^4]: http://www.sdss.org
|
---
abstract: |
The iterative refinement method (IRM) has been very successfully applied in many different fields for examples the modern quantum chemical calculation and CT image reconstruction. It is proved that the refinement method can create a exact inverse from an approximate inverse with a few iterations. The IRM has been used in CT image reconstruction to lower the radiation dose. The IRM utilize the errors between the original measured data and the recalculated data to correct the reconstructed images. However if it is not smooth inside the object, there often is an over-correction along the boundary of the organs in the reconstructed images. The over-correction increase the noises especially on the edges inside the image. One solution to reduce the above mentioned noises is using some kind of filters. Filtering the noise before/after/between the image reconstruction processing. However filtering the noises also means reduce the resolution of the reconstructed images. The filtered image is often applied to the image automation for examples image segmentation or image registration but diagnosis. For diagnosis, doctor would prefer the original images without filtering process.
In the time these authors of this manuscript did the work of interior image reconstruction with local inverse method, they noticed that the local inverse method does not only reduced the truncation artifacts but also reduced the artifacts and noise introduced from filtered back-projection method without truncation. This discovery lead them to develop the sub-regional iterative refinement (SIRM) image reconstruction method. The SIRM did good job to reduce the artifacts and noises in the reconstructed images.
The SIRM divide the image to many small sub-regions. To each small sub-region the principle of local inverse method is applied. After the image reconstruction, the reconstructed image has grids on the border of sub-region inside the object. If do not consider the grids, the noise and artifacts are reduced compare the original reconstructed image. To eliminate the grids, these authors have to add the margin to the sub-region. they did not think the margin is important issue in the beginning. However when they considering the size of sub-region tending to only one pixel, they found that the margin play a important role. This limit situation of sub-regional iterative refinement is referred as local-region regional iterative refinement(LIRM). If the margin is very large for example as half the image size, the SIRM and the LIRM become no iterative refinement method (NIRM), i.e. normal filtered back-projection method. If the size of the margin tend to 0, the LIRM become the IRM, which is also referred as traditional iterative refinement method (TIRM). For SIRM and LIRM, if margin is too small, the image is rich in noise like IRM, if the margin is too big the image is rich in artifacts like NIRM. If a suitable margin is taken, for example the margin is around 20 pixel for a 512[\*]{}512 image, the summation of the noise and artifacts can be minimized.
SIRM and LIRM can be seen as local inverse applied to the image reconstruction of full field of veiw. SIRM and LIRM can be seen as generalized iterative refinement method(GIRM). SIRM and LIRM do not minimize the noise or artifacts but minimize the summation of the noise and artifacts. Even the SIRM and LIRM are developed in the field of CT image reconstruction, these authors believe they are a general methods and can be applied widely in physics and applied mathematics.
author:
- 'Kang Yang$^{1}$, Kevin Yang$^{1}$, Xintie Yang$^{2}$, Shuang-Ren Zhao$^{1}$'
title: 'Image Reconstruction Image reconstruction by using local inverse for full field of view.'
---
$^{1}$Imrecons.com North York, Ontario, Canada
$^{2}$Northwestern Poly Technical University Xi’an China
[*Keyword*]{}: artifact, noise, Local, inverse, iteration, iterative, reprojection, image reconstruction, filtered back-projection, FBP, x-ray, CT, parallel-beam, fan-beam, cone-beam, sub-regional, local-region, iterative refinement, LFOV, ROI, Tomography, filter
Introduction
============
Iterative refinement method (IRM) in applied mathematics, physics and image recovery / image reconstruction
------------------------------------------------------------------------------------------------------------
The IRM[@iterativeRefinement] is widely applied to physics and applied mathematics for example ref.[@W-Moench-Freiberg; @Xinyuan-Wu-Jun-Wu]. The advantage of the IRM is that it can produce an exact inverse from approximate inverse[@Shin-ichi; @Robert-Bridsony]. A recent very important application of the IRM in quantum physics can be found[@Anders-M-N-Niklasson] in which the Eigen values can be calculated dependent linearly with the size of matrix. The IRM is also applied to image recovery[@Haricharan-Lakshman] and MR image reconstruction[@Lin-He].
IRM in CT image reconstruction
------------------------------
Many reconstruction algorithms have been developed for parallel-beam[@Kak-A-C], fan-beam[@NooF; @Parker; @Kak-A-C; @Zhao-S-R-1993; @ref_Shuangren_zhaoFourierFan1; @ref_Shuangren_zhaoFourierFan2], and cone-beam tomographic systems[@Ref-11-L-A-Feldkamp; @Ref-13-B-D-Smith; @Ref-14-P-Grangeat; @Ref-15-Defrise; @Ref-16-Shuangren-Zhao]. The above reconstruction methods which has no iteration are referred as direct reconstruction. On the other hand, image can be reconstructed by IRMs[@Chang; @Zeng; @Riddell; @OSullivan; @Delaney]. In the IRM, the reconstructed image is re-projected and the errors between the measured projections and re-projections are calculated. These errors are utilized to correct the first reconstruction. In order to distinguish other IRM which will be introduced later in this article, this IRM is referred as traditional iterative refinement reconstruction method (TIRM). The direct reconstruction without iterative refinement is referred as non iterative refinement reconstruction method (NIRM). The TIRM is known that it can reduce artifacts especially beam-harden artifacts at a price of increasing the noises. Moreover, TIRM is more time-consuming compared to the direct reconstruction (NIRM), since it has a iteration. Time-consuming in image reconstruction is not a big issue in our generation since the improvement of the computer hardware. A recent iterative refinement reconstruction was the work of Johan Sunnegardh[@JohanSunnegardh]. Siemens AG has claimed that Johan’s method is their new generation CT image reconstruction method. This means that big companies begin to notice the importance of the IRM in image reconstruction. It is noticed that Johan’s iterative refinement reconstruction has a pre-filtering and post-filtering process which are used to reduce the noises of their IRM. For CT image reconstruction the IRM can reduce the artifacts of reconstructed image hence reduce the dose required for CT image. However the IRM often has an over-corrections on the image edges which increases the noises. Johan’s iterative refinement reconstruction has to include filtering process to reduce the noises. Pre-filtering and post-filtering process does not only reduce the noises, but it also decreases the information contained in the original projection data. these authors have also roughly introduced a IRM in the past which is referred as sub-regional iterative refinement method (SIRM)[@Ref-25-shuangRenZhao]. SIRM dose not include pre-fltering and post filtering processes, since their noise level is not increased. This article offers the details and also the history how this method has been discovered. The limit situation when the size of the sub-region approaches to $1$ is also discussed, in this case the SIRM method becomes local-region iterative refinement method (LIRM).
The history about developing the SIRM and LIRM
----------------------------------------------
One of the important task in image reconstruction is to reduce the truncation artifacts. The truncation artifacts are caused by LFOV (limited field of view) of the detector. The truncation artifacts can be reduced through extrapolation of the missing rays[@Ref-3-Maria-Magnusson-Seger; @Ref-8-F-Rashid-Farrokhi; @MattiasNilsson] or local tomography[@Ref-7Adel-Faridani; @Ref-8-F-Rashid-Farrokhi; @Ref-17-Alexander-Katsevich; @Ref-18-Alexander-Katsevich; @MattiasNilsson]. Extrapolation often produces an over-correction or an under-correction to the reconstructed image[@Ref-24-Shuangren-Zhao]. Local tomography cups the reconstructed image. Iterative reconstruction and re-projection algorithm [@Ref-4-Nassi; @Ref-5-Paul-S-Cho; @Ref-20-J-H-Kim] was developed for the reconstruction of limited angle of view. Another iterative reconstruction re-projection algorithm has been developed for LFOV[@Ref-22-Shuangren-Zhao]. The iterative algorithm for LFOV reduces the truncation artifacts remarkably. This algorithm led to a truncation free solution local inverse for LFOV[@Ref-24-Shuangren-Zhao]. The fast extrapolation used in [@Ref-24-Shuangren-Zhao] is done in [@Ref-23-ShuangrenZhao]. During the work with the iterative algorithm for LFOV, it was occasionally found that this algorithm did not only reduce the truncation artifacts but also it reduced the normal artifacts. Here the normal artifacts means that the artifacts exist in the direct reconstruction$\:$(NIRM) for example FBP (filtered back projection) method with full field of view (FFOV). It is well know that even FBP method is a exact method, it is actually not exact because the numerical calculations. This finding led to the SIRM for FFOV[@Ref-25-shuangRenZhao].
SIRM was adjusted to suit the case of FFOV. This was done by taking away the extrapolation process. The first reconstruction in the iteration was FBP method. For the second reconstruction, the region of the object was divided to many sub-regions. The iterative reconstruction was done for each sub-region adding a margin. The part of the object from the 1st reconstruction outside a sub-region (including its margin) was reprojected. These re-projections were subtracted from the original projections. The resulted projections were utilized to make the second reconstruction inside the sub-region. Finally the reconstructed images on all sub-regions were put together. It is worthwhile to say that the margin was added in the beginning to eliminate the cracks on the border of the sub-regions. It was not thought as an important issue. However it was found that it was the margin that really plays the role to reduce artifacts [@Ref-25-shuangRenZhao] and decreasing the noises.
Other image reconstruction methods
----------------------------------
It is worth to mention that Many fanbeam and conebeam image reconstruction algorithms have been derived in recent 20 years. Amongst these derivations are the following examples: the references[@key-23; @key-24; @key-25; @key-26; @key-40; @key-43; @key-45; @key-49; @key-67; @key-68; @Ref-22-Shuangren-Zhao; @Ref-23-ShuangrenZhao; @Ref-24-Shuangren-Zhao; @Ref-25-shuangRenZhao] are reconstruction algorithms for truncated projections, super short scan, interior and exterior reconstruction. The references[@key-27; @key-57; @key-58] are reconstruction algorithms in Fourier domain. The references[@key-29; @key-30; @key-32; @key-41; @key-50; @key-52; @key-53; @key-55; @key-63] are reconstruction algorithms with FBP method. The references[@key-36; @key-47; @key-48; @key-69] are reconstruction algorithms with total variation (TV) minimization or compressed sensing method. The references[@key-35; @key-37; @key-39; @key-51] are reconstruction algorithms with iterations. The references[@key-33; @key-46] are reconstruction algorithms with GPU accelerate. The reference[@key-28] is reconstruction with neural network algorithm. The reference[@key-31] is the reconstruction algorithm with multiresolution. The reference[@key-42] is reconstruction with dual-source. The reference[@key-66] introduces the reconstruction with a Laplace operator. The extrapolation method[@key-76; @key-77; @MattiasNilsson] and adapted extrapolation method[@key-70; @key-71; @key-74; @key-75] to solve LFOV problem.
These author’s contributions
----------------------------
Two generalized iterative refinement methods (IRM), i.e. sub-regional iterative refinement method (SIRM) and local-region iterative refinement method (LIRM) for inverse problem or CT image reconstruction with FFOV are introduced, which can reduce the artifacts and keep the noise not increased too much. In SIRM the reconstructed image has been divided to small pieces and re-projected, reconstructed to produce the final reconstruction image. SIRM was used to overcome the problem of the TIRM (i.e. the artifacts reduction is at the price of increasing the noises).
Among NIRM, TIRM, LIRM and SIRM. there are two parameters which are the margin size of the sub-region and the size of sub-region. The margin size is originally introduced to eliminate the grids on the reconstructed image of the SIRM. These authors found that if the margin size is between 0 and the size of image and the sub-region is chosen as small as one pixel (the size of sub-region equals 1), the SIRM becomes LIRM. If the margin size also closes to 0 then the LIRM becomes TIRM. If the margin size is large enough such as same size of image, the LIRM becomes NIRM. Hence, the relationship among the SIRM, LIRM, TIRM and NIRM are summarized. These authors have proved that LIRM is a special situation of SIRM and proved that NIRM and TIRM are special situation of LIRM. Hence LIRM and SIRM are two generalized IRM.
The examples of LIRM in simple inverse problem is introduced to learn the concept of LIRM, which are also suitable to SIRM. these authors did not implement LIRM for CT image reconstruction, since it is very time consuming. A CT reconstruction example using SIRM is done.
Arrangement in this article
---------------------------
In section 2 these authors review the reconstruction method without iterative refinement (NIRM). In section 3 TIRM is discussed. In section 4 LIRM is discussed . In section 5 the SIRM is discussed.
Local inverse to solve under determinate equation
=================================================
Assume the determinate equation is
$$T\,z=h$$ where $T$ is a $m\times m$ matrix, $z$ is a unknown vector with length $m$. $z\in Z$ $h$ is a known vector of length $m$. $h\in H$. $Z$ and $H$ are definite region of $z$ and $h$. Assume $$T=\left[\begin{array}{cc}
A & B\\
C & D
\end{array}\right]$$ $A$, $B$, $C$, $D$ are submatrix of $T$, $$h=\left[\begin{array}{c}
f\\
g
\end{array}\right]$$ $$z=\left[\begin{array}{c}
x\\
y
\end{array}\right]$$ $f$ and $g$ are subvector of $h$. $x$ and $y$ are subvector of $z$. We assume $x\in X$, $y\in Y$. Here $X+Y=Z$. We assume $X$ is the interior region or region of interest (ROI). $Y$ is the outside region or outer side of ROI. We assume $f\in F$, $g\in G$, $G+F=H$. $F$ is called field of view (FOV). $G$ is called outside of FOV.
$$\left[\begin{array}{cc}
A & B\\
C & D
\end{array}\right]\left[\begin{array}{c}
x\\
y
\end{array}\right]=\left[\begin{array}{c}
f\\
g
\end{array}\right]$$
In the case only $f$ is known, the above equation can be written as
$$\left[\begin{array}{cc}
A & B\end{array}\right]\left[\begin{array}{c}
x\\
y
\end{array}\right]=f$$ This is referred to under determinate equation which has infinite solution. One of the important solution is minimal norm solution.
$$\left[\begin{array}{c}
x^{(0)}\\
y^{(0)}
\end{array}\right]=\left[\begin{array}{cc}
A & B\end{array}\right]^{+}f$$ The subscript “$^{+}$” means generalized inverse which gives the the solution corresponding to minimal norm solution. subscript “$^{(0)}$” corresponding to the solution of first iteration.
The minimal normal solution is not the best solution. The following we try to further improve the result with so called local inverse method.
Assume we are more interest to know “$x$”. We can subtract the contribution of $y^{(0)}$from $f$,
$$f^{(0)}=f-B\,y^{(0)}$$ We can calculate the solution $x^{(1)}$ the following way, $$x^{(1)}=A^{+}f^{(0)}$$
We found that the above solution has no any improvement, i.e.
$$x^{(1)}=x^{(0)}$$ However we can change the formula of $f^{(0)}$ to $$f^{(0)}=f-B\,\mathrm{modify}(y^{(0)})$$ The above function “$\mathrm{modify}$” is just modify the mean value or flop according some priori. In this way we can improve the result a lot. The following is a example.
The local inverse is introduced with very simple example. Assume
$$A=\left[\begin{array}{cccc}
3 & 4 & 7 & 6\\
8 & 5 & 8 & 7\\
3 & 6 & 9 & 8\\
4 & 2 & 8 & 9
\end{array}\right]$$
$$B=\left[\begin{array}{cccc}
4 & 5 & 6 & 7\\
8 & 7 & 6 & 5\\
6 & 3 & 4 & 6\\
4 & 8 & 4 & 10
\end{array}\right]$$ $$x=\left[\begin{array}{c}
3\\
4\\
3\\
6
\end{array}\right]$$
$$y=\left[\begin{array}{c}
2\\
6\\
4\\
8
\end{array}\right]$$ Assume
$$z=\left[\begin{array}{c}
x\\
y
\end{array}\right]$$ We can calculate $$A\,x+B\,y=f$$ Now we assume we know vector $f$ and matrix $A$, $B$, we need to solve $x$ and $y$. We can solve it with least square method so that can be write as $$z^{(0)}=[A+B]^{+}f$$
$$x^{(0)}=first4(z^{(0)})$$
$$y^{(0)}=last4(z^{(0)})$$ We can calculate the errors $$error_{x}^{(0)}=\sum|x^{(0)}-x|$$
$$error_{y}^{(0)}=\sum|y^{(0)}-y|$$ $$f_{x}^{(0)}=f-B\,k_{y}\,y^{(0)}$$ $modified(y^{0})=k_{y}\,y^{(0)}$ $$f_{y}^{(0)}=f-B\,k_{x}\,x^{(0)}$$ $modified(x^{(0)})=k_{x}\,x^{(0)}$ $$x_{1}=A^{+}f_{x0}$$
$$y_{1}=B^{+}f_{y0}$$
$$error_{x}^{(1)}=\sum|x^{(1)}-x|$$
$$error_{y}^{(1)}=\sum|y^{(1)}-y|$$
Nor iterative refinement reconstruction method (NIRM)
=====================================================
Assume the parallel-beam projections are known as $p=p(\theta,u)$, where $\theta$ is projection angle and $u$ is the index of detector elements. Assume the projection operator $P$ is known, which is 2D Radon transform. Assume the non-iteration reconstruction operator $R$ is also known, which is for example FBP reconstruction. However, the object or original image $X_{o}(x)$ is unknown. Here $x$ is the coordinates of the image pixel. The forward equation of the problem can be write as
$$P\,X=p=p_{o}+p_{n}\label{eq:0-10}$$
where $X$ is the unknown image of the above equation. $p_{o}$ is projections without noises $$p_{o}=P\,X_{o}\label{eq:0-10a}$$ $p_{n}$ indicates noises in projections. The image can be reconstructed as
$$X^{(0)}=R\,p\label{eq:0-20}$$
$X^{(0)}$ is the first iteration of the reconstructed image using the reconstruction operator $R$. The superscripts $(0)$ indicates that $X^{(0)}$ is the first reconstruction in the iteration which will appear in the following paragraph. The above two formulas can be written as
$$X^{(0)}=J\,X=J\:X_{o}+R\:p_{n}\label{eq:0-30}$$
The first term of the right of the above equation is the signal dependent reconstructed image. The second term is noise dependent reconstructed image. In the above equation, $J$ is the projection-reconstruction operator which is defined as
$$J\equiv R\,P\label{eq:0-40}$$
where $x$ is the index of the image pixel before projection-reconstruction operation. $y$ is the index of pixel of the image after the projection-reconstruction operation. The operation of Eq.(\[eq:0-40\]) is defined as
$$J\,X=\sum_{x}j(y,x)\,X(x)\label{eq:0-55}$$
Here $j(y,x)$ is the kernel of operator $J$. $J$ can be written as
$$J=\sum_{x}j(y,x)\,\bullet\label{eq:6-0}$$
Where “$\bullet$” is multiplication.
The identical operator is defined in the following $$I=\sum_{x}\delta(y,x)\,\bullet\label{eq:6-2}$$ where
$$\delta(y,x)=\left\{ \begin{array}{c}
1\,\,\,\,\,\,\,\,\textrm{if}\,\,\,\,y=x\\
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,else
\end{array}\right.\label{eq:0-50}$$
From the above definition there is,
$$\sum_{y}\:\delta(y,x)=1\label{eq:6}$$
In the ideal reconstruction the reconstructed image is same as the original image that means $J\rightarrow I$. “$\rightarrow$” is “close to”. $J$ should satisfy the unitary condition same as $I$
$$\sum_{y}j(y,x)=1\label{eq:6-1}$$
Assume $x'$ is any pixel in the image. The above equation can be rewritten as
$$\sum_{y}\sum_{x}\:j(y,x)\:\delta(x-x')=1\label{eq:6A}$$
The above equation means that the delta function $\delta(x-x')$ after projection and reconstruction operation and sum operation $\sum_{y}$, $1$ is obtained.
The unitary condition guarantees that the dc-composition is not changed after the operation $J=R\:P$. $j(y,x)$ is referred as the resolution function, $J$ is referred as the resolution operator. In the above situation it is the resolution function of NIRM. The error of NIRM is defined as
$$Err^{(0)}\equiv X^{(0)}-X_{o}=(J-I)\:X_{o}+R\:p_{n}\label{eq:0-51}$$
The first term is corresponding to artifacts of the method. The second term is corresponding to noises of the method. Since the resolution function $J$ is not an exact unitary operator $I$, even if the noises $p_{n}=0$, it is true according to Eq.(\[eq:0-51\]) in general that $$X^{(0)}\neq X_{o}\label{eq:0-60}$$ One of the problem of the image reconstruction is how to improve the results of the reconstruction $X^{(0)}$ to let the reconstructed image more close to the original object function $X_{o}$ ? This can be rephrased as to find an algorithm with its resolution function more close to $\delta(y,x)$ function than the resolution function of NIRM $j(x,y)$ in the same time the noise term does not increase heavily. The IRMs (TIRM, LIRM and SIRM) deals this problem in the following sections. These methods have different noise characters and computation complicity.
Traditional iterative refinement reconstruction method (TIRM)
=============================================================
In order to further improve the reconstruction results $X^{(0)}$, TIRM was proposed. TIRM is iterative algorithm with the reconstruction and re-projection processes. It was obtained in the same way as the IRM to solve linear equation with an inaccurate known inverse operator. It was used so many times in different fields and hence it is difficult to find all of the sources of it. A few examples can be seen [@Chang; @Zeng; @Riddell; @OSullivan; @Delaney]. The processes of TIRM is given in the following. The errors i.e. the differences between the projections $p$ and re-projections $P\,X^{(0)}$ are calculated: $Err=(p-P\,X^{(0)})$. The errors are utilized to correct the reconstruction $X^{(1)}=X^{(0)}+R\,Err$. The algorithm can be summarized in the following $$X^{(1)}=X^{(0)}+R\,(p-P\,X^{(0)})\label{eq:2}$$ where $X^{(0)}$ is obtained in Eq.(\[eq:0-20\]). Substituting Eq.(\[eq:0-20\]) to the Eq.(\[eq:2\]), considering Eq.(\[eq:0-40\]) the above algorithm can be seen as a filtering algorithm
$$X^{(1)}=F_{TIRM}X^{(0)}\label{eq:3}$$
Where the filtering function is defined as
$$F_{TIRM}\equiv2\,I-J\label{eq:4}$$
Considering Eq.(\[eq:6\], \[eq:6-1\]) the filtering function $F_{TIRM}$ satisfies the unitary condition
$$\sum_{y}f_{TIRM}(y,x)=1\label{eq:7}$$
$f_{TIRM}(y,x)$ is kernel function of operator $F_{TIRM}$. The results of TIRM and NIRM will be utilized as contrast to LIRM which will be discussed in the next paragraph. The resolution operator of TIRM is $F_{TIRM}\:J$, which is really more close to identical operator $I$ than the resolution operator of NIRM $J$ does. However TIRM is rarely directly applied to clinical situation since it is sensitive to noises. This can be seen by further modify the Eq.(\[eq:2\]) using Eq.(\[eq:0-10\]), Eq.(\[eq:0-10a\]) and Eq.(\[eq:0-20\])
$$X^{(1)}=F_{TIRM}\:J\:X_{o}+F_{TIRM}R\:p_{n}\label{eq:3a}$$
Hence the errors
$$Err^{(1)}\equiv X^{(1)}-X_{o}=(F_{TIRM\:}J-I)\:X_{o}+F_{TIRM}R\:p_{n}\label{eq:3aA}$$
The first term of the above formula is a signal dependent image, which is referred to as artifacts. The second term of the formula is the noise dependent image. It can be compared the above formula with Eq.(\[eq:0-51\]). Since normally $F_{TIRM}\:J$ is closer to $I$ than $J$, the first term $F_{TIRM}\:J\:X_{o}$ is closer to $X_{o}$ than $J\:X_{0}$ does. This means that TIRM can get a more accurate reconstruction corresponding the first term. However the second term $F_{TIRM}R\:p_{n}$ is normally larger than $R\:p_{n}$. This means the TIRM is more noise sensitive than NIRM.
Recently a modified TIRM by Johan can be found in reference[@JohanSunnegardh]. In Johan’s iterative refinement reconstruction (JIRM), a pre-filtering process, and two regularization filtering processes are added to TIRM. Johan has compared the resolution of his method JIRM with NIRM, Johan’s result the Figure 5.7 of [@JohanSunnegardh]. a) in Figure 5.7 of [@JohanSunnegardh] shows the resolution function of NIRM. b) in Figure 5.7 of [@JohanSunnegardh] shows the resolution function JIRM without pre-filtering. c) in Figure 5.7 of [@JohanSunnegardh] shows the resolution function of JIRM with pre-filtering. It can be seen that in b) in Figure 5.7 of [@JohanSunnegardh], there is a black ring surrounding the white dot. This black ring is called over-correction. Over-correction increases the resolution but it leads a noise increase too. Johan used the pre-filtering process to overcome the problem of over-correction and noises. He first pre-filtered the projection data and applied the filtered data to his reconstruction. It is well know that the pre-filtering process does not only smooth the noise data but also reduced the information contained in the data. This is why many doctor prefer to see the noise data than the smoothed data after a filtering process. Pre-filtering process often is not accepted. The following section will shows the method to overcome the problem of over-correction in TIRM.
Local-region iterative refinement reconstruction method (LIRM)
==============================================================
The method
----------
In this section another IRM is defined which is a special case of next section. First two operators $K$ and $H$ are define as following $$K\:X=\sum_{x}k(y,x)\:X(x)\label{eq:sec4-01}$$ $$H\:X=\sum_{x}h(y,x)\:X(x)\label{eq:sec4-02}$$ where
$$k(y,x)=\begin{cases}
\begin{array}{cc}
j(y,x) & \mathrm{if}\:|x-y|\leq r\\
0 & \mathrm{else}
\end{array}\end{cases}\label{eq:sec4-10}$$
$$h(y,x)=\begin{cases}
\begin{array}{cc}
0 & \mathrm{if}\:|x-y|<r\\
j(y,x) & \mathrm{else}
\end{array}\end{cases}\label{eq:sec4-20}$$
where $r$ is a constant parameter. It defines a region $[y-r,y+r]$ close to $y.$ $r$ is called the margin size. The margin will be explained in next section. In this section it is assumed that the image is 1-dimensional. But the result is easy to extent to 2 or 3 dimensional situation. It is clear that there is the relation:
$$J=K+H\label{eq:sec4-30}$$
The Local-region iterative refinement method (LIRM) is defined as following,
$$X^{(1)}=\eta\:R\:p-R\:(P\:\Omega)\:X^{(0)})\label{eq:sec4-40}$$
where $\eta$ is a normalized parameter which will be decided later and the operator $\Omega$ is defined as:
$$\Omega(y,x)=\begin{cases}
\begin{array}{cc}
0 & \mathrm{if}\:|x-y|\leq r\\
1 & \mathrm{else}
\end{array}\end{cases}\label{eq:sec4-50}$$
In this article all operators are linear operator except the operator $\Omega$ which is a normal multiplication operator, i.e.:
$$(J\:\Omega)\:X=\sum_{x}(j(y,x)\:\Omega(y,x))\:X(x)\label{eq:sec4-60}$$
Considering Eq.(\[eq:0-20\]) and Eq.(\[eq:0-40\]), Eq.(\[eq:sec4-40\]) can be rewritten as
$$X^{(1)}=\eta\:X^{(0)}-(J\:\Omega)\:X^{(0)}\label{eq:sec4-70}$$
considering $$J\:\Omega=H\label{eq:sec4-71}$$ the LIRM can rewritten as
$$X^{(1)}=\eta\:X^{(0)}-H\:X^{(0)}\label{eq:sec4-80}$$
The above formula is rewritten as
$$X^{(1)}=F_{LIRM}X^{(0)}\label{eq:sec4-90}$$
where
$$F_{LIRM}=\eta\:I\:-H=\eta\:I-J+K\label{eq:sec40-100}$$
$F_{LIRM}$ is the filtering function of the LIRM. Because $J$ can implemented as $P\:R$, $K$ can be implemented as a small matrix when $r$ is small. That is why the right part of the above formula is easier to be implemented than the middle part of the formula. Similar to Eq.$\:$(\[eq:7\]) $F_{LIRM}$ should meet the unitary condition:
$$\sum_{y}\sum_{x}f_{LIRM}(y,x)\:\delta(x-x')=1\label{eq:sec4-110}$$
This unitary condition means that if the input is $\delta(x-x')$, $x'$ is any point in the image, the whole output of the filter $F_{LIRM}$ is $1$, since $F_{LIRM}$ should looks like $\delta(y,x)$ function. The above condition can be write as $$\sum_{y}f_{LIRM}(y,x)=1\label{eq:sec4-111}$$ Considering Eq.$\:$(\[eq:sec40-100\] and \[eq:sec4-111\]) implies that
$$\eta\equiv\eta(x)=1+\sum_{y}h(y,x)=2-\sum_{y}k(y,x)\label{eq:sec4-120}$$
The LIRM Eq.(\[eq:sec4-40\]) can be interpreted as following. The reconstructed image $X^{(0)}$ is re-projected except the vicinity of the pixel where the iterative Reconstruction is calculated $(J\:\Omega)\:X^{(0)}$. The measured projection is reconstructed with $R$, scaled a little to adjust the dc composition $\eta\:R\:p$. The difference of the above two projections are used to produced a reconstruction $(\eta\:R\:p-(J\:\Omega)\:X^{(0)})$. Inside of the vicinity, only the center point is kept, which is the pixel where the iterative method is done. Other pixel can be obtained in the same way. The LIRM always concerns in a local region, which is the vicinity of the pixel. Hence it is referred as the local region iterative refinement method.
In the limit case
-----------------
In this article the discussed IRM are generalized inverse methods. CT image reconstruction can be seen as a example. It is easy to understand the principle of method to consider the object $X_{o}$ as one dimensional image. Assume $x\in[-\frac{L}{2},\frac{L}{2}]$ is the coordinates of the image pixel. Here $L$ is the number of pixels of the image. if $r\rightarrow L$, then according to Eq.(\[eq:sec4-10\], \[eq:sec4-20\], \[eq:sec4-30\] and \[eq:sec4-120\]) there are
$$K\rightarrow J,\quad H\rightarrow0,\quad\eta\rightarrow1\label{eq:sec4-130}$$
Considering the above formula and Eq.(\[eq:sec40-100\]) there is $$F_{LIRM}=\eta\:I\:-H\rightarrow I=F_{NIRM}\label{eq:sec4-131}$$ i.e. in this case there is LIRM $\rightarrow$NIRM.
if $r\rightarrow0$, the similarly according to Eq.(\[eq:sec4-10\], \[eq:sec4-20\], \[eq:sec4-30\]) there are
$$k(y,x)\rightarrow j(y,x)\delta(y,x)\label{eq:sec4-140}$$
That is
$$K\rightarrow j(y,y)\:I\label{eq:sec4-141-0}$$
considering Eq.(\[eq:sec4-120\]), there is $$\eta(x)\rightarrow2-j(x,x)\label{eq:sec4-141}$$ and $$\eta\:I-J+K\rightarrow(2-j(y,y))\:I-J+j(y,y)\:I=2\:I-J\label{eq:sec4-150}$$ Considering the above formula and Eq.(\[eq:sec40-100\]) and Eq.(\[eq:4\]), there is $$F_{LIRM}\rightarrow F_{TIRM}\label{eq:sec4-151}$$ i.e. in this case LIRM $\rightarrow$TIRM. This means that the NIRM and TIRM are two special cases of LIRM as $r\rightarrow L$ or $r\rightarrow0$.
A simple example
----------------
In following example, the original image is 1-dimensional for simplicity. CT reconstruction is not taken which is at least two dimension. Assume the forward operator is $P=T\:V$. Here $T$ is assumed as discrete Fourier transform. The inverse operator is taken as $R=T^{-1}$ , $T^{-1}$ is the inverse discrete Fourier transform. Assume the size of the image is $L=128$. The blurring operator $V$ is assumed as
$$V(x)=\frac{1}{[(\frac{x}{\alpha L})^{2}+\beta]^{\rho}}\label{eq:sec4-190}$$
where $\alpha=0.01$, $\beta=0.2$, $\rho=0.9$. In this example the resolution function
$$J=R\:P=T^{-1}\:T\:V=V\label{eq:sec4-200}$$
In this example $J(x,y)=J(x-y)=V(x-y)$. $F_{NIRM}$, $F_{TIRM}$ and $F_{LIRM}$ are summarized as $F_{a}$. Corresponding 3 IRMs can be written as following,
$$X^{(1)}=F_{a}\:J\:X_{o}+F_{a}R\:p_{n}\label{eq:sec4-210}$$
The error function can be written as,
$$Err_{a}^{(1)}\equiv X^{(1)}-X_{o}=(F_{a}\:J-I)\:X_{o}+F_{a}R\:p_{n}\label{eq:sec4-220}$$
Here $a$ is corresponding to NIRM, TIRM and LIRM, and $F_{a}$ has 3 format, $F_{NIRM}=I$, $F_{TIRM}=2\:I-J$, $F_{LIRM}=\eta\:I-J+K$ The noises and artifacts depends on the shape of the operator $F_{a}$
Advantages and disadvantages
----------------------------
In general, the errors of LIRM are less than TIRM meaning that
$$||Err_{LIRM}||\ll||Err_{TIRM}||\label{eq:sec4-180}$$
The noises of LIRM are less than TIRM meaning that
$$\sigma_{y}^{2}\{Err_{LIRM}\}\ll\sigma_{y}^{2}\{Err_{TIRM}\}\label{eq:sec4-181}$$
Here $\sigma_{y}^{2}$ is defined local variance as
$$\sigma_{y}^{2}\{X(x)\}=\frac{1}{N-1}\sum_{x\in\omega_{y}}(X(x)-E(X(x)))^{2}\label{eq:4-182}$$
Where $\omega_{y}$ is the local region close to $y$. $E$ is the local mean defined as $$E_{y}\{X(x)\}=\frac{1}{N}\sum_{x\in\omega_{y}}X(x)\label{eq:4-183}$$ Where $N$ is number of pixel(voxel) inside the local region $\omega_{y}$. $N$ can be found experimentally. these authors are not going to prove the above formula Eq(\[eq:sec4-180\], \[eq:sec4-181\]), instead an example in following section will show the results.
Figure \[fig:fig01\] shows the filtering function of $F_{a}$ for different methods. I is can be seen that $F_{TIRM}$ has a large negative values at $|x|<r$. This is corresponding the black ring of the resolution function in b) of Figure 5.7 of the reference[@JohanSunnegardh]. This is also referred as over-correction. The over-correction can increase the resolution yet it also increases the noises. On the other hand, the filtering function $F_{LIRM}$ has no large negative value at $|x|<r$. When $|x|>r$ $F_{LIRM}\simeq F_{TIRM}$ , hence $F_{LIRM}$ can also reduce the artifacts for example beam harden artifacts in CT reconstruction. In Figure \[fig:fig01\] these authors choose $r=3$. $r$ is a parameter can be adjusted according to the size of the image. Usually the large the image size, the large the $r$ should be. From Figure \[fig:fig01\] it can be seen that if $r\rightarrow0$, there is $F_{LIRM}\rightarrow F_{TIRM}$. When $r\rightarrow L$, there is $F_{LIRM}\rightarrow F_{NIRM}$.
In the following it is assumed that $X_{o}(x)=\mathrm{sign}(x)$. The image edge is at the place $x=0$. It is also assumed that the size of image is $L=2048$, and $r=122$. Here these authors have increased the size of image to show the results more clearly. Measured data is simulated with $p_{o}=P\:X_{o}$. Matlab is used to created the simulated noise: $p_{n}(x)=0.004*randint(1,$[\[]{}-5,5[\]]{}). Noise $p_{n}$ is added to data $p=p_{o}+p_{n}$. Assume the forward operator is $P=T\:V$ . Here $T$ is assumed as discrete Fourier transform. The inverse operator is taken as $R=T^{-1}$ , $T^{-1}$ is the inverse discrete Fourier transform. $V$ is defined in Eq.(\[eq:sec4-190\]). The three reconstruction results using the methods NIRM, TIRM, LIRM are compared.
\[fig:fig02-d\]
In Figure \[fig:fig02\] it can be seen that in the place of image edges the TIRM has the smallest error. In the place far away from the edges, The LIRM has lowest errors. Parameter $r$ can be adjusted so the image is optimal at reducing the noise and increasing of the accuracy. The errors of LIRM are less than NIRM in both place of image edges and the place far away from the edges. The errors of TIRM are less than NIRM in the place of image edges but it is at the same level with NIRM at the place far away from the image edges. Normally it is acceptable that there are errors at image edges, but the errors at the place far away from image edges should be as small as possible. Thus LIRM is better than TIRM and NIRM in decrease errors and artifacts.
It can be seen that the results are not dependent on the operator of $T$. It is dependent only with $J$. Here it is assumed that $T$ is Fourier Transform to make things easy. Actually if $J=R\:P=V$ and $V\neq I$ (here $I$ is identical operator), the results of all IRMs (TIRM, LIRM) are not related to the operator $R$ and $P$, but it is dependent to there product $J=V=R\:P$.
For example if a image $G$ is filtered with convolution by operator $F$. Assume the filtered image $G'=F\star G$ is known, “$\star$” means convolution. The original image $G$ is required to be recovered from $G'$ and the known operator $F$. Assume that the Fourier transform of $F$ is known which is $\tilde{F}=\mathcal{F}\{F$}, $\mathcal{F}$ is Fourier transform. The recover operator $Q=\mathcal{F^{\mathrm{-1}}}(\frac{1}{\tilde{F}})$ can be defined. $\mathcal{F^{\mathrm{-1}}}$ is inverse Fourier transform. $Q$ can fully recover the original image, since if $\tilde{Q}=\mathcal{F}\{Q\}$, $\tilde{Q}\:\tilde{G'}=\tilde{Q}\:\tilde{F}\:\tilde{G}=\frac{1}{\tilde{F}}\:\tilde{F}\:\tilde{G}=\tilde{G}$, and hence $Q\star G'=G$. However if $\tilde{F}$ has $0$ or very close to $0$ some where. $\tilde{Q}=\frac{1}{\tilde{F}}$ will have “$\frac{1}{0}$”. In this situation, the above image recovery method can not be implemented. Thus a regularization is required. For example $Q=\mathcal{F}(\frac{1}{\tilde{F}+\alpha})$ can be defined, here $\alpha$ small number which is a regularization factor. In this case the recovered image $Q\star G'=\mathcal{F}(\frac{\tilde{F}}{\tilde{F}+\alpha})\star G$. $J=Q\star F=\mathcal{F}(\frac{\tilde{F}}{\tilde{F}+\alpha})$. $V=\mathcal{F}(\frac{\tilde{F}}{\tilde{F}+\alpha})$ can be referred as the blurring function. The recovery method can be improved by the IRMs (NIRM, TIRM and LIRM). The results of IRMs are only related to the resolution function $J=V$. If $V$ is same as Eq.(\[eq:sec4-190\]), the recovered image $G'$ will has the same results as the Fig.$\:$\[fig:fig01\] and \[fig:fig02\].
Since to implement the LIRM with the example of CT image reconstruction is very time-consuming, these authors only study the simple examples in this section which is 1-D inverse problem. Note that although in this section, the example of image reconstruction has not been done, the analysis results are still suitable to the situation of CT image reconstruction. In the next section these authors will study another IRM, i.e. SIRM, which is close to LIRM, but is easier to implement for CT image reconstruction.
Sub-regional iterative refinement method (SIRM)
===============================================
History of SIRM
---------------
During the work of iterative reconstruction for LFOV[@Ref-22-Shuangren-Zhao; @Ref-23-ShuangrenZhao], these authors have noticed that not only the truncation artifacts are reduced but the normal artifacts are also reduced. Here the normal artifacts means the artifacts appeared in the reconstruction of FFOV (full field of view) instead of LFOV (limited field of view). Figure \[fig:fig4-new\] shows the normal artifacts appearing with the FBP method, this results is from Matlab.
If the local inverse reconstruction for LFOV [@Ref-22-Shuangren-Zhao; @Ref-23-ShuangrenZhao] is applied in the situation of FFOV , the algorithm can be summarized as the following,
$$\begin{aligned}
p^{(1)} & = & p-P\,\Omega\,R\,p\label{eq:5-3a}\\
X^{r} & = & R\,p^{(1)}\label{eq:5-3}\end{aligned}$$
Where $p=P\,X$. $\Omega$ is the reverse truncation operator, which is defined in the following, $$\Omega(x)=\left\{ \begin{array}{cc}
0 & \textrm{if}\;x\in\textrm{ROI}\\
1 & \textrm{if}\;x\notin\textrm{ROI}
\end{array}\right.\label{eq:5-4}$$ Here ROI is the region of interest that is any small arbitrary sub-region where a reconstruction can be made, and $x$ is the coordinates of the pixel of the reconstructed image. In the following example it is assumed that the ROI of the object is a centric disk-shape region and its radius is half of the radius of the image. Here the disk-shape region is chosen because these authors start this kind reconstruction from LFOV which require a disk-shaped region.
Eq.(\[eq:5-3\]) is a simplification of the algorithm[@Ref-22-Shuangren-Zhao]. The extrapolation process is take away because this is FFOV instead LFOV. There is no truncation and extrapolation is not necessary. In order to simplify the algorithm Eq.(\[eq:5-3a\],\[eq:5-3\]) further, the truncation operator can be defined as:
$$T(x)=\left\{ \begin{array}{cc}
1 & \textrm{if}\;x\in\textrm{ROI}\\
0 & \textrm{if}\;x\notin\textrm{ROI}
\end{array}\right.\label{eq:5-5}$$
The relation between the truncation and the reverse truncation operator is given in the following
$$\Omega(x)=1(x)-T(x)\label{eq:5-6}$$
Here $1(x)$ is unit operator with its value as $1$ everywhere on the image. Considering the above Eq.(\[eq:5-6\]), Eq.(\[eq:5-3a\],\[eq:5-3\]) can be rewritten as $$\begin{aligned}
p^{(1)} & = & P\,T\,R\,p+(p-P\,R\,p)\label{eq:5-7a}\\
X^{r} & = & T\,R\,p^{(1)}\label{eq:5-7}\end{aligned}$$ In order further improved the iterative reconstruction, a truncation operator $T$ is added to the reconstructed image of the second formula of Eq.(\[eq:5-7\]). The truncation operator set zeros outside the ROI. This helps to delete unwanted image outside the ROI.
Figure \[fig:fig2\] offers the reconstruction results for FBP algorithm and the above iterative algorithm. Figure \[fig:fig2\_a\] is the image of the Shepp-Logan head phantom. Figure \[fig:fig2\_b\] is the crop of the image of the phantom corresponding to the ROI which is a centered disk. Figure \[fig:fig2\_c\] is the reconstruction with FBP algorithm from the simulated parallel beam projections obtained from Matlab. The number of projections is 360 for the half circle scan (180 degree). The space between the two elements of the detector is taken as the same as the space between the two pixels of image. The data size of image of phantom is $512\times512$. Figure \[fig:fig2\_d\] is the reconstruction with iterative algorithm of Eq.(\[eq:5-7a\], \[eq:5-7\]). The projections and all parameters are same as Figure \[fig:fig2\_c\]. It is difficult to see the differences between Figure \[fig:fig2\_c\] and Figure \[fig:fig2\_d\] if you do not see them carefully. An error function is defined as following
$$Err=T\,|X^{r}-X|^{2}\label{eq:9}$$
Figure \[fig:fig2\_e\] and Figure \[fig:fig2\_f\] are error functions corresponding to Figure \[fig:fig2\_c\] and Figure \[fig:fig2\_d\]. Figure \[fig:fig2\_e\] and Figure \[fig:fig2\_f\] use the same scale of brightness. It is clear that Figure \[fig:fig2\_e\] is bright than Figure \[fig:fig2\_f\] meaning that the reconstruction errors of FBP algorithm is larger than the iterative algorithm of Eq.(\[eq:5-7a\], \[eq:5-7\]). It is can be seen that the values of reconstruction with FBP algorithm are little bit lower than the values of the phantom. However the values with iterative reconstruction of Eq.(\[eq:5-7a\], \[eq:5-7\]) are much close to the values of the phantom. This also shows the improvement of the iterative algorithm Eq.(\[eq:5-7a\], \[eq:5-7\]).
In the following example the modified Shepp-Logan head phantom is taken in consideration. A massive object is added outside the region of interest. This massive object represents the bone of a human arm. This object can introduce more artifacts for the reconstruction process. The improvement of the iterative reconstruction can be seen more clearly from this example. The stripe artifacts of FBP algorithm are remarkably reduced for this example; see Figure \[fig:fig4\].
The method
----------
SIRM is sourced from the iterative reconstruction and re-projection algorithm or local inverse method[@Ref-22-Shuangren-Zhao], [@Ref-23-ShuangrenZhao] for LFOV. However it requires an improvement to reconstruct the whole image. First, the region of interest (ROI) can be made in any shapes. It is not required that ROI is a round disk-shape region. In the past ROI was chosen as round disk-shape region, this is because of the situation of LFOV. In the following, ROI will be chosen as many small square. The iterative algorithm[@Ref-22-Shuangren-Zhao] is used to every square. The extrapolation is taken away because of FFOV. The algorithm was posted on-line in Chinese roughly, see reference [@Ref-25-shuangRenZhao] and it is summarized more details in the following, $$\begin{aligned}
p_{i}^{(1)} & = & p-P\,H_{i}\,X^{(0)}\label{eq:20-2}\\
X^{(1)} & = & \sum_{i=1}^{M}T_{i}\,R\,p_{i}^{(1)}\label{eq:20-3}\end{aligned}$$ where $X^{(0)}$ is obtained in Eq.(\[eq:0-20\]); where the subscript $i$ is the index of sub-region which is a small square box; $M$ is the number of sub-regions. $X^{(1)}$ is the iterative reconstruction. Superscript $(1)$ is corresponding to first iteration (the second reconstruction). $p_{i}^{(1)}$ is iterative re-projection for $\omega_{i}$. $\omega_{i}$ is the $i^{th}$ sub-region. After the parts of the object in all sub-regions are reconstructed, all the parts of image are put together to form the reconstructed image $X^{(1)}$. Two truncation operators above are defined in the following $$\begin{aligned}
H_{i}(x) & = & \left\{ \begin{array}{cc}
0 & \textrm{if}\:x\in\omega_{i}\\
1 & \textrm{if}\:x\notin\omega_{i}
\end{array}\right.\label{eq:30}\end{aligned}$$ $$\begin{aligned}
T_{i}(x) & = & \left\{ \begin{array}{cc}
1 & \textrm{if}\:x\in\omega_{i}\\
0 & \textrm{if}\:x\notin\omega_{i}
\end{array}\right.\label{eq:40}\end{aligned}$$ It is worthwhile to say that the operation $T_{i}(x)$ and $H_{i}(x)$ are normal multiplication. They are different from the operation of $J$ which is corresponding to matrix multiplication. $T_{i}(x)$ is defined $1$ on a sub-region. $H_{i}(x)$ is defined $1$ in the whole region except the sub-region. Hence $H_{i}(x)$ is a hole-shape function. Define unitary operator
$$\begin{aligned}
1(x) & \equiv & 1\label{eq:50}\end{aligned}$$
Thus, there is
$$\begin{aligned}
1(x) & =H_{i}(x) & +T_{i}(x)\label{eq:60}\end{aligned}$$
Using above equation, Eq.(\[eq:20-2\]) can be written as
$$\begin{aligned}
p_{i}^{(1)} & = & P\,T_{i}\,X^{(0)}+p-P\,X^{(0)}\label{eq:70}\end{aligned}$$
However the results of Eq.(\[eq:20-2\],\[eq:20-3\]) shows cracks between sub-regions. The cracks can be seen in the reconstructed image, see Figure \[Flo:fig25a\]. In order to eliminate the cracks, the Eq.(\[eq:20-2\] or \[eq:70\]) is upgraded as
$$\begin{aligned}
p_{i}^{(1)} & = & P\,T_{i}^{+}\,X^{(0)}+p-P\,X^{(0)}\label{eq:80}\end{aligned}$$
where
$$\begin{aligned}
T_{i}^{+}(x) & = & \left\{ \begin{array}{cc}
1 & \textrm{if}\:x\in\omega_{i}+\mathrm{Margin})\\
0 & \textrm{if}\:x\notin\omega_{i}+\mathrm{Margin})
\end{array}\right.\label{eq:90}\end{aligned}$$
$T_{i}^{+}$ is the image truncation operator with margin, see Figure \[Flo:fig25b\]. It was found that if margin size $r$ is taken as $4\sim10$ pixels, the cracks can be eliminated. But the margin can be chosen as for example 40 if the image size is big. Considering
$$\sum_{i=1}^{M}T_{i}\,f=f\label{eq:91}$$
and substituting Eq.(\[eq:80\]) to Eq.(\[eq:20-3\]), there is
$$X^{(1)}=(\sum_{i=1}^{M}T_{i}\,R\,P\,T_{i}^{+}X^{(0)})+R\,(p-P\,X^{(0)})\label{eq:100}$$
or
$$X^{(1)}=[(\sum_{i=1}^{M}T_{i}\,R\,P\,T_{i}^{+})+(I-J)]X^{(0)}\label{eq:110}$$
where $J=R\:P$ defined in Eq.(\[eq:0-40\]). The above formula can be rewritten as,
$$X^{(1)}=F_{SIRM}\,X^{(0)}\label{eq:120}$$
$F_{SIRM}$ is a filtering function corresponding to SIRM. The filter function is defined as
$$F_{SIRM}\equiv U+(I-J)\label{eq:130}$$
where $U$ is the sub-region projection and reconstruction operator
$$U\equiv\sum_{i=1}^{M}T_{i}\,J\,T_{i}^{+}\label{eq:140}$$
Corresponding to Eq.(\[eq:130\]), there is,
$$f_{SIRM}(y,x)=u(y,x)+\delta(y,x)-j(y,x)\label{eq:150-a}$$
The above filtering function does not satisfy the unitary condition which keeps the dc value unchanged after the reconstruction compare to the original image. Hence it is required to be upgraded as
$$f_{SIRM}(y,x)=u(y,x)+\eta(x)\,\delta(y,x)-j(y,x)\label{eq:150}$$
$\eta(x)$ is normalization function similar to which used in LIRM. Considering the unitary condition
$$\sum_{y}f_{SIRM}(y,x)=1\label{eq:160}$$
implies
$$\eta(x)=2-\sum_{y}u(y,x)\label{eq:170}$$
Here the summation is taken on the definition area of the variable $y\,\in V$. $V=\sum_{i}\omega_{i}$ is the region of whole image. The Eq.(\[eq:130\]) can be replaced as
$$F_{SIRM}\equiv\eta I-J+U\label{eq:130A}$$
Usually $\eta(x)$ can be taken as
$$\eta(x)\approx1\label{eq:180}$$
Only if the sub-region ($\omega_{i}$) is very small, $\eta(x)$ is possible to be significantly different from $1$. This is same as the case of the LIRM discussed in last section. The resolution function of SIRM algorithm is $F_{SIRM}\:J$.
The biggest difference of the form of SIRM from TIRM is the operator $U$ in Eq.(\[eq:140\]) comparing Eq.(\[eq:130A\]) and Eq.(\[eq:4\]). In order to have a good understanding of this operator, the process of this operator is shown in Fig. \[fig:fig8\]. Fig. \[fig:fig8\_a\] illustrates the first reconstruction $X^{(0)}=R\,p$. Fig. \[fig:fig8\_b\] shows that the image of Fig. \[fig:fig8\_a\] is divided into sub-regions $T_{i}^{+}\,X^{(0)}\;i=1,2,....M$. Fig. \[fig:fig8\_c\] shows that the sub-region images of (b) are reprojected $P\,T_{i}^{+}\,X^{(0)}\;i=1,2,....M$. Fig. \[fig:fig8\_d\] shows that the sub-region image is reconstructed from Fig. \[fig:fig8\_c\] by using $R\,P\,T_{i}\,X^{(0)}\;i=1,2,....M$. Fig. \[fig:fig8\_e\] shows that the sub-region images are put together to form a whole image by using $U\,X^{(0)}=\sum_{i=1}^{M}T_{i}\,R\,P\,T_{i}^{+}X^{(0)}$.
The original image Fig. \[fig:fig8\_a\] is chosen as the modified Shepp-Logan head phantom with data size $512\times512$. The modification is adding a massive small disk to the bottom of the image. The massive small disk will increase the normal artifacts, which will be utilized to test the algorithms in the next paragraph. The projection operator $P$ is parallel beam and defined in Matlab. The projections data is created by the operator $P$ to the above modified Shepp-Logan head phantom. The number of projections is 360 for the half circle scan (180 degree); the space between the two elements of the detector is taken equal to the space between the two pixels of image. The operator $R$ is corresponding to NIRM which is filtered back projection method which is defined in Matlab.
The process of operator $U$ shown in Fig. \[fig:fig8\] looks mediocre. Actually, it is really not mediocre because the margins in the algorithm plays an important role in eliminating artifacts and decreasing the noises.
In the limit case $\omega_{i}$ is small as only one pixel,
$$\sum_{i=1}^{M}T_{i}=1(x)\label{eq:181}$$
$$J\,T_{i}^{+}=K\label{eq:182}$$
Here $K$ is defined in Eq.(\[eq:sec4-01\]), Hence considering Eq.(\[eq:140\]), there is
$$U=K\label{eq:183}$$
Considering Eq.(\[eq:130A\]) Eq.(\[eq:sec40-100\]) there is SIRM$\rightarrow$LIRM in the case $\omega_{i}\rightarrow$one pixel.
The iterative algorithm with more loops
---------------------------------------
In the above discussion, two algorithms are iterated with only one loop. If one loop does not satisfy, more loops can be utilized, this can be written as
$$X_{l}^{(n)}=F_{l}^{n}\,X^{(0)}\label{eq:190}$$
where $X_{l}^{(n)}$ is the reconstructed image with more loops of iteration. $n$ is the iteration number. $F_{l}^{n}=(F_{l})^{n}$ is the filtering operator for iteration number $n$. $l$ indicates different algorithm, $l=\{TIRM,\,LIRM,\,SIRM\}$. The resolution function with more loops is
$$J_{l}^{(n)}=F_{l}^{n}J\label{eq:195}$$
For the reconstructions, the error is defined as
$$Err_{l}^{(n)}\equiv X_{l}^{(n)}-X_{o}\label{eq:200}$$
$X_{l}^{(n)}$ is defined in Eq.(\[eq:190\]). $X_{o}$ is the object or the original image. Considering Eq.(\[eq:0-20\]) and Eq.(\[eq:190\]) the error can be defined as
$$Err_{l}^{(n)}=(F_{l}^{n}J-I)X_{o}+F_{l}^{n}\,R\,p_{n}\label{eq:205}$$
The first item of the above formula is corresponding to artifacts which is related to the original image $X_{o}$; the second item is corresponding to noises which is related to noises in the first reconstruction $R\,p_{n}$. The error for NIRM method is $Err^{(0)}=(J-I)X_{o}-R\:p_{n}$. The artifact transfer function can be defined as $$A_{l}^{n}=F_{l}^{n}J-I\label{eq:206}$$ and the noise transfer function can be defined as $$N_{l}^{n}=F_{l}^{n}\:R\label{eq:207}$$ The noise transfer function and the artifact transfer function have different forms, which gives the possibility to optimize the algorithm by balancing the artifacts and the noises and adjusting the filtering function.
The absolute error $|Err_{l}^{(n)}|$ will be used to study reconstruction results. The distance between the reconstructed image $X_{l}^{(n)}$ and the original image $X$ can be used also to compare the reconstruction. The distance is defined in the following,
$$d_{l}^{(n)}=\frac{\sum_{x}(X_{l}^{(n)}(x)-X_{o}(x))^{2}}{\sum_{x}(X(x)-\bar{X}_{o}(x))^{2}}\label{eq:210}$$
where $\bar{X}$ is the average of the image $X(x)$. See reference[@Ref-5-Paul-S-Cho] for details of the definition of the distance.
Results
-------
Figure \[fig:fig6\] shows the comparison of SIRM with NIRM and TIRM. NIRM is implemented in Eq(). TIRM is implemented with Eq.(\[eq:2\]). $\eta$ in Eq.(\[eq:150\]) is chosen as $1$ for simplification. SIRM is implemented with Eq.(\[eq:110\]). The region of the object is divided according to $4\,\times\,4$ grid for SIRM. Hence, there are $M=16$ sub-regions. The margin is chosen as $10$ pixels. The iteration for Eq.(\[eq:190\]) are done with only one loop, i.e. $n=1$ .
The original image $X_{o}$, the projection projector $P$ and the reconstruction operator $R$ are chosen the same as in Fig. \[fig:fig8\], see section 5.2. The projections $p$ is obtained through the simulation with $p=P\,X$. In this example additional noises $p_{n}$ are not added to the projections. However, since there are always calculation errors, $p_{n}\neq0$ in general.
SIRM yielded the best reconstruction results than the NIRM and the TIRM. The stripe artifacts shown on Figure \[fig:fig6\_a\] are reduced remarkably on Figure \[fig:fig6\_c\]. The absolute errors $|Err^{(0)}|$ shown in Figure \[fig:fig6\_d\] $|Err_{l=NRRM}^{(0)}|$ and Figure \[fig:fig6\_e\] $|Err_{l=TRRM}^{(1)}|$ are larger than in Figure \[fig:fig6\_f\] $|Err_{l=SRRM}^{(1)}|$. The results of TIRM (Figure \[fig:fig6\_b\],\[fig:fig6\_e\]) are similar to the results of NIRM method (Figure \[fig:fig6\_a\],\[fig:fig6\_d\]).
It is important to mention that: A) in Figure \[fig:fig6\_f\] the absolute errors on the two sub-regions containing the massive disk are little bit larger than the errors in other sub-regions. This drawback can be eliminated through increasing the number of sub-regions, for example using $16\times16$ grid instead of $4\times4$ grid. In practice, the smaller sub-regions are required to be used only in the two sub-regions containing the massive disk. B) the above results of the iterative algorithm are only done with one loop of iteration and further more loops using Eq.(\[eq:190\]) can also improve the results, but the improvement is limited. C) If LIRM is implemented, since the sub-region becomes as small as only one pixel(voxel), the result should be better (in the meaning of reducing the artifacts and decreasing the noises) than SIRM if the same margin $r$ is used. LIRM is more time-consuming, to implement it more modern technologies for example GPU parallel calculation and fast back-projection techniques are required. The implementation of LIRM will be left for the future work.
Methods: Distance:
----------------------- -----------
NIRM Eq.(\[eq:0-20\]) 0.0177
TIRM Eq.(\[eq:2\]) 0.0134
SIRM Eq.(\[eq:110\]) 0.0172
: The distance for different methods[]{data-label="tab:1"}
The distances for the above three algorithms have been calculated. Table \[tab:1\] tells that the distance from the reconstruction of SIRM to the phantom is smaller than the distance from the reconstruction of NIRM to the phantom. However, the smallest distance is obtained through TIRM. Do these results mean that the reconstruction results from the TIRM is better than SIRM? The following details of the profiles give the answer.
Table \[tab:1\] tells that the TIRM has the smallest distance. However, TIRM has an over correction at the image edges. The over correction can be seen in Fig. \[fig:fig10\_a\]. Here the dotted line is far away from the solid line compared to dashed line and dash-dot line. The dashed line and dash-dot line are close to each other. Fig. \[fig:fig10\_a\] shows that TIRM has a over correction at the image edges. The areas close to the edge of the different image structures are strongly relayed to the distance defined in Eq.(\[eq:210\]). This kind of over correction can reduce the distance, but it causes the reconstructed image to be oscillated at the image edges. In a clinical cases the over correction can not be accepted. The over correction is easy to be thought as some kind of real structure, it is dangerous to clinical situation. Even though TIRM has the smallest distance, the reconstructed image through TIRM is noisier than other two algorithms. TIRM is rarely used directly in clinics. The reference [@JohanSunnegardh] is the example of indirectly using TIRM. It is TIRM plus pre-filtering and post-filtering in the reconstruction. Pre-filtering can cause the lose of information.
In contrast, the SIRM reduces the oscillation at the place close to the edges and reduces the artifacts at the place far away from the edges simultaneously, which can be seen in Figure \[fig:fig10\_b\]. Here the dash-dot line is the closest line to the solid line. The dash-dot line is corresponding to SIRM.
According to the above discussion, SIRM and LIRM have better quality compared with NIRM and TIRM in image reconstruction with FFOV.
Conclusions and future work
===========================
Two generalized iterative refinement methods LIRM and SIRM have been introduced. As an example, simple inverse problem to ** utilize LIRM has been given. The LIRM eliminates the over correction and it is less noise sensitive comparing to TIRM.
SIRM has been applied to the CT image reconstruction from untruncated parallel-beam projections. The simulations shown that it can reduce the normal artifacts remarkably, which exists in the reconstruction with FBP algorithm. SIRM has been compared to the TIRM and NIRM. The result shows that the SIRM has less artifacts in reconstructed image and TIRM is more sensitive to noises. The distance of SIRM is smaller than NIRM. The smallest distance is obtained through TIRM. However the smallest distance is achieved through an over-correction in the places close to the image edges, which can not be accepted.
These authors have shown that LIRM is a special case of SIRM. NIRM and TIRM are special cases of LIRM. Hence LIRM and SIRM are two generalized iterative refinement reconstruction methods. SIRM and LIRM can be seen as local inverse applied to the image reconstruction of full field of veiw. Hence, SIRM and LIRM can be seen as generalized iterative refinement method(GIRM) and local inverse method for FFOV. SIRM and LIRM do not minimize the noise or artifacts alone but minimize the total values of the noise and artifacts. Even the SIRM and LIRM are developed in the field of CT image reconstruction, these authors believe they are a general methods and can be applied widely in physics and applied mathematics where IRM can be applied.
The future work of these authors is to implement the LIRM in CT image reconstruction. these authors also plan to implement SIRM and LIRM in fan-beam and cone-beam geometries.
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abstract: 'An equiangular tight frame (ETF) yields a type of optimal packing of lines in a Euclidean space. ETFs seem to be rare, and all known infinite families of them arise from some type of combinatorial design. In this paper, we introduce a new method for constructing ETFs. We begin by showing that it is sometimes possible to construct multiple ETFs for the same space that are “mutually unbiased" in a way that is analogous to the quantum-information-theoretic concept of mutually unbiased bases. We then show that taking certain tensor products of these mutually unbiased ETFs with other ETFs sometimes yields infinite families of new complex ETFs.'
author:
- 'Matthew Fickus, , Benjamin R. Mayo[^1]'
title: Mutually Unbiased Equiangular Tight Frames
---
Welch bound, equiangular tight frames, mutually unbiased bases, relative difference sets
Introduction
============
For any $N\geq D\geq1$, $N>1$, Welch [@Welch74] gives the following bound on the *coherence* of $N$ unit vectors ${\{{{\boldsymbol{\varphi}}_n}\}}_{n=1}^{N}$ in ${\mathbb{C}}^D$: $$\label{eq.Welch}
{\operatorname{coh}}({\{{{\boldsymbol{\varphi}}_n}\}}_{n=1}^{N})
:=\max_{n\neq n'}{|{{\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}}|}
\geq{\bigl[{\tfrac{N-D}{D(N-1)}}\bigr]}^{\frac12}.$$ It is well known [@StrohmerH03] that ${\{{{\boldsymbol{\varphi}}_n}\}}_{n=1}^{N}$ achieves equality in if and only if ${\{{{\boldsymbol{\varphi}}_n}\}}_{n=1}^{N}$ is an *equiangular tight frame* (ETF) for ${\mathbb{C}}^D$, that is, if and only if the value of ${|{{\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}}|}$ is constant over all $n\neq n'$ (equiangularity) and there exists $C>0$ such that C${\|{{\mathbf{y}}}\|}^2=\sum_{n=1}^{N}{|{{\langle{{\boldsymbol{\varphi}}_n},{{\mathbf{y}}}\rangle}}|}^2$ for all ${\mathbf{y}}\in{\mathbb{C}}^D$ (tightness).
The coherence of any unit vectors is the cosine of the smallest principal angle between any two of the lines (one-dimensional subspaces) they individually span. By achieving equality in , an ETF yields $N$ lines in ${\mathbb{C}}^D$ whose smallest pairwise principal angle is as large as possible, namely an optimal way to pack $N$ points on the projective space that consists of all lines in ${\mathbb{C}}^D$. Due to their optimality, ETFs arise in various applications including waveform design for wireless communication [@StrohmerH03], compressed sensing [@BajwaCM12; @BandeiraFMW13], quantum information theory [@Zauner99; @RenesBSC04] and algebraic coding theory [@JasperMF14].
Much of the ETF literature is devoted to the *existence problem*: for what $D$ and $N$ does there exist an ${\operatorname{ETF}}(D,N)$, that is, an $N$-vector ETF for ${\mathbb{C}}^D$? Here, one key subproblem is to resolve *Zauner’s conjecture* that an ${\operatorname{ETF}}(D,D^2)$ exists for any $D\geq1$ [@Zauner99; @RenesBSC04]. In quantum information theory, such an ETF is called a *symmetric, informationally complete, positive operator-valued measure* (SIC-POVM), and a finite, but remarkable number of these have already been found [@FuchsHS17]. Another key subproblem is to characterize the existence of *real* ${\operatorname{ETF}}(D,N)$, that is, ETFs where ${\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}\in{\mathbb{R}}$ for all $n,n'$. Real ETFs equate to a subclass of *strongly regular graphs* (SRGs) [@vanLintS66; @Seidel76; @HolmesP04; @Waldron09], which are a mature subject in and of themselves [@Brouwer07; @Brouwer17; @CorneilM91]. In general, the existence problem remains poorly understood, with lists of known ETFs [@FickusM16] falling far short of known necessary conditions, namely that an ${\operatorname{ETF}}(D,N)$ with $1<D<N-1$ can only exist if $N\leq\min{\{{D^2,(N-D)^2}\}}$ [@HolmesP04] (a generalization of *Gerzon’s bound* [@LemmensS73]) and that an ${\operatorname{ETF}}(3,8)$ does not exist [@Szollosi14].
All known positive existence results for ${\operatorname{ETF}}(D,N)$ with $1<D<N-1$ are due to explicit construction involving some type of combinatorial design; see [@FickusM16] for a survey. ETFs whose *redundancy* $\frac ND$ is either nearly or exactly $2$ arise from the related concepts of Hadamard matrices, conference matrices, Gauss sums and Paley tournaments [@StrohmerH03; @HolmesP04; @Renes07; @Strohmer08]. The equivalence between real ETFs and certain SRGs has been partially generalized to the complex case using roots of unity [@BodmannPT09; @BodmannE10], abelian distance-regular antipodal covers of complete graphs [@CoutinkhoGSZ16; @FickusJMPW19], and association schemes [@IversonJM16]. *Harmonic ETFs* equate to *difference sets* in finite abelian groups [@Konig99; @XiaZG05; @DingF07]. *Steiner ETFs* arise from *balanced incomplete block designs* (BIBDs) [@GoethalsS70; @FickusMT12]. Nontrivial generalizations of the Steiner ETF construction yield other ETFs arising from projective planes containing hyperovals [@FickusMJ16], Steiner triple systems [@FickusJMP18], and group divisible designs [@FickusJ19].
In this paper, we provide a new method for constructing ETFs. It is inspired by an ETF-based perspective [@FickusJKM18; @FickusS20] of a classical factorization [@GordonMW62] of the complement of a *Singer* difference set in terms of a *relative difference set* (RDS). The main idea is to take tensor products of vectors in a given ${\operatorname{ETF}}(D_1,N_1)$ with those belonging to a collection of $N_1$ distinct ${\operatorname{ETF}}(D_2,N_2)$ that are *mutually unbiased* in the quantum-information-theoretic sense. We show that this technique, for example, yields (complex) ${\operatorname{ETF}}(D,N)$ with $$\label{eq.new ETF from 2 pos}
D=\tfrac{Q-1}{Q+1}(\tfrac{Q-1}{2}Q^{2J-1}-1),
\quad
N=\tfrac{Q-1}{Q+1}(Q^{2J}-1),$$ for any prime power $Q\geq4$ and $J\geq2$, as well as ones with $$\label{eq.neq ETF from GQ}
D=\tfrac{Q^3+1}{Q^4-1}(\tfrac{Q^3+1}{Q+1}Q^{4J-3}-1),
\
N=\tfrac{Q^3+1}{Q^4-1}(Q^{4J}-1),$$ for any prime power $Q\geq2$ and $J\geq2$. Remarkably, all such ETFs seem to be new. For example, taking $J=2$ and $Q=4,5$ in yields ${\operatorname{ETF}}(57,153)$ and ${\operatorname{ETF}}(166,416)$, neither of which were previously known [@FickusM16].
In the next section we establish notation, and review known concepts that we will need later on. In Section \[sec.MUETFs\], we explain what it means for several ${\operatorname{ETF}}(D,N)$ to be mutually unbiased (Definition \[def.MUETF\]), and give a necessary condition on their existence (Theorem \[thm.Gerzon\]). We moreover construct mutually unbiased ETFs from RDSs, both in general (Theorem \[thm.RDS gives MUETF\]) and using a classical family (Corollary \[cor.Singer MUETFF\]). In Section \[sec.new ETFs\], we discuss the aforementioned tensor-product-based technique (Theorem \[thm.tensor\]). Combining it with Corollary \[cor.Singer MUETFF\] yields our main result (Theorem \[thm.main result\]). In special cases where the initial ${\operatorname{ETF}}(D_1,N_1)$ is *positive* or *negative* in the sense of [@FickusJ19], our main result yields several infinite families of new (complex) ETFs (Corollary \[cor.pos neg\]).
Background {#sec.background}
==========
Equiangular tight frames and Naimark complements
------------------------------------------------
Let ${\mathbb{F}}$ be either ${\mathbb{R}}$ or ${\mathbb{C}}$. For any $N$-element set of indices ${\mathcal{N}}$, equip ${\mathbb{F}}^{\mathcal{N}}:={\{{{\mathbf{x}}:{\mathcal{N}}\rightarrow{\mathbb{F}}}\}}$ with the complex dot product ${\langle{{\mathbf{x}}_1},{{\mathbf{x}}_2}\rangle}:=\sum_{n\in{\mathcal{N}}}[{\mathbf{x}}_1(n)]^*{\mathbf{x}}_2(n)$ which, like all inner products in this paper, is conjugate-linear in its first argument. For any finite sequence ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ of vectors in a Hilbert space ${\mathbb{H}}$ over ${\mathbb{F}}$, the corresponding *synthesis operator* is ${\boldsymbol{\Phi}}:{\mathbb{F}}^{\mathcal{N}}\rightarrow{\mathbb{H}}$, ${\boldsymbol{\Phi}}{\mathbf{x}}:=\sum_{n\in{\mathcal{N}}}{\mathbf{x}}(n){\boldsymbol{\varphi}}_n$. Its adjoint ${\boldsymbol{\Phi}}^*:{\mathbb{H}}\rightarrow{\mathbb{F}}^{\mathcal{N}}$, $({\boldsymbol{\Phi}}^*{\mathbf{y}})(n)={\langle{{\boldsymbol{\varphi}}_n},{{\mathbf{y}}}\rangle}$ is called the *analysis operator*. We sometimes identify a vector ${\boldsymbol{\varphi}}\in{\mathbb{H}}$ with its synthesis operator ${\boldsymbol{\varphi}}:{\mathbb{F}}\rightarrow{\mathbb{H}}$, ${\boldsymbol{\varphi}}(x):=x{\boldsymbol{\varphi}}$, an operator whose adjoint is the linear functional ${\boldsymbol{\varphi}}^*:{\mathbb{H}}\rightarrow{\mathbb{F}}$, ${\boldsymbol{\varphi}}^*{\mathbf{y}}={\langle{{\boldsymbol{\varphi}}},{{\mathbf{y}}}\rangle}$. In the special case where ${\mathbb{H}}={\mathbb{F}}^{\mathcal{D}}$ for some $D$-element set ${\mathcal{D}}$, ${\boldsymbol{\Phi}}$ is just the ${\mathcal{D}}\times{\mathcal{N}}$ matrix whose $n$th column is ${\boldsymbol{\varphi}}_n$, and ${\boldsymbol{\Phi}}^*$ is its ${\mathcal{N}}\times{\mathcal{D}}$ conjugate-transpose.
In general, the *frame operator* of ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ is the composition ${\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^*:{\mathbb{H}}\rightarrow{\mathbb{H}}$ of its synthesis and analysis operators, namely ${\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^*=\sum_{n\in{\mathcal{N}}}{\boldsymbol{\varphi}}_n^{}{\boldsymbol{\varphi}}_n^*$, ${\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^*{\mathbf{y}}=\sum_{n\in{\mathcal{N}}}{\langle{{\boldsymbol{\varphi}}_n},{{\mathbf{y}}}\rangle}{\boldsymbol{\varphi}}_n$. The reverse composition is the ${\mathcal{N}}\times{\mathcal{N}}$ *Gram matrix* that has $({\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}})(n,n')={\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}$ as its $(n,n')$th entry. This matrix has ${\operatorname{rank}}({\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}})={\operatorname{rank}}({\boldsymbol{\Phi}})=\dim({\operatorname{span}}{\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}})$ and is positive-semidefinite. Conversely, any positive-semidefinite ${\mathcal{N}}\times{\mathcal{N}}$ matrix ${\mathbf{G}}$ factors as ${\mathbf{G}}={\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}}$ where ${\boldsymbol{\Phi}}$ is the synthesis operator of a sequence ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ that spans ${\mathbb{H}}$ where $\dim({\mathbb{H}})={\operatorname{rank}}({\mathbf{G}})$. Here, ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ and ${\mathbb{H}}$ are only unique up to unitary transformations, meaning we can take ${\mathbb{H}}={\mathbb{F}}^D$ if so desired, where $D={\operatorname{rank}}({\mathbf{G}})$.
We say ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ is a ($C$-)*tight frame* for ${\mathbb{H}}$ if ${\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^*=C{\mathbf{I}}$ for some $C>0$. By the polarization identity, this equates to having $\sum_{n\in{\mathcal{N}}}{|{{\langle{{\boldsymbol{\varphi}}_n},{{\mathbf{y}}}\rangle}}|}^2={\|{{\boldsymbol{\Phi}}^*{\mathbf{y}}}\|}^2=C{\|{{\mathbf{y}}}\|}^2$ for all ${\mathbf{y}}\in{\mathbb{H}}$. An ${\mathcal{N}}\times{\mathcal{N}}$ self-adjoint matrix ${\mathbf{G}}$ is the Gram matrix ${\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}}$ of a $C$-tight frame ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ for some space ${\mathbb{H}}$ if and only if ${\mathbf{G}}^2=C{\mathbf{G}}$, namely when $\frac1C{\mathbf{G}}$ is an orthogonal projection operator. In particular, ${\operatorname{Tr}}({\mathbf{G}})=CD$ where $D={\operatorname{rank}}({\mathbf{G}})$.
We say ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ is a *unit norm tight frame* (UNTF) for ${\mathbb{H}}$ if it is a tight frame for ${\mathbb{H}}$ and ${\|{{\boldsymbol{\varphi}}_n}\|}=1$ for all $n$. Here, we necessarily have $N={\operatorname{Tr}}({\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}})=CD$ where $D=\dim({\mathbb{H}})$. As such, a sequence ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ of $N$ unit vectors in ${\mathbb{H}}$ is a UNTF for ${\mathbb{H}}$ if and only if $${\|{{\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^*-\tfrac ND{\mathbf{I}}}\|}_{{{\operatorname{Fro}}}}^2
={\operatorname{Tr}}[({\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^*-\tfrac ND{\mathbf{I}})^2]
={\|{{\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}}}\|}_{{{\operatorname{Fro}}}}^2-\tfrac{N^2}{D}$$ is zero. That is, any $N$ unit vectors ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ in ${\mathbb{H}}$ satisfy $$\label{eq.FP}
\tfrac{N^2}{D}
\leq{\|{{\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}}}\|}_{{{\operatorname{Fro}}}}^2
=\sum_{n\in{\mathcal{N}}}\sum_{n'\in{\mathcal{N}}}{|{{\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}}|}^2,$$ and achieve equality here if and only if they form a UNTF for ${\mathbb{H}}$. When ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ is a UNTF for ${\mathbb{H}}$, $\frac DN{\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}}$ is an orthogonal projection operator with constant diagonal entries, implying ${\mathbf{I}}-\frac DN{\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}}$ is another such operator of rank $N-D$. In particular, when this occurs with $N>D$, $\frac{N}{N-D}{\mathbf{I}}-\frac{D}{N-D}{\boldsymbol{\Phi}}^*{\boldsymbol{\Phi}}$ is the Gram matrix of a UNTF for a space of dimension $N-D$. Such a sequence is called a *Naimark complement* of ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$. Up to unitary transformations, it is uniquely defined according to $$\label{eq.Naimark}
{\langle{\tilde{{\boldsymbol{\varphi}}}_n},{\tilde{{\boldsymbol{\varphi}}}_{n'}}\rangle}
=\left\{\begin{array}{cl}
1,&\ n=n',\\
-\tfrac{D}{N-D}{\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle},&\ n\neq n'.\end{array}\right.$$
Returning to , we now note that bounding the off-diagonal terms of this sum by their maximum value gives $$\begin{aligned}
\label{eq.Welch derivation 1}
\tfrac{N^2}{D}
&\leq\sum_{n\in{\mathcal{N}}}\sum_{n'\in{\mathcal{N}}}{|{{\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}}|}^2\\
\label{eq.Welch derivation 2}
&\leq N+N(N-1)\max_{n\neq n'}{|{{\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}}|}^2,\end{aligned}$$ which equates to the Welch bound . Moreover, equality in is equivalent to equality in both and , namely to when ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ is a UNTF for ${\mathbb{H}}$ that also happens to be equiangular, namely an ETF for ${\mathbb{H}}$. In particular, equality holds in if and only if ${|{{\langle{{\boldsymbol{\varphi}}_n},{{\boldsymbol{\varphi}}_{n'}}\rangle}}|}^2=\frac{N-D}{D(N-1)}$ for all $n\neq n'$, and in this case, ${\{{{\boldsymbol{\varphi}}_n}\}}_{n\in{\mathcal{N}}}$ is necessarily a UNTF for ${\mathbb{H}}$. By , the Naimark complement of an ${\operatorname{ETF}}(D,N)$ is an ${\operatorname{ETF}}(N-D,N)$, a fact we will use often.
Harmonic frames and relative difference sets
--------------------------------------------
A *character* of a finite abelian group ${\mathcal{G}}$ is a homomorphism $\gamma:{\mathcal{G}}\rightarrow{\mathbb{T}}:={\{{z\in{\mathbb{C}}: {|{z}|}=1}\}}$. The set of all such characters is known as the *(Pontryagin) dual* $\hat{{\mathcal{G}}}$ of ${\mathcal{G}}$, which is itself a group under pointwise multiplication. In fact, since ${\mathcal{G}}$ is finite, $\hat{{\mathcal{G}}}$ is known to be isomorphic to ${\mathcal{G}}$. The synthesis operator ${\boldsymbol{\Gamma}}$ of the characters of ${\mathcal{G}}$ is a square ${\mathcal{G}}\times\hat{{\mathcal{G}}}$ matrix having ${\boldsymbol{\Gamma}}(g,\gamma)=\gamma(g)$ for all $g$ and $\gamma$. ${\boldsymbol{\Gamma}}$ is often called the *character table* of ${\mathcal{G}}$, and its adjoint ${\boldsymbol{\Gamma}}^*:{\mathbb{C}}^{{\mathcal{G}}}\rightarrow{\mathbb{C}}^{\hat{{\mathcal{G}}}}$, $({\boldsymbol{\Gamma}}^*{\mathbf{y}})(\gamma)={\langle{\gamma},{{\mathbf{y}}}\rangle}$ (the analysis operator of the characters) is the *discrete Fourier transform* (DFT) over ${\mathcal{G}}$. Since ${\mathcal{G}}$ is finite, it is known that its characters form an equal-norm orthogonal basis for ${\mathbb{C}}^{\mathcal{G}}$, and so ${\boldsymbol{\Gamma}}^{-1}=\frac1G{\boldsymbol{\Gamma}}^*$ where $G=\#({\mathcal{G}})$. In particular ${\boldsymbol{\Gamma}}{\boldsymbol{\Gamma}}^*=G{\mathbf{I}}$.
For any ${\mathcal{D}}\subseteq{\mathcal{G}}$ with $D=\#({\mathcal{D}})>0$, let ${\boldsymbol{\Psi}}$ be the synthesis operator of the corresponding *harmonic frame* ${\{{{\boldsymbol{\psi}}_\gamma}\}}_{\gamma\in\hat{{\mathcal{G}}}}$, that is, the normalized restrictions of the characters of ${\mathcal{G}}$ to ${\mathcal{D}}$: $$\label{eq.harmonic frame}
{\boldsymbol{\Psi}}\in{\mathbb{C}}^{{\mathcal{D}}\times\hat{{\mathcal{G}}}},
\quad
{\boldsymbol{\Psi}}(d,\gamma)
={\boldsymbol{\psi}}_\gamma(d)
:=D^{-\frac12}\gamma(d).$$ Any such frame is automatically a UNTF: for any $d_1,d_2\in{\mathcal{D}}$, $$({\boldsymbol{\Psi}}{\boldsymbol{\Psi}}^*)(d_1,d_2)
=\tfrac1D\sum_{\gamma\in\hat{{\mathcal{G}}}}[\gamma(d_1)]^*\gamma(d_2)
=\tfrac1D({\boldsymbol{\Gamma}}{\boldsymbol{\Gamma}}^*)(d_1,d_2),$$ and so ${\boldsymbol{\Psi}}{\boldsymbol{\Psi}}^*=\tfrac1D{\boldsymbol{\Gamma}}{\boldsymbol{\Gamma}}^*=\tfrac GD{\mathbf{I}}$. Meanwhile, for any $\gamma_1,\gamma_2\in\hat{{\mathcal{G}}}$, the corresponding entry of the Gram matrix is $$\begin{aligned}
{\langle{{\boldsymbol{\psi}}_{\gamma_1}},{{\boldsymbol{\psi}}_{\gamma_2}}\rangle}
&=\tfrac1D\sum_{d\in{\mathcal{D}}}[\gamma_1(d)]^*\gamma_2(d)\\
&=\tfrac1D\sum_{g\in{\mathcal{G}}}[(\gamma_1^{}\gamma_2^{-1}(g)]^*{\boldsymbol{\chi}}_{\mathcal{D}}(g)\\
&=\tfrac1D({\boldsymbol{\Gamma}}^*{\boldsymbol{\chi}}_{\mathcal{D}})(\gamma_1^{}\gamma_2^{-1}),\end{aligned}$$ where ${\boldsymbol{\chi}}_{\mathcal{D}}\in{\mathbb{C}}^{\mathcal{G}}$ is the ${\{{0,1}\}}$-valued characteristic (indicator) function of ${\mathcal{D}}$. In particular, $$\label{eq.RDS 0}
{|{{\langle{{\boldsymbol{\psi}}_{\gamma_1}},{{\boldsymbol{\psi}}_{\gamma_2}}\rangle}}|}^2
=\tfrac1{D^2}{|{({\boldsymbol{\Gamma}}^*{\boldsymbol{\chi}}_{\mathcal{D}})(\gamma_1^{}\gamma_2^{-1})}|}^2,
\quad\forall\,\gamma_1,\gamma_2\in\hat{{\mathcal{G}}}.$$ To continue, we exploit the fact that the DFT ${\boldsymbol{\Gamma}}^*$ distributes over *convolution*: for any ${\mathbf{y}}_1,{\mathbf{y}}_2\in{\mathbb{C}}^{\mathcal{G}}$, defining ${\mathbf{y}}_1*{\mathbf{y}}_2\in{\mathbb{C}}^{\mathcal{G}}$ by $({\mathbf{y}}_1*{\mathbf{y}}_2)(g):=\sum_{g'\in{\mathcal{G}}}{\mathbf{y}}_1(g'){\mathbf{y}}_2(g-g')$, we have $$[{\boldsymbol{\Gamma}}^*({\mathbf{y}}_1*{\mathbf{y}}_2)](\gamma)
=({\boldsymbol{\Gamma}}^*{\mathbf{y}}_1)(\gamma)({\boldsymbol{\Gamma}}^*{\mathbf{y}}_2)(\gamma),
\quad\forall\,\gamma\in\hat{{\mathcal{G}}}.$$ (When considering such ${\mathcal{G}}$ in general, we default to writing the group operation on ${\mathcal{G}}$ and its dual $\hat{{\mathcal{G}}}$ as addition and multiplication, respectively.) Meanwhile, the DFT of the *involution* $\tilde{{\mathbf{y}}}\in{\mathbb{C}}^{\mathcal{G}}$ of ${\mathbf{y}}\in{\mathbb{C}}^{\mathcal{G}}$, $\tilde{{\mathbf{y}}}(g):=[{\mathbf{y}}(-g)]^*$ is $({\boldsymbol{\Gamma}}^*\tilde{{\mathbf{y}}})(\gamma)=[({\boldsymbol{\Gamma}}^*{\mathbf{y}})(\gamma)]^*$ for all $\gamma\in\hat{{\mathcal{G}}}$. Combined, we have $${|{({\boldsymbol{\Gamma}}^*{\boldsymbol{\chi}}_{\mathcal{D}})(\gamma)}|}^2
=[{\boldsymbol{\Gamma}}^*({\boldsymbol{\chi}}_{\mathcal{D}}*\tilde{{\boldsymbol{\chi}}}_{\mathcal{D}})](\gamma),
\quad\forall\,\gamma\in\hat{{\mathcal{G}}}.$$ Here, ${\boldsymbol{\chi}}_{\mathcal{D}}*\tilde{{\boldsymbol{\chi}}}_{\mathcal{D}}$ is the *autocorrelation* of ${\boldsymbol{\chi}}_{\mathcal{D}}$, which counts the number of distinct ways that any given $g\in{\mathcal{G}}$ can be written as a difference of members of ${\mathcal{D}}$: $$\begin{aligned}
({\boldsymbol{\chi}}_{\mathcal{D}}*\tilde{{\boldsymbol{\chi}}}_{\mathcal{D}})(g)
&=\sum_{g'\in{\mathcal{G}}}{\boldsymbol{\chi}}_{\mathcal{D}}(g')\tilde{{\boldsymbol{\chi}}}_{\mathcal{D}}(g-g')\\
&=\sum_{g'\in{\mathcal{G}}}{\boldsymbol{\chi}}_{\mathcal{D}}(g'){\boldsymbol{\chi}}_{g+{\mathcal{D}}}(g')\\
&=\#{\{{{\mathcal{D}}\cap(g+{\mathcal{D}})}\}}\\
&=\#{\{{(d,d')\in{\mathcal{D}}\times{\mathcal{D}}: g=d-d'}\}}.\end{aligned}$$ Altogether, we see that there is a relationship between the combinatorial properties of the differences $d-d'$ of members of ${\mathcal{D}}$ and the magnitudes of the inner products of vectors that belong to the corresponding harmonic frame. This relationship has long been exploited [@Turyn65] to characterize certain types of ${\mathcal{D}}$ including, as we now explain, relative difference sets:
\[def.RDS\] Let ${\mathcal{H}}$ be an $H$-element subgroup of an abelian group ${\mathcal{G}}$ of order $G$. A $D$-element subset ${\mathcal{D}}$ of ${\mathcal{G}}$ is an ${\mathcal{H}}$-RDS for ${\mathcal{G}}$ if there exists a constant $\Lambda$ such that $$\label{eq.RDS 1}
{\boldsymbol{\chi}}_{\mathcal{D}}*\tilde{{\boldsymbol{\chi}}}_{\mathcal{D}}=\Lambda({\boldsymbol{\chi}}_{\mathcal{G}}-{\boldsymbol{\chi}}_{\mathcal{H}})+D{\boldsymbol{\delta}}_0,$$ namely if no nonzero member of ${\mathcal{H}}$ is a difference of two members of ${\mathcal{D}}$ while every member of ${\mathcal{H}}^{\mathrm{c}}$ can be written as a difference of members of ${\mathcal{D}}$ in exactly $\Lambda$ ways.
In the literature, an RDS with these parameters is usually denoted as an “${\operatorname{RDS}}(N,H,D,\Lambda)$" where $N=\frac{G}{H}$. In the special case where ${\mathcal{H}}={\{{0}\}}$, an ${\mathcal{H}}$-RDS for ${\mathcal{G}}$ is simply called a *difference set* for ${\mathcal{G}}$. To proceed, we use the *Poisson summation formula*, namely that ${\boldsymbol{\Gamma}}^*{\boldsymbol{\chi}}_{\mathcal{H}}=H{\boldsymbol{\chi}}_{{\mathcal{H}}^\perp}$ where ${\mathcal{H}}^\perp:={\{{\gamma\in\hat{{\mathcal{G}}}: \gamma(h)=1,\ \forall\,h\in{\mathcal{H}}}\}}$ is the *annihilator* of ${\mathcal{H}}$, which is a subgroup of $\hat{{\mathcal{G}}}$ that is isomorphic to ${\mathcal{G}}/{\mathcal{H}}$. (“The DFT of a comb is a comb.") In particular, taking the DFT of gives that ${\mathcal{D}}$ is an ${\mathcal{H}}$-RDS for ${\mathcal{G}}$ if and only if $$\label{eq.RDS 2}
{|{({\boldsymbol{\Gamma}}^*{\boldsymbol{\chi}}_{\mathcal{D}})(\gamma)}|}^2
=\Lambda[G{\boldsymbol{\delta}}_1(\gamma)-H{\boldsymbol{\chi}}_{{\mathcal{H}}^\perp}(\gamma)]+D,
\ \forall\,\gamma\in\hat{{\mathcal{G}}}.$$ Here, evaluating at $\gamma=1$ gives $D^2=\Lambda(G-H)+D$, namely that $\Lambda=\frac{D(D-1)}{G-H}=\frac{D(D-1)}{H(N-1)}$; this also follows from a simple counting argument. As such, $$D-\Lambda H
=D-\tfrac{D(D-1)}{N-1}
=\tfrac{D(N-D)}{N-1}$$ and so equates to having $${|{({\boldsymbol{\Gamma}}^*{\boldsymbol{\chi}}_{\mathcal{D}})(\gamma)}|}^2
=\left\{\begin{array}{cl}
\frac{D(N-D)}{N-1},&\ \gamma\in{\mathcal{H}}^\perp,\gamma\neq1,\smallskip\\
D,&\ \gamma\notin{\mathcal{H}}^\perp.
\end{array}\right.$$ In light of , we see that ${\mathcal{D}}$ is an ${\mathcal{H}}$-RDS for ${\mathcal{G}}$ if and only if the corresponding harmonic frame satisfies $$\label{eq.RDS 3}
{|{{\langle{{\boldsymbol{\psi}}_{\gamma_1}},{{\boldsymbol{\psi}}_{\gamma_2}}\rangle}}|}^2
=\left\{\begin{array}{cl}
\frac{N-D}{D(N-1)},&\ \gamma_1^{}\gamma_2^{-1}\in{\mathcal{H}}^\perp,\ \gamma_1\neq\gamma_2,\smallskip\\
\frac1D,&\ \gamma_1^{}\gamma_2^{-1}\notin{\mathcal{H}}^\perp.
\end{array}\right.$$ In the special case where ${\mathcal{H}}={\{{0}\}}$, the second condition above becomes vacuous, and this result reduces to the equivalence between difference sets and harmonic ETFs given in [@XiaZG05; @DingF07].
Positive and negative ETFs {#subsec.pos neg}
--------------------------
In Section \[sec.new ETFs\], we show that, in certain circumstances, one can construct an ${\operatorname{ETF}}(D_1D_2,N_1N_2)$ from an ${\operatorname{ETF}}(D_1,N_1)$ and $N_1$ mutually unbiased ${\operatorname{ETF}}(D_2,N_2)$. It turns out that this technique applies to many distinct types of ${\operatorname{ETF}}(D_1,N_1)$, including Naimark complements of Steiner ETFs [@FickusMT12] and Tremain ETFs [@FickusJMP18], as well as polyphase BIBD ETFs [@FickusJMPW19]. Here, to prevent duplication of effort, it helps to have the following concepts from [@FickusJ19], which unite the $(D_1,N_1)$ parameters of these disparate ETFs into a common framework:
\[def.pos neg ETFs\] For any ${\operatorname{ETF}}(D,N)$ with $N>D>1$, let $$L\in{\{{1,-1}\}},
\quad
S:={\bigl[{\tfrac{D(N-1)}{N-D}}\bigr]}^{\frac12},
\quad
K:=\tfrac{NS}{D(S+L)}.$$ When $S,K\in{\mathbb{Z}}$, we say this ETF is *type* $(K,L,S)$. In this case, depending on whether $L$ is $1$ or $-1$, we also refer to such an ETF as being ($K$-)*positive* or ($K$-)*negative*, respectively.
We caution that some ETFs are both positive and negative: for example, the well-known ${\operatorname{ETF}}(3,9)$ is both $2$-positive and $6$-negative, being both of type $(2,1,2)$ and $(6,-1,2)$. Regardless, for any ${\operatorname{ETF}}(D,N)$ of type $(K,L,S)$, Theorem 3.1 of [@FickusJ19] gives expressions for $(D,N)$ in terms of $(K,L,S)$: $$\begin{aligned}
\label{eq.pos neg D}
D
&=\tfrac{S}{K}[S(K-1)+L]\\
\label{eq.pos neg N}
N
&=(S+L)[S(K-1)+L].\end{aligned}$$ As summarized in [@FickusJ19], almost all currently known constructions of ${\operatorname{ETF}}(D,N)$ with $N>2D>2$ are either positive or negative, with the only exceptions being certain SIC-POVMs, harmonic ETFs, and examples where $N=2D+1$. As summarized in Theorems 1.2, 4.1 and 4.2 of [@FickusJ19], an ETF of type $(K,L,S)$ exists whenever either:
1. $(K,L,S)=(1,1,S)$ where $S\geq 2$ (regular simplices);
2. $(K,L,S)=(K,1,S)$ and a ${\operatorname{BIBD}}(V,K,1)$ exists where $V=(K-1)S+1$ (Steiner ETFs [@FickusMT12]), including:
1. when $K=2,3,4,5$, $S\geq K$, $S\equiv 0,1\bmod K$,
2. when $K\mid S(S-1)$ and $S$ is sufficiently large;
3. $(K,L,S)=(Q,1,Q)$ where $Q$ is any prime power (Naimark complements of polyphase BIBD ETFs [@FickusJMPW19]);
4. $(K,L,S)=(2,-1,S)$ where $S\geq 3$ (${\operatorname{ETF}}(D,N)$ with $D=\frac12(N+\sqrt{N})$);
5. $(K,L,S)=(3,-1,S)$ where $S\geq2$, $S\equiv0,2\bmod 3$ (Tremain ETFs [@FickusJMP18]);
6. $(K,L,S)=(Q+1,-1,Q+1)$ where $Q$ is an even prime power (hyperoval ETFs [@FickusMJ16]);
7. $(K,L,S)=(4,-1,S)$ where $S\equiv3\bmod8$ [@FickusJ19].
This is not a comprehensive list: for the sake of brevity and clarity, we have omitted some infinite families of negative ETFs that are either also (postive) Steiner ETFs or are overly technical (see, for example, Theorems 1.2, 1.3 and 4.4 of [@FickusJ19] for some additional $K$-negative ETFs with $K=4,5,6,7,10,12,15$), as well as a finite number of positive and/or negative ETFs for which, it turns out, our theory below does not apply.
Mutually Unbiased ETFs {#sec.MUETFs}
======================
Let ${\mathcal{M}}$, ${\mathcal{N}}$ and ${\mathcal{D}}$ be sets of cardinality $M$, $N$ and $D$, respectively, where $M\geq1$, $N\geq D\geq1$. For each $m\in{\mathcal{M}}$, let ${\{{{\boldsymbol{\psi}}_{m,n}}\}}_{n\in{\mathcal{N}}}$ be an ETF for ${\mathbb{F}}^{\mathcal{D}}$ with synthesis operator ${\boldsymbol{\Psi}}_m$. When $N=D$, this equates to a collection of $M$ orthonormal bases for ${\mathbb{F}}^{\mathcal{D}}$; in quantum information theory, one says that such bases are *mutually unbiased* if whenever $m\neq m'$. As we now explain, this same condition in general ensures that the concatenation of these ETFs has minimal coherence. This concatenation is an $MN$-vector UNTF for ${\mathbb{F}}^{\mathcal{D}}$, as its synthesis operator ${\boldsymbol{\Psi}}$ satisfies $${\boldsymbol{\Psi}}{\boldsymbol{\Psi}}^*
=\sum_{m\in{\mathcal{M}}}\sum_{n\in{\mathcal{N}}}{\boldsymbol{\psi}}_{m,n}^{}{\boldsymbol{\psi}}_{m,n}^*
=\sum_{m\in{\mathcal{M}}}{\boldsymbol{\Psi}}_m^{}{\boldsymbol{\Psi}}_m^*
=\tfrac{MN}{D}{\mathbf{I}}.$$ Thus gives . Here, ${\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}$ is an ${\mathcal{M}}\times{\mathcal{M}}$ block matrix whose $(m,m')$th block is the ${\mathcal{N}}\times{\mathcal{N}}$ *cross-Gram* matrix ${\boldsymbol{\Psi}}_m^*{\boldsymbol{\Psi}}_{m'}^{}$ whose $(n,n')$th entry is $({\boldsymbol{\Psi}}_m^*{\boldsymbol{\Psi}}_{m'}^{})(n,n')={\langle{{\boldsymbol{\psi}}_{m,n}},{{\boldsymbol{\psi}}_{m',n'}}\rangle}$. Thus, $$\tfrac{M^2N^2}{D}
={\|{{\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}}\|}_{{{\operatorname{Fro}}}}^2
=\sum_{m\in{\mathcal{M}}}\sum_{m'\in{\mathcal{M}}}{\|{{\boldsymbol{\Psi}}_m^*{\boldsymbol{\Psi}}_{m'}^{}}\|}_{{\operatorname{Fro}}}^2.$$ Moreover, for any $m\in{\mathcal{M}}$, ${\{{{\boldsymbol{\psi}}_{m,n}}\}}_{n\in{\mathcal{N}}}$ is a UNTF for ${\mathbb{F}}^{\mathcal{D}}$ and so gives ${\|{{\boldsymbol{\Psi}}_m^*{\boldsymbol{\Psi}}_m^{}}\|}_{{\operatorname{Fro}}}^2=\tfrac{N^2}{D}$. Subtracting these $M$ diagonal-block terms from the previous equation gives $$\begin{aligned}
\tfrac{M(M-1)N^2}{D}
&=\sum_{m\in{\mathcal{M}}}\sum_{m'\neq m}{\|{{\boldsymbol{\Psi}}_m^*{\boldsymbol{\Psi}}_{m'}^{}}\|}_{{\operatorname{Fro}}}^2\\
&\leq M(M-1)N^2
\max_{\substack{m\neq m'\\n,n'\in{\mathcal{N}}}}{|{{\langle{{\boldsymbol{\psi}}_{m,n}},{{\boldsymbol{\psi}}_{m',n'}}\rangle}}|}^2,\end{aligned}$$ where equality holds if and only if ${|{{\langle{{\boldsymbol{\psi}}_{m,n}},{{\boldsymbol{\psi}}_{m',n'}}\rangle}}|}^2=\frac1D$ whenever $m\neq m'$. Since for every $m\in{\mathcal{M}}$ we further have that ${|{{\langle{{\boldsymbol{\psi}}_{m,n}},{{\boldsymbol{\psi}}_{m,n'}}\rangle}}|}^2=\frac{N-D}{D(N-1)}\leq\frac1D$ for all $n\neq n'$, this is actually a lower bound on the coherence of the concatenation of any $M$ ${\operatorname{ETF}}(D,N)$ for ${\mathbb{F}}^{\mathcal{D}}$. This motivates the following:
\[def.MUETF\] Let ${\mathcal{M}}$, ${\mathcal{N}}$ and ${\mathcal{D}}$ be sets of cardinality $M\geq1$ and $N\geq D\geq1$, respectively. A sequence ${\{{{\boldsymbol{\psi}}_{m,n}}\}}_{m\in{\mathcal{M}},n\in{\mathcal{N}}}$ of unit vectors in ${\mathbb{F}}^{\mathcal{D}}$ is a *mutually-unbiased-equiangular tight frame* (MUETF) for ${\mathbb{F}}^{\mathcal{D}}$ if $$\label{eq.def of MUETF}
{|{{\langle{{\boldsymbol{\psi}}_{m,n}},{{\boldsymbol{\psi}}_{m',n'}}\rangle}}|}^2
=\left\{\begin{array}{cl}
\tfrac{N-D}{D(N-1)},&\ m=m', n\neq n',\smallskip\\
\frac1D,&\ m\neq m'.\end{array}\right.$$
We often denote an MUETF that consists of $M$ mutually unbiased ${\operatorname{ETF}}(D,N)$ as an “${\operatorname{MUETF}}(D,N,M)$". In the special case where $N=D$, such an MUETF equates to a collection of $M$ *mutually unbiased bases* (MUBs) for ${\mathbb{F}}^{\mathcal{D}}$. If instead $N=D+1$, this equates to $M$ *mutually unbiased simplices* (MUSs) for ${\mathbb{F}}^{\mathcal{D}}$ [@FickusS20; @Schmitt19].
We now derive an upper bound on the number $M$ of mutually unbiased ${\operatorname{ETF}}(D,N)$ that can exist. Any unit vector ${\boldsymbol{\varphi}}$ in ${\mathbb{F}}^{\mathcal{D}}$ “lifts" to a rank-one orthogonal projection operator ${\boldsymbol{\varphi}}{\boldsymbol{\varphi}}^*$, and the Frobenius inner product of any two such operators is $${\langle{{\boldsymbol{\varphi}}_{1}^{}{\boldsymbol{\varphi}}_{1}^*},{{\boldsymbol{\varphi}}_{2}^{}{\boldsymbol{\varphi}}_{2}^*}\rangle}_{{{\operatorname{Fro}}}}
={\operatorname{Tr}}({\boldsymbol{\varphi}}_{1}^{}{\boldsymbol{\varphi}}_{1}^*{\boldsymbol{\varphi}}_{2}^{}{\boldsymbol{\varphi}}_{2}^*)
={|{{\langle{{\boldsymbol{\varphi}}_{1}},{{\boldsymbol{\varphi}}_{2}}\rangle}}|}^2.$$ In particular, if ${\{{{\boldsymbol{\psi}}_{m,n}}\}}_{m\in{\mathcal{M}},n\in{\mathcal{N}}}$ is an MUETF for ${\mathbb{F}}^{\mathcal{D}}$, then the Gram matrix of the lifted vectors ${\{{{\boldsymbol{\psi}}_{m,n}^{}{\boldsymbol{\psi}}_{m,n}^*}\}}_{m\in{\mathcal{M}},n\in{\mathcal{N}}}$ is the entrywise-modulus-squared ${|{{\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}}|}^2$ of the Gram matrix ${\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}$ of ${\{{{\boldsymbol{\psi}}_{m,n}}\}}_{m\in{\mathcal{M}},n\in{\mathcal{N}}}$. Here, implies that every off-diagonal block of ${|{{\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}}|}^2$ is $\frac1D{\mathbf{J}}_{\mathcal{N}}$ (where ${\mathbf{J}}_{\mathcal{N}}$ denotes an all-ones ${\mathcal{N}}\times{\mathcal{N}}$ matrix) and that every diagonal block of ${|{{\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}}|}^2$ is $\tfrac{(D-1)N}{D(N-1)}{\mathbf{I}}_{\mathcal{N}}+\tfrac{N-D}{D(N-1)}{\mathbf{J}}_{\mathcal{N}}$. This equates to having $${|{{\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}}|}^2
=\tfrac{(D-1)N}{D(N-1)}{\mathbf{I}}-[\tfrac{D-1}{D(N-1)}{\mathbf{I}}-\tfrac1D{\mathbf{J}}_{{\mathcal{M}}}]\otimes{\mathbf{J}}_{{\mathcal{N}}}.$$ Diagonalizing this matrix reveals that it has eigenvalues $\tfrac{(D-1)N}{D(N-1)}$, $0$ and $\tfrac{MN}{D}$ with multiplicities $M(N-1)$, $M-1$ and $1$, respectively. In particular, ${|{{\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}}|}^2$ has rank $M(N-1)+1$. At the same time, the operators ${\{{{\boldsymbol{\psi}}_{m,n}^{}{\boldsymbol{\psi}}_{m,n}^*}\}}_{m\in{\mathcal{M}},n\in{\mathcal{N}}}$ lie in the real inner product space of all self-adjoint ${\mathcal{D}}\times{\mathcal{D}}$ matrices, meaning the rank of their Gram matrix ${|{{\boldsymbol{\Psi}}^*{\boldsymbol{\Psi}}}|}^2$ is at most the dimension of this space, namely $D^2$ when the underlying field ${\mathbb{F}}$ is ${\mathbb{C}}$, and $\frac12D(D+1)$ when ${\mathbb{F}}={\mathbb{R}}$. That is, $$\label{eq.Gerzon}
M(N-1)+1\leq\left\{\begin{array}{cl}D^2,&\ {\mathbb{F}}={\mathbb{C}},\\\frac12D(D+1),&\ {\mathbb{F}}={\mathbb{R}}. \end{array}\right.$$ When $M=1$, this reduces to a necessary condition on ${\operatorname{ETF}}(D,N)$ known as Gerzon’s bound. More generally, solving for $M$ in the above inequality gives the following result:
\[thm.Gerzon\] If an ${\operatorname{MUETF}}(D,N,M)$ exists and $N>1$ then $$M\leq\left\{\begin{array}{cl}
\lfloor\frac{D^2-1}{N-1}\rfloor,&\ {\mathbb{F}}={\mathbb{C}},\smallskip\\
\lfloor\frac{(D-1)(D+2)}{2(N-1)}\rfloor,&\ {\mathbb{F}}={\mathbb{R}}.
\end{array}\right.$$
In the special case where $N=D$, this reduces to the classical upper bound on the maximal number of MUBs, namely that $M\leq N+1$ when ${\mathbb{F}}={\mathbb{C}}$ and that $M\leq\lfloor\frac D2\rfloor+1$ when ${\mathbb{F}}={\mathbb{R}}$. In the special case where $N=D+1$, this reduces to a recently derived upper bound on the maximal number of MUSs [@Schmitt19]. As detailed below, in at least these two special cases, there are an infinite number of values of $D$ for which these bounds can be achieved.
Constructing MUETFs from relative difference sets
-------------------------------------------------
From Section \[sec.background\], recall that restricting and then normalizing the characters of any abelian group ${\mathcal{G}}$ of order $G$ to a nonempty $D$-element subset ${\mathcal{D}}$ of ${\mathcal{G}}$ yields a harmonic UNTF ${\{{{\boldsymbol{\psi}}_\gamma}\}}_{\gamma\in\hat{{\mathcal{G}}}}$, for ${\mathbb{C}}^{\mathcal{D}}$. Further recall that for any subgroup ${\mathcal{H}}$ of ${\mathcal{G}}$ of order $H$, such a subset ${\mathcal{D}}$ is an ${\mathcal{H}}$-${\operatorname{RDS}}(N,H,D,\Lambda)$ (Definition \[def.RDS\]) if and only if this harmonic UNTF satisfies where $N=\frac{G}{H}$. Comparing to immediately gives that such a harmonic UNTF yields a (harmonic) MUETF, where each individual ETF is indexed by a coset of ${\mathcal{H}}^\perp$. To formalize this connection, we abuse notation, letting the indexing “$\alpha\in\hat{{\mathcal{G}}}/{\mathcal{H}}^\perp$" denote letting $\alpha$ vary over any particular transversal (set of coset representatives) of ${\mathcal{H}}^\perp$ with respect to $\hat{{\mathcal{G}}}$. Doing so permits us to uniquely factor any $\gamma\in\hat{{\mathcal{G}}}$ as $\gamma=\alpha\beta$ where $\beta\in{\mathcal{H}}^\perp$, at which point comparing to gives:
\[thm.RDS gives MUETF\] Letting ${\mathcal{H}}$ and ${\mathcal{D}}$ be a subgroup and nonempty subset of a finite abelian group ${\mathcal{G}}$, respectively, the sequence $${\{{{\boldsymbol{\psi}}_{\alpha,\beta}}\}}_{\alpha\in\hat{{\mathcal{G}}}/{\mathcal{H}}^\perp,\,\beta\in{\mathcal{H}}^\perp}
\subseteq{\mathbb{C}}^{\mathcal{D}},
\quad{\boldsymbol{\psi}}_{\alpha,\beta}(d):=D^{-\frac12}\alpha(d)\beta(d),$$ is an ${\operatorname{MUETF}}(D,N,H)$ for ${\mathbb{C}}^{\mathcal{D}}$ (Definition \[def.MUETF\]) if and only if ${\mathcal{D}}$ is an ${\mathcal{H}}$-${\operatorname{RDS}}(N,H,D,\Lambda)$ for ${\mathcal{G}}$ (Definition \[def.RDS\]).
In general, we refer to any MUETF created by Theorem \[thm.RDS gives MUETF\] as a *harmonic* MUETF. In the special case where ${\mathcal{H}}={\{{0}\}}$, Theorem \[thm.RDS gives MUETF\] reduces to the known equivalence between harmonic ETFs and difference sets [@XiaZG05; @DingF07]. Meanwhile, in the special case where $D=N$, Theorem \[thm.RDS gives MUETF\] converts any ${\mathcal{H}}$-${\operatorname{RDS}}(D,H,D,\Lambda)$ into $H$ MUBs for ${\mathbb{C}}^{\mathcal{D}}$ in a manner identical to that of [@GodsilR09]. If instead $N=D+1$, Theorem \[thm.RDS gives MUETF\] yields $H$ MUSs for ${\mathbb{C}}^{\mathcal{D}}$ [@FickusS20; @Schmitt19]. As we shall see, Theorem \[thm.RDS gives MUETF\] is a true generalization of these previously known results, yielding infinite numbers of MUETFs that do not belong to any one of these three special categories.
Moving forward, it helps to note that if ${\mathcal{D}}$ is any ${\mathcal{H}}$-RDS for ${\mathcal{G}}$, then by a simple counting argument, quotienting it by any subgroup ${\mathcal{K}}$ of ${\mathcal{H}}$ produces an $({\mathcal{H}}/{\mathcal{K}})$-RDS for ${\mathcal{G}}/{\mathcal{K}}$, namely ${\mathcal{D}}/{\mathcal{K}}:={\{{d+{\mathcal{K}}\in{\mathcal{G}}/{\mathcal{K}}: d\in{\mathcal{D}}}\}}$ [@Pott95]. In particular, if ${\mathcal{K}}$ has order $K$, doing so transforms an ${\operatorname{RDS}}(N,H,D,\Lambda)$ into an ${\operatorname{RDS}}(N,\frac{H}{K},D,K\Lambda)$. As an extreme case, quotienting an ${\mathcal{H}}$-RDS by ${\mathcal{K}}={\mathcal{H}}$ yields a difference set for ${\mathcal{G}}/{\mathcal{H}}$. From the perspective of Theorem \[thm.RDS gives MUETF\], these are special cases of more general ideas, namely that any subcollection of $M$ mutually unbiased ${\operatorname{ETF}}(D,N)$ are still mutually unbiased, and that any single one of them is an ETF.
\[rem.harmonic MUETFF\] On a related note, the particular ${\operatorname{ETF}}(D,N)$ that arises from Theorem \[thm.RDS gives MUETF\] by taking $\alpha=1$ is identical to the harmonic ETF that arises from the difference set ${\mathcal{D}}/{\mathcal{H}}$ for ${\mathcal{G}}/{\mathcal{H}}$. To elaborate, for any $\beta\in{\mathcal{H}}^\perp$ and $d\in{\mathcal{D}}$, $$\label{eq.RDS 4}
{\boldsymbol{\psi}}_{1,\beta}(d)
=D^{-\frac12}1(d)\beta(d)
=D^{-\frac12}\beta(d).$$ At the same time, ${\mathcal{H}}^\perp$ is naturally identified with the Pontryagin dual of ${\mathcal{G}}/{\mathcal{H}}$ via the isomorphism that maps any given $\gamma\in{\mathcal{H}}^\perp$ to the character $g+{\mathcal{H}}\mapsto\gamma(g)$. Under this identification, evaluating the $\beta$th member of the harmonic ETF arising from ${\mathcal{D}}/{\mathcal{H}}$ at $d+{\mathcal{H}}$ gives . For this reason, for any ${\mathcal{H}}$-RDS ${\mathcal{D}}$ for ${\mathcal{G}}$, we usually regard ${\{{{\boldsymbol{\psi}}_\beta}\}}_{\beta\in{\mathcal{H}}^\perp}$, ${\boldsymbol{\psi}}_\beta:={\boldsymbol{\psi}}_{1,\beta}$, as the “prototypical" ETF that arises from it. Indeed, for any $\alpha\in\hat{{\mathcal{G}}}$, letting ${\boldsymbol{\Psi}}_\alpha$ be the synthesis operator for ${\{{{\boldsymbol{\psi}}_{\alpha,\beta}}\}}_{\beta\in{\mathcal{H}}^\perp}$, we have ${\boldsymbol{\Psi}}_\alpha={\boldsymbol{\Delta}}_\alpha{\boldsymbol{\Psi}}$ where ${\boldsymbol{\Psi}}:={\boldsymbol{\Psi}}_1$ is the synthesis operator of ${\{{{\boldsymbol{\psi}}_\beta}\}}_{\beta\in{\mathcal{H}}^\perp}$ and where ${\boldsymbol{\Delta}}_\alpha$ is the ${\mathcal{D}}\times{\mathcal{D}}$ unitary diagonal matrix whose $d$th diagonal entry is $\alpha(d)$: $${\boldsymbol{\Psi}}_\alpha(d,\beta)
=D^{-\frac12}\alpha(d)\beta(d)
=({\boldsymbol{\Delta}}_\alpha{\boldsymbol{\psi}}_\beta)(d)
=({\boldsymbol{\Delta}}_\alpha{\boldsymbol{\Psi}})(d,\beta).$$
Constructions of harmonic MUETFs
--------------------------------
As noted in [@GodsilR09], in the special case where $N=D$, applying Theorem \[thm.RDS gives MUETF\] to an ${\mathcal{H}}$-${\operatorname{RDS}}(D,H,D,\Lambda)$ actually implies the existence of $H+1$ MUBs for ${\mathbb{C}}^{\mathcal{D}}$: since ${|{{\boldsymbol{\psi}}_{\alpha,\beta}(d)}|}=D^{-\frac12}$ for all $\alpha$, $\beta$ and $d$, every orthonormal basis ${\{{{\boldsymbol{\psi}}_{\alpha,\beta}}\}}_{\beta\in{\mathcal{H}}^\perp}$ is also unbiased to the standard basis. This is especially significant since, for any prime power $Q$, there is a classical construction of an ${\operatorname{RDS}}(Q,Q,Q,1)$ [@Pott95] which in turn yields $Q+1$ (the maximal number of) MUBs in ${\mathbb{C}}^Q$. When $Q$ is odd, the construction is shockingly simple: let $${\mathcal{G}}={\mathbb{F}}_Q\times{\mathbb{F}}_Q,
\
{\mathcal{H}}={\{{0}\}}\times{\mathbb{F}}_Q,
\
{\mathcal{D}}={\{{(x,x^2): x\in{\mathbb{F}}_Q}\}},$$ where here and throughout, ${\mathbb{F}}_Q$ denotes the finite field of order $Q$. Indeed, if $(x,x^2)-(y,y^2)\in{\mathcal{H}}={\{{0}\}}\times{\mathbb{F}}_Q$ then $x=y$ and so $(x,x^2)-(y,y^2)=(0,0)$. Meanwhile, for any $(a,b)\in{\mathcal{H}}^{\mathrm{c}}$ we have $a\neq0$, and so there exists exactly one pair $(x,x^2),(y,y^2)$ such that $$(a,b)
=(x,x^2)-(y,y^2)
=(x-y,(x-y)(x+y)),$$ namely the pair arising from $x=\frac12(\frac ba+a)$ and $y=\frac12(\frac ba-a)$. The construction is more complicated when $Q$ is even [@Pott95]. This is not surprising since in that case the characters of ${\mathbb{F}}_Q\times{\mathbb{F}}_Q$ are real-valued, and there are at most $\frac Q2+1$ MUBs in ${\mathbb{R}}^Q$.
The proof of our main result (Theorem \[thm.main result\]) relies on mutually unbiased ETFs that are not MUBs, and which arise from another classical RDS construction [@Pott95]. For any prime power $Q$ and $J\geq2$, regard ${\mathbb{F}}_{Q^J}$ as a $J$-dimensional vector space over its subfield ${\mathbb{F}}_Q$. Let ${\operatorname{tr}}:{\mathbb{F}}_{Q^J}\rightarrow{\mathbb{F}}_Q$, ${\operatorname{tr}}(x):=\sum_{j=0}^{J-1}x^{Q^j}$ be the field trace, which is a nontrivial linear functional. Let be the (cyclic) multiplicative group of ${\mathbb{F}}_{Q^J}$, let ${\mathcal{H}}={\mathbb{F}}_Q^\times$, and consider the affine hyperplane $$\label{eq.Bose RDS}
{\mathcal{D}}={\{{x\in{\mathbb{F}}_{Q^J}^\times : {\operatorname{tr}}(x)=1}\}},$$ which has cardinality $Q^{J-1}$. For any , $$\label{eq.RDS 5}
{\mathcal{D}}\cap(y{\mathcal{D}})={\{{x\in{\mathbb{F}}_{Q^J}^\times : {\operatorname{tr}}(x)=1={\operatorname{tr}}(y^{-1}x)}\}}.$$ For any $y\in{\mathbb{F}}_Q^\times$, $y\neq1$, the linearity of the trace gives ${\operatorname{tr}}(y^{-1}x)=y^{-1}{\operatorname{tr}}(x)$, implying ${\mathcal{D}}\cap(y{\mathcal{D}})$ is empty. Meanwhile, for any $y\notin{\mathbb{F}}_Q^\times$, we can take ${\{{1,y^{-1}}\}}$ as the first two vectors in a basis ${\{{z_j}\}}_{j=1}^{J}$ for ${\mathbb{F}}_{Q^J}$ over ${\mathbb{F}}_Q$. Since the trace is nontrivial, the “analysis operator" of this basis, namely $${\mathbf{L}}:{\mathbb{F}}_{Q^J}\rightarrow{\mathbb{F}}_Q^J,
\quad
{\mathbf{L}}(x)=({\operatorname{tr}}(z_1x),\dotsc,{\operatorname{tr}}(z_Jx)),$$ has a trivial null space and is thus invertible. Letting ${\mathbf{A}}$ be the $2\times J$ matrix whose rows are the first two rows of ${\mathbf{I}}$, we thus have that the mapping $x\mapsto{\mathbf{A}}{\mathbf{L}}(x)=({\operatorname{tr}}(x),{\operatorname{tr}}(y^{-1}x))$ has rank two. This implies is an affine subspace of codimension $2$ and so has cardinality $Q^{J-2}$. Altogether, $$\#[{\mathcal{D}}\cap(y{\mathcal{D}})]
=\left\{\begin{array}{cl}
Q^{J-1},&y=1,\smallskip\\
0, &y\in{\mathbb{F}}_Q^\times,\ y\neq 1,\smallskip\\
Q^{J-2},&y\notin{\mathbb{F}}_Q^\times,
\end{array}\right.$$ and so ${\mathcal{D}}$ is an ${\operatorname{RDS}}(\frac{Q^J-1}{Q-1},Q-1,Q^{J-1},Q^{J-2})$ for ${\mathbb{F}}_{Q^J}^\times$.
Quotienting this ${\mathcal{H}}$-RDS by ${\mathcal{H}}$ gives a difference set ${\mathcal{D}}/{\mathbb{F}}_Q^\times$ for ${\mathbb{F}}_{Q^J}^\times/{\mathbb{F}}_Q^\times$; since the trace is linear over ${\mathbb{F}}_Q^\times$, it is $$\begin{aligned}
{\mathcal{D}}/{\mathbb{F}}_Q^\times
={\{{x{\mathbb{F}}_Q^\times\in{\mathbb{F}}_{Q^J}^\times/{\mathbb{F}}_Q^\times : {\operatorname{tr}}(x)=1}\}}\\
={\{{x{\mathbb{F}}_Q^\times\in{\mathbb{F}}_{Q^J}^\times/{\mathbb{F}}_Q^\times : {\operatorname{tr}}(x)\neq0}\}},\end{aligned}$$ namely the complement of the well-known *Singer* difference set for ${\mathbb{F}}_{Q^J}^\times/{\mathbb{F}}_Q^\times$, defined as ${\{{x{\mathbb{F}}_Q^\times\in{\mathbb{F}}_{Q^J}^\times/{\mathbb{F}}_Q^\times : {\operatorname{tr}}(x)=0}\}}$.
Applying Theorem \[thm.RDS gives MUETF\] to this RDS yields $Q-1$ mutually unbiased ${\operatorname{ETF}}(Q^{J-1},\frac{Q^J-1}{Q-1})$. To make this construction more explicit, let $\alpha$ be a generator of ${\mathbb{F}}_{Q^J}^\times$, and consider the isomorphism $\alpha^t\mapsto t$ from this group onto ${\mathbb{Z}}_{Q^J-1}$. Under this isomorphism, ${\mathcal{G}}$, ${\mathcal{H}}$ and ${\mathcal{D}}$ become $${\mathcal{G}}={\mathbb{Z}}_{Q^J-1},\
{\mathcal{H}}=\langle \tfrac{Q^J-1}{Q-1}\rangle,\
{\mathcal{D}}={\{{d\in{\mathbb{Z}}_{Q^J-1}: {\operatorname{tr}}(\alpha^d)=1}\}}.$$ This allows us to also regard $\hat{{\mathcal{G}}}$ as ${\mathbb{Z}}_{Q^J-1}$, isomorphically identifying $s\in{\mathbb{Z}}_{Q^J-1}$ with the character $t\mapsto\exp(\frac{2\pi{\mathrm{i}}st}{Q^J-1})$. Under this identification, ${\mathcal{H}}^\perp$ becomes $$\begin{aligned}
{\mathcal{H}}^\perp
&={\{{s\in{\mathbb{Z}}_{Q^J-1}:
\exp(\tfrac{2\pi{\mathrm{i}}st}{Q^J-1})=1,\ \forall\, t\in\langle\tfrac{Q^J-1}{Q-1}\rangle}\}}\\
&=\langle Q-1 \rangle.\end{aligned}$$ Any $s\in{\mathbb{Z}}_{Q^J-1}$ can be uniquely written as $s=m+(Q-1)n$ where $m$ lies in the transversal ${\{{0,1,2,\dotsc,Q-2}\}}$ of $\langle Q-1 \rangle$ and . Under these identifications, the $n$th member of the $m$th mutually unbiased ${\operatorname{ETF}}(Q^{J-1},\frac{Q^J-1}{Q-1})$ for ${\mathbb{C}}^{\mathcal{D}}$ produced by Theorem \[thm.RDS gives MUETF\] becomes $$\begin{aligned}
{\boldsymbol{\psi}}_{m,n}(d)
&=Q^{-\frac{J-1}2}\exp(\tfrac{2\pi{\mathrm{i}}[m+(Q-1)n]d}{Q^J-1})\\
&=\exp(\tfrac{2\pi{\mathrm{i}}md}{Q^J-1})
Q^{-\frac{J-1}2}\exp(\tfrac{2\pi{\mathrm{i}}(Q-1)nd}{Q^J-1}).\end{aligned}$$ Here, by Definition \[def.MUETF\], ${|{{\langle{{\boldsymbol{\psi}}_{m,n}},{{\boldsymbol{\psi}}_{m',n'}}\rangle}}|}^2$ has value $\frac1{Q^{J-1}}$ whenever $m\neq m'$, and otherwise has value $$\begin{aligned}
[\tfrac{Q^{J}-1}{Q-1}-Q^{J-1}][Q^{J-1}(\tfrac{Q^{J}-1}{Q-1}-1)]^{-1}
=\tfrac1{Q^J}.\end{aligned}$$ Combining these facts with the perspective of Remark \[rem.harmonic MUETFF\] gives:
\[cor.Singer MUETFF\] For any prime power $Q$ and any integer $J\geq2$, let $\alpha$ be a generator of ${\mathbb{F}}_{Q^J}^\times$ and let $${\mathcal{D}}={\biggl\{{d\in{\mathbb{Z}}_{Q^J-1}: {\operatorname{tr}}(\alpha^d)=\sum_{j=0}^{J-1}\alpha^{dQ^j}=1}\biggr\}}
\subseteq{\mathbb{Z}}_{Q^J-1}.$$ Letting ${\boldsymbol{\Delta}}$ be the ${\mathcal{D}}\times{\mathcal{D}}$ unitary diagonal matrix whose $d$th diagonal entry is $\exp(\tfrac{2\pi{\mathrm{i}}d}{Q^J-1})$, $${\{{{\boldsymbol{\psi}}_{m,n}}\}}_{m=0,}^{Q-2}\,_{n=0}^{\frac{Q^J-1}{Q-1}-1}\subseteq{\mathbb{C}}^{\mathcal{D}},
\quad
{\boldsymbol{\psi}}_{m,n}:={\boldsymbol{\Delta}}^m{\boldsymbol{\psi}}_n,$$ is an ${\operatorname{MUETF}}(Q^{J-1},\frac{Q^J-1}{Q-1},Q-1)$ for ${\mathbb{C}}^{\mathcal{D}}$ where $${\{{{\boldsymbol{\psi}}_n}\}}_{n=0}^{\!\frac{Q^J-1}{Q-1}-1}\subseteq{\mathbb{C}}^{\mathcal{D}},
\quad
{\boldsymbol{\psi}}_n(d):=Q^{-\frac{J-1}2}\exp(\tfrac{2\pi{\mathrm{i}}(Q-1)nd}{Q^J-1}),$$ is the harmonic ${\operatorname{ETF}}(Q^{J-1},\frac{Q^J-1}{Q-1})$ for ${\mathbb{C}}^{\mathcal{D}}$ arising from the complement of a Singer difference set. In particular, $${|{{\langle{{\boldsymbol{\psi}}_{m,n}},{{\boldsymbol{\psi}}_{m',n'}}\rangle}}|}^2
=\left\{\begin{array}{cl}
\frac1{Q^J},&\ m=m',\, n\neq n',\smallskip\\
\frac1{Q^{J-1}},&\ m\neq m'.
\end{array}\right.$$
\[ex.MUETF(4,5,3)\] When $Q=4$ and $J=2$, $X^4+X+1$ is a primitive polynomial in ${\mathbb{F}}_2[X]$, meaning $\alpha=X+\langle X^4+X+1\rangle$ generates the multiplicative group of $$\begin{aligned}
{\mathbb{F}}_{16}
&={\mathbb{F}}_2[X]/\langle X^4+X+1\rangle\\
&={\{{a+b\alpha+c\alpha^2+d\alpha^3: a,b,c,d\in{\mathbb{F}}_2,\ \alpha^4=\alpha+1}\}}.\end{aligned}$$ As such, elements of ${\mathbb{F}}_{16}$ can be represented as the powers of the $2\times 2$ companion matrix ${\mathbf{A}}$ of $\alpha$, whose entries lie in ${\mathbb{F}}_2$: $${\mathbf{A}}=\left[\begin{smallmatrix}
0&0&0&1\\
1&0&0&1\\
0&1&0&0\\
0&0&1&0
\end{smallmatrix}\right].$$ This representation facilitates the computation of $$\begin{aligned}
{\mathcal{D}}&={\{{d\in{\mathbb{Z}}_{15}: {\operatorname{tr}}(\alpha^d)=1}\}}\\
&={\{{d\in{\mathbb{Z}}_{15}: {\mathbf{A}}^d+{\mathbf{A}}^{4d}={\mathbf{I}}}\}}\\
&={\{{1,2,8,4}\}}.\end{aligned}$$ This ${\operatorname{RDS}}(5,3,4,1)$ yields three mutually unbiased ${\operatorname{ETF}}(4,5)$ with synthesis operators ${\boldsymbol{\Psi}}$, ${\boldsymbol{\Delta}}{\boldsymbol{\Psi}}$, ${\boldsymbol{\Delta}}^2{\boldsymbol{\Psi}}$ where $\omega={\mathrm{e}}^{\frac{2\pi{\mathrm{i}}}{15}}$, $${\boldsymbol{\Delta}}=\left[\begin{array}{cccc}
\omega &0&0&0\\
0&\omega^2&0&0\\
0&0&\omega^8&0\\
0&0&0&\omega^4
\end{array}\right],
\
{\boldsymbol{\Psi}}=\frac12\left[\begin{array}{rrrrr}
1&\omega^{ 3}&\omega^{ 6}&\omega^{ 9}&\omega^{12}\\
1&\omega^{ 6}&\omega^{12}&\omega^{ 3}&\omega^{ 9}\\
1&\omega^{ 9}&\omega^{ 3}&\omega^{12}&\omega^{ 6}\\
1&\omega^{12}&\omega^{ 9}&\omega^{ 6}&\omega^{ 3}\\
\end{array}\right].$$ Here, ${\boldsymbol{\Psi}}$ is also the synthesis operator of the harmonic ETF that arises from the difference set ${\{{1,2,3,4}\}}$ in ${\mathbb{Z}}_5$ that itself arises by quotienting the ${\mathcal{D}}={\{{1,2,8,4}\}}\subseteq{\mathbb{Z}}_{15}$ by ${\mathcal{H}}=\langle 5\rangle$.
As seen in this example, in the special case where $J=2$, Corollary \[cor.Singer MUETFF\] yields $Q-1$ mutually unbiased ${\operatorname{ETF}}(Q,Q+1)$, namely $Q-1$ mutually unbiased $Q$-simplices. Such MUSs recently arose [@FickusS20] in a study of harmonic ETFs that are a disjoint union of regular simplices [@FickusJKM18]. In the next section, we reverse some of the analysis of [@FickusJKM18; @FickusS20], using the MUETFs constructed in Corollary \[cor.Singer MUETFF\] to produce new ETFs. Before doing so, note that by Theorem \[thm.Gerzon\], the maximal number of complex mutually unbiased ${\operatorname{ETF}}(D,D+1)$ is at most $$\lfloor\tfrac{D^2-1}{(D+1)-1}\rfloor
=\lfloor D-\tfrac1D\rfloor
=D-1.$$ Altogether, we see that for any prime power $Q$, the maximal number of mutually unbiased complex ${\operatorname{ETF}}(Q,Q+1)$ is exactly $Q-1$. Meanwhile, when $J\geq3$, there is a gap between the number $Q-1$ of mutually unbiased complex ${\operatorname{ETF}}(Q^{J-1},\frac{Q^J-1}{Q-1})$ produced by Corollary \[cor.Singer MUETFF\] and the upper bound on this number provided by Theorem \[thm.Gerzon\], namely $$\lfloor\tfrac{D^2-1}{N-1}\rfloor
=\lfloor\tfrac{(Q-1)(Q^{J-1}+1)}{Q}\rfloor
=Q^{J-2}(Q-1).$$ This gap persists even if, following [@GodsilR09], we refine the analysis of so as to account for the fact that the MUETF vectors ${\boldsymbol{\psi}}_{m,n}$ constructed by Corollary \[cor.Singer MUETFF\] have entries of constant modulus $D^{-\frac12}$. To elaborate, in this case each ${\boldsymbol{\psi}}_{m,n}$ lifts to an orthogonal projection operator ${\boldsymbol{\psi}}_{m,n}^{}{\boldsymbol{\psi}}_{m,n}^*$ which lies in real $(D^2-D+1)$-dimensional space of complex Hermitian ${\mathcal{D}}\times{\mathcal{D}}$ matrices with constant diagonal entries. As such, refines to $M(N-1)+1\leq D^2-D+1$, namely $$M
\leq\lfloor\tfrac{D(D-1)}{N-1}\rfloor
=\Bigl\lfloor\tfrac{Q^{J-1}(Q^{J-1}-1)}{Q(\frac{Q^{J-1}-1}{Q-1})}\Bigr\rfloor
=Q^{J-2}(Q-1).$$
Though this might seem like an esoteric issue, in the next section, we introduce a way of constructing new ETFs from MUETFs, and this method very much depends on the number of mutually unbiased ETFs available. And remarkably, the RDS literature itself warns that Corollary \[cor.Singer MUETFF\] can be improved upon: when $Q=4$ and $J=3$, Corollary \[cor.Singer MUETFF\] yields $3$ mutually unbiased ${\operatorname{ETF}}(16,21)$ arising from an ${\operatorname{RDS}}(21,3,16,4)$ when in fact, an ${\operatorname{RDS}}(21,6,16,2)$ for ${\mathbb{Z}}_{126}$ exists [@Pott95], and applying Theorem \[rem.harmonic MUETFF\] to it yields $6$ mutually unbiased ${\operatorname{ETF}}(16,21)$. We leave a deeper investigation of this issue for future work.
New ETFs from MUETFs {#sec.new ETFs}
====================
In this section, we show how to combine the MUETFs of the previous section with certain other known ETFs to produce yet more ETFs. Though MUETFs are an exotic ingredient, the recipe itself is simple:
\[thm.tensor\] If ${\{{{\boldsymbol{\varphi}}_{n_1}}\}}_{n_1\in{\mathcal{N}}_1}$ is an ${\operatorname{ETF}}(D_1,N_1)$ for ${\mathbb{F}}^{{\mathcal{D}}_1}$ and ${\{{{\boldsymbol{\psi}}_{n_1,n_2}}\}}_{n_1\in{\mathcal{N}}_1,n_2\in{\mathcal{N}}_2}$ is an ${\operatorname{MUETF}}(D_2,N_2,N_1)$ for ${\mathbb{F}}^{{\mathcal{D}}_2}$, and these parameters satisfy $$\label{eq.consistency}
\tfrac{N_1-D_1}{D_1(N_1-1)}
=\tfrac{N_2-D_2}{N_2-1},$$ then ${\{{{\boldsymbol{\varphi}}_{n_1}\otimes{\boldsymbol{\psi}}_{n_1,n_2}}\}}_{(n_1,n_2)\in{\mathcal{N}}_1\times{\mathcal{N}}_2}$ is an ${\operatorname{ETF}}(D_3,N_3)$ for ${\mathbb{F}}^{{{\mathcal{D}}_1}\times{{\mathcal{D}}_2}}$, where $D_3=D_1D_2$, $N_3=N_1N_2$ and $$\begin{aligned}
\label{eq.tensor 1}
N_3-D_3&=D_1N_1(N_2-D_2)+(N_1-D_1),\\[2pt]
\label{eq.tensor 2}
\tfrac{N_3-D_3}{D_3(N_3-1)}&=\tfrac{N_2-D_2}{D_2(N_2-1)},\\[2pt]
\label{eq.tensor 3}
\tfrac{D_3(N_3-D_3)}{N_3-1}&=D_1^2\tfrac{D_2(N_2-D_2)}{N_2-1}.\end{aligned}$$ Moreover, if $\frac{D_2(N_2-D_2)}{N_2-1},\frac{(N_3-D_3)(N_3-1)}{D_3}\in{\mathbb{Z}}$ then $\frac{N_1}{D_1}\in{\mathbb{Z}}$.
Since ${\{{{\boldsymbol{\varphi}}_{n_1}}\}}_{n_1\in{\mathcal{N}}_1}$ is an ${\operatorname{ETF}}(D_1,N_1)$, $${|{{\langle{{\boldsymbol{\varphi}}_{n_1}},{{\boldsymbol{\varphi}}_{n_1'}}\rangle}}|}^2
=\left\{\begin{array}{cl}
1,&n_1=n_1',\smallskip\\
\tfrac{N_1-D_1}{D_1(N_1-1)},& n_1\neq n_1'.
\end{array}\right.$$ Also ${\{{{\boldsymbol{\psi}}_{n_1,n_2}}\}}_{n_1\in{\mathcal{N}}_1,n_2\in{\mathcal{N}}_2}$ is an ${\operatorname{MUETF}}(D_2,N_2,N_1)$, and so Definition \[def.MUETF\] gives $${|{{\langle{{\boldsymbol{\psi}}_{n_1,n_2}},{{\boldsymbol{\psi}}_{n_1',n_2'}}\rangle}}|}^2
=\left\{\begin{array}{cl}
1,&(n_1,n_2)=(n_1',n_2'),\smallskip\\
\tfrac{N_2-D_2}{D_2(N_2-1)},&n_1=n_1', n_2\neq n_2',\smallskip\\
\frac1{D_2},&n_1\neq n_1'.\end{array}\right.$$ For any $(n_1,n_2),(n_1',n_2')\in{\mathcal{N}}_1\times{\mathcal{N}}_2$, multiplying these expressions gives $$\begin{gathered}
{|{{\langle{{\boldsymbol{\varphi}}_{n_1}\otimes{\boldsymbol{\psi}}_{n_1,n_2}},{{\boldsymbol{\varphi}}_{n_1'}\otimes{\boldsymbol{\psi}}_{n_1',n_2'}}\rangle}}|}^2\\
\label{eq.pf of tensor 1}
=\left\{\begin{array}{cl}
1,&(n_1,n_2)=(n_1',n_2'),\smallskip\\
\tfrac{N_2-D_2}{D_2(N_2-1)},&n_1=n_1', n_2\neq n_2',\smallskip\\
\tfrac{N_1-D_1}{D_2D_1(N_1-1)},&n_1\neq n_1'.\end{array}\right.\end{gathered}$$ In particular, equates to ${\{{{\boldsymbol{\varphi}}_{n_1}\otimes{\boldsymbol{\psi}}_{n_1,n_2}}\}}_{(n_1,n_2)\in{\mathcal{N}}_1\times{\mathcal{N}}_2}$ being equiangular. To prove that this sequence of vectors is a tight frame for ${\mathbb{F}}^{{\mathcal{D}}_1\times{\mathcal{D}}_2}$ note that for any $n_1\in{\mathcal{N}}_1$, the fact that ${\{{{\boldsymbol{\psi}}_{n_1,n_2}}\}}_{n_2\in{\mathcal{N}}_2}$ achieves the Welch bound for $N_2$ vectors in ${\mathbb{F}}^{{\mathcal{D}}_2}$ implies it is necessarily a UNTF for ${\mathbb{F}}^{{\mathcal{D}}_2}$, i.e., that its synthesis operator ${\boldsymbol{\Psi}}_{n_1}$ satisfies ${\boldsymbol{\Psi}}_{n_1}^{}{\boldsymbol{\Psi}}_{n_1}^*=\frac{N_2}{D_2}{\mathbf{I}}_{{\mathcal{D}}_2}$. Similarly, the synthesis operator ${\boldsymbol{\Phi}}$ of ${\{{{\boldsymbol{\varphi}}_{n_1}}\}}_{n_1\in{\mathcal{N}}_1}$ satisfies ${\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^*=\frac{N_1}{D_1}{\mathbf{I}}_{{\mathcal{D}}_1}$. Thus, the frame operator of ${\{{{\boldsymbol{\varphi}}_{n_1}\otimes{\boldsymbol{\psi}}_{n_1,n_2}}\}}_{(n_1,n_2)\in{\mathcal{N}}_1\times{\mathcal{N}}_2}$ is $$\begin{gathered}
\sum_{n_1\in{\mathcal{N}}_1}{\boldsymbol{\varphi}}_{n_1}^{}{\boldsymbol{\varphi}}_{n_1}^*\otimes
{\biggl({\,\sum_{n_2\in{\mathcal{N}}_2}{\boldsymbol{\psi}}_{n_1,n_2}^{}{\boldsymbol{\psi}}_{n_1,n_2}^*}\biggr)}\\
=\sum_{n_1\in{\mathcal{N}}_1}{\boldsymbol{\varphi}}_{n_1}^{}{\boldsymbol{\varphi}}_{n_1}^*\otimes(\tfrac{N_2}{D_2}{\mathbf{I}}_{{\mathcal{D}}_2})
=\tfrac{N_1N_2}{D_1D_2}{\mathbf{I}}_{{{\mathcal{D}}_1}\times{{\mathcal{D}}_2}}.\end{gathered}$$ Thus, ${\{{{\boldsymbol{\varphi}}_{n_1}\otimes{\boldsymbol{\psi}}_{n_1,n_2}}\}}_{(n_1,n_2)\in{\mathcal{N}}_1\times{\mathcal{N}}_2}$ is an ${\operatorname{ETF}}(D_3,N_3)$ for ${\mathbb{F}}^{{\mathcal{D}}_1\times{\mathcal{D}}_2}$ where $D_3=D_1D_2$ and $N_3=N_1N_2$. By , the Welch bound of this ${\operatorname{ETF}}(D_3,N_3)$ equals that of our mutually unbiased ${\operatorname{ETF}}(D_2,N_2)$, giving . (Alternatively, follows directly from .) Next, multiplying by $D_3(N_3-1)=D_1D_2(N_1N_2-1)$ and using gives : $$\begin{aligned}
N_3-D_3
&=\tfrac{D_1(N_1N_2-1)(N_2-D_2)}{N_2-1}\\
&=D_1N_1(N_2-D_2)+\tfrac{D_1(N_1-1)(N_2-D_2)}{N_2-1}\\
&=D_1N_1(N_2-D_2)+(N_1-D_1).\end{aligned}$$ Meanwhile, multiplying by $D_3^2=D_1^2D_2^2$ immediately gives . For the final conclusion, note that if $$\tfrac{D_2(N_2-D_2)}{N_2-1},\ \tfrac{(N_3-D_3)(N_3-1)}{D_3}$$ are integers then their product is as well; by and , this product is $$\begin{aligned}
\tfrac{D_2(N_2-D_2)}{N_2-1}\tfrac{(N_3-D_3)(N_3-1)}{D_3}
&=\tfrac{D_2^2(N_3-D_3)}{D_3(N_3-1)}\tfrac{(N_3-D_3)(N_3-1)}{D_3}\\
&=(\tfrac{N_3-D_3}{D_1})^2\\
&=[N_1(N_2-D_2)+\tfrac{N_1}{D_1}-1]^2,\end{aligned}$$ and so $D_1$ necessarily divides $N_1$.
Since $(D_1,N_1)=(2,3)$ and $(D_2,N_2)=(4,5)$ satisfy , i.e., $\frac{3-2}{2(3-1)}=\frac14=\frac{5-4}{5-1}$, we can apply Theorem \[thm.tensor\] to an ${\operatorname{ETF}}(2,3)$ and the ${\operatorname{MUETF}}(4,5,3)$ of Example \[ex.MUETF(4,5,3)\] to produce an ${\operatorname{ETF}}(8,15)$. In particular, we take as before, and let ${\{{{\boldsymbol{\varphi}}_n}\}}_{n=0}^2$ be the ${\operatorname{ETF}}(2,3)$ with $${\boldsymbol{\Phi}}=\left[\begin{array}{ccc}
{\boldsymbol{\varphi}}_0&{\boldsymbol{\varphi}}_1&{\boldsymbol{\varphi}}_2
\end{array}\right]
=\frac1{\sqrt{2}}\left[\begin{array}{ccc}
1&\omega^{ 5}&\omega^{10}\\
1&\omega^{10}&\omega^{ 5}
\end{array}\right].$$ Taking ${\boldsymbol{\Delta}}$ and ${\boldsymbol{\Psi}}$ as in Example \[ex.MUETF(4,5,3)\], Theorem \[thm.tensor\] gives that $$\left[\begin{array}{ccc}
({\boldsymbol{\varphi}}_0\otimes{\boldsymbol{\Psi}})
&({\boldsymbol{\varphi}}_1\otimes{\boldsymbol{\Delta}}{\boldsymbol{\Psi}})
&({\boldsymbol{\varphi}}_2\otimes{\boldsymbol{\Delta}}^2{\boldsymbol{\Psi}})
\end{array}\right]$$ is the synthesis operator of an ${\operatorname{ETF}}(8,15)$.
In fact, it turns out that perfectly shuffling the columns of this matrix—converting three collections of five vectors into five collections of three vectors—yields a harmonic ETF arising from a difference set for ${\mathbb{Z}}_{15}$ that is itself the sum of a difference set ${\{{5,10}\}}$ for the subgroup ${\mathcal{H}}={\{{0,5,10}\}}$ of ${\mathbb{Z}}_{15}$ and the ${\mathcal{H}}$-RDS ${\{{1,2,8,4}\}}$ for ${\mathbb{Z}}_{15}$ from Example \[ex.MUETF(4,5,3)\], namely $$\label{eq.GMW 8x15}
{\{{6,11,7,12,13,3,9,14}\}}
={\{{5,10}\}}+{\{{1,2,8,4}\}}.$$
A little later on, we explain that while not every ETF produced by Theorem \[thm.tensor\] is harmonic, this does occur infinitely often. In such special cases, it turns out that the construction of Theorem \[thm.tensor\] is a reversal of some of the ideas from [@FickusJKM18; @FickusS20]. To elaborate, [@FickusJKM18] considers ETFs that contain subsequences which are regular simplices for their span. There, it was shown that certain Singer-complement-harmonic ETFs partition into such subsequences, and moreover that their spans form a type of optimal packing of subspaces known as an *equi-chordal tight fusion frame* (ECTFF). (The techniques of [@FickusJKM18] show, for example, that the harmonic ${\operatorname{ETF}}(8,15)$ arising from partitions into three ${\operatorname{ETF}}(4,5)$, yielding an ECTFF that consists of three $4$-dimensional subspaces of ${\mathbb{C}}^8$.) This analysis was refined in [@FickusS20], showing that these subspaces are actually *equi-isoclinic* when the underlying difference set factors in a manner similar to .
Theorem \[thm.tensor\] reverses this idea, concatenating $N_1$ subsequences, each of which is an ${\operatorname{ETF}}(D_2,N_2)$ for its span, to form an ${\operatorname{ETF}}(D_1D_2,N_1N_2)$. Indeed, for any $n_1\in{\mathcal{N}}_1$, ${\boldsymbol{\varphi}}_{n_1}\otimes{\boldsymbol{\Psi}}_{n_1}=({\boldsymbol{\varphi}}_{n_1}\otimes{\mathbf{I}}){\boldsymbol{\Psi}}_{n_1}$ is the synthesis operator of , meaning this subsequence is the embedding of the ETF into a $D_2$-dimensional subspace of ${\mathbb{F}}^{{\mathcal{D}}_1\times{\mathcal{D}}_2}$ via the isometry ${\boldsymbol{\varphi}}_{n_1}\otimes{\mathbf{I}}$. In the special case where $N_2=D_2+1$, the ETFs constructed by Theorem \[thm.tensor\] partition into a union of $N_1$ regular $D_2$-simplices.
We now apply Theorem \[thm.tensor\] with the only MUETFs we know of that are not MUBs, namely those described in Corollary \[cor.Singer MUETFF\]:
\[thm.main result\] If an ${\operatorname{ETF}}(D,N)$ exists where $D<N<2D$ and $Q=\frac{D(N-1)}{N-D}$ is a power of a prime, then for every positive integer $J$, there exists an ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ where $$\begin{aligned}
\label{eq.main 1}
D^{(J)}&=DQ^{J-1},\\
\label{eq.main 2}
N^{(J)}&=N(\tfrac{Q^J-1}{Q-1}),\\
\label{eq.main 3}
N^{(J)}-D^{(J)}&=DN(\tfrac{Q^{J-1}-1}{Q-1})+(N-D).\end{aligned}$$ Here, for any $J\geq 2$, $$\begin{aligned}
\label{eq.main 4}
\tfrac{D^{(J)}(N^{(J)}-1)}{N^{(J)}-D^{(J)}}&=Q^J,\\[2pt]
\label{eq.main 5}
\tfrac{(N^{(J)}-D^{(J)})(N^{(J)}-1)}{D^{(J)}}&=D^2Q^{J-2},\\[2pt]
\label{eq.main 6}
\tfrac{(N^{(J)}-D^{(J)})(N^{(J)}-1)}{D^{(J)}}&\notin{\mathbb{Z}},\\[2pt]
\label{eq.main 7}
D^{(J)}+1&<N^{(J)}<2D^{(J)},\end{aligned}$$ and so no ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ can be real-valued.
When $J=1$, , and become $D^{(1)}=D$, $N^{(1)}=N$ and $N^{(1)}-D^{(1)}=N-D$, respectively, and the given ${\operatorname{ETF}}(D,N)$ is an ${\operatorname{ETF}}(D^{(1)},N^{(1)})$. As such, assume $J\geq 2$. Here, Corollary \[cor.Singer MUETFF\] provides an ${\operatorname{MUETF}}(Q^{J-1},\frac{Q^J-1}{Q-1},Q-1)$. Moreover, since $N$ and $D$ are integers, having $N<2D$ implies $N-D\leq D-1$ and so $$Q-1
=\tfrac{D(N-1)}{N-D}-1
=\tfrac{N(D-1)}{N-D}
\geq N.$$ Thus, we can take just $N$ of these mutually unbiased ETFs to form an . To apply Theorem \[thm.tensor\] with this MUETF, note that $(N_2,D_2)=(Q^{J-1},\frac{Q^J-1}{Q-1})$ satisfies $$\begin{aligned}
\label{eq.pf of main 0}
N_2-D_2
&=\tfrac{Q^J-1}{Q-1}-Q^{J-1}
=\tfrac{Q^{J-1}-1}{Q-1},\\
\label{eq.pf of main 1}
N_2-1
&=\tfrac{Q^J-1}{Q-1}-1
=Q(\tfrac{Q^{J-1}-1}{Q-1}).\end{aligned}$$ When combined with our assumption that $Q=\frac{D(N-1)}{N-D}$, this implies that is satisfied when $(N_1,D_1)=(D,N)$: $$\tfrac{N_2-D_2}{N_2-1}
=\tfrac1Q
=\tfrac{N-D}{D(N-1)}
=\tfrac{N_1-D_1}{D_1(N_1-1)}.$$ As such, Theorem \[thm.tensor\] yields an ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ where $$\begin{aligned}
D^{(J)}&=D_3=D_1D_2=DQ^{J-1},\\
N^{(J)}&=N_3=N_1N_2=N(\tfrac{Q^J-1}{Q-1}),\end{aligned}$$ as claimed in and , respectively. Moreover, combining with immediately yields . Next, and follow from combining and with , and the fact that $D_2=Q^{J-1}$. Continuing, since $J\geq 2$, gives $N^{(J)}-D^{(J)}>N-D\geq1$, namely one half of . For the remaining half, note that since $Q=\frac{D(N-1)}{N-D}$ and $N<2D$, $$\tfrac{N^{(J)}}{D^{(J)}}
=\tfrac{N}{D}\tfrac{Q^J-1}{Q^{J-1}(Q-1)}
<\tfrac{N}{D}\tfrac{Q}{Q-1}
=\tfrac{N-1}{D-1}
\leq\tfrac{(2D-1)-1}{D-1}
=2.$$ Moreover, since $J\geq2$, $$\tfrac{D_2(N_2-D_2)}{N_2-1}=\tfrac{Q^{J-1}}{Q}=Q^{J-2}\in{\mathbb{Z}},$$ while $1<\frac{N}{D}<2$ and so $\frac{N_1}{D_1}=\frac{N}{D}\notin{\mathbb{Z}}$. As such, the final statement of Theorem \[thm.tensor\] gives . In particular, when $J\geq 2$, and imply that $(D^{(J)},N^{(J)})$ violates a well-known necessary condition on the existence of real ETFs [@SustikTDH07]. (We caution that and are not necessary when $J=1$. In fact, some of the most fruitful applications of these ideas are scenarios where $(D^{(1)},N^{(1)})=(D,N)$ either satisfies $\frac{(N-D)(N-1)}{D}\in{\mathbb{Z}}$ or $N=D+1$.)
By our earlier remarks, any ETF produced by Theorem \[thm.main result\] partitions into $N$ subsequences, each consisting of $\tfrac{Q^J-1}{Q-1}$ vectors that form an ETF for their $Q^{J-1}$-dimensional span. In particular, taking $J=2$ yields ETFs that are disjoint unions of regular $Q$-simplices. We also note that in light of and , Theorem \[thm.main result\] can be applied to any ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ that it itself produces; the interested reader can verify that doing so only yields a proper subset of those that arise by applying it to the original ${\operatorname{ETF}}(D,N)$.
By , any ETF produced by Theorem \[thm.main result\] with $J\geq2$ satisfies the most basic necessary condition on the existence of harmonic ETFs, namely that the *index* $$D-\Lambda
=D-\tfrac{D(D-1)}{N-1}
=\tfrac{D(N-D)}{N-1}$$ of its underlying difference set is an integer. At the same time, the fact that such an ETF satisfies makes it unusual. In fact, most known ${\operatorname{ETF}}(D,N)$ have the property that both $\frac{D(N-1)}{N-D}$ and $\frac{(N-D)(N-1)}{D}$ are integers, including all real ETFs, SIC-POVMs, positive and negative ETFs, ETFs of redundancy two, and all harmonic ETFs that arise from either Hadamard, McFarland, Spence, Davis-Jedwab and Chen difference sets [@SustikTDH07; @FickusJ19]. Prior to Theorem \[thm.main result\], the only known exceptions seemed to be certain harmonic ETFs (arising for example from Singer, Paley, cyclotomic, Hall and twin-prime-power difference sets) and ${\operatorname{ETF}}(D,N)$ with $N=2D\pm1$ [@Renes07; @Strohmer08]. To be clear, Theorem \[thm.main result\] recovers some of these previously known unusual ETFs: the next subsection is devoted to the special relationship between Theorem \[thm.main result\] and Singer difference sets; moreover, applying Theorem \[thm.main result\] to any ${\operatorname{ETF}}(D,N)$ where $N=2D-1$ and $D$ is an even prime power yields another ETF of this same type. However, in other cases, Theorem \[thm.main result\] yields ETFs with new parameters:
\[ex.ETF(6,9)\] The Naimark complement of a SIC-POVM for ${\mathbb{C}}^3$ is an ${\operatorname{ETF}}(6,9)$. Since $Q=\frac{6(9-1)}{9-6}=16$ is a prime power, Theorem \[thm.main result\] can be applied to it. We walk through its proof in the special case where $J=2$. Here, Corollary \[cor.Singer MUETFF\] provides $15$ mutually unbiased ${\operatorname{ETF}}(16,17)$ (regular $16$-simplices). Since $\frac{9-6}{6(9-1)}=\frac1{16}=\frac{17-16}{17-1}$, condition of Theorem \[thm.tensor\] is met, and so taking tensor products of the members of this ${\operatorname{ETF}}(6,9)$ with the members of any $9$ of these $15$ mutually unbiased ${\operatorname{ETF}}(16,17)$ yields an ${\operatorname{ETF}}(D,N)$ with $(D,N)=(6(16),9(17))=(96,153)$. This ETF is new [@FickusM16]. ETFs with these parameters cannot be real, SIC-POVMs, positive or negative since $\frac{(N-D)(N-1)}{D}=\frac{361}{4}=(\frac{19}{2})^2$ is not an integer. Moreover, no difference set of cardinality $96$ (or equivalently, $57$) exists in either ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_3\times{\mathbb{Z}}_{17}$ or ${\mathbb{Z}}_9\times{\mathbb{Z}}_{17}$ [@Gordon19], despite the fact that $\frac{D(N-D)}{N-1}=36$ is an integer.
For another “small" example, an ${\operatorname{ETF}}(10,16)$ exists [@FickusM16] and $Q=\frac{10(16-1)}{16-10}=25$ is a prime power. As such, when $J=2$ for example, Theorem \[thm.main result\] yields an ${\operatorname{ETF}}(D,N)$ with $(D,N)=(10(25),16(26))=(250,416)$. This ETF is also new [@FickusM16], and $\frac{(N-D)(N-1)}{D}=(\frac{83}{5})^2$ is not an integer. The existence of a difference set of cardinality 250 (or equivalently, 166) in a group of order $416$ is an open problem [@Gordon19].
In the next two subsections, we evaluate the novelty of the ETFs produced by Theorem \[thm.main result\] in general. It turns out that an infinite number of them are new, and that another infinite number of them are not new.
Recovering known ETFs with Theorem \[thm.main result\]
------------------------------------------------------
Theorem \[thm.main result\] applies to any harmonic ETF that arises from the complement of a Singer difference set, but doing so just yields another ETF of this same type. To elaborate, when $K\geq 2$ and $(D,N)=(Q^{K-1},\frac{Q^K-1}{Q-1})$ for some prime power $Q$, we have $$1<\tfrac{N}{D}=\tfrac{Q^K-1}{Q^K-Q^{K-1}}<2,\quad
\tfrac{D(N-1)}{N-D}=Q^K.$$ Theorem \[thm.main result\] thus provides an ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ with $$\begin{aligned}
D^{(J)}
&=D(Q^K)^{J-1}
=Q^{K-1}Q^{K(J-1)}
=Q^{JK-1},\\
N^{(J)}
&=N\tfrac{(Q^K)^J-1}{Q^K-1}
=\tfrac{Q^K-1}{Q-1}\tfrac{Q^{JK}-1}{Q^K-1}
=\tfrac{Q^{JK}-1}{Q-1}.\end{aligned}$$
As we now explain, this relates to the fact that Theorem \[thm.main result\] can be regarded as a generalization of a classical factorization of the complement of a Singer difference set due to Gordon, Mills and Welch [@GordonMW62]. Denoting the field trace from ${\mathbb{F}}_{Q^{JK}}$ to ${\mathbb{F}}_{Q^K}$ as “${\operatorname{tr}}_{JK:K}$," the freshman’s dream gives $${\operatorname{tr}}_{JK:1}(x)
={\operatorname{tr}}_{K:1}({\operatorname{tr}}_{JK:K}(x)),\quad\forall x\in{\mathbb{F}}_{Q^{JK}}.$$ We claim that ${\mathcal{E}}_1={\mathcal{E}}_2{\mathcal{E}}_3$ where $$\begin{aligned}
{\mathcal{E}}_1&={\{{x_1\in{\mathbb{F}}_{Q^{JK}}^\times: {\operatorname{tr}}_{JK:1}(x_1)=1}\}},\\
{\mathcal{E}}_2&={\{{x_2\in{\mathbb{F}}_{Q^K}^{\times}: {\operatorname{tr}}_{K:1}(x_2)=1}\}},\\
{\mathcal{E}}_3&={\{{x_3\in{\mathbb{F}}_{Q^{JK}}^\times: {\operatorname{tr}}_{JK:K}(x_3)=1}\}}.\end{aligned}$$ Indeed, for any $x_2\in{\mathcal{E}}_2$ and $x_3\in{\mathcal{E}}_3$, $${\operatorname{tr}}_{JK:1}(x_2x_3)
={\operatorname{tr}}_{K:1}(x_2{\operatorname{tr}}_{JK:K}(x_3))
={\operatorname{tr}}_{K:1}(x_2)
=1,$$ and conversely, any $x_1\in{\mathcal{E}}_1$ factors as $x_1=x_2x_3$ where $x_2:={\operatorname{tr}}_{JK:K}(x_1)\in{\mathcal{E}}_2$ and $x_3:=x_1^{}x_2^{-1}\in{\mathcal{E}}_3$. Since the hyperplanes ${\mathcal{E}}_1$, ${\mathcal{E}}_2$ and ${\mathcal{E}}_3$ have cardinality $Q^{JK-1}$, $Q^{K-1}$ and $(Q^K)^{J-1}=Q^{JK-K}$, respectively, this “$x_1=x_2x_3$" factorization is unique. Applying the quotient homomorphism $x\mapsto\overline{x}:=x{\mathbb{F}}_{Q}^\times$ to ${\mathcal{E}}_1={\mathcal{E}}_2{\mathcal{E}}_3$ gives ${\mathcal{D}}_1={\mathcal{D}}_2{\mathcal{D}}_3$ where $$\begin{aligned}
\nonumber
{\mathcal{D}}_1
&=\overline{{\mathcal{E}}_1}
={\{{\overline{x}_1\in{\mathbb{F}}_{Q^{JK}}^\times/{\mathbb{F}}_{Q}^\times: {\operatorname{tr}}_{JK:1}(x_1)\neq0}\}},\\
\label{eq.GMW 1}
{\mathcal{D}}_2
&=\overline{{\mathcal{E}}_2}
={\{{\overline{x}_2\in{\mathbb{F}}_{Q^K}^{\times}/{\mathbb{F}}_{Q}^\times: {\operatorname{tr}}_{K:1}(x_2)\neq0}\}},\\
\nonumber
{\mathcal{D}}_3
&=\overline{{\mathcal{E}}_3}
={\{{\overline{x}_3\in{\mathbb{F}}_{Q^{JK}}^\times/{\mathbb{F}}_{Q}^\times:{\operatorname{tr}}_{JK:K}(x_3)\in{\mathbb{F}}_{Q}^\times}\}}.\end{aligned}$$ Here, ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are the complements of Singer difference sets in ${\mathbb{F}}_{Q^{JK}}^\times/{\mathbb{F}}_{Q}^\times$ and ${\mathbb{F}}_{Q^K}^{\times}/{\mathbb{F}}_{Q}^\times$, respectively, while ${\mathcal{D}}_3$ is an $${\operatorname{RDS}}(\tfrac{Q^{JK}-1}{Q^K-1},\tfrac{Q^K-1}{Q-1},Q^{(J-1)K},(Q-1)Q^{(J-2)K})$$ obtained by quotienting the RDS ${\mathcal{E}}_3$ (of type where “$Q$" is $Q^K$) by . When written additively, we have ${\mathcal{D}}_1={\mathcal{D}}_2+{\mathcal{D}}_3$ where ${\mathcal{D}}_1$, ${\mathcal{D}}_2$ and ${\mathcal{D}}_3$ are subsets of ${\mathbb{Z}}_{N_1}$ where $N_1=\tfrac{Q^{JK}-1}{Q-1}$. In fact, gives that ${\mathcal{D}}_2$ is a subset of the subgroup of ${\mathbb{Z}}_{N_1}$ of order $N_2=\frac{Q^K-1}{Q-1}$, namely $\langle N_3\rangle$ where $N_3=\frac{N_1}{N_2}=\frac{Q^{JK}-1}{Q^K-1}$. As such, any $d_1\in{\mathcal{D}}_1$ can be written as $d_1=N_3d_2+d_3$ where $N_3d_2\in{\mathcal{D}}_2$, $d_3\in{\mathcal{D}}_3$. This in turn implies that the value of any given character of ${\mathbb{Z}}_{N_1}$ at $d_1\in{\mathcal{D}}_1$ is $$\begin{aligned}
\exp(\tfrac{2\pi{\mathrm{i}}nd_1}{N_1})
&=\exp(\tfrac{2\pi{\mathrm{i}}n(N_3d_2+d_3)}{N_1})\\
&=\exp(\tfrac{2\pi{\mathrm{i}}nd_2}{N_2})\exp(\tfrac{2\pi{\mathrm{i}}nd_3}{N_3}).\end{aligned}$$ From this, we see that each vector in the harmonic ETF arising from ${\mathcal{D}}_1$ is a tensor product of a vector in the harmonic ETF arising from ${\mathcal{D}}_2$ with a vector in the harmonic tight frame arising from the RDS ${\mathcal{D}}_3$.
Overall, we see that the classical factorization of the complements of certain Singer difference sets [@GordonMW62] indeed leads to a special case of the “ETF-tensor-MUETF" construction of Theorem \[thm.tensor\] where, as in the proof of Theorem \[thm.main result\], the MUETF in question arises from an RDS. In particular, the harmonic ETF that arises from the complement of the Singer difference set in ${\mathbb{F}}_{Q^{JK}}^\times/{\mathbb{F}}_{Q}^\times$ partitions into copies of the harmonic ETF that arises from the complement of the Singer difference set in , each isometrically embedded into a $Q^{(J-1)K}$-dimensional subspace of a common space of dimension $Q^{JK-1}$. In the special case where $J=2$, this partitions a harmonic ${\operatorname{ETF}}(Q^{2K-1},\frac{Q^{2K}-1}{Q-1})$ into embedded regular $Q^K$-simplices, recovering a result of [@FickusJKM18].
Constructing new ETFs with Theorem \[thm.main result\]
------------------------------------------------------
Theorem \[thm.main result\] requires an ${\operatorname{ETF}}(D,N)$ where $$\label{eq.necessary}
\tfrac{D(N-1)}{N-D}\ \text{is a prime power and}\ D<N<2D.$$ Some examples of such ETFs are regular simplices, that is, have $N=D+1$. All other examples have $D+1<N<2D$, meaning the parameters $(N-D,N)$ of its Naimark complement satisfy $N>2(N-D)>2$. As discussed in Section \[sec.background\], this means such an ${\operatorname{ETF}}(D,N)$ is either the Naimark complement of a SIC-POVM, satisfies $N=2D-1$ where $D$ is even, is a harmonic ETF, or is the Naimark complement of a positive and/or negative ETF (Definition \[def.pos neg ETFs\]).
In light of the previous subsection, we ignore ${\operatorname{ETF}}(D,N)$ whose parameters match those of one that arises from the complement of a Singer difference set, as applying Theorem \[thm.main result\] to them only recovers ETFs with known parameters. With a little work, one finds that this includes all ${\operatorname{ETF}}(D,N)$ that satisfy and are either regular simplices, have $N=2D-1$ where $D$ is even, or are harmonic ETFs arising from the complements of difference sets of the following types: Singer, Paley, cyclotomic, Hall and twin-prime-power. Moreover, every ${\operatorname{ETF}}(D,N)$ that satisfies and whose Naimark complement is a SIC-POVM is an ${\operatorname{ETF}}(6,9)$, and every ${\operatorname{ETF}}(3,9)$ is both $2$-positive and $6$-negative. Some harmonic ETFs are also positive or negative, including those that arise from Hadamard, McFarland, Spence, and Davis-Jedwab difference sets [@FickusJ19].
Our search thus reduces to the following: find ${\operatorname{ETF}}(D,N)$ that satisfy and are either Naimark complements of positive or negative ETFs or are harmonic ETFs that arise from (complements of) difference sets which are not one of the aforementioned types. In the latter case, after searching the literature [@JungnickelPS07], the only potential candidates that we found are difference sets due to Chen [@Chen97], whose complements yield ${\operatorname{ETF}}(D,N)$ where $$D = Q^{2J-1}[2(\tfrac{Q^{2J}-1}{Q-1})-1],
\quad
N = 4Q^{2J}(\tfrac{Q^{2J}-1}{Q-1}),$$ where $J\geq 2$ and $Q$ is either a power of $3$ or an even power of an odd prime. Here, $D<N<2D$ and $$\tfrac{D(N-1)}{N-D}=[2(\tfrac{Q^{2J}-1}{Q-1})-1]^2,$$ is sometimes a prime power and sometimes is not. In cases where it is, applying Theorem \[thm.main result\] to the corresponding ${\operatorname{ETF}}(D,N)$ seems to yield new ETFs with enormous parameters; we leave a deeper investigation of them for future work.
The only known ETFs that remain to be considered are ${\operatorname{ETF}}(D,N)$ whose Naimark complements are either positive or negative. Taking the complementary parameters of those in and , this means there exists $L\in{\{{1,-1}\}}$ and integers $K\geq1$ and $S\geq2$ such that $$\begin{aligned}
D&=[S(\tfrac{K-1}{K})+L][S(K-1)+L],\\
N&=(S+L)[S(K-1)+L].\end{aligned}$$ As noted in [@FickusJ19], such ETFs satisfy $D<N<2D$ provided we exclude $1$-positive ETFs, $2$-negative ETFs, and ETFs of type $(3,-1,2)$ or $(3,-1,3)$. That is, in terms of the partial list (a)–(g) of known families of positive and negative ETFs given in Subsection \[subsec.pos neg\], we exclude all ETFs from (a) and (d), the first two members of (e), and the first member of (f). The remaining ETFs on this list satisfy if and only if $$\begin{aligned}
\tfrac{D(N-1)}{N-D}
&=\tfrac{K}{S}[S(\tfrac{K-1}{K})+L][S^2(K-1)+KLS]\\
&=[S(K-1)+KL]^2\end{aligned}$$ is a prime power, namely if and only if $$\label{eq.pos neg necessary}
Q=S(K-1)+KL\ \text{is a prime power.}$$ Here, $Q\equiv L\bmod(K-1)$, and substituting into the above expressions for $D$, $N$ and gives $$\begin{aligned}
D&=\tfrac{Q}{K}[Q-(K-1)L],\\
N&=\tfrac{Q-L}{K-1}[Q-(K-1)L],\\
\tfrac{D(N-1)}{N-D}&=Q^2.\end{aligned}$$
Many ETFs from (b) satisfy . When $K=2,3,4,5$ for example, we only need prime powers $Q\equiv 1\bmod(K-1)$ such that $S=\frac{Q-K}{K-1}$ satisfies $S\geq K$ and $S\equiv 0,1\bmod K$, that is, such that $Q\geq K^2$ and $Q\equiv K,2K-1\bmod K(K-1)$. An infinite number of such $Q$ exist, including all powers of $K$. More generally, for any $K\geq 2$, we need prime powers $Q\equiv 1\bmod(K-1)$ such that $S=\frac{Q-K}{K-1}$ is sufficiently large and has the property that $K\mid S(S-1)$. An infinite number of such $Q$ exist: in fact, since $K$ and $K-1$ are relatively prime, their product is relatively prime to their sum, at which point Dirichlet’s theorem implies there are an infinite number of primes $Q$ such that $Q\equiv 2K-1\bmod K(K-1)$, implying $S=\frac{Q-K}{K-1}\equiv 1\bmod K$.
All ETFs from (c) satisfy : when $(K,L,S)=(P,1,P)$ for some prime power $P$, $Q=S(K-1)+KL=P^2$ is also a prime power. (In contrast, Steiner ETFs arising from projective planes of order $P$ are of type $(P+1,1,P+1)$ [@FickusJ19], and in this case $Q=(P+1)^2$ is only sometimes a prime power, such as when $P$ is a Mersenne prime.)
An infinite number of the (nonexcluded) ETFs from (e) satisfy : having $(K,L,S)=(3,-1,S)$ where $S\geq5$, $S\equiv0,2\bmod 3$ implies $Q=S(K-1)+KL=2S-3$ can be any prime power such that $Q\geq 7$, $Q \equiv 1,3 \bmod 6$, including all powers of $3$ apart from $3$ itself.
None of the (nonexcluded) ETFs from (f) satisfy : when $(K,L,S)=(2^J+1,-1,2^J+1)$ for some $J\geq 2$, $Q=S(K-1)+KL=(2^J+1)(2^J-1)$ is not a prime power.
An infinite number of the ETFs from (f) satisfy : when $(K,L,S)=(4,-1,S)$ where $S\equiv3\bmod8$, we have that $Q=S(K-1)+KL=3S-4$ can be any prime power $Q$ such that $Q \equiv 5 \bmod 24$.
Applying Theorem \[thm.main result\] to the Naimark complements of these positive and negative ETFs yields the following result:
\[cor.pos neg\] Let $K\geq 2$, $L\in{\{{1,-1}\}}$, and let $Q$ be any prime power such that either:
1. $Q\geq K^2$ and $Q\equiv K,2K-1\bmod K(K-1)$ where $K=2,3,4,5$ and $L=1$,
2. $Q=K$ where $L=1$,
3. $Q\geq 7$ and $Q \equiv 1,3 \bmod 6$ where $K=3$ and $L=-1$,
4. $Q \equiv 5 \bmod 24$ where $K=4$ and $L=-1$.
Then for any $J\geq 1$, an ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ exists where $$\begin{aligned}
D^{(J)}
&=\tfrac1K[Q-(K-1)L]Q^{2J-1},\\[2pt]
N^{(J)}
&=\tfrac{Q-(K-1)L}{K-1}(\tfrac{Q^{2J}-1}{Q+L}),\\[2pt]
N^{(J)}-D^{(J)}
&=\tfrac{Q^{2J-1}[Q-(K-1)L]^2-K[Q-(K-1)L]}{K(K-1)(Q+L)}.\end{aligned}$$
In the special case of (i) where $K=2$, Corollary \[cor.pos neg\] yields ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ whose Naimark complements have parameters for any prime power $Q\geq4$ and $J\geq 1$. These ETFs arise by applying Theorem \[thm.main result\] to the Naimark complements of Steiner ETFs that themselves arise from BIBDs consisting of all $2$-element subsets of a $(Q-1)$-element vertex set [@FickusMT12; @FickusJ19]. Meanwhile, case (ii) yields ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ whose Naimark complements have parameters for any prime power $Q$.
Remarkably, all of the ${\operatorname{ETF}}(D^{(J)},N^{(J)})$ produced by Corollary \[cor.pos neg\] with $J\geq2$ seem to be new: though implies that they satisfy one necessary condition of harmonic ETFs, some of them are not harmonic (Example \[ex.ETF(6,9)\]), and none of them seem to arise from known difference sets [@JungnickelPS07]; since these ETFs satisfy , we know that neither they nor their Naimark complements are real, SIC-POVMs, positive or negative, or have redundancy two.
Conclusions and Future Work
===========================
Theorem \[thm.tensor\] produces an ETF by taking certain tensor products of the vectors in a given ETF and MUETF with compatible parameters. Theorem \[thm.main result\] is the special case of this idea in which the MUETF arises from a certain classical RDS. It is a generalization of the classical Gordon-Mills-Welch factorization of the complement of a Singer difference set [@GordonMW62], and a reversal of some of the analysis of [@FickusJKM18; @FickusS20]. When applied to various families of known positive and negative ETFs [@FickusJ19], Theorem \[thm.main result\] yields many infinite families of new ETFs, some of which are summarized in Corollary \[cor.pos neg\].
Of course, it would be nice to be able to apply Theorem \[thm.tensor\] more broadly. Doing so requires a better fundamental understanding of MUETFs. Since harmonic MUETFs equate to RDSs (Theorem \[thm.RDS gives MUETF\]), one approach is to devote more time and energy to their study; see [@Pott95] for some interesting, important open problems concerning RDSs. More generally, for what $(D,N,M)$ does an ${\operatorname{MUETF}}(D,N,M)$ exist? In the special case where $D=N$, MUETFs reduce to MUBs, and though some MUBs arise from RDSs [@GodsilR09], not all seemingly do: some arise from tensor products of other MUBs, and yet others arise from mutually orthogonal Latin squares [@WocjanB05]. Moreover, some MUBs are real [@BoykinSTW05]. To what extent do these facts generalize to the non-MUB case? We leave these questions for future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
[WW]{}
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abstract: 'The $Kepler$ spacecraft provides new opportunities to search for long term frequency and amplitude modulations of oscillation modes in pulsating stars. We analyzed nearly two years of uninterrupted data obtained with this instrument on the DBV star KIC 08626021 and found clear signatures of nonlinear resonant mode coupling affecting several triplets. The behavior and timescales of these amplitude and frequency modulations show strong similarities with theoretical expectations. This may pave the way to new asteroseismic diagnostics, providing in particular ways to measure for the first time linear growth rates of pulsation modes in white dwarf stars.'
author:
- 'W. Zong$^1$, S. Charpinet$^1$ and G. Vauclair$^1$'
title: Evidence of Resonant Mode Coupling in the Pulsating DB White Dwarf Star KIC 08626021
---
Introduction
============
The $Kepler$ spacecraft is a magnificent instrument to search for long term frequency and amplitude modulations of oscillation modes in pulsating stars. Among the 6 pulsating white dwarfs present in the $Kepler$ field, KIC 08626021 is the unique DB pulsator. It has a rotation period P$_{rot}\sim1.7$ days, estimated from the observed frequency spacings of 3 $g$-mode triplets [@os11]. It has been observed by $Kepler$ for 23 months in short cadence (SC) mode without interruption. Thus, it is a suitable candidate to investigate the resonant mode coupling mechanisms that could induce long term amplitude and frequency modulations of the oscillation modes. Such resonant couplings are predicted to occur in triplets where the rotationally shifted components have frequencies $\nu_1$ and $\nu_2$ such that $\nu_1\,+\nu_2\,\sim\,2\,\nu_0$, where $\nu_0$ is the frequency of the central component. The theoretical exploration of those mechanisms was extensively developed long before the era of space observations [@bu95; @bu97] but was almost interrupted more than a decade ago because of the lack of clear observational evidence of such phenomena, due to the difficulty of capturing amplitude or frequency variations that occur on months to years timescales from ground based observatories. Resonant coupling within triplets was proposed for the first time as the explanation for the frequency and amplitude long term variations observed in the GW Vir pulsator PG 0122+200 [@va11]. We present the analysis of KIC 08626021 in which two triplets exhibit amplitude and frequency variations during the 23-month of observation. Such time modulations pave the way to new asteroseismic diagnostics, providing in particular ways to measure for the first time linear growth rates of pulsation modes in white dwarf stars.
Frequency and amplitude modulations
===================================
The white dwarf star KIC 08626021 has been continuously observed by $Kepler$ since quarter Q10.1 up to Q17.2. The high precision photometric data cover $\sim$684.2 days (23 months), with a duty circle of $\sim$89 %. We used a dedicated software, FELIX, to extract frequencies (details of the program can be found in @ch10). In this study, we concentrate on rotationally split triplets and investigate the variation of amplitude and frequency between components of these triplets and their relationship. We point out that the frequencies near 3682$\mu$Hz reported by @os11 were in fact resolved into several close peaks with the 23-month light curve. It is therefore probably not a real triplet contrary to the other structures found at 4310 and 5073$\mu$Hz (see below). In order to study the variability with time of these modes, we constructed a filter window covering 200 days and slid the filter window along the whole light curve by time steps of 20 days, thus constructing a time-frequency diagram. We also prewhithened the frequencies “chunk by chunk”, i.e., the 23-month light curve of KIC 08626021 was divided into 20 chunks, each containing 6-month of data except the last 3 chunks being at the end of the observations. The results for the two triplets are discussed below.
The amplitude and frequency modulations of the triplet near 4310 $\mu$Hz are shown in Figure\[pf\_mod\] . The grey scale (or color scale for electronic version) in the upper left panel represents the amplitude. The vertical dashed line in the lower left panel is the average value of the frequency over the entire run. The right panel shows the amplitude modulations of each component forming the triplet. Both the amplitudes and frequencies show clear signatures of quasi periodic modulations with the same timescale of $\sim$750 days. The frequencies and amplitudes of the side components evolve in phase and are antiphased with the central component. Figure\[sf\_mod\] shows the modulations observed in the other triplet at 5073$\mu$Hz. The frequencies in this triplet appear to be stable during the nearly two years of monitoring, while the amplitudes show modulations. Note that the amplitude of the $m$ = -1 component went down below the 4$\sigma$ detection threshold and was essentially lost in the noise during the last half of the observations.
Discussion
==========
The frequency and amplitude modulations observed in the triplets of KIC 086226021 can be related to nonlinear resonant mode coupling mechanisms. The first triplet at 4310$\mu$Hz (Figure\[pf\_mod\]) behaves like if it is in the intermediate regime of the resonance, in which the oscillation modes undergo periodic amplitude and frequency modulations. Theory suggests that the time scale of these modulations should be roughly a few times the inverse of the growth rate of the pulsating mode [@go98]. Therefore this periodicity could in principle be used to measure the growth rate. In addition if we compare the second order effect of rotational splitting as estimated directly from the measured mean frequencies : $$\delta\nu = \nu_1 + \nu_2 - 2 \nu_0 \qquad ~\textrm{(1)}$$ with the estimated value following @dz92, $$~~\delta\nu = 4 C \frac{\Omega^2}{\nu_0} ~~~~~~~~~~~~\qquad \textrm{(2)}$$ where $C$ is the first order Ledoux constant ($\sim 0.5$ for dipole $g$-modes) and $\Omega\,=\,2\pi/P_{rot}$ is the angular frequency of the stellar rotation, i.e ., $\delta\nu$$\sim$0.018$\mu$Hz, both are found to be very similar. In the intermediate regime, the expected periodic modulation timescale is $P_{mod} \sim 1/\delta\nu$ [@go98], which with the values given above leads to $P_{mod}\,\sim\,
650$ days, i.e., very similar to the amplitude and frequency modulation timescale of 750 days roughly estimated from Figure\[pf\_mod\]. This further supports our interpretation that nonlinear resonant coupling is indeed at work in this star. The triplet at 5073$\mu$Hz (Figure\[sf\_mod\]) would be in a different regime, likely in the nonresonant regime. The amplitude of the components show clear modulations while the frequency are relatively stable during the observation. This means the ratio of the real part over the imaginary part of the coupling coefficients is large in that case. This ratio roughly measures nonlinear nonadiabaticities in the star. Hence our result shows that two neighbor triplets can belong to different resonant regimes (frequency lock, time dependent or nonresonant), as it was also suggested in the white dwarf star GD 358 [@go98].
Conclusion
==========
Frequency and amplitude modulations of oscillation modes have been found in several rotationally split multiplets detected in the DB pulsator KIC 08626021, thanks to the high quality and long duration photometric data obtained with the $Kepler$ spacecraft. These modulations show signatures pointing toward nonlinear resonant coupling mechanisms occuring among the multiplet components. This is the first time that such signatures are identified so clearly in white dwarf pulsating stars. Periodic modulations of frequency and amplitude that occur in the intermediate resonant regime may allow for new asteroseismic diagnostics, providing in particular a way to measure for the first time linear growth rates of pulsation modes in white dwarf stars. Such results should motivate further theoretical work on nonlinear resonant mode coupling mechanisms and revive interest in nonlinear stellar pulsation theory in general. Finally, we mention that similar modulations are also found in hot B subdwarf stars according to $Kepler$ data.
WKZ acknowledges the financial support from the China Scholarship Council. This work was supported in part by the Programme National de Physique Stellaire (PNPS, CNRS/INSU, France) and the Centre National d’Etudes Spatiales (CNES, France).
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address: 'Tong Zhang is Professor, Statistics Department, Rutgers University, Piscataway, New Jersey 08816, USA .'
author:
-
title: 'Discussion of “Is Bayes Posterior just Quick and Dirty Confidence?” by D. A. S. Fraser'
---
Confidence Region Estimation
============================
The author has written an interesting article on the relationship of confidence distribution and Bayesian posterior distribution. Confidence distribution has its origin from Fisher’s fiducial distribution, and in this discussion we refer to it simply as the “confidence distribution approach.” It allows frequentists to assign confidence intervals (or, more generally, confidence regions) to the outcome of estimation procedures.
The idea can be simply described as follows. Consider a statistical model with a family of distributions $p_\theta(y)$, where $y$ is the observation and $\theta
$ is the model parameter. We assume that the observed $y$ is generated according to a true parameter $\theta_*$ which is unknown to the statistician. If we can find a real-valued quantity $U(y;\theta)$ that depends on $\theta$ and $y$ such that for all $\theta$, when $y$ is generated from $p_\theta(y)$, $U(y;\theta)$ is uniformly distributed in $(0,1)$, then we can estimate the confidence interval of $\theta$ given an observation $y$ as the set $I_{\alpha,\beta}(y)=\{\theta\dvtx U(y;\theta) \in(\alpha,\beta
)\}$ for some $0 \leq\alpha\leq\beta\leq1$. An interpretation of this confidence region is that no matter what is the true underlying $\theta_*$ that generates $y$, the region $I_{\alpha,\beta}(y)$ contains the true parameter $\theta_*$ with probability $\beta-\alpha$ (when $y$ is generated according to $\theta_*$).
Indeed, the above interpretation is a very natural definition of confidence region in the frequentist setting. It does not assume that $\theta_*$ is generated according to any prior, and the interpretation holds universally true for all possible $\theta_*$ in the model. This interpretation can be compared to a confidence region from the Bayesian posterior calculation that assumes that $\theta_*$ is generated according to a specific prior which has to be known to the statistician. If the statistician chooses the wrong prior, then the confidence region calculated from the Bayesian approach will be incorrect in that it may not contain the true parameter $\theta_*$ with the correct probability.=1
The paper takes this interpretation of confidence region, and goes on to provide several examples showing that the Bayesian approach does not lead to correct confidence estimates for all $\theta_*$. The author then argued that the confidence distribution approach is the more “correct” method for obtaining confidence intervals and the Bayesian approach is just a quick and dirty approximation.
One question that needs to be addressed in the confidence distribution approach is how to construct a statistics $U(y_0;\theta)$ with the desired property. The author considered the quantity $U(y_0;\theta)=\int_{y \leq y_0} p_\theta(y) \,d y$, which is well-defined if the observation $y$ is a real-valued number. This corresponds to the proposal in Fisher’s fiducial distribution. The idea of fiducial distribution received a number of discussions throughout the years, and is known to be adequate for unconstrained location families (for which the fiducial confidence distribution matches the Bayesian confidence distribution using a flat prior). However, the general concept is controversial, and largely regarded as a major blunder by Fisher.
In this discussion article we will explain why the idea of confidence distribution with $$U(y_0;\theta)=\int_{y \leq y_0} p_\theta(y)\, d y$$ has not received more attention for general statistical estimation problems, although it does give confidence region estimates that fit the frequentist intuition.
Suboptimality
=============
The purpose of confidence distribution is to provide a confidence region that is consistent with the frequentist definition. However, one flaw of this approach is that the result it produces may not be optimal. While this issue was pointed out in the article, it was not explicitly discussed. In my opinion, this is the main reason why the idea of confidence distribution hasn’t become more popular in statistics. Therefore, this section provides a more detailed discussion on this issue.
To understand this point, we shall first consider a simple illustration. Let $U(y,\theta)$ be a uniform random variable in $(0,1)$ that is independent of $y$ and $\theta$. By definition, given any $\theta_*$, the confidence region $I_{\alpha
,\beta}(y)=\{\theta\dvtx U(y;\theta) \in(\alpha,\beta)\}$ contains $\theta_*$ with probability $\beta-\alpha$. Since this applies to the parameter that generates $y$, the confidence region obtained this way is consistent with the frequentist intuition of what a confidence region should mean. However, this estimate is not useful statistically because the method just randomly guesses either the entire domain of $\theta$ when $U \in(\alpha,\beta)$ or the empty region otherwise; the decision does not even depend on $y$.
While the above example is extreme, it does show that a confidence region merely consistent with the frequentist semantics is not necessarily a useful estimate. Statistically, this is because the confidence region obtained is suboptimal. In fact, this claim also applies to the confidence distribution approach this article considers. More specifically, for nonlinear problems that this paper focused on, the method can produce confidence regions that are quite suboptimal. By “optimal” (or even “good”), we mean that the confidence region a method produces should be small by some measure. In particular, if another method provides confidence regions that also fit in the frequentist semantics but is no larger on average for all $\theta$ and smaller for some $\theta$, then it can be regarded as a better method. This corresponds to the notion of admissibility in decision theory.
Consider the following simple nonlinear location estimation model: $y$ is generated either from $N(0,\sigma_0^2)$ when $\theta=0$, or from $N(1,\sigma_1^2)$ when $\theta=1$. There are only two possible positions $\theta=0$ or $\theta=1$ for the unknown location parameter $\theta$, and we assume that the variance parameters $\sigma_0^2$ and $\sigma_1^2$ are known quantities that are not necessarily equal. Note that the restriction of $\theta$ to two positions is only for simplicity, which is not critical for our illustration—we can extend the example to allow all locations in $R$.
For this example, the confidence distribution approach gives the following $U(y_0,\theta)$: $$U(y_0,\theta) =
%
\cases{
\Phi(y_0/\sigma_0), & $\theta=0,$ \vspace*{2pt}\cr
\Phi\bigl((y_0-1)/\sigma_1\bigr), & $\theta=1 ,$}
%$$ where $\Phi(z)$ denotes the cdf of the standard Gaussian $N(0,1)$.
Let’s consider the confidence region $I_{\delta,1-\delta}(y)$ for some $\delta\in(0,0.25)$, which we simplify as $I(y)$. By definition, the estimated confidence region $I(y)$ contains the position $\theta=0$ if and only if $y \in\Omega_0$ with $\Omega_0= (\sigma_0 \Phi^{-1}(\delta), -\sigma_0 \Phi^{-1}(\delta
))$, and $I(y)$ contains the position $\theta=1$ if and only if $y \in
\Omega_1$ with $\Omega_1= (1 + \sigma_1 \Phi^{-1}(\delta), 1- \sigma_1
\Phi^{-1}(\delta))$. For convenience, we also define $$\begin{aligned}
\mu_0&=& P(y \in\Omega_1|\theta=0)\\
&=& \int_{1 + \sigma_1 \Phi^{-1}(\delta
)}^{1-\sigma_1 \Phi^{-1}(\delta)} \frac{1}{\sqrt{2\pi}\sigma_0} \exp
\biggl(- \frac{y^2}{2 \sigma_0^2}\biggr) \,d y .\end{aligned}$$ In order to show that the confidence distribution approach is suboptimal, we can, for simplicity, consider the case $\sigma_0 \gg1$ and $\sigma_1 \ll1$, so that $1 - \sigma_1 \Phi^{-1}(\delta) < -\sigma_0 \Phi^{-1}(\delta)$ and $\mu_0 < 2 \delta$. The first condition implies that $\Omega_1 \subset\Omega_0$. Therefore, when the parameter $\theta=1$, with probability $1-P(y \in
\Omega_1|\theta=1)=1-2\delta$ over $y \sim N(1,\sigma_1^2)$, we have $y
\in\Omega_1$ and, thus, $|I(y)|=2$ \[i.e., $I(y)$ contains both $\theta=0$ and $\theta=1$\]. Therefore, we have (note that we have assumed that $\delta<0.25$) $$E_{y|\theta=1} |I(y)| > 2 (1-2\delta) > 1 . \label{eq:conf-size-theta-1}$$ Moreover, we have $$\begin{aligned}
E_{y|\theta=0} |I(y)| &=& P(y \in\Omega_0|\theta=0) + P(y \in\Omega
_1|\theta=0)
\\&=& 1-2\delta+ \mu_0 .\end{aligned}$$
Now we would like to construct a better confidence region estimator by using the condition (which we made earlier) that $P(y \in\Omega_1|\theta=0) = \mu_0 < 2 \delta$. Therefore, we can pick $\Omega_0'$ such that $\Omega_0' \cap\Omega_1
=\varnothing$ and $P(y \in\Omega_0'|\theta=0)=1-2\delta$. This means that we can choose the following confidence region estimate $I'(y)$: $I'(y)$ contains the position $\theta=0$ if and only if $y \in\Omega_0'$ and $I'(y)$ contains the position $\theta=1$ if and only if $y \in\Omega_1$. This estimate obeys the frequentist definition because $P(\theta\in
I'(y)|\theta)=1-2\delta$ both when $\theta=0$ and $\theta=1$. Moreover, we have $$E_{y|\theta=0} |I'(y)| = 1-2\delta+ \mu_0 ,\quad
E_{y|\theta=1} |I'(y)| \leq1 .$$ The second inequality is due to the fact that $|I'(y)| \leq1$ for all $y$ because $\Omega_0' \cap\Omega_1=\varnothing$. In comparison to (\[eq:conf-size-theta-1\]), we know that when $\theta=1$, the confidence distribution approach gives a confidence region $I(y)$ with a larger average size. This means that for this simple problem, the confidence distribution approach gives a suboptimal estimate of confidence region $I(y)$ that is dominated by a better method $I'(y)$. The difference can be significant when $\delta\approx0$.
Conclusion
==========
The confidence distribution approach is a rather general method to obtain confidence regions for parameter estimation problems consistent with the frequentist semantics. The method can also be easily generalized to the multivariate situation where $y$ is a vector instead of a real number. Nevertheless, the confidence region it estimates can be rather suboptimal in the sense that the region obtained by this method can be significantly larger than what can be done with more sophisticated methods. Although we have only illustrated this phenomenon with a relatively simple example, the conclusion holds more generally.
At the root of this suboptimality, we note that whether a model parameter $\theta_0$ belongs to the confidence region obtained by the confidence distribution approach only depends on the distribution $p(y|\theta=\theta_0)$ at the parameter $\theta_0$ itself, without considering the alternative models at $\theta\neq\theta_0$. This unnatural behavior is what causes its suboptimality for general nonlinear models. For example, in order to achieve good performance for the simple two-position location estimation example given in the previous section, the confidence region estimate $I'(y)$ at $\theta=0$ has to be modified in order to take advantage of the alternative model $\theta=1$ (so that $\Omega_0'\cap\Omega_1=\varnothing$). Such adaptation does not occur in the confidence distribution approach. As noted by the author during the discussion of the bounded parameter example, the confidence distribution estimate does not change when we restrict the model space, and this phenomenon is rather odd. The author dismissed this problem as a secondary issue because it does not change the semantics of the confidence region in the frequentist interpretation. However, if we are interested in achieving (near) optimality for the estimated confidence region, then this issue becomes a more serious concern because it means that this simple method ignores a significant amount of available information that could have been used in more complicated algorithms. In conclusion, while the confidence distribution approach is simple to apply, the simplicity is achieved by ignoring some useful information. Therefore, we have to keep the limitations of this method in mind whenever it is applied to complex statistical models.
|
---
abstract: 'A simple conservation law formula for field equations with a scaling symmetry is presented. The formula uses adjoint-symmetries of the given field equation and directly generates all local conservation laws for any conserved quantities having non-zero scaling weight. Applications to several soliton equations, fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein gravitational field equations are considered.'
address: |
Department of Mathematics\
Brock University, St. Catharines, ON Canada
author:
- 'Stephen C. Anco'
title: 'Conservation laws of scaling-invariant field equations '
---
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Introduction
==============
Conservation laws are central to the analysis of physical field equations by providing conserved quantities, such as energy, momentum, and angular momentum. For a given field equation, local conservation laws are well-known to arise through multipliers [@Olver-book], analogous to integrating factors of ODEs [@ourbook], with the product of the multiplier and the field equation being a total divergence expression. Such divergences correspond to a conserved current vector for solutions of the field equation whenever the multiplier is non-singular. If a field equation possesses a Lagrangian, Noether’s theorem [@Olver-book] shows that the multipliers for local conservation laws consist of symmetries of the field equation such that the action principle is invariant (to within a boundary term). Moreover, the variational relation between the Lagrangian and the field equation yields an explicit formula for the resulting conserved current vector. This characterization of multipliers for a Lagrangian field equation has a generalization to any field equation by means of adjoint-symmetries [@Anco-Bluman1; @Anco-Bluman2], whether or not a Lagrangian formulation exists.
Recall, geometrically, symmetries are tangent vector fields on the solution space of a field equation and thus are determined as field variations satisfying the linearization of the field equation on its entire solution space. Adjoint-symmetries are defined to satisfy the adjoint equation of the symmetry determining equation on the solution space of a field equation [@trivial]. (As such, unlike for symmetries, there is no obvious geometrical motion or invariance associated with adjoint-symmetries.) Through standard results in the calculus of variations [@Olver-book], it is known that the multipliers for local conservation laws are precisely adjoint-symmetries of the field equation subject to a certain adjoint invariance condition [@Anco-Bluman1; @invcondition]. This allows a system of determining equations for multipliers to be formulated in terms of the adjoint-symmetry determining equation augmented by extra determining equations [@Anco-Bluman2; @deteq]. In addition, the resulting conservation laws are yielded by means of a homotopy integral expression [@Olver-book; @Anco-Bluman1; @Anco-Bluman2] involving just the field equation and the multiplier, which is derived from the adjoint invariance condition, analogously to the line integral formula for first integrals of ODEs [@ourbook].
The purpose of this paper is to show that in the physically interesting situation where a field equation possesses a scaling symmetry, then the adjoint invariance condition and homotopy integral formula can be completely by-passed for obtaining conservation laws. In particular, a simple algebraic formula that directly generates conservation laws in terms of adjoint-symmetries for any such field equation is presented. Most important, when applied to a multiplier, the formula recovers the corresponding conservation law determined by the multiplier, to within a proportionality factor. This factor turns out to be the scaling weight of the conserved quantity defined from the conservation law. Consequently, all conserved quantities with non-zero scaling weight are obtainable from this formula [@linear].
In , the conservation law formula is derived. Examples and applications of this formula are presented in . As new results, first, a recursion formula is obtained for the local higher-order conservation laws of the sine-Gordon equation and a vector generalization of the Korteweg-de Vries equation; second, a simple proof is given for closing a gap in the classification of local conservation laws of the Yang-Mills equations and Einstein gravity equations. Some concluding remarks are made in .
Conservation Law Formula {#formula}
==========================
Consider a general system of field equations \[ueq\] (x,u,,,…) =0 for field variables $\u{a}(x)$ depending on a total of $n\geq 2$ time and space variables $\x{\alpha}{}$, with $\ujet{k}$ denoting partial derivatives $\uder{a}{\alpha_1\cdots\alpha_k}
=\parderu{a}{k}{\x{\alpha_1}{}}{\x{\alpha_k}{}}$, up to some finite differential order. (The coordinate indices $\alpha,\beta,\gamma$ run $0$ to $n-1$; the field index $a$ runs $1$ to $N$; the equation index $A$ runs $1$ to $m$. Summation is assumed over any repeated indices. This formalism allows the number of components of the fields $\u{a}$ and equations $\ueq{A}{}$ to be different.) For simplicity of presentation, the differential order of system will be restricted to $k\leq 2$.
Symmetries of the field equations are the solutions $\delta\u{a}=\symm{}{a}(x,u,\ujet{},\ujet{2},\ldots)$ of the linearized equations \[usymmeq\] = +() +() =0 for all $\u{a}(x)$ satisfying system , where $\D{\alpha}$ is the total derivative with respect to $\x{\alpha}{}$, and where $\ueqder{a}{}{A}$, $\ueqder{a}{\alpha}{A}$, / denote partial derivatives $\partial\ueq{A}{}/\partial\u{a}$, $\partial\ueq{A}{}/\partial\uder{a}{\alpha}$, /. The adjoint of equation is given by \[uadsymmeq\] = -( ) +( ) =0 whose solutions $\adsymm{A}(x,u,\ujet{},\ujet{2},\ldots)$ for all $\u{a}(x)$ satisfying system are the adjoint-symmetries of the field equations . Note the operators $\linop{\ueq{}{}}$ and $\adlinop{\ueq{}{}}$ are related by the identity \[mainid\] - = ¶(,;) with \[curr\] ¶(,;) = ( -( ) ) - () . Hence, this expression yields a local conservation law $\D{\alpha} \P{\alpha}{}(\adsymm{},\symm{}{}) =0$ for any pair $\adsymm{A},\symm{}{a}$, on all solutions of the field equations .
Now suppose the field equations are invariant under a scaling of the variables \[scaling\] \^ ,ă \^ ă , with $\p{\alpha}=\const$, $\q{a}=\const$. From the corresponding scaling symmetry, given by \[scsymm\] ă=(x,u,) = ă - , the expression $\P{\alpha}{}(\adsymm{},\scsymm{}{})$ produces a conserved current in terms of any adjoint-symmetry $\adsymm{A}$.
[**Proposition 2.1**]{}: For scaling invariant field equations , every adjoint-symmetry generates a conserved current on all solutions of by the formula ¶ = && ( ă - ) ( -( ) )\
&& +( (-) + ) . \[currformula\]
Consider, now, a multiplier $\Q{}{A}(x,u,\ujet{},\ujet{2},\ldots)$ for a local conservation law \[conslaw\] = that is assumed to be homogeneous [@homogeneous] under the scaling symmetry, so \[Qscaling\] = = - with scaling weight $\r{A}=\const$. Let $\s{A}=\const$ be the scaling weight of the field equations, \[eqscaling\] = = - . Due to the scaling homogeneity of $\curr{\alpha}{\Q{}{}}$, the constant $\r{A}+\s{A}$ is independent of the index $A$. Then, the following important relation holds between the conserved currents $\curr{\alpha}{\Q{}{}}$ and $\P{\alpha}{\Q{}{}}$.
[**Theorem 2.2**]{}: In terms of the scaling weights of the field equations and the multiplier , every local conservation law satisfies the scaling relation \[scalingformula\] ¶ , = + +\_ for all $\u{a}(x)$ satisfying the field equations, where “$\simeq$” denotes equality to within a trivial conserved current [@conslaw] $\D{\beta} \triv{\alpha\beta}$ for some local expression $\triv{\alpha\beta}=-\triv{\beta\alpha}$. Moreover, $\w{\Q{}{}}$ is simply the scaling weight of the flux integral $\int \curr{\alpha}{\Q{}{}} \n{}{\alpha} \rmd^{n-1}x$ defined on any $(n-1)$-dimensional hypersurface $\x{\alpha}{}\n{}{\alpha} =\const$ (with normal vector $\n{}{\alpha}$), \[scalingweight\] \^[n-1]{}x = \^[n-1]{}x .
[**Definition 2.3**]{}: A conservation law will be called (non)critical with respect to the scaling if the scaling weight of the corresponding conserved quantity is (non)zero.
[**Corollary 2.4**]{}: As all multipliers necessarily are given by adjoint-symmetries, the conservation law formula consequently generates all noncritical conservation laws of the field equations .
The proof of relation starts from the identity with $\symm{}{a}=\scsymm{}{a}$. We substitute the condition $\adlin{\adsymm{}}{\ueq{}{}}{a} = - \adlin{\ueq{}{}}{\adsymm{}}{a}$ on $\adsymm{A}$ (holding for all $\u{a}(x)$ without use of the field equations), which is necessary and sufficient [@Olver-book] for an adjoint-symmetry to be a multiplier $\Q{}{A}=\adsymm{A}$. Here, $\adlinop{\adsymm{}}$ is the adjoint of the linearization operator $\linop{\adsymm{}}$ defined analogously to $\adlinop{\ueq{}{}}$ and $\linop{\ueq{}{}}$. We next use the adjoint relation = - (,;) . Substituting the scaling relations , followed by integrating by parts, we obtain = ( ¶ + (,;) ) where, note, the last term in this divergence vanishes when $\ueq{A}{}=0$. Then conservation law equation leads to the scaling relation .
Examples and Applications {#examples}
===========================
Soliton equations
-------------------
For applications of the main conservation law formulas , consider, firstly, soliton field equations in 1+1 dimensions.
[*Korteweg-de Vries equation.*]{} The KdV equation in physical form for scalar field $u(t,x)$ is given by \[kdveq\] (u,,,) = +u + =0 . This field equation is invariant under the scaling $t\rightarrow \lambda^3 t$, $x\rightarrow \lambda x$, $u\rightarrow \lambda^{-2} u$. The corresponding scaling symmetry u= = -2u-3t-x is a solution of the linearized field equation \[kdvsymmeq\] = + +u + =0 for all $u(t,x)$ satisfying the KdV equation. The adjoint of equation is given by \[kdvadsymmeq\] = - -u - =0 whose solutions $\adsymm{}$ for all $u(t,x)$ satisfying are the adjoint-symmetries of the KdV equation. Here, we see $\adlinop{\ueq{}{}} \neq \linop{\ueq{}{}}$, reflecting the fact that the KdV equation lacks a local Lagrangian formulation in terms of $u(t,x)$. Now, for any adjoint-symmetry $\adsymm{}$, the conservation law formula gives the conserved density \[kdvformula\] ¶[t]{} = -(3t +x+2u) . From the obvious solution $w=u$ of equation , we consider the infinite sequence of KdV adjoint-symmetries $\adsymm{(k)} = (\adR)^k u$, $k=0,1,2,\ldots$, generated by the operator = +u +(u) which is the adjoint of the well-known KdV recursion operator [@Olver] = +u +. Note that, under the KdV scaling symmetry, $\adsymm{(k)} \rightarrow \lambda^{-2(1+k)} \adsymm{(k)}$.
Each adjoint-symmetry $\adsymm{(k)}$ is known to be a multiplier for a local conservation law of the form \[kdvconslaw\] (u,,,…) + (u,,,…) =0 on KdV solutions $u(t,x)$, through lengthy calculations. For instance, originally the KdV conservation laws were derived one by one via the Miura transformation [@KdV], from which the multipliers can be calculated. An alternative approach has involved extracting the conserved densities one by one through a residue method using a formal symmetry (pseudo-differential operator) [@Olver-book] for the KdV equation. More recently, in the conserved densities were obtained one at a time from a homotopy integral formula in terms of the adjoint-symmetries, after a verification of the adjoint invariance condition on each one.
Here, by-passing such cumbersome steps, formula yields the resulting conserved densities directly in terms of $\adsymm{(k)}$, \[kdvcurrs\] ¶[t]{} () -2u +3t(u +) -3() which follows by means of properties of $\adR$. This leads to a simple explicit recursion formula for all of the KdV local conservation laws \[kdvscalingformula\] = = () , to within a trivial conserved density $\D{x}\triv{}$, as a result of the scaling formula .
[*Sine-Gordon equation.*]{} The sine-Gordon equation is given by \[sgeq\] (u,) = -u =0 for scalar field $u(t,x)$. This is a Lagrangian field equation with the scaling invariance $t\rightarrow \lambda^{-1} t$, $x\rightarrow \lambda x$, $u\rightarrow u$. The corresponding scaling symmetry is $\delta u=\scsymm{}{} = t\Du{t}-x\Du{x}$ which is a solution of the linearized field equation \[sgsymmeq\] = -(u) =0 for all $u(t,x)$ satisfying the sine-Gordon equation. Here, as we have $\adlinop{\ueq{}{}} = \linop{\ueq{}{}}$, symmetries are the same as adjoint-symmetries, $\adsymm{}=\symm{}{}$. Hence, for any symmetry $\symm{}{}$, the conservation law formula gives the conserved density \[sgformula\] ¶[t]{} = (x -t) . We now consider the well-known infinite sequence of symmetries $\symm{}{(k)} = (\R)^k \Du{x}$, $k=0,1,2,\ldots$, generated by the sine-Gordon recursion operator [@Olver] = +() , starting from the translation symmetry $\symm{}{}= \Du{x}$. Under the sine-Gordon scaling, note $\symm{}{(k)} \rightarrow \lambda^{-2k} \symm{}{(k)}$. Each symmetry $\symm{}{(k)}$ is a multiplier for a local conservation law of the form on sine-Gordon solutions $u(t,x)$. These conservation laws were found originally by an application of Noether’s theorem [@Dodd-Bullough] and subsequently were derived by the same techniques used for the KdV equation [@Olver-book], yielding the conserved densities one at a time through lengthy calculations. Here, similarly to the KdV case, formula directly leads instead to a simple explicit expression for all of the sine-Gordon local conservation laws, \[sgconslaws\] ¶[t]{} -( ) = - ( )/ = ¶[t]{}[(k)]{} and hence \[sgscalingformula\] ¶[t]{}[(k)]{} = ( -)/ from the scaling formula . Here $\nlR$ stands for the nonlocal part of $\R$. If the relation $-(\D{x}\P{t}{(k)})/\Du{x} = \D{x}\symm{}{(k)}$ is substituted into expression to get = -¶[t]{}[(k)]{} -( ¶[t]{}[(k)]{} ) then equation yields an explicit conservation law recursion formula = ( ) with = \^2 +\^2 +( ( +\^2) ) representing a recursion operator on conserved densities.
[*Modified Korteweg-de Vries vector equation.*]{} It is known that the recursion operator and symmetry hierarchy of the sine-Gordon equation are closely related to that of the modified Korteweg-de Vries equation $\Du{t} +\frac{3}{2} u^2\Du{x} +\Du{xxx} =0$. This scalar field equation has an interesting generalization [@Wolf] \[mkdveq\] (,,,) = + + =0 for an $N$-dimensional vector field $\vecu(t,x)$, with any $N\geq 1$. The vector mKdV equation is invariant under the scaling $t\rightarrow \lambda^3 t$, $x\rightarrow \lambda x$, $\vecu\rightarrow \lambda^{-1} \vecu$. Its symmetries are the solutions of the linearized field equation \[mkdvsymmeq\] = +3 + + =0 for all mKdV solutions $\vecu(t,x)$, while its adjoint-symmetries are the solutions of the adjoint of equation \[mkdvadsymmeq\] = - - +3 -3 - =0 . The vector mKdV equation admits the recursion operator \[mkdvRop\] = ++( ) -( ) , where “$\wedge$” denotes the antisymmetric outer product of two vectors and “$\intprod$” denotes the interior product (/ contraction) of a vector against a tensor, namely ${\vec c}\intprod({\vec a}\wedge{\vec b}) =
({\vec c}\cdot{\vec a}) {\vec b} - ({\vec c}\cdot{\vec b}) {\vec a}$. This expression is a manifestly $SO(N)$-invariant version of the vector mKdV recursion operator first derived in . The adjoint of $\R$ is given by the similar operator = +( ) + ( ) + ( ) (closely resembling the form of the sine-Gordon recursion operator in the scalar case $N=1$, when the “$\wedge$” terms vanish). We now consider the infinite sequence of mKdV adjoint-symmetries generated by $\vecadsymm{(k)} = (\adR)^k \vecu$, $k=0,1,2,\ldots$, starting from the obvious solution of equation , $\vecadsymm{}=\vecu$. By means of the scaling symmetry = = --3t-x , the conservation law formula yields the conserved density \[mkdvformula\] ¶[t]{} = -(3t +x+) -() = ¶[t]{}[(k)]{} . Note this can be expressed in terms of the recursions operator $\adR$ by ¶[t]{}[(k)]{} = -()/ . As $\vecadsymm{(k)} \rightarrow \lambda^{-(1+2k)} \vecadsymm{(k)}$ under the mKdV scaling symmetry, we see from the scaling formula that each adjoint-symmetry $\vecadsymm{(k)}$ is a multiplier for a local conservation law on mKdV solutions, given by \[mkdvconslaw\] (,,,…) + (,,,…) =0 with \[mkdvscalingformula\] ¶[t]{}[(k)]{} . However, in contrast to the scalar KdV and sine-Gordon cases, here expressions do not lead in any immediate way to a recursion operator for the mKdV conservation laws (since $\D{x}\vecadsymm{(k)}$ cannot be expressed directly in terms of $\P{t}{(k)}$, $\vecu$, and their derivatives). Nevertheless we have an explicit recursion formula = (( ()\^k )) for generating all of the local conservation laws (to within a trivial conserved density $\D{x}\triv{}$).
Other soliton field equations, like the nonlinear Schrodinger equation, Tzetzeica equation, Harry-Dym equation, Boussinesq equation, and their variants [@solitoneqs], as well as more general multi-component scalar/vector field equations [@Wolf], can be treated in a similar way to the preceeding examples.
Fluid flow and wave propagation
---------------------------------
Secondly, field equations for fluid flow and nonlinear wave propagation in 2+1 and 3+1 dimensions will be considered.
[*Euler equations.*]{} The field equations for an incompressible inviscid fluid in two or three spatial dimensions are given by \[fluideq\] (,,,P) = ĭ +j ĭ +P =0 ,() = ĭ = 0 for fluid velocity $\u{i}(t,\vecx)$ and pressure $P(t,\vecx)$, with constant density $\dens$. This system is invariant under the family of scalings $t\rightarrow \lambda^p t$, $\x{i}{}\rightarrow \lambda \x{i}{}$, $\u{i}\rightarrow \lambda^{1-p} \u{i}$, $P\rightarrow \lambda^{2-2p} P$, for arbitrary $p=\const$. The fluid symmetries are solutions $(\symm{}{i},\symm{}{})$ of the linearized field equations \[fluidsymmeq\] = +j +ĭ + =0 , = =0 for all $\vecu(t,\vecx),P(t,\vecx)$ satisfying the Euler equations. The adjoint of equations is given by \[fluidadsymmeq\] = - -j +j - =0 , =- =0 whose solutions $(\adsymm{i},\adsymm{})$ for all $\vecu(t,\vecx),P(t,\vecx)$ satisfying are the adjoint-symmetries of the Euler equations. In addition to scaling symmetries, && ĭ= = (1-p)ĭ-ĭ-ptĭ ,\
&& P= = (2-2p)P -P-ptP , the Euler equations are well-known to possess the Galilean group [@Landau-Lifshitz; @Olver-book] of symmetries, comprising time translations \[fluidtKVsymm\] ĭ= =ĭ = -j ĭ -P ,P== P , Galilean boosts, with velocity $\v{i}=\const$, \[fluidboostsymm\] ĭ= =tǰĭ -ǐ ,P= = tǰP , and space translations and rotations \[fluidKVsymm\] ĭ= =ĭ = ĭ -(j) ,P= = P = P , where $\kv{i}{}(x)$ is a Killing vector of the Euclidean space in which the fluid flow takes place, / $\coder{(i}\kv{j)}{}=0$. (In particular, $\kv{i}{}=a^{i}=\const$ yields translations, and $\kv{i}{}=b^{ij}\x{}{j}$, $b^{ij}=-b^{ji}=\const$, yields rotations.) Corresponding adjoint-symmetries are given by the relations && = ,\
&& = dĭ - +P , yielding && =j , =ĭj +P , \[fluidtKVadsymm\]\
&& = , =j , \[fluidKVadsymm\]\
&& = tǰ , = ǐ (tj -) . \[fluidboostadsymm\] Now, typically, local conservation laws (,,P) + (,,P) =0 on solutions of the Euler equations are derived through consideration of Newton’s laws applied to fluid elements [@Landau-Lifshitz] or by Noether’s theorem in a Hamiltonian formulation [@Olver-book; @fluidnoether]. In contrast, the conservation law formula in terms of any adjoint-symmetry $(\adsymm{i},\adsymm{})$ directly yields a conserved density \[fluidformula\] ¶[t]{}[(,)]{} =( (1-p)ĭ -ĭ-ptĭ ) ( (ptj -)ĭ +(1-p)ĭ ) to within a trivial conserved density $\D{i}\triv{i}$. Here, this formula easily leads to momentum and angular momentum \[fluidmomentum\] = j ¶[t]{}/ from the Killing vector adjoint-symmetries , and Galilean momentum \[fluidgalileanmomentum\] = tǐ j ¶[t]{}/ from the boost adjoint-symmetry , as well as energy \[fluidenergy\] = ĭ j ¶[t]{}/ from the fluid velocity adjoint-symmetry , to within proportionality factors. (A useful identity in these calculations is $\u{j} = \D{i}( \x{j}{}\u{i} )$ on fluid solutions.) From the scaling formula , in three dimensions, it follows that $\w{\veckv}=4-p$ when $\veckv$ is a translation, $\w{\veckv}=5-p$ when $\veckv$ is a rotation, $\w{\vecu}=5-2p$ and $\w{\vecv}=4$, representing the scaling weights of, respectively, the integrals for momentum and angular momentum $\int \veckv{\cdot}\vecu\ \rmd^3x$, energy $\int \frac{1}{2}|\vecu|^2 \rmd^3x$, and Galilean momentum $\int t\vecv{\cdot}\vecu\ \rmd^3x$. Note, for the dilation scaling $p=1$, all the scaling weights are positive and hence the conservation laws , , are noncritical. These weights decrease by $1$ in two dimensions, leading to the same conclusions.
The Euler equations also are known to possess a vorticity conservation law [@vorticity], which is unrelated to symmetries in contrast with the energy and momentum conservation laws [@fluidnoether]. Here, a derivation will be given by formula directly in terms of fluid adjoint-symmetries.
In three dimensions, vorticity is the curl of the fluid velocity = k satisfying the vorticity equations [@Landau-Lifshitz] +j - ĭ =0 , =0 , where $\cross{i}{jk}$ is the cross-product operator (/ $\cross{}{ijk} = \id{li}{} \cross{l}{jk} = \cross{}{[ijk]}$ is the totally antisymmetric symbol). We observe these equations have precisely the form of the adjoint-symmetry equations and hence = , = 0 yields a corresponding fluid adjoint-symmetry. Then the conserved density formula leads to \[fluidvorticity\] = ĭ k ¶[t]{}/ where, from scaling formula , $\w{\vecvort}=4-2p$ is the scaling weight of the vorticity integral $\int \vecu \cdot( \grad\times\vecu ) \rmd^3x$. Physically, this conserved quantity describes the total helicity (degree of knottedness) of vortex filaments. Note its scaling weight is noncritical provided $p\neq 2$.
The situation in two dimensions is slightly different. The role of fluid vorticity is played by the scalar curl = k where $\cross{}{jk} = \cross{}{[jk]}$ is the antisymmetric symbol. This scalar vorticity satisfies the conservation equation \[fluidvorteq\] +j =0 . By taking a curl, we immediately see that = , = -/2 satisfy the fluid adjoint-symmetry equations . The conserved density formula now yields \[fluidvorticity’\] = ( k)\^2 ¶[t]{}/ with the proportionality factor $\w{\vort{}}=2-2p$ given by the scaling weight of the conserved vorticity integral $\int ( \grad\times\vecu )^2 \rmd^2x$ (where $\grad\times\vecu$ denotes the scalar curl $\cross{}{jk} \coder{j} \u{k}$). Note that this vorticity quantity is noncritical if $p\neq 1$, / other than for a dilation scaling. More generally, any function $f(\vort{})$ in two dimensions is also a conserved density due to conservation of the vorticity $\vort{}=\grad\times\vecu$, but the resulting vorticity integral $\int f(\grad\times\vecu) \rmd^2x$ will have a well-defined scaling weight only if $p=0$, namely for a spatial dilation scaling. In this case the conserved density = f( k) ¶[t]{}[f]{}/ again arises directly from formula , through the adjoint-symmetry = f” , = f- f’ . Here the proportionality factor is simply $\w{f}=2$, due to the scaling invariance of $f(\vort{})$, and consequently the vorticity integral is noncritical. The same conclusion holds for the vorticity integral in three dimensions if a spatial dilation scaling, $p=0$, is considered.
The Navier-Stokes equations and polytropic gas dynamics equations can be treated analogously.
[*Nonlinear wave equation.*]{} The scalar field equation for $u(t,\vecx)$ \[waveeq\] (u,u,u) = u -u u\^ = - u u\^=0 describes a nonlinear wave with interaction strength depending on a positive integer $\pow> 1$ (where $\g{}{\alpha\beta}$ is the Minkowski metric tensor and $\x{\alpha}{}=(t,\vecx)$ are Minkowski spacetime coordinates). This is a Lagrangian wave equation invariant under the scaling $\x{\alpha}{}\rightarrow \lambda \x{\alpha}{}$, $u\rightarrow \lambda^{q} u$, for $q=2/(1-\pow) \neq 0$. The corresponding scaling symmetry u= = q u -u is a solution of the linearized field equation \[wavesymmeq\] = - u u\^[-1]{} =0 for all $u(t,\vecx)$ satisfying the wave equation . Here, symmetries are the same as adjoint-symmetries, since $\adlinop{\ueq{}{}} = \linop{\ueq{}{}}$. The additional spacetime symmetries (Poincaré group) of the wave equation for arbitrary $\pow>0$ are given by translations, rotations, and boosts, \[waveKVsymm\] u= = u = u where $\kv{\alpha}{}(x)$ is a Killing vector of Minkowski space, $\coder{(\alpha}\kv{\beta)}{}=0$. For certain interaction powers, $\pow=5$ in 2+1 dimensions and $\pow=3$ in 3+1 dimensions, the wave equation also admits [@Strauss] inversion symmetries \[waveCKVsymm\] u= = u + u (where $\div\kv{}{}=\der{\alpha}\kv{\alpha}{}$) associated with conformal Killing vectors $\kv{\alpha}{} =
c^\beta \x{}{\beta} \x{\alpha}{}
-\frac{1}{2} c^\alpha \x{}{\beta} \x{\beta}{}$, $c^\beta=\const$, satisfying $\coder{(\alpha}\kv{\beta)}{} =\Omega\g{\alpha\beta}{}$ for a conformal factor $\Omega(x)$. Through Noether’s theorem, all spacetime symmetries are known to be multipliers [@Strauss] for local conservation laws on solutions of the wave equation , (t,,u,u,u)=0 with = - (u,) -( uu - () u\^2 ) given in terms of the conserved stress-energy tensor \[waveT\] (u,) = uu -( uu u\^[+1]{} ) . The conservation $\der{\alpha}\T{\alpha}{\beta}(u,\ujet{})=0$ of this tensor on solutions of the wave equation provides a well-known alternative derivation of the conservation laws associated with spacetime Killing vectors, = - (u,) Here, the resulting conserved densities instead will be obtained from the conservation law formula directly in terms of the corresponding symmetries . This yields ¶ && = ( (q-1)u -u ) ( u + u)\
&&- ( q u - u ) ( u + u + u + () u )\
&& - \[waveformula\] to within a trivial conserved density ($\D{\beta}\triv{\alpha\beta}$, with $\triv{\alpha\beta}=-\triv{\beta\alpha}$). The proportionality factor $\w{\bdkv}$ is, by the scaling formula , the scaling weight of the flux integrals $\int \curr{\alpha}{\kv{}{}} \n{}{\alpha} \rmd\Sigma$, on a $t=\const$ spatial hypersurface $\Sigma$ with normal vector $\n{}{\alpha}=\der{\alpha}t$. For translations $\kv{\alpha}{}=a^{\alpha}=\const$, we have =4/(1-) in 2+1 dimensions, while in 3+1 dimensions, =(5-)/(1-) ; the weight $\w{\bdkv}$ increases by $1$ for rotations and boosts $\kv{\alpha}{}=b^{\alpha\beta}\x{}{\beta}$, $b^{\alpha\beta}=-b^{\beta\alpha}=\const$, and increases by $1$ again for inversions so thus $\w{\bdkv}=1$ in the case of proper conformal Killing vectors. Hence, a critical case $\w{\bdkv}=0$ only occurs for translations in 3+1 dimensions when $\pow=5$, and for rotations and boosts in 3+1 dimensions when $\pow=3$ as well as in 2+1 dimensions when $\pow=5$, corresponding to scaling invariance of the energy-momentum integral $\int -\T{\alpha}{\beta}(u,\ujet{})a^\beta \n{}{\alpha} \rmd\Sigma$ and of the angular-boost momentum integral $\int -\T{\alpha}{\beta}(u,\ujet{})b^{\beta\gamma}\x{}{\gamma}
\n{}{\alpha} \rmd\Sigma$ in these cases. Consequently, $\P{\alpha}{\symm{}{}} \simeq 0$ is trivial only for these critical interaction powers [@critical] and Killing vectors. In this situation, all local conservation laws are produced nevertheless from the more general formula directly in terms of pairs of Killing vector symmetries , (\_1,\_2) -(u,) where $\kv{\alpha}{} =[\kv{}{1},\kv{}{2}]^\alpha$ is the commutator of the Killing vectors. The same result holds even for the non-critical cases where $\w{\bdkv}\neq 0$.
Other nonlinear wave equations, such as sigma models and wavemap equations [@Shatah], can be treated in the same way.
Gauge theories
----------------
Finally, Yang-Mills fields and gravitational fields in 3+1 dimensions will be considered.
[*Yang-Mills theory.*]{} The Yang-Mills field on Minkowski space $(\Rnum^4,\g{}{\alpha\beta})$ is a vector potential $\A{a}{\alpha}(x)$ that takes values in an internal Lie algebra $\liealg=(\Rnum^N,\c{a}{bc})$. Associated with $\A{a}{\alpha}$ is the Yang-Mills covariant derivative = +a[bc]{} and the Yang-Mills field strength tensor = +a[bc]{} where $\c{a}{bc}$ denotes the structure constants of the Lie algebra $\liealg$. The Yang-Mills equation with gauge group based on $\liealg$ is then given by \[ymeq\] (A,,) = =0 which is invariant under the scaling $\x{\alpha}{} \rightarrow \lambda\x{\alpha}{}$, $\A{a}{\alpha} \rightarrow \lambda^{-1}\A{a}{\alpha}$. Whenever the gauge group is semisimple, the Yang-Mills equation arises from a Lagrangian (see / ), and in this situation both the symmetries and adjoint-symmetries of this field equation are given by solutions $\delta\A{a}{\alpha} = \symm{\alpha}{a}$ of the linearized Yang-Mills equation = ( + a[bc]{} ) =0 for all Yang-Mills solutions $\A{a}{\alpha}(x)$. Note, here, $\ymder{\alpha} = \D{\alpha} +\c{a}{bc} \A{b}{\alpha}$ acts as a total derivative operator. The well-known local symmetries of the Yang-Mills equation are comprised by gauge symmetries \[ymgaugesymm\] = involving any Lie-algebra valued scalar function $\sf{a}(x,A,\Ajet{},\ldots)$, and spacetime symmetries \[ymkvsymm\] = 2 = - ( ) where $\kv{\beta}{}(x)$ is any conformal Killing vector on Minkowski space, $\coder{(\alpha}\kv{\beta)}{} =\frac{1}{4}\g{\alpha\beta}{} \div\kv{}{}$. In the case of a dilation Killing vector, $\kv{\beta}{}=\x{\beta}{}$, the spacetime symmetry reduces to a sum of the Yang-Mills scaling symmetry and a gauge symmetry, \[ymscsymm\] = = - - = -2 - ( ) . A recent classification analysis [@Pohjanpelto] has proved that these are in fact the only nontrivial local symmetries admitted by the Yang-Mills equation if the Lie algebra $\liealg$ is real and simple. However, for a simple Lie algebra $\liealg$ with a complex structure, the same analysis found that the Yang-Mills equation also admits complexified spacetime symmetries \[ymcomplexkvsymm\] = 2 where $\j{a}{b}$ is the complex structure map on $\liealg$, satisfying the properties = - , b[cd]{} = a[ed]{} , =0 with $\ck{}{ab} = \c{c}{ad} \c{d}{bc}$ being the Cartan-Killing metric on $\liealg$. Through Noether’s theorem, these symmetries are multipliers for local conservation laws $\D{\alpha} \curr{\alpha}{}(x,A,\Ajet{}) =0$ consisting of, respectively, = 0 related to the Bianchi identity on $\F{a}{\alpha\beta}$, and \[ymkvconslaw\] = (F) , = (F) , given by the conserved Yang-Mills stress-energy tensor (F) = ( - ) and its complexification (F) = ( - ) . These conservation laws yield (complexified) energy-momentum, angular and boost momentum for translation, rotation and boost Killing vectors $\coder{(\alpha}\kv{\beta)}{} =0$, and additional quantities for dilation and inversion Killing vectors $\coder{(\alpha}\kv{\beta)}{}
=\frac{1}{4}\div\kv{}{}\g{\alpha\beta}{} \neq 0$.
However, the previous results do not fully settle the classification of local conservation laws of the Yang-Mills equation , since it leaves open the question of whether any trivial symmetries could yield nontrivial conservation laws by Noether’s theorem. For Lagrangian field equations whose principal part (/ highest derivative terms) is nondegenerate, it is known that there is a one-to-one correspondence between nontrivial variational symmetries and nontrivial conservation laws [@Olver-book; @Anco-Bluman2]. But this correspondence is not automatic for a field equation with gauge symmetries, due to the resulting degeneracy of the field equation’s principal part, in contrast to the previous examples in this section. Here, through an application of formula , the gap in the classification of Yang-Mills conservation laws will be addressed.
We consider local symmetries \[ymsymm\] = (x,A,,…) assumed to be homogeneous with respect to the Yang-Mills scaling , \[ymsymmweight\] = r - with scaling weight $r=\const$. As noted in , there is no loss of generality in such a homogeneity restriction. Now, formula yields a conserved current generated from any such symmetry, ¶(,A) 2( - ( + ) ) , with the current being linear and homogeneous in $\symm{\alpha}{a}$ and $\ymder{\mu}\symm{\alpha}{a}$. If $\symm{\alpha}{a}=\Q{a}{\alpha}$ is a multiplier for a local conservation law of the Yang-Mills equation , \[ymconslaw\] = where the current (x,A,,…) can be assumed homogeneous under the Yang-Mills scaling , then the scaling formula gives the relation ¶(,A) (r+1) to within a trivial conserved current. Hence, for a variational symmetry that is trivial, so $\symm{\alpha}{a} =0$ for solutions of , we see that, if $r\neq -1$, ¶(0,A) =0 is a trivial noncritical current on Yang-Mills solutions. Note the factor $r+1$ here is precisely the scaling weight of the flux integral of the current $\curr{\alpha}{\Q{}{}}$. This establishes the following classification result.
[**Proposition 3.1**]{}: No nontrivial conservation laws that are noncritical — / whose associated conserved quantity $\int \curr{\alpha}{} \der{\alpha}t \rmd\Sigma$, on a spatial hypersurface $\Sigma$ given by $t=\const$, has non-zero scaling weight — may arise from variational trivial symmetries of the Yang-Mills equation .
[*General Relativity.*]{} The vacuum gravitational field equation [@Wald] on a 4-dimensional spacetime manifold is given by \[greq\] (g,,) ==0 where $\G{\alpha\beta}{}$ is the Einstein tensor (/ trace-reversed Ricci tensor $\curv{\alpha\beta}{}$) for the spacetime metric $\g{}{\alpha\beta}(x)$. Its linearized field equation is given by \[grsymmeq\] = -( -2 + ) =0 whose solutions for all $\g{}{\alpha\beta}(x)$ satisfying the Einstein equation are the symmetries $\delta\g{\alpha\beta}{}=\symm{}{\alpha\beta}(x,g,\gjet{},\ldots)$ of the gravitational field, where $\covder{\mu}$ is the covariant total derivative operator associated with the metric, $\covder{\mu}\g{}{\alpha\beta}=0$, and “bar” denotes trace-reversal on a given tensor. Here, since the gravitational field equation comes from a Lagrangian [@Wald], adjoint-symmetries are the same as symmetries. It is known that the only admitted nontrivial local symmetries [@Anderson-Torre] consist of a constant conformal scaling \[grscaling\] = and diffeomorphism gauge symmetries \[grdiffeo\] = = 2 for any local vector field $\vf{\beta}{}(x,g,\gjet{},\ldots)$. If we consider local conservation laws of the gravitational field equation , \[grconslaw\] (x,g,,…) =0 , then through Noether’s theorem the constant conformal scaling is not a multiplier, while the diffeomorphisms yield a trivial conservation law $\curr{\alpha}{}=\vf{\beta}{} \G{\beta}{\alpha} =0$ on solutions of .
asserts that in fact there exist no nontrivial local conservation laws , but does not provide a full statement of the proof. Here a complete classification proof will be given for diffeomorphism-covariant conservation laws, by use of a covariant version of the conservation law formula . A suitable dilation scaling symmetry is provided by the diffeomorphism symmetry specialized to a homothetic vector field $\vf{\alpha}{}=\scvf{\alpha}{}$, namely \[grhomothetic\] = ,=0 . Now, a local conservation law is diffeomorphism-covariant whenever it is of the form \[grnaturalconslaw\] (g,R,R,…)=0 on solutions of the Einstein equation , with $\curr{\alpha}{}$ satisfying the natural transformation property $\delta\curr{\alpha}{}= \lieder{\vf{}{}}\curr{\alpha}{}$ under all diffeomorphism symmetries . Correspondingly, without loss of generality we consider diffeomorphism-covariant local symmetries \[grnaturalsymm\] =(g,R,R,…) , which are necessarily homogeneous with respect to the dilation scaling = . Then for any such symmetry , formula yields a conserved current ¶(,g) = - ( - ) + + whose dependence on $\symm{}{\alpha\beta}$ and $\covder{\mu}\symm{}{\alpha\beta}$ is linear, homogeneous. It now follows from the scaling formula in covariant form that if $\symm{}{\alpha\beta}=\coQ{\alpha\beta}$ is a multiplier for a local conserved current, $\Q{}{\alpha\beta}\G{}{\alpha\beta} = \covder{\alpha}\curr{\alpha}{\Q{}{}}$, then to within a trivial conserved current, \[grformula\] ¶(,g) , w=0 . Hence, for a trivial multiplier that is diffeomorphism-covariant , we have ¶(0,g) =0 on solutions of the Einstein equation , since $\symm{}{\alpha\beta}=0$. Therefore, the following result holds.
[**Proposition 3.2**]{}: No nontrivial diffeomorphism-covariant conservation laws may arise from variational trivial symmetries of the gravitational field equation .
Solutions of the Einstein equation with a Killing vector $\lieder{\kv{}{}}\g{}{\alpha\beta} = 2\covder{(\alpha} \kv{}{\beta)}=0$ are known to possess an unexpected extra symmetry [@Geroch1] \[grkvsymm\] = -2 , = - , where $\twist$ is the scalar twist and $\dualkv{}{\beta}$ is the dual of $\kv{\beta}{}$, defined by the equations = ,2 = . Note this symmetry is nonlocal due to its dependence on $\twist,\dualkv{}{\beta}$ (in terms of $\g{}{\alpha\beta},\kv{}{\alpha}$). Here, a corresponding nonlocal conservation law will be derived through the conservation law formula in covariant form using the constant conformal scaling , \[grcurr\] ¶(,g) = - . This yields, from , ¶(,g) = +2( ) 2 . Moreover, on solutions of the Einstein equation with two commuting Killing vectors $\kv{\alpha}{1}$ and $\kv{\alpha}{2}$, the symmetries are known to generate an infinite-dimensional algebra of nonlocal symmetries (the Geroch group [@Geroch2]). Then the covariant formula leads to a corresponding infinite sequence of conservation laws, related to the complete integrability of the system $\G{\alpha\beta}{} =0$, $\lieder{\kv{}{1}}\g{}{\alpha\beta} = \lieder{\kv{}{2}}\g{}{\alpha\beta} =0$.
A similar classification treatment of local and nonlocal conservation laws of the self-dual Einstein equation and self-dual Yang-Mills equation will be given elsewhere.
Conclusion {#conclude}
============
The conservation law expression derived in Proposition 2.1 is a generalization of a formula mentioned in in the case of linear PDEs and extends a similar formula considered for self-adjoint PDEs in . A variant of this expression has been central to a recently obtained classification of local conserved currents for linear massless spinorial field equations of spin $s>0$ [@maxwell; @spins]. Numerous applications to nonlinear ODEs are presented in .
Apart from its main use in generating all noncritical local conservation laws in an algebraic manner — by-passing the standard homotopy integral formula and (adjoint-) invariance conditions — in terms of local (adjoint-) symmetries admitted by any given PDEs, the conservation law expression is able to produce nonlocal conservation laws from any admitted nonlocal (adjoint-) symmetries. Such examples and applications will be considered in a forthcoming paper [@nonlocal].
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Throughout, “symmetries” will refer to local (point or generalized) symmetries in evolutionary form (see ). A symmetry, or adjoint-symmetry, is trivial if it vanishes on the solution space of the field equations. Two symmetries, or adjoint-symmetries, are considered equivalent if they differ by one that is trivial.
The adjoint invariance condition holds for only certain adjoint-symmetries, if any, in an equivalence class.
The determining equation for adjoint-symmetries is the same as that for symmetries when and only when a field equation is self-adjoint, which is also the necessary and sufficient condition for the field equation to have a Lagrangian formulation. In this case, adjoint-symmetries are symmetries, and the adjoint invariance condition is equivalent to invariance of the action principle.
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This entails no essential loss of generality, as scaling invariance of the field equations implies that any multiplier is, formally, a Laurent series of homogeneous multipliers.
A conservation law is trivial if, on the solution space of the field equations, it has the form of the divergence of an antisymmetric tensor. Two conservation laws are equivalent if they differ by a trivial conservation law.
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abstract: 'We determined on the temperature-pressure-magnetic field ($T$-$p$-$H$) phase diagram of the ferromagnet LaCrGe$_3$ from electrical resistivity measurements on single crystals. In ferromagnetic systems, quantum criticality is avoided either by a change of the transition order, becoming of the first order at a tricritical point, or by the appearance of modulated magnetic phases. In the first case, the application of a magnetic field reveals a wing-structure phase diagram as seen in itinerant ferromagnets such as ZrZn$_2$ and UGe$_2$. In the second case, no tricritical wings have been observed so far. Our investigation of LaCrGe$_3$ reveals a double-wing structure indicating strong similarities with ZrZn$_2$ and UGe$_2$. But, unlike these, simpler systems, LaCrGe$_3$ is thought to exhibit a modulated magnetic phase under pressure which already precludes it from a pressure-driven paramagnetic-ferromagnetic quantum phase transition in zero field. As a result, the $T$-$p$-$H$ phase diagram of LaCrGe$_3$ shows both the wing structure as well as the appearance of new magnetic phases, providing the first example of this new possibility for the phase diagram of metallic quantum ferromagnets.'
author:
- 'Udhara S.'
- 'Sergey L.'
- 'Paul C.'
- 'Valentin [^1]'
title: 'Tricritical wings and modulated magnetic phases in LaCrGe$ _{3}$ under pressure'
---
Suppressing a second-order, magnetic phase transition to zero temperature with a tuning parameter (pressure, chemical substitutions, magnetic field) has been a very fruitful way to discover many fascinating phenomena in condensed matter physics. In the region near the putative quantum critical point (QCP), superconductivity has been observed in antiferromagnetic [@Mathur1998Nature] as well as ferromagnetic systems [@Saxena2000Nature; @Aoki2001Nature; @Huy2007PRL]. One peculiarity of the clean ferromagnetic systems studied so far is that the nature of the paramagnetic-ferromagnetic (PM-FM) phase transition always changes before being suppressed to zero temperature [@Brando2016RMP]: in most cases, the transition becomes of the first order [@Goto1997PRB; @Huxley2000PB; @Uhlarz2004PRL; @Colombier2009PRBYbCu2Si2; @Araki2015JPSJ; @Shimizu2015PRB]. Recently, another possibility, where a modulated magnetic phase (AFM$_Q$) appears (spin-density wave, antiferromagnetic order), has been observed in CeRuPO [@Kotegawa2013JPSJ; @Lengyel2015PRB], MnP [@Cheng2015PRL; @Matsuda2016PRB] and LaCrGe$_3$ [@Taufour2016PRL]. When a FM transition becomes of the first order at a tricritical point (TCP) in the temperature $T$ pressure $p$ plane, the application of a magnetic field $H$ along the magnetization axis reveals a wing structure phase diagram in the $T$-$p$-$H$ space. This is seen in UGe$_2$ [@Taufour2010PRL; @Kotegawa2011JPSJ] and ZrZn$_2$ [@Kabeya2012JPSJ] and is schematically represented in Fig.\[fig:diagabcd\]a. This phase diagram shows the possibility of a new kind of quantum criticality at the quantum wing critical point (QWCP). In contrast with the conventional QCP, symmetry is already broken by the magnetic field at a QWCP. In the more recently considered case where the transition changes to a AFM$_Q$ phase, no wing structure phase diagram has been reported, but it is found that the AFM$_Q$ is suppressed by moderate magnetic field [@Kotegawa2013JPSJ; @Lengyel2015PRB]. This second possible $T$-$p$-$H$ phase diagram has been schematically presented in a recent review [@Brando2016RMP] and reproduced in Fig.\[fig:diagabcd\]b.
 a) Schematic $T$-$p$-$H$ phase diagram of a quantum ferromagnet: the paramagnetic-ferromagnetic (PM-FM) transition becomes of the first order at a tricritical point (TCP) after which there is a quantum phase transition (QPT) at $0$ K. Tricritical wings emerge from the TCP under magnetic field and terminate at quantum wing critical points (QWCP). b) Schematic $T$-$p$-$H$ phase diagram of a quantum ferromagnet when a modulated magnetic phase (SDW/AFM) emerges from the Lifshitz point (LP). c) New possible schematic $T$-$p$-$H$ phase diagram for which tricritical wings as well as a new magnetic phase are observed. d) $T$-$p$ phase diagram of LaCrGe$_3$ from electrical resistivity measurements [@Taufour2016PRL] showing two FM regions (FM1 and FM2) separated by a crossover.](schematicsABCD.eps){width="8.6cm"}
Here, we report electrical resistivity measurements on LaCrGe$_3$ under pressure and magnetic field. We determine the $T$-$p$-$H$ phase diagram and find that it corresponds to a third possibility where tricritical wings emerge in addition to the AFM$_Q$ phase. This new type of phase diagram is illustrated in Fig.\[fig:diagabcd\]c: it includes both the tricritical wings and the AFM$_Q$ phase. In addition, the phase diagram of LaCrGe$_3$ shows a double wing structure similar to what is observed in the itinerant ferromagnets UGe$_2$ [@Taufour2011JPCS] and ZrZn$_2$ [@Kimura2004PRL], but with the additional AFM$_Q$ phase. LaCrGe$_3$ is the first example showing such a phase diagram.
Recently, we reported on the $T$-$p$ phase diagram of LaCrGe$_3$ [@Taufour2016PRL], which is reproduced in Fig.\[fig:diagabcd\]d. At ambient pressure, LaCrGe$_{3}$ orders ferromagnetically at $T_\textrm{C}=86$K. Under applied pressure, $T_\textrm{C}$ decreases and disappears at $2.1$ GPa. Near $1.3$ GPa, there is a Lifshitz point at which a new transition line appears. The new transition corresponds to the appearance of a modulated magnetic phase (AFM$_Q$) and can be tracked up to $5.2$ GPa. Muon-spin rotation ($\mu$SR) measurements show that the AFM$_Q$ phase has a similar magnetic moment as the FM phase but without net macroscopic magnetization [@Taufour2016PRL]. In addition, band structure calculations suggest that the AFM$_Q$ phase is characterized by a small wave-vector $Q$ and that several small $Q$ phases are nearly degenerate. Below the PM-AFM$_Q$ transition line, several anomalies marked as gray cross in Fig.\[fig:diagabcd\]d can be detected in $\rho(T)$ [@Taufour2016PRL]. These other anomalies within the AFM$_Q$ phase are compatible with the near degeneracy of different $Q$-states (shown as AFM$_Q$ and AFM$_{Q'}$) with temperature and pressure driven transitions between states with differing wavevectors.
Results {#results .unnumbered}
=======
In this article, we determine the three dimensional $T$-$p$-$H$ phase diagram of LaCrGe$_3$ by measuring the electrical resistivity of single crystals of LaCrGe$_3$ under pressure and magnetic field. The sample growth and characterization has been reported in Ref. [@Lin2013PRB]. The pressure techniques have been reported in Ref. [@Taufour2016PRL]. The magnetic field dependent resistivity was measured in two Quantum Design Physical Property Measurement Systems up to $9$ or $14$ T. The electrical current is in the $ab$-plane, and the field is applied along the $c$-axis, which is the easy axis of magnetization [@Cadogan2013SSP; @Lin2013PRB].
Whereas most of the features in Fig.\[fig:diagabcd\]d were well understood in Ref. [@Taufour2016PRL], we also indicate the pressure dependence of $T_x$ ( d$\rho$/d$T_\text{max}$) at which a broad maximum is observed in $d\rho/dT$ below $T_\textrm{C}$ and shown as orange triangles in Fig.\[fig:diagabcd\]d. At ambient pressure, $T_x\approx71$ K. No corresponding anomaly can be observed in magnetization [@Taufour2016PRL], internal field [@Taufour2016PRL] or specific heat [@Lin2013PRB]. Under applied pressure, $T_x$ decreases and cannot be distinguished from $T_\textrm{C}$ (d$\rho$/d$T_\text{mid}$) above $1.6$ GPa. As will be shown, application of magnetic field allows for a much clearer appreciation and understanding of this feature.
 Temperature dependence of the resistivity (black line) and its derivative (blue line) of (a) LaCrGe$_3$ at $1.14$GPa and (b) UGe$_2$ at $0$GPa from Ref. [@Taufour2010PRL]. The crossover between the two ferromagnetic phases (FM1 and FM2) is inferred from the maximum in d$\rho$/d$T$ ($T_x$) and marked by a red triangle, whereas the paramagnetic-ferromagnetic transition is inferred from the middle point of the sharp increase in d$\rho$/d$T$ ($T_\textrm{C}$) and indicated by a blue circle.](1_14GPa){width="8.5cm"}
Figure \[1\_14GPa\]a shows the anomalies at $T_x$ and $T_\textrm{C}$ observed in the electrical resistivity and its temperature derivative at $1.14$ GPa. For comparison, Fig. \[1\_14GPa\]b shows ambient pressure data for UGe$_2$ [@Taufour2010PRL] where a similar anomaly at $T_x$ can be observed. In UGe$_2$, this anomaly was studied intensively [@Pfleiderer2002PRL; @Hardy2009PRB; @PalacioMorales2016PRB]. It corresponds to a crossover between two ferromagnetic phases FM1 and FM2 with different values of the saturated magnetic moment [@Pfleiderer2002PRL; @Hardy2009PRB]. Under pressure, there is a critical point at which the crossover becomes a first-order transition, which eventually vanishes where a maximum in superconducting-transition temperature is observed [@Saxena2000Nature]. In the case of LaCrGe$_3$, we cannot locate where the crossover becomes a first order transition, since the anomaly merges with the Curie temperature anomaly near $1.6$ GPa, very close to the TCP. However, as we will show below, the two transitions can be separated again with applied magnetic field above $2.1$ GPa. This is similar to what is observed in UGe$_2$ where the PM-FM1 and FM1-FM2 transition lines separate more and more as the pressure and the magnetic field are increased. Because of such similarities with UGe$_2$, we label the two phases FM1 and FM2 and assume that the anomaly at $T_x$ corresponds to a FM1-FM2 crossover. A similar crossover was also observed in ZrZn$_2$ [@Kimura2004PRL]. In Refs. [@Sandeman2003PRL; @Wysokinski2014PRB], a Stoner model with two peaks in the density of states near the Fermi level was proposed to account for the two phases FM1 and FM2, reinforcing the idea of the itinerant nature of the magnetism in LaCrGe$_3$.
 (a) Field dependence of the electrical resistivity at $2$K, $13.5$K, and $30$K at $2.39$GPa. Continuous and dashed lines represent the field increasing and decreasing respectively. (b) Corresponding field derivatives (d$\rho$/d$H$). The curves are shifted by $15$$\mu\Omega$cmT$^{-1}$ for clarity. Vertical arrows represent the minima. The transition width is determined by the full width at half minimum as represented by horizontal arrows. The temperature dependence of the hysteresis width of $H_{\textrm{min}1}$ and $H_{\textrm{min}2}$ are shown in (c) and (d)(left axes). The hysteresis width gradually decreases with increasing temperature and disappears at [ $T_{\textrm{WCP}}$]{}. The right axes show the temperature dependence of the transition widths. The width is small for the first-order transition and becomes broad in the crossover region. The blue-color shaded area represents the first order transition region whereas the white color area represents the crossover region. These allow for the determination of the wing critical point of the FM1 transition at $13.5$ K, $2.39$ GPa and $5.1$ T and the one for the FM2 transition at $12$ K, $2.39$ GPa and $7.7$ T.](2.39_GPa1.eps){width="8.5cm"}
In zero field, for applied pressures above $2.1$ GPa, both FM1 and FM2 phases are suppressed. Upon applying a magnetic field along the $c$-axis, two sharp drops of the electrical resistivity can be observed (Fig.\[2.39\_GPa\]a) with two corresponding minima in the field derivatives (Fig.\[2.39\_GPa\]b). At $2$ K, clear hysteresis of $\Delta H\sim0.7$ T can be observed for both anomalies indicating the first order nature of the transitions. The emergence of field-induced first-order transitions starting from $2.1$ GPa and moving to higher field as the pressure is increased is characteristic of the ferromagnetic quantum phase transition: when the PM-FM transition becomes of the first order, a magnetic field applied along the magnetization axis can induce the transition resulting in a wing structure phase diagram such as the one illustrated in Fig.\[fig:diagabcd\]a. In the case of LaCrGe$_3$, evidence for a first order transition was already pointed out because of the very steep pressure dependence of $T_\textrm{C}$ near $2.1$ GPa and the abrupt doubling of the residual ($T=2$ K) electrical resistivity [@Taufour2016PRL]. In UGe$_2$ or ZrZn$_2$, the successive metamagnetic transitions correspond to the PM-FM1 and FM1-FM2 transitions. In LaCrGe$_3$, due to the presence of the AFM$_Q$ phase at zero field, the transitions correspond to AFM$_Q$-FM1 and FM1-FM2.
As the temperature is increased, the hysteresis decreases for both transitions, as can be seen in Figs. \[2.39\_GPa\]c and d and disappears at a wing critical point (WCP). Also, the transition width is small and weakly temperature dependent below the WCP and it broadens when entering in the crossover regime. Similar behavior has been observed in UGe$_2$ [@Kotegawa2011JPSJ]. At $2.39$ GPa for example, we locate the WCP of the first-order FM1 transition around $13.5$ K and the one of the first-order FM2 transition around $12$ K. At this temperature and pressure, the transitions occur at $5.1$ and $7.7$ T respectively. This allows for the tracking of the wing boundaries in the $T$-$p$-$H$ space up to our field limit of $14$ T. At low field, near the TCP, the wing boundaries are more conveniently determined as the location of the largest peak in d$\rho$/d$T$ (Supplementary Information).
 Projection of the wings in (a) $T$-$H$, (b) $T$-$p$ and (c) $H$-$p$ planes. Black solid squares and green solid circles represents the FM$_1$-wing and FM$_2$-wing respectively. Red lines (represented in the $T$-$p$-$H$ space in Fig.\[Diag3DPaper2\]) are guides to the eyes and open symbols represent the extrapolated QWCP. (d) $H$-$p$ phase diagram at $2$K. The arrow represents the pressure $p_{c}=2.1$GPa.](Wing1_2.eps){width="8.5cm"}
The projections of the wings lines $T_\textrm{WCP}(p,H)$ in the $T$-$H$, $T$-$p$ and $H$-$p$ planes are shown in Figs. \[Wing1\]a, b and c respectively. The metamagnetic transitions to FM1 and FM2 start from $2.1$ GPa and separate in the high field region as the pressure is further increased. For the FM1 wing, the slope d$T_w$/d$H_w$ is very steep near $H=0$ (Fig. \[Wing1\]a) whereas d$H_w$/d$p_w$ is very small (Fig. \[Wing1\]c). This is in agreement with a recent theoretical analysis based on the Landau expansion of the free energy which shows that d$T_w$/d$H_w$ and d$p_w$/d$H_w$ are infinite at the tricritical point [@Taufour2016PRB]. This fact was overlooked in the previous experimental determinations of the wing structure phase diagram in UGe$_2$ [@Taufour2010PRL; @Kotegawa2011JPSJ] and ZrZn$_2$ [@Kabeya2012JPSJ], but appears very clearly in the case of LaCrGe$_3$. In the low field region, there are no data for the FM2 wing since the transition is not well separated from the FM1 wing, but there is no evidence for an infinite slope near $H=0$. The wing lines can be extrapolated to quantum wing critical points (QWCPs) at $0$ K in high magnetic fields of the order of $\sim30$ T (Fig. \[Wing1\]a) and pressures around $\sim3$ GPa (Fig. \[Wing1\]b). Figure \[Wing1\]d shows the $p$-$H$ phase diagram at low temperature ($T=2$ K). Identical $H$-$p$ phase diagram in Fig. \[Wing1\]b and Fig. \[Wing1\]c reveals the near vertical nature of the wings.
 $T$-$p$-$H$ phase diagram of LaCrGe$_{3}$ based on resistivity measurements. Red solid lines are the second order phase transition and blue color planes are planes of first order transitions. Green color areas represent the AFM$_Q$ phase.](Diag3DPaper2.eps){width="1\linewidth"}
The resulting three-dimensional $T$-$p$-$H$ phase diagram of LaCrGe$_3$ is shown in Fig. \[Diag3DPaper2\] which summarizes our results (Several of the constituent $T$-$H$ phase diagrams, at various pressures, are given in Supplementary Information). The double wing structure is observed in addition to the AFM$_Q$ phase. This is the first time that such a phase diagram is reported. Other materials suggested that there is either a wing structure without any new magnetic phase [@Taufour2010PRL; @Kotegawa2011JPSJ; @Kabeya2012JPSJ], or a new magnetic phase without wing structure [@Kotegawa2013JPSJ; @Lengyel2015PRB]. The present study illustrates a third possibility where all such features are observed. Moreover, the existence of the two metamagnetic transitions (to FM1 and FM2) suggests that this might be a generic feature of itinerant ferromagnetism. Indeed, it is observed in ZrZn$_2$, UGe$_2$, and LaCrGe$_3$, although these are very different materials with different electronic orbitals giving rise the the magnetic states. We note that a wing structure has also been determined in the paramagnetic compounds UCoAl [@Aoki2011JPSJUCoAl; @Combier2013JPSJ; @Kimura2015PRB] and Sr$_3$Ru$_2$O$_7$ [@Wu2011PRB], implying that a ferromagnetic state probably exists at negative pressures in these materials. Strikingly, two anomalies could be detected upon crossing the wings in UCoAl (two kinks of a plateau in electrical resistivity [@Aoki2011JPSJUCoAl], two peaks in the ac susceptibility [@Kimura2015PRB]), as well as in Sr$_3$Ru$_2$O$_7$ (two peaks in the ac susceptibility [@Wu2011PRB]). These double features could also correspond to a double wing structure.
To conclude, the $T$-$p$-$H$ phase diagram of LaCrGe$_3$ provides an example of a new possible outcome of ferromagnetic quantum criticality. At zero field, quantum criticality is avoided by the appearance of a new modulated magnetic phase, but the application of magnetic field shows the existence of a wing structure phase diagram leading towards QWCP at high field. These experimental findings reveal new insights into the possible phase-diagram of ferromagnetic systems. The emergence of the wings reveals for the first time a theoretically predicted tangent slope [@Taufour2016PRB] near the tricritical point, a fact that was overlooked in previous experimental determination of phase diagrams of other compounds because of the lack of data density in that region. In addition, the double nature of the wings appears to be a generic feature of itinerant ferromagnetism, as it is observed in several, a priori, unrelated materials. This result deserves further theoretical investigations and unification.
[**Acknowledgements**]{} We would like to thank S. K. Kim, X. Lin, V. G. Kogan, D. K. Finnemore, E. D. Mun, H. Kim, Y. Furukawa, R. Khasanov for useful discussions. This work was carried out at the Iowa State University and the Ames Laboratory, US DOE, under Contract No. DE-AC02-07CH11358. This work was supported by the Materials Sciences Division of the Office of Basic Energy Sciences of the U.S. Department of Energy.\
[**Supplementary Information**]{} is attached below.\
[**Author Contributions**]{} V.T. and P.C. initiated this study. U.K. and P.C. prepared the single crystals. U.K, V.T. and S.B. performed the pressure measurements. U.K, V.T, S.B. and U.K analysed and interpreted the pressure data. U.K. and V.T. wrote the manuscript with the help of all authors.\
Supplementary Information
=========================
Determination of the location of the tricritical point
------------------------------------------------------
 (a)-(b) Temperature dependence of d$\rho$/d$T$ at various magnetic fields at $1.67$ and $1.83$GPa. Arrow in panel (a) represent the $T_C$ and panel (b) represent the [ $T_{\textrm{WCP}}$]{}. (c) The variation of d$\rho$/d$T_{peak}$ as a function of external field for $p$$<$[$p_{\textrm{TCP}}$]{}, $p$$\approx$[$p_{\textrm{TCP}}$]{} and [$p_{\textrm{TCP}}$]{}$<$$p$$<$[$p_{\textrm{c}}$]{} .](H_sweep.eps){width="8.5cm"}
In Ref. [@Taufour2016PRL], the position of the tricritical point TCP was estimated near $40$ K and $1.75$GPa based on a discontinuity in the resistivity as a function of pressure $\rho(p)$. Here, we use measurements under magnetic field to locate the TCP. When the paramagnetic-ferromagnetic (PM-FM)transition is of the second order, the magnetic field applied along the magnetization axis ($c$-axis) breaks the time reversal symmetry, so that no phase transition can occur. Instead, a crossover is observed resulting in a broadening and disappearing of the anomalies. Supplementary Fig. \[H\_sweep\]a, shows the peak in the temperature derivative of resistivity d$\rho$/d$T$ at various magnetic fields at $1.67$ GPa. The peak amplitude decreases showing that the transition is of the second order. This is in contrast with the behavior at $1.83$ GPa (Supplementary Fig. \[H\_sweep\]b) where the peak first increases under magnetic field indicating the first order nature of the transition. The evolution of the value of d$\rho$/d$T$ at the peak position as a function of magnetic field is shown in Supplementary Fig. \[H\_sweep\]c for various pressures. We can distinguish two regimes: for pressures below $\sim1.75$ GPa, the peak size monotonically decreases with applied magnetic field; for pressure above $1.75$ GPa, the peak size first increases with field, reach a maximum at a field $H_{WCP}$ and then decreases. With this procedure, we find the TCP to be near $1.75$ GPa, at which pressure the transition temperature is $40$ K.
For $p>p_{\textrm{TCP}}$, the location of the maximum value of d$\rho$/d$T$ at the peak position serves to locate the wing critical point as a function of temperature, pressure and magnetic field.
Determination of the three-dimensional $T$-$p$-$H$ phase diagram
----------------------------------------------------------------
In Supplementary Fig.\[compildiag\], we show several $T$-$H$ phase diagrams at various pressures (as illustrated in Supplementary Fig\[fig:2KDiag\]) . For each pressure, anomalies in the temperature and field dependence of the electrical resistivity are located and serve to outline the phase boundaries. To be complete, and for future reference, we also indicate the location of broad maxima or kinks in d$\rho$/d$T$ which do not seem to correspond to phase transitions at this point and are most likely related to crossover anomalies.
{width="17cm"}
The $T$-$p$-$H$ phase diagram shown as Fig.\[Diag3DPaper2\] in the main text is constructed by combining all the $T$-$H$ phase diagrams at various pressures.
![\[fig:2KDiag\]$p$-$H$ phase diagram of LaCrGe$_3$ at $2$ K. the black dashed lines indicate the position of the pressures for the diagrams shown in Supplementary Fig. \[compildiag\]. Note: This is an expanded view of diagram shown in Figs. \[Wing1\]d in main text](2KDiag.eps){width="8.5cm"}
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[^1]: Present address University of California, Davis
|
---
abstract: 'We construct a theoretical framework to study Population III (Pop III) star formation in the post-reionization epoch ($z\lesssim 6$) by combining cosmological simulation data with semi-analytical models. We find that due to radiative feedback (i.e. Lyman-Werner and ionizing radiation) massive haloes ($M_{\rm halo}\gtrsim 10^{9}\ \rm M_{\odot}$) are the major ($\gtrsim 90$%) hosts for potential Pop III star formation at $z\lesssim 6$, where dense pockets of metal-poor gas may survive to form Pop III stars, under inefficient mixing of metals released by supernovae. Metal mixing is the key process that determines not only when Pop III star formation ends, but also the total mass, $M_{\rm PopIII}$, of *active* Pop III stars per host halo, which is a crucial parameter for direct detection and identification of Pop III hosts. Both aspects are still uncertain due to our limited knowledge of metal mixing during structure formation. Current predictions range from early termination at the end of reionization ($z\sim 5$) to continuous Pop III star formation extended to $z=0$ at a non-negligible rate $\sim 10^{-7}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$, with $M_{\rm PopIII}\sim 10^{3}-10^{6}\ \rm M_{\odot}$. This leads to a broad range of redshift limits for direct detection of Pop III hosts, $z_{\rm PopIII}\sim 0.5-12.5$, with detection rates $\lesssim 0.1-20\ \rm arcmin^{-2}$, for current and future space telescopes (e.g. HST, WFIRST and JWST). Our model also predicts that the majority ($\gtrsim 90$%) of the cosmic volume is occupied by metal-free gas. Measuring the volume filling fractions of this metal-free phase can constrain metal mixing parameters and Pop III star formation.'
author:
- |
Boyuan Liu^[{width="2.5mm"}](https://orcid.org/0000-0002-4966-7450)^[^1]$^{1}$ and Volker Bromm$^{1}$\
$^{1}$Department of Astronomy, University of Texas, Austin, TX 78712, USA\
bibliography:
- 'ref.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'When did Population III star formation end?'
---
\[firstpage\]
early universe – dark ages, reionization, first stars – galaxies: dwarf
Introduction {#s1}
============
The ‘standard model’ of early star formation predicts that the first generation of stars, the so-called Population III (Pop III), started to form at redshift $z\sim 20-30$ in minihaloes of $M_{\rm halo}\sim 10^{6}\ \rm M_{\odot}$ [@abel2002formation; @bromm2002; @bromm2013], driven by cooling from $\rm H_{2}$ and $\rm HD$ molecules in primordial gas [@johnson2006; @lithium]. Although the properties of Pop III stars are still uncertain in the absence of direct observations, current theoretical models (e.g. @stacy2013constraining [@susa2014mass; @hirano2015primordial; @machida2015accretion; @stacy2016building; @hirano2017formation; @hosokawa2020]), and indirect observational constraints (e.g. @frebel2015near [@ji2015preserving; @hartwig2015; @magg2019observational; @ishigaki2018initial; @yuta2020]) converge on the picture that Pop III stars are characterized by a top-heavy initial mass function (IMF), covering a few to a few hundred $\rm M_{\odot}$. As a result, they have distinct features compared with present-day stars formed in metal-enriched enrionments, such as bluer spectra with narrow [He <span style="font-variant:small-caps;">ii</span>]{} emission lines and higher efficiencies of producing supernovae (SNe) and (binary) black holes (e.g. @nagao2008photometric [@whalen2013finding; @kinugawa2014possible; @sobral2015evidence; @belczynski2017likelihood; @ishigaki2018initial]).
A fundamental question regarding Pop III stars is when in cosmic history this special mode of star formation was terminated, which is intricately linked to our understanding of the feedback mechanisms that not only regulate Pop III star formation, but also drive cosmic thermal and chemical evolution. An important goal is to provide guidance to observational campaigns searching for Pop III stars at lower, more accessible, redshifts ($z\lesssim 5$), possibly even extending to recent times. If detected at lower redshifts, Pop III systems could be studied in detail, directly measuring their IMF, which is out of reach at high redshifts, even with the next generation of telescopes, such as the [*James Webb Space Telescope*]{} (JWST). The challenge is to identify any such pockets of Pop III star formation at more recent epochs, which would be extremely rare.
There are three main physical processes that regulate Pop III star formation: (i) metal enrichment, (ii) Lyman-Werner (LW) radiation, and (iii) reionization. Locally, once the metallicity is above some critical value $Z_{\rm crit}\sim 10^{-6}-10^{-3.5}\ \rm Z_{\odot}$ (e.g. @bromm2003formation [@omukai2005thermal; @smith2009three]), star formation is shifted to the low-mass Population II/I (Pop II/I) mode. This threshold is typically exceeded within atomic cooling haloes ($M_{\rm halo}\gtrsim 10^{7-8}\ \rm M_{\odot}$), which host the first galaxies (e.g. @wise2011birth [@pawlik2013first; @jeon2019signature]), and minihaloes externally enriched by nearby SNe [@wise2014birth; @smith2015first; @jeon2017]. However, metal enrichment is highly inhomogeneous, driven by complex interactions between SN blast waves and the ambient medium, accretion of primordial gas, and turbulent mixing during structure formation (e.g. @greif2010first [@pan2013modeling; @ritter2015metal]). As a result, even if the mean metallicity is above $Z_{\rm crit}$ at lower redshifts ($z\lesssim 10$), extremely metal-poor gas may still be available for Pop III star formation, as implied by observed quasar spectra with little metal absorption (e.g. @simcoe2012extremely).
Globally, LW radiation can photo-dissociate molecular coolants, thus delaying and relocating Pop III star formation to occur in more massive haloes (e.g. @machacek2001simulations [@oshea2006; @safranek2012star; @xu2013population]). Similarly, after reionization, star formation is significantly reduced in low-mass haloes ($M_{\rm halo}\lesssim 10^{9}\ \rm M_{\odot}$, e.g. @pawlik2015spatially [@pawlik2017aurora; @benitez2020detailed]), where hot ionized gas cannot collapse. In general, LW and ionizing photons can suppress and even terminate Pop III star formation in low-mass haloes, but have little effect on massive haloes, where metal mixing is the key.
Taking into account all or some of these processes, Pop III star formation at lower redshifts ($z\lesssim 10$) has been studied with semi-analytical models and cosmological simulations (e.g. @tornatore2007population [@karlsson2008uncovering; @trenti2009formation; @muratov2013revisiting; @pallottini2014simulating; @jaacks2019legacy]). While semi-analytical models predict the termination of Pop III star formation at $z\sim 5-16$ [@scannapieco2003detectability; @yoshida2004era; @greif2006two; @hartwig2015; @mebane2018persistence; @chatterjee2020], sharp cut-offs of Pop III star formation are not seen in simulations for $z\gtrsim 2.5$, where the inhomogeneous nature of metal enrichment is better captured. High-resolution simulations, which at least marginally resolve minihaloes, generally agree that Pop III star formation continues down to $z\sim 4-8$ at a level of a few $ 10^{-6}$ to $10^{-4}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$, without a sharp decrease towards lower redshifts [@wise2011birth; @johnson2013first; @xu2016late; @sarmento2018following]. Nevertheless, different sub-grid models for star formation, stellar feedback, and metal mixing are adopted with simplifying assumptions, leading to uncertainties and discrepancies in the detailed histories and environments of Pop III star formation. Besides, it is interesting to investigate Pop III star formation at even lower redshifts ($z\lesssim 4$), when it may finally end. However, it is still computationally prohibitive to run a cosmological hydrodynamics simulation that can well resolve minihaloes ($\lesssim 5\times 10^{4}\ \rm M_{\odot}$ for dark matter) down to $z=0$ in a representative volume ($V_{\rm C}\gtrsim 10^{6}\ \rm Mpc^{3}$ for $\nu\lesssim 2$ peaks). In this regime, we have to rely on extrapolation of what is learned in simulations at higher redshifts, together with semi-analytical techniques.
In light of this, we construct a theoretical framework of Pop III star formation in the post-reionization epoch ($z\lesssim 6$), by combining simulation data and semi-analytical modelling of turbulent metal mixing and the reionization process, which may not be fully captured in simulations. Although we cannot provide a definitive answer to the question of when Pop III star formation was terminated, our work serves as a flexible platform to address this challenge. Specifically, individual elements of the framework, such as the metal mixing efficiency and ionization history, can be constrained by separate studies. Among them are high-resolution, zoom-in simulations, focusing on metal transport in the wake of SN feedback. Furthermore, our approach can be applied to different cosmological simulations. It also enables us to explore the observational signatures of possible late Pop III star formation at more recent cosmic times.
The paper is structured as follows. Section \[s2\] briefly describes the sub-grid models and setup of our cosmological simulations, for which details are available in @liu2020 (LB20, henceforth). In Section \[s3\], we demonstrate how the key feedback processes that regulate Pop III star formation arise during the evolution of the universe. Section \[s4\] presents our framework of Pop III star formation after reionization, including its possible termination and the corresponding observational signatures. We summarize our findings and conclusions in Section \[s5\].
Simulating Early Star Formation {#s2}
===============================
Our cosmological simulations are conducted with the <span style="font-variant:small-caps;">gizmo</span> code [@hopkins2015new], using the Lagrangian meshless finite-mass (MFM) hydro solver, with a number of neighbours $N_{\mathrm{ngb}}=32$, and the Tree+PM gravity solver from <span style="font-variant:small-caps;">gadget-3</span> [@springel2005cosmological]. The properties of baryons are computed with a non-equilibrium primordial chemistry and cooling network for 12 primordial species, supplemented by metal-line cooling of [C <span style="font-variant:small-caps;">ii</span>]{}, [O <span style="font-variant:small-caps;">i</span>]{}, [Si <span style="font-variant:small-caps;">ii</span>]{} and [Fe <span style="font-variant:small-caps;">ii</span>]{} [@jaacks2018baseline]. Sub-grid models for Pop III and Pop II star formation, stellar feedback, black hole formation, accretion, dynamics, feedback and reionization are employed (LB20). Specifically, Pop III and Pop II are distinguished by a threshold metallicity $Z_{\rm th}=10^{-4}\ \rm Z_{\odot}$ ($\rm Z_{\odot}=0.02$). The former is characterized by a modified Larson IMF, $dN/dM\propto M^{-\alpha}\exp(-M^{2}_{\mathrm{cut}}/M^{2})$, with $\alpha=0.17$ and $M^{2}_{\mathrm{cut}}=20\ \mathrm{M}_{\odot}^{2}$ in the mass range $1-150\ \mathrm{M}_{\odot}$ [@jaacks2018baseline], while the latter by a Chabrier IMF in the mass range $0.08-100\ \mathrm{M}_{\odot}$ [@jaacks2019legacy]. LW radiation is modelled with a background term derived from the simulated star formation rate density (SFRD) and a local term under the optically thin assumption, taking into account self-shielding for photo-dissociation. Ionization and SN feedback from massive stars are implemented in terms of their long-term (‘legacy’) thermal and chemical impact [@jaacks2018baseline; @jaacks2019legacy], as well as SN-driven winds (@springel2003wind). Reionization is modelled with UV background (UVB) heating based on the photo-ionization rate from @faucher2009new, assuming a characteristic scale of $1\ \rm kpc$ for shielding in the intergalactic medium (IGM). Table \[t1\] summarizes the key parameters of SN feedback. We refer the reader to LB20 for detailed descriptions and calibrations of the sub-grid models.
[ccccc]{} Type & $E_{\rm SN}$ & $M_{Z}$ & $t_{\star}$ & $r_{\rm final}$\
Pop III & $\sum_{i}N_{\star,i}E_{i}$ & $\sum_{i}y_{i}M_{\star,i}$ & 3 Myr & $\propto E_{\rm SN}^{0.383}$\
& $\sim 7\times 10^{51}\ \rm erg$ & $\sim 39\ \rm M_{\odot}$ & & $\sim 650\ \rm pc$\
\
Pop II & $\langle E_{\rm SN}\rangle m_{\star}$ & $\langle y_{Z}\rangle m_{\star}$ & 10 Myr & $\propto E_{\rm SN}^{0.383}$\
& $\simeq 6.7\times 10^{51}\ \rm erg$ & $\simeq 10\ \rm M_{\odot}$ & & $\simeq 640\ \rm pc$\
\[t1\]
In this work, we focus on the most representative run `FDbox_Lseed` in LB20 (referred as the/our simulation, henceforth), where the simulated region is a periodic cubic box of side length $L=4\ h^{-1}\rm Mpc$. The initial positions and velocities of simulation particles are generated with the <span style="font-variant:small-caps;">music</span> code [@hahn2011multi] at an initial redshift $z_{i}=99$ for the *Planck* $\Lambda$CDM cosmology [@planck]: $\Omega_{m}=0.315$, $\Omega_{b}=0.048$, $\sigma_{8}=0.829$, $n_{s}=0.966$, and $h=0.6774$. The chemical abundances are initialized with the results in @galli2013dawn, following @liu2019global (see their Table 1). The (initial) mass of gas (dark matter) particles is $9.4\times 10^{3}\ \rm M_{\odot}$ ($5.2\times 10^{4}\ \rm M_{\odot}$). The basic unit of star formation (i.e. the mass of stellar particles) is set to $m_{\star}\simeq 600\ \rm M_{\odot}$, for both Pop III and Pop II stars. This choice reflects the typical mass ($500-1000\ \mathrm{M}_{\odot}$) of a Pop III star cluster, based on high-resolution simulations of Pop III star formation in individual minihaloes (e.g. @stacy2013constraining [@susa2014mass; @machida2015accretion; @stacy2016building; @hirano2017formation; @hosokawa2020]), and constraints from the observed global 21-cm absorption signal [@schauer2019constraining]. We have verified that the choice of $m_{\star}$ has little impact on processes involving Pop II stars. The simulation stops at $z=4$, when the simulation volume is marginally representative for all $\nu\lesssim 2$ peaks. The simulation data are analysed with <span style="font-variant:small-caps;">yt</span>[^2] [@turk2010yt], and dark matter haloes are identified by the <span style="font-variant:small-caps;">rockstar</span>[^3] halo finder [@behroozi2012rockstar].
Key Feedback Effects {#s3}
====================
In this section, we discuss the build-up of global LW and ionizing radiation fields as well as metal enrichment in the simulation, which are the key feedback effects for regulating Pop III star formation. We also evaluate the uncertainties in our simulation results, by comparison with other simulations and observational constraints. In the next section, we introduce semi-analytical corrections to account for potentially underestimated feedback effects.
Radiation feedback {#s3.1}
------------------
Fig. \[radbg\] shows the evolution of cosmic radiation backgrounds, in terms of LW intensity $J_{\rm LW,bg}$ (left) and production rate density of ionizing photons $\dot{n}_{\rm ion}$ (right). Individual contributions from Pop III and Pop II stars are also shown. For LW radiation (see Sec. 2.2.2 of LB20), the Pop II component exceeds that from Pop III at $z\sim 20$ shortly after the onset of Pop II star formation ($z\sim 23$), and dominates the LW background for $z\lesssim 15$. The fractional contribution from Pop III decreases with decreasing redshift, and becomes negligible ($\lesssim 1$%) in the post-reionization era ($z\lesssim 6$).
To evaluate the strength of the LW feedback, we also plot the ‘critical’ LW intensity for significant suppression of Pop III star formation in low-mass haloes. Two definitions are employed for this ‘critical’ intensity. The first is based on the increase of threshold halo mass for star formation caused by LW feedback, given by the fitting formula [@machacek2001simulations; @fialkov2014] $$\begin{aligned}
\hat{M}_{\rm th}^{\rm mol}(J_{\rm LW})=M_{\rm th}^{\rm mol}\left[1+6.96(4\pi J_{\rm LW,21})^{0.47}\right]\mbox{\ ,}\end{aligned}$$ where $\hat{M}_{\rm th}^{\rm mol}$ and $M_{\rm th}^{\rm mol}$ are the mass thresholds with and without LW feedback, $J_{\rm LW,21}=J_{\rm LW}/(10^{-21}\rm erg\ s^{-1}\ cm^{-2}\ Hz^{-1}\ sr^{-1})$, and we regard an increase by a factor in the range of $10$ to $M_{\rm th}^{\rm atom}/M_{\rm th}^{\rm mol}$ as ‘critical’ (shaded region). Here we adopt the threshold masses $M_{\rm th}^{\rm mol}$ and $M_{\rm th}^{\rm atom}$ in @trenti2009formation, and further impose a lower limit of $10^{6}\ \rm M_{\odot}$ for $M_{\rm th}^{\rm mol}$, based on the simulations of @Anna2018 for the typical case of $1\sigma$ baryon-dark-matter streaming velocity. The second is based on the $\rm H_{2}$ formation and destruction balance[^4] at a typical state of collapsing primordial gas with a temperature $T=250\ \rm K$, a density $n=100\ \rm cm^{-3}$, and an electron abundance $x_{\rm e}=10^{-5}$ (dotted horizontal line). It turns out that $J_{\rm LW,bg,21}\sim 0.1-1$ at $z\lesssim 19$, residing in the ‘critical’ range given by the first definition, and exceeding the ‘critical’ value according to the second definition at $z\sim 13-7$ (with $J_{\rm LW,bg,21}\sim 1$). In Sec. \[s4.1\], we will show that Pop III star formation is indeed shifted to more massive haloes (i.e. $M_{\rm halo}\gtrsim M_{\rm th}^{\rm atom}$) at $z\sim 13-7$.
For ionizing radiation, we derive the total production rate density of ionizing photons as the summation of Pop III and Pop II contributions: $$\begin{aligned}
\dot{n}_{\rm ion}&=\dot{n}_{\rm ion,PopIII} + \dot{n}_{\rm ion,PopII}\ ,\\
\dot{n}_{\mathrm{ion},k}&=\dot{\rho}_{\star, k}t_{\star,k}\langle \dot{N}_{\mathrm{ion},k}\rangle\ ,\quad k=\rm PopIII,\ PopII \ ,\end{aligned}$$ where $\dot{\rho}_{\star}$ is the simulated SFRD, $t_{\star}$ the lifetime of massive stars (3 Myr for Pop III and 10 Myr for Pop II), and $\langle \dot{N}_{\mathrm{ion}}\rangle$ the luminosity of ionizing photons per unit stellar mass. Following @visbal2015 [@schauer2019constraining], we adopt $\langle \dot{N}_{\mathrm{ion}}\rangle\sim 10^{48}\ \rm s^{-1}\ M_{\odot}^{-1}$ for Pop III, based on @schaerer2002properties (see their table 4), and $\langle \dot{N}_{\mathrm{ion}}\rangle\sim 10^{47}\ \rm s^{-1}\ M_{\odot}^{-1}$ for Pop II, based on @samui2007probing (see their table 1), given a typical metallicity $Z\sim 0.05\ \rm Z_{\odot}$ (see Fig. \[metal1\]). Similar to the case of LW radiation, the Pop II contribution starts to dominate at $z\sim 20$, and that of Pop III drops below 1% at $z\lesssim 10$. For comparison, we estimate the production rate density required for reionization, based on the ‘critical’ SFRD [@johnson2013first] $$\begin{aligned}
\dot{\rho}_{\star}&=0.05\ \mathrm{M_{\odot}\ yr^{-1}\ Mpc^{-3}}\notag\\ &\times\left(\frac{C}{6}\right)\left(\frac{f_{\rm esc}}{0.1}\right)^{-1}\left(\frac{1+z}{7}\right)^{3}\ ,\end{aligned}$$ where $C$ is the clumping factor, and $f_{\rm esc}$ is the escape fraction. We consider the typical range of escape fraction $f_{\rm esc}\sim 0.1-0.7$ [@so2014fully; @paardekooper2015first], and clumping factor evolution from @chen2020scorch (for $\Delta<200$, see their equ. 13 and table. 2). The ‘critical’ production rate is reached in our simulation at $z\sim 10-4$, which is consistent with our implementation of the UVB heating and the resulting ionization history.
This is demonstrated in Fig. \[fion\], which shows the mass- and volume-weighted hydrogen ionized fractions in the simulation. Both fractions start to rise at $z\sim 10$, reaching 0.6 (mass-weighted) and 0.97 (volume-weighted) at the end of the simulation ($z=4$). We further fit the volume-weighted ionized fraction to the widely used tanh form [@lewis2008] $$\begin{aligned}
\hat{f}_{\rm ion}=\frac{1}{2}(1-x_{\rm e}^{\rm rec})\left[1+\tanh\left(\frac{y_{\rm re}-y}{\Delta y}\right)\right]+x_{\rm e}^{\rm rec}\ ,\label{e10}\end{aligned}$$ where $x_{\rm e}^{\rm rec}$ is the ionized fraction left over from recombination, $y(z)=(1+z)^{3/2}$, $\Delta y=1.5\sqrt{1+z_{\rm re}}\Delta z$, $z_{\rm re}$ is the redshift at which $\hat{f}_{\rm ion}=0.5$, and $\Delta z$ describes the duration of reionization. We keep $x_{\rm e}^{\rm rec}=2\times 10^{-4}$ fixed, and fit for $z_{\rm re}$ and $\Delta z$. As shown in Fig. \[fion\], the fit is excellent at $z\lesssim 9$, with best-fit parameters $z_{\rm re}\simeq 7.6$ and $\Delta z\simeq 1.6$.
In our simulation, reionization is delayed compared to what is inferred from *Planck* data, where $z_{\rm re}=8.8_{-1.4}^{+1.7}$ [@planck], but the difference is still within $1\sigma$. This implies that we may underestimate the effect of reionization feedback, as is also evident in the mass-weighted ionized fraction, which is only 0.6 at $z=4$. Note that the mass fraction of collapsed objects (i.e. haloes, with overdensities $\Delta\gtrsim 200$) is $\sim 0.1<1-f_{\rm ion}$ at $z\sim 4$, indicating that self-shielding in the IGM is significant. Indeed, we will show in Sec. \[s4.1\] that in our simulation the effect of reionization on star formation becomes important only at $z\lesssim 4.5$. This is unphysically late, likely caused by this overestimation of self-shielding against external UV radiation. Actually, the characteristic scale of $1\ \rm kpc$ assumed for IGM self-shielding leads to an optical depth $\tau_{\rm ion}>10$ for $\Delta\gtrsim 5$ (i.e. structures that have passed turnover in the spherical collapse model) at $z\gtrsim 3$. Therefore, we conclude that the effect of reionization is not fully captured in our simulation, and should be modelled separately (see Sec. \[s4.2\]). Meanwhile, to correct for the overestimated self-shielding, we reset $\Delta z=0.94$ for further applications of $\hat{f}_{\rm ion}$, based on the combined constraints ($z_{\rm re}=7.44 \pm 0.76$, $\Delta z<0.94$) from the cosmic microwave background (CMB) and fast radio bursts (FRBs; @dai2020).
![Evolution of mass- (solid) and volume- (dashed) weighted ionized fractions of hydrogen. The volume-weighted ionized fraction is fitted with a tanh function (Equ. \[e10\]), which has two free parameters: the location $z_{\rm re}$ and width $\Delta z$ of reionization. The best-fit parameters are $z_{\rm re}\simeq 7.6$ and $\Delta z\simeq 1.6$ in our case (dashed-dotted). To correct for overestimated self-shielding, $\Delta z=0.94$ is adopted in further applications of the best-fit ionized fraction $\hat{f}_{\rm ion}$ (dotted), based on observational CMB and FRB constraints [@dai2020], shown with the shaded region.[]{data-label="fion"}](fion_z.pdf){width="1\columnwidth"}
Metal enrichment {#s3.2}
----------------
Fig. \[metal0\] shows the global metal enrichment process in terms of the (mass-weighted) mean gas metallicity $\langle Z\rangle$ (left) and volume-filling fractions of gas $\mathcal{F}$ at different metallicity levels (right), derived from the zoom-in run `FDzoom_Hseed` in LB20. The mean metallicity exceeds the critical metallicity for transitioning to Pop II star formation, $Z_{\rm crit}\sim 10^{-6} - 10^{-3.5}\ \rm Z_{\odot}$ at $z\sim 20-10$. We further estimate the mean metallicity in collapsed metal-enriched structures, i.e. haloes with $M_{\rm halo}>M_{\rm th}^{\rm mol}$ (dotted curve), assuming that all metals are confined, i.e. $\langle Z\rangle_{\rm col}=\langle Z\rangle/f_{\rm col}$, where $f_{\rm col}=\int_{M_{\rm th}^{\rm mol}}^{\infty}n_{\rm h}(M)dM/\rho_{m}$ is the mass fraction of such structures, given the halo mass function $n_{\rm h}$ (calculated by @murray2013hmfcalc), cosmic mean matter density $\rho_{m}$, and molecular cooling threshold $M_{\rm th}^{\rm mol}$ for star formation. The resulting mean metallicity $\langle Z\rangle_{\rm col}$ is always above $Z_{\rm crit}$ by at least one order of magnitude. Meanwhile, the volume-filling fraction remains below 10%. This outcome reflects the inhomogeneous nature of metal enrichment, such that regions close to star formation sites are rapidly enriched, overshooting $Z_{\rm crit}$, while others remain extremely metal-poor [@scannapieco2003detectability]. Therefore, the volume filling fraction of significantly enriched gas $\mathcal{F}(Z>Z_{\rm th}=10^{-4}\ \rm Z_{\odot})$ is a better indicator of the effect of metal-enrichment on Pop III star formation than the mean metallicity $\langle Z\rangle$. We also compare the volume filling fraction of metal-enriched mass to that of haloes with $M_{\rm halo}>M_{\rm th}^{\rm mol}$. The latter is estimated as $\mathcal{F}_{\rm col}\simeq f_{\rm col}/200$. We find that $\mathcal{F}_{\rm col}<\mathcal{F}$, even for $Z>10^{-1}\ \rm Z_{\odot}$ at $z\lesssim 11$, which is a sign of metal enrichment of the IGM driven by galactic outflows. Similar trends are also seen in previous studies [@wise2011birth; @johnson2013first; @pallottini2014simulating; @xu2016late].
Similar to radiation feedback, the Pop III contribution to the mean metallicity of gas also decreases towards lower redshifts, dropping to $\sim 1$% at $z=4$. However, the Pop III contribution to the volume-filling fraction of significantly metal-enriched gas is always non-negligible (40%-70%). The reason is that Pop III star formation tends to occur in low-density regions (away from previous star formation and metal enrichment activity) where SN bubbles can expand to larger volumes. Actually, in the extreme case where all Pop III stars end in pair-instability SNe (PISNe), the volume-filling fraction (estimated by rescaling the Pop III contribution based on the boost of SN energy by PISNe) will be dominated ($\gtrsim 90$%) by Pop III stars and reaches 10% at $z=4$. This feature is also seen in the recent cosmological simulation from @takanobu2020, which finds that metals of Pop III origin dominate in low-density regions ($\delta \lesssim 10$) at $z\sim 3$. Particularly, their models (b) and (c), which resemble our Pop III IMF, predict a range of mean (Pop III) metallicity $\sim 10^{-5}-10^{-3.5}\ \rm Z_{\odot}$ across regions of different overdensities (see their fig. 5), consistent with our result $\langle Z\rangle\simeq 7\times 10^{-5}\ \rm Z_{\odot}$ (corresponding to a mean metal mass density of $\sim 10^{-34}\ \rm g\ cm^{-3}$) at $z= 4$. We further compared our volume-filling fractions with literature results [@wise2011birth; @johnson2013first; @pallottini2014simulating; @xu2016late]. For instance, our $\mathcal{F}(Z>10^{-4}\ \rm Z_{\odot})$ agrees well with that in @pallottini2014simulating at $z\lesssim 5$, but is higher by up to a factor of 10 at $z\gtrsim 6$ than those in @johnson2013first [@pallottini2014simulating]. Meanwhile, the simulations in @wise2011birth [@xu2016late] with the <span style="font-variant:small-caps;">enzo</span> code predict $\mathcal{F}(Z>10^{-4}\ \rm Z_{\odot})\sim 3-6$% at $z\sim 7-8$, higher than our results by up to a factor of 10. Our extreme model, where all Pop III stars end in PISNe, places an upper limit on the volume-filling fraction of 10% at $z=4$, which is also consistent with the value 15% from the corresponding model (a) in @takanobu2020. In general, the discrepancies in different simulations are significant (up to two orders of magnitude), reflecting the uncertainties in sub-grid models for SN feedback and metal transport. It remains uncertain to what extent the fine-grain metal mixing process (at a scale of $\sim 10^{-3}$ pc; @spitzer2006physics [@sarmento2016following]) is captured in such cosmological simulations with limited resolution ($\Delta x\gtrsim 10$ pc). Therefore, the late-time ($z\lesssim 6$) Pop III star formation in our simulation may not be adequately captured, due to the imperfect treatment of metal transport.
To be more specific, we assume that metals are instantaneously mixed into the gas particles enclosed by the final radius of SN shell expansion, while in reality, the timescale for complete mixing can be non-negligible ($\sim 1-10$ Myr) at the scale of gas particles ($m_{\rm gas}\sim 10^{4}\ \rm M_{\odot}$), marking the resolution limit of the simulation[^5]. That is to say, even if the mean metallicity of a gas particle is above $Z_{\rm crit}$, there could still be a significant fraction of gas with $Z<Z_{\rm crit}$. It is shown in @sarmento2016following [@sarmento2018following] that taking into account such unresolved metal-poor gas will enhance the Pop III SFRD by a factor of 2. On the other hand, we do not smooth gas metallicities with any kernels, such that the metallicity of a gas particle is only affected by nearby SNe. Our approach is valid at the scale of $m_{\rm gas}\sim 10^{4}\ \rm M_{\odot}$, as the MFM method sets the mass fluxes across mesh boundaries explicitly to zero. However, turbulent metal diffusion at unresolved scales across different gas particles is not captured, which may reduce the fraction of metal-poor gas in a halo. Besides, the mesh itself in <span style="font-variant:small-caps;">gizmo</span> is actually smoothed. Therefore, in Sec. \[s4.2\], we use semi-analytical models of metal-mixing to correct for the potentially overestimated amount of metal-poor gas in the simulated Pop III SFRD. It is also possible to implement passive scalar diffusion in simulations (see @hopkins2017anisotropic). We defer such investigations to future work.
![Stellar metallicity-mass relation. Simulated star-forming haloes at $z=4$ are shown with green dots, and the ones undergoing recent Pop III star formation within 3 Myr are labelled with triangles. Systems with $Z< 10^{-6}\ \rm Z_{\odot}$ are plotted at $Z= 10^{-6}\ \rm Z_{\odot}$. For comparison, we also plot the results for FIRE simulation [@ma2016origin], AMAZE observations [@maiolino2008amaze], and local group dwarfs (LGDFs; @simon2019faintest) with blue, red and orange shaded regions. Individual LGDFs are labelled with stars.[]{data-label="metal1"}](Zsbh_Ms_23.pdf){width="1.\columnwidth"}
Although our treatment of metal transport is idealized, the observed stellar metallicity-mass relation [@maiolino2008amaze; @simon2019faintest] is well reproduced by our simulation in a broad mass range $M_{\star}\sim 10^{3}-10^{8}\ \rm M_{\odot}$, especially for local group dwarfs (LGDFs), as shown in Fig. \[metal1\]. We find that the scatter in stellar metallicity increases with decreasing stellar mass, in particular for $M_{\star}\lesssim 5\times 10^{5}\ \rm M_{\odot}$, also consistent with the trend in observations. However, at a fixed stellar mass, the stellar metallicity is higher in our simulation compared with the FIRE simulations [@ma2016origin], again demonstrating the uncertainties in cosmological simulations regarding metal enrichment.
![Co-moving Pop III SFRD. The results from our fiducial run and the corresponding fit are shown with thin and thick solid curves. We further plot the results from other cosmological simulations in @johnson2013first (JCS13; with LW feedback), @xu2016late (XH16), @sarmento2018following (SR18) and @jaacks2019legacy (JJ19), with the dashed, dashed-dotted, dotted and densely dashed-dotted curves, which demonstrate the range of Pop III star formation histories in current models. Note that the XH16 results are based on a zoom-in simulation for a low-density region ($\langle\delta\rangle=-0.26$ at $z=8$), which should be regarded as lower limits. The SR18 results include two cases with (upper) and without (lower) unresolved inefficient metal mixing. For comparison, we plot the (extrapolated) Pop II/I ($\approx$ total) SFRD $\dot{\rho}_{\star}=0.015(1+z)^{2.7}/\{1+[(1+z)/2.9]^{5.6}\}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$ (with 0.2 dex scatters) from @madau2014araa (shaded region), inferred by UV and IR galaxy surveys, such as @finkelstein2016observational (FS16; data points). The corresponding simulated total SFRD is shown with the long-dashed curve.[]{data-label="sfrd"}](popIIIsfrd.pdf){width="1\columnwidth"}
Pop III star formation after reionization {#s4}
=========================================
In this section, we demonstrate our framework for Pop III star formation after reionization ($z\lesssim 6$). We first characterize a representative sample of simulated haloes with recent Pop III star formation at $z\sim 4-6$, considering their mass and metallicity distributions, as well as the masses and locations of active Pop III stars within them (Sec. \[s4.1\]). Based on this sample, we then employ semi-analytical models for metal mixing and reionization to extrapolate the Pop III SFRD to $z=0$ (Sec. \[s4.2\]). Finally, we discuss the observational constraints and possible signatures of Pop III star formation in the post-reionization epoch ($z\lesssim 6$), as well as its potential termination (Sec. \[s4.3\]).
The starting point of our framework is the simulated (co-moving) Pop III SFRD, which is shown in Fig. \[sfrd\], in comparison with literature results [@johnson2013first; @xu2016late; @sarmento2018following; @jaacks2019legacy]. Our Pop III SFRD peaks at $z\sim 10$ with $\sim 10^{-4}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$, and drops to $\sim 2\times 10^{-5}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$ at $z=4$. We fit the simulated Pop III SFRD to the form [@madau2014araa] $$\begin{aligned}
\frac{\dot{\rho}_{\star,\mathrm{PopIII}}^{\rm sim}(z)}{\rm M_{\odot}\ yr^{-1}\ Mpc^{-3}}=\frac{a(1+z)^{b}}{1+[(1+z)/c]^{d}}\ ,\label{fit}\end{aligned}$$ which leads to best-fit parameters $a=765.7$, $b=-5.92$, $c=12.83$ and $d=-8.55$. For the post-reionization epoch ($z\lesssim 6$), this is approximately equivalent to a power-law extrapolation $\propto (1+z)^{b-d}\simeq (1+z)^{2.6}$, as $d<0$ in our case. Interestingly, the power-law index here is similar to that of the Pop II/I SFRD from @madau2014araa. Integrating $\dot{\rho}_{\star,\mathrm{PopIII}}^{\rm sim}(z)$ across cosmic history gives the density of all Pop III stars ever formed, $\sim 10^{5}\ \rm M_{\odot}\ Mpc^{-3}$ (in which 55% comes from $z>6$), consistent with the constraints in @visbal2015, set by *Planck* data.
Our Pop III SFRD agrees well with @johnson2013first at $z\gtrsim 7$ and @sarmento2018following at $z\gtrsim 9$, but is lower (higher) compared with that in @jaacks2019legacy (@xu2016late) at $z\gtrsim 7$. This can be explained with the fact that @jaacks2019legacy did not include mechanical SN feedback, while @xu2016late targeted a low-density region ($\langle\delta\rangle=-0.26$ at $z=8$), whose results should be regarded as lower limits. In general, our Pop III SFRD is approximately the median value among various simulation results [@tornatore2007population; @wise2011birth; @johnson2013first; @xu2016late; @sarmento2018following]. In Fig. \[sfrd\], we also plot the simulated total SFRD (dominated by Pop II/I at $z\lesssim 22$), which is consistent with observations within a factor of 2 [@madau2014araa; @finkelstein2016observational]. We refer the reader to Section 3 of LB20 for more detailed comparisons between our simulations and observations.
Host haloes of Pop III stars {#s4.1}
----------------------------
![Cosmic web from the fiducial run at $z=4$, in terms of the projected distribution of dark matter (in co-moving coordinates, with a thickness of $4\ h^{-1}\rm Mpc$). Pop III stellar particles with ages $\tau<3$ and $\sim 3-10$ Myr are labelled with orange triangles and red filled circles. Their host haloes are also shown with empty circles whose sizes reflect their virial radii. Note that small haloes ($M_{\rm halo}\lesssim 10^{10}\ \rm M_{\odot}$) have been covered by the labels of Pop III stars.[]{data-label="popIIIdis"}](popIIIpos_23_1.png){width="1\columnwidth"}
{width="1.4\columnwidth"}
A fundamental question for Pop III star formation at late times is where Pop III stars could possibly continue to form. As an example, Fig. \[popIIIdis\] shows the locations of (active) Pop III stellar particles with ages $\tau<3$ Myr and $\tau\sim 3-10$ Myr, on top of the cosmic web in the last simulation snapshot at $z=4$. We identify the host of a Pop III particle as the most massive halo that encloses the Pop III particle within its virial radius. Massive ($M_{\rm halo}\gtrsim 10^{10}\ \rm M_{\odot}$) host haloes of active Pop III star formation are also shown in Fig. \[popIIIdis\], constituting 50% of the host halo population at $z=4$, thus indicating that formation of Pop III stars in massive haloes is important. The reason is that metal mixing is inefficient in our simulation, such that metal-poor gas in dense filaments (i.e. cold accretion flows) can still form Pop III stars, even though the densest regions within the halo have been significantly enriched by previous SNe. This process is illustrated in Fig. \[gasdis\], where (active) Pop III particles are plotted on top of the projected distributions of dark matter (left), metal-poor gas ($Z<10^{-4}\ \rm Z_{\odot}$, middle) and metal-enriched gas ($Z>10^{-4}\ \rm Z_{\odot}$, right), for one of the most massive haloes at $z=4$ that host active Pop III stars with $M_{\rm halo}\simeq 3\times 10^{10}\ \rm M_{\odot}$. This halo is still under assembly with a few (groups) of sub-haloes separated by a few (physical) kpc, where Pop III stars are formed on the edges of such sub-structures. Actually, Pop III stars tend to form at the ‘connection points’ of (metal-enriched) sub-structures and dense filaments rich in metal-poor gas. This trend is consistent with the ‘Pop III wave’ scenario [@tornatore2007population], which is also seen in previous simulations (e.g. @pallottini2014simulating [@xu2016late]). Besides, recent work by @bennett2020 found that inflows of cold dense gas are significantly enhanced with better resolution of shocks, leading to metal-poor star formation in primordial filaments, for even more massive haloes ($M_{\rm halo}\sim 10^{12}\ \rm M_{\odot}$).
To characterize the host haloes of Pop III stars after reionization ($z\lesssim 6$), we combine 4 snapshots at $z=4$, 4.5, 5 and $6$ to construct a sample of 145 (52) haloes that have recent Pop III star formation within 10 (3) Myr (the representative sample, henceforth). As mentioned in Sec. \[s3.1\], and to be further discussed below, reionization feedback is not well captured in our simulation, while the effect of LW feedback is treated more realistically. Therefore, the representative sample from our simulation effectively corresponds to the case under a moderate LW background, but *without* reionization feedback. In the next subsection, additional corrections are made to fully take into account the effect of reionization.
We divide the Pop III host haloes into three groups, based on the atomic cooling threshold $M_{\rm th}^{\rm atom}$ and the (dark matter+baryonic) Jeans mass of fully ionized gas $$\begin{aligned}
M_{\rm J,ion}&\simeq 6.7\times 10^{8}\ \mathrm{M_{\odot}}\notag\\
&\times \left[\frac{(1+z)^{3}\Delta}{5^3\times 125}\right]^{-1/2}\left(\frac{T_{\rm b}}{20000\ \rm K}\right)^{3/2} \mbox{\ ,}\label{mj}\end{aligned}$$ where $\Delta$ is the overdensity and $T_{\rm b}$ the temperature of ionized gas. We use the Jeans mass (Equ. \[mj\]) to approximate the halo mass threshold below which star formation is significantly suppressed due to reionization (i.e. the filtering mass, @gnedin2000effect). We adopt $\Delta=125$ and $T_{\rm b}=20,000$ K in accordance with more complex calculations and simulations [@pawlik2015spatially; @pawlik2017aurora; @benitez2020detailed; @hutter2020astraeus]. For simplicity, we evaluate $M_{\rm th}^{\rm atom}$, $M_{\rm J,ion}$ at $z=4$, and apply $M_{\rm th}^{\rm atom}\simeq 1.2\times 10^{8}\ \rm M_{\odot}$, $M_{\rm J,ion}\simeq 6.7\times 10^{8}\ \mathrm{M_{\odot}}$ to the entire representative sample at $z\sim 4-6$.
The first group refers to the ‘classical’ formation sites of Pop III stars with $M_{\rm halo}<M_{\rm th}^{\rm atom}$, the minihaloes, where molecular (hydrogen) cooling dominates, and which are particularly important at high-$z$. This group itself is interesting, as it reflects how feedback regulates Pop III star formation. In Fig. \[fiso\], we plot the fraction of active Pop III stars in molecular cooling haloes, $f_{\rm mol}$, in comparison with the fraction of newly star-forming haloes[^6], $f_{\rm new}$, for $z\sim 4-20$ (i.e. isolated Pop III star formation). In general, $f_{\rm new}> f_{\rm mol}$, especially for $z\lesssim 13$, which indicates that at lower redshifts, the majority of isolated Pop III star formation occurs in atomic cooling haloes. $f_{\rm mol}$ drops from close to 1 to a few percent when $z$ decreases from $\sim 20$ to $\sim 13$, resulting from the suppression/delay of star formation in molecular cooling haloes by LW radiation. Actually, $f_{\rm mol}$ anti-correlates with the background LW intensity $J_{\rm LW,bg}$, shown in the left panel of Fig. \[radbg\]. For instance, $f_{\rm mol}$ remains a few percent at $z\sim 13-7$ when $J_{\rm LW,bg}$ is above the ‘critical’ value ($J_{\rm LW,bg,21}\gtrsim 1$). Similar trends are also seen in the recent simulation of @danielle2020 (see their fig. 5). Both $f_{\rm new}$ and $f_{\rm mol}$ decrease rapidly at $z\lesssim 4.5$, where reionization starts to take effect. This is later than expected, for the reason explained in Sec. \[s3.1\]. In the next subsection, for the purpose of post-processing, we use a smoothed version of $f_{\rm mol}$, assuming that $f_{\rm mol}=0.2$ at $z<6$, which again reflects the case under a moderate LW background ($J_{\rm LW,bg,21}\sim 0.1-1$), but *without* reionization feedback.
![Fractions of active Pop III stars in molecular cooling haloes ($f_{\rm mol}$) and new star-forming haloes ($f_{\rm new}$), for $\tau<3$ (solid and dashed-dotted) and 10 (dashed and dotted) Myr. We also show a smoothed version of $f_{\rm mol}$ with the thick gray curve, in which $f_{\rm mol}=0.2$ at $z<6$ is assumed to denote the case under a moderate LW background but *without* reionization. The effect of reionization will be modelled separately in Sec. \[s4.2\] given the smoothed $f_{\rm mol}$ as a starting point.[]{data-label="fiso"}](fiso_z.pdf){width="1\columnwidth"}
The other two groups refer to haloes with $M_{\rm halo}\in [ M_{\rm th}^{\rm atom}, M_{\rm J,ion}]$ and $M_{\rm halo}>M_{\rm J,ion}$. The former, together with molecular cooling haloes, is not expected to form stars after reionization. Therefore, their contributions to the Pop III SFRD are removed for the Pop III SFRD models in the next subsection. To evaluate the relative importance of the three groups, we plot the halo mass distribution of the representative sample in Fig. \[mhdis\], where haloes are weighted by enclosed mass of active Pop III stars, $M_{\rm PopIII}$, such that the distribution is proportional to $dM_{\rm PopIII}/d\log M_{\rm halo}$. It turns out that the ratio of the contributions from the three groups to Pop III star formation is approximately 2 : 1 : 1. Besides, the distribution at $M_{\rm halo}\gtrsim M_{\rm J,ion}$ can be approximated with a power-law of index $\alpha_{m}\sim 0.5$ (solid), while that at $M_{\rm th}^{\rm atom}\lesssim M_{\rm halo}\lesssim M_{\rm J,ion}$ can be described by another power-law with $\alpha_{m}\sim -1$.
![Halo mass distribution (in log scale) of the representative halo sample with recent Pop III star formation within 3 (orange histograms) and 10 (red dashed contour) Myr. Haloes are weighted by enclosed masses of active Pop III stars, such that the distribution here is proportional to $dM_{\rm PopIII}/d\log M_{\rm halo}$. The atomic cooling threshold $M_{\rm th}^{\rm atom}$ and Jeans mass of haloes with fully ionized gas $M_{\rm J,ion}$ are shown with the dotted and dashed-dotted vertical lines. The distribution at $M_{\rm halo}\gtrsim M_{\rm J,ion}$ can be approximated with a power-law of index $\alpha_{m}\sim 0.5$ (solid), while that at $M_{\rm th}^{\rm atom}\lesssim M_{\rm halo}\lesssim M_{\rm J,ion}$ can be described by another power-law with $\alpha_{m}\sim -1$. The total masses of active Pop III stars in these two groups of haloes are almost identical (with $<10\%$ difference).[]{data-label="mhdis"}](mh_dis.pdf){width="1\columnwidth"}
Besides the host mass, another crucial property of late-time Pop III star formation is the distribution of Pop III stars in their host haloes. We define the relative distance, $r_{\rm PopIII}$, from an active Pop III particle to the halo center as the ratio of the physical distance $R_{\rm PopIII}$ to the virial radius $R_{\rm vir}$, i.e. $r_{\rm PopIII}\equiv R_{\rm PopIII}/R_{\rm vir}$. The distribution of $r_{\rm PopIII}$ is shown in Fig. \[rratdis\] for the active Pop III particles in haloes with $M_{\rm halo}>M_{\rm J,ion}$ from the representative sample. This distribution, i.e. $dM_{\rm PopIII}/d\log r_{\rm PopIII}\propto r_{\rm PopIII}^{3}\rho_{\rm PopIII}(r_{\rm PopIII})$, given the density profile of Pop III stars, $\rho_{\rm PopIII}$, can be approximated with a power-law of index $\alpha_{r}\sim 0.3$. The result for all atomic cooling haloes ($M_{\rm halo}>M_{\rm th}^{\rm atom}$) is similar. This indicates that the (quasi-natal) distribution of Pop III stars is less concentrated than that of Pop II/I stars and dark matter (with $\rho\propto r^{-4}$ and $r^{-3}$, i.e. $\alpha_{r}\sim -1-0$, at the outskirts). About half (47-61%) of the Pop III particles occur at the outskirts of haloes ($r_{\rm PopIII}\gtrsim 0.1$), consistent with the ‘Pop III wave’ theory [@tornatore2007population]. However, a few percent of Pop III particles with ages $\tau\sim 3-10$ Myr are still found in halo centers ($r_{\rm PopIII}\lesssim 10^{-2}$), which are expected to be polluted by metals. One explanation is that for haloes during assembly (mergers), the mass center of a halo as a whole may not be close to any sub-haloes with recent star formation activities (i.e. sources of metal enrichment), as shown in Fig. \[gasdis\]. Nevertheless, outflows driven by SN winds may have enriched the halo center (or even the entire halo) in reality, so that we may have overestimated Pop III star formation.
![Distribution of relative distances $r_{\rm PopIII}\equiv R_{\rm PopIII}/R_{\rm vir}$ (in log scale) for the active Pop III particles with ages $\tau<3$ (orange histograms) and 10 (red dashed contour) Myr, in haloes above the Jeans mass of ionized gas ($M_{\rm halo}>M_{\rm J,ion}$) from the representative sample. The distribution is fitted to a power-law form, resulting in a power-law index of $\alpha_{r}\sim 0.3$, such that the enclosed mass of active Pop III stars follows $M_{\rm PopIII}(<r)\propto r^{\alpha_{r}}\sim r^{0.3}$. []{data-label="rratdis"}](rrat_dis.pdf){width="1\columnwidth"}
In light of this, we further look into the extreme case in which metals are fully mixed in the entire halo (i.e. within $R_{\rm vir}$) by measuring the mean (gas-phase and stellar) metallicities of Pop III host haloes in the representative sample. The cumulative distribution functions of the halo mean metallicities for different groups of haloes are shown in Fig. \[cumZ\], together with the metallicities of active Pop III particles themselves. The latter is meant to explore the dependence of Pop III star formation on the critical metallicity ($Z_{\rm crit}\sim 10^{-6}-10^{-3.5}\ \rm Z_{\odot}$) for the Pop III to Pop II/I transition. Atomic cooling haloes ($M_{\rm halo}>M_{\rm th}^{\rm atom}$, representative before reionization) and haloes above the filtering mass ($M_{\rm halo}>M_{\rm J,ion}$, representative after reionization) are considered separately. To better capture the natal environments of Pop III stars, we focus on the gas-phase metallicity for haloes[^7] with $\tau<3$ Myr, but stellar metallicity for haloes with $\tau<10$ Myr. It is shown that if the Pop III mode is restricted to metal-free gas (equivalent to $Z_{\rm crit}\lesssim 10^{-6}\ \rm Z_{\odot}$ in our case), about 50% of Pop III star formation will be shifted to Pop II/I. If metals are fully mixed inside haloes and $Z_{\rm crit}\lesssim 10^{-5}\ \rm Z_{\odot}$, $\sim 10-25$% of Pop III star formation remains before reionization (for $M_{\rm halo}>M_{\rm th}^{\rm atom}$), while only $\lesssim 3$% remains after reionization (for $M_{\rm halo}>M_{\rm J,ion}$).
![Cumulative metallicity distribution functions for active Pop III particles (thick) and their host haloes (thin), from the representative sample. The results for atomic cooling haloes are shown with solid and dashed curves, while those for haloes above the Jeans mass of ionized gas with dashed-dotted and dotted curves, for $\tau<3$ and 10 Myr, respectively. Again, haloes are weighted by enclosed masses of active Pop III stars. To better capture the natal environments of Pop III stars, gas-phase metallicity is adopted for host haloes with $\tau<3$ Myr, while stellar metallicity is used for $\tau<10$ Myr. []{data-label="cumZ"}](ZpopIII_dis.pdf){width="1\columnwidth"}
Finally, a parameter of particular importance for direct detection of Pop III stars is the total mass of active Pop III stars $M_{\rm PopIII}$ per halo. This parameter is the product of the ‘quantum’ of Pop III star formation, i.e. the typical Pop III stellar mass formed per local (cloud-scale) star formation event, and the number of Pop III star-forming clouds coexisting in a few Myr. Fig. \[MpopIII\] shows the distribution of $M_{\rm PopIII}$ for the entire representative sample. We find no clear correlation between $M_{\rm PopIII}$ and $M_{\rm halo}$, such that the distribution remains similar when only haloes with $M_{\rm halo}>M_{\rm J, ion}$ are considered (i.e. after reionization). The average mass of active Pop III stars per halo is $\langle M_{\rm PopIII}\rangle \simeq 10^{3}\ \rm M_{\odot}$ with large scatter. More than 50% of haloes only have one active Pop III particle (i.e. $M_{\rm PopIII}=m_{\star}\simeq 600\ \rm M_{\odot}$), and less than 10% of haloes have $M_{\rm PopIII}\sim 2\times 10^{3}-10^{4}\ \rm M_{\odot}$, consistent with theoretical and observational upper limits of $M_{\rm PopIII}\lesssim 10^{6}\ \rm M_{\odot}$ [@yajima2017upper; @bhatawdekar2020][^8]. Our results also (marginally) agree with a recently discovered strongly lensed Pop III candidate Lyman-$\alpha$ (Ly$\alpha$) emitter at $z\simeq 6.6$, which has $M_{\rm PopIII}\sim 10^{4}\ \rm M_{\odot}$ [@vanzella2020candidate], residing at the high mass end of our prediction. Note that the simulations of @xu2016late [@danielle2020] also find typically $M_{\rm PopIII}\lesssim 10^{3}\ \rm M_{\odot}$, while other simulations with lower resolution or different star formation routines predict higher values, e.g. $M_{\rm PopIII}\gtrsim 10^{5}\ \rm M_{\odot}$ [@pallottini2014simulating; @sarmento2018following]. As the observational constraints are still weak/unclear, the total mass of active Pop III stars per halo/galaxy is uncertain, especially for massive haloes at late times ($M_{\rm halo}\gtrsim 10^{9}\ \rm M_{\odot}$, $z\lesssim 6$), depending on resolution and sub-grid models for star formation and stellar feedback, particularly chemical feedback from SNe.
![Distribution of enclosed mass of active Pop III stars for the representative sample of recent Pop III star formation within 3 (orange histograms) and 10 (red dashed contour) Myr. There is no clear correlation between $M_{\rm PopIII}$ and $M_{\rm halo}$, such that the distribution remains similar when only haloes with $M_{\rm halo}>M_{\rm J, ion}$ are considered (i.e. after reionization). The average mass of active Pop III stars per halo $\langle M_{\rm PopIII}\rangle \simeq 10^{3}\ \rm M_{\odot}$ is shown with the vertical dotted line.[]{data-label="MpopIII"}](MpopIII_dis){width="1\columnwidth"}
Extrapolating Pop III star formation to the present day {#s4.2}
-------------------------------------------------------
Based on what is learned from the representative sample, we now extrapolate Pop III star formation to $z=0$ by introducing corrections to the fit of simulated Pop III SFRD $\dot{\rho}_{\star,\rm PopIII}^{\rm sim}$ (Equ. \[fit\]) for reionization and metal mixing. We decompose the Pop III SFRD into two components: one from molecular cooling haloes ($M_{\rm halo}<M_{\rm th}^{\rm atom}$) and the other from atomic cooling haloes ($M_{\rm halo}>M_{\rm th}^{\rm atom}$). Both components are subject to reionization corrections, while we only consider additional metal mixing for the latter. Actually, in our simulation, most molecular cooling haloes only experience one episode of Pop III star formation before merging into more massive haloes[^9], such that internal enrichment is not important. Although external enrichment may play a role [@wise2014birth; @smith2015first; @jeon2017], we neglect this effect for simplicity. Note that star formation in molecular cooling haloes is prohibited after reionization, and the contribution of molecular cooling haloes is only a few percent during reionization ($z\sim 7-13$) due to strong LW feedback (see Fig. \[fiso\]). We write the Pop III SFRD after such corrections as $$\begin{aligned}
\dot{\rho}_{\star,\rm PopIII}^{\rm cor}=\dot{\rho}_{\star,\rm PopIII}^{\rm sim}(\hat{f}_{\rm atom}\langle f_{\rm mp}\rangle+\hat{f}_{\rm mol})\ ,\label{e1}\end{aligned}$$ where $\hat{f}_{\rm mol}$ and $\hat{f}_{\rm atom}$ are the terms for reionization correction, while $\langle f_{\rm mp}\rangle$ captures the effect of additional metal mixing. The reionization terms are calculated with $$\begin{aligned}
&\hat{f}_{k}(z)=f_{k}(z)\times \{f_{k,0}+f_{k,1}[1-\hat{f}_{\rm ion}(z)]\}\ ,\label{e9}
$$ where $f_{k}(z)$ is derived from the simulation for $k=\rm mol,\ atom$, with $f_{\rm mol}+f_{\rm atom}=1$. To be specific, we use a smoothed version of $f_{\rm mol}$ based on simulation data (see Fig. \[fiso\] and the left panel of Fig. \[radbg\]), in which $f_{\rm mol}=1$ for $z>19$ with negligible LW feedback, $f_{\rm mol}=0.05$ for $12.5>z>6$ under a strong LW background ($J_{\rm LW,bg,21}\gtrsim 1$), $f_{\rm mol}=0.2$ for $z<6$ under a moderate LW background ($J_{\rm LW,bg,21}\sim 0.1-1$), and these three plateaus are connected with two linear functions of $z$. Within each component $k$, $f_{k,0}$ is the fraction of Pop III star formation unaffected by reionization (i.e. in massive haloes $M_{\rm halo}>M_{\rm J,ion}$), and $f_{k,1}$ is that suppressed by reionization. Note that $f_{k,0}+f_{k,1}=1$. We set $f_{\rm mol,0}=0$ and $f_{\rm atom,0}=0.5$, based on the representative sample (see Fig. \[mhdis\]).
For (additional) metal mixing, $\langle f_{\rm mp}\rangle$ is defined as the fraction of Pop III star formation remaining, when more sufficient metal mixing is considered than captured in our simulation: $$\begin{aligned}
&\langle f_{\rm mp}\rangle=\int_{M_{1}}^{M_{2}}f_{\rm mp}(z, M)w(M)dM/\int_{M_{1}}^{M_{2}}w(M)dM\ ,\label{e2}\\
&f_{\rm mp}(z,M)=\max\{1-\left[R_{\rm mix}/R_{\rm vir}\right]^{\alpha_{r}}, 0\}\ ,\label{e3}\end{aligned}$$ where $f_{\rm mp}(z,M)$ is the metal-poor fraction of potential Pop III forming gas as a function of halo virial radius $R_{\rm vir}$ and metal mixing radius $R_{\rm mix}$, for a halo of mass $M$ at $z$. In the second line (Equ. \[e3\]), we have assumed spherical symmetry and locate the halo center as the source of enrichment. We adopt $M_{1}=M_{\rm th}^{\rm atom}$ and $M_{2}=\max[10M_{\rm th}^{\rm atom},M_{\rm crit}(\nu=2)]$, corresponding to the mass range of haloes in our simulation, where $M_{\rm crit}(\nu=2)$ is the critical mass for 2-sigma peaks. The weight function is written as $w(M_{\rm halo}) = A M_{\rm halo}^{\alpha_{m}-1}\propto M_{\rm halo}^{-1}dM_{\rm PopIII}/d\log M_{\rm halo}$, where $\alpha_{m}\sim 0.5$ and $A=0.4$ for $M_{\rm halo}\ge M_{\rm J,ion}$, while $\alpha_{m}\sim-1$ and $A=1-\hat{f}_{\rm ion}(z)$ for $M_{\rm halo}<M_{\rm J,ion}$. The power-law indices $\alpha_{m}$ and normalization factors $A$ are derived from the simulated distribution of (active) Pop III mass in haloes, $dM_{\rm PopIII}/d\log M_{\rm halo}$, as shown in Fig. \[mhdis\]. Note that the reionization effect has been absorbed into $A$ for low-mass haloes ($M_{\rm halo}<M_{\rm J,ion}$). We use $\alpha_{r}\sim 0.3$, based on the radius distribution of Pop III particles (Fig. \[rratdis\]). The metal mixing radius $R_{\rm mix}$ is estimated by tracking the halo growth history with the gravity-driven turbulent diffusion model based on @karlsson2008uncovering, $$\begin{aligned}
R_{\rm mix}(z, M)&=\left[6\int_{z_{i}}^{z}D_{\rm turb}(z')\left |\frac{dt}{dz'}\right|dz'\right]^{1/2}\ ,\label{e4}\\
D_{\rm turb}(z')&\equiv\langle v_{\rm turb}\rangle l_{\rm turb}/3=\beta_{\rm mix}v'_{\rm vir}R'_{\rm vir}/3\ ,\label{e5}\\
v'_{\rm vir}&=\sqrt{\frac{GM'}{R'_{\rm vir}}}\ ,\quad R'_{\rm vir}=\left[\frac{3 M'}{4\pi \Delta\rho_{m}(z')}\right]^{1/3}\ .\label{e6}\end{aligned}$$ Here $\Delta=200$, and $\beta_{\rm mix}$ is an adjustable parameter that reflects the strength of metal mixing, in terms of how the turbulent diffusion coefficient $D_{\rm turb}(z')\equiv D_{\rm turb}(z'|z,M)$ depends on halo dynamics. Evidently, $f_{\rm mp}(z,M)$ decreases with increasing $\beta_{\rm mix}$. The onset of internal metal enrichment $z_{i}\equiv z_{i}(z,M)$ is derived by $M'(z_{i}|z, M)=M_{\rm th}^{\rm atom}(z_{i})$ for $z_{i}<20$. The halo growth history is obtained by solving for $M'\equiv M'(z'|z, M)$, which is the progenitor mass at $z'>z$ of a halo at $z$ with mass $M$. This is done by integrating the (average) halo growth rate formula from @fakhouri2010merger, which is derived from simulations for $\Lambda$CDM cosmology: $$\begin{aligned}
\frac{dM}{dz}&=\dot{M}(z,M)\left |\frac{dt}{dz}\right|\ ,\notag\\
&\simeq 46\ \mathrm{M_{\odot}\ yr^{-1}}\left(\frac{M}{10^{12}\ \mathrm{M_{\odot}}}\right)^{1.1}\xi(z)\left |\frac{dt}{dz}\right|\ ,\notag\\
\xi(z)&=\left[1.1(1+z)-0.11\right]\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}\ .\label{e7}\end{aligned}$$ In general, our model predicts that $f_{\rm mp}(z,M)$ increases with increasing mass $M$ and increasing redshift $z$.
We also consider a more simulation-based model, where the metal-poor fraction is expressed with $$\begin{aligned}
\hat{f}_{\rm mp}^{\rm sim}(z) = f_{\rm mp}^{\rm post} + (f_{\rm mp}^{\rm pre}-f_{\rm mp}^{\rm post})[1-\hat{f}_{\rm ion}(z)]\ .\label{e8}\end{aligned}$$ Here $f_{\rm mp}^{\rm post}$ and $f_{\rm mp}^{\rm pre}$ are the fractions of metal-poor gas for Pop III star formation in atomic cooling haloes after and and before reionization. If metal mixing is actually efficient at the halo scale ($R_{\rm mix}\gtrsim R_{\rm vir}$), but not fully captured in our simulation, Pop III particles in haloes with (mass-weighted) *mean* metallicities above $Z_{\rm crit}$ should be removed. This pessimistic case can be evaluated with the distributions of halo mean metallicities for Pop III host haloes, as shown in Fig. \[cumZ\] for the representative sample. As an upper limit, we use $f_{\rm mp}^{\rm pre}\simeq 0.25$ and $f_{\rm mp}^{\rm post}\simeq 0.03$, given a critical metallicity $Z_{\rm crit}\lesssim 3\times 10^{-5}\ \rm Z_{\odot}$, based on the halo stellar metallicity distributions of Pop III particles with ages $\tau<10$ Myr, in all atomic cooling haloes ($M_{\rm halo}>M_{\rm th}^{\rm atom}$, before reionization) and only massive haloes above the Jeans mass of ionized gas ($M_{\rm halo}>M_{\rm J,ion}$, after reionization).
Finally, examples of the Pop III SFRD models with the above reionization and metal-mixing corrections, $\dot{\rho}_{\star,\rm PopIII}^{\rm cor}$, are shown in Fig. \[sfrdex\], on top of observational constraints and the Pop II/I counterpart [@madau2014araa]. We consider the upper bounds on Pop III SFRD ($\sim 10^{-6}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$ at $z\sim 0$ and $\sim 10^{-4}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$ at $z\sim 2-4$), inferred from the observed rate densities of super-luminous SNe (as PISN candidates, see @gal2012luminous [@cooke2012superluminous]), assuming a Pop III PISN efficiency of $\epsilon_{\rm PISN}=10^{-3}\ \rm M_{\odot}$ (for typical top-heavy IMFs). Our model SFRDs are always lower than these upper bounds (by at least a factor of 10), even for the optimistic case with $\langle f_{\rm mp}\rangle=1$. We also plot the Pop III SFRD values, inferred from observations of narrow [He <span style="font-variant:small-caps;">ii</span>]{} line emitters as candidates of Pop III systems [@nagao2008photometric; @prescott2009discovery; @cassata2013he], which are generally lower (by up to a factor of 4) than the optimistic model ($\langle f_{\rm mp}\rangle=1$) and approximately correspond to the metal mixing models with $\beta_{\rm mix}\lesssim 0.03$. Note that such observational constraints are highly sensitive to the Pop III IMF, escape fraction of ionizing photons and potential selection effects.
![Pop III SFRD models based on extrapolation of the simulation results (thick short dashed), with corrections for reionization (Equ. \[e9\]) and different models of metal mixing. The optimistic model ($\langle f_{\rm mp}\rangle=1$) is plotted with the thick solid curve. The pessimistic model based on the metal-poor fraction $\hat{f}_{\rm mp}^{\rm sim}$ with halo-scale metal mixing of *simulated* haloes (Equ. \[e8\]) is shown with the (normal) long dashed curve of downward arrows. Semi-analytical metal mixing models (Equ. \[e2\]-\[e7\]) for $\beta_{\rm mix}=0.01$, 0.03, 0.1, 0.3 and 1 are shown with the dotted curves (from top to bottom). The thin horizontal line shows the physically-motivated ‘critical’ Pop III SFRD for a typical halo ($M_{\rm halo,*}\sim 2\times 10^{12}\ \rm M_{\odot}$) at $z=0$ to form one typical Pop III star cluster of $\sim 600\ \rm M_{\odot}$ within one dynamical timescale $t_{\rm dyn}\sim 0.1/H_{0}$ (see Sec. \[s4.3\] for details). The orange shaded region shows the upper bounds inferred from observations of super-luminous SNe (see main text). Constraints from narrow [He <span style="font-variant:small-caps;">ii</span>]{} line emitters (HeIIEs) as candidates of Pop III systems [@nagao2008photometric; @prescott2009discovery; @cassata2013he] are shown with the triangles. The Pop II/I SFRD from @madau2014araa is also shown for comparison (grey shaded region). []{data-label="sfrdex"}](sfrd_extra.pdf){width="1\columnwidth"}
Termination of Pop III star formation {#s4.3}
-------------------------------------
The precise definition of what it means to terminate Pop III star formation is non-trivial, in the absence of abrupt cut-offs in the Pop III SFRD, akin to a cosmic phase transition such as reionization. Our theoretical models here indeed do not exhibit any precipitous drop, and are instead characterized by a more gradual tapering off. We first consider a physically motivated definition based on the average Pop III star formation rate[^10] (SFR) for a halo of mass $M$ at $z$, given a Pop III SFRD $\dot{\rho}_{\rm \star,PopIII}$: $$\begin{aligned}
\dot{M}_{\rm PopIII}&(M,z|\dot{\rho}_{\rm \star,PopIII})=V_{\rm eff}(M,z)\dot{\rho}_{\star,\rm PopIII}\ ,\notag\\
V_{\rm eff}&(M,z) = w(M)\left[n_{\rm h}(M,z)\int_{M_{1}}^{M_{2}}w(M)dM\right]^{-1}\ ,\label{mdot}\end{aligned}$$ where $n_{\rm h}$ is the halo mass function (calculated by @murray2013hmfcalc), $w(M)$, $M_{1}$ and $M_{2}$ refer to the weight function and mass range of Pop III hosts (see Equ. \[e2\] and the description thereafter), which embodies the distribution of Pop III mass in haloes, i.e. $w(M_{\rm halo})\propto M_{\rm halo}^{-1}dM_{\rm PopIII}/d\log M_{\rm halo}$ (see Fig. \[mhdis\]).
We then focus on a typical halo at $z=0$ with a mass $M_{\rm halo,*}\sim 2\times 10^{12}\ \rm M_{\odot}$ and define the ‘critical’ Pop III SFRD as the one required to form one typical/minimum Pop III star cluster with $M_{\rm PopIII,*}\sim 600\ \rm M_{\odot}$ in one dynamical timescale $t_{\rm dyn}\sim 0.1/H_{0}$. This leads to $\dot{\rho}_{\star, \rm crit}\equiv M_{\rm PopIII,*}/[t_{\rm dyn}V_{\rm eff}(M=M_{\rm halo,*},z=0)]\sim 3\times 10^{-8}\ \rm M_{\odot} \rm\, yr^{-1} \rm\, Mpc^{-3}$. Finally, the termination of Pop III star formation is defined via $\dot{\rho}_{\star,\rm PopIII}=\dot{\rho}_{\star,\rm crit}$. In our case, Pop III star formation will be terminated at $z\sim 5$ with complete halo-scale metal mixing (i.e. $\langle f_{\rm mp}\rangle\sim 0$, achieved with $\beta_{\rm mix}\gtrsim 0.18$). For the simulation-based pessimistic model (i.e. $\langle f_{\rm mp}\rangle=\hat{f}_{\rm mp}^{\rm sim}$, see Equ. \[e8\]), Pop III star formation ends at $z\gtrsim 1.5$, which approximately corresponds to the case of $\beta_{\rm mix}\gtrsim 0.15$. While for inefficient metal mixing with $\beta_{\rm mix}\lesssim 0.03$, there is no termination at $z>0$, according to this definition.
We can also consider the termination of Pop III star formation from the observational perspective. Given the host properties and Pop III SFRD models in Sec. \[s4.1\] and \[s4.2\], we can now predict the detection rates of Pop III stars and their PISNe as functions of the horizon redshift $z_{\rm PopIII}$, as shown in Fig. \[Drate\]. Here, we adopt a field of view (FoV, i.e. survey area) of $10\ \rm arcmin^{2}$ for direct observation of Pop III stars, which is relevant to JWST and Hubble deep-field campaigns, and $\rm FoV=10\ \rm deg^{2}$ for detection of Pop III PISNe, achievable with the *Vera C. Rubin Observatory*, specifically its *Legacy Survey of Space and Time* (LSST).
In the optimistic case, where metal mixing is inefficient ($\beta_{\rm mix}\sim 0$, $\langle f_{\rm mp}\rangle\sim 1$), direct detection of Pop III systems would reach $\sim 10\ \rm arcmin^{-2}$ for $z_{\rm PopIII}\sim 2$, and up to 2000 per $\rm arcmin^{2}$ for $z_{\rm PopIII}\sim 10$, assuming that all Pop III stars are grouped into systems of $M_{\rm PopIII}= 10^{3}\ \rm M_{\odot}$. However, as the simulated Pop III systems are not massive ($M_{\rm PopIII}\lesssim 2\times 10^{3}\ \rm M_{\odot}$), we infer $z_{\rm PopIII}\sim 0.5$ for the *Hubble Space Telescope* (HST) and JWST, leading to a detection rate $\sim 0.1$ per $\rm arcmin^{2}$ even in the optimistic case. Here, in the calculation of $z_{\rm PopIII}$, we consider the HST WFC3 filter F555W with a limiting (AB) magnitude of 30 (for $\rm SNR>5$ in a 10-hour exposure), and the JWST NIRCam filter F150W with a limiting magnitude of 31.4. Optimistic magnitudes for Pop III stellar systems are derived with the Stellar Population Synthesis (SPS) code <span style="font-variant:small-caps;">yggdrasil</span>[^11] [@zackrisson2011spectral], under their (instantaneous-burst) Pop III.1 model (with an extremely top-heavy Salpeter IMF in the range of $50-500\ \rm M_{\odot}$) based on @schaerer2002properties, and optimal parameters for nebular emission and Ly$\alpha$ transmission (i.e. $f_{\rm cov}=1$, $f_{\rm Ly\alpha}=0.5$).
Using their Pop III.2 model with a moderately top-heavy IMF from @raiter2010predicted will reduce the flux by a factor of 3. If lensing pushes the limiting magnitude to 33, we can reach $z_{\rm PopIII}\sim 4$ with JWST, where up dozens of Pop III host systems will reside in one $\rm arcmin^{2}$, but the fraction of lensed systems may still be too low for promising detection. For the *Wide Field Infrared Survey Telescope* (WFIRST), given a sensitivity similar to that of HST in the optical and a large FoV of $0.3\ \rm deg^{2}\approx 10^{3}\ arcmin^{2}$, one exposure of 10 hours can detect $\sim 100$ Pop III systems with $\rm SNR>5$ (for the R062 filter) at $z\lesssim 0.5$ in the optimistic case. However, in the pessmimistic model with halo-scale metal mixing of *simulated* haloes (i.e. $\langle f_{\rm mp}\rangle=\hat{f}_{\rm mp}^{\rm sim}$), the detection rate will be reduced by a factor of 30. Furthermore, as discussed below, it is non-trivial to identify Pop III host systems at such low redshifts when Pop II/I star formation is dominating and prevalent. This is particularly challenging at low redshifts ($z\lesssim 1$), where no powerful instrument currently exists in the rest-frame UV to search for distinct features of Pop III (e.g. bluer spectra and [He <span style="font-variant:small-caps;">ii</span>]{} emission lines).
Beside direct observation of active Pop III stars themselves, detection of Pop III PISNe is another important channel to probe late-time Pop III star formation [@scannapieco2005detectability]. However, we find that the scarcity of Pop III stars remains the main obstacle to detection of their PISNe, as seen in previous studies (e.g. @hummel2012). Even for a FoV as large as $10\ \rm deg^{2}$, the detection rate only reaches 1 per year at $z_{\rm PopIII}\sim 7$ in the optimistic model. While the estimated horizon redshift is $z_{\rm PopIII}\sim 0.75-2$ for LSST[^12], and that for JWST NIRSpec ($\rm SNR>10$) is $z_{\rm PopIII}\sim 5$, based on the properties of the recently discovered PISN candidate SN2016aps [@nicholl2020]. For LSST, in the optimistic case with $z_{\rm PopIII}=2$, where all Pop III PISNe are massive interacting events similar to SN2016aps [@nicholl2020] and luminous for long ($\sim 1$ yr), a survey area of $\gtrsim 100\ \rm deg^{2}$ is required to detect one Pop III PISN. For JWST, although it can reach $z_{\rm PopIII}\sim 5$, it cannot afford a large survey area. For instance, the FoV considered by the First Lights at REionization (FLARE) project is only $\sim 0.1\ \rm deg^{2}$, such that detection of Pop III PISNe is unlikely to be achieved in a survey time of a few years [@wang2017first; @regHos2020detecting]. In our calculation of the PISN rates, we assume a Pop III PISN efficiency $\epsilon_{\rm PISN}=10^{-3}\ \rm M_{\odot}^{-1}$ for typical top-heavy Pop III IMFs. We adopt $\epsilon_{\rm PISN}=6\times 10^{-5}\ \rm M_{\odot}^{-1}$ for Pop II/I as an optimal estimation, based on a Salpeter IMF from 0.1 to 200 $\rm M_{\odot}$, neglecting mass loss from stellar winds. Given such assumptions and the Pop II/I SFRD from @madau2014araa, we find that the Pop III contribution to the total PISN rate remains below $10^{-3}$ at $z\lesssim 7$. Tuning the Pop III IMF and normalization of the Pop III SFRD within observational constraints (e.g. @visbal2015 [@inayoshi2016gravitational]) can enhance the Pop III PISN rate by at most a factor of 30, such that the dominance of Pop II/I remains, unless the (average) PISN efficiency for Pop II/I stars is much lower in reality than our optimal estimation. Note that the PISN efficiency is highly sensitive to the upper mass limit of a stellar population, which is still uncertain, especially for Pop III and II stars with low metallicities. For metal-enriched stars (Pop II/I), strong stellar winds may drive the upper mass limit below the PISN threshold. In that case, [*only*]{} Pop III would contribute to the PISN rate.
In general, our results indicate that detection of Pop III stars and PISNe in the post-reionization epoch is extremely challenging, even for the optimistic model with continuous Pop III star formation at a rate $\sim 10^{-7}-10^{-4}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$ down to $z\sim 0$. Actually, this prospect will be rendered even more difficult, if we further consider the fact that most Pop III stars formed at late times ($z\lesssim 6$) would reside in massive systems, where (young massive) Pop II/I stars are also present. As long as $M_{\rm PopIII}\lesssim 10^{3}\ \rm M_{\odot}$, any galaxy with a Pop II/I SFR $\dot{M}_{\star}\gtrsim 10^{-2}\ \rm M_{\odot}\ yr^{-1}$ in the past 10 Myr or a total stellar mass $M_{\star}\gtrsim 10^{7}\ \rm M_{\odot}$ will be dominated by the light from Pop II/I stars, even if it has experienced recent Pop III star formation. In other words, the active Pop III to total stellar mass ratio $M_{\rm PopIII}/M_{\star}$ must be above $\sim 10^{-4}-0.01$ for the Pop III to Pop II/I flux ratio to exceed one ($F_{\rm PopIII}\gtrsim F_{\rm PopII/I}$). We regard such systems as Pop III-bright[^13]. In Fig. \[focc\], we explore the probability of identifying Pop III-bright systems in dwarf galaxies ($M_{\star}\lesssim 10^{8.5}\ \rm M_{\odot}$) by considering the occupation fraction of Pop III hosts in star forming galaxies (left), and the cumulative distribution function of the active Pop III to total stellar mass ratio (right), for the representative sample. Before reionization ($z\gtrsim 10$), $\sim 5$ (10)% of all atomic cooling haloes have recent Pop III activities within 3 (10) Myr. However, after reionization ($z\sim 4-6$), only $\sim 1$ (2)% percent of star-forming haloes with $M_{\rm halo}>M_{\rm J,ion}$ host active Pop III stars for $\tau<3$ (10) Myr, and the occupation fraction will decrease with decreasing redshift, similar to the trend in Pop III SFRD. Moreover, given the Pop III-bright criterion $M_{\rm PopIII}/M_{\star}\gtrsim 10^{-4}-0.01$, $35-60$% of the atomic cooling haloes with recent Pop III star formation are Pop III-bright, while only $\lesssim 10$% of Pop III host haloes with $M_{\rm halo}>M_{\rm J,ion}$ are Pop III-bright[^14]. As a result, Pop II/I stars will dominate in most ($\gtrsim 99.9\%$) massive haloes ($M_{\rm halo}\gtrsim M_{\rm J,ion}\sim 10^{9}\ \rm M_{\odot}$) after reionization ($z\lesssim 6$), according to the star formation main sequence and assembly histories of such haloes [@pawlik2013first; @sparre2015star; @yajima2017growth], whereas before reionization ($z\gtrsim 10$), $\sim 2.5-6$% of all dwarf galaxies in atomic cooling haloes will be Pop III-bright.
However, given $M_{\rm PopIII}< 10^{4}\ \rm M_{\odot}$, such galaxies must form less than $10^{6}\ \rm M_{\odot}$ Pop II/I stars within 10 Myr, such that they cannot be reached by JWST at $z\gtrsim 6.5$. Nevertheless, as mentioned in Sec. \[s4.2\], the total mass of active Pop III stars per halo/galaxy itself is still uncertain, which depends on resolution and the sub-grid models for star formation and stellar feedback, particularly chemical feedback from SNe. In the optimal case where $M_{\rm PopIII}\sim 10^{5}\ \rm M_{\odot}$ , Pop III-bright galaxies would be detectable by JWST (HST/WFIRST) up to $z\sim 12\ (4)$. We thus arrive at the conclusion that right before reionization ($z\sim 10$) is the optimal epoch to search for Pop III-bright systems, consistent with @sarmento2018following, which predict a Pop III-bright[^15] fraction of $\sim 2.5-16$% at $z\sim 9-10$. In this way, our optimistic Pop III SFRD model predicts that JWST (HST/WFIRST) is able to find $\sim 10$ (0.1) such Pop III-bright systems per $\rm arcmin^{2}$. Again, in more realistic models with enhanced metal mixing, those detection rates would be significantly suppressed.
Summary and Conclusions {#s5}
=======================
We construct a theoretical framework to study Pop III star formation in the post-reionization epoch ($z\lesssim 6$) by combining cosmological simulation data with semi-analytical models. To be specific, we closely look into a representative sample of haloes hosting active Pop III stars at $z\sim 4-6$ from a cosmological simulation in LB20 (Sec. \[s2\] and \[s4.1\]). Based on this, we extrapolate the Pop III SFRD to $z=0$ with additional semi-analytical modelling of turbulent metal mixing and reionization (Sec. \[s4.2\]), which may not be fully captured in the simulation. In this way, we evaluate the key physical processes that shape Pop III star formation at late times and the corresponding observational prospects (Sec. \[s4.3\]). Although many of these processes are currently not well understood, future theoretical and observational efforts will reduce the uncertainties and shed light on the fundamental question of the termination of Pop III star formation. Our main findings are summarized below.
- Both radiative and chemical feedback play important roles in regulating Pop III star formation. The former, in terms of LW feedback and reionization, shifts (potential) Pop III star formation to massive haloes (i.e. atomic cooling haloes and haloes above the filtering mass, $M_{\rm halo}\gtrsim 10^{7-9}\ \rm M_{\odot}$). The latter, in terms of mixing of metals released from SNe into the interstellar/circumgalactic medium (ISM/CGM), then determines whether Pop III star formation is possible or not in such massive haloes, which is particularly important in the post-reionization epoch.
In our optimistic model (without additional metal mixing beyond that captured by the simulation), the contribution of minihaloes (i.e. the ‘classical’ site of Pop III star formation) to the overall Pop III SFRD drops to a few percent at $z\lesssim 13$ due to LW feedback (see Fig. \[fiso\]), and decreases exponentially with redshift (to $\lesssim 10^{-5}$ at $z=0$) after reionization ($z\lesssim 6$). Late-time Pop III star formation is dominated by massive haloes ($M_{\rm halo}\gtrsim 10^{9}\ \rm M_{\odot}$), where the densest regions have been significantly metal enriched, but pockets of dense metal-poor gas (e.g. in cold accretion flows) may still exist to form Pop III stars due to inefficient metal mixing (see Fig. \[gasdis\]), consistent with the ‘Pop III wave’ theory [@tornatore2007population].
However, limited by resolution, treatments of metal mixing are imperfect in cosmological simulations, such that metal mixing can be more efficient in reality than in our optimistic model (see Sec. \[s3.2\] for details). For instance, if we assume that metals are fully mixed within the halo virial radius, the Pop III SFRD would be reduced by more than a factor of 30 at $z\lesssim 6$.
- Next to the global Pop III SFRD, the metal mixing process is also important for another key parameter, the total mass, $M_{\rm PopIII}$, of *active* Pop III stars per host halo. Note that we here focus on *active* Pop III stars, and the relevant timescale is short (a few Myr) by nature, such that the signals of short-lived massive Pop III stars can add up in observation. Therefore, $M_{\rm PopIII}$ is equivalent to the (instantaneous) Pop III SFR measured at a timescale of a few Myr. In general, $M_{\rm PopIII}$ is the product of the ‘quantum’ of Pop III star formation, i.e. the typical Pop III stellar mass formed per local (cloud-scale) star formation event, and the number of dense ($n_{\rm H}\gtrsim 100\ \rm cm^{-3}$) metal-poor ($Z\lesssim 10^{-6}-10^{-3.5}\ \rm Z_{\odot}$) star-forming gas clouds in the ISM/CGM of a halo, coexisting on a timescale of a few Myr. The former is well constrained to $500-1000\ \rm M_{\odot}$ for $\Lambda$CDM[^16] by high-resolution simulations and constraints from the timing of the global 21-cm absorption signal [@stacy2013constraining; @susa2014mass; @machida2015accretion; @stacy2016building; @hirano2017formation; @Anna2018; @hosokawa2020]. While the latter is highly sensitive to the metal mixing process. Therefore, in $\Lambda$CDM, $M_{\rm PopIII}$ reflects the number of newly-formed Pop III star clusters in a halo. In our simulation, we find that only a few Pop III clusters can be formed within a few Myr per halo, i.e. $M_{\rm PopIII}<10^{4}\ \rm M_{\odot}$ and the average is $\langle M_{\rm PopIII}\rangle\simeq 10^{3}\ \rm M_{\odot}$ (see Fig. \[MpopIII\]). Interestingly, we also find that $M_{\rm PopIII}$ is independent of halo mass and total stellar mass, quite different from the case of Pop II/I stars where SFR is correlated with stellar mass (i.e. the star formation main sequence). This further indicates the importance of metal mixing for Pop III star formation.
- The total mass $M_{\rm PopIII}$ is particularly important for direct detection of Pop III stars. For instance, if $M_{\rm PopIII}\sim 10^{3}\ \rm M_{\odot}$, as shown in our simulation, direct detection of Pop III stars is only possible at very low redshifts ($z\lesssim 0.5$), considering the sensitivities of space telescopes at present or in the near future (e.g. HST, JWST and WFIRST). If Pop III star formation were to extend to such low redshifts, as predicted by our optimistic model, WFIRST, with its large FoV, could detect $\sim 100$ galaxies with active Pop III stars in one exposure of 10 hours. However, as long as $M_{\rm PopIII}\sim 10^{3}\ \rm M_{\odot}$, only the faintest hosts of Pop III stars can be identified as Pop III-bright (where the Pop III flux exceeds that of Pop II/I), while the emission from the majority ($\gtrsim 99.9$%) of luminous hosts will be dominated by Pop II/I stars, unless we observe in the rest-frame UV. Unfortunately, no powerful UV instrument currently exists to search for distinct features of Pop III (e.g. bluer spectra and [He <span style="font-variant:small-caps;">ii</span>]{} emission lines) in the rest-frame UV at such low redshifts. Detection of Pop III-bright systems would still be challenging for WFIRST.
Nevertheless, as metal mixing is not well understood, $M_{\rm PopIII}$ is still uncertain (see @xu2016late [@danielle2020; @pallottini2014simulating; @sarmento2018following]). Our value lies at the lower end, while the upper limit is $\sim 10^{6}\ \rm M_{\odot}$, derived from theoretical calculations of collapsing primordial gas [@yajima2017upper] and the recent non-detection of Pop III features in the Hubble Frontier Fields at $z\sim 6-9$ [@bhatawdekar2020]. If metal-mixing is overestimated in our simulation and $M_{\rm PopIII}\sim 10^{5-6}\ \rm M_{\odot}$ in reality, Pop III-bright galaxies will be detectable by JWST (HST/WFIRST) up to $z\sim 12.5\ (5)$. In this way, our optimistic Pop III SFRD model predicts that JWST (HST/WFIRST) is able to find up to $\sim 10$ (0.1) Pop III-bright systems per $\rm arcmin^{2}$.
- Finally, our simulations, similar to previous cosmological simulations [@wise2011birth; @johnson2013first; @pallottini2014simulating; @xu2016late], predicts that the overall volume-filling fraction of metal-enriched gas is only a few percent when the universe has expanded to 10-20% of its current size. As it is more difficult to enrich large volumes of gas when the Universe expands further, such simulation results imply that the majority ($\gtrsim 90$%) of the IGM in the observable Universe is occupied by metal-free gas, likely at very low column-densities, undetectable by current instruments. Detecting and quantifying this metal-free phase of the IGM will constrain theoretical models of metal mixing and, therefore, late-time Pop III star formation. This is complemented by similar observations of metal-absorption lines at high redshifts, where bright gamma-ray burst afterglows could serve as background sources [@wang2012]. Our semi-analytical modelling for gravity-driven turbulent metal mixing in virialized systems can be easily extended to describe IGM metal enrichment, governed by the same mixing strength parameter, $\beta_{\rm mix}$ (and additional parameters if necessary). Therefore, it is possible to directly relate late-time Pop III star formation to the volume-filling fraction of metal-free gas. We defer such exploration to future work.
When did Pop III star formation end? The current answer is uncertain. In the optimistic case, Pop III star formation would extend to $z\sim 0$ at a low yet non-negligible rate of $\sim 10^{-7}\ \rm M_{\odot}\ yr^{-1}\ Mpc^{-3}$, while in the pessimistic case, Pop III star formation may already be terminated by the end of reionization ($z\sim 5$). To answer this fundamental question, we must better understand cosmic chemical evolution in terms of mixing of metals released by SNe into the ISM/CGM/IGM during structure formation. On the theory side, we need cosmological simulations with proper resolution and complete representations of the halo population (from minihaloes $\sim 10^{6}\ \rm M_{\odot}$ to galaxy clusters $\sim 10^{14}\ \rm M_{\odot}$) across the entire cosmic history, equipped with advanced sub-grid models for metal mixing and zoom-in simulation techniques (e.g. @pan2013modeling [@hopkins2017anisotropic; @stephen2020]). For observations, stronger constraints will soon come from JWST and WFIRST for (potential) high-$z$ and low-$z$ sources, on both the overall Pop III SFRD and the typical total mass of active Pop III stars per halo. Gravitational wave observations of the binary black hole mergers originated from Pop III stars can also constrain the Pop III SFRD (e.g. @sesana2009observing [@kinugawa2014possible; @hartwig2016; @belczynski2017likelihood; @liu2020]). Meanwhile, we advocate for new UV space telescopes to search for galaxies with distinct Pop III features in the rest-frame UV at low redshifts ($z\lesssim 1$), and programs designed to measure the volume-filling fraction of metal-free gas in the IGM.
Overall, many aspects regarding Pop III star formation are still uncertain, as discussed here via our framework. Nevertheless, with improved theoretical and observational efforts, particularly on the metal mixing process[^17], we will arrive at a more complete picture of Pop III star formation, from onset to termination, thus further elucidating the most elusive population of stars.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank María Emilia De Rossi for insightful discussion regarding stellar population synthesis models for Pop III stars, and acknowledge the Texas Advanced Computing Center (TACC) for providing HPC resources under XSEDE allocation TG-AST120024.
Data availability {#data-availability .unnumbered}
=================
The data underlying this article will be shared on reasonable request to the corresponding authors.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: <https://yt-project.org/doc/index.html>
[^3]: <https://bitbucket.org/gfcstanford/rockstar/src>
[^4]: We have taken into account self-shielding against LW radiation in calculating the $\rm H_{2}$ destruction rate, with the same method used in the simulation (see Sec. 2.2.2 of LB20), based on @wolcott2011photodissociation.
[^5]: According to @pan2013modeling [@sarmento2016following], the dynamical mixing timescale is $\sim 1-10$ Myr, for our gas particles at a scale of $\Delta x\sim (m_{\rm gas}/\rho_{\rm gas})^{1/3}\sim 70$ pc, under the typical conditions in SN remnants with a mean matallicity $\bar{Z}\sim 10^{2}-10^{4}\ Z_{\rm crit}$, a density $n\sim 10^{-3}-1\ \rm cm^{-3}$, a temperature $T\sim 2\times 10^{4}$ K and a Mach number $\mathcal{M}\sim 1$.
[^6]: A halo with active Pop III stars is called a newly star-forming halo, if it has not experienced any star formation activities prior to the recent Pop III star formation.
[^7]: For a Pop III particle, $\tau$ is the age of the underlying stellar population. For a halo with recent Pop III star formation, $\tau$ is the age of the youngest Pop III stellar particles within it.
[^8]: @yajima2017upper derived $M_{\rm PopIII}\lesssim 10^{6}\ \rm M_{\odot}$ from a semi-analytical model for the collapse of primordial gas under the effect of angular momentum loss via Lyman-$\alpha$ (Ly$\alpha$) radiation drag and the gas accretion onto a galactic centre. The lack of evidence for Pop III dominated systems in the Hubble Frontier Fields at $z\sim 6-9$ [@bhatawdekar2020] implies $M_{\rm PopIII}\lesssim 4-7\times 10^{5}\ \rm M_{\odot}$, given a limiting rest-frame UV (absolute) AB magnitude $M_{\rm UV}=-13.5$ (see Sec. \[s4.3\] for the Pop III stellar population synthesis model adopted to derive $M_{\rm PopIII}$ from $M_{\rm UV}$).
[^9]: In the representative sample, $\sim 80$% of molecular cooling haloes hosting active Pop III stars ($\tau<3$ Myr) have not experienced any previous star formation (and internal enrichment).
[^10]: With the SFR formula (Equ. \[mdot\]), our models (see Fig. \[sfrdex\]) can predict the *average* formation rate of Pop III stars in a present-day halo of a mass similar to that of the Milky Way halo ($M_{\rm halo}\simeq 1.5\times 10^{12}\ \rm M_{\odot}$). The result for the optimistic model ($\langle f_{\rm mp}\rangle=1$) is $\dot{M}_{\rm PopIII}\sim 10^{-6}\ \rm M_{\odot}\ yr^{-1}$. Given $M_{\rm PopIII}\sim 10^{3}\ \rm M_{\odot}$, this *average* SFR implies that the probability of such a halo to have recent Pop III star formation (within 3 Myr) is $\sim 0.003$, which serves as a rough estimation for the Milky Way. More accurate estimation for the chance of finding active Pop III stars in the real Milky Way halo needs to consider the detailed assembly history and metal mixing process.
[^11]: <https://www.astro.uu.se/~ez/yggdrasil/yggdrasil.html>
[^12]: The range of $z_{\rm PopIII}$ for LSST reflects the uncertainty in how PISN blast waves interact with the circumstellar medium [@nicholl2020].
[^13]: We also use <span style="font-variant:small-caps;">Yggdrasil</span> to derive the magnitudes of Pop II/I stars, with a Kroupa IMF in the range of $0.1-100\ \rm M_{\odot}$, a metallicity $Z=0.02\ \rm Z_{\odot}$ and a constant SFR over 10 Myr, based on the Starburst99 Padova-AGB tracks [@leitherer1999starburst99; @vazquez2005optimization]. Again, optimal parameters for nebular emission and Ly$\alpha$ transmission are adopted (i.e. $f_{\rm cov}=1$, $f_{\rm Ly\alpha}=0.5$).
[^14]: Note that our simulation is limited in volume such that only dwarf galaxies ($M_{\star}\lesssim 10^{8.5}\ \rm M_{\odot}$) at $z\sim 4-6$ are considered in this analysis. The fraction of Pop III-bright systems is expected to be lower at lower redshifts, where more massive galaxies will be the (potential) hosts of Pop III stars. Such massive galaxies are more likely have dominant Pop II/I components, where feedback from the central massive black holes can also regulate Pop III star formation.
[^15]: Note that @sarmento2018following adopts a more strict definition for ‘Pop III-bright’ as $F_{\rm PopIII}>3F_{\rm PopII/I}$. Therefore, the values quoted here should be regarded as lower limits for our definition ($F_{\rm PopIII}>F_{\rm PopII/I}$).
[^16]: The picture can be different for other dark matter models (see e.g. @gao2007lighting [@hirano2017first]).
[^17]: Metal mixing is also crucial for inferring the properties of Pop III stars from observations of extremely metal-poor stars in the local Universe, i.e. ‘stellar archaeology’ (e.g. @frebel2015near [@ji2015preserving; @hartwig2015; @ishigaki2018initial; @magg2019observational; @magg2020]).
|
QMUL-PH-05-08\
hep-th/0504226
**Inflation from Geometrical Tachyons**\
**Steven Thomas** [^1] **and John Ward**[^2]\
*Department of Physics\
Queen Mary, University of London\
Mile End Road, London\
E1 4NS U.K\
*
**Abstract**
We propose an alternative formulation of tachyon inflation using the geometrical tachyon arising from the time dependent motion of a BPS $D3$-brane in the background geometry due to $k$ parallel $NS$5-branes arranged around a ring of radius $R $. Due to the fact that the mass of this geometrical tachyon field is $\sqrt{2/k} $ times smaller than the corresponding open-string tachyon mass, we find that the slow roll conditions for inflation and the number of e-foldings can be satisfied in a manner that is consistent with an effective 4-dimensional model and with a perturbative string coupling. We also show that the metric perturbations produced at the end of inflation can be sufficiently small and do not lead to the inconsistencies that plague the open string tachyon models. Finally we argue for the existence of a minimum of the geometrical tachyon potential which could give rise to a traditional reheating mechanism.
Introduction.
=============
Despite making many promising advances toward a complete theory of quantum gravity, string/M theory has perhaps not made similar progress in the area of inflationary cosmology where there is an abundance of 4D theoretical models and experimental data [@carroll; @linde]. The main focus has effectively been on two types of models. The first is where there is some kind of spontaneous compactification of the extra dimensions of spacetime, leaving us with our familiar four dimensional universe and the standard model. The second model is that the universe itself is located on the world volume of a 3-brane perhaps in a higher dimensional bulk, with the standard model living on the world-volume. Both approaches have proven to be illuminating, yet the problem of inflation in string theory models still proves elusive.
One of the simplest proposals for inflation in a string theory context, has been that of tachyonic inflation [@gibbons], where the tachyon plays the role of the inflaton. In the past few years there has been great progress in understanding the nature and role of tachyons in string theory [@sen], in particular it has emerged that many of the features of the tachyon can be captured surprisingly well by an effective DBI action [@tachyon_action]. Tachyons arise in several contexts in the theory, most notably in the latter stages of brane-antibrane annihilation [@carroll] but also as open string degrees of freedom on a Non-BPS brane [@sen]. This has stimulated several papers on tachyon inflation and cosmology [@cline; @piao; @joris; @li; @fairbairn].
However there are several problems associated with open-string tachyon cosmology [@kofman2; @shiu] which appears to cast doubt over the tachyon’s role in inflation. The first, and most serious, is that the potential for the field is a runaway exponential, tending to its asymptotic minimum at $T=\pm \infty$. [^3]
Thus, not only is this far too steep to generate the required number of e-foldings but there is no minimum for the tachyon to oscillate around and generate reheating [@kofman1; @sami]. The second problem is that at the start of inflation the de Sitter radius of the universe is actually smaller than the string length and thus an effective theoretical description breaks down, a consequence of this is that the string coupling is large in this region and so perturbative analysis cannot be used. There is also the additional problem of the tachyonic energy, which dominates during inflation and therefore dominates for all time, although arguably this is not as problematic as the other two objections. Whilst there have been several ingenious attempts to bypass these problems, most notably [@cline; @joris; @li; @bento], it seems unlikely that the open-string tachyon could be responsible for inflation, although it may still play a role as dark matter fluid [@sen] or as part of a pre-inflationary phase [@kofman1; @linde].
Recent work on time dependent solutions in the linear dilaton background of coincident $NS$5-branes, has shed new light on a possible geometrical description of the open string tachyon [@time_dependence; @kutasov; @thomas]. In particular, it is conjectured that radion fields on a probe $Dp$-brane become tachyonic when the probe moves in a bounded, compact space. In addition, the mass scale of this geometrical tachyonic mode is substantially smaller than that of the open string tachyon, and therefore may resolve some of the problems associated with inflation.
In this note we will examine the effective action for a $D3$-brane with a geometrical tachyon on its world volume, described by a cosine potential [@thomas]. The cosine potential arises naturally in the context of ’Natural Inflation’ [@freese] due to the creation of pseudo Nambu-Goldstone bosons arising from symmetry breaking, whereas in our case the cosine potential arises due to the geometry of the background brane configuration. It is tempting to identify parameters in the geometrical picture with that of Natural Inflation, however the non-linear form of the tachyon effective action makes this difficult. We will see how our theory fits in with the inflationary paradigm, and whether conventional reheating is possible. We begin by reviewing the construction of the geometrical tachyon solution and the basics of tachyon cosmology. We will then check the consistency of the theory with regards to the slow roll and e-folding approximations before attempting to calculate various perturbation amplitudes and provide a discussion of reheating. We close with some remarks and possible implications for future work.
String Background.
==================
We begin this note with a brief description of the string theory background associated with our model. We will consider the $NS$5-brane background of type II string theory, where we have $k$ parallel but static fivebranes localised on a circle of radius $R$. The branes will be assumed to be unresolvable for the moment, which can be interpreted as a smearing of the charge of $k$ branes around a ring configuration. The background solutions for these fivebranes are given by the CHS solutions [@CHS] $$\begin{aligned}
\label{eq:ring}
ds^2 &=& \eta_{\mu \nu} dx^{\mu} dx^{\nu} + H(x^n)dx^m dx^m \\
e^{2(\phi-\phi_{0})} &=& H(x^n) \nonumber \\
H_{mnp} &=& -\varepsilon^q_{mnp} \partial_q \phi \nonumber,\end{aligned}$$ where $\mu, \nu $ parameterize the directions parallel to the fivebranes and $m, n$ are the transverse directions and $\phi$ is the dilaton. $H(x^n)$ is the harmonic function which describes the orientation of the fivebranes in the transverse space, which in our simplistic case [^4] is given by [@sfetsos] $$\label{eq:ring2}
H = 1+ \frac{kl_s^2}{\sqrt{(R^2+\rho^2 + \sigma^2)^2-4R^2\rho^2}}.$$ In the above solution we have switched to polar coordinates $$\begin{aligned}
x^6 &=& \rho \cos(\theta), \hspace{1cm} x^7=\rho \sin(\theta) \\
x^8 &=& \sigma \cos(\psi), \hspace{0.9cm} x^9 =\sigma\sin(\psi), \nonumber\end{aligned}$$ and now the harmonic function describes a ring oriented in the $x^6-x^7$ plane and we have an $SO(2) \times SO(2)$ symmetry in the transverse space. In our analysis, $l_s$ is the string length and we will be working entirely in string frame. Now the above ring configuration is supersymmetric in 10D, however the introduction of a probe $Dp$-brane breaks this supersymmetry entirely and the probe brane will experience a gravitational force pulling it toward the fivebranes. To avoid further complications we will always assume that $3 \le p \le 5$, since for $p <3$ there is a divergence due to the emission of closed string modes which will render the classical theory useless [@sahakyan]. Note that we are also able to switch between IIA and IIB theory because the $NS$5 brane background in insensitive to T-duality, as the harmonic function couples only to the transverse parts of the metric. We now insert a probe $Dp$-brane in this background whose low energy effective action is the DBI action, which we write as $$S=-\tau_{p} \int d^{p+1} \zeta e^{-(\phi-\phi_0)} \sqrt{-det(G_{\mu \nu}+B_{\mu \nu}+\lambda F_{\mu\nu})},$$ where both $G_{\mu \nu}$ and $B_{\mu \nu}$ are the pullbacks of the space-time tensors to the brane, $\lambda = 2\pi l_s^2$ is the usual coupling for the open string modes, $F_{\mu \nu}$ is the field strength of the $U(1)$ gauge field on the world volume, and $\tau_p$ is the tension of the brane. We will assume that the transverse scalars are time dependent only, and set the gauge field and Kalb-Ramond field to zero for simplicity. Upon insertion of the $NS$5-brane background solution, we see the action in static gauge reduces to the simple form $$S=-\tau_p \int d^{p+1} \zeta \sqrt{H^{-1}-\dot{X}^2},$$ with $X^m$ parameterizing the transverse scalar fields. In order to find our geometrical tachyon, we consider motion of the probe brane in the plane of the ring ($ i.e. \, \sigma = 0 $ and in the interior of the ring and map the above action to a form that is familiar from the non-BPS action for open-string tachyons. The result is that we have [@tachyon_action; @kutasov; @thomas] $$\label{eq:gtachyon}
S= -\int d^{p+1}\zeta V(T) \sqrt{1-\dot{T}^2},$$ where the potential is given by $$\begin{aligned}
\label{eq:cosine}
V(T)&=& \tau_p^u \cos\left(\frac{T}{\sqrt{kl_s^2}}\right) \hspace{0.2cm} \\
\tau_p^u &=& \frac{\tau_p R}{\sqrt{kl_s^2}}, \nonumber\end{aligned}$$ and the tachyon can be expressed as a function of the coordinate $\rho$ as $$T(\rho) = \sqrt{kl_s^2} \arcsin(\rho/R).$$ In obtaining the above, we have used the throat approximation for the harmonic function, which means neglecting the factor of unity in (\[eq:ring2\]). This may not be necessary, but it does allow an exact expression to be obtained. Under this assumption, we see that taking $\rho = \sigma=0$ in (\[eq:ring2\]) (i.e. the centre of the ring) requires that $ \sqrt{k} l_s >> R$, which is the first constraint we find on the parameters $k, l_s $ and $R$. Later on we will use numerical techniques to arrive at a form of the potential $V(T)$ which will use the exact form of the harmonic function as calculated in [@sfetsos]. In principle we can then relax the throat approximation which leads to the cosine potential in (\[eq:cosine\]) so that the previous inequality may not be needed.
To avoid confusion with the open-string tachyon, from now on (unless otherwise stated) we will use tachyon to refer to the geometrical tachyon in (\[eq:gtachyon\]).
We see that the tachyon potential is symmetric about the origin, which arises as a consequence of the background geometry. It should be noted that this mapping is non-trivial in the sense that we began by probing a gravitational background, and have ended up with a solution in flat Minkowski space. This tells us that there are two equivalent ways of visualizing the theory. Firstly there is the bulk viewpoint, where there is actually a ring of $NS$5-branes and the solitary probe brane universe moving in the throat geometry. Alternatively, we could view the problem as a single brane moving in flat space-time with a highly non-trivial field condensing on its world volume. In what follows we will find it useful to switch between these two pictures in order to better understand the physics. In fact the bulk viewpoint is even more complicated as we know [@kutasov; @thomas] that the tachyon field has a geometrical interpretation as a BPS brane in a confined, bounded space, but we could also describe it as a non-BPS brane which has a soliton kink stretched across the interior of the ring [@thomas].
Clearly we see that the geometrical tachyon varies between $T=\pm \pi \sqrt{kl_s^2}/2$ in contrast to the usual open string tachyon which is valued between $\pm \infty$. This is due to the probe brane being confined *inside* the ring. Expanding the potential about the unstable vacuum yields a tachyonic mass of $M_T^2 = -1/kl_s^2$. For sufficiently large $k$ this can be made much smaller than the open string tachyonic mass $M^2=-1/2$ (in units where the string length is unity). It is this different mass scale and profile of the potential which suggest that the geometrical tachyon may be useful in describing inflation on a $D3$-brane. The energy momentum tensor can be calculated in the usual way, and has non-zero components $$\begin{aligned}
T_{00} &=& \frac{V(T)}{\sqrt{1-\dot{T}^2}} \\
T_{ij} &=& -\delta_{ij} V(T) \sqrt{1-\dot{T}^2} \nonumber\end{aligned}$$ from which we see that the pressure of the tachyon fluid tends to zero as the tachyon rolls toward the zero of $ V(T)$.
In [@thomas] we found that there was also probe brane motion through the ring in the $x^8-x^9$ plane. Although this did not lead to the creation of a geometrical tachyon, one could imagine a scenario where the probe oscillates in this direction through the ring, radiating energy as it does so. Eventually the probe would settle at the origin, which is an unstable point in the ring plane corresponding to $T=0$, and we then recover our geometrical tachyon solution.
Tachyon Cosmology.
==================
In a cosmological context, the condensing tachyon will generate a gravitational field on the probe $D3$-brane and therefore we must include this minimal coupling in the action. We will also assume that there is no coupling to any other string mode in order to keep the analysis simple, however we should be aware that there is no reason why other modes should not be included [@choudhury]. Our Lagrangian density can thus be written $$\mathcal{L} = \sqrt{-g} \left( \frac{R}{16\pi G}-V(T)
\sqrt{1+g^{\mu\nu}\partial_{\mu}T \partial_{\nu} T }\right),$$ where $g^{\mu \nu}$ is the metric and $R$ is the usual scalar curvature. For simplicity we will assume that there is a FLRW metric of the form $$ds^2= -dt^2+a(t)^2 dx_i^2,$$ with $i$ running over the spatial directions. We have implicitly assumed here that we have a flat universe, which is acceptable because any curvature is negligible in the very early stages of the universe. The effect of the scale factor is to modify the energy density, $u$ for the flat background such that it is no longer conserved in time [@gibbons], instead we find $$u=\frac{a^3 V(T)}{\sqrt{1-\dot{T}^2}},$$ which prevents us from obtaining an exact solution for the tachyon in the presence of the gravitational field in the usual manner. From this we can determine the late time behaviour of the tachyon condensate. If we assume that $u$ is constant, then the pressure will vary as $p = -V(T)^2 u$ and will tend to zero as $V(T)$ reaches its minimum. Using the standard equation of state $p=\omega u$, we find that $\omega =-(1-\dot{T}^2)$ which implies $-1 \le \omega \le 0$. From the Lagrangian density, we can also obtain the Friedman and Raychaudhuri equations for the tachyon condensate $$\begin{aligned}
H^2 &=& \left(\frac{\dot{a}}{a}\right)^2= \frac{\kappa^2 V(T)}{3 \sqrt{1-\dot{T}^2}} \\
\frac{\ddot{a}}{a} &=& \frac{\kappa^2 V(T)}{3\sqrt{1-\dot{T}^2}} \left(1-\frac{3\dot{T}^2}{2} \right),\end{aligned}$$ where $\kappa^2 = 8\pi G = M_{p}^{-2}$, $M_p = 2.2\times 10^{18} GeV$ and the cosmological constant term is set to zero. There is also a useful relationship between the 4D Planck mass and the string scale obtained via dimensional reduction [@jones] $$\label{eq:reduction}
M_p^2 = \frac{v M_s^2}{g_s^2},$$ where $M_s = l_s^{-1}$ is the fundamental string scale and the quantity $v$ is given by $$v = \frac{(M_s r)^d}{\pi},$$ with $r, d$ being the radius and number of compactified dimensions respectively (typically $d$=6 for the superstring). Note that for our effective theory to hold we require $v >> 1$. Obviously we are assuming there is some unknown mechanism which stabilizes the compactification manifold, and freezes the moduli so that they do not interfere with our tachyon solution.
The evolution of the universe is effectively determined by the Raychaudhuri equation which shows that inflation will cease when $\dot{T}^2 = 2/3$ and the universe will then decelerate as the tachyon velocity increases. Upon variation of the action, we find the equation of motion for the tachyon field can be written $$\label{eq:eom}
\frac{V(T) \ddot{T}}{1-\dot{T}^2}+3HV(T)\dot{T} + V'(T) = 0,$$ where a prime denotes differentiation with respect to $T$, and $H$ is the Hubble parameter. Note that in deriving this equation we must also use the conservation of entropy of the tachyon fluid [@gibbons]. We see that $3HV(T)\dot{T}$ acts as a friction term, in much the same way as in standard inflationary models, except that this term may vanish for the open string tachyon as the field rolls to $\pm \infty$ where its potential vanishes. For a scalar field to be a candidate for the inflaton it must satisfy the usual slow roll parameters as well as providing enough e-foldings during rolling. The tachyon is no exception [@steer], and so we use the conventions employed in [@li; @hwang] to write the slow roll parameters $$\begin{aligned}
\epsilon(T) &=& \frac{2}{3}\left(\frac{H'(T)}{H^2(T)} \right)^2 \\
\eta(T) &=& \frac{1}{3} \left(\frac{H''(T)}{H^3(T)} \right). \nonumber\end{aligned}$$ Where we assume that the acceleration of the tachyon is negligible, and require that $\epsilon <<1$ and $|\eta|<<1$ in order to generate inflation. There appears to be little or no consensus on the correct slow roll equations to use for the tachyon, however we see that the same general behaviour is obtained even if we use the conventions in [@piao] or [@fairbairn]. The number of e-folds produced between $T_o$ and $T_e$ is given by $$\label{eq:efoldings}
N(T_o, T_e) = \int_{T_o}^{T_e} dT \frac{H}{\dot{T}},$$ which must satisfy $N \sim 60$ to agree with observational data [@carroll]. $T_o$ and $T_e$ are two arbitrary points on the potential where the slow roll conditions are still satisfied.
Geometrical tachyon inflation.
==============================
One of the many problems associated with tachyon inflation is that the mass scale of the open string tachyon is simply too large. However we have seen that this is not necessarily the case where we have a geometrical tachyon, and so we may enquire whether inflation is possible in this instance. We first consider the geometrical tachyon starting very close to the top of its potential with a small initial velocity to ensure that it will roll. Note that if $T=0$ then spontaneous symmetry breaking will cause the universe to fragment into small domains which will each have differing values of the tachyon field. Inflation can occur only if $H^2 >> |M_T^2|$ near the top of the potential, which translates into the constraint $$\frac{\tau_3 R}{3 M_p^2} >> \frac{1}{\sqrt{kl_s^2}}.$$ Where $\tau_3$ is the tension of a stable $D3$-brane. Since we are considering the large $k$ limit, the RHS is very small and so we find that this condition is satisfied. [^5] Furthermore this also suggests that the effective theory for geometrical tachyons is valid because we can clearly see that $$(kl_s^2)^{1/2} >> H^{-1},$$ and so the de-Sitter horizon may be larger than the string length for large $k$. This is in contrast to the open string scenario where we find that the horizon is smaller than the string length, and thus should not be described by an effective theory.
In order to check that this is correct, we must try and get a handle on the size of the string coupling. In the open string tachyon case we find that in order to satisfy $H^2>>|M_T^2|$ at the top of the potential, we have the constraint $$g_s >> 260 v.$$ As already mentioned, the effective theory is only valid for $v>>1$, which implies that we are in the strong coupling regime and therefore an effective theory may not yield reliable results. In contrast, a similar calculation for the geometrical tachyon implies $$\label{eq:top}
g_s >> \frac{24 \pi^3 v }{\sqrt{k} R M_s} \approx 744 \frac{v}{\sqrt{k} R M_s},$$ thus by fixing appropriate values for $k$ and $R$ we may ensure that $v>>1$ and also that $g_s << 1$. Earlier we saw the throat condition $ \sqrt{k}l_s >> R $ which means that $ \sqrt{k} R M_s << k $. Thus, in fact it is large values for $ k$ that will essentially allow a satisfactory solution to (\[eq:top\]). For example, assuming $v = 10$ a value of $k\approx 10^5 $ would allow for perturbative $g_s$ to solve (\[eq:top\]). Relaxing the throat approximation may allow much smaller values of $k$. This is interesting because we see that the weak coupling arises entirely due to a parameters describing the background brane solution in the bulk picture. In contrast, there is not a clear explanation to account for the origin of the weak self-coupling of the inflaton in standard single scalar field [@steer] inflation.
Despite this apparent success we may be concerned that the effective theory may still not be a valid description at the top of the potential [@kofman2]. In order to check this we should compare the effective tension of the unstable brane to the Planck scale. After some algebra, and using the equation for weak coupling we find $$\frac{\tau_3^{u}}{M_p^4} \sim \frac{3}{k^2}\left(\frac{24 \pi^3}{R M_s} \right)^2 v.$$ Again we see that for a certain range of background parameters (and assuming $R M_s >> 1$), the effective tension need not be Super-Planckian and therefore the DBI can still be a good approximation to 4D gravity.
As there is an obvious similarity with Natural Inflation [@freese] we could demand that the height of the potential to be of the order of $M_{GUT}^4$ (where $M_{GUT} \sim 10^{16}$ GeV) in order to generate inflation, however we will try to keep this arbitrary for the moment. Using the potential, we immediately see that the slow roll conditions for our geometrical tachyon can be written as $$\begin{aligned}
\epsilon &=& \frac{M_p^2}{2\tau_3 R \sqrt{kl_s^2}} \frac{\tan^2(T/\sqrt{kl_s^2})}{\cos(T/\sqrt{kl_s^2})}\\
\eta &=& \frac{-M_p^2}{4\tau_3 R \sqrt{kl_s^2}} \frac{\left(1+\cos^2(T/\sqrt{kl_s^2})\right)}{\cos^3(T/\sqrt{kl_s^2})}.\end{aligned}$$ Slow roll will only be a valid approximation when the tachyon is near the top of its potential and thus we may effectively neglect the contribution from the trigonometric functions as they will only be terms of $\mathcal O(1)$. In fact for small $T$ we see that $\epsilon$ is already extremely small. Dropping all numerical factors of order unity gives us the primary constraint for slow roll; $$M_p^2 << \tau_3^u R \sqrt{kl_s^2},$$ which must be satisfied by both equations. Given that the reduced Planck mass in string theory is typically of the order of $2.4 \times 10^{18}$ GeV, this means that $k$ and $R/l_s$ must be large. Generally the slow roll conditions will be satisfied due to the mass scale of the geometrical tachyon, for large $k$. In the open string models, the larger mass implies that the tachyon may only have been involved in some pre-inflationary phase. Of course, the analysis in both cases is classical and quantum corrections may well prove to be important is determining the exact behaviour near the top of the potential.
We can estimate the number of e-foldings using (\[eq:efoldings\]), however this turns out to be sensitive to the value of the tachyon velocity near the top of the potential. To remedy this we use the equations of motion and the slow roll approximation, which allows us to re-write this equation in terms of the potential and its derivative. In fact this is the method most commonly used in standard inflationary analysis. After some algebra we obtain $$\begin{aligned}
\label{eq:efoldings2}
N_e &=& -3 \int dT \frac{H^2 V(T)}{V'(T)} \\
&=& \frac{\tau_3 R \sqrt{kl_s^2}}{M_p^2} \left\lbrace\cos\left(\frac{T_e}{\sqrt{kl_s^2}}\right)-
\cos\left(\frac{T_o}{\sqrt{kl_s^2}}\right)+\ln\left(\frac{\tan(T_e/2\sqrt{kl_s^2})}{\tan(T_o/2\sqrt{kl_s^2})}\right) \right\rbrace \nonumber\end{aligned}$$ Using the constraint from the slow-roll equations we see that the leading term must be large. If we demand that $T_o$ and $T_e$ are reasonably close, then the contribution from the other terms will be small, and so the number of e-foldings will depend on the ratio $$\nu = \frac{\tau_3R \sqrt{kl_s^2}}{M_p^2},$$ where $\nu \ge 60$ in order for there to be enough inflation. However if we dont impose this restriction, but allow inflation to begin near the top of the potential and end near the bottom, then there can be significant contribution to the number of e-foldings from the additional terms. This has the effect of reducing the value of $\nu$ - however it must still satisfy the slow roll constraint of being larger that unity. We can write the unstable brane tension in terms of the string coupling, string mass scale and the parameter $\nu$, thus we have the height of the potential given by $$M_{infl}^4 = \frac{M_p^2 M_s^2 \nu}{k},$$ which defines our effective inflation scale $M_{infl}$. The exact value of $M_s$ depends on the particular string model but it is usually assumed to lie in the range 1 Tev - $10^{16}$ GeV. So as an example, if $\nu \sim 60$, $M_s \sim 10^{16}$ GeV and $k \sim 10^5 $ we find $M_{infl} \sim 10^{16}$ GeV.
Numerical Analysis.
-------------------
We can also check the consistency of our analytic solutions by numerically solving for the Hubble parameter. We can write the Hubble equation as a function of $T$ rather than time (since the tachyon field is monotonic with respect to time - at least initially), and then using the Friedmann equation we obtain the following first order differential equation [@fairbairn] $$\label{eq:diff1}
H^{\prime 2}(T) - \frac{9}{4}H^4(T)+ \frac{1}{4M_p^4}V(T)^2=0,$$ where a prime denotes differentiation with respect to $T$. Solving this for the Hubble term gives us a constraint on the velocity of the tachyon field $$\label{eq:velocity}
\dot{T}^2 = 1 - \left(\frac{V(T)}{3M_p^2 H(T)^2} \right)^2.$$ It will be convenient to work with dimensionless variables in our numerical analysis, so we define the dimensionless tachyon field and Hubble parameter as follows, $$y=\frac{T}{\sqrt{kl_s^2}}, \hspace{0.5cm} h(y)=\sqrt{kl_s^2}H(y).$$ We can solve (\[eq:diff1\]) to obtain $h(y)$ and then substitute this into (\[eq:velocity\]) to determine the velocity of the tachyon field. We choose the initial velocity of the field to be zero, and the initial value of $T_o$ to be very small. As in [@fairbairn], the general behaviour is dependent upon the dimensionless ratio $X_0$, where $X_0^2=\nu$. Some results are plotted in Figure 1. We find that the velocity (strictly speaking this is the square of the velocity) of the tachyon field is very small over a large range, only becoming large as it nears the bottom of the potential. In inflationary terms this implies that universe will be inflating for almost the entire duration of the rolling of the field. For increasing values of $X_0$, inflation ends at larger values of $T$. However even for the case of $X_0=2$, which barely satisfies the slow roll constraints, we expect inflation to end reasonably close to the bottom of the potential.
We can also make a numerical check on the smallness of the slow roll parameters $\epsilon, \eta $ (see figs 2 and 3). Using our numerical solution for $h$ we can also determine the amount of e-foldings during inflation via (\[eq:efoldings2\]). It turns out that in order to generate at least 60 e-foldings we only need to take $\nu \sim 30$
Finally, we can also use numerical techniques to try and reconstruct the tachyon potential by using the full form of the ring harmonic function as derived in [@sfetsos] without assuming the approximation that lead to the cosine potential (\[eq:cosine\]). Recall that this approximation was that the $NS$5 branes were unresolvable as point sources arranged uniformly around the ring. As our tachyon field rolls from near the top of the cosine potential down towards the value $T/{\sqrt{k}l_s} = \pi/2$, the geometric picture of this process is that we start from near the centre of the ring at $\rho = 0$ and move towards the ring located at $\rho = R $. As $T/{\sqrt{k}l_s}$ nears $\pi/2$, even for large $k$, the approximation of a continuous distribution of $NS$5 branes around the ring will break down and individual sources will be resolvable. It is at this point that we expect the true potential $V(T)$ to deviate from the cosine form. Fig 4 shows the shape of the potential one obtains for the case $k=1000$, by numerically implementing the tachyon map discussed earlier, using the exact form of the ring harmonic function. In this plot we have chosen the angular variable $\theta$ that appears in the exact form of the harmonic function to be fixed at $\pi/{2k} $ for simplicity. Details of the harmonic function relevant to fully resolvable $NS$5 branes are given later on in Section 5. What is perhaps most apparent about this potential is the existence of a minimum very close to the ring location. It also turns out that our previous cosine potential is an excellent approximation to this numerical plot for values of $T $ to the left of the minimum. Later on in section 5, we shall see how analytic methods can be used to verify the existence of this minimum.
{width="10cm" height="6cm"}
Perturbations.
--------------
So far, so good, but we must also consider the perturbation fluctuations generated at the end of inflation. One of the generic difficulties associated with open-string tachyonic inflation is the fact that the tension of the $D3$-brane must be significantly larger than the Planck mass. This indicates that the effective action cannot adequately describe 4D gravity, as it will have metric fluctuations that are always too large [@kofman2]. This is not the case for our geometrical tachyon as we seen there are additional scales in the theory which can reduce the overall effect of these fluctuations. There are two main perturbations to consider, the scalar, and the gravitational (tensor) ones which we will denote by $\mathcal{P_T}$ and $\mathcal{P_G}$ respectively. (Strictly speaking, $\mathcal{P}$ corresponds to the amplitude of the perturbation). Constraints from observational data imply the relation [^6] $$|\mathcal{P_T}| + |\mathcal{P_G}| \le 10^{-5}.$$ During inflation, gravitational waves are produced whose amplitude is given by $\mathcal{P_G} \sim \frac{H}{M_p}$, but observational data of the anisotropy of the CMB [@kofman2; @li; @fairbairn] implies that at the end of inflation $$\label{eq:tensor}
\frac{H_{end}}{M_p} \le 3.6 \times 10^{-5},$$ and we must ensure that this condition is consistently satisfied in order for us to consider the geometrical tachyon as a possible candidate for the inflaton. In order to verify this we will first consider the scalar perturbation and use the solution from that to determine our mass scales for the metric perturbations, as it is generally more important to see whether the tachyon action allows for small metric fluctuations. For simplicity we will assume that inflation ends when the following constraint is satisfied [@kofman2] $$H \sim |M_T| \sim \frac{M_s}{\sqrt{k}},$$ and we will also assume that the tachyon velocity at this time is given by $\dot{T}=\sqrt{2/3}$. The scalar perturbations are determined in the usual manner using $$|\frac{\delta \rho}{\rho}| \sim \frac{H\delta T}{\dot{T}},$$ where $\delta T$ satisfies the following constraint near the top of the potential [@fairbairn; @parameters] $$\delta T \sim \frac{H^2}{2\pi \sqrt{V(T)}}.$$ Combining the last two equations we write the amplitude for the scalar perturbation as $$\label{eq:scalar}
\mathcal{P}_T \sim \frac{H^2}{2\pi \dot{T} \sqrt{V(T)}} \le 10^{-5}.$$ (We should actually calculate the values of $H$ and $\dot{T}$ during inflation in order to determine the ratio of the perturbations, however since we expect $T$ to be a slowly varying field (\[eq:scalar\]) should remain constant over a large range of wavelengths [@linde].) If we assume that $T$ is small then the cosine part of the potential is close to unity, and upon substitution of the Hubble term we find $$\label{eq:scalar_result}
\mathcal{P}_T \sim \frac{M_s}{M_p}\sqrt{\frac{3}{8\pi^2k \nu}} \le 10^{-5},$$ We can use this to determine a constraint upon the string scale/Planck scale ratio as follows $$\label{eq:ratio}
\frac{M_s}{M_p} \le \sqrt{\frac{8 \pi^2k\nu}{3}} \times 10^{-5}.$$ As an example, for $k \sim 10^3$ and $\nu \sim 28$ (\[eq:ratio\]) implies $ M_s \leq 10^{16} GeV$
Solving the equation for the metric perturbation leaves us with $$\mathcal{P}_{G} \sim \frac{H}{M_p} \sim \frac{M_s}{M_p \sqrt{k}} \le 3.6 \times 10^{-5}$$ which is explicitly dependent upon this ratio. We can establish the absolute upper bound on the perturbation using (\[eq:ratio\]) $$\mathcal{P}_{G} \le 2\pi \times 10^{-5} \sqrt{\frac{2\nu}{3}}.$$ If $\nu$ is $\mathcal{O}(30)$ then this implies the maximum perturbation will be of the order of $10^{-4}$ which is slightly too large. However in general we may expect that the metric perturbations will be acceptably small by assuming a smaller string scale than the one that saturates (\[eq:ratio\]) for given $k$. This is encouraging since the open string tachyon always admits large metric fluctuations, and therefore cannot be responsible for the last 60 e-foldings of inflation [@kofman1]. In our case these fluctuations can be suppressed when $k$ is sufficiently large, and we can find inflationary behaviour leading to the correct amount of structure formation.
Reheating.
==========
Perhaps the most problematic aspect of tachyon inflation is the shape of the potential itself. The open string tachyon potential is exponentially decaying at large field values with its minimum at asymptotic infinity. Thus even if it were possible to satisfy all the inflationary conditions, the lack of minimum effectively kills the model as there will be no reheating [@kofman1] in the classical sense. (As mentioned previously, gravitational reheating is far too weak in these models to account for the particle abundance we see today.) It is possible to obtain reheating if the tachyon is coupled to several gauge fields [@cline], and is a direction that certainly needs further consideration. It is also certainly possible that the potential vanishes for finite $T$, leading to small oscillations about the minimum [@fairbairn] which could provide a mechanism for reheating. In any case, the issue does not seem to be resolved in a satisfactory manner.
Our geometrical tachyon is no exception to these criticisms. Although the minimum is not at infinity, the effective theory breaks down when the tachyon rolls to its maximum value and we are unable to proceed. In the 10D gravitational picture this is due to the probe brane hitting the ring of smeared fivebranes. However even with the simple form of the DBI action in this instance, we see that outside the ring the potential is approximately exponential [@thomas] and it is suggestive that it may somehow smoothly map onto the cosine at $\rho=R$. One may well enquire what happens if we consider a case where the fivebranes are not smeared around the ring, rather that they appear resolved thus allowing a probe brane to pass between them. In this case, we would not expect the effective DBI action to break down and we can find corrections to the truncated cosine potential and thus obtain a minimum. This is exactly what we found following the numerical analysis in section 4. Let us now see how the existence of such a minimum can be seen analytically. In order to proceed, we refer the reader back to the full harmonic function describing $k$ branes at arbitrary points on the circle [@sfetsos], with the interbrane distance, $x$, given by $$x = \frac{2\pi R}{k}.$$ The full form of the function in the throat region is given by $$H \sim \frac{kl_s^2}{2R\rho \sinh(y)} \frac{\sinh(ky)}{(\cosh(ky)-\cos(k\theta))},$$ where $\rho, \theta$ parameterize the coordinates in the ring plane, and the factor $y$ is given by $$\cosh(y) = \frac{R^2+\rho^2}{2R\rho}.$$ We clearly see that as $k\to \infty$ we recover the expression for the smeared harmonic function which we used in the previous sections to derive the tachyon potential. Furthermore we see that when $\rho = R$ the function reduces to $$H \sim \frac{k^2l_s^2}{2R^2} \frac{1}{(1-\cos(k\theta))}$$ which is clearly finite provided that $\theta \ne 2n\pi/k$, which are the locations of the $NS$5 branes. In order to look for a minima we must expand about the point $\rho = R$ using $\rho = R +\xi$, where $\xi$ is a small parameter which can be positive or negative. Using the expansion properties of hyperbolic functions we power expand the harmonic function for an arbitrary fixed angle $\theta$, and we find to leading order[^7] $$H \sim \frac{k^2l_s^2}{2R^2}\frac{1}{1-\cos(k\theta)} \left(1-|\frac{\xi}{R}|+(5/6- k^2 \frac{2+\cos(k\theta)}{6(1-\cos(k\theta))} )\frac{\xi^2}{R^2}+\ldots \right),$$ where we have used the fact that $k\xi << R$ and have neglected any higher order correction terms. Note that the inter brane distance is given by $2 \pi R /k$ and so our expansion will only be valid for distances much smaller than the brane separation. Of course we must be careful not to take $k$ to be too small since our effective action for the geometrical tachyon will be invalid. We can clearly see that if the trajectory is at an angle $\theta = (2n+1)\pi/2k$, then the harmonic function will reduce to the form (again to leading order in large k) $$H \sim \frac{k^2l_s^2}{2R^2}\left(1-\frac{k^2\xi^2}{3 R^2}+\ldots \right)$$ We now perform the tachyon map to determine the value of the tachyon as a function of $\xi$. Note that we expect this ’tachyonic’ field to have positive mass squared since it is near the minimum of its potential. Up to constants we find that $$T(\xi) \sim \sqrt{\frac{k^2l_s^2}{2(1-\cos(k\theta))}}\left(\frac{\xi}{R} - \frac{\xi^2}{2R^2}+\ldots \right)$$ If we assume that the $\xi^2$ term is negligible then we can invert our solution and calculate the perturbed tachyon potential. Note that keeping higher order terms here does not lead to a simple analytic solution, and so we would hope that a numerical analysis would be more appropriate. After some manipulation we find $$\begin{aligned}
V(T) &\sim & \frac{\tau_3}{kl_s} \left( 2R^2(1-\cos(k\theta ))\right)^{1/2}
[ 1+ \frac{T}{2kl_s}\sqrt{2(1-\cos(k\theta))}+ \nonumber \\
&&T^2 (\, \frac{2+\cos(k\theta )}{6l_s^2} - \frac{(1-\cos(k\theta )\, )}{12k^2l_s^2}\, ) + \ldots ] \end{aligned}$$ which shows that the potential is approximately linear around the minimum as it interpolates between the cosine and the exponential functions, however this linear term is suppressed by a factor of $1/k$ and we would expect it be negligible in the large $k$ limit, thus we can see that there is an approximately quadratic minimum. We see that the minimum of the potential in the tachyonic direction will be $$\label{eq:minimum}
V(T(\xi=0))= \frac{\tau_3R}{k l_s}\sqrt{2(1-\cos(k\theta))}$$ which can obviously be made small in the large $k$ limit, and will clearly go to zero when the $D3$-brane trajectory is such that it hits one of the $NS$5-branes. The local maximum will occur at the bisection angle $\theta=\pi/k$, which we suspect will be an unstable point. All this fits nicely with our earlier numerical analysis. Figure 4 in section 4 showed the result of numerical methods used to plot the potential using the exact form of the ring harmonic function. Numerical solutions to the tachyon map inside and outside the ring were matched together to obtain this plot. The minimum can be seen to be quadratic for small distances before mapping onto an exponential function outside the ring as expected from [@thomas]. This is because the numerical analysis includes all the higher order correction terms, which produces a curved potential at the minimum.
The condensing tachyon field may oscillate about the minimum of this potential, assuming that the energy of the tachyon is such that it will not overshoot and return back up the potential toward $T=0$. This assumption seems to be valid because as we have just seen the potential no longer has to vanish at $\rho=R$, so the friction term in (\[eq:eom\]) will not vanish as the tachyon condenses. However in order for reheating to occur we must ensure that this term sufficiently damps the motion, confining the field to very small oscillations about this minimum.
From standard inflationary models we know that the oscillations about the minimum can be thought of as being a condensate of zero momentum particles of (mass)$^2$ = V$^{\prime \prime}$(T). The decay of the oscillations leads to the creation of new fields coupled to the tachyon condensate via the reheating mechanism. The temperature of this reheating can be approximated as the difference between the maximum and minimum of the potential, and so we find $$T_{RH}^4 \sim M_{infl}^4\left(1-\frac{1}{\sqrt{k}}\sqrt{2(1-\cos(k\theta))}\right)$$ and so if we assume that the conversion of the tachyon energy is almost perfectly efficient then we will have an upper bound for the reheating temperature given by the effective inflation scale $M_{infl}$.
We must now consider the more general case where we perturb $\theta$ away from its bisection value of $\pi/k$. Since we are assuming that the $NS$5-branes are somehow resolvable, we must also be aware that a single brane does not form an infinite throat [@kutasov]. As such, a passing probe brane will feel the gravitational effect of the fivebranes, but because we expect it to be moving relativistically we expect that its trajectory will only suffer a slight deflection as it passes by. In this instance, the perturbed harmonic function at $\rho =R$ reduces to $$H \sim \frac{k^2l_s^2}{2R^2}\frac{1}{(1+\cos(k\delta))},$$ where $\delta$ represents the angular perturbation. Now, we know that the harmonic function becomes singular when our probe brane hits a five brane so the function needs to be minimized to ensure a stable trajectory, This is clearly accomplished by sending $\delta \to 0$. So the value $\pi/k$ corresponds to an unstable *maximum* from the viewpoint of the tachyon potential. Of course, we could also see this directly from (\[eq:minimum\]) by considering perturbations about the bisection angle. For small angular deflection we may write $$H \sim \frac{k^2l_s^2}{4R^2} \left( 1+\frac{k^2 \delta^2}{4}+\ldots \right),$$ and so we see that provided $k \delta << 1$ the trajectory of the probe will be relatively unaffected by the presence of the fivebranes and therefore we may expect that it will pass between them. On the other hand, for larger values of $k\delta$, this will not be true and the probe brane may be pulled into the fivebranes. In any case, we expect that our analysis of the geometrical tachyon will be invalid in this instance.
The analysis will also be true for a $D3$-brane in a ring $D5$-brane background using S-duality, the only difference will be to replace $$kl_s^2 \to 2 g_sk l_s^2,$$ where $g_s$ is the string coupling and we again consider $k$ branes on the ring. The overall effect of switching to the $D5$-brane background is to allow for a weaker coupling at the top of the potential. In fact the analogue of (\[eq:top\]) in this picture becomes $$g_s >> \left(\frac{24 \pi^3 v }{R \sqrt{2k}M_s}\right)^{2/3}$$ The situation is made slightly more complicated due to the presence of background RR charge, but this can be neglected when the tachyon is purely time dependent. Thus we would expect similar results to those obtained in the last two sections. Of course, we should remember that fundamental strings can end on the $D5$-branes and consequently there can be additional open string tachyons in the theory.
Discussion
==========
In this note we have examined the cosmological consequences of the rolling geometrical tachyon in the early universe. Because of the different mass scale compared to the open string tachyon, the geometrical tachyon resolves some of the problems besieging tachyon inflationary models.
The effective theory is self consistent and appears to be valid description of 4D gravity due to the weak string coupling. This weak coupling arises because we can select a specific region of our moduli space associated with the background geometry. Furthermore we have seen that the form of the potential satisfies all the slow roll and e-folding conditions, whilst providing acceptable levels of metric perturbations at the end of inflation. In addition, we have shown that the potential has a minimum which will be approximately quadratic for small perturbations in the tachyon field and may therefore be used to describe traditional reheating [@kofman1].[^8] We have not in any way discussed how reheating can occur in this model, but we have attempted to show that the potential may well have a metastable minimum which could allow for the kind of field theoretic inflaton oscillation required for standard reheating.
There are still potential problems associated with this model. Firstly it seems unlikely that we can have an analytic expression for the tachyon valid for any point on the potential due to the complicated nature of the full harmonic function. Thus we have been forced to make approximations or resort to numerical methods. Secondly there is still the issue of fine tuning to deal with because we need specific values of $k$ and $ R$ such that a probe brane passes between fivebranes in the bulk picture without causing the tachyon solution to collapse. Furthermore we effectively require the $D3$-brane to exhibit one dimensional motion, so that it continually passes between the $NS$5 branes on the ring as it rolls in the minimum of the potential. More general motion would imply that the probe brane will eventually hit one of the source branes and cause the effective theory to break down. Thirdly we must ensure that the probe brane is not too energetic, otherwise it may overshoot the minimum and return to the origin. This requires the friction term to sufficiently damp the motion of the field as it approaches the minimum of the potential. Although we have argued that this may occur it is not clear whether this requires any fine tuning or not. Finally there is the issue of coupling to other string modes, which will be essential in generating the standard model fields after reheating, and have been neglected in this and other notes on tachyon inflation.
A more detailed investigation is required before we can rule out this model, however we have found some promising results that circumvent many of the problems associated with tachyon inflation and it is suggestive that other geometrical tachyon modes may well be viable candidates for the inflaton. It would also be useful to extend this work to include assisted inflation, and perhaps non-commutative geometrical tachyons, for example [@calcagni]. In addition, it would be useful to have more understanding of these geometrical tachyon modes and their relationship to the open string tachyon. In particular the case of $k=2$ is important [@kutasov], as it relates observables in Little String Theory (LST) to those in 10D supergravity. Because the geometrical tachyon fields are dependent upon the background brane geometry, we may imagine that all these tachyon solutions correspond to specific points or cross sections of some larger moduli space. It would therefore be useful to consider other background geometries and how they are inter-related to some of the known open string tachyon solutions. We will hopefully return to some of these issues at a future date.
**Acknowledgements**
Many thanks to D. J. Mulryne and N. Pidokrajt for their useful comments and observations. JW is supported by a QMUL studentship. This work was in part supported by the EC Marie Curie Research Training Network MRTN-CT-2004-512194.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: Note however that in the case of non-BPS branes in superstring theory, there is evidence that the tachyon potential for static fields develops localized minima at finite values of $T$, see e.g. [@sen2]
[^4]: The exact form of the harmonic potential for $k$ $NS$5 branes arranged around a ring, as computed in [@sfetsos], is rather more complicated than the expression in (\[eq:ring2\]). The latter form emerges as an approximation valid for distances $ r >> {2\pi R}/k $ i.e. effectively a large $k$ approximation. In this limit the $NS$5-branes appear as smeared around the ring. [@thomas]
[^5]: In the finite $k$ case we will have to be careful to ensure that this constraint is fulfilled.
[^6]: Thanks to D. J. Mulryne for pointing this out.
[^7]: Thanks to Shinji Tsujikawa for pointing out an algebraic error in a previous draft of this note.
[^8]: Although there may be some objections to this due to the non-linearity of the tachyonic action [@sami].
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address: |
$^{\text{\sf 1}}$Computer Science Department, Saarland University, Saarbruecken, 66123, Germany and\
$^{\text{\sf 2}}$Department of Biological Sciences, Saarland University, Saarabruecken, 66123, Germany.
author:
- 'Charalampos Kyriakopoulos$^{\text{\sfb 1,}*}$, Pascal Giehr$^{\text{\sf 2}}$ and Verena Wolf$^{\text{\sfb 1,}*}$'
bibliography:
- 'document.bib'
subtitle: Genome analysis
title: 'H(O)TA: estimation of DNA methylation and hydroxylation levels and efficiencies from time course data'
---
Introduction
============
DNA methylation refers to the transfer of a methyl group to the C-5 position of cytosine (C) to produce 5-methylcytosine (5mC). In mammals it is predominantly found in the symmetrical CpG context and, as a major epigenetic modification, it plays an essential role in the regulation of gene expression. Moreover, DNA methylation contributes to a wide range of cellular processes such as development, X-inactivation, and imprinting ([@bourc2004meiotic]) and aberrant methylation patterns have been linked to several human diseases including cancer ([@herman1999hypermethylation]). The oxidized form, 5-hydroxymethylcytosine (5hmC), has recently gained attention as it is not only involved in gene regulation but also seems to play a major role in active and passive DNA demethylation. It is hypothesized that CpGs can traverse an iterative cycle of methylation and demethylation through oxidation and base excision repair ([@zhang2012thymine]). DNA methylation is commonly measured by using bisulfite genomic sequencing (BS-seq) during which C is converted to uracil ([@frommer1992genomic]) while both 5mC and 5hmC are read as Cs and can therefore not be discriminated (see Fig. \[fig:converror\]). As opposed to this, oxidative bisulfite sequencing (oxBS-seq) converts 5hmC to 5-formylcytosine (5fC) and conversion of the newly formed 5fC to uracil allows to discriminate between 5hmC and 5mC but not between C and 5hmC ([@booth2012quantitative]). Hence 5hmC levels must be inferred by a simultaneous estimation based on both BS-seq and oxBS-seq data. Standard BS-seq or oxBS-seq can only capture the modification state of one individual DNA strand at a time. To overcome this limitation hairpin BS-seq has been developed, which allows to determine the state of both cytosines of a CpG dyad. Thus, nine different possible states (pairs of the three possible states C, 5mC and 5hmC) can be implicitly measured ([@laird2004hairpin]). Moreover, while most cell types display relatively stable DNA methylation patterns, the dynamically changing gene expression program during mammalian development is accompanied by an alteration of methylation patterns. Given measurements at different times, (time-dependent) methylation efficiencies can be inferred and provide useful information about the mechanisms that control the developmental program ([@arand2012vivo]).
Here, we present airpin (xidative) bisulfite sequencing ime course nalyzer (H(O)TA) - a tool that accurately infers (hydroxy-)methylation levels and determines the efficiencies of the involved enzymes at a certain DNA locus. The procedure for estimating levels and efficiencies is based on the construction of two coupled Hidden Markov Models (HMMs) and gets as input time course measurements from hairpin BS-seq and oxBS-seq, respectively. Using the ML approach proposed in [@giehr2016], the evolution of the HMMs is determined by time-dependent methylation and (hydroxy-)methylation efficiencies and takes into account all relevant conversion errors (see dashed arrows in Fig. \[fig:converror\]).
![Conversion scheme for BS-seq and oxBS-seq. The first (last) line determines the hidden (observable) states. The conversion rates $c,$ $d$, $e$, and $f$ are taken into account by adjusting the emission probabilities of the models. []{data-label="fig:converror"}](convErrors.eps){width="40.00000%"}
H(O)TA Software
===============
H(O)TA has been developed in MATLAB and its execution requires the installation of the free Matlab runtime environment (MRE). The tool and the MRE can be downloaded as a single installation file available for Linux, MacOS, and Windows operating systems. As opposed to methods for single time point data, H(O)TA performs an analysis that considers the transient probability distribution over the set $\{u,m,h\}^2$ of nine hidden states of the two cytosines of a CpG dyad, where $u, m$ and $h$ describe C, 5mC, and 5hmC, respectively. Thus, besides the states $uu$ and $mm$, which correspond to the blue and red bars in the bar plots of the hidden states’ probabilities in Fig. \[fig:main\], the model considers hemimethylated sites (states $um$, $mu$, green bars) as well as fully hydroxylated sites (state $hh$) and hemihydroxylated sites (states $uh$, $hu$) and combinations of 5mC and 5hmC (states $mh$, $hm$), whose levels are given by orange bars and refined in detailed plots on the right of each bar plot in Fig. \[fig:main\]. The observable states reflect the possible outcomes of hairpin BS-seq and hairpin oxBS-seq, respectively, that is, $\{T,C\}^2$ (cf. last line in Fig. \[fig:converror\] and upper middle line plots). Users can provide BS-seq and oxBS-seq time course data. For each observation time point, estimates of the methylation and hydroxylation levels are computed, as well as, linear functions for the methylation (maintenance or de novo) and hydroxylation efficiencies, i.e., the probability of a methylation or a hydroxylation event between two cell divisions. In addition, an estimation is provided for the probability that no maintenance is performed when the current state is $mh$ or $hm$, which hints on the existence of a passive demethylation mechanism induced by hydroxylation. The user must specify the number of cell divisions between two observation time points and provide conversion errors of BS-seq and oxBS-seq (see dashed arrows in Fig. \[fig:converror\]). For all the estimated parameters confidence intervals are computed and a statistical test is carried out in order to verify certain hypotheses about the efficiencies. For a derivation of the likelihood and details about the optimization as well as the statistical validation of the results, we refer to ([@giehr2016]).
The graphical user interface of H(O)TA consists of two windows: a dialogue window for loading the input files of a DNA locus and running the analysis and the main window (Fig. \[fig:main\]) for visualizing the output. The tool can automatically aggregate data of different CpGs of a locus and compute average (hydroxy-)methylation levels as well as average efficiencies. In addition, the analysis can be performed for each CpG individually.
Users can provide three input .txt files. The first file contains BS-seq data, the second one oxBS-seq data and the third file contains the conversion errors of the two experiments as well as a string that describes how many cell divisions take place between two observation time points. Only the file with the BS-seq data is mandatory and the other two are optional. If only BS-seq data is given, then the tool will predict only the methylation levels of the region (merged with the unknown hydroxylation levels). For a detailed documentation of the input files we refer to the tool webpage.
![The main window of the graphical user interface of H(O)TA.\[fig:main\] ](mainWindow.eps){width="50.00000%"}
The main window has two subpanels. The right panel shows the output of the analysis either for the aggregated data or for each of the previously chosen CpG sites. The upper left and middle plots show the fit between the data and the model prediction for the observable states TT, TC, CT, CC for BS and oxBS experiments. The upper right plot presents the efficiencies of the enzymes responsible for maintenance methylation (dark red), de novo methylation (blue) and hydroxylation (yellow) as well as the total methylation (light red) on hemimethylated CpGs (see the tool webpage). The lower left plot shows the (hydroxy-)methylation levels of the current region and the lower right plot shows the exact distribution of the different hydroxylation states. In the upper left corner of the right panel the user can choose between the plots of different CpG sites (or the aggregated data). In the lower right corner there are several options for exporting the estimation results in a desirable format. In the left panel of the main window the (hydroxy-)methylation levels and the efficiencies of all individual CpGs are plotted such that they can be compared with each other and with the chosen plots in the right panel.
|
---
abstract: 'Torsional oscillator experiments show evidence of mass decoupling in solid $^4$He. This decoupling is amplitude dependent, suggesting a critical velocity for supersolidity. We observe similar behavior in the elastic shear modulus. By measuring the shear modulus over a wide frequency range, we can distinguish between an amplitude dependence which depends on velocity and one which depends on some other parameter like displacement. In contrast to the torsional oscillator behavior, the modulus depends on the magnitude of stress, not velocity. We interpret our results in terms of the motion of dislocations which are weakly pinned by $^3$He impurities but which break away when large stresses are applied.'
author:
- James Day
- Oleksandr Syshchenko
- John Beamish
bibliography:
- 'nonlinear.bib'
title: 'Non-linear Elastic Response in Solid Helium: critical velocity or strain?'
---
Helium is a uniquely quantum solid and recent torsional oscillator (TO) measurements[@Kim04-1941; @Kim06-115302; @Rittner07-175302; @Kondo07-695; @Penzev07-677; @Aoki07-015301; @Keiderling09-032040; @Hunt09-632] provide evidence for a supersolid phase in hcp $^4$He. At temperatures below about 200 mK the TO frequency increases, suggesting that some of the $^4$He decouples from the oscillator. This evidence of “non-classical rotational inertia” (NCRI) inspired searches for other unusual thermal or mechanical behavior in solid helium. Heat capacity measurements[@Lin07-449; @Lin09-125302] do show a small peak near the onset temperature of decoupling, supporting the idea of a phase transition in solid $^4$He. Mass flow is an obvious possible signature of supersolidity but experiments in this temperature range[@Day06-105304; @Sasaki06-1098; @Sasaki07-205302] show no pressure-induced flow through the solid (although recent measurements[@Ray08-235301; @Ray09-224302] at higher temperatures showed intriguing behavior).
We recently made low frequency measurements[@Day07-853; @Day09-214524] of the shear modulus of hcp $^4$He and found a large stiffening, with the same temperature dependence as the frequency changes seen in TO experiments. This modulus increase also had the same dependence on measurement amplitude and on $^3$He concentration and, like the TO decoupling, its onset was accompanied by a dissipation peak. It is clear that the shear stiffening and the TO decoupling are closely related. Subsequent experiments[@West09-598] with $^3$He showed similar elastic stiffening in the hcp phase below 0.4 K, but not in the bcc phase (the bcc phase exists only over a temperature range in $^4$He). TO measurements with hcp $^3$He did not, however, show any sign of a transition in the temperature range where the stiffening occurred, nor was a transition seen with bcc $^3$He. The stiffening appears to depend on crystal structure (appearing in the hcp but not the bcc phase) while the TO frequency and dissipation changes occur only for the bose solid, $^4$He.
Although it is clear that solid $^4$He shows unusual behavior below 200 mK, the interpretation in terms of supersolidity rests almost entirely upon torsional oscillator experiments. In addition to frequency changes which imply mass decoupling, two other features of the TO experiments are invoked as evidence of superflow. One is the “blocked annulus” experiment[@Kim04-1941; @Rittner08-155301] in which NCRI is greatly reduced by inserting a barrier into the flow path, thus indicating that long-range coherent flow is involved. The other is the reduction of the NCRI fraction when the oscillation amplitude exceeds a critical value[@Kim04-1941]. In analogy to superfluidity in liquid helium, this is interpreted in terms of a critical velocity v$_c$ (of order 10 $\mu$m/s), above which flow becomes dissipative. However, torsional oscillators are resonant devices and measurements made at a single frequency cannot distinguish an amplitude dependence which sets in at a critical velocity from one which begins at a critical displacement or a critical acceleration. A recent experiment[@Aoki07-015301] used a compound torsional oscillator which operated in two modes, allowing measurements to be made on the same solid $^4$He sample at two different frequencies (496 and 1173 Hz). The amplitude dependence scaled somewhat better with velocity than with acceleration or displacement, supporting the superflow interpretation of TO experiments. However, recent measurements[@Kojima09-private] in which one mode was driven at large amplitude while monitoring the low-amplitude response of the other mode gave unexpected results. The suppression of NCRI, as seen in the low amplitude mode, appeared to depend on the acceleration generated by the high amplitude mode, rather than on the velocity. To settle the question of whether the amplitude dependence seen in torsional oscillators reflects a critical velocity for superflow, measurements over a wider frequency range are needed.
We have made direct measurements[@Day07-853] of the amplitude dependence of the shear modulus, $\mu$, of hcp $^4$He in a narrow gap of thickness D. An AC voltage, V, with angular frequency $\omega$ = 2$\pi$f, is applied to a shear transducer (with piezoelectric coefficient d$_{15}$) to generate a displacement $\delta$x=d$_{15}V$ at its surface. This produces a quasi-static shear strain in the helium, $\epsilon$ = $\delta$x/D. The resulting stress, $\sigma$, generates a charge, q, and thus a current, I = $\omega$q, in a second transducer on the opposite side of the gap. The shear modulus $\mu$=$\sigma$/$\epsilon$ is then proportional to I/fV. In contrast to torsional oscillator measurements, this technique is non-resonant and so we could measure the shear modulus over a wide frequency range, from a few Hz (a limit set by preamplifier noise) to a maximum of 2500 Hz (due to interference from the first acoustic resonance in our cell, around 8 kHz). The technique is very sensitive, allowing us to make modulus measurements at strains as low as $\epsilon$ $\approx$ 10$^{-9}$, corresponding to stresses $\sigma$ $\approx$ 0.02 Pa. The drive voltage can be increased substantially without heating the sample, so measurements can be made at strains up to $\epsilon$ $\approx$ 10$^{-5}$, allowing us to study the amplitude dependence, including hysteretic effects, at low temperatures.
Figure 1 shows the temperature dependence of the normalized shear modulus $\mu$/$\mu$$_o$ for an hcp $^4$He sample at a pressure of 38 bar and a frequency of 2000 Hz. The crystal was grown from standard isotopic purity $^4$He (containing about 0.3 ppm $^3$He) using the blocked capillary method. A piezoelectric shear stack[@piceramic] was used to generate strains in a narrow gap (D = 500 $\mu$m) between it and a detecting transducer. Other experimental details are the same as in refs. 16 to 18. The curves in Fig. 1 correspond to different transducer drive voltages, i.e. to different strains, and were measured during cooling from a temperature of 0.7 K. The shear modulus increases at low temperature, with $\Delta$$\mu$/$\mu$$_o$ $\approx$ 17$\%$ at the lowest amplitude. Similar stiffening was seen in all hcp $^4$He crystals[@Day07-853; @Day09-214524], although the onset temperature was lower in crystals of higher isotopic purity and the magnitude of the modulus change varied by a factor of about 2 from sample to sample (the TO NCRI varies much more, by a factor of 1000, although its temperature dependence is always essentially the same[@West09-598]). The amplitude dependence of the shear modulus stiffening is essentially the same as that of the TO NCRI. At the lowest drive voltages and temperatures, both are independent of amplitude but they decrease at high amplitudes. The onset of stiffening shifts to lower temperatures at high amplitudes, as does the onset of TO decoupling. However, in the case of elastic measurements, it is more natural to think of this behavior in terms of a critical stress (proportional to the strain, i.e. to transducer displacement), rather than a critical velocity.
![Temperature and amplitude dependence of the shear modulus in a solid $^4$He sample at 38 bar, grown by the blocked capillary method. The modulus was measured at 2000 Hz for transducer drive voltages (peak to peak) from 10 mV to 3 V. Values are normalized by $\mu$$_o$ the low temperature value at the lowest drive voltage.[]{data-label="fig:Figure1.EPS"}](Figure1.EPS){width="\linewidth"}
The amplitude dependence of the shear modulus is shown in more detail in Fig. 2. Open circles show the modulus at 48 mK, taken from the temperature sweeps (i.e. the points marked by open circles in Fig. 1). The solid circles are the modulus measured when the drive voltage was reduced at fixed temperature (48 mK), after cooling from high temperature at the highest drive amplitude (3 V$_{pp}$). Figure 2 also shows the corresponding amplitude dependence at temperatures well above the shear modulus anomaly (open squares are data at 700 mK from the temperature sweeps of Fig. 1; solid squares are from amplitude sweeps at 800 mK). The data taken with the two protocols agree very well. The critical drive voltage (where the modulus becomes amplitude dependent) is around 30 mV$_{pp}$ (corresponding to $\sigma$ $\approx$ 4x10$^{-8}$, $\sigma$ $\approx$ 0.8 Pa) and at the highest drive levels (3 V$_{pp}$ corresponding to $\sigma$ $\approx$ 80 Pa) the stiffening is almost completely suppressed. The amplitude dependence of the modulus closely resembles that of the NCRI in TO experiments, behavior which is attributed to a superflow critical velocity (typically around 10 $\mu$m/s). However, even in TO measurements there are inertial stresses in the helium, produced by its acceleration, which could lead to the observed amplitude dependence.
![Low and high temperature shear modulus for the crystal of Fig. 1, as a function of drive voltage (bottom axis) or strain (top axis). The open symbols are taken from temperature sweeps at different drive levels (the data of Fig. 1). The solid symbols are taken while decreasing the drive voltage at fixed temperature.[]{data-label="fig:Figure2.EPS"}](Figure2.EPS){width="\linewidth"}
A number of experiments[@Aoki07-015301; @Rittner08-155301; @Shimizu09; @Clark08-184531] have shown that the TO amplitude dependence is hysteretic at low temperatures. If a sample is cooled at high oscillation amplitude, the apparent NCRI is small. When the amplitude is reduced at low temperature, the TO frequency (NCRI) rises, becoming constant below some critical amplitude. When the drive is then increased at low temperature, the NCRI does not begin to decrease at the critical amplitude - it remains essentially constant at substantially larger drives. This hysteresis between data taken while decreasing and increasing the drive amplitude disappears at temperatures above about 70 mK.
Figure 3 shows the corresponding hysteresis in the shear modulus. At 120 mK the maximum stiffening is about half as large as at 36 mK and already depends on amplitude at the lowest strains shown. The modulus measured when the amplitude is reduced (open circles) and when it is subsequently increased (solid circles) agree. Hysteresis appears when the sample is cooled below 60 mK and is nearly temperature independent below 45 mK. Figure 3 shows this hysteresis at 36 mK. The sample was cooled from high temperature while driving at high amplitude (3 V$_{pp}$). The amplitude was then lowered at 36 mK (open circles) which resulted in a shear modulus increase $\Delta$$\mu$/$\mu$$_o$ of about 15$\%$. When the amplitude was then raised (solid circles), the modulus remained constant to much higher amplitude, the same behavior seen for the NCRI in TO experiments. At very high amplitudes (above about 1 V$_{pp}$, corresponding to $\epsilon \approx$ 10$^{-6}$, $\sigma \approx$ 20 Pa) the modulus decreased, nearly closing the hysteresis loop. After each change we waited 2 minutes for the modulus to stabilize at the new amplitude. The only region where we observed further time dependence was while increasing the amplitude at drive levels above 1 V$_{pp}$. The modulus decrease was sharper when we waited longer at each point. In acoustic resonance measurements[@Day09-214524], we found that even larger stresses ($\sigma \approx$700 Pa) produced irreversible changes which only disappeared after annealing above 0.5 K.
![Hysteresis between the shear modulus measured while decreasing and while increasing the drive amplitude at low temperature (36 mK). At 120 mK the amplitude dependence is reversible.[]{data-label="fig:Figure3.EPS"}](Figure3.EPS){width="\linewidth"}
The only obvious mechanism which can produce shear modulus changes as large as those shown in Figs. 1 to 3 involves the motion of dislocations[@Nowick72]. At low temperatures, dislocations are pinned by $^3$He impurities and the intrinsic modulus is measured[@Day09-214524]. As the temperature is raised, $^3$He impurities thermally unbind from the dislocations, allowing them to move and reducing the modulus. At high amplitudes, elastic stresses can also tear the dislocations away from the impurities. The hysteresis seen in Fig. 3 can be understood if, when a crystal is cooled at high strain amplitudes, the rapid motion of dislocations prevents $^3$He atoms from attaching to them. When the drive is reduced at low temperatures, impurities can bind, thus pinning the dislocations and increasing the modulus. Once the $^3$He impurities bind, the pinning length of dislocations is smaller and larger stresses are required to unpin them so the modulus retains its intrinsic value to much higher amplitudes. The critical amplitude for this “stress-induced breakaway” can be estimated[@Iwasa80-1722; @Paalanen81-664] if the dislocation length and impurity binding energy are known. In single crystals of helium, a typical dislocation network pinning length[@Iwasa79-1119] is (L $\sim$ 5 $\mu$m), and dislocations would break away from a $^3$He impurity at strain $\epsilon \approx$3x10$^{-7}$. Our crystals are expected to have higher dislocation densities and smaller network lengths, so breakaway would occur at higher strains as the amplitude is increased. Measurements with different $^3$He concentrations would be useful to confirm that the amplitude dependence is due to this mechanism.
This interpretation of the shear modulus behavior involves elastic stress (which is proportional to strain) rather than velocity, and so is at odds with the interpretation of the TO amplitude dependence in terms of a superfluid-like critical velocity. Since we can make modulus measurements over a wide frequency range, we can unambiguously distinguish between an amplitude dependence which scales with stress (or strain) and one which depends on velocity. Figure 4 shows the modulus at 18 mK, measured at three different frequencies (2000, 200 and 20 Hz) as the drive amplitude was reduced from its maximum value. In Fig. 4a the modulus is plotted vs. shear strain $\epsilon$ (calibrated using the low temperature piezoelectric coefficient of the shear stack, d$_{15}$=1.25 nm/V) and in Fig. 4b the same data is plotted versus the corresponding velocities (v=$\omega$$\epsilon$D). The amplitude dependence scales much better with strain than with velocity (and the scaling with acceleration is even less satisfactory). The critical strain appears be slightly larger at lower frequency.
![Scaling of the shear modulus’ amplitude dependence with (a) strain and (b) velocity for three different frequencies: 20 Hz (triangles), 200 Hz (squares) and 2000 Hz (circles).[]{data-label="fig:Figure4.EPS"}](Figure4.EPS){width="\linewidth"}
It is clear that the amplitude dependence of the shear modulus is most closely associated with stress (or strain) amplitude, rather than with a superfluid-like critical velocity. The many similarities to the TO behavior (e.g. the dependence on temperature, $^3$He concentration and frequency, the amplitude dependence and its hysteresis) suggest that the apparent velocity dependence of the NCRI may have a similar origin, e.g. in inertial stresses which exceed the critical value for the shear modulus. However, estimates of the inertial stress corresponding to TO critical velocities give values significantly lower than the critical stress for the shear modulus. For an annular TO geometry, the maximum inertial stress can be estimated as $\sigma$=$\rho$t$\omega$v/2, where $\rho$ is the helium density, t is the width of the annular channel, $\omega$ is the angular frequency of the TO and v is the oscillation velocity. For the oscillator of ref. 1, we estimate $\sigma \approx$ 0.002 to 0.015 Pa at the apparent critical velocities of 5 to 38 $\mu$m/s. Measurements in an open cylindrical TO[@Clark08-184531] show amplitude dependence at velocities corresponding to inertial stresses below 0.01 Pa, also much smaller than the stresses at which we observe amplitude dependence in the shear modulus. There is also other evidence that the TO frequency changes and dissipation are not just mechanical consequences of the modulus changes. First, the apparent NCRI is too large to be explained simply by mechanical stiffening of the torsional oscillator[@West09-598; @Clark08-184531]. Secondly, comparable modulus changes in hcp $^3$He are not reflected in the corresponding TO behavior[@West09-598]. The existence of a critical velocity for superflow in solid helium can only be definitively shown if TO measurements can be made over a wide range of frequency and/or TO geometries.
This work was supported by the Natural Sciences and Engineering Research Council of Canada.
|
---
abstract: |
In this paper, we consider the computation of the degree of the Dieudonné determinant of a linear symbolic matrix $
A = A_0 + A_1 x_1 + \cdots + A_m x_m,
$ where each $A_i$ is an $n \times n$ polynomial matrix over $\KK[t]$ and $x_1,x_2,\ldots,x_m$ are pairwise “non-commutative" variables. This quantity is regarded as a weighted generalization of the non-commutative rank (nc-rank) of a linear symbolic matrix, and its computation is shown to be a generalization of several basic combinatorial optimization problems, such as weighted bipartite matching and weighted linear matroid intersection problems. Based on the work on nc-rank by Fortin and Rautenauer (2004), and Ivanyos, Qiao, and Subrahmanyam (2018), we develop a framework to compute the degree of the Dieudonné determinant of a linear symbolic matrix. We show that the deg-det computation reduces to a discrete convex optimization problem on the Euclidean building for ${\rm SL}(\KK(t)^n)$. To deal with this optimization problem, we introduce a class of discrete convex functions on the building. This class is a natural generalization of L-convex functions in discrete convex analysis (DCA). We develop a DCA-oriented algorithm (steepest descent algorithm) to compute the degree of determinants. Our algorithm works with matrix computation on $\KK$, and uses a subroutine to compute a certificate vector subspace for the nc-rank, where the number of calls of the subroutine is sharply estimated. Our algorithm enhances some classical combinatorial optimization algorithms with new insights, and is also understood as a variant of the combinatorial relaxation algorithm, which was developed earlier by Murota for computing the degree of the (ordinary) determinant.
author:
- |
Hiroshi HIRAI\
Department of Mathematical Informatics,\
Graduate School of Information Science and Technology,\
The University of Tokyo, Tokyo, 113-8656, Japan.\
`[email protected]`
title: Computing the degree of determinants via discrete convex optimization on Euclidean buildings
---
Keywords: non-commutative rank, Dieudonné determinant, skew field, discrete convex analysis, mixed matrix, combinatorial relaxation algorithm, submodular function, L-convex function, Euclidean building, uniform modular lattice.
Introduction
============
A [*linear symbolic matrix*]{} or [*linear matrix*]{} $A$ is a matrix each of whose entries is a linear (affine) function in variables $x_1,x_2,\ldots, x_m$. Namely $A$ admits the form of $$\label{eqn:A}
A = A_0 + A_1 x_1 + \cdots + A_m x_m,$$ where each $A_i$ is a matrix over a field $\mathbb{K}$ and $A$ is viewed as a matrix over the polynomial ring $\mathbb{K}[x_1,x_2,\ldots,x_m]$ or the rational function field $\mathbb{K}(x_1,x_2,\ldots,x_m)$. In this paper, we address the symbolic rank computation of linear matrices and its generalization. This problem is a fundamental problem in discrete mathematics and computer science. In a classical paper [@Edmonds67] in combinatorial optimization, Edmonds noticed that the maximum matching number of a bipartite graph is represented as the rank of such a matrix: For each edge $e = ij$, introduce a variable $x_e$ and a matrix $E_{e}$ having $1$ for the $(i,j)$-entry and zero for the others. Then the rank of linear matrix $A = \sum_{e} E_e x_e$ (with $A_0 = 0$ in (\[eqn:A\])) is equal to the maximum matching number of the graph. In the same paper, Edmonds asked for a polynomial time algorithm to compute the rank of a general linear matrix. Clearly the rank computation is easy if we substitute an actual number for each variable. According to random substitution, Lovász [@Lovasz79] developed a randomized polynomial time algorithm to compute the rank of linear matrices. Designing a deterministic polynomial time algorithm is a big challenge in theoretical computer science, since it would lead to a breakthrough in circuit complexity theory [@KabanetsImpagliazzo04]. Currently, such deterministic algorithms are known for very restricted classes of linear matrices. Most of them are connected to polynomially-solvable combinatorial optimization problems, such as matching, matroid intersection, and their generalizations; see [@Lovasz89]. The rank of linear matrices also plays important roles in engineering applications. A [*mixed matrix*]{} due to Murota and Iri [@MurotaIri85] is a linear matrix including constant matrix $A_0$ in the above bipartite graph example. Mixed matrices are applied to the analysis of a linear control system including physical parameters (such as positions, temperatures) that cannot be measured exactly, where these parameters are modeled as variables $x_i$. Then the rank describes several fundamental characteristics (such as the controllability) of the system. See [@MurotaMatrix] for the theory of mixed matrices and its applications. The (infinitesimal) [*rigidity*]{} of a bar-and-joint structure is also characterized by the rank of a linear matrix, called the [*rigidity matrix*]{}, where variables $x_i$ represent “generic" positions of the structure. Polynomial-time rank computation is known for mixed matrices (see [@MurotaMatrix]) and some classes of rigidity matrices (see e.g., [@Lovasz89]).
Recently there are significant developments in the rank computation of linear matrices. In the above paragraph, we assumed that $A$ is a matrix over polynomial ring $\KK[x_1,x_2,\ldots, x_m]$ with (commutative) indeterminates $x_1,x_2,\ldots,x_m$, and the rank is considered in rational function field $\KK(x_1,x_2,\ldots, x_m)$. However $A$ can also be viewed as a matrix over the free ring $\KK\langle x_1,x_2,\ldots,x_m \rangle$ generated by $x_1,x_2,\ldots,x_m$, where variables are supposed to be pairwise non-commutative, i.e., $x_i x_j \neq x_j x_i$. It is shown by Amitsur [@Amitsur1966] that there is a non-commutative analogue $\KK(\langle x_1,x_2,\ldots,x_m \rangle)$ of the rational function field, called the [*free skew field*]{}, to which $\KK\langle x_1,x_2,\ldots,x_m \rangle$ is embedded. Now we can define the rank of $A$ over this skew field $\KK(\langle x_1,x_2,\ldots,x_m \rangle)$. This rank concept of $A$ is called the [*non-commutative rank*]{} ([*nc-rank*]{}) of $A$. Fortin and Rautenauer [@FortinReutenauer04] proved a formula of the nc-rank, which says that the nc-rank is equal to the optimal value of an optimization problem over the lattice of all vector subspaces of $\KK^n$. Garg, Gurvits, Oliveira, and Wigderson [@GGOW15] proved that the nc-rank of $A$ can be computed in deterministic polynomial time if $\KK = \QQ$. They showed that Gurvits’ [*operator scaling algorithm*]{} [@Gurvits04], which was earlier developed for the rank computation of a special class (called [*Edmonds-Rado class*]{}) of linear matrices, can be a polynomial time algorithm for the nc-rank. Ivanyos, Qiao, and Subrahmanyam [@IQS15a; @IQS15b] developed a polynomial time algorithm to compute the nc-rank over an arbitrary field. Their algorithm is a vector-space analogue of the augmenting path algorithm for bipartite matching, and utilizes an invariant theoretic result by Derksen and Makam [@DerksenMakam17] for the complexity estimate. Independent of this line of research, Hamada and Hirai [@HamadaHirai17] investigated (a variant of) the optimization problem for the nc-rank which they called the [*maximum vanishing subspace problem (MVSP)*]{}. Their motivation comes from a canonical form of a matrix under block-restricted transformations [@ItoIwataMurota94; @IwataMurota95]. They also developed a polynomial time algorithm for the nc-rank based on the fact that MVSP is viewed as a submodular function optimization on the modular lattice of all vector subspaces of $\KK^n$.
In this paper, we consider a “weighted analogue" of the nc-rank. Our principal motivation is to capture the weighted versions of combinatorial optimization problems from the non-commutative points of view. Consider, for example, the weighted matching problem on a bipartite graph, where two color classes are supposed to have the same cardinality, and each edge $e$ has (integer) weight $c_e$. By introducing new indeterminate $t$, modify the above bipartite-graph linear matrix $A$ as $A := \sum_{e} t^{c_e} x_{e} E_e$. Then the maximum weight of a perfect matching is equal to the degree of the determinant of $A$ with respect to $t$. This well-known example suggests that such a weighted analogue is the degree of the determinant. This motivates us to consider the degrees of the determinants of linear matrices in the non-commutative setting. The main contribution of this paper is to develop a computational framework for the degrees of the determinants in the non-commutative setting, which captures some of classical weighted combinatorial optimization problems. Our results and their features are summarized as follows:
- For a determinant concept for matrices over skew field $\FF$, we consider the [*Dieudonné determinant*]{} [@Dieudonne1943]. Although the value of the Dieudonné determinant is no longer an element of the ground field, in the skew field $\FF(t)$ of rational functions ([*Ore quotient ring*]{} of polynomial ring $\FF[t]$), its degree is well-defined, see e.g., [@Taelman06]. In this paper, we use the notation ${\mathop{\rm Det} }$ for the Dieudonné determinant, whereas $\det$ is used for the ordinary determinant.
Our target is the degree $\deg {\mathop{\rm Det} }A$ of the Dieudonné determinant ${\mathop{\rm Det} }A$ of a linear matrix $A = A_0+ A_1 x_1 + \cdots + A_m x_m$, where each $A_i$ is a square polynomial matrix over $\KK[t]$ and $A$ is viewed as a matrix over the rational function skew field $\FF(t)$ of the free skew field $\FF = \KK(\langle x_1,x_2,\ldots,x_m \rangle)$.
- We establish a duality theorem for $\deg {\mathop{\rm Det} }$, which is a natural generalization of the Fortin-Rautenauer formula for the nc-rank. In fact, a weak duality relation was previously observed by Murota [@Murota95_SICOMP] for $\deg \det$, and is now a strong duality for $\deg {\mathop{\rm Det} }$. Analogously to the Fortin-Rautenauer formula saying that the nc-rank is equal to the optimal value of an optimization problem (MVSP) over the lattice of all vector subspaces of $\KK^n$, our formula says that $\deg {\mathop{\rm Det} }$ is equal to the optimal value of an optimization problem over the lattice of all full-rank $\KK(t)^-$-submodules of $\KK(t)^n$, where $\KK(t)^-$ is the valuation ring of $\KK(t)$ with valuation $\deg$. In the literature of group theory, this lattice structure is known as the [*Euclidean building*]{} for ${\rm SL}(\KK(t)^n)$ [@BruhatTits], whereas the lattice of all vector subspaces is the [*spherical building*]{} for ${\rm SL}(\KK^n)$ [@Tits].
- We approach this optimization problem on the Euclidean building from [*Discrete Convex Analysis (DCA)*]{} [@MurotaBook] with its recent generalization [@HH17survey; @HH16L-convex]. Although DCA was originally a theory of discrete convex functions on $\ZZ^n$ that generalizes matroids and submodular functions, recent study [@HH17survey; @HH16L-convex] shows that DCA-oriented concepts and algorithm design are effective and useful for optimization problems on certain discrete structures beyond $\ZZ^n$. [*L-convexity*]{}, which is one of the central concepts of DCA, is particularly important for us. L-convex functions are generalization of submodular functions and arise naturally from representative combinatorial optimization problems such as minimum-cost network flow and weighted bipartite matching. L-convex functions admit a simple minimization algorithm, called the [*steepest descent algorithm (SDA)*]{}, on which our algorithm for deg Det will be built. We introduce an analogue of an L-convex function on the building. The previous work [@HH17survey; @HH16L-convex] introduced L-convexity on Euclidean buildings of type C, whereas our building here is of type A. We show that the established formula of $\deg {\mathop{\rm Det} }$ gives rise to an L-convex function, analogously to the submodular function in MVSP for the nc-rank. Consequently $\deg {\mathop{\rm Det} }$ is computed via an L-convex function minimization on the Euclidean building.
- We develop an algorithm to compute $\deg {\mathop{\rm Det} }A$ for linear polynomial matrices $A$ over $\KK[t]$. Our algorithm requires a subroutine to solve MVSP over $\KK$, and is described in terms of matrix computation over $\KK$. However it can be viewed as the steepest descent algorithm (SDA) applied to the L-convex function on the Euclidean building. Geometrically, SDA traces the 1-skeleton of the building with decreasing the value of the L-convex function. Each move is done by solving an optimization problem on the spherical building associated with the local structure of the Euclidean building. This local problem coincides with MVSP. By utilizing the recent analysis on SDA [@MurotaShioura14], we show that the number of the moves is sharply estimated by the [*Smith-McMillan form*]{} of $A$.
Our algorithm can also be interpreted as a variant of the [*combinatorial relaxation algorithm*]{}, which was developed earlier for $\deg \det$ of matrices (without variables) by Murota [@Murota90_SICOMP; @Murota95_SICOMP] and was further extended to mixed polynomial matrices by Iwata and Takamatsu [@IwataTakamatsu13]; see [@MurotaMatrix Section 7.1] and recent work [@IwataOkiTakamatsu17]. This interpretation sheds building-theoretic insights on the combinatorial relaxation algorithm.
- We study a class of linear matrices $A$ for which $\deg \det A = \deg {\mathop{\rm Det} }A$ holds. In the case of the nc-rank, it is known from the results in [@IvanyosKarpinskiOiaoSantha15; @Lovasz79] that if each $A_i$ other than $A_0$ is a rank-1 matrix over $\KK$, then it holds ${\mathop{\rm rank} }A = {\mathop{\rm nc\mbox{-}rank} }A$. We show a natural extension: if each $A_i$ other than $A_0$ is a rank-1 matrix over $\KK(t)$, then it holds $\deg \det A = \deg {\mathop{\rm Det} }A$. This property implies that some of classical combinatorial optimization problems, represented as $\deg \det$, fall into our framework of $\deg {\mathop{\rm Det} }$. Examples include weighted bipartite matching and weighted linear matroid intersection. In these examples, the optimal value is interpreted as $\deg \det$ as well as $\deg {\mathop{\rm Det} }$. A [*mixed polynomial matrix*]{} [@MurotaMatrix] is also such an example. We explain how our SDA framework works for these examples, and discuss connections to some of classic algorithms, such as the Hungarian method, the matroid greedy algorithm, and the matroid intersection algorithm by Lawler [@Lawler75] and Frank [@Frank81]. For mixed polynomial matrices, our framework brings a new algorithm, which is faster than the previous one [@IwataOkiTakamatsu17; @IwataTakamatsu13] in terms of time complexity. One of motivating applications of mixed polynomial matrices is the analysis of [*differential algebraic equations (DAE)*]{}; see [@MurotaBook Chapter 6]. We present a possible application of our result to the mixed-matrix DAE analysis.
The rest of this paper is organized as follows. In Section \[sec:skewfield\], we summarize basic facts on skew field, nc-rank, Fortin-Rautenauer formula, MVSP, and Dieudonné determinant. In Section \[sec:L-convex\], we introduce L-convex functions on Euclidean buildings and show their basic properties. Instead of dealing with Euclidean buildings in the usual axiom system, we utilize an elementary lattice-theoretic equivalent concept, called [*uniform modular lattices*]{} [@HH18a]. This class of modular lattices admits the L-convexity concept in a straightforward way. In Section \[sec:computing\], we establish a formula for the degree of the Dieudonné determinant of a linear matrix and present an algorithm. In Section \[sec:rank-1\], we study linear matrices with rank-1 summands.
Closing the introduction, let us mention a recent work by Kotta, Belikov, Halás and Leibak [@KottaBelikovHalasLeibak2017] in control theory. They showed that the order of the minimal state-space realization of a (linearized) nonlinear control system is described by the degree of the Dieudonné determinant of the non-commutative polynomial matrix associated with the system, where the matrix involves elements in the skew polynomial ring of non-commuting variables $t,\delta$. This result generalizes the basic fact on the degrees of the (ordinary) determinants in linear time-invariant control systems; see e.g., [@Kailath80]. They gave a primitive algorithm to compute $\deg {\mathop{\rm Det} }$ based on (symbolic) Gaussian elimination. It would be an interesting research direction to extend our framework to deal with this type of deg-Det computation.
Skew field {#sec:skewfield}
==========
A [*skew field*]{} (or [*division ring*]{}) is a ring $\FF$ such that every nonzero element $x \in \FF$ has inverse element $x^{-1} \in \FF$ with $x^{-1}x = x x^{-1} = 1$. The product $\FF^n$ of $\FF$ will be treated as a right $\FF$-vector space of column vectors as well as a left $\FF$-vector space of row vectors; which one we suppose will be clear in the context. The set of all $n \times n'$ matrices over $\FF$ is denoted by $\FF^{n \times n'}$. The [*rank*]{} of matrix $A \in \FF^{n \times n'}$ is defined as the dimension of the right $\FF$-vector space spanned by columns of $A$, which is equal to the dimension of the left $\FF$-vector space spanned by rows of $A$. Let $\ker_{\rm R} A$ denote the right kernel $\{x \in {\FF}^{n'} \mid Ax = {\bf 0}\}$, and let $\ker_{\rm L} A$ denote the left kernel $\{x \in {\FF}^n \mid x A = {\bf 0}\}$. Then the rank of $A \in \FF^{n \times n'}$ is equal to $n - \dim \ker_{\rm L} A = n' - \dim \ker_{\rm R} A$. A square matrix $A \in \FF^{n \times n}$ is called [*nonsingular*]{} if its rank is equal to $n$, or equivalently if it has the inverse, which is denoted by $A^{-1}$, i.e., $A A^{-1} = A^{-1} A = I$. These properties are easily seen from the Bruhat normal form of $A$; see Lemma \[lem:Bruhat\] in Section \[subsec:Det\]. An $n \times n'$ matrix $A \in \FF^{n \times n'}$ is viewed as a map $\FF^n \times \FF^{n'} \to \FF$ by $$\label{eqn:bilinear}
A(x,y) := x A y \quad (x \in \FF^n, y \in \FF^{n'}).$$ Then $A$ is bilinear in the sense that $A(\alpha x +\alpha' x', y)
= \alpha A(x, y) + \alpha' A(x', y)$ and $A(x, y\beta + y'\beta') = A(x,y)\beta + A(x,y')\beta'$. Conversely, any bilinear map on the product of a left $\FF$-vector space $U$ and a right $\FF$-vector space $V$ is identified with a matrix over $\FF$ by choosing bases of $U$ and $V$. Let ${\cal S}_{\rm R}(\FF^n)$ and ${\cal S}_{\rm L}(\FF^n)$ denote the families of all right and left $\FF$-vector subspaces of $\FF^n$, respectively. If $\FF$ is commutative, then $\rm R$ and $\rm L$ are omitted, such as ${\cal S}(\FF^n)$.
Free skew field, nc-rank, and MVSP
----------------------------------
Let $\KK$ be a field. Let $\KK[x_1,x_2,\ldots,x_m]$ and $\KK(x_1,x_2,\ldots,x_m)$ denote the ring of polynomials and the field of rational functions with variables $x_1,x_2,\ldots,x_m$, respectively, where variables are supposed to commute each other, i.e., $x_i x_j = x_j x_i$. Let $\KK \langle x_1,x_2,\ldots,x_m \rangle$ be the free ring generated by pairwise non-commutative variables $x_1,x_2,\ldots,x_m$ over $\KK$. It is known that the free ring $\KK \langle x_1,x_2,\ldots,x_m \rangle$ is embedded to the universal skew field of fractions, called the [*free skew field*]{}, which is denoted by $\KK (\langle x_1,x_2,\ldots,x_m \rangle)$, or simply, by $\KK(\langle x \rangle)$. Elements of $\KK (\langle x_1,x_2,\ldots,x_m \rangle)$ are equivalence classes of all rational expressions constructed from $x_i$, $x_i^{-1}$, and elements of $\KK$, under an equivalence relation obtained by substitutions of nonsingular matrices for variables $x_i$; see [@Amitsur1966; @CohnAlgebra3; @CohnSkewField] for details. We do not go into the detailed construction of $\KK (\langle x_1,x_2,\ldots,x_m \rangle)$.
As mentioned in the introduction, a [*linear symbolic matrix*]{} or [*linear matrix*]{} on $\KK$ is a matrix of form $A = A_0 + A_1 x_1 + A_2 x_2 + \cdots + A_m x_m$, where each summand $A_i$ is a matrix over $\KK$. A linear matrix is viewed as a matrix over $\KK(x_1,x_2,\ldots,x_m)$ as well as over $\KK(\langle x_1,x_2,\ldots,x_m \rangle)$. The (commutative) rank of $A$ is defined as the rank of $A$ regarded as a matrix over $\KK(x_1,x_2,\ldots,x_m)$. The [*non-commutative rank*]{} ([*nc-rank*]{}) of $A$, denoted by nc-${\mathop{\rm rank} }A$, is defined as the rank of $A$ regarded as a matrix over $\KK(\langle x_1,x_2,\ldots,x_m \rangle)$. The nc-rank is not less than the (commutative) rank.
Let $A = A_0 + A_1 x_1 + A_2 x_2 + \cdots + A_m x_m$ be a linear $n \times n'$ matrix. We consider an upper bound of nc-${\mathop{\rm rank} }A$. For nonsingular matrices $S \in \KK^{n \times n},T \in \KK^{n' \times n'}$ (not containing variables), if $SAT$ has a zero submatrix of $r$ rows and $s$ columns, then nc-${\mathop{\rm rank} }A$ is at most $n + n' - r - s$. This gives rise to the following optimization problem (MVSP): $$\begin{aligned}
{\rm MVSP}: \quad {\rm Max.} && r + s \\
{\rm s.t.} && \mbox{$SAT$ has a zero submatrix of $r$ rows and $s$ columns},\\
&& S \in \KK^{n \times n}, T \in \KK^{n' \times n'}: \mbox{nonsingular}.\end{aligned}$$ This upper bound was observed by Lovász [@Lovasz89] for the usual rank. The name MVSP becomes clear below. Fortin and Reutenauer [@FortinReutenauer04] showed that this upper bound is actually tight for the nc-rank.
\[thm:FortinReutenauer\] For a linear $n \times n'$ matrix $A = A_0 + A_1 x_1 + A_2 x_2 + \cdots + A_m x_m$, [nc]{}-${\mathop{\rm rank} }A$ is equal to $n+n'$ minus the optimal value of MVSP.
As mentioned in the introduction, Garg, Gurvits, Oliveira, and Wigderson [@GGOW15] showed that nc-${\mathop{\rm rank} }A$ can be computed in polynomial time when $\KK = \QQ$. One drawback of this algorithm is not to output optimal matrices $S,T$ in MVSP. Such an algorithm was developed by Ivanyos, Qiao, and Subrahmanyam [@IQS15a; @IQS15b] for an arbitrary field.
MVSP can be solved in polynomial time.
Hamada and Hirai [@HamadaHirai17] also gave a “polynomial time" algorithm based on submodular optimization (see Lemma \[lem:submo\]), though the bit-length required in the algorithm is not bounded if $\KK = \QQ$. Note that their formulation is slightly different from that presented here; see Section \[subsec:HH\] in Appendix for the relation.
As [@HamadaHirai17; @HH16DM] did, MVSP is formulated as the following optimization problem ([*maximum vanishing subspace problem*]{}) of the space of all vector subspaces of $\KK^n$: $$\begin{aligned}
{\rm MVSP}: \quad {\rm Max.} && \dim X + \dim Y \nonumber \\
{\rm s.t.} && A_i (X,Y) = \{0\} \quad (i=0,1,2,\ldots,m),\nonumber \\
&& X \in {\cal S}(\KK^n), Y \in {\cal S}(\KK^{n'}). \label{eqn:MVSP}\end{aligned}$$ Recall the notation ${\cal S}(\KK^n)$ for all vector subspaces of $\KK^n$, and that each matrix $A_i$ is viewed as a bilinear form $\KK^n \times \KK^{n'} \to \KK$ as in (\[eqn:bilinear\]). We call an optimal $(X,Y)$ a [*maximum vanishing subspace*]{} or an [*mv-subspace*]{}. Notice that we can eliminate variable $X$ by substituting $X = A(Y)^{\bot}$ ($=$ the orthogonal complement of the image of $Y$ by $A$), and obtain the formulation of [@GGOW15; @IQS15a; @IQS15b] — the [*minimum shrunk subspace problem*]{}.
The nc-${\mathop{\rm rank} }A$ is also obtained via MVSP over $\KK(\langle x \rangle)$: $$\begin{aligned}
\overline{\rm MVSP}: \quad {\rm Max.} && \dim X + \dim Y \\
{\rm s.t.} && A (X,Y) = \{0\},\\
&& X \in {\cal S}_{\rm L}(\KK(\langle x\rangle)^n),
Y \in {\cal S}_{\rm R}(\KK(\langle x\rangle)^{n'}).\end{aligned}$$ Indeed, $\overline{\rm MVSP}$ has obvious optimal solutions $( \KK(\langle x\rangle)^n, \ker_{\rm R} A)$ and $(\ker_{\rm L} A, \KK(\langle x\rangle)^{n'})$. Notice that $\KK(\langle x \rangle)^n$ is a scalar extension of $\KK^n$, i.e., $\KK(\langle x \rangle)^n \simeq \KK(\langle x \rangle) \otimes \KK^n$. Therefore a feasible solution in MVSP is embedded into a feasible solution in $\overline{\rm MVSP}$ by $(X,Y) \mapsto (\KK(\langle x \rangle)
\otimes X, Y\otimes \KK(\langle x \rangle))$. Then Theorem \[thm:FortinReutenauer\] also says that MVSP is an exact inner approximation of $\overline{\rm MVSP}$:
\[lem:innerapprox\] Any mv-subspace of MVSP is also an mv-subspace of $\overline{\mbox{MVSP}}$.
Dieudonné determinant {#subsec:Det}
---------------------
Here we introduce a determinant concept for matrices over skew field $\FF$, known as [*Dieudonné determinant*]{} [@Dieudonne1943]. Our reference of Dieudonné determinant is [@CohnAlgebra3 Section 11.2]. Our starting point is the following normal form of matrices over skew field $\FF$.
\[lem:Bruhat\] Any matrix $A$ over $\FF$ is represented as $$\label{eqn:Bruhat}
A = L D P U,$$ for a diagonal matrix $D$, a permutation matrix $P$, a lower-unitriangular matrix $L$, and an upper-unitriangular matrix $U$, where $D P$ is uniquely determined.
Here a [*lower(upper)-unitriangular matrix*]{} is a lower(upper)-triangular matrix such that all diagonals are $1$. The proof is done by the Gaussian elimination, and is essentially the LU-decomposition (without pivoting).
Let $\FF^{\times} := \FF \setminus \{0\}$ denote the multiplicative group of $\FF$, and let $[\FF^{\times}, \FF^{\times}]$ denote the derived group of $\FF^{\times}$, i.e., $[\FF^{\times}, \FF^{\times}]$ is the group generated by all commutators $aba^{-1}b^{-1}$. The [*abelianization*]{} $\FF^{\times}_{\rm ab}$ of $\FF^{\times}$ is defined by $\FF^{\times}_{\rm ab} := \FF^{\times} / [\FF^{\times}, \FF^{\times}]$. For a nonsingular matrix $A$, the [*Dieudonné determinant*]{} ${\rm Det} A$ of $A$ is defined by $${\rm Det} A := {\rm sgn} (P) d_1 d_2 \cdots d_n \mod [\FF^{\times}, \FF^{\times}],$$ where $A$ is represented as (\[eqn:Bruhat\]) for permutation matrix $P$ and diagonal matrix $D$ with nonzero diagonals $d_1,d_2,\ldots,d_n$. If $\FF$ is a field, then $[\FF^\times,\FF^\times] = \{1\}$, $\FF^{\times}_{\rm ab} = \FF^{\times}$, and ${\mathop{\rm Det} }= \det$.
\[ex:2x2\] Consider the case of a $2$ by $2$ matrix $\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right)$. If $a \neq 0$, then the Bruhat normal form is given by $$\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) = \left(
\begin{array}{cc}
1 & 0 \\
ca^{-1} & 1
\end{array}
\right) \left( \begin{array}{cc}
a & 0 \\
0 & d - c a^{-1}b
\end{array}
\right)\left(
\begin{array}{cc}
1 & a^{-1}b \\
0 & 1
\end{array}\right).$$ Also, if $a = 0$ and $b \neq 0$, then $$\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) = \left(
\begin{array}{cc}
1 & 0 \\
d b^{-1} & 1
\end{array}
\right) \left( \begin{array}{cc}
b & 0 \\
0 & c
\end{array}
\right)\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right)
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right).$$ Hence the Dieudonné determinant is given by $${\mathop{\rm Det} }\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right)
= \left\{ \begin{array}{ccc}
a(d - c a^{-1}b) & \mbox{mod $[\FF^{\times}, \FF^{\times}]$} & {\rm if}\ a \neq 0, \\
- bc & \mbox{mod $[\FF^{\times}, \FF^{\times}]$} & {\rm if}\ a = 0.
\end{array}
\right.$$
The Dieudonné determinant has the desirable properties of the determinant, though its value is no longer an element of $\FF$.
\[lem:DetAB\] For nonsingular matrices $A,B \in \FF^{n \times n}$, it holds ${\mathop{\rm Det} }A B = {\mathop{\rm Det} }A {\mathop{\rm Det} }B$.
Next we introduce the polynomial ring $\mathbb{F}[t]$ and its skew field $\mathbb{F}(t)$ of fractions (or the [*Ore quotient ring*]{} of $\FF[t]$), where the indeterminate $t$ commutes every element in $\mathbb{F}$. Here $\mathbb{F}[t]$ consists of polynomials $\sum_{i=0}^k a_i t^i$, where $k \geq 0$ and $a_i \in \FF$. By using commuting rule $x t = t x$, the addition and multiplication in $\FF[t]$ are naturally defined. The resulting ring $\mathbb{F}[t]$ is called a polynomial ring over $\mathbb{F}$ with indeterminate $t$. In the notation of [@CohnAlgebra3; @CohnSkewField; @NoetherianBook], $\mathbb{F}[t]$ is the skew polynomial ring $\FF[t; 1,0]$. The [*degree*]{} $\deg p$ of polynomial $p = \sum_{i=0}^k a_i t^i$ with $a_k \neq 0$ is defined by $\deg p := k$.
The polynomial ring $\mathbb{F}[t]$ is an [*Ore domain*]{}, i.e., any two nonzero polynomials $p,q \in \FF[t]$ admit a common multiple $pu = q v$ for some nonzero $u,v \in \mathbb{F}[t]$. See [@CohnAlgebra3 Section 9.1] and [@NoetherianBook Chapter 6] for the details of Ore domains and their fields of fractions. This enables us to introduce addition and multiplication on the set $\mathbb{F}(t)$ of all fractions $p/q$ for $p \in \mathbb{F}[t], q \in \mathbb{F}[t] \setminus \{0\}$. Here $p/q$ is the equivalence class of $(p,q) \in \mathbb{F}[t] \times \mathbb{F}[t] \setminus \{0\}$ under the equivalence relation: $(p,q) \sim (p',q')$ $\Leftrightarrow$ $(pu,qu) = (p'v,q'v)$ for some nonzero $u,v \in \mathbb{F}[t]$. Addition $p/q + p'/q'$ is defined as $(pu + p'v)/qu$ by choosing $u,v \in \mathbb{F}[t]$ with $qu= q'v$. Multiplication $(p/q)(p'/q')$ is defined as $pu/q'v$ by choosing $u,v \in \mathbb{F}[t]$ with $qu = p'v$. They are well-defined. The inverse of nonzero element $p/q$ (i.e., $p \neq 0$) is given by $q/p$. In this way, $\mathbb{F}(t)$ becomes a skew field into which $\mathbb{F}[t]$ is embedded by $p \mapsto p/1$. Element $1/t^k$, which is denoted by $t^{-k}$, commutes each element of $\mathbb{F}(t)$.
The degree $\deg p/q$ of $p/q$ is defined as $\deg p - \deg q$. As was observed by Taelman [@Taelman06], the degree of the Dieudonné determinant is well-defined, since the degree is zero on commutators.
\[ex:2x2-deg\] We see from Example \[ex:2x2\] that the degree of the determinant of a $2 \times 2$ matrix over $\mathbb{F}(t)$ is similar to the commutative case: $$\deg {\mathop{\rm Det} }\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) \leq \max\{\deg(a) + \deg(d), \deg(b) + \deg(c) \}.$$ The equality holds if $\deg(a) + \deg(d) \neq \deg(b) + \deg(c)$.
We let $\deg {\mathop{\rm Det} }A := - \infty$ if $A$ is singular. From the definition and Lemma \[lem:DetAB\], we have:
\[lem:DegAB\] For $A,B \in \mathbb{F}(t)^{n \times n}$, it holds $\deg {\mathop{\rm Det} }AB = \deg {\mathop{\rm Det} }A + \deg {\mathop{\rm Det} }B$.
By using the Dieudonné determinant, we can formulate the [*Smith-McMillan form*]{} of a matrix over $\mathbb{F}(t)$; see [@MurotaMatrix Section 5.1.2] for the commutative case. An element $p/q \in \mathbb{F}(t)$ is said to be [*proper*]{} if $\deg(p/q) \leq 0$. Let $\mathbb{F}(t)^-$ denote the ring of proper elements of $\mathbb{F}(t)$. A matrix over $\mathbb{F}(t)^-$ is also called proper. A proper matrix is called [*biproper*]{} if it is nonsingular and its inverse is also proper. For integer vector $\alpha \in \ZZ^n$, let $(t^{\alpha})$ denote the diagonal $n \times n$ matrix such that the $(i,i)$-entry is $t^{\alpha_i}$ for $i=1,2,\ldots,n$.
\[prop:SmithMcMillan\] For a nonsingular matrix $A \in \mathbb{F}(t)^{n \times n}$, there are biproper matrices $S,T$ and integer vector $\alpha \in \ZZ^n$ with $\alpha_1 \geq \alpha_{2} \geq \cdots \geq \alpha_n$ such that $$S A T = (t^{\alpha}).$$ The integers $\alpha_k$ are uniquely determined by $$\alpha_k = \delta_k - \delta_{k-1} \quad (k = 1,2,\ldots,n),$$ where $\delta_k$ is the maximum degree of the Dieudonné determinants of $k \times k$ submatrices of $A$, and let $\delta_0 := 0$.
A part of the statement is given as an exercise in [@CohnAlgebra3 pp. 459–460] in a general setting of a valuation ring. The proof is given in Appendix \[app:SM\], which goes in almost the same way as in the commutative case.
Any proper element $p/q$ is written as $u + p'/q$, where $u \in \mathbb{F}$ and $\deg p' < \deg q$. Indeed, if $p/q = (a t^k + p'')/(b t^k + q')$ for $a,b \in \FF$, $b \neq 0$, $\deg p'' < k$, and $\deg q' < k$, then $u = ab^{-1}$. This element $u$ is uniquely determined (independent of expression $p/q$); see [@NoetherianBook Exercise 6F]. Thus any proper matrix $A$ is uniquely written as $A = A^0 + t^{-1} A'$, where $A^0$ is a matrix over $\mathbb{F}$ and $A'$ is proper.
\[lem:key\] Let $A$ be a square proper matrix. Then the degree of each diagonal of the Smith-McMillan form of $A$ is nonpositive, and $\deg {\mathop{\rm Det} }A \leq 0$. In addition, the following conditions are equivalent:
- $\deg {\mathop{\rm Det} }A = 0$.
- $A^0$ is nonsingular over $\FF$.
- $A$ is biproper.
- $A$ is written as $Q_1 Q_2 \cdots Q_k$ $(k \geq 0)$, where each $Q_i$ is a permutation matrix, proper unitriangular matrix, or diagonal matrix with degree-zero elements.
The former part is immediate from Lemma \[prop:SmithMcMillan\] with the fact that $\alpha_1$ is the maximum degree of entries of $A$, and is now nonpositive; then $\deg {\mathop{\rm Det} }A = \delta_n = \sum_{k=1}^n \alpha_k \leq 0$.
We show the equivalence.
\(4) $\Rightarrow$ (3) follows from the fact that each $Q_i$ is biproper.
\(3) $\Rightarrow$ (2). If $B$ is the inverse of $A$ and is represented as $B = B^0 + t^{-1}B'$ for proper $B'$, then $B^0$ must be the inverse of $A^0$.
\(2) $\Rightarrow$ (1). Consider the Smith-McMillan form $A = S(t^{\alpha})T$. From $\alpha \leq 0$ and $S^0 (t^{\alpha})^0 T^0 = A^0$, if $\deg {\mathop{\rm Det} }A = \sum_{k} \alpha_k <0$, then $\alpha_k < 0$ for some $k$, and $A^0$ must be singular.
\(1) $\Rightarrow$ (4). We see in the proof of Proposition \[prop:SmithMcMillan\] (Appendix \[app:SM\]) that $S,T$ in the Smith-McMillan form of $A$ are taken as the product of those matrices.
For $A \in \mathbb{F}(t)^{n \times n}$, it holds $\deg {\mathop{\rm Det} }A \leq n \alpha_1$. In addition, if $A$ is a nonsingular polynomial matrix, then $\deg {\mathop{\rm Det} }A \geq 0$.
The latter part is contained in [@Taelman06 Theorem 1.1].
The first statement follows from the Smith-McMillan form and $\deg {\mathop{\rm Det} }S = \deg {\mathop{\rm Det} }T = 0$ for biproper matrices $S,T$ (Lemma \[lem:key\]). The polynomial ring $\FF[t]$ is a (left and right) Euclidean domain. Therefore, by elementary row and column operations on $\FF[t]$ with row and column permutations, $A$ is diagonalized so that the diagonal entries are polynomials in $\FF[t]$ (such as the Smith form); see [@CohnAlgebra3 Section 9.2]. Namely $PAQ$ is a diagonal polynomial matrix for some matrices $P,Q$ with $\deg {\mathop{\rm Det} }P = \deg {\mathop{\rm Det} }Q = 0$. This implies $\deg {\mathop{\rm Det} }A = \deg {\mathop{\rm Det} }PAQ \geq 0$.
Finally we note a useful discrete convexity property of the degree of the Dieudonné determinant. A [*valuated matroid*]{} [@DressWenzel_greedy; @DressWenzel] on a set $E$ is a function $\omega: 2^E \to \RR \cup \{-\infty\}$ satisfying the following condition:
[(EXC)]{}
: For any $X,Y \subseteq E$ with $\omega(X),\omega(Y) \neq - \infty$ and $e \in X \setminus Y$, there is $f \in Y \setminus X$ such that $$\omega(X) + \omega(Y) \leq \omega(X \cup \{f\} \setminus \{e\}) + \omega(Y \cup \{e\} \setminus \{f\}).$$
It is well-known in the literature that the deg-det function gives rise to a valuated matroid [@DressWenzel_greedy; @DressWenzel]; see [@MurotaMatrix Chapter 5]. This is also the case for the deg-Det function.
\[prop:valuated\] Let $A$ be an $n \times m$ matrix over $\FF(t)$. The following function $\omega_A: 2^{\{1,\ldots,m\}} \to \RR \cup \{-\infty\}$ is a valuated matroid: $$\omega_A(X)
:= \left\{ \begin{array}{ll}
\deg {\mathop{\rm Det} }A[X] & {\rm if}\ |X| = n, \\
- \infty & {\rm otherwise}
\end{array}\right.
\quad (X \subseteq \{1,2,\ldots,m\}),$$ where $A[X]$ denotes the submatrix of $A$ consisting of the $i$-th columns over $i \in X$.
For the verification we use a local characterization [@MurotaMatrix Theorem 5.2.25] of valuated matroids, which says that function $\omega:2^{\{1,2,\ldots,m\}} \to \RR \cup \{-\infty\}$ is a valuated matroid if and only if $\{ X \subseteq \{1,2,...,m\} \mid \omega(X) \neq - \infty\}$ is the base family of a matroid and $\omega$ satisfies (EXC) for all pairs $X,Y$ with $|X \setminus Y| = |Y \setminus X| = 2$. The first condition for $\omega_A$ follows from the observation that $\omega_A(X) \neq \infty$ if and only if $A[X]$ is nonsingular, i.e., $X$ forms a basis of (right) vector space $\FF(t)^n$ over skew field $\FF(t)$. Therefore the family $\{ X \subseteq \{1,2,...,m\} \mid \omega(X) \neq - \infty\}$ is the base family of a representable matroid over skew field $\FF(t)$. We next consider the latter condition, i.e., (EXC) for $X,Y$ with $|X \setminus Y| = |Y \setminus X| = 2$, where we can assume that $\omega_{A}(X) \neq -\infty$. Let $A'$ denote the $n \times (n+4)$ submatrix of $A$ consisting columns in $X \cup Y$. By Lemma \[lem:DegAB\], elementary row operations (or multiplying a nonsingular matrix from left) to $A$ does not change $\omega_A$ other than constant addition. Also we can arrange columns of $A'$ so that $X \cap Y$ forms the first $n-2$ columns. Hence we can assume that $A'$ is the form $\left( \begin{array}{cc} I & C \\ O & B \end{array} \right)$, where $I$ is the unit matrix of size $n-2$, $B$ is a $2 \times 4$ matrix, and $C$ is an $(n-2) \times 4$ matrix. Then $\omega_{A'} ( (X \cap Y) \cup \{i, j\}) = \omega_B (\{i,j\})$ holds for $i,j \in (X \setminus Y) \cup (Y \setminus X)$. This can be seen from the definition of the Dieudonné determinant as: By elementary column operations (or multiplying an upper-unitriangular matrix from right), $A[(X \cap Y) \cup \{i,j\}]$ becomes $\left( \begin{array}{cc} I & O \\ O & B[\{i,j\}] \end{array} \right)$, and then ${\mathop{\rm Det} }A[(X \cap Y) \cup \{i,j\}] = {\mathop{\rm Det} }B[\{i,j\}]$.
Therefore this reduces our problem to the verification of (EXC) for an arbitrary $2 \times 4$ matrix $A$. Then (EXC) is equal to:
[(4PT)]{}
: the maximum of $\omega(12) + \omega(34)$, $\omega(13) + \omega(24)$, $\omega(14) + \omega(23)$ is attained at least twice,
where $\omega_A (\{i,j\})$ is simply written as $\omega(ij)$.
We may assume that $\omega(12) \neq - \infty$ and the $(1,1)$-entry is nonzero (by column permutation). By row operations, we can make $A$ so that the $(2,1)$-entry is zero. If the $(2,3)$-entry is nonzero, then make $A$ so that $(1,3)$-entry is zero. Then we may consider two cases: $$\left( \begin{array}{cccc}
a & c & d & e \\
0 & b & 0 & f
\end{array}
\right) \ {\rm and}\
\left( \begin{array}{cccc}
a & c & 0 & e \\
0 & b & d & f
\end{array}
\right).$$ Recall Example \[ex:2x2-deg\]. For the former case, $\omega(12) + \omega(34) = \omega(14) + \omega(23) = \deg(a) + \deg (b) + \deg(d) + \deg (f)$, and $\omega(13) + \omega(24) = - \infty$. For the latter case, $\omega(12) + \omega(34) = \deg(a) + \deg (b) + \deg(d) + \deg (e)$, $\omega(14) + \omega(23) = \deg(a) + \deg (f) + \deg(c) + \deg (d)$, and $\omega(13) + \omega(24) \leq \deg (a) + \deg (d) +\max \{ \deg (c) + \deg(f), \deg(e) + \deg(b)\}$ with equality if $\deg (c) + \deg(f) \neq \deg(e) + \deg(b)$. Thus (4PT) holds for all cases.
L-convex function on Euclidean building {#sec:L-convex}
=======================================
In this section, we introduce L-convex function on Euclidean building. It will turn out in Section \[sec:computing\] that the deg-Det computation reduces to an L-convex function minimization on the Euclidean building for ${\rm SL}(\KK(t)^n)$. Our approach is lattice-theoretic. First we set up basic lattice terminologies. Then we introduce a uniform modular lattice, which is a lattice-theoretic counterpart of a Euclidean building of type A, and we introduce L-convexity on it.
Lattice
-------
A [*lattice*]{} is a partially ordered set ${\cal L}$ such that every pair $x,y$ of elements has the minimum common upper bound $x \vee y$ and the maximum common lower bound $x \wedge y$; the former is called the [*join*]{} and the latter is called the [*meet*]{}. The partial order is denoted by $\preceq$, where $x \prec y$ is meant as $x \preceq y$ and $x \neq y$. A totally ordered subset of ${\cal L}$ is called a [*chain*]{}, which is written as $x^0 \prec x^1 \prec \cdots \prec x^k$. The [*length*]{} of chain $C$ is defined as $|C|-1$. For $x,y \in {\cal L}$ with $x \preceq y$, the [*interval*]{} $[x,y]$ is defined as the set of elements $z$ with $x \preceq z \preceq y$. If $[x,y] = \{x,y\}$, we say that $y$ [*covers*]{} $x$. In this paper, we only consider lattices in which every chain of every interval has a finite length. A [*sublattice*]{} ${\cal L}'$ is a subset of ${\cal L}$ such that $x,y \in {\cal L}'$ implies $x \wedge y, x \vee y \in {\cal L}'$. An interval is a sublattice. The [*opposite*]{} $\check{\cal L}$ of lattice ${\cal L}$ is the lattice obtained from ${\cal L}$ by reversing the partial order of ${\cal L}$. The direct product ${\cal L} \times {\cal L}'$ of two lattices ${\cal L},{\cal L}'$ becomes a lattice by the product order: $(x,x') \preceq (y,y')$ $\Leftrightarrow$ $x \preceq y$ and $x' \preceq y'$.
A [*modular lattice*]{} is a lattice ${\cal L}$ such that for every triple $x,y,z \in {\cal L}$ with $x \preceq z$, it holds $x \vee (y \wedge z) = (x \vee y) \wedge z$. The opposite of a modular lattice is also a modular lattice. A useful criterion for the modularity is given as follows. A [*valuation*]{} on a lattice ${\cal L}$ is a function $v: {\cal L} \to \RR$ such that
- $v(x) + v(y) = v(x \wedge y) + v(x \vee y)$ for all $x,y \in {\cal L}$, and
- $v(x) < v(y)$ for all $x,y \in {\cal L}$ with $x \prec y$.
If a lattice ${\cal L}$ admits a valuation, then ${\cal L}$ is a modular lattice.
A [*unit valuation*]{} is a valuation $v$ such that $v(x) = v(y) - 1$ provided $y$ covers $x$. A modular lattice (having the minimum element) is said to be [*complemented*]{} if every element is the join of atoms ($=$ elements covering the minimum element).
Let $\FF$ be a skew field. The families ${\cal S}_{\rm R}(\FF^n)$ and ${\cal S}_{\rm L}(\FF^n)$ of vector subspaces of $\FF^n$ are complemented modular lattices, where the partial order is the inclusion relation. The join and meet are given by $+$ and $\cap$, respectively. Also $X \mapsto \dim X$ is a unit valuation.
The family of chains of ${\cal S}_{\rm R}(\FF^n) \setminus \{\emptyset, \FF^n \}$ (or ${\cal S}_{\rm L}(\FF^n) \setminus \{\emptyset, \FF^n \}$) is known as the [*spherical building*]{} of ${\rm SL}(\FF^n)$. More generally, the family of chains of a complemented modular lattice is equivalent to a spherical building of type A. See [@Tits].
A function $f$ on lattice ${\cal L}$ is called [*submodular*]{} if it satisfies $$f(x) + f(y) \geq f(x \wedge y) + f(x \vee y) \quad (x,y \in {\cal L}).$$ As was noticed in [@HamadaHirai17; @HH16DM; @ItoIwataMurota94], MVSP is viewed as a submodular optimization on a modular lattice. For a matrix $A \in \FF^{n \times n'}$ regarded as a bilinear form (\[eqn:bilinear\]), define $r_A:{\cal S}_{\rm L}(\FF^{n})
\times {\cal S}_{\rm R}(\FF^{n'}) \to \ZZ$ by $$r_A(X,Y) := \mbox{the rank of the restriction of $A$ to $X \times Y$.}$$
\[lem:submo\] Let $A \in \FF^{n \times n'}$ be a matrix over $\FF$. Then $r_{A}$ is submodular on ${\cal S}_{\rm L}(\FF^{n}) \times \check{\cal S}_{\rm R}(\FF^{n'})$, i.e., $$r_A(X,Y) + r_A(X',Y') \geq
r_A(X + X',Y \cap Y') + r_A(X \cap X',Y + Y').
$$
The vanishing condition $A(X,Y) = \{0\}$ is equivalent to $r_A(X,Y) = 0$. By including $r_A$ in the objective as a penalty term, MVSP is formulated as an unconstrained submodular optimization over modular lattice ${\cal S}(\KK^{n}) \times \check{\cal S}(\KK^{n'})$: $$\begin{aligned}
{\rm MVSP}': \quad {\rm Min.} && - \dim X - \dim Y + C \sum_{i=1}^m r_{A_i}(X,Y) \nonumber \\
{\rm s.t.} && X \in {\cal S}(\KK^n), Y \in {\cal S}(\KK^{n'}), \nonumber\end{aligned}$$ where $C > 0$ is a large constant. The approach by Hamada and Hirai [@HamadaHirai17] is based on this idea.
Uniform modular lattice and Euclidean building {#subsec:uniform}
----------------------------------------------
The [*ascending operator*]{} of a lattice ${\cal L}$ is a map $(\cdot)^+:{\cal L} \to {\cal L}$ defined by $$(x)^+ := \bigvee \{ y \in {\cal L} \mid \mbox{$y$ covers $x$} \} \quad (x \in {\cal L}).$$ A [*uniform modular lattice*]{} [@HH18a] is a modular lattice ${\cal L}$ such that the ascending operator is defined and is an automorphism on ${\cal L}$. Suppose that ${\cal L}$ is a uniform modular lattice. The rank ($=$ the length of a maximal chain) of $[x, (x)^+]$ is independent of $x$, and is called the [*uniform-rank*]{} of ${\cal L}$. The inverse of $(\cdot)^+$ is given by $x \mapsto$ the meet of elements covered by $x$. In particular, the opposite $\check{\cal L}$ of ${\cal L}$ is also uniform modular. The product of two uniform modular lattices is also uniform modular.
$\ZZ^n$ becomes a lattice with respect to vector order $\leq$, where $x \vee y$ equals $\max(x,y)$ (componentwise maximum of $x,y$) and $x \wedge y$ equals $\min(x,y)$ (componentwise minimum of $x,y$). Now $\ZZ^n$ is a uniform modular lattice, where $z \mapsto \sum_{i=1}^n z_i$ is a unit valuation, the ascending operator is given by $x \mapsto x + {\bf 1}$ for all one-vector $\bf 1$, and the uniform-rank is equal to $n$.
A [*$\ZZ^n$-skeleton*]{} of ${\cal L}$ is a sublattice $\varSigma$ such that $\varSigma$ is isomorphic to $\ZZ^n$ and the restriction of the ascending operator of ${\cal L}$ to $\varSigma$ is the same as the ascending operator of $\varSigma$. A chain $x_0 \prec x_1 \prec \cdots \prec x_m$ is said to be [*short*]{} if $x_m \preceq (x_0)^+$.
\[lem:skeleton\] Let ${\cal L}$ be a uniform modular lattice with uniform-rank $n$.
- For two short chains $C,D$, there is a $\ZZ^n$-skeleton $\varSigma$ containing them.
- If two $\ZZ^n$-skeletons $\varSigma,\varSigma'$ contain short chains $C,D$, there is an order-preserving bijection from $\varSigma$ to $\varSigma'$ such that it is the identity on $C \cup D$.
(B1) and (B2) are essentially the apartment axiom of [*Euclidean building of type A*]{} [@BruhatTits]; see [@Garrett]. The paper [@HH18a] shows that the family of all short chains in a uniform modular lattice actually forms a Euclidean building of type A, and that every Euclidean building of type A is obtained in this way. An [*apartment system*]{} of ${\cal L}$ is a family of $\ZZ^n$-skeletons such that a $\ZZ^n$-skeleton in (B1) can be chosen from the family. A $\ZZ^n$-skeleton in an apartment system is simply called an [*apartment*]{}.
Next we consider an important example of a uniform modular lattice arising from a skew field with a discrete valuation. Let $\FF$ be a skew field, and let $\FF(t)$ be the skew field of rational functions over $\FF$. Let $\FF(t)^-$ be the ring of proper elements of $\FF(t)$. Consider the $n$-product $\FF(t)^n$, which is regarded as a left $\FF(t)^-$-module of row vectors as well as a right $\FF(t)^-$-module of column vectors. Let ${\cal L}_{\rm L} (\FF(t)^n)$ denote the family of all full-rank free $\FF(t)^-$-submodules[^1] of $\FF(t)^n$, where $\FF(t)^n$ is regarded as a left $\FF(t)^-$-module of row vectors. Let ${\cal L}_{\rm R} (\FF(t)^n)$ be defined as the right analogue. By definition, an element $L \in {\cal L}_{\rm L}(\FF(t)^n)$ is represented as $\langle Q \rangle_{\rm L} := \{ \lambda Q \mid \lambda \in (\FF(t)^-)^n \}$ for a nonsingular matrix $Q$ over $\FF(t)$. Similarly, an element $L \in {\cal L}_{\rm R}(\FF(t)^n)$ is written as $\langle P \rangle_{\rm R} := \{ P\lambda \mid \lambda \in (\FF(t)^-)^n \}$ for a nonsingular matrix $P$ over $\FF(t)$. For $L \in {\cal L}_{\rm L}(\FF(t)^n)$ or ${\cal L}_{\rm R}(\FF(t)^n)$, define $\deg L$ by $$\deg L := \deg {\mathop{\rm Det} }P$$ for a nonsingular matrix $P$ with $L = \langle P \rangle_{\rm L}$ or $\langle P \rangle_{\rm R}$. This is well-defined; if $\langle P \rangle_{\rm R} = \langle P' \rangle_{\rm R}$, then $P' = P S$ for some biproper matrix $S$, and $\deg {\mathop{\rm Det} }P' = \deg {\mathop{\rm Det} }P$ by Lemmas \[lem:DegAB\] and \[lem:key\]. We give three lemmas on the family ${\cal L}_{\rm R}(\FF(t)^n)$ below. They hold when $\rm R$ is replaced by $\rm L$. The first one is shown in [@HH18a] for the case where $\FF$ is a field.
\[lem:L(F(t)\^n)\] ${\cal L}_{\rm R}(\FF(t)^n)$ is a uniform modular lattice, where $\wedge = \cap$, $\vee = +$, $L \mapsto \deg L$ is a unit valuation, the uniform-rank is equal to $n$, and the ascending operator is given by $L \mapsto tL$.
We give in the appendix a proof by adapting the argument in [@HH18a] for our non-commutative setting.
For an integer vector $z \in \ZZ^n$, recall that $(t^z)$ denotes the diagonal matrix with diagonals $t^{z_1},t^{z_2},\ldots,t^{z_n}$. For a nonsingular matrix $Q$, let $\varSigma_{\rm R}(Q)$ denote the sublattice of ${\cal L}_{\rm R}(\FF^n)$ consisting of $\langle Q(t^z) \rangle_{\rm R}$ for all $z \in \ZZ^n$. Similarly, define $\varSigma_{\rm L}(Q)$ by $\varSigma_{\rm L}(Q) := \{ \langle (t^z)Q \rangle_{\rm L} \mid z \in \ZZ^n \}$.
\[lem:apartment\] The family of sublattices consisting of $\varSigma_{\rm R}(Q)$ for all nonsingular $Q \in \FF(t)^{n \times n}$ forms an apartment system in ${\cal L}_{\rm R}(\FF(t))$, where $z \mapsto \langle Q(t^z) \rangle_{\rm R}$ is an isomorphism between $\ZZ^n$ and $\varSigma_{\rm R}(Q)$.
The proof is given in the appendix. Next we study the lattice structure of interval $[L, (L)^+] = [L,tL]$, which is a complemented modular lattice and is turned out to be the spherical building at the link of $L$. For $M \in {\cal L}_{\rm R}(\FF(t)^n)$ with $L \subseteq M \subseteq tL$, the quotient module $M/L$ becomes a right $\FF$-vector space by $(u + L)\alpha := u \alpha + L$ for $\alpha \in \FF$. For an $\FF$-vector subspace $X$ of $tL/L$, define submodule $L \circ X$ of $tL$ by $$L \circ X := \{ u \in tL \mid u + L \in X \}.$$
\[lem:\[L,tL\]\] Let $L \in {\cal L}_{\rm R}(\FF(t)^n)$.
- $tL/L$ is a right $\FF$-vector space with dimension $n$.
- $[L,tL]$ is isomorphic to ${\cal S}_{\rm R}(tL/L)$ by $M \mapsto M/L$ with inverse $X \mapsto L \circ X$.
- For $X \in {\cal S}_{\rm R}(tL/L)$, it holds $\deg L \circ X = \deg L + \dim X$.
- If $L = \langle P \rangle_{\rm R}$, then $[L,tL]$ is given by $$[L,tL] = \{ \langle P S (t^{{\bf 1}_{\leq k}}) \rangle_{\rm R} \mid 0 \leq k \leq n,\ S \in \FF^{n \times n}: \mbox{nonsingular} \},$$ where ${\bf 1}_{\leq k}$ denotes the 0,1-vector such that the first $k$ elements are $1$ and others are zero.
(1). Suppose that $\{ p_1,p_2,\ldots,p_n\}$ is a basis of $L$. Then $\{tp_1,tp_2,\ldots,tp_n\}$ is a basis of $tL$. We show that $\{ tp_1 + L,tp_2+ L,\ldots,tp_n + L\}$ is a basis of $tL/L$. Every element $u \in tL$ is written as $u = \sum_{i=1}^n t p_i \lambda_i$ for $\lambda_i \in \FF(t)^-$. Here $\lambda_i$ is written as $\lambda_i = \lambda_i^0 + t^{-1} \lambda_i'$ for $\lambda_i^0 \in \FF$ and $\lambda_i' \in \FF(t)^-$. This means that $u \in \sum_{i=1}^n t p_i \lambda_i^0 + L$. Thus $\{tp_1 + L,tp_2+ L,\ldots,tp_n + L\}$ spans $tL/L$. We show the linear independence. Suppose that $\sum_{i=1}^n t p_i \alpha_i = u \in L$ for $\alpha_i \in \FF$. Since $\{p_1,p_2,\ldots,p_n\}$ is a basis of $L$, it must be $\alpha_i t \in \FF(t)^-$, and $\alpha_i = 0$.
(2). It suffices to verify that $L \circ X$ is a full-rank free submodule. Since $tL$ is a free $\FF(t)^-$-module and $\FF(t)^-$ is PID, the submodule $L \circ X$ of $tL$ is a free module containing $L$, and hence has rank $n$.
\(3) follows from (2) and the fact that $\deg$ and $\dim$ are unit valuations.
(4). By the proof of (1), the column vectors of $tP$ modulo $L$ become an $\FF$-basis of $tL/L$. Therefore any vector subspace $X \subseteq tL/L$ is spanned by $\FF$-linear combinations of column vectors of $tP$ modulo $L$. Thus, if $\dim X =k$, then for some nonsingular matrix $S$ over $\FF$, $X$ is spanned by the first $k$ columns of $tPS$ (modulo $L$). Then $L \circ X = \langle PS(t^{{\bf 1}_{\leq k}}) \rangle_{\rm R}$ must hold. Indeed, $\supseteq$ is obvious, and the equality follows from $\deg \langle PS(t^{{\bf 1}_{\leq k}}) \rangle_{\rm R}
= \deg L + k = \deg L + \dim X = \deg L \circ X$ (by (3)).
Consider the simplest case of ${\cal L}_{\rm R}(\FF(t)^2)$ with $\FF = \ZZ/2\ZZ$. Then $L \in {\cal L}_{\rm R}(\FF(t)^2)$ is spanned by two vectors $p_1,p_2 \in \FF(t)^2$; we simply write it as $L = \langle p_1, p_2 \rangle_{\rm R}$. According to the proof of Lemma \[lem:\[L,tL\]\], $e_1 := tp_1 + L$ and $e_2 := tp_2 + L$ form an $\FF$-basis of $tL/L$. In particular, $tL/L$ is isomorphic to $\FF^2$ by $\FF^2 \ni
\left( \begin{array}{c}
a_1 \\
a_2
\end{array} \right)
\mapsto a_1 e_1 + a_2 e_2$. There are five subspaces in $\FF^2$: $$X_0 = \{0\},\
X_1 = \FF \left( \begin{array}{c}
1 \\
0
\end{array} \right), \
X_2 = \FF \left( \begin{array}{c}
0 \\
1
\end{array} \right), \
X_3 = \FF \left( \begin{array}{c}
1 \\
1
\end{array} \right), \ X_4 = \FF^2.$$ Then $[L, tL]$ consists of $L = L \circ X_0$, $t L = L \circ X_4$, and $$L \circ X_1 = \langle t p_1, p_2 \rangle_{\rm R},\ L \circ X_2 = \langle t p_2, p_1 \rangle_{\rm R},\
L \circ X_3 = \langle t (p_1+ p_2), p_2 \rangle_{\rm R}.$$
L-convex function
-----------------
We first review L-convex functions on $\ZZ^n$; see [@MurotaBook Chapter 7] for details. A function $g:\ZZ^n \to \RR \cup \{\infty\}$ is called [*L-convex*]{} if it satisfies:
[(SUB$^{\ZZ}$)]{}
: $g(x) + g(y) \geq g(\min(x,y)) + g(\max(x,y))$ for $x,y \in \ZZ^n$.
[(LIN$^{+\mathbf{1}}$]{})
: There is $r \in \RR$ such that $g(x+ {\bf 1}) = g(x) + r$ for $x \in \ZZ^n$.
We treat the infinity element $\infty$ as $\infty + c = \infty$ for $c \in \RR \cup \{\infty\}$ and $b < \infty$ for $b \in \RR$. Also we assume that any function $g: \ZZ^n \to \RR \cup \{\infty\}$ has a point $x$ with $g(x) < \infty$.
\[ex:L-convex\_ex\] The following function $g:\ZZ^N \to \RR \cup \{\infty\}$ is known to be L-convex: $$g(x) = \sum_{1 \leq i,j \leq N} \phi_{ij}(x_i - x_j) \quad (x \in \ZZ^N),$$ where $\phi_{ij}:\ZZ \to \RR \cup \{\infty\}$ is a $1$-dimensional convex function for each $i,j$. This L-convex function arises from the dual of minimum-cost network flow.
We are interested in minimization of an L-convex function. Note that $r = 0$ in (LIN$^{+\mathbf{1}})$ is a necessary condition for the existence of a minimizer. We tacitly assume $r = 0$ in the sequel. The following optimality property is basic.
\[lem:opt\] Let $g:\ZZ^n \to \RR \cup \{\infty\}$ be an L-convex function. A point $x \in \ZZ^n$ is a minimizer of $g$ if and only if $$g(x) \leq g(x+ u) \quad (u \in \{0,1\}^n).$$
This property naturally leads to the following simple descent algorithm, called the [*steepest descent algorithm*]{}.
Steepest Descent Algorithm (SDA$(\ZZ^n)$)
:
Input:
: An L-convex function $g:\ZZ^n \to \RR \cup \{\infty\}$.
Output:
: A minimizer of $g$.
Step 0:
: Choose $x^0 \in \ZZ^n$ with $g(x^0) < \infty$. Let $i := 0$.
Step 1:
: Find a minimizer $y$ of $g$ over $x^i + \{0,1\}^n$.
Step 2:
: If $g(y) < g(x^i)$, then let $i \leftarrow i+1$, $x^{i} \leftarrow y$, and go to step 1.
Step 3:
: Otherwise, output $x^i$ as a minimizer.
The point $y$ in Step 1 is called a [*steepest direction*]{} at $x^i$. The function $u \mapsto g(x + u)$ is submodular on Boolean lattice $\{0,1\}^n$. Hence a steepest direction can be found by a submodular function minimization on the Boolean lattice. It is a fundamental fact in combinatorial optimization that any submodular function on the Boolean lattice can be minimized in polynomial time; see e.g., [@KorteVygen Section 14.3]. An intriguing property of SDA is the following bound of the number of the iterations.
\[thm:bound\_ZZ\^n\] Let $g:\ZZ^n \to \RR \cup \{\infty\}$ be an L-convex function, and let $k$ be the minimum $l_{\infty}$-distance between $x^0$ and minimizers $y^* \geq x^0$, i.e., $$k := \min \{\|x^0 - y^* \|_{\infty} \mid \mbox{$y^*$ is a minimizer of $g$ with $y^* \geq x^0$ } \}.$$ In SDA, the $k$-th point $x^k$ is a minimizer of $g$.
Next we introduce L-convex functions on a uniform modular lattice, and generalize the above properties. The argument goes in a straightforward way. Let ${\cal L}$ be a uniform modular lattice with uniform-rank $n$. A function $g:{\cal L} \to \RR \cup \{\infty\}$ is called [*L-convex*]{} if it satisfies:
[(SUB)]{}
: $g(x) + g(y) \geq g(x \wedge y) + g(x \vee y)$ for all $x,y \in {\cal L}$.
[(LIN$^+$)]{}
: There is $\alpha \in \RR$ such that $g((x)^+) = g(x) + \alpha$ for all $x \in {\cal L}$.
Fix an arbitrary apartment system of ${\cal L}$. Every apartment of ${\cal L}$ is a sublattice isomorphic to $\ZZ^n$ and preserves the ascending operation. Thus the L-convexity is characterized by the L-convexity on each apartment.
\[lem:restriction\] A function $g: {\cal L} \to \RR \cup \{\infty\}$ is L-convex if and only if the restriction of $g$ to every apartment $\varSigma$ is L-convex, where $\varSigma$ is identified with $\ZZ^n$.
The optimality criterion (Lemma \[lem:opt\]) is generalized as follows.
\[lem:opt’\] Let $g:{\cal L} \to \RR \cup \{\infty\}$ be an L-convex function. A point $x \in {\cal L}$ is a minimizer of $g$ if and only if $$g(x) \leq g(y) \quad (y \in [x, (x)^+]).$$
Suppose that $x$ is not a minimizer. Consider a minimizer $y^*$ of $g$. Choose an apartment $\varSigma$ containing $x$ and $y^*$. By $\varSigma \simeq \ZZ^n$ and Lemmas \[lem:opt\] and \[lem:restriction\], there is $y \in [x,x+{\bf 1}] \subseteq [x,(x)^+]$ with $g(y) < g(x)$.
The steepest descent algorithm is formulated as follows.
Steepest Descent Algorithm (SDA$({\cal L})$)
:
Input:
: An L-convex function $g:{\cal L} \to \RR \cup \{\infty\}$.
Output:
: A minimizer of $g$.
Step 0.
: Choose $x^0 \in {\cal L}$ with $g(x^0) < \infty$. Let $i := 0$.
Step 1.
: Find a minimizer $y$ of $g$ over $[x^i, (x^i)^+]$.
Step 2.
: If $g(y) < g(x^i)$, then let $i \leftarrow i+1$, $x^{i} \leftarrow y$, and go to step 1.
Step 3.
: Otherwise, output $x^i$.
The interval $[x^i, (x^i)^+]$ is a complemented modular lattice, and $g$ is submodular on $[x^i, (x^i)^+]$. In particular, Step 1 reduces to a submodular function minimization on the complemented modular lattice. In the building-theoretic view, $[x^i, (x^i)^+]$ is the spherical building at the link of point $x^i$, and Step 1 is an optimization on the spherical building. To generalize the iteration bound (Theorem \[thm:bound\_ZZ\^n\]), we introduce the $l_{\infty}$-distance on ${\cal L}$. For two elements $x,y \in {\cal L}$, choose an apartment $\varSigma$ containing $x,y$, identify $\varSigma$ with $\ZZ^n$, and define the [*$l_{\infty}$-distance*]{} $d_{\infty}(x,y)$ by $$d_{\infty}(x,y) := \|x - y\|_{\infty}.$$ One can see from the property (B2) in Lemma \[lem:skeleton\] that $d_{\infty}$ is independent of the choice of the apartment.
\[thm:bound\_L\] Let $g:{\cal L} \to \RR \cup \{\infty\}$ be an L-convex function, and let $k$ be the minimum $l_{\infty}$-distance between $x^0$ and minimizers $y^* \succeq x^0$, i.e., $$k := \min \{ d_{\infty}(x^0,y^*) \mid \mbox{$y^*$ is a minimizer of $g$ with $y^* \succeq x^0$ } \}.$$ In SDA$({\cal L})$, the $k$-th point $x^k$ is a minimizer of $g$.
We may suppose that a minimizer exists. It suffices to show that the distance $k$ decreases by $1$ on the update $x^0 \to x^1$. Let $y^*$ be a minimizer of $g$ with $y^* \succeq x_0$ and $d_{\infty}(x^0,y^*) = k$. Choose apartment $\varSigma$ containing $y^*$ and short chain $\{ x^0,x^1\}$. Identify $\varSigma$ with $\ZZ^n$. Then $x^0$ is not a minimizer of $g$ over $\varSigma$. Then the update $x^0 \to x^1$ is viewed as the update of the first iteration of SDA$(\ZZ^n)$. Thus, by Theorem \[thm:bound\_ZZ\^n\], the distance decreases on $\varSigma$ and on ${\cal L}$.
Let $\FF$ be a skew field, and let $\FF(t)$ be the skew field of rational functions. For (nonzero) $A \in \FF(t)^{n \times n}$, define $\deg A
: {\cal L}_{\rm L}(\FF(t)^n) \times {\cal L}_{\rm R}(\FF(t)^n) \to \ZZ$ by $$\deg A(L,M) := \max \{ \deg u \mid u \in A(L,M) \}.$$ Notice that if $L = \langle P \rangle_{\rm L}$ and $M = \langle Q \rangle_{\rm R}$ then $\deg A(L,M)$ is equal to the maximum degree of an entry of $PAQ$. An affine analogue of Lemma \[lem:submo\] is the following:
\[lem:L-convexity\] Let $A \in \FF(t)^{n \times n}$. Then the function $(L,M) \mapsto \infty \cdot \deg A(L,M)$ is L-convex on ${\cal L}_{\rm L}(\FF(t)^n) \times \check{\cal L}_{\rm R}(\FF(t)^n)$, where $\infty \cdot c$ is defined as $\infty$ if $c > 0$ and $0$ if $c \leq 0$.
Here ${\cal L}_{\rm L}(\FF(t)^n) \times \check{\cal L}_{\rm R}(\FF(t)^n)$ is viewed as a uniform modular lattice with ascending operator $(L,M) \mapsto (tL,t^{-1}M)$.
By Lemmas \[lem:restriction\] and \[lem:apartment\], it suffices to show the L-convexity on the apartment $\varSigma_{\rm L}(P) \times \check\varSigma_{\rm R}(Q)$ of ${\cal L}_{\rm L}(\FF(t)^n) \times \check{\cal L}_{\rm R}(\FF(t)^n)$. Suppose that the row vectors of $P$ are $p_1,p_2, \ldots, p_n$ and column vectors of $Q$ are $q_1,q_2, \ldots, q_n$. Then the apartment $\varSigma_{\rm L}(P) \times \check\varSigma_{\rm R}(Q)$ is isomorphic to $\ZZ^{n} \times \ZZ^n$ by $(z,w) \mapsto (\langle (t^z) P \rangle_{\rm L}, \langle Q(t^{-w}) \rangle_{\rm R})$. Now $\deg A(\langle (t^z) P \rangle_{\rm L}, \langle Q(t^{-w}) \rangle_{\rm R}) = \max_{1 \leq i,j \leq n} \deg (t^{z_i} p_i A t^{- w_i} q_j)
= \max_{1 \leq i,j \leq n} z_i - w_j + \deg p_i Aq_j$. Thus $$\infty \cdot \deg A(\langle (t^z) P \rangle_{\rm L}, \langle Q(t^{-w}) \rangle_{\rm R}) = \sum_{1 \leq i,j \leq n} \infty \cdot ( z_i - w_j + \deg p_i Aq_j).$$ Notice that $x \mapsto \infty \cdot (x + b)$ is convex on $\ZZ$. By Example \[ex:L-convex\_ex\] (with $N = 2n$), $(z,w) \mapsto \infty \cdot \deg A(\langle (t^z) P \rangle_{\rm L}, \langle Q(t^{-w}) \rangle_{\rm R})$ is L-convex on $\ZZ^n \times \ZZ^n = \ZZ^{2n}$. This means that $\infty \cdot \deg A$ is L-convex on ${\varSigma}_{\rm L}(P) \times \check{\varSigma}_{\rm R}(Q)$.
Computing the degree of determinants {#sec:computing}
====================================
The goal of this section is to establish a formula and algorithm for the degree of the Dieudonné determinant of a linear symbolic matrix. Let $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ be a linear matrix over $\KK(t)$. Now $A$ is viewed as a matrix over the skew field $\KK(\langle x_1,x_2,\ldots,x_m \rangle) (t)$ of rational functions over the free field $\KK(\langle x_1,x_2,\ldots,x_m \rangle)$. As in the case of the nc-rank, we first give an upper bound of $\deg {\mathop{\rm Det} }A$. The following upper bound of $\deg {\mathop{\rm Det} }A$ is observed by Murota [@Murota95_SICOMP] for $\deg \det A$, which was a basis of the combinatorial relaxation algorithm.
\[lem:bound\_degDet\] For nonsingular matrices $P,Q$ over $\KK(t)$, if $P A_i Q$ is a proper matrix over $\KK(t)$ for $i=0,1,2,\ldots,m$, then $\deg {\mathop{\rm Det} }A \leq - \deg \det P - \deg \det Q$.
$P A Q$ is a proper matrix over $\KK(\langle x \rangle) (t)$. Also $\deg \det P = \deg {\mathop{\rm Det} }P$ and $\deg \det Q = \deg {\mathop{\rm Det} }Q$. Thus the claim follows from Lemmas \[lem:DetAB\] and \[lem:key\].
This gives rise to the following optimization problem ([*maximum vanishing submodule problem (MVMP)[^2]*]{}): $$\begin{aligned}
{\rm MVMP:} \quad {\rm Max.} && \deg \det P + \deg \det Q \\
{\rm s.t.} && \mbox{$P A_i Q$: proper} \quad (i =0,1,\ldots,m), \\
&& P,Q \in \KK(t)^{n \times n}: \mbox{nonsingular}.\end{aligned}$$ Just as MVSP is formulated as an optimization over the lattice of vector subspaces of $\KK^n$ (see (\[eqn:MVSP\])), MVMP is reformulated as an optimization over the lattice of submodules of $\KK(t)^n$. Recall notions in Section \[subsec:uniform\]. Then the above problem is rephrased as the following: $$\begin{aligned}
{\rm MVMP:} \quad {\rm Max.} && \deg L + \deg M \\
{\rm s.t.} && \deg A_i(L,M) \leq 0 \quad (i =0,1,\ldots,m), \\
&& L \in {\cal L}_{\rm L}(\KK(t)^n),\ M \in {\cal L}_{\rm R}(\KK(t)^n).\end{aligned}$$ The following theorem states that this upper bound is tight for $\deg {\mathop{\rm Det} }$, which is an extension of the Fortin-Rautenauer formula (Theorem \[thm:FortinReutenauer\]). The proof is given later.
\[thm:degDet\_formula\] Let $A = A_0 + A_1 x_1 + A_2 x_2 + \cdots + A_m x_m$ be an $n \times n$ linear matrix over $\KK(t)$. Then $\deg {\mathop{\rm Det} }A$ is equal to the negative of the optimal value of MVMP.
Then $\deg {\mathop{\rm Det} }$ is an upper bound of $\deg \det$, analogously to the relation between ${\mathop{\rm rank} }$ and ${\mathop{\rm nc\mbox{-}rank} }$.
\[cor:deg\_leq\_Deg\] $\deg \det A \leq \deg {\mathop{\rm Det} }A$.
Let $(\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$ be an optimal solution for MVMP. By Theorem \[thm:degDet\_formula\], it holds $\deg {\mathop{\rm Det} }A = - \deg \det P - \deg \det Q$. Now $P A Q$ is also a proper matrix over $\KK(x)(t)$. By Lemma \[lem:key\] we have $\deg \det A \leq - \deg \det P - \deg \det Q = \deg {\mathop{\rm Det} }A$.
We also give an algorithm to solve MVMP for a polynomial matrix $A$.
\[thm:degDet\_algo\] Let $A = A_0 + A_1 x_1 + A_2 x_2 + \cdots + A_m x_m$ be an $n \times n$ linear matrix over $\KK[t]$. MVMP can be solved in $O(\ell n \gamma + \ell^2 m n^{\omega+2})$ time, and in $O((\ell- \alpha_n) \gamma + (\ell - \alpha_n)^2 m n^{\omega})$ time if $A$ is nonsingular, where
- $\gamma$ is the time complexity of solving MVSP for an $n \times n$ linear matrix over $\KK$,
- $\ell (= \alpha_1)$ is the maximum degree of entries in $A$,
- $\alpha_n$ is the minimum degree of the Smith-McMillan form of $A$ in $\KK(\langle x\rangle)(t)$, and
- $\omega$ is the exponent of the time complexity of matrix multiplication of $n \times n$ matrices.
Here we make a strong assumption that arithmetic operations on $\KK$ can be done in constant time. The bit-length consideration of our algorithm for the case $\KK = \QQ$ is left to future work. On this direction, Oki [@OkiHJ19] devised an alternative algorithm for $\deg {\mathop{\rm Det} }$, which works with a bounded bit-length. Theorems \[thm:degDet\_formula\] and \[thm:degDet\_algo\] are proved in the subsequent subsections.
The maximum degree of the subdeterminants of $n \times n'$ linear matrix $A$ can be computed by combining the above result with the valuated-matroid property (Proposition \[prop:valuated\]). Indeed, consider the expanded matrix $\tilde A := (I\ A)$ and the valuated matroid $\omega$ obtained from column vectors of $\tilde A$. Then the maximum degree of the subdeterminants of $A$ is equal to the maximum value of $\omega(X)$ over $X \subseteq \{1,2,\ldots,n+n'\}$ with $|X| = n$. By the greedy algorithm [@DressWenzel_greedy] (see also [@MurotaMatrix Section 5.2.4]), it is obtained by $O((n+n')^2)$ evaluations of $\omega$, where the evaluation is done by the algorithm in Theorem \[thm:degDet\_algo\].
Optimality
----------
Here we establish an optimality criterion for MVMP, and prove Theorem \[thm:degDet\_formula\]. We first note that MVMP can be viewed as L-convex function minimization on a uniform modular lattice: $$\begin{aligned}
{\rm Min.} && - \deg L - \deg M + \sum_{i=0}^m \infty \cdot \deg A_i(L,M) \nonumber \\
{\rm s.t.} && (L,M) \in {\cal L}_{\rm L}(\KK(t)^n) \times \check{\cal L}_{\rm R}(\KK(t)^n). \label{eqn:L-convex-min}\end{aligned}$$ Recall Lemma \[lem:L-convexity\] for the notation $\infty \cdot \deg A_i(L,M)$. Then the objective function is actually L-convex. Indeed, by Lemma \[lem:L-convexity\], the functions in the summation are L-convex. Recall Lemma \[lem:L(F(t)\^n)\] that $\deg$ is a unit valuation on a uniform modular lattice. Then $(L,M) \mapsto - \deg L - \deg M$ is L-convex on ${\cal L}_{\rm L}(\KK(t)^n) \times \check{\cal L}_{\rm R}(\KK(t)^n)$ with $- \deg t L - \deg t^{-1} M = - \deg L - \deg M$. Notice from the definition that the sum of L-convex functions is L-convex.
This fact and Lemma \[lem:opt’\] motivate us to consider the restriction of MVMP to interval $[(L,M),(L,M)^+] = [(L,M),(tL,t^{-1}M)]$ for a feasible solution $(L, M)$ of MVMP. Since $\deg A_i (L,M) \leq 0$, it holds $\deg A_i (L',M') \leq 1$ for $(L',M') \in [(L,M),(L,M)^+]$; see Lemma \[lem:\[L,tL\]\] (4). To study the feasibility of MVMP on $[(L,M),(L,M)^+]$, we may consider the coefficient of $t$ in $A_i(tu,v)$ for $u \in L, v \in M$. For each $A_i$, define a bilinear map $A_i^{L,M}: tL/L \times M/t^{-1}M \to \KK$ by $$\begin{aligned}
A_i^{L,M}(tu+L, v + t^{-1}M) & := & A_i(u,v)^0 \\
& = & \mbox{the coefficient of $t$ in $A_i(tu,v)$ } \quad (u \in L, v \in M).\end{aligned}$$ This is well-defined (by $A(L,M) \subseteq \KK(t)^-$). Define MVSP$^{L,M}$ by $$\begin{aligned}
{\rm MVSP}^{L,M}: \quad {\rm Max.} && \dim X + \dim Y \\
{\rm s.t.} && A^{L,M}_i(X,Y) = \{0\} \quad (i=0,1,\ldots,m), \\
&& X \in {\cal S}(tL/L), Y \in {\cal S}(M/t^{-1}M).\end{aligned}$$ Recall Lemma \[lem:\[L,tL\]\] for notation $L \circ X$ for $X \in {\cal S}(tL/L)$. Also, for $Y \in {\cal S}(M/t^{-1}M)$, define $M \bullet Y := t^{-1} M \circ Y$. Then the following lemma verifies that MVSP$^{L,M}$ is the restriction of MVMP to $[(L,M),(L,M)^+] (\simeq {\cal S}(tL/L) \times \check{\cal S}(M/t^{-1} M))$:
\[prop:augment\_MVMP\] Let $(L,M)$ be a feasible solution of MVMP. For $(X,Y) \in {\cal S}(tL/L) \times {\cal S}(M/t^{-1}M)$, we have the following:
- $\deg L \circ X + \deg M \bullet Y = \deg L + \deg M + (\dim X + \dim Y - n)$.
- $(L \circ X, M \bullet Y)$ is feasible to MVMP if and only if $(X,Y)$ is feasible to MVSP$^{L,M}$.
\(1) follows from Lemma \[lem:\[L,tL\]\] (3). (2) follows from: $$\begin{aligned}
&& \deg A_i (L \circ X, M \bullet Y) \leq 0 \\
&\Leftrightarrow& \deg A_i (tu, v) \leq 0 \quad (u \in L: tu+ L \in X,\ v \in M: v+ t^{-1}M \in Y ) \\
&\Leftrightarrow& A_i (u, v)^0 = 0 \quad (u \in L: tu+ L \in X,\ v \in M: v+ t^{-1}M \in Y ) \\
&\Leftrightarrow& A_i^{L,M}(X,Y) = \{0\}.
\end{aligned}$$
Suppose that $L$ and $M$ are given as $L = \langle P \rangle_{\rm L}$ and $M = \langle Q \rangle_{\rm R}$. Then MVSP$^{L,M}$ is also written as $$\begin{aligned}
{\rm MVSP}^{P,Q}: \quad {\rm Max.} && \dim X + \dim Y \\
{\rm s.t.} && (PA_iQ)^0 (X,Y) = \{0\} \quad (i=0,1,\ldots,m), \\
&& X \in {\cal S}(\KK^n), Y \in {\cal S}(\KK^n).\end{aligned}$$ Now we have the following optimality criterion.
\[prop:opt\_MVMP\] For a feasible solution $(L,M) = ( \langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$ of MVMP, the following conditions are equivalent:
- $(L,M)$ is optimal to MVMP.
- The optimal value of MVSP$^{L,M}$ is at most $n$ (is equal to $n$).
- The linear matrix $$(PAQ)^0 = (PA_0 Q)^0 + (PA_1 Q)^0 x_1 + \cdots + (PA_m Q)^0 x_m$$ is nonsingular on $\KK(\langle x \rangle)$.
- $\deg {\mathop{\rm Det} }A$ is equal to $- \deg \det P- \deg \det Q$.
\(4) $\Rightarrow$ (1) follows from Lemma \[lem:bound\_degDet\]. (2) $\Rightarrow$ (3) follows from Theorem \[thm:FortinReutenauer\]. Indeed, the optimal value of MVSP$^{L,M}$ is equal to $2n - {\mathop{\rm nc\mbox{-}rank} }(PAQ)^0$. If the value is at most $n$, then ${\mathop{\rm nc\mbox{-}rank} }(PAQ)^0 \geq n$, and hence ${\mathop{\rm nc\mbox{-}rank} }(PAQ)^0 = n$, i.e., $(PAQ)^0$ is nonsingular on $\KK(\langle x \rangle)$. (3) $\Rightarrow$ (4) follows from Lemma \[lem:key\]. (1) $\Rightarrow$ (2) follows from Proposition \[prop:augment\_MVMP\].
We may assume that MVMP is bounded. Take a feasible solution $(L,M) = (\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$. If $(PAQ)^0$ is nonsingular, then $\deg {\mathop{\rm Det} }A = - \deg L - \deg M$. Otherwise, we obtain another feasible solution $(L',M')$ of MVMP with $\deg L' + \deg M' > \deg L + \deg M$. Let $(L,M) \leftarrow (L',M')$. Repeating this procedure finitely many times, we obtain $\deg {\mathop{\rm Det} }A = - \deg L - \deg M$.
Notice that the proof is also obtained directly from Proposition \[prop:opt\_MVMP\] (1) $\Leftrightarrow$ (4).
Steepest descent algorithm
--------------------------
The above proof of Theorem \[thm:degDet\_formula\] is algorithmic, and naturally leads to the following algorithm, which can be viewed as the steepest descent algorithm for the L-convex function in (\[eqn:L-convex-min\]).
Steepest Descent Algorithm for $\deg {\mathop{\rm Det} }$ (coordinate-free version)
:
Input:
: A linear matrix $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ over $\KK(t)$.
Output:
: The degree $\deg {\mathop{\rm Det} }A$ of the Dieudonné determinant of $A$.
Step 0:
: Choose a feasible solution $(L,M)$ of MVMP.
Step 1:
: Solve MVSP$^{L,M}$ to obtain an mv-subspace $(X,Y)$.
Step 2:
: If $\dim X + \dim Y \leq n$, then $(L,M)$ is optimal to MVMP and output $- \deg L - \deg M$.
Step 3:
: Let $(L,M) \leftarrow (L \circ X, M \bullet Y)$, and go to step 1.
In step 1, the algorithm chooses an mv-subspace $(X,Y)$, and therefore $(L \circ X, M \bullet Y)$ is actually a steepest direction at $(L,M)$. The input is allowed to be a linear rational matrix $A$. If $A$ is nonsingular, then the algorithm outputs the correct answer after finitely many iterations; the exact number of iterations will be given in Lemma \[lem:iterations\]. In the case of singular $A$, we do not know when the algorithm should output $-\infty$.
We next consider the case of a linear polynomial matrix, and prove Theorem \[thm:degDet\_algo\]. We specialize the above algorithm with a matrix form.
Steepest Descent Algorithm for $\deg {\mathop{\rm Det} }$ (matrix version)
:
Input:
: A linear matrix $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ over $\KK[t]$, where $\ell$ is the maximum degree of entries of $A$.
Output:
: The degree $\deg {\mathop{\rm Det} }A$ of the Dieudonné determinant of $A$.
Step 0:
: Let $A_i \leftarrow A_i t^{-\ell}$ for $i=0,1,2,\ldots,m$, and $D^* \leftarrow n\ell$.
Step 1:
: Solve MVSP in the matrix form $$\begin{aligned}
{\rm Max.} && r + s \\
{\rm s.t.} && \mbox{$S A^{0}_i T$ has a zero submatrix in first $r$ rows and first $s$ columns,}\\
&& S, T \in \KK^{n \times n}: \mbox{nonsingular},\end{aligned}$$ and obtain optimal matrices $S,T$.
Step 2:
: If the optimal value $r+s$ is at most $n$, then output $D^*$.
Step 3:
: Let $A_i \leftarrow (t^{{\bf 1}_{\leq r}})S A_i T
(t^{- {\bf 1}_{> s}} )$ for $i=0,1,2,\ldots,m$, and $D^* \leftarrow D^* - (r+s - n)$. If $D^* < 0$, then output $- \infty$. Go to step 1 otherwise.
Here ${\bf 1}_{>s} := {\bf 1} - {\bf 1}_{\leq s}$. Notice that the matrix version changes the input linear matrix $A$ in each iteration. In step 0, we suppose feasible module $(L,M) = (\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})= (I, t^{-\ell} I)$ with $\deg L + \deg M = - D^* = - n \ell$. The update in step 3 can be understood as the movement from $(L,M) = (\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$ to a steepest direction $(\langle (t^{{\bf 1}_{\leq r}}) SP \rangle_{\rm L}, \langle QT(t^{- {\bf 1}_{> s}} ) \rangle_{\rm R})$ in $[(L,M), (tL,t^{-1}M)]$; see Lemma \[lem:\[L,tL\]\] (4). Since the input is a polynomial matrix, $\deg {\mathop{\rm Det} }A$ is guaranteed to be nonnegative if $A$ is nonsingular (Lemma \[lem:bound\_degDet\]). Also $D^*$ is always an upper bound of $\deg {\mathop{\rm Det} }A$. Thus $D^* < 0$ in step 3 implies $\deg {\mathop{\rm Det} }A = - \infty$.
The following modification of step 3 is natural.
Step 3$'$:
: Choose the minimum integer $\kappa \geq 1$ such that $((t^{\kappa{\bf 1}_{\leq r}})S A T
(t^{- \kappa {\bf 1}_{>s}}))^0$ has a nonzero submatrix in first $r$ rows and $s$ columns. Let $A_i \leftarrow (t^{\kappa{\bf 1}_{\leq r}})S A_i T
(t^{- \kappa {\bf 1}_{>s}} )$ for $i=0,1,2,\ldots,m$, and let $D^* := D^* - \kappa(r+s - n)$. If $\kappa$ is unbounded or $D^* < 0$, then output $\deg {\mathop{\rm Det} }A = -\infty$. Go to step 1 otherwise.
The coordinate-free formulation cannot incorporate this modification, since it depends on basis matrices for the current $(L,M)$ and the mv-subspace in step 2. The modified SDA using step 3$'$ is considered in Section \[subsec:classical\].
We next estimate the number of iterations by using L-convexity (Theorem \[thm:bound\_L\]). For this purpose, we consider the master problem $\overline{\rm MVMP}$ of MVMP: $$\begin{aligned}
\overline{\rm MVMP}: \quad {\rm Max.} && \deg L + \deg M \\
{\rm s.t.} && \deg A(L,M) \leq 0, \\
&& L \in {\cal L}_{\rm L}(\KK(\langle x \rangle)(t)^n),\ M \in {\cal L}_{\rm R}(\KK(\langle x \rangle)(t)^n),\end{aligned}$$ where the linear matrix $A$ is regarded as a bilinear form on $\KK(\langle x \rangle)(t)^n \times \KK(\langle x \rangle)(t)^n$. Solving $\overline{\rm MVMP}$ is theoretically easy. Choose biproper matrices $P,Q$ so that $PAQ$ is the Smith-McMillan form. Now $PAQ$ is the diagonal matrix $(t^{\alpha})$ for $\alpha \in \ZZ$. Consider $L^* := \langle (t^{- \alpha^-})P \rangle_{\rm L}$ and $M^* := \langle Q(t^{- \alpha^+}) \rangle_{\rm R}$ for $\alpha^+ := \max ({\bf 0}, \alpha)$ and $\alpha^- := \min ( {\bf 0}, \alpha)$. Then $(L^*,M^*)$ is feasible to $\overline{\rm MVMP}$. Also $\deg L^* + \deg M^* = - \sum_{i=1}^n \alpha_i = - \deg {\mathop{\rm Det} }A$, and hence $(L^*,M^*)$ is an optimal solution.
As for MVSP embedded to $\overline{\rm MVSP}$, MVMP is embedded to $\overline{\rm MVMP}$ by the scalar extension $(L, M) \mapsto (\KK(\langle x \rangle)(t)^- \otimes L, M \otimes \KK(\langle x \rangle)(t)^-)$. In particular MVMP is an exact inner approximation of $\overline{\rm MVMP}$. We further show that the steepest descent algorithm for MVMP is viewed as that for $\overline{\rm MVMP}$. Let $(L,M) = (\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$ be a feasible solution of MVMP and of $\overline{\rm MVMP}$. Consider MVSP$^{P,Q}$, and then $\overline{\rm MVSP}^{P,Q}$, which is given by $$\begin{aligned}
\overline{\rm MVSP}^{P,Q}: \quad {\rm Max.} && \dim X + \dim Y \\
{\rm s.t.} && (PAQ)^0 (X,Y) = \{0\}, \\
&& X \in {\cal S}_{\rm L}(\KK(\langle x \rangle)^n),
Y \in {\cal S}_{\rm R}(\KK(\langle x \rangle)^n).\end{aligned}$$ By Lemma \[lem:innerapprox\], any mv-subspace of MVSP$^{P,Q}$ is also an mv-subspace of $\overline{\rm MVSP}^{P,Q} = \overline{{\rm MVSP}^{P,Q}}$. Thus we have:
A steepest direction for MVMP at $(L,M)$ is also a steepest direction for $\overline{\mbox{MVMP}}$ at $(L,M)$.
We next show the exact number of the iterations of SDA, where by the number of the iterations we mean the number of the updates of $(L,M)$ (or $A$).
\[lem:iterations\] If $A$ is nonsingular, then the number of the iterations of the steepest descent algorithm is equal to $\alpha_1 - \alpha_n$, where $\alpha_1$ and $\alpha_n$ are the maximum and minimum degrees, respectively, of the Smith-McMillan form of $A$.
Notice that $\ell = \alpha_1$. By the initial update $A \leftarrow At^{-\alpha_1}$, we can assume that $\alpha_1 = 0 \geq \alpha_n$ and the initial point $(L,M)$ is $(\langle I \rangle_{\rm L}, \langle I \rangle_{\rm R})$. An optimal solution $(L^*,M^*)$ of $\overline{\rm MVMP}$ with $(L^*,M^*) \succeq (L,M)$ is given by $(\langle (t^{-\alpha}) P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$ for $\alpha = (\alpha_1,\alpha_2,\ldots,\alpha_n)$ and biproper $P,Q$. By $(\langle I \rangle_{\rm L}, \langle I \rangle_{\rm R}) = (\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$, both the initial point and the optimum belong to the apartment $\varSigma_{\rm L}(P) \times \check \varSigma_{\rm R}(Q)$ in ${\cal L}_{\rm L}(\KK(\langle x \rangle)(t)^n) \times \check{\cal L}_{\rm R}(\KK(\langle x \rangle)(t)^n)$. Hence the $\ell_{\infty}$-distance from initial point to optimal solutions is at most $-\alpha_n$. By Theorem \[thm:bound\_L\], the number of iteration is at most $\alpha_n$. The algorithm terminates when $\alpha_n=0$, i.e., $A$ becomes biproper (Lemma \[lem:key\]). Thus it suffices to show that $\alpha_n (< 0)$ increases by at most one in the update $A \leftarrow (t^{{\bf 1}_{\leq r}})S A T
(t^{- {\bf 1}_{> s}})$. Obviously $\delta_n = \deg {\mathop{\rm Det} }A$ increases by $r+s-n$. Also $\delta_{n-1}$ increases by $r+s - n$, $r+s -1 -n$, or $r+s + 1- n$; then $\alpha_n = \delta_n - \delta_{n-1}$ increases by most one, as required.
The increase of $\delta_{n-1}$ can be seen as follows. Notice first that the update $A \leftarrow SAT$ does not change $\delta_{n-1}$. In the next update $A \leftarrow (t^{{\bf 1}_{\leq r}}) A
(t^{- {\bf 1}_{> s}})$, the degree of an $(n-1) \times (n - 1)$ submatrix of $A$ increases by $r+s +1 - n$ if the submatrix has all of the first $r$ rows and misses one of the last $n - s$ columns, by $r+s -1 -n$ if the submatrix misses one of the first $r$ rows and has all of the last $n - s$ columns, and by $r+s -n$ otherwise.
We verify the time complexity of SDA (matrix form). After the initialization (step 0), each matrix $A_i$ is kept in the form $$\label{eqn:expression}
A_i^0 + A_i^{(1)}t^{-1} + \cdots + A_i^{(d)} t^{-d},$$ where $A_i^{(j)}$ is a matrix over $\KK$, and $d := \ell$. Step 1 can be done in $\gamma$ time. The update of expression (\[eqn:expression\]) in Step 3 can be done in $O(d mn^{\omega})$ time. The total number of iterations is $n\ell$ if $A$ is singular, and $\ell - \alpha_n$ if $A$ is nonsingular (Lemma \[lem:iterations\]). In each iteration, $d$ increases by one. Thus the total is $O(\ell n \gamma + \ell^2 mn^{\omega+2})$ time if $A$ is singular, and is $O( (\ell - \alpha_n) \gamma + (\ell - \alpha_n)^2 mn^{\omega})$ if $A$ is nonsingular.
Combinatorial relaxation algorithm
----------------------------------
The steepest descent algorithm changes basis matrices $P,Q$ in each iteration. It is a natural idea to optimize on the apartment $\varSigma_{\rm L} (P) \times \varSigma_{\rm R}(Q)$ in each iteration. This modification can be expected to reduce matrix operations, and leads to the following algorithm, which is viewed as a generalization of the [*combinatorial relaxation algorithm*]{} previously developed for $\deg \det$ [@IwataOkiTakamatsu17; @IwataTakamatsu13; @Murota90_SICOMP; @Murota95_SICOMP].
Combinatorial Relaxation Algorithm for $\deg {\mathop{\rm Det} }$
:
Input:
: A linear matrix $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ over $\KK[t]$, where $\ell$ is the maximum degree of entries of $A$.
Output:
: The degree $\deg {\mathop{\rm Det} }A$ of the Dieudonné determinant of $A$
Step 0:
: Let $A_i \leftarrow A_i t^{-\ell}$ for $i=0,1,2,\ldots,m$, and $D^* \leftarrow n\ell$.
Step 1:
: If $A^0$ is nonsingular, then output $D^* = \deg {\mathop{\rm Det} }A$.
Step 2:
: Find nonsingular matrices $S,T \in \KK^{n \times n}$ such that each $S A^{0}_i T$ $(i=0,1,2,\ldots,m)$ has a zero submatrix in first $r$ rows and first $s$ columns with $r+s > n$.
Step 3:
: Solve the following problem: $$\begin{aligned}
{\rm MVMP}(\varSigma): \quad {\rm Max.} && \sum_{i} p_i - \sum_{i} q_i \\
{\rm s.t.} && \mbox{$(t^{p}) S A T (t^{-q})$ is proper},\\
&& p,q \in \ZZ^n_+
\end{aligned}$$ to obtain optimal vectors $p,q \in \ZZ^n_+$. Let $A_i \leftarrow (t^{p})S A_i T (t^{- q})$ for $i=0,1,2,\ldots,m$, and let $D^* \leftarrow D^* - \sum_{i} p_i + \sum_{i} q_i$. If $D^* < 0$ or [MVMP]{}$(\varSigma)$ is unbounded, then output $\deg {\mathop{\rm Det} }A := -\infty$. Otherwise, go to step 1.
The condition $r+s > n$ in step 1 guarantees that $D^*$ strictly decreases. Hence the algorithm terminates after $\ell n$ steps. If the current solution $(L,M)$ is given by $(\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$, then MVMP$(\varSigma)$ is viewed as the restriction of MVMP to the apartment $\varSigma_{\rm L}(SP) \times \varSigma_{\rm R}(QT)$. Moreover, MVMP$(\varSigma)$ is the dual of the weighted matching problem in a bipartite graph. Indeed, the condition that $(t^{p})S A T(t^{-q})$ is proper is written as $$p_i - q_j + d_{ij} \leq 0 \quad (1 \leq i,j \leq n),$$ where $d_{ij}(\leq 0)$ is the maximum degree of the $(i,j)$-entry of $SAT$. Thus MVMP$(\varSigma)$ is the dual of the following weighted perfect matching problem: $$\begin{aligned}
{\rm Max.} && \sum_{i=1}^n d_{i \sigma(i)} \\
{\rm s.t.} && \sigma:\mbox{permutation on $\{1,2,\ldots,n\}$}, \end{aligned}$$ which can be efficiently solved by the Hungarian method to obtain optimal solution $p,q$ of the dual.
The combinatorial relaxation algorithm is seemingly more efficient than the steepest descent algorithm, although we do not know any nontrivial iteration bound. The meaning of “relaxation" is explained as follows. Step 3 can be viewed as a relaxation process that the linear matrix $A$ is “relaxed" into another linear matrix $\tilde{A}$ by replacing each leading term $a_{ij} t^{d_{ij}}$ $(a_{ij} \in \KK)$ of $A$ with $x_{ij} t^{d_{ij}}$ for a new variable $x_{ij}$. The optimal value of MVMP$(\varSigma)$ is the negative of $\deg {\mathop{\rm Det} }\tilde A$, and $\deg {\mathop{\rm Det} }A \leq \deg {\mathop{\rm Det} }\tilde A$. Step 1 tests whether the relaxation is tight or not.
Linear symbolic matrix with rank-$1$ summands {#sec:rank-1}
=============================================
In this section, we study a class of linear matrices $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ for which $\deg \det A = \deg {\mathop{\rm Det} }A$ holds. In the case of (nc-)rank, Lovász [@Lovasz89] showed that if each summand $A_i$ is a rank-1 matrix, then the rank of $A$ is given by MVSP, i.e., ${\mathop{\rm rank} }A = {\mathop{\rm nc\mbox{-}rank} }A$. Ivanyos, Karpinski, and Saxena [@IKS10] extended this result to the case where each $A_i$ other than $A_0$ has rank one.
\[thm:rank-1\] Let $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ be a linear matrix over field $\KK$. If $A_1,A_2,\ldots,A_m$ are rank-$1$ matrices, then ${\mathop{\rm rank} }A = {\mathop{\rm nc\mbox{-}rank} }A$.
We remark that the rank computation of such a matrix reduces to linear matroid intersection [@Lovasz89; @Soma14].
We show that Theorem \[thm:rank-1\] is naturally extended to the degree of the determinant of linear matrix $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ over $\KK(t)$.
\[thm:rank-1\_degdet\] Let $A = A_0 + A_1 x_1 + \cdots + A_m x_m$ be a linear matrix over $\KK(t)$. If $A_1,A_2,\ldots,A_m$ are rank-$1$ matrices, then $\deg \det A = \deg {\mathop{\rm Det} }A$.
Consider an optimal module $(\langle P \rangle_{\rm L}, \langle Q \rangle_{\rm R})$ for MVMP. By Proposition \[prop:opt\_MVMP\], the linear matrix $(PAQ)^0$ is nonsingular as a matrix over $\KK(\langle x \rangle)$. Notice that each $(PA_iQ)^0$ for $i=1,2,\ldots,m$ has rank one. Indeed, $PA_iQ$ is a rank-1 matrix over $\KK(t)$, and hence is written as $\tilde u \tilde v^{\top}$ for (nonzero) $\tilde u, \tilde v \in \KK(t)^n$. Consider the maximum degrees $c$ and $d$ of the components of $\tilde u$ and $\tilde v$, respectively. Necessarily $c+d \leq 0$ (since $PA_iQ$ is proper), and $PA_iQ$ is written as $t^{c}u t^{d}v^{\top}$ for $u,v \in (\KK(t)^-)^n$. Then $(PA_iQ)^0 = u^0(v^0)^\top$ if $c+d = 0$ and zero if $c+d < 0$.
By Theorem \[thm:rank-1\], the linear matrix $(PAQ)^0$ is also nonsingular as a matrix over $\KK(x)$. By Lemma \[lem:key\], we have $\deg \det A = - \deg \det P - \deg \det Q = \deg {\mathop{\rm Det} }A$.
Observe from the Fortin-Rautenauer formula (Theorem \[thm:FortinReutenauer\]) that the nc-${\mathop{\rm rank} }A$ is a property of the matrix vector subspace ${\cal A} \subseteq \KK^{n \times n'}$ spanned by $A_0, A_1,\ldots,A_m$ over $\KK$. Therefore, ${\mathop{\rm rank} }A= {\mathop{\rm nc\mbox{-}rank} }A$ still holds if the matrix subspace ${\cal A}' \subseteq \KK^{n \times n'}$ spanned by $A_1,\ldots,A_m$ admits a rank-1 basis $B_1, B_2,\ldots, B_{m'}$. Indeed, the constraint $A_i(X,Y) = \{0\}$ in MVSP can be replaced by $B_i(X,Y) = \{0\}$. See [@Gurvits04; @IvanyosKarpinskiOiaoSantha15] for the rank computation of such a linear matrix with a hidden rank-1 basis.
In the case of $\deg {\mathop{\rm Det} }$, instead of the matrix vector space, we may consider the matrix submodule ${\cal A} \subseteq \KK(t)^{n \times n}$ generated by $A_1,A_2,\ldots,A_m$ over $\KK(t)^-$. Since $\KK(t)^-$ is a PID and ${\cal A}$ is a submodule of a free module of matrices with bounded degree entries, ${\cal A}$ is also free, and has a $\KK(t)^-$-basis. Analogously to the above, $\deg \det A = \deg {\mathop{\rm Det} }A$ holds if the matrix module ${\cal A}'$ generated by $A_1,\ldots,A_m$ admits a rank-1 basis $B_1, B_2,\ldots, B_{m'}$; the constraint $\deg A_i(L,M) \leq 0$ can be replaced by $\deg B_i(L,M) \leq 0$.
An important example of a linear matrix with possibly ${\mathop{\rm rank} }< {\mathop{\rm nc\mbox{-}rank} }$ is a skew-symmetric linear matrix $A = \sum_{i=1}^m A_i x_i$ with rank-2 skew-symmetric summands $A_i$; see the next example. The problem of computing the (usual) rank of such a matrix is a generalization of the nonbipartite matching problem, and is equivalent to the [*linear matroid parity problem*]{}; see [@Lovasz89]. Recently Iwata and Kobayashi [@IwataKobayashi17] developed a polynomial time algorithm for the [*weighted*]{} linear matroid parity problem by considering $\deg \det$ and using the idea of the combinatorial relaxation method. It is an interesting future direction to refine our non-commutative framework for skew-symmetric linear matrices to capture nonbipartite matching and its generalizations.
Consider the following $3$ by $3$ linear skew-symmetric matrix $A = A_1x_1 + A_2 x_2 + A_3x_3$ (with rank-2 summands): $$A = \left(
\begin{array}{ccc}
0 & x_1 & x_2 \\
-x_1 & 0 & x_3 \\
-x_2 & -x_3 & 0
\end{array} \right).$$ Then it is obvious that ${\mathop{\rm rank} }A = 2$. However ${\mathop{\rm nc\mbox{-}rank} }A = 3$. Indeed, it holds that $(1\ u \ v)A_i(1\ u'\ v')^{\top} = 0$ $(i=1,2,3)$ implies $u= u'$ and $v = v'$ and $(1\ u \ v)A_i(0\ u'\ v')^{\top} = 0$ $(i=1,2,3)$ implies $u' = v' = 0$. From this, we see that there is no vanishing subspace $(X,Y)$ with $(\dim X, \dim Y) = (2,1)$ or $(1,2)$. Therefore mv-subspaces are trivial ones $(K(\langle x \rangle)^3, 0)$ and $(0, K(\langle x \rangle)^3)$, and ${\mathop{\rm nc\mbox{-}rank} }A = 3$.
Next consider a weighted version $$A = \left(
\begin{array}{ccc}
0 & t^{c_1}x_1 & t^{c_2} x_2 \\
- t^{c_1} x_1& 0 & t^{c_3} x_3 \\
- t^{c_2} x_2 & - t^{c_3}x_3 & 0
\end{array} \right).$$ for weights $c_1,c_2, c_3 \in \ZZ$. Then, for $\alpha := (c_1+ c_2 -c_3, c_1- c_2 + c_3, - c_1+ c_2 + c_3)$, it holds $$( t^{- \alpha/2}) A ( t^{- \alpha/2})
= \left(
\begin{array}{ccc}
0 & x_1 & x_2 \\
-x_1 & 0 & x_3 \\
-x_2 & -x_3 & 0
\end{array} \right).$$ By Proposition \[prop:opt\_MVMP\], we have $\deg {\mathop{\rm Det} }A = c_1 + c_2 + c_3$. On the other hand, it obviously holds $\deg \det A = - \infty$.
Some classical examples in combinatorial optimization {#subsec:classical}
-----------------------------------------------------
As mentioned in the introduction, some of classical combinatorial optimization problems are formulated as the computation of the degree of the determinant of a linear matrix with the rank-1 property. Here we consider representative three examples (bipartite matching, linear matroid greedy algorithm, linear matroid intersection), and explain how the steepest descent algorithm works on these problems. This gives some new insights on classical algorithms in combinatorial optimization.
For a subset $J \subseteq \{1,2,\ldots,n\}$, let $\QQ^J \subseteq \QQ^n$ denote the coordinate subspace spanned by unit vectors $e_i$ for $i \in J$, and let ${\bf 1}_{J} := \sum_{i \in J} e_i$.
### Bipartite matching {#subsub:bipartite}
Let $G = (U,V; E)$ be a bipartite graph with color classes $U,V$. Vertices of $U$ (resp. $V$) are numbered as $1,2,\ldots,n$ (resp. $1,2,\ldots,m$). As mentioned in the introduction, the maximum size $\nu(G)$ of a matching of $G$ is written as the rank of an $n \times m$ linear matrix $A = \sum_{e = ij \in E} x_e E_{ij}$, where $E_{ij}$ is the matrix having $1$ at $(i,j)$-entry and zero at others, and $x_e$ $(e \in E)$ are variables. Each $E_{ij}$ of $A$ has rank $1$. It holds that ${\mathop{\rm rank} }A =$ nc-${\mathop{\rm rank} }A$. By Theorem \[thm:FortinReutenauer\], $\nu(G)$ is equal to $n+m$ minus the dimension of an mv-subspace $(X,Y)$. Observe that any feasible subspace $(X,Y)$ of MVSP is of the form of $(\QQ^J, \QQ^K)$ for $J \subseteq \{1,2,\ldots,n\}, K \subseteq \{1,2,\ldots,m\}$ such that there is no edge between $J$ and $K$, i.e., $J \cup K$ is a stable set of $G$, and is the complement of a vertex cover. Thus Theorem \[thm:FortinReutenauer\] is nothing but König’s formula for the maximum matching.
Next we consider the weighted situation. Suppose $|U| = |V| = n$ for simplicity, and that each edge $e \in E$ has weight $c_e \in \ZZ$. Consider a linear matrix $A := \sum_{e = ij \in E} t^{c_{e}} x_e E_{ij}$ over $\QQ (t)$. Then the maximum weight of a perfect matching of $G$ is equal to $\deg \det A$, and is equal to $\deg {\mathop{\rm Det} }A$ (by Theorem \[thm:rank-1\_degdet\]). We explain how the steepest descent algorithm works in this case. We use the modified step 3$'$. Suppose for explanation that $c_e \leq 0$ for each $e \in E$. Linear matrix $A^0$ corresponds to the subgraph $G^0$ consisting of edges with $c_e = 0$. A steepest direction is given by $(\QQ^J, \QQ^K)$ for a maximum stable set $J \cup K$ of $G^0$. In step 3$'$, $\kappa$ is chosen as the maximum of $- c_e (> 0)$ for edges $e \in E$ belonging to $J \cup K$. Then $A$ is updated to $(t^{\kappa {\bf 1}_J}) A (t^{- \kappa ({\bf 1}- {\bf 1}_K)})$. SDA repeats this process, which is viewed as a cut-canceling algorithm. The resulting optimal solution is a form of $(\langle(t^{p}) \rangle_{\rm R}, \langle(t^q)\rangle_{\rm L})$ for $p,q \in \ZZ^n$. Here vectors $p,q$ are dual optimal solutions of the LP-formulation of the weighted matching problem. If we always choose a maximum stable set $J \cup K$ with minimal $J$ (and maximal $K$) in each iteration, then SDA coincides with the Hungarian method. Indeed, $J$ is the subset reachable from vertices in $U$ not covered by a maximum matching in the corresponding residual graph of $G^0$.
### Maximum weight base in linear matroid
Let $a_1,a_2,\ldots,a_m$ be $n$-dimensional vectors of $\QQ^n$. Consider $m$ variables $x_1,x_2,\ldots,x_m$, and linear matrix $A = \sum_{i=1}^m x_i a_i a_i^{\top}$. Then ${\mathop{\rm rank} }A = {\mathop{\rm nc\mbox{-}rank} }A = {\mathop{\rm rank} }(a_1\ a_2\ \cdots a_m)$. Let $W \subseteq \QQ^n$ be the vector space spanned by $a_1,a_2,\ldots,a_m$. Then $(W^{\bot},\RR^n)$ is an mv-subspace, where $W^{\bot}$ denotes the orthogonal subspace of $W$.
As in Section \[subsub:bipartite\], consider the weighted situation. Let $c_i \in \ZZ$ be the weight on $a_i$ for each $i$. Consider linear matrix $A = \sum_{i=1}^m t^{c_i} x_i a_i a_i^{\top}$. Then $\deg \det A = \deg {\mathop{\rm Det} }A$ is equal to the maximum of $\sum_{i \in B} c_i$ over all $B \subseteq \{1,2,\ldots,m\}$ such that $\{ a_i \mid i \in B\}$ forms a basis of $\QQ^n$. Namely $\deg \det A$ is equal to the maximum weight of a base of the matroid represented by vectors $a_1,a_2,\ldots,a_m$.
In this case, the steepest descent algorithm is viewed as the greedy algorithm. Suppose that each $c_i$ is nonpositive. Then $A^0$ is the linear matrix $\sum_{i \in I_0} x_i a_i a_i^{\top}$, where $I_0$ is the set of indices $i$ with $c_i = 0$. Consider the subspace $W_1$ spanned by $a_i$ $(i \in I_0)$. Then $(W_1^{\bot},\QQ^n)$ is an mv-subspace. Consider a nonsingular matrix $Q \in \QQ^{n \times n}$ such that the first $k_1$ rows form a basis of $W_1^{\bot}$. In step 3, $A$ is updated as $(t^{{\bf 1}_{\leq k_1}}) Q A$, or feasible module $(L,M)$ moves from $(\langle I \rangle_{\rm L}, \langle I \rangle_{\rm R})$ to $(\langle (t^{{\bf 1}_{\leq k_1}})Q \rangle_{\rm L}, \langle I \rangle_{\rm R})$. The exponent of term $t^{c_i} x_i Q a_ia_i^{\top}$ increases if and only if $a_i$ does not belong to $W_1$. Thus, in step 3$'$, SDA can augment $A$ as $(t^{\alpha_1 {\bf 1}_{\leq k_1}}) Q A$ until $c_i + \alpha_1$ becomes zero for some $a_i \not \in W_1$. Then $I_0$ increases, and the next subspace $W_2$ spanned by $a_i$ $(i \in I_0)$ increases. Consequently $W_2^{\bot} \subset W_1^{\bot}$. We can modify $Q$ so that it also includes a basis of $W_2^{\bot}$. SDA moves $(L,M)$ to $(\langle (t^{\alpha_1 {\bf 1}_{\leq k_1} + \alpha_2 {\bf 1}_{\leq k_2}}Q) \rangle_{\rm L}, \langle I \rangle_{\rm R})$, and obtain $W_3^{\bot} \subset W_2^{\bot}$ as above. Repeat the same process. Eventually SDA reaches an optimal module $(\langle (t^{\sum_{j=1}^h \alpha_j {\bf 1}_{\leq k_j}})Q \rangle_{\rm L}, \langle I \rangle_{\rm R})$, where $Q$ consists of bases of vector spaces $W_1^{\bot} \supset W_2^{\bot} \supset \cdots \supset W_h^{\bot}$. This process simply chooses vectors $a_i$ from largest weights. It is nothing but the matroid greedy algorithm, where we need no explicit computation of $Q$. The obtained $\alpha_k$ can be interpreted as an optimal dual solution of the LP-formulation of the maximum weight base problem, where $\alpha_k$ is the dual variable corresponding to the flat $\{i \mid a_i \in W_k\}$.
### Linear matroid intersection
In addition to $a_1,a_2,\ldots,a_m$ above, we are given vectors $b_1,b_2,\ldots,b_m \in \QQ^n$. Consider a linear matrix $A = \sum_{i=1}^m x_i a_i b_i^{\top}$ with variables $x_1,x_2,\ldots,x_m$. We have ${\mathop{\rm rank} }A = {\mathop{\rm nc\mbox{-}rank} }A$, they are equal to the maximum cardinality of a subset $I \subseteq \{1,2,\ldots,m\}$ such that both $\{ a_i \mid i \in I\}$ and $\{b_i \mid i \in I\}$ are independent. Namely, ${\mathop{\rm rank} }A$ is the maximum cardinality of a common independent set of two matroids ${\bf M}_1$ and ${\bf M}_2$ represented by $a_1,a_2,\ldots,a_m$ and $b_1,b_2,\ldots,b_m$, respectively. For $I \subseteq \{1,2,\ldots,m\}$, let $\rho(I)$ and $\rho'(I)$ denote the dimension of vector spaces spanned by $a_i$ $(i \in I)$ and by $b_i$ $(i \in I)$, respectively. By the matroid intersection theorem, ${\mathop{\rm rank} }A$ is the minimum of $\rho(I) + \rho'(J)$ over all bi-partitions $I,J$ of $\{1,2,\ldots,m\}$. Then an mv-subspace $(X,Y)$ is given by $X = \{a_i \mid i \in I\}^{\bot}$ and $Y = \{b_j \mid j \in J\}^{\bot}$ for bi-partition $I,J$ attaining the minimum. This fact is noted in [@Lovasz89].
Suppose that we are further given weights $c_i \in \ZZ$ for each $i=1,2,\ldots,m$. Consider a linear matrix $A = \sum_{i=1}^m t^{c_i} x_i a_i b_i^{\top}$ over $\QQ(t)$. Then $\deg \det A = \deg {\mathop{\rm Det} }A$ is equal to the maximum weight $\sum_{i \in B} c_i$ of a common independent set $B \subseteq \{1,2,\ldots,m\}$ with $|B| = n$ of matroids ${\bf M}_1$ and ${\bf M}_2$. Namely, the problem of finding $\deg \det A$ is the weighted linear matroid intersection problem. Let us explain the behavior of the steepest descent algorithm applied to this case. Suppose that we are given a feasible module $(L,M)$ of form $L = \langle (t^{\alpha}) S \rangle_{\rm L}$ and $M = \langle T (t^\beta) \rangle_{\rm R}$ for nonsingular matrices $S,T \in \QQ^{n \times n}$ and integer vectors $\alpha, \beta \in \ZZ^n$. It may appear that a naive choice of a steepest direction $(X,Y)$ at $(L,M)$ would violate this form in the next step, but, in fact, such a situation can naturally be avoided.
Let $R_1,R_2,\ldots,R_\mu$ be the partition of $\{1,2,\ldots,n\}$ such that $i,j$ belong to the same part if and only if $\alpha_i = \alpha_j$. Similarly, let $C_1,C_2,\ldots,C_\nu$ be the partition such that $i,j$ belong to the same part if and only if $\beta_i = \beta_j$. Regard matrix $SAT$ as a block matrix, where columns and rows are partitioned by $R_1,R_2,\ldots,R_\mu$ and $C_1,C_2,\ldots,C_\nu$. In $(t^\alpha) SAT (t^\beta)$, the $(k,\ell)$-th block is uniformly multiplied by $t^{\alpha_i + \beta_j}$ for $i \in R_k$, $j \in C_\ell$. Consider the linear matrix $((t^\alpha) SAT (t^\beta))^0$ (to obtain a steepest direction). Then each summand $((t^\alpha) S a_ib_i^{\top} T (t^\beta))^{0}x_i$ has at most one nonzero block, where the nonzero block (if it exists) has rank $1$. Now $((t^\alpha) SAT (t^\beta))^0$ is essentially in the situation of a [*partitioned matrix with rank-1 blocks*]{} [@HH16DM]. See also Section \[subsec:HH\] in Appendix. By the partition structure, any mv-subspace $(X,Y)$ is of the form of $(X_1 \oplus X_2 \oplus \cdots \oplus X_{\mu}, Y_1 \oplus Y_2 \oplus \cdots \oplus Y_{\nu})$ for $X_k \subseteq \QQ^{R_k}$ and $Y_\ell \subseteq \QQ^{C_\ell}$ $(k=1,2,\ldots,\mu,\ell = 1,2,\ldots,\nu)$. Then basis matrices $S',T'$ for $X,Y$ are taken as block diagonal form so that $S'(t^{\alpha}) = (t^{\alpha})S'$ and $T'(t^{\beta}) = (t^{\beta})T'$. In the next iteration, $(L,M)$ is $(\langle (t^{\alpha'}) S'S \rangle_{\rm L}, \langle TT'(t^{\beta'}) \rangle_{\rm R})$. Consequently the obtained optimal solution is of the form of $(\langle (t^{\alpha}) S \rangle_{\rm L}, \langle T(t^{\beta}) \rangle_{\rm R})$. In particular, exponent vectors $\alpha$ and $\beta$ can be dealt with as numerical vectors. We here note that this algorithm is viewed as a variant of the primal dual algorithm for weighted matroid intersection problem by Lawler [@Lawler75]. His algorithm keeps and updates chains of flats in ${\bf M}_1$ and ${\bf M}_2$ and their weights. Observe that module $\langle T (t^{\beta})_{\rm L} \rangle$ can be identified with a chain $\emptyset \neq X_1 \subset X_2 \subset \cdots \subset X_n = \QQ^n$ of subspaces and coefficients $\lambda_i$ $(i=1,2,\ldots,n)$ such that $\lambda_i \geq 0$ for $i < n$. Indeed, arrange $\beta$ as $\beta_1 \geq \beta_2 \geq \cdots \geq \beta_n$, and define $X_i$ as the subspace spanned the first $i$ rows and $\lambda_i$ as $\beta_i - \beta_{i+1}$ (with $\beta_{n+1} = 0$). This correspondence is unique if subspaces $X_i$ with $\lambda_i = 0$ are omitted. In this way, module $(L,M)$ can be kept as a pair of weighted chains of subspaces. If these subspaces are orthogonal complements of the subspaces spanned by some subsets of $a_1,a_2,\ldots,a_m$ and $b_1,b_2,\ldots,b_m$, then $(L,M)$ can further be kept by a pair of weighted chains of flats of matroids ${\bf M}_1$ and ${\bf M}_2$, as in Lawler’s algorithm.
Moreover, if we choose an mv-subspace $(X,Y)$ with minimal $X$ (and maximal $Y$) in each iteration and use the modified step 3$'$, then SDA coincides with the [*weight splitting algorithm*]{} by Frank [@Frank81] (applied to linear matroids), where $(\alpha, \beta)$ corresponds to a weight splitting; see also [@KorteVygen Section 13.7] for the weight splitting algorithm. The detail of this correspondence is given in [@FurueHirai19].
Mixed polynomial matrix
-----------------------
A [*mixed polynomial matrix*]{} is a polynomial matrix $A = \sum_{k=0}^\ell (Q_k + T_k) t^k$ with indeterminate $t$ such that $Q_k$ is a matrix over $\QQ$, each entry of $T_k$ is zero or one of variables $x_1,x_2,\ldots,x_m$, and each variable $x_i$ appears as one entry of one of $T_1,T_2,\ldots,T_k$. In the case of $\ell =0$, $A$ is called a [*mixed matrix*]{}. See [@MurotaMatrix] for detail of mixed (polynomial) matrices. A mixed polynomial matrix is viewed as a linear matrix over $\QQ(t)$ with rank-$1$ summands, since the coefficient matrix of $x_k$ is written as $E_{ij}$. Therefore it holds that $\deg \det A = \deg {\mathop{\rm Det} }A$. It is shown in [@IwataOkiTakamatsu17; @IwataTakamatsu13] that the combinatorial relaxation algorithm computes $\deg \det A$ in $O(\ell^2 n^{\omega+2})$ time. This estimate seems very rough, since it is based on a trivial bound $\ell n$ of the number of iterations. In the case of the steepest descent algorithm, we obtain a sharper estimate.
\[thm:mixedpoly\] Let $A$ be an $n \times n$ mixed polynomial matrix with maximum degree $\ell$. By the steepest descent algorithm, $\deg \det A$ can be computed in $O(\ell^2 n^{\omega+2})$ time, and in $O((\ell - \alpha_n) n^3 \log n+ (\ell - \alpha_n)^2 n^{\omega} )$ time if $A$ is nonsingular, where $\alpha_n$ is the minimum degree of diagonals of the Smith-McMillan form of $A$.
To prove Theorem \[thm:mixedpoly\], we will work on a mixed matrix of a special form, as in [@IwataOkiTakamatsu17]. A [*layered mixed (polynomial) matrix*]{} [@MurotaBook] is a mixed (polynomial) matrix $A$ of form $$\label{eqn:LM0}
A = \left( \begin{array}{c}
Q \\
T
\end{array}\right),$$ where $Q$ is a matrix over $\QQ[t]$ and $T$ is a variable matrix as above. It is well-known in the mixed-matrix literature that the rank and deg-det computation of a mixed matrix $Q+T$ reduce to those of a layered one $$\label{eqn:LM}
\left(\begin{array}{cc}
Q & I \\
T & D
\end{array}\right),$$ where $D$ is a diagonal matrix of new variables.
To compute a steepest direction, we need an mv-subspace of a layered mixed (nonpolynomial) matrix, which is naturally obtained from a min-max formula of the rank. Let $A$ in (\[eqn:LM0\]) be an $n \times n'$ layered mixed matrix. For $J \subseteq \{1,2,\ldots,n'\}$, let $Q[J]$ denote the submatrix of $Q$ consisting of $j$-th columns for $j \in J$, and let $\varGamma(J)$ denote the set of indices $i$ with $T_{ij} \neq 0$ for some $j$.
For an $n \times n'$ layered mixed matrix $A$ in $(\ref{eqn:LM0})$. $$\label{eqn:LM-rk-formula}
{\mathop{\rm rank} }A = \min_{J \subseteq \{1,2,\ldots,n'\}}
\{ {\mathop{\rm rank} }Q[J] + |\varGamma(J)| - |J| \} + n'.$$
Let $R_Q$ and $R_T$ denote the sets of row indices of matrices $Q$ and $T$, respectively.
\[lem:J\] Let $J$ be a minimizer of $(\ref{eqn:LM-rk-formula})$. Let $X := \ker_{\rm L} Q[J] \oplus \QQ^{R_T \setminus \varGamma(J)}$ and $Y := \QQ^{J}$. Then $(X,Y)$ is an mv-subspace.
This follows from $n + n' - \dim X - \dim Y = n + n' - |R_Q| + {\mathop{\rm rank} }Q[J] - |R_T| + |\varGamma(J)| - |J| = {\mathop{\rm rank} }Q[J] + |\varGamma(J)| - |J| + n'$.
For a mixed matrix $Q+T$, we compute the degree of the determinant of the corresponding layered mixed matrix (\[eqn:LM\]). As an initialization (step 0), $(P, Q)$ is defined as $P = I$ and $Q = (t^{- \ell {\bf 1}_{\leq n}})$, and let $A \leftarrow PAQ$. In step 1, SDA computes an mv-subspace of layered mixed matrix $A^0$. A minimizer $J$ of (\[eqn:LM-rk-formula\]) is obtained by Cunningham’s matroid intersection algorithm [@Cunningham86] in $O(n^3 \log n)$ time. Namely $\gamma = O(n^3 \log n)$. In step 3, the matrix multiplication is needed only for the $Q$-part of $A$, which eliminates $m$ in the time complexity of Theorem \[thm:degDet\_algo\].
#### Application to DAE.
A motivating application of mixed polynomial matrices is analysis of linear [*differential algebraic equations (DAE)*]{} with constant coefficients, where each coefficient is an accurate number or one of (inaccurate) parameters $x_1,x_2,\ldots, x_m$, and no parameter appears as distinct coefficients; see [@MurotaBook Chapter 6]. By the Laplace transformation, the analysis of such a DAE reduces to linear equation $A x = b$, where $A$ is a mixed polynomial matrix over $\RR[s]$ with variables $x_1,x_2,\ldots,x_m$. Suppose the case where matrix $A$ is a square matrix. The [*index*]{} is a barometer of “difficulty" of DAE $Ax = b$, and is defined as $- \alpha_n + 1$, where $\alpha_n$ is the minimum degree of the Smith-McMillan form of $A$. A DAE with high index ($\geq 2$) is difficult to solve numerically, and suggests an inconsistency of the mathematical modeling in deriving this DAE. Therefore it is meaningful to decide whether the index of given a DAE is at most the limit $\varDelta$. Here $\varDelta (\simeq 2)$ is the allowable upper bound for the index of DAE-models of the system we want to analyze. The steepest descent algorithm can decide in $O((\ell + \varDelta) n^3 \log n + (\ell+ \varDelta)^2 n^{\omega})$ time whether DAE $Ax = b$ has index at most $\varDelta$. Indeed, apply SDA to $A$. Index $- \alpha_n + 1$ is obtained from the number $\ell - \alpha_n$ of required iterations (Lemma \[lem:iterations\]). If SDA terminates before $\ell + \varDelta$ iterations, then the DAE has index within $\varDelta$. Otherwise the index is over the limit $\varDelta$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks Kazuo Murota, Satoru Iwata, Yuni Iwamasa, Taihei Oki and Koyo Hayashi for careful reading and helpful comments, and thanks Mizuyo Takamatsu for remarks. Also the author thanks the referees for helpful comments. This work was supported by JSPS KAKENHI Grant Numbers JP26280004, JP17K00029.
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Appendix
========
Relation to the formulation by Hamada and Hirai {#subsec:HH}
-----------------------------------------------
Hamada and Hirai [@HamadaHirai17] actually studied the following variant of MVSP: We are given a matrix $A \in \KK^{m \times n}$ partitioned into submatrices as $$A = \left(
\begin{array}{ccccc}
A_{11} & A_{12} &\cdots & A_{1\nu} \\
A_{21} & A_{22} &\cdots & A_{2\nu} \\
\vdots & \vdots & \ddots & \vdots \\
A_{\mu1}&A_{\mu2} &\cdots & A_{\mu \nu}
\end{array}\right),$$ where $A_{\alpha \beta} \in \KK^{m_\alpha \times n_\beta}$ for $\alpha=1,2,\ldots,\mu$ and $\beta=1,2,\ldots,\nu$. The goal is to find a collection of vector subspaces $X_\alpha \subseteq \KK^{m_{\alpha}}$, $Y_{\beta} \subseteq \KK^{n_{\alpha}}$ $( \alpha=1,2,\ldots,\mu; \beta=1,2,\ldots,\nu)$ such that $$\label{eqn:vanishing}
A_{\alpha \beta} (X_{\alpha}, Y_{\beta}) = \{0\} \quad
(\alpha=1,2,\ldots,\mu; \beta=1,2,\ldots,\nu),$$ and the sum of their dimensions $$\label{eqn:dimension}
\sum_{\alpha = 1}^{\mu} \dim X_{\alpha} + \sum_{\beta = 1}^{\nu} \dim Y_{\beta}$$ is maximum. Hamada and Hirai refer to this problem as MVSP. We here call it [*block-MVSP*]{}
Block-MVSP reduces to MVSP (in our sense). Indeed, introduce a new variable $x_{\alpha \beta}$ for each $\alpha,\beta$, and consider MVSP for the linear matrix $$\tilde A = \left(
\begin{array}{ccccc}
A_{11}x_{11} & A_{12}x_{12} &\cdots & A_{1\nu}x_{1\nu} \\
A_{21}x_{21} & A_{22}x_{22} &\cdots & A_{2\nu}x_{2\nu} \\
\vdots & \vdots & \ddots & \vdots \\
A_{\mu1}x_{\mu1}&A_{\mu2}x_{\mu2} &\cdots & A_{\mu \nu}x_{\mu \nu}
\end{array}\right),$$ Observe that any mv-subspace $(X,Y)$ is necessarily a form of $(X_1 \oplus X_2 \oplus \cdots \oplus X_{\mu}, Y_1 \oplus Y_2 \oplus \cdots \oplus Y_{\nu})$, where $X_{\alpha}, Y_{\beta}$ satisfy (\[eqn:vanishing\]) and maximize (\[eqn:dimension\]). (Consider the projections $X_{\alpha}$ of $X$ and $Y_{\beta}$ of $Y$ to the coordinate subspaces corresponding to the partitions. Then $(\oplus_{\alpha} X_\alpha, \oplus_{\beta} Y_\beta)$ is also feasible to MVSP with $X \subseteq \oplus_{\alpha} X_\alpha$ and $Y \subseteq \oplus_{\beta} Y_\beta$.)
The converse reduction is also possible. For a linear $n \times n'$ matrix $A = A_0 + \sum_{i=1}^{m} A_i x_i$, consider the block matrix $$\label{eqn:converse}
\left(
\begin{array}{ccccc}
A_{0} & I & & \\
A_{1} & I & \ddots & \\
\vdots & & \ddots & I \\
A_{m} & & & I
\end{array}\right),$$ where the unfilled blocks are zero matrices and $I$ is the $n$ by $n$ unit matrix. For a solution $(X,Y)$ of MVSP for $A$, the collection of subspaces $X_{\alpha}$, $Y_{\beta}$ defined by $$\begin{aligned}
&& X_{\alpha} := X \quad (\alpha =1,\ldots, m+1), \\
&& Y_1 := Y,\ Y_{\beta} := X^{\bot} \quad (\beta =2,\ldots, m+1)
\end{aligned}$$ is a solution of the block-MVSP for the matrix (\[eqn:converse\]), and has the sum of dimensions $$(m+1) \dim X + \dim Y + m (n- \dim X) = \dim X + \dim Y + mn.$$
In fact, we can always choose an optimal solution with such a form in this block-MVSP. Let $X_{\alpha}, Y_{\beta}$ ($\alpha,\beta =1,\ldots,m+1$) be an optimal solution of the block-MVSP. Necessarily $Y_{\alpha+1} = X_{\alpha}^{\bot} \cap X_{\alpha+1}^{\bot}$ holds for $\alpha = 1,2,\ldots,m$. Then it holds $$\begin{aligned}
&& \dim X_{\alpha} + \dim X_{\alpha+1} + \dim Y_{\alpha+1} \\
&& = \dim X_{\alpha} \cap X_{\alpha+1} + \dim (X_{\alpha} + X_{\alpha+1}) + \dim X_{\alpha}^{\bot} \cap X_{\alpha+1}^{\bot} \\
&& = \dim X_{\alpha} \cap X_{\alpha+1} + \dim (X_{\alpha} + X_{\alpha+1}) + \dim (X_{\alpha} + X_{\alpha+1})^{\bot} \\
&& = \dim X_{\alpha} \cap X_{\alpha+1} + n \\
&& = 2 \dim X_{\alpha} \cap X_{\alpha+1} + \dim (X_{\alpha} \cap X_{\alpha+1})^{\bot}.
\end{aligned}$$ Hence $(X_{\alpha}, X_{\alpha+1}, Y_{\alpha+1})$ can be replaced by $( X_{\alpha} \cap X_{\alpha+1}, X_{\alpha} \cap X_{\alpha+1}, (X_{\alpha} \cap X_{\alpha+1})^{\bot})$. This implies the existence of an optimal solution in which all $X_{\alpha}$ are equal. In particular, $(X,Y) := ( X_1 \cap \cdots \cap X_{m+1},Y_1)$ is an mv-subspace of MVSP for $A$.
As was noted in [@HamadaHirai17 Remark 3.13], without such a reduction, their approach and algorithm are quickly adapted to MVSP (in our sense).
Proof of Lemma \[lem:L(F(t)\^n)\]
---------------------------------
Let ${\cal L} := {\cal L}_{\rm R}(\FF(t)^n)$. We omit ${\rm R}$ from $\langle \cdot \rangle_{\rm R}$. We first consider the join and meet of two $L, M \in {\cal L}$. Suppose that $L = \langle P \rangle$ and $M = \langle Q \rangle$. Then $L \subseteq M$ if and only if $Q^{-1}P$ is proper. From we see that $t^{- \ell} L \subseteq L \cap M \subseteq L + M \subseteq t^\ell L$ for a large $\ell> 0$. Both $L \cap M$ and $L + M$ are submodules of the free module $t^{\ell}L$ over PID $\FF(t)^-$. Hence both $L \cap M$ and $L + M$ are free. Since they contain the full-rank free module $t^{- \ell} L$, they are also full-rank, and hence belong to ${\cal L}$. Necessarily $L \wedge M = L \cap M$ and $L \vee M = L + M$.
We show the modularity $L + (M \cap L') = (L + M) \cap L'$ for $L' \in {\cal L}$ with $L \subseteq L'$. It suffices to show $(\supseteq)$. Let $z = x + y \in (L+ M) \cap L'$ with $x \in L$ and $y \in M$. By $z,x \in L'$, it holds $y = z - x \in L'$, and $y \in M \cap L'$. Thus $z = x+y \in L + (M \cap L')$, as required. Note that this argument is standard for proving that normal subgroups of a group forms a modular lattice.
We next show that $\deg$ is a unit valuation. Here we can assume that $M = \langle Q \rangle = \langle P (t^{\alpha}) \rangle$, where $(t^{\alpha})$ is the Smith-McMillan form of $P^{-1} Q$. Indeed, If $S^{-1}P^{-1}QT = (t^{\alpha})$ for biproper $S,T$, then $P$ and $Q$ can be replaced by $PS$ and $QT$, respectively, since $\langle P \rangle =\langle PS \rangle$ and $\langle Q \rangle =\langle QT \rangle$. Therefore, if $L$ is covered by $M$, then $\alpha = e_1$, and $\deg M - \deg L = \deg {\mathop{\rm Det} }P(t^{e_1})- \deg {\mathop{\rm Det} }P = 1$．
Also $\langle t P \rangle$ is the join of $\langle P (t^{e_i}) \rangle$ for $i=1,2,\ldots,n$, and the ascending operator is given by $L \mapsto t L$, which is clearly an automorphism on ${\cal L}$. This concludes that ${\cal L}$ is a uniform modular lattice.
Proof of Lemma \[lem:apartment\]
--------------------------------
We continue the above notation of omitting ${\rm R}$. Consider two short chains $C,D$. By (B1) in Lemma \[lem:skeleton\], there is a $\ZZ^n$-skeleton $\varSigma$ containing them. Identify $\varSigma$ with $\ZZ^n$. If $L \in \varSigma$ corresponds to $x \in \ZZ^n$, then we write $L \equiv x$. We may assume that $C,D$ belong to interval $[0,k]^n \subseteq \ZZ^n$. It suffices to show that the interval $[0,k]^n$ belongs to $\varSigma(P)$ for some nonsingular matrix $P$. Consider $L \in \varSigma$ with $L \equiv {\bf 0} \in \ZZ^n$. We show by an inductive argument that there are $p_1,p_2,\ldots,p_n \in L$ such that $L = \langle (p_1\ p_2 \cdots\ p_n) \rangle$ and $\langle(p_1\ \cdots\ t^{\ell} p_{i}\ \cdots\ p_n) \rangle \equiv \ell e_i$ for each $i$ and $\ell \leq k$. Then $P = (p_1\ p_2\ \cdots\ p_n)$ is a desired matrix. Indeed, for $\alpha \in [0,k]^n$, the join $M$ of $\langle(p_1\ \cdots\ t^{\alpha_i} p_{i}\ \cdots\ p_n)\rangle$ over $i=1,2,\ldots, n$ corresponds to $\alpha$. Obviously $\langle P (t^{\alpha}) \rangle \subseteq M$. Since $M$ and $\langle P (t^{\alpha}) \rangle$ have the same height $\deg L + \alpha_1 + \alpha_2 + \cdots + \alpha_n = \deg \det P (t^{\alpha})$, it must hold $\langle P (t^{\alpha}) \rangle = M \equiv \alpha$. Thus the interval $[0,k]^n$ belongs to $\varSigma(P)$.
Suppose that $L = \langle (q_1\ q_2 \cdots\ q_n) \rangle$. By Lemma \[lem:\[L,tL\]\] (4), for each $i$ there is $p_i \in L$ such that $e_i \equiv p_i t \FF(t)^- + L$. Then $p_1,p_2,\ldots,p_n$ form an $\FF(t)^-$-basis of $L$, i.e., $L = \langle (p_1\ p_2 \cdots\ p_n) \rangle$. Indeed, by $\sum_{i} p_i t \FF(t)^- + L \equiv e_1 + e_2 + \cdots + e_n = {\bf 1} \equiv t L$, each $tq_j$ is written as an $\FF(t)^-$-linear combination of $tp_1,tp_2,\ldots,tp_n$ and $q_1,q_2,\ldots,q_n$. Consequently, $(q_1\ q_2\ \cdots\ q_n )(I + t^{-1}A) = (p_1\ p_2\ \cdots \ p_n)B$ holds for some square matrices $A,B$ over $\FF(t)^-$. Here $I + t^{-1}A$ is biproper (Lemma \[lem:key\]). This means that each $q_j$ is written as an $\FF(t)^-$-linear combination of $p_1,p_2,\ldots,p_n$. Then $L = \langle (p_1\ p_2 \cdots\ p_n) \rangle$ and $e_i \equiv \langle (p_1\ \cdots\ tp_i\ \cdots\ p_n) \rangle = p_i t \FF(t)^- + L$ for each $i$.
Suppose (by induction) that $\langle (t^\ell p_1\ p_2\ \cdots\ p_n) \rangle \equiv \ell e_1$ holds for $\ell \leq k-1 (\geq 1)$. We show that $p_1$ can be replaced by some $p'_1$ such that $\langle (t^\ell p'_1\ p_2\ \cdots\ p_n) \rangle \equiv \ell e_1$ holds for $\ell \leq k$. Consider $L' \in \varSigma$ with $L' \equiv k e_1$. By Lemma \[lem:\[L,tL\]\] (4), $L'$ is generated by vectors obtained by replacing one of $t^{k-1} p_1,p_2,\ldots, p_n$ with $r = t (t^{k-1} p_1 \lambda_1 + \sum_{i=2}^{n} p_i \lambda_i)$ for $\lambda_i \in \FF$. Now $\lambda_1 \neq 0$. Indeed, if $\lambda_1 =0$ and $\lambda_2 \neq 0$ (say), then $L' = \langle (t^{k-1} p_1\ r\ \cdots\ p_n) \rangle \subseteq
t \langle(t^{k-2} p_1\ p_2\ \cdots\ p_n)\rangle$, implying a contradiction $k e_1 \leq (k-2) e_1 + {\bf 1}$. Let $p'_1 := t^{-k} r$. Thus $L' = \langle (t^{k} p'_1\ p_2\ \cdots\ p_n)\rangle$. Also observe $\langle (t^{\ell} p'_1\ p_2\ \cdots\ p_n) \rangle
= \langle (t^{\ell} p_1\ p_2\ \ldots\ p_n) \rangle$ for $\ell \leq k-1$. Replace $p_1$ by $p'_1$, and apply the same replacement for $p_2,p_3,\ldots,p_n$. Then we obtain desired $p'_1,p'_2,\ldots,p'_n$.
Proof of Proposition \[prop:SmithMcMillan\] {#app:SM}
-------------------------------------------
The degree of the determinant $\deg {\mathop{\rm Det} }$ is a [*matrix valuation*]{} in the sense of [@CohnSkewField Section 9] (with min and max interchanged); see Theorem 9.3.4 of the reference. In particular, the $\deg {\mathop{\rm Det} }$ function satisfies the following property ((MV.4) in [@CohnSkewField Section 9.3]):
[(MV)]{}
: For nonsingular $A \in \FF(t)^{n \times n}$ and a vector $b \in \FF(t)^n$ regarded as a row (or column) vector, let $B$ be the matrix obtained from $A$ by replacing the first row (or column) by $b$, and let $C$ be the matrix obtained from $A$ by adding $b$ to the first row (or column) vector. Then it holds $$\deg {\mathop{\rm Det} }C \leq \max \{ \deg {\mathop{\rm Det} }A, \deg {\mathop{\rm Det} }B \}.$$ The strict inequality holds only if $\deg {\mathop{\rm Det} }A = \deg {\mathop{\rm Det} }B$.
In [@CohnSkewField], only the column version is proved but the row version can be proved in the same way. Indeed, by column permutation, we can make $A$ (and $B,C$) so that the cofactor $A'$ of $(1,1)$-entry is nonsingular. Then we have $$A =
\left(\begin{array}{cc}
a_{11} & a' \\
0 & A'
\end{array}\right)E, \
B =
\left(\begin{array}{cc}
b_{11} & b' \\
0 & A'
\end{array}\right)E,\
C =
\left(\begin{array}{cc}
c_{11} & c' \\
0 & A'
\end{array}\right)E,$$ where $E$ is the product of permutation matrices and upper unitriangular matrices, and $(c_{11}\ c') = (a_{11}\ a') + (b_{11}\ b')$. Thus $\deg {\mathop{\rm Det} }A = \deg a_{11} + \deg {\mathop{\rm Det} }A'$, $\deg {\mathop{\rm Det} }B = \deg b_{11} + \deg {\mathop{\rm Det} }A'$, and $\deg {\mathop{\rm Det} }C = \deg c_{11} + \deg {\mathop{\rm Det} }A' = \deg (a_{11}+ b_{11}) + \deg {\mathop{\rm Det} }A'$. From $\deg (a_{11} + b_{11}) \leq \max \{ \deg a_{11}, \deg b_{11} \}$, we obtain (MV).
Now let us start to prove Proposition \[prop:SmithMcMillan\]. For $u \in \FF(t)$ and $k,\ell \in \{1,2,\ldots,n\}$ with $k \neq \ell$, define $E(k,\ell;u) \in \FF(t)^{n \times n}$ by $$E(k,\ell;u)_{ij} = \left\{
\begin{array}{ll}
1 & {\rm if}\ i=j, \\
u & {\rm if}\ i=k, j = \ell, \\
0 & {\rm otherwise}.
\end{array}\right.$$ $E(k,\ell;u)$ is called an [*elementary matrix*]{}. Observe that $E(k,\ell;u)$ is nonsingular with $E(k,\ell;u)^{-1} = E(k,\ell;-u)$. In particular, $E(k,\ell;u)$ is biproper if and only if $u \in \FF(t)^-$.
A required diagonalization is obtained as follows. First, by multiplying permutation matrices to the left and the right of $A$, modify $A$ so that $A_{{11}}$ has the maximum degree among all entries of $A$. By multiplying elementary matrices $E(1,\ell;u)$ from right and $E(\ell',1; u')$ from left, modify $A$ so that all entries except $A_{11}$ in the first row and column are zero. Here $u,u'$ can be taken from $\FF(t)^-$ by the maximality. Therefore $E(1,\ell;u)$ and $E(\ell,1';u')$ are biproper, and the degree of entries of $A$ does not increase. Now $A_{11}$ is written as $t^{\alpha_1} v$ for $\alpha_1 = \deg A_{11}$ and $v \in \FF(t)^-$ with $\deg v = 0$. Multiply a biproper diagonal matrix whose $(1,1)$-entry is $v^{-1}$ and other diagonals are $1$. Then $A_{11}$ is now $t^{\alpha_n}$. Repeat the same process to the submatrix from the second row and column. Eventually $A$ is diagonalized to $(t^{\alpha})$ with $\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_n$. By construction, $P,Q$ are the product of proper elementary matrices and permutation matrices, and are biproper.
Next we show that $\delta_k$ is invariant throughout the above procedure, which implies the latter part of the claim. It is obvious that $\delta_k$ is invariant under any row and column permutation. Consider the case of multiplying elementary matrix $E(i,j;u)$ from the right. This operation corresponds to adding the $i$-th columns multiplied by $u$ to the $j$-th column. Consider a $k \times (k+1)$ submatrix having the $i$- and $j$-th columns, and consider the change of its $k \times k$ minors by the multiplication of $E(i,j;u)$. Obviously any $k \times k$ minor containing the $i$-th column does not change. Consider the $k \times k$ minor not containing the $i$-th column. From the property (MV), the degree of this minor is at most the degree of the original or $\deg u (\leq 0)$ plus the degree of the minor not containing $j$. From this, we see that the maximum degree of a $k \times k$ minor of this matrix does not change. Consequently $\delta_k$ does not change. The proof for the left multiplication is the same.
[^1]: In the literature of building, such a module is called a [*lattice*]{}. We do not use this term for avoiding confusion.
[^2]: The second M means subModule.
|
---
abstract: 'In the Dictionary-based String Matching (DSM) problem, a retrieval system has access to a source sequence and stores the position of a certain number of strings in a posting table. When a user inquires the position of a string, the retrieval system, instead of searching in the source sequence directly, relies on the the posting table to answer the query more efficiently. In this paper, the Statistical DSM problem is a proposed as a statistical and information-theoretic formulation of the classic DSM problem in which both the source and the query have a statistical description while the strings stored in the posting sequence are described as a code. Through this formulation, we are able to define the efficiency of the retrieval system as the average cost in answering a users’ query in the limit of sufficiently long source sequence. This formulation is used to study the retrieval performance for the case in which (i) all the strings of a given length, referred to as $k$-grams , and (ii) prefix-free codes.'
address: |
Electrical and Computer Engineering Department,\
National Chiao Tung University (NCTU), Taiwan
bibliography:
- 'posting\_bib.bib'
title: |
The statistical dictionary-based\
string matching problem
---
Introduction {#sec:intro}
============
Let us define a Dictionary-based String Matching (DSM) problem is defined as follows. The *retrieval system* has access to a *source sequence* and constructs a *posting table* in which it stores the position of a set of source substrings, referred to as a *posting code*. More precisely, each row in the posting table contains the *posting list* corresponding to a given codeword, consisting of a list of positions in which the codeword appears in the source sequence, as selected by the *posting function*. At a later time, a user submits a string to the retrieval system, termed a *query*, and the retrieval system is tasked with providing the positions of the query in the source sequence, called *matches*. If the query does not appear in the source sequence, a empty message is returned to the user. If the retrieval system is not able to retrieve some of the matches, an error message is returned.
We consider a variation of the classic DSM in which we assume that (i) the source sequence and the query have a statistical description and that (ii) the cost of a query is proportional to the product of the length of the entries in the posting table visited by the retrieval system in answering a query. Under these two assumptions, we consider the problem of designing the posting code that minimizes the expected cost of retrieving the positions of a query, from the information in the posting sequence in the limit of an infinitely long source sequence. We term this problem as the Statistical DSM (SDSM). We are interested in the study of the SDSM as we wish to determine the ultimate information searching efficiency in the posting table. As such, this paper represents a stepping stone toward the development of a universal and dynamic SDSM in which the source sequence is any stationary sequence while the query distribution is unknown at the retrieval system. [**Literature Review:**]{} The DSM problem has been studied in a number of context and modeled through various assumptions so that a vast literature is available on the topic. To the best of our knowledge, no formulation has explicitly considered either the distribution of source and queries, or the performance in the limit of large source length. In the information retrieval context, the DSM problem is referred to as “inverted index” problem and the concern is with the respect to the memory required to store the entries of the posting table [@baeza1999modern]. The distribution of the queries is used in [@baeza2003three] to design a three level memory organization for a search engine inverted file index. In computation lingustics and natural language processing, the DSM problem has been studied to determine robust retrieval methods [@baeza1998fast; @mihov2004fast] to search in a text affected by errors. In the context of genomics, bioinformatics and computational biology, the problem is sometimes referred to as off-line or indexed pattern matching: here the focus is on the retrieval of sequences that approximatively match the given query [@apostolico1997pattern; @navarro2001indexing]. More generally, the source distribution implicitly appears in the literature concerned with the compression of the entries of the posting table, such as [@anh2005inverted], [@yan2009inverted] and [@chen2010inverted], [**Contributions & Organization:**]{} The remainder of the paper is organized as follows
$\bullet$ [**Sec. \[sec:Problem Formulation\]- Problem formulation:**]{} We propose formulation of the SDSM problem which accounts for source and query along with the cost of accessing the entries in the posting table. We define the efficiency of a retrieval system as the expected cost of retrieving a query in the limit for large source length. Through this performance measure, we formulate an optimization problem that helps us determine the code with the optimal memory utilization.
$\bullet$ [**Sec. \[sec:examples\]- Relevant examples:**]{} To validate the propose model, we study in detail the performance of two retrieval system: one storing (i) $k$-grams, all possible source sequences of length $k$ and (ii) prefix-free codes, codes in which no codeword is a prefix of another codeword. For illustrative purposes, we consider the simple case of binary i.i.d. source and query distributions. $\bullet$ [**Sec. \[sec:Numerical Evaluations\]- Numerical Evaluations:**]{} we numerically investigate the design of the optimal code for the case of binary i.i.d. source and query distributions.
*Notation:* With $\xv= [x_1, \ldots, x_N] \subseteq \Xcal^N$ we indicate a sequence of elements from $\Xcal$ with length $N$. The notation $\xv_i^j$ indicates the substring $[x_i, \ldots, x_j]$. Given the sequence $\xv$, $l(\xv)$ indicates the length of the sequence $\xv$, $w(\xv)$ indicates the Hamming weight, respectively. The notation $\av.\bv$ indicates the vector concatenation operation. The notation $\av \preceq \bv$ indicates that $\av$ is a substring of $\bv$. Let $\Pcal(\Xcal)$ indicate the power set of $\Xcal$. Define $\xo=1-x$.
Problem Formulation {#sec:Problem Formulation}
===================
The SDSM problem is comprised of a *source sequence*, a *retrieval system* and a *user*. The source sequence is defined as the random sequence $X^N$ with support $\Xcal^N$ and distribution $P_{X^N}(\xv)$ and let the *query* be defined as the Random Variable (RV) $Q$ with support $\Pcal(\Xcal) \setminus \emptyset$ and with distribution $P_{Q}(\qv)$. A retrieval system is comprised of a *posting code*, a *posting table*, a *storing function* and a *retrieval function*. A posting code of size $M$ is defined as the set $\Ccal=\{\cv_i\}_{i=1}^M$ with $\cv_i \in \Pcal(\Xcal^N)$. The set of *source matches* of the codeword $\cv_i$ is the set $\Ical(\cv_i) = \{W_m(i)\}_{m \in \Nbb}$ for which The *posting list* of the codeword $\cv_i$ is defined as $\Tcal(\cv_i,X^N)$ and is such that $\Tcal(\cv_i,X^N) \subseteq \Ical(\cv_i)$. The *posting table* is defined as the tuple $T(\Ccal,X^N)=\{\Tcal(\cv_i,X^N) \}_{i=1}^M$. The *storing function* is the mapping $f_S(\Ical(\cv_i),X^{N})$ which produces the posting list from the set of source matches for each codeword $\cv_i$, i.e. A user provides a *query* $Q$ with distribution $P_Q(\qv)$ to the retrieval system: upon receiving a query $Q$, the retrieval system produces a *covering* of length $V$ of the query $Q=\qv$, defined as the tuple $\Scal(\qv)=\{\cv^{(V)},\nv^{(V)}\}$ such that If a covering of the query does not exists, a *retrieval error* is declared. Once a covering is produced, the retrieval system fetches the position of the codewords in the covering from the posting table. Finally, the *retrieval function*, $f_R$, is the mapping where $\mv$ is such that $X_{m_i}^{l(\qv)}=\qv$ for all $m_i \in \mv$.
The average size of the posting list and the average size of the posting table are defined as [ ]{} respectively. If a covering for the query exists, than the cost of a covering $\Scal(\qv)$ is defined as If a covering for the query does not exist, than the cost of the query is infinite. The *minimum expected cost* for a given query $Q=\qv$, $c(\qv)$, is defined as
Finally, we are now ready to state the optimization problem of our interest. For given source, query distributions, and the size of the posting code, the maximal efficiency $\eta$ in the SDSM problem is defined as
The cost function in is chosen so as to approximate the complexity of finding the positions in the entries of the posting lists in $\Scal(\qv)$ corresponding to contiguous codewords in the covering. See [@buttcher2016information; @baeza2004fast; @baeza2005experimental].
The Pre-fix free coded, Complete and Parsed (PCP) SDSM problem
--------------------------------------------------------------
In the above formulation, the SDSM problem is presented in the greatest possible generality. In the following, we focus on a specific formulation of the SDSM problem, the Pre-fix free coded, Complete and Parsed (PCP) SDSM problem, which can be more readily analyzed. In particular, we consider the case in which (i) the posting code is a complete pre-fix free code (see [@yeung2012first]), (ii) the posting table stores all the matches, (iii) queries are covered by non-overlapping codewords. While property (i) and (ii) are straightforward, for (iii), we resort to the following definition. A covering is defined as a *parsing* if there exists an $K$ such that $n_j+l(\cv_{(j)})=n_{j+1}$ for $j\in [1:K-1]$, while for $j \in [K:V]$ and no codeword outside the set $\{\cv_{j}\}_{K}^V$ satisfies . In other words, a parsing of a query is a covering with no overlapping over the codewords, apart from the tail of the query. In the tail of the query, codewords start from the same position $n_K$ and overflow the end of the sequence. The parsing between $K$ and $V$ contains all codewords that contain $Q_{n_K}^{l(Q)}$ as a prefix. The string $Q_{n_K}^{l(Q)}$ is referred to as the tail of the query. Our interest in the PCP-SDSM problem is motivated by the next theorem.
\[thm:PCP-SDSM\] In the PCP-SDSM problem, the following holds: $-$ no retrieval error occurs,
$-$ the minimum covering cost is always finite,
$-$ there exists only one parsing of any query, thus this parsing is the optimal covering,
$-$ the number of entries in the posting table is always equal to $N$.
Relevant Examples {#sec:examples}
=================
In the remainder of the paper, we evaluate the efficiency for two example codes. In both cases, we consider the scenarios of binary i.i.d. sources and queries distribution. In particular, the source distribution is obtained as
[**PCP-SDSM problem with a $k$-gram code:**]{} Perhaps the simplest choice of posting code for the binary i.i.d. setting is the case in which $\Ccal$ contains all possible binary sequences of length $k$ such that $M=2^{k}$. Such a posting code is usually referred to as $k$-gram code and is typical employed in genomic research for indexing DNA sequences, such as in the well-known BLAST algorithm [@altschul1990basic].
In the regime of large blocklength, the length of the posting table is obtained by constructing a Markov chain with $M$ states, each corresponding to a possible $k$-gram. The $i$-th window of the source sequence, $K_i=X_{i}^{i+k}$, can be represented as a state in the Markov chain: as the window slides by one position, yielding $K_{i+1}=X_{i+1}^{i+k+1}$, this corresponds to a state transition of the Markov chain. The length of the posting sequence of each codeword in the codebook can then be obtained as the average time spent in the corresponding state of the Markov chain. By considering the structure of the Markov chain and transition probability matrix, we obtain the steady state distribution $\pi_{i} = \po^{k-w(\cv_{i})}p^{w(\cv_{i})}$ for $i=1,2\ldots,M$ and the average time spent in the state $K_i=\cv_i$. Let us next consider the cost of each query: let us assume that the queries are obtained as $\sum_l P_L(l) P_{Q|L=l}$ and that that is, given that the query length is $l$, the query is an i.i.d. sequence of Bernoulli distribution with parameter $q$ of length $l$. The RV determining the length of the query can always be expressed as quotient and remainder of the division by $k$, i.e. where $Z \in \Nbb$ and $R=[0:k-1]$. The minimum expected cost of the query is then that is, the cost of the query is the cost of parsing the query with $Z$ $k$-grams along with covering the tail and accounting for its cost. If the tail has length zero, than the cost of the tail is zero, otherwise the tail of the query is composed of all $2^{k-R}$ codewords with prefix $Q_{k Z}^{kZ+R}$.
\[lem: k mer\] For the PCP-SDSM problem with a $k$-gram posting code and source and query distribution in and , the efficiency is obtained as for $Z$ and $R$ are defined as in .
[**PCP-SDSM problem for Run-Length Encoding (RLE):**]{} RLE is very simple form of lossless data compression to encode binary data in which one symbol occurs with much higher frequency than the other. This coding is useful, for instance, when encoding line drawings, as the black pixels are sparse. For such a setting, we consider the problem of identifying a specific binary pattern that can itself be described as a set of $B$ run lengths. For this reason, we consider a posting code of the form where $\zeros(x)$ is the vector of all zeros of length $x$, so that $l(\cv_i)=i$ for $i\in[1:M-1]$ and $l(\cv_M)=M-1$. In other words, the retrieval system stores the successive occurrences of zeros before a one appears, up to length $M-1$. As argued for the case of $k$-grams, the length of each entry in the posting table can obtained from the average time spent in the state $K_i=\cv_i$ in the Markov chain corresponding to the windowing of the source sequence. Accordingly, in the regime of sufficiently large $N$, the length of each entry in the posting sequence converges to since each codeword apart from $\cv_M$ has unitary weight. Let us next define the distribution queries: queries are of the form $S_1.S_2\ldots S_B$, where $S_j$ is a run length of length $k_j$ and $B$ is the number of run lengths. The distribution of the $S_j$ is obtained Similar to , the length of success-run $S_j$ can be expressed as quotient and remainder of the division by $M-1$, i.e. so that the cost of the query is $Z_j$-times the cost of the all zero codeword plus the cost of the codeword equal to $R_j$.
\[lem: run length coding\] For the PCP-SDSM problem with a RLE code and source and query distribution in and , the efficiency is obtained as where $B$ is the number of run lengths in the query.
Numerical Evaluations {#sec:Numerical Evaluations}
=====================
We performed two of sets of preliminary, small-scale simulations to evaluate the the proposed model and to gain some insight into the performance of the codes discussed in the previous section. For the first simulation, prefix-free codes with maximum codeword lengths of 8 were generated by probabilistic splitting of nodes in the code trees. The efficiency of these randomly generated prefix-free codes was compared with that of $k$-gram codes, with a maximum $k$ of 8 and two sets of $p$ and $q$. We observe that values for $M$ in the neighborhood of 128, the prefix-free codes have a better efficiency than the $k$ gram for $k=7$ as the minimum expected costs are lower in this region. We also see that, as $M$ approaches 256, the efficiency of the prefix-free codes approaches that of the $k$-gram for $k=8$. In the case of run-length codes, from Lem. \[lem: run length coding\] we see that as the source sequence grows large, the efficiency approaches the expected number of run-lengths in the query. This behavior can be observed in Fig. \[fig:run length conv\]. Queries with up to 4 run-lengths ($B=4$), were simulated and the distribution of $B$ was chosen to be geometric with a success probability of $0.8$.
\[fig:run length conv\]
Conclusion {#sec:Conclusion}
==========
In the paper we propose a statistical and information-theoretic formulation of the dictionary-based string matching (SDSM) problem. In the SDSM problem, a retrieval system has access to a source sequence and it stores the position of a certain number of strings, in a table called the posting table. Upon receiving a query from a user, the retrieval system access the entries in the table to efficiently determine the position of the matches in the source sequence. For this problem, we assume that source and query distributions are described as random processes and we propose a cost function for the query retrieval. Through this formulation, we are able to define an optimal posting code as the code which attains the smallest expected cost in retrieving a query. For the proposed model, we provide some relevant examples and preliminary numerical evaluations.
Appendix A: Proof of Th. \[thm:PCP-SDSM\] {#appendix-a-proof-of-th.-thmpcp-sdsm .unnumbered}
-----------------------------------------
Any (posting) code can be represented by a $|\Xcal|$-ary tree in which codewords are represented as nodes in the tree and each branch outgoing from an edge is labeled with one of the elements in $\Xcal$. The codeword associated with each node is obtained as the sequence of labeled visited in the path from the root of the three to the node. A prefix-free code is represented by a tree in which all codewords are leaves. A code is complete if all the leaves in the three representing the code are codewords. Let us prove each of the properties of the PCP SDSM in Th. \[thm:PCP-SDSM\].
$-$ [**no retrieval error occurs:**]{} since the posting code is complete, any sequence can be parsed using such code. This follows because, starting from the beginning of the source sequence, the first codeword ends in a leaf of the tree. The next symbol in the source sequence will, consequently, start from the root of the coding tree and the second codeword will again end in a leaf. By repeating this argument, the desired property is shown.
$-$ [**the minimum covering cost is always finite:**]{} since queries are parsed with posting codewords and given that the posting code is complete, it follows that a parsing of a query always exists
$-$ [**there exists only one parsing of any query, thus this parsing is the optimal covering:**]{} again following from the completeness of the posting code, it follows that there exists a unique parsing of any codeword.
$-$ [**the number of entries in the posting table is always equal to $N$:**]{} at each position in the source sequence, a codeword exists. Following from the completeness of the storing function, such codeword is stored in the posting table.
Appendix B: Proof of Lem. \[lem: k mer\] {#appendix-b-proof-of-lem.-lem-k-mer .unnumbered}
----------------------------------------
To construct the postings table, the source sequence $X^N$ is parsed into overlapping $k$-grams such that each $k$-gram has an overlap of $k-1$ bits with its adjacent $k$-grams and the positions of each $k$-gram in the sequence are recorded in postings lists. Let us first derive the average length of each posting list: this can be determined by observing that the transition from a $k$-gram to its adjacent overlapping $k$-gram can be described through a Markov chain. Consider the Markov chain with $M$ states, each corresponding to a possible $k$-gram, and each labelled with the decimal representation of the corresponding $k$-gram. The transition between two states corresponds to the sliding of the $k$-gram of a position forward, i.e. $K_i=X_{i}^{i+k}$ to $K_{i+1}=X_{i+1}^{i+k+1}$. The transition matrix can be constructed by observing that $$P_{ij}=
\begin{cases}
\po, & \text{if } i \leq 2^{k-1} \text{ and } j = 2i-1\\
p, & \text{if } i \leq 2^{k-1} \text{ and } j = 2i\\
\po, & \text{if } i > 2^{k-1} \text{ and } j = 2(i-2^{k-1})-1\\
p, & \text{if } i > 2^{k-1} \text{ and } j = 2(i-2^{k-1})\\
0, & \text{otherwise },
\end{cases}$$ since the sliding removes the most significant bit in the $k$-grams and introduces a least significant bit. The transition probability matrix can be represented has a block matrix structure has $$P = \begin{pmatrix}
\po & p & 0 & 0 & \cdots & 0 & 0 & 0 & 0 \\
0 & 0 & \po & p & \cdots &0 & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & \po & p & 0 & 0 \\
0 & 0 & 0 & 0 & \cdots & 0 & 0 & \po & p \\
\po & p & 0 & 0 & \cdots & 0 & 0 & 0 & 0 \\
0 & 0 & \po & p & \cdots & 0 & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & \po & p & 0 & 0 \\
0 & 0 & 0 & 0 & \cdots & 0 & 0 & \po & p \\
\end{pmatrix}.$$ In order to calculate the average length of the posting list of a given $k$-gram, we make use of some results on Markov chains. For a Markov chain with transition matrix $P$ and initial state $i$, the steady state distribution is denoted by $\pi$, so that the following holds The number of visits to state $i$ before time $n$ is Under suitable conditions, we can determine the average time spent in a state using the following result and so the average length of the postings list of codeword $i$ is given by The steady-state distribution is obtained as the left eigenvector corresponding the eigenvalue at one. We denote component $i$ of the eigenvector $u$ by $u_{i}$. This eigenvector has an eigenvalue of 1. Based on the structure of the transition matrix $P$, we guess the following where $|w_{i}|$ is the hamming weight of the k-gram $w_{i}$. If $u$ is the eigenvector corresponding to eigenvalue 1, it will satisfy the following equation If holds for the choice in , this must indeed be the eigenvector as eigenvectors are unique. Eq. \[eq:eigenvector\] can be expressed component wise as Observe that there are only two non-zero entries in every column of $P$ and that these entries are always separated by $2^{k-1}$ rows. This implies that $w_{i+2^{k-1}} = w_{i}+1$. Using this observation, the RHS of the preceding equation for $j=0,2,..,M$ can be written as When $j=1,3,..,M-1$, the multiplication factor $(1-p)$ is replaced with $p$. For the last step, note that $w_{\frac{j-1}{2}} = w_{j} - 1$ and so $w_{\frac{j-1}{2} + 2^{k-1}} = w_{j}$. The eigenvector needs to be normalized to make it a steady-state distribution. To do this end, we calculate the sum $S$ of the elements of the eigenvector as follows Dividing $u$ by $S$ we obtain the steady-state distribution $\pi_{i} = (\po)^{k-w_{i}}p^{w_{i}} $ which yields for $i=1,2, \ldots,M$ and where $\stackrel{\Pcal}{=}$ indicates equality in probability. Note that the average time spent in a state only depends on the Hamming weight of the state. Next we move to the analysis of servicing a query. A query $\qv$ with length $l(\qv)$ is parsed into successive $k$-grams to service it using the inverted index of $k$-grams. Since, in a PCP-SDSM problem there exists only one parsing, this parsing is also optimal. The number of $k$-grams in the parsing of length $L$ in is $Z+1_{\{R>0\}} 2^{k-R}$ since the number of codewords to cover a tail of length $R$ is $2^{k-R}$. For each $k$-grams, since symbols in a $k$-grams are iid, we conclude that holds. Finally, we have that $Q_1^K$ is a Binomial random varible so that where we have used and where $M_W(q,k,t)$ indicates the moment generating function of a Binomial random variable with parameters $q$ and $k$ with independent variable $t$.
Appendix C: Proof of Lem. \[lem: run length coding\] {#appendix-c-proof-of-lem.-lem-run-length-coding .unnumbered}
-----------------------------------------------------
Also, for the case of RLE, the structure of the codes allows obtaining a compact expression for the average cost of servicing a query. As argued for the case of $k$-grams, the length of each entry in the posting table can obtained from the average time spent in the state $K_i=\cv_i$ in the Markov chain corresponding to the windowing of the source sequence. The expected cost of each run length is then obtained as
Since the source symbols are iid, the expected cost of a run length, decomposed as in is
In this case, we normalize the cost term to factor out the effect of the sequence length, and then take the expected value
This is generalized to the case of $b$ run lengths by using the fact that run lengths are i.i.d.
Taking the log and re-normalizing, we obtain the expression
|
---
abstract: 'We study extensions of the minimal supersymmetric standard model (MSSM) with new degrees of freedom that couple sizably to the MSSM Higgs sector and lie in the TeV range. After integrating out the physics at the TeV scale, the resulting Higgs spectrum can significantly differ from typical supersymmetric scenarios, thereby providing a window beyond the MSSM (BMSSM). Taking into account current LEP and Tevatron constraints, we perform an in-depth analysis of the Higgs collider phenomenology and explore distinctive characteristics of our scenario with respect to both the standard model and the MSSM. We propose benchmark scenarios to illustrate specific features of BMSSM Higgs searches at the Tevatron and the LHC.'
author:
- Marcela Carena
- Eduardo Pontón
- José Zurita
title: BMSSM Higgs Bosons at the Tevatron and the LHC
---
Introduction {#sec:intro}
============
There has been a recent surge of interest in extensions of the minimal supersymmetric standard model (MSSM) by higher-dimension operators [@Strumia:1999jm; @Brignole:2003cm; @Dine:2007xi; @Antoniadis:2007xc; @Antoniadis:2009rn; @Randall:2007as; @Batra:2008rc; @Carena:2009gx; @Blum:2009na; @Casas:2003jx; @Cassel:2009ps]. These can have an important impact on the Higgs sector, alleviating in particular the tension present in the MSSM that results from the LEP Higgs bounds. Such effective field theory (EFT) studies allow a model-independent description of a large class of extensions of the MSSM, and permit one to quantify the sense in which the Higgs sector can serve as a window beyond the MSSM (BMSSM).
This point of view was clearly put forward in Ref. [@Dine:2007xi], where it was emphasized that at leading order in $1/M$ –where $M$ is the scale of the physics that is integrated out– the MSSM is extended by only two parameters. The surprisingly large effects of such higher-dimension operators can be understood from the fact that the MSSM Higgs potential is rather restricted at tree-level. The nonrenormalizable operators in the superpotential induce renormalizable (quartic) operators in the Higgs potential that are not present in the MSSM limit (at tree-level), so that in spite of the fact that their coefficients are “small” –of order $\mu/M$– they correspond to qualitatively new effects. In fact, the operators thus induced can easily “destabilize” the MSSM-like minimum and lead to new minima that exist only as a direct result of the higher-dimension operators (i.e. the heavy physics). It was emphasized in [@Batra:2008rc] that such minima can be phenomenologically viable, can be studied within the EFT framework, and can explain the distinct properties induced by the heavy physics on the Higgs sector.
If the BMSSM physics is sufficiently heavy, the leading order analysis at order $1/M$ can suffice. However, it is perfectly possible that $M$ is not too far from the electroweak (EW) scale, and that nevertheless the heavy physics may not be easy to see directly at the LHC, even if it is within its kinematic reach (e.g. heavy singlets that couple only through the Higgs sector). In such cases, the EFT approach is still useful to describe the properties of the MSSM Higgs sector. It turns out that the effects of order $1/M^{2}$ are more important than naively expected. This observation also finds a simple explanation in the structure of the MSSM tree-level Higgs potential [@Carena:2009gx] together with the smallness of the MSSM tree-level quartic couplings (the root cause for a Higgs state lighter than the $Z$ mass in the MSSM at tree-level). The crucial point is that the leading order contributions to a subset of the quartic Higgs operators, from the heavy physics, first enter at order $1/M^{2}$. Thus, these end up correcting a coefficient of order $g^{2}$ instead of a $1/M$ effect, and can give a relevant correction even if the expansion parameter is relatively small. Nevertheless, it is important to appreciate that the fact that the first two orders in the expansion in $1/M$ can even result in comparable contributions to the Higgs masses, in no way implies a breakdown of the EFT.
In Ref. [@Carena:2009gx] a detailed study of the consequences for the Higgs masses and couplings up to order $1/M^{2}$ was given, and a selected number of phenomenological observations were already made, such as: enhanced gluon fusion production cross sections in a large number of cases, and the presence of “exotic” decay modes with more than one Higgs boson in the decay chain. In this work we analyze the constraints from LEP and the Tevatron on the neutral Higgs bosons, as well as the charged Higgs bounds from LEP [^1]. We also expand on the associated collider phenomenology, emphasizing the type of signals that can be expected at both the Tevatron and the LHC. We point out that due to the large corrections to the Higgs masses (especially to the CP-even Higgs bosons) the production and decay patterns can be markedly different from those in the MSSM. Examples include models where both CP even Higgs bosons have significant branching fractions into gauge bosons, thus giving rise to spectacular signals such as two clearly defined peaks in the di-lepton invariant mass distribution. In addition, we observe new decay chains that allow for production of the “nonstandard” Higgs bosons without large $\tan\beta$ enhancements. It is possible that the full two-Higgs-doublet-model (2HDM) content can be mapped in detail, thus providing a clear and definite signal for physics beyond the standard model, and a rather detailed understanding of the mechanism of electroweak symmetry breaking (EWSB). If, in addition, relatively light superparticle signals are observed, as might be expected in these scenarios, a clear case for BMSSM physics could be established. Apart from the collider phenomenology induced indirectly by the heavy physics, higher-dimension operators have also been studied in the context of dark matter [@Cheung:2009qk; @Berg:2009mq; @Bernal:2009jc], cosmology [@Bernal:2009hd] and EW baryogenesis [@Grojean:2004xa; @Bodeker:2004ws; @Delaunay:2007wb; @Noble:2007kk; @Blum:2008ym], and it may be interesting to further explore the connections with collider physics.
This paper is organized as follows. In Sec. \[sec:review\], we summarize the most relevant aspects of the models under study. In Sec. \[sec:spectra\] we discuss the modifications of the Higgs spectrum, which are the dominant factor in determining the Higgs collider phenomenology. In Sec. \[sec:results\] we discuss in detail the range of signatures uncovered by our survey, separating the analysis into the low and large $\tan\beta$ regimes. We conclude in Sec. \[sec:conclu\].
Extended SUSY Higgs Sectors at a Glance {#sec:review}
=======================================
As already mentioned, when considering BMSSM scenarios where the non-MSSM degrees of freedom have masses parametrically larger than the weak scale, an EFT approach is very useful. The fact that at leading order only two parameters are added to those in the MSSM makes this a rather economic extension [@Dine:2007xi], that nevertheless can significantly change the MSSM Higgs phenomenology. However, the same reason that makes these $1/M$ suppressed effects rather important also implies that the next order in the $1/M$ expansion can be phenomenologically relevant, without implying a breakdown of the EFT [@Carena:2009gx]. At order $1/M^{2}$ there are several SUSY-preserving and SUSY-violating operators in the [ ]{}potential, the most important of which, in relation to the Higgs phenomenology, were listed in Ref. [@Carena:2009gx]. We refer the reader to this reference for the detailed form of such operators and how they affect the expressions for the Higgs masses and couplings. Here we restrict ourselves to a few general remarks that summarize the most relevant features for the present study (full details were given in the above reference).
First, it has to be pointed out that the higher-dimension operators to order $1/M^{2}$ can be easily generated from UV completions that include a combination of Higgs singlets, $SU(2)$ Higgs triplets, heavy W primes and Z primes. As argued in [@Carena:2009gx] the upshot is that the coefficients of the higher-dimension operators, from a low-energy point of view, can be chosen in an uncorrelated manner. Although the EFT description to order $1/M^{2}$ introduces a large number of parameters, which makes the framework more involved compared to the truncation at order $1/M$, one should notice that this same feature gives additional handles to infer properties of the heavy sector from the properties of the low-energy degrees of freedom. In any case, since our goal is to survey the collider signal possibilities in a model-independent way (in a supersymmetric framework), we focus on a low-energy study based on the EFT at order $1/M^{2}$, as described in [@Carena:2009gx]. [^2]
A random scan over parameter space was performed, and a set of points satisfying several constraints was selected. The set of points in this study satisfy:
- All the dimensionless coefficients parametrizing the higher-dimension operators are taken to be at most of order one, i.e. it is assumed that the heavy physics at $M$ is weakly coupled.
- *Global minimum*: since the scalar potential can present several minima, we make sure that the vacuum under study is the global one (at least within the EFT). We also check that there are no charge/color breaking minima, and for simplicity we restrict to the CP conserving case (checking that the global minimum does not break CP spontaneously).
- *Robustness*: there are no accidental cancellations that can render (not computed) higher orders in the $1/M$ expansion more important than expected.
- *“Light” SUSY spectrum*: given that generically, and unlike in the MSSM, these models satisfy the LEP bounds on the Higgs mass at tree-level, there is no need for large radiative corrections. Naturalness suggests that in these models the SUSY spectrum would be expected to be light (in the few hundred GeV range, consistent with direct bounds).
- Agreement with EW precision constraints, in particular in regards to the Peskin-Takeuchi $T$ parameter [@Peskin:1990zt]. These arise from three sources: a subset of the higher-dimension operators (as generated, for instance, by Higgs triplets), the details of the MSSM Higgs spectrum, and potential custodially-violating mass splittings in the sparticle spectrum. We emphasize that mild cancellations allow for higher-dimension operator effects that can have a non-negligible impact on the Higgs collider phenomenology.
All of the above constraints were described in detail in [@Carena:2009gx]. In addition, we impose the current collider bounds from LEP and the Tevatron using the code HiggsBounds v1.2.0 [@Bechtle:2008jh; @Bechtle:2009ic]. [^3] To this we add the LEP bounds on charged Higgs production [@:2001xy], and the newest combined result from the Tevatron in the WW channel [@Aaltonen:2010yv], and in the inclusive tau search [@:tau_inclusive], that are not included in the currently available version of this code. We use HiggsBounds in the “effective coupling” mode, which requires effective couplings defined by g\_[X]{}\^2 &=& , \[geff\] where $\phi = h, H, A$ is any of the neutral Higgs states, $\Gamma(\phi \to X)$ is the partial width in our model into any of the final states $X =
s\bar{s},c\bar{c},b\bar{b},\tau\bar{\tau},WW,ZZ,\gamma \gamma$ or $g
g$ (when applicable), and $\Gamma_{SM}(\phi \to X)$ is the partial width for a SM Higgs of the corresponding mass. Together with the total widths in our model (and in the SM), these effective couplings encode the information about branching fractions into these decay channels in our model.
We have implemented our tree-level expressions for the spectrum and Higgs couplings in HDECAY v3.4 [@Djouadi:1997yw]. This allows us to compute the Higgs partial decay widths, taking into account the QCD radiative corrections, that are known to be sizable (for a review, see [@Spira:1997dg]). In addition, we include the radiative corrections derived from the 1-loop RG improved effective potential due to supersymmetric particles [@Carena:1995bx], and the SUSY QCD/EW corrections to the Yukawa couplings [@Carena:1994bv; @Carena:1999py]. Loop contributions from the heavy physics that has been integrated out are suppressed by both a loop factor and by powers of $M$, hence they are expected to be negligible.
In all the plots that follow, we have fixed the following dimensionful parameters: $M=1~{\rm TeV}$, $\mu = m_{S} = 200~{\rm
GeV}$, [^4] and for simplicity, we use a common value $M_{SUSY} = 300~{\rm GeV}$ and $A_{t} = A_{b} = 0$ in the MSSM sparticle sector.[^5] The light superparticle spectrum implies that the loop contributions to the Higgs masses are modest, while the loop contributions to the Higgs couplings are more important and sensitive to the details of this spectrum [@Carena:1994bv; @Carena:1999py; @Dawson:1996xz]. The above choice of $M_{SUSY} = 300~{\rm GeV}$ is simply meant to illustrate the possible loop effects arising from relatively light superparticles. In particular, one can expect the first two generation squarks to be somewhat heavier to satisfy direct collider bounds [@:2007ww; @Aaltonen:2008rv] or the sleptons could be somewhat lighter, without changing our generic conclusions regarding the Higgs collider phenomenology. Note also that the neutralino/chargino sector depends on parameters not affecting the Higgs sector directly, and in particular that we do not impose constraints from dark matter (in this work, we remain agnostic as to the identity of the DM candidate, but see [@CMR]). We have also not imposed indirect constraints, such as those arising from $b
\rightarrow s \gamma$, $B_{s} \to \mu^+ \mu^{-}$ and $g_{\mu} - 2$, that have the potential to put important restrictions, but depend on the flavor structure of the soft SUSY-breaking parameters.
We consider two representative values of $\tan\beta$: $\tan\beta = 2$ and $\tan\beta = 20$. The CP-odd mass was varied in the range $20-400~{\rm GeV}$. The upper bound is taken to ensure a proper separation between the light and heavy scales, as required by the EFT analysis. The very low mass range is expected to be severely constrained, but we defer the study of such region to future work. We turn next to a detailed description of the most important physical characteristics of the set of models in the scan, starting with the Higgs spectrum.
Masses of Low-energy Higgs Bosons {#sec:spectra}
=================================
In this section we study the spectra of these models, analyzing the modifications with respect to the MSSM. Compared to the results already presented in [@Carena:2009gx], we include the 1-loop supersymmetric corrections to the Higgs quartic couplings as given in [@Carena:1995bx] (a minor effect for the relatively low SUSY spectrum we have in mind), as well as the constraints coming from collider data (LEP and Tevatron).
{width="8.0cm"} {width="8.0cm"}
In Fig. \[fig:mlvsma\] we show the mass of the lightest CP-even Higgs ($h$) as a function of $m_A$, for both $\tan \beta=2$ (left panel) and $\tan \beta=20$ (right panel). The green points represent models ruled out by LEP, while the magenta ones are excluded by current data from CDF and D0. We divide the remaining allowed models into two subsets. First, one has those models that will be probed at the Tevatron at 95 % C.L (red points), assuming $10~{\rm fb}^{-1}$ per experiment and $50\%$ efficiency improvements [@Moriond_Fisher] (see [@Draper:2009fh] for detailed projections in the MSSM context). These comprise the future reach of two search channels: $h/H \to b \bar{b}$ with the Higgs being produced in association with electroweak gauge bosons, and $gg \to h/H
\to W^+ W^-$. Second, the blue points are those that will be out of the Tevatron reach under the previous assumptions. For reference, we also show the MSSM (dashed) curve, assuming the same light SUSY spectrum. This color code will be employed in all the plots.
The corrections to $m_h$ due to the new physics are most important in the low $\tan \beta$ regime. Nevertheless, it is clear that they can also be relevant at large $\tan \beta$. The higher-dimension operators affect $m_h$ in such a way that it can easily be above the MSSM value. In the left plot, where $\tan \beta=2$, all the points lie above the MSSM curve; $m_h$ can reach values as high as $250~{\rm
GeV}$.
Moreover, the left panel of Fig. \[fig:mlvsma\] shows in a clear way how the Tevatron probes these models. For high enough values of $m_A
$ one distinguishes mostly uniformly single colored horizontal stripes. The magenta one, where $m_h \sim160-170~{\rm GeV}$, corresponds to $h$ being excluded by the current Tevatron search in the $WW$ channel [@Aaltonen:2010yv]. Note that this range is slightly larger than the SM one ($162 - 166~{\rm GeV}$). This is due to the fact that, in our models, the gluon fusion cross section can be mildly enhanced with respect to the SM one. By the same token, one understands the presence of a few red points within the magenta stripe as those corresponding to models whose gluon fusion cross section is below the SM value. The two red stripes ($m_{h}$ in the ranges $140 -
160~{\rm GeV}$ and $180 - 190~{\rm GeV}$) represent the future Tevatron reach of the $h \to W^+ W^-$ channel. Notice also the presence of a thin stripe of red points, with $m_h$ around $120~{\rm
GeV}$, that extends along a wide range of $m_A$: these models can be probed by the $h \to b\bar{b} $ channel, that is effective only for relatively low values of $m_h$. No points are excluded by the $H \to
b \bar{b}$ decay mode, since $H$ is always much heavier than $120~{\rm
GeV}$.
The two blue stripes correspond to points where there is no reach from the Tevatron in the $WW$ channel. This can be explained either by a low signal due to the reduced branching fraction into gauge bosons, or simply because the $gg$ parton luminosity is not enough to produce such a heavy Higgs boson. Note however that in the high $m_{h}$ blue region the $ZZ \to 4 l$ channel becomes kinematically accessible, so that this Higgs could be observed in the *gold plated* four-lepton mode at the LHC. We will postpone further comments on this region to the next section.
Regarding the LEP constraints, one sees that there are a few currently allowed (blue and red) points below the SM LEP bound of $114.4~{\rm
GeV}$[@:2001xx; @Schael:2006cr]. The nonexclusion is due to the fact that the coupling of $h$ to the gauge bosons is reduced with respect to the SM value. However, all the red points below the LEP-bound can potentially be excluded in the $H \to WW$ channel.
For the remaining points ($m_A < 160~{\rm GeV}$, $114.4 ~{\rm GeV}
\lesssim m_h \lesssim 170~{\rm GeV}$), the situation is more complex, and magenta, blue and red points coexist in this region. In particular, there is a region of allowed (blue) points with $m_h \sim130~{\rm GeV} - 140~{\rm GeV}$ and relatively low $m_A$. These points have suppressed branching fractions into both $WW$ and $b\bar{b}$, with $AA$ being the dominant decay channel.
In the case of $\tan \beta=20$, the deviations from the MSSM are far less dramatic. Ultimately, this is explained by the fact that several higher-dimension operators are $\tan \beta$ suppressed. However, $m_h$ can reach values as high as $160~ {\rm GeV}$. In this case, since $h$ is SM-like, the LEP bound is very strict, forcing $m_h$ to be above $\sim 110~{\rm GeV}$. Regarding the Tevatron searches, we see that there are two small and disjoint currently excluded (magenta) regions. The region with $m_h$ around $160~{\rm GeV}$ corresponds, as in the low $\tan\beta$ case, to exclusion based on the $h \to WW$ decay mode. The second magenta region has lower values of $m_h~(114-130~{\rm GeV}$) and $m_A ~(100-135 ~{\rm GeV}$). This latter set of models are currently excluded by the inclusive tau search with 2.2 fb$^{-1}$, using the combination from CDF and D0 [@:tau_inclusive]. This channel becomes important here, since the $H/A$ –and in some cases the $h$– coupling to down-type fermions is $\tan \beta$ enhanced.[^6] Turning to the red points (i.e. those within future Tevatron sensitivity), a closer inspection reveals that all of them can be excluded due to the decay modes of the lightest Higgs. In more detail, the $h \to b\bar{b}$ channel probes points with $m_{h}$ below $126~{\rm GeV}$, while the rest are probed by the $h \to WW$ search. This can be understood from the fact that, as in the MSSM, in the large $\tan \beta$ limit $H$ tends to be non SM-like. In contrast to the low $\tan \beta$ case, all the Tevatron allowed (blue) points correspond to somewhat heavy values of $m_A$ (above $140~{\rm GeV}$).
{width="7.9cm"} {width="7.9cm"}
It is also interesting to study the relation between the CP-even Higgs masses. In Fig. \[fig:mhvsml\] we show $m_H$ as a function of $m_h$, for $\tan \beta=2$ (left panel) and $\tan \beta=20$ (right panel). For most of the points these masses are not correlated. For instance, if in the left plot one takes $m_h$ in the $120-200~{\rm
GeV}$ range, then $M_H$ can vary between $200$ and $400~{\rm GeV}$. For $\tan \beta=2$, one has not only the (now vertical) stripes corresponding to exclusion due to $h$ that we have found in Fig. \[fig:mlvsma\]: there are also horizontal stripes, corresponding to $m_H$ ranges where the Tevatron is excluding models by means of the $H \to WW$ decay channel. This sheds some light into the region already mentioned in the discussion of Fig. \[fig:mlvsma\] with $m_A < 160~{\rm GeV}$ and $114.4 ~{\rm GeV}
\lesssim m_h \lesssim 170~{\rm GeV}$. In this region both $h$ and $H$ can couple to $WW$, typically resulting in some suppression with respect to the SM for one or the other CP-even Higgs boson. This constitutes an interesting example of how the $h$ and $H$ signals can complement each other. The right panel confirms what we have anticipated from our discussion of Fig. \[fig:mlvsma\]: in the large $\tan \beta$ regime, $m_H$ tends to be heavy, and the decays of $H$ are less restrictive than the ones from $h$, hence there are no horizontal stripes in this plot. As mentioned before, the $h$ search channels give rise to all the red points.
{width="7.9cm"} {width="7.9cm"}
Finally, we show in Fig. \[fig:mhiggsvsma\] the masses of the heavy CP-even and charged Higgs bosons as a function of $m_A$. The deviations from the MSSM value are much less dramatic than for $h$. This is particularly true in the large $m_{A}$ limit and for large $\tan
\beta$. Nonetheless, in this region the contribution from the new physics effects to the masses is of ${\cal O}(10~{\rm GeV)}$, which cannot be neglected. For low values of $\tan \beta$ (left plots) we see that in the region of blue points with low values of $m_A$, both $m_{H^{\pm}}$ and $m_H$ are above the MSSM value. Notice that this effect is more important for $m_H$ than for $m_{H^{\pm}}$. As a direct consequence, in the low $\tan \beta$ regime, new *exotic* channels like $H \to AA$ and $H^{\pm} \to A W^{\pm}$ can be open, with large BRs, as we will see in the next section. This does not happen for $\tan \beta=20$ since, as stated before, there are no allowed (blue and red) points with $m_A$ below $140~{\rm GeV}$ and the mass splittings do not allow the previous decay modes.
Having analyzed the modifications in the spectra due to the higher-dimension operators, we will devote the next section to study the collider phenomenology of these models.
BMSSM Collider Phenomenology {#sec:results}
============================
In this section we study the phenomenology of the BMSSM Higgs sector, including all of the effects and constraints described in Section \[sec:review\]. We consider the low and large $\tan \beta$ cases separately.
Low $\tan\beta$ searches: general features {#sec:lowtb}
------------------------------------------
We start with the low $\tan \beta$ regime, fixing $\tan \beta=2$. As we have described in the previous section, the main modification introduced by the higher-dimension operators is to shift the Higgs spectrum with respect to the MSSM one. However, the couplings of the Higgs bosons also get corrected. The combination of these two effects can give rise to sizable modifications in both the Higgs production cross sections and the branching fractions.
We compute the production cross sections in the following way. For the Higgs-strahlung and vector boson fusion processes, we simply scale the corresponding SM cross section by the (square of) the Higgs-$W$-$W$ coupling in our scenario, normalized to the SM coupling, i.e. by the effective coupling as defined in Eq. (\[geff\]) (for all practical purposes this ratio coincides with the normalized Higgs-$Z$-$Z$ coupling [^7]). For the gluon fusion cross section, we shall argue that the NLO K-factor in our scenario is expected to agree with the NLO K-factor in the SM within 20%. This implies that to this accuracy \[eq:ggheff\] && , since the ratio of cross sections equals the ratio of widths at leading order in $\alpha_{s}$ [@Georgi:1977gs; @Spira:1995rr; @Dawson:1996xz]. The right-hand side of Eq. (\[eq:ggheff\]) is computed using our modified version of HDECAY [@Djouadi:1997yw], which includes the tree-level expressions for masses and couplings in the presence of the higher-dimension operators.
The K-factor in our scenario differs from the SM one in two respects. First, the contribution to the gluon fusion cross section from bottom loops cannot be neglected, specially in the large $\tan \beta$ regime. Second, one has to consider the presence of a relatively light SUSY spectrum. We discuss separately these two effects. To assess the impact of the bottom loop we use the code HIGLU [@Spira:1995mt], that includes both the LO and NLO results for both the SM and the MSSM [@Spira:1995rr] (but, at present, does not include SUSY particles in the loop), to compute the K-factors in these two models. We find that at low $\tan\beta$ and for a wide range of Higgs masses, the NLO K-factors for $h$, $H$ and $A$ coincide within 5% with the SM NLO K-factor for a Higgs of the corresponding mass. At larger $\tan\beta$ ($\sim 30$) the differences are larger, as expected, but still smaller than about 20%. We expect that the same will hold in our extended SUSY scenarios. The changes in the NLO K-factor due to relatively light sparticles in the loop, again in the MSSM context, were studied in [@Spira:1997dg], where the effect was found to be less than $3 \%$ for $\tan \beta=1.5$. Therefore, we conclude that at low $\tan\beta$ Eq. (\[eq:ggheff\]) holds to an accuracy of better than $10 \%$, and allows us to obtain a sufficiently precise estimate for the NLO gluon fusion cross section in our scenario. Note that this uncertainty is below the one obtained by comparing the NLO and NNLO/NNLL results in the SM calculation [@Harlander:2002wh; @Anastasiou:2002yz; @Ravindran:2003um; @Catani:2003zt; @Anastasiou:2008tj; @deFlorian:2009hc]. It is also important to note that the bulk of the effects of the light SUSY spectrum is taken into account in the LO cross section, and that these effects are fully implemented in HDECAY, which is used to compute the right-hand side of Eq. (\[eq:ggheff\]). This also includes radiative effects that correct the bottom Yukawa coupling, which can be important at large $\tan\beta$ [@Carena:1999py].
{width="7.9cm"} {width="7.9cm"}
We discuss next a number of general features regarding the BMSSM Higgs signals. In both the SM and the MSSM, the dominant decay channel for a Higgs boson whose mass is greater than 140 GeV is into W pairs. Therefore, an important observable at a hadron collider is the production cross section times the branching fraction in the WW channel. In Fig. \[fig:ggtohtoWWtb2\] we show this quantity for $h$ and $H$, normalized to the SM result, as a function of the corresponding Higgs mass. In the left panel we clearly see the Tevatron exclusion in the $h \rightarrow WW$ channel: the V-shaped magenta and red regions around $m_{h} \sim 160~{\rm GeV}$ correspond to the *stripes* that were already discussed in Section \[sec:spectra\] (see Figs. \[fig:mlvsma\] and \[fig:mhvsml\]). We stress that the blue points with $m_h$ above $180~{\rm GeV}$ have a slightly enhanced $WW$ signal compared to the SM. In turn, this mass range will be explored at the LHC via the $h
\to ZZ \to 4l$ channel (recall that, for all practical purposes, the CP-even Higgs normalized couplings to $WW$ and $ZZ$ are the same, hence the plot can be directly applied to the $ZZ$ channel). Thus, an enhanced signal in this region is an interesting feature: for these points, the Higgs cannot escape detection. For the blue points with $WW$ signal reduced by a factor of 10 or more ($m_h < 160~{\rm GeV}$), one may have to rely on other search channels.
Note that this figure exhibits currently allowed (blue and red) points with $m_{h}$ below the LEP bound of $114.4~{\rm GeV}$. These correspond to models where the coupling to gauge bosons is below the SM value. We also notice a group of red points whose signal is around the SM value, and with a mass slightly above the LEP bound ($114.4~{\rm GeV} \le m_h \lesssim 120~{\rm GeV}$): these are within the Tevatron reach in the $h \to b\bar{b}$ channel, assuming an accumulated luminosity of $10~{\rm fb}^{-1}$ per experiment and a 50% efficiency improvement in this channel.
Turning our attention to the right panel of Fig. \[fig:ggtohtoWWtb2\], we see that in the case of $H$ it is hard to differentiate regions where a single color is predominant, as was possible in the left panel. We can identify a mostly green region with $m_{H}$ above the LEP bound (and above the MSSM curve). These points are excluded by the LEP bound on $m_{h}$ rather than on $m_{H}$, and serve as a reminder that the constraints may come from observables not related to those shown in a given plot. This is not to say that there are no points where the exclusion is through $H$ directly instead of $h$: for instance, the magenta and red points in the upper left side of the plot correspond to the V-shape exclusion from $H \to WW$ at the Tevatron. This is the only region where the signal is enhanced with respect to both the SM and the MSSM. These correspond to models where $H$ is SM-like, while $h$ decays mainly into $b \bar{b}$ and $\tau \bar{\tau}$.
Aside from the $WW$ channel, there are other important decay modes for light Higgs bosons, in particular $b \bar{b}$, $\tau \bar{\tau}$ and $\gamma \gamma$. In the first case, the huge QCD backgrounds render this channel very difficult to measure at a hadron collider. This does not mean, however, that this decay mode is completely useless. For instance, in the Higgstrahlung process, $q \bar{q} \to Z^{*}/W^{*}
\to Z/W + \textrm{Higgs}$, the gauge boson can be fully reconstructed from its decay modes, and then $\textrm{Higgs} \to b \bar{b}$ becomes a feasible option. Another example to search for a SM-like Higgs boson decaying into $b\bar{b}$ is the Higgs associated production together with a top quark pair. This has the problem of being quite challenging at hadron colliders. The di-photon channel, on the other hand, constitutes the most promising decay channel for a relatively light SM-like Higgs at the LHC since, in spite of its tiny BR of ${\cal O} (10^{-3})$, an excellent energy resolution can be achieved and the background is under good experimental control. The other important search channel at the LHC in the low Higgs mass range is the vector boson fusion with the subsequent decay of the Higgs into a $\tau\bar{\tau}$ pair.
{width="7.9cm"} {width="7.9cm"}
In Fig. \[fig:htobgamtb2\] we show the branching fraction of $h$ into $b \bar{b}$ (left plot) and $\gamma \gamma$ (right plot), for $\tan \beta =2$. Notice that the $b \bar{b}$ channel can be suppressed with respect to the SM one, as in the blue points with masses in the $120-150~{\rm GeV}$ range and ${\rm BR}(h \to b\bar{b}) < 10^{-1}$. This is an interesting feature, since it can lead to enhancements in other search channels. One can also see currently allowed (blue and red) points with BRs into $b\bar{b}$ above the SM curve: those have a reduced BR into W’s, as we have previously identified in Fig. \[fig:ggtohtoWWtb2\]. In the case of $H$ (not shown here), the BR into $b\bar{b}$ is typically higher than the SM value. With respect to the MSSM, we find that there is no definite tendency: ${\rm
BR}(H \to b\bar{b})$ can be either increased or suppressed by an order of magnitude. It is worth mentioning that the branching fraction in the $\tau \bar{\tau}$ channel follows closely the $b \bar{b}$ behavior. This is as expected, since the extended Higgs sectors under consideration do not distinguish between the down-type fermions, in the sense that the Yukawa coupling normalized to the SM value is the same for bottoms and taus, while differences due to the SUSY QCD and top Yukawa interactions, that arise at loop level, are not significant at small $\tan\beta$.
Turning our attention to the right panel of Fig. \[fig:htobgamtb2\], we see that most models present a suppressed branching fraction in the di-photon channel. However, it is worth noticing the group of points above the SM curve, where an enhancement of up to a factor of 2 can be achieved.
{width="7.9cm"} {width="7.9cm"}
Decays of the CP-even Higgs bosons into pairs of $A$ bosons can become the dominant decay mode. Such a scenario has been previously considered in the literature (see, for instance, [@Carena:2007jk] for a model-independent analysis, and [@Dermisek:2008uu] for NMSSM studies). In Fig. \[fig:htoAA\] we show the branching fraction of $h$ and $H$ into $AA$, for $\tan \beta =2$. The left panel shows that the branching fraction in this channel can reach ${\cal O} (1)$ values, thus becoming the most relevant decay mode of $h$. The Tevatron allowed (blue) points in this figure present a reduced branching fraction in both the $b \bar{b}$ and the $WW$ channel, and were already mentioned in the context of Fig. \[fig:htobgamtb2\]. In the case of $H$ (right panel), the branching fractions vary considerably, but the $AA$ channel may still become the primary decay mode in some models.
{width="7.9cm"} {width="7.9cm"}
The main modification to the decay phenomenology of $A$ and $H^{\pm}$ with respect to the MSSM is due to the shift in the overall Higgs spectrum. The channels that change the most are those that involve a Higgs decaying into either a pair of Higgs bosons, or a Higgs boson plus a gauge boson. As an example of the latter, we take the decay of $H^{\pm}$ into a W boson and a neutral Higgs. In the MSSM, since $h$ tends to be rather light, one has that $H^{\pm} \to h W^{\pm}$ can be an important decay channel. On the other hand, since $m_A$ and $m_{H^\pm}$ tend to be rather degenerate, one finds that $H^{\pm} \to
A W^{\pm}$ is generally highly suppressed. On the contrary, in the context of the BMSSM, one can find points where the mass hierarchy suffers an inversion, i.e. $m_A$ can be well below $m_h$ and split from $m_{H^{\pm}}$. In this case, one finds that $A$ and $h$ interchange their roles with respect to the above described situation in the MSSM, as can be seen in Fig. \[fig:Ctb2\]. The left panel shows points where the $A W^{\pm}$ channel has a BR greater than $0.1$, while the right panel shows that the $h W^{\pm}$ decay mode is highly suppressed. In this case, the process $H^{\pm} \to A W^{\pm}
\to b \bar{b} W^{\pm}$ can give rise to an interesting signal at the LHC, possibly allowing the discovery of two nonstandard Higgs bosons. Note that we have already encountered another example of an inversion between $A$ and $h$ in the context of Fig. \[fig:htoAA\]: the potentially open MSSM channels $A \to hh$ and $H \to hh$ are replaced by $h \to AA$ and $H \to AA$ in the BMSSM context.
Finally, turning our attention to $A$, we have found that the $A \to h
Z$ decay channel is significantly reduced with respect to the MSSM value $\sim 0.3$ for values of $m_A$ below $250~{\rm GeV}$, due to the shift in $m_h$ that disfavors this decay mode. [^8] This reduction brings an enhancement in both the $b \bar{b}$ and $\tau \bar{\tau}$ channels. As in the MSSM, the former is the dominant decay channel below the $t\bar{t}$ threshold, and the latter stays almost constant at about 10%.
Low $\tan\beta$ searches: benchmark points {#sec:lowtb:benchmarks}
------------------------------------------
Up to this point we have analyzed each observable almost independently of the others. We would like to understand, however, how the different features that we have singled out are correlated with each other. We shall consider benchmark scenarios currently allowed by LEP and Tevatron data and explore two possibilities: a) models that can be probed at 95 % C.L at the Tevatron in the near future, from now on referred to as *Tevatron covered* (red) points, and b) models that are beyond the expected Tevatron reach and will be explored at the LHC, from now on referred to as *Tevatron uncovered* (blue) points. We will also indicate the importance of the $1/M^2$ effects, as measured by their impact on $m_{h}$.
### Scenarios within the Tevatron reach {#sec:TevatronPoints}
The Tevatron covered models can be divided into three subsets, according to which channel can exclude the point: $h \to b \bar{b}$, $h \to WW$ and $H \to WW$. It is interesting to ask whether a given model can be probed by more than one channel at the Tevatron. We find, however, that the previous subsets are disjoint. The disjointness between the subsets probed by $h \to b \bar{b}$ and $h
\to WW$ can be understood in terms of the relevant mass ranges, since the $b \bar{b}$ search is most sensitive to the $m_h \lesssim 120~{\rm
GeV}$ range, while the di-boson channel probes the region $165 \pm
20~{\rm GeV}$. In principle, they do not have to be mutually exclusive, but one would need an enhancement of 3.4 over the SM in the $h \to WW$ signal in order to probe a $120~{\rm GeV}$ Higgs in this channel,[^9] which is not achievable within these models: the increase in the $gg \to h \to WW$ cross section is always below a factor of $2$. In the case of $h$ and $H$ decaying into W bosons, even if both of them are in the favorable mass region ($\sim 150-170~{\rm GeV})$, the MSSM sum rule $g^2_{hWW} +
g^2_{HWW}=1$ is valid within 5 % accuracy, and it is not possible for $h$ and $H$ to have large enough couplings to W’s for both signals to simultaneously be within the Tevatron reach. The subsets probed by the $h \to b \bar{b}$ and $H \to WW$ searches are disjoint because both processes require a sizable coupling of the Higgs to $W$’s (for production and decay, respectively), but this does not happen when $m_{h}$ and $m_{H}$ are sufficiently different, as would be required for simultaneous searches in these two channels.
*Point A: MSSM-like scenarios*
**POINT A**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
--------------------------- --------------------- ---------------------------------- -----------------------------
239 118 246 245
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.992 0.008 1.06 0.55
channel BMSSM (SM) channel BMSSM (SM)
$h \to b \bar{b}$ 0.78 (0.73) $ h \to WW $ 0.08 (0.11)
$ h \to \tau \bar{\tau} $ 0.08 (0.08) $ h \to \gamma \gamma / 10^{-3}$ 1.42 (2.30)
$H \to b \bar{b}$ 0.15 $ H \to WW $ 0.22
$ H \to ZZ $ 0.11 $ H \to hh $ 0.50
$A \to b \bar{b}$ 0.89 $ H^{+} \to t \bar{b}$ 0.99
$ A \to \tau \bar{\tau} $ 0.08 $ A \to Z h$ 0.24
: [*Masses and branching fractions in the BMSSM (and in the SM for $h$) for point A. We only show the main decay modes. The effective couplings $g^{2}_{\phi X}$ were defined in Eq. (\[geff\]).* ]{}[]{data-label="tab:pointA"}
We start our analysis with the points that can be probed via the $h
\to b \bar{b}$ decay mode. We find that these do not differ greatly from the decoupling limit of the MSSM with rather heavy sparticles ($\sim$ a few TeV). In this case, the observation of a light SUSY spectrum (in the few hundred$~{\rm GeV}$ range) would be the smoking gun of BMSSM physics, since such a light SUSY spectrum would be in conflict with the LEP limits on the MSSM lightest CP-even Higgs boson. We illustrate the main features of this subset by showing point A [^10] in Table \[tab:pointA\], where we include the mass spectrum and the branching fractions of the most important decay channels for each Higgs boson. For reference, in the case of $h$ we also indicate between parentheses the SM values. We also note that for this point, the $1/M^2$ operators contribute about $25~{\rm
GeV}$ to $m_{h}$.
Generically, the branching fractions of $h$ do not deviate much from the SM ones. One finds a small increase in the $h \to b \bar{b}$ channel, while ${\rm BR}(h \to \gamma \gamma)$ is slightly suppressed with respect to the SM (by at most a factor of 3). Since $g_{hWW}^2 $ and $g_{hgg}^2$ are close to one, the production cross sections by Higgs-strahlung and gluon fusion are, for all practical purposes, the same as in the SM. Thus, the change in the signal is given by the ratio of the branching fractions in our scenario to those in the SM. For point A, the production rate in $h \to b \bar{b}$ is 6% above the SM result. In this case, the Tevatron could claim a hint on a SM Higgs boson, while at the LHC the direct detection of $h$ would proceed in the di-photon channel, since the $gg \to h \to \gamma \gamma$ cross section is $0.65$ of the SM value. Some of the remaining Higgs bosons may also be observed. For $H$ and $A$, the $H \to h h \to \gamma
\gamma b \bar{b}$ and $A \to Z h \to l l b \bar{b}$ searches provide the best prospects for discovery [@atlasphystdr]. For a charged Higgs with a mass above $m_t$, the ATLAS update of 2009 [@Aad:2009wy] found that the $tb$ channel is rather challenging and that the low $\tan\beta$ region cannot be covered.
*Point B: Light Higgs spectra*
**POINT B**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
------------------------------------- --------------------- ---------------------------------- -----------------------------
101 129 141 135
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.8 0.2 1.72 0.06
channel BMSSM (SM) channel BMSSM (SM)
$h \to b \bar{b} $ 0.01 (0.56) $h \to \tau \bar{\tau} $ 0.001 (0.06)
$ h \to WW$ 0.63 (0.28) $ h \to ZZ$ 0.08 (0.04)
$ h \to {\rm jets} $ 0.26 (0.06) $ h \to \gamma \gamma / 10^{-3}$ 3.97 (2.38)
$H \to b \bar{b}/ \tau \bar{\tau}$ 0.84 / 0.09 $ H \to WW $ 0.05
$A \to b \bar{b} / \tau \bar{\tau}$ 0.89 / 0.09 $ H^{+} \to \tau \nu_{\tau}$ 0.87
: [*Masses and branching fractions in the BMSSM (and in the SM for $h$) for point B.* ]{}[]{data-label="tab:pointB"}
We turn now our attention to the models that can be excluded at the Tevatron by the $h \to WW$ channel. Those can be further split into two categories, according to whether $m_h$ is high ($\gtrsim 170~{\rm
GeV}$) or low ($\lesssim 160~{\rm GeV}$), corresponding to the two red stripes defined in the context of Fig. \[fig:mlvsma\]. As a general feature of the lower red stripe, the branching fraction of $h$ into $b
\bar{b}$ can be sizably reduced with respect to the SM, as we pointed out in the left panel of Fig. \[fig:htobgamtb2\]. This implies that the remaining channels are enhanced, which is interesting for the $h
\to WW$ and $h \to \gamma \gamma$ decay modes. We present as an example point B [^11] in Table \[tab:pointB\]. Here, one sees that the Higgs spectrum is relatively light. $H$ is the heaviest Higgs, while $h$ is lighter than $H^{\pm}$, but heavier than $A$. It turns out that for this point, the $1/M^2$ effects result in a slight net reduction of $m_{h}$ by a couple of GeV.
Since $h$ is SM-like, we give in parentheses the corresponding branching fractions in the SM. Here we clearly observe that $h$ presents an increase in the gluon fusion cross section, and in the branching fractions into photons and W bosons, accompanied by a sizable reduction in the down-type fermion decay modes. Note that the $gg \to h \to \gamma \gamma$ and $gg \to h \to WW$ signals are larger than in the SM by factors of $2.86$ and $3.82$ respectively, which would facilitate the search of $h$ at the LHC as well. $H$ decays mainly into bottoms and taus, and its production cross section by gluon fusion is strongly reduced with respect to the SM case. The most promising discovery channel at the LHC would be $q q H \to q q
\tau \bar{\tau}$, where the signal is reduced with respect to the SM by a factor of two. The CP-odd $A$ decays as in the MSSM, while for the charged Higgs the $\tau \nu_{\tau}$ channel is the dominant one.
*Point C: The heavy CP-even $H$ as the SM-like Higgs*
In the high $m_h$ region that can be probed at the Tevatron in the $h
\to WW$ channel one finds an unusual SUSY spectrum. Typically, one runs into the previously mentioned *inversions* between $h$ and $A$. Moreover, $h$ can also be heavier than the charged Higgs, which is a feature that is not present in the region where $m_h$ is below $150~{\rm GeV}$. We illustrate this with point C [^12] in Table \[tab:pointC\]. For this point, the $1/M^2$ operators contribute about $30~{\rm GeV}$ to $m_{h}$.
**POINT C**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
------------------------------- --------------------- ------------------------- -----------------------------
135 174 186 164
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.11 0.89 1.05 0.65
channel BMSSM (SM) channel BMSSM (SM)
$h \to b \bar{b} $ 0.12 (0.01) $ h \to WW$ 0.84 (0.96)
$ H \to WW $ 0.81 (0.82) $ H \to ZZ $ 0.17 (0.17)
$A \to b \bar{b} $ 0.90 $A \to \tau \bar{\tau}$ 0.10
$ H^{+} \to \tau \nu_{\tau} $ 0.59 $ H^{+} \to t \bar{b}$ 0.38
: [*Masses and branching fractions in the BMSSM (and in the SM for $h$ and $H$) for point C.* ]{}[]{data-label="tab:pointC"}
Here we see that the two CP-even Higgs bosons have masses well above the maximum value for $m_h$ that can be obtained in the $m_h$ max scenario within the MSSM context. In this case, it makes sense to compare both $h$ and $H$ with the SM Higgs. For this particular point, $h$ has not been yet excluded by the Tevatron search since it is not SM-like and its branching fraction into $WW$ is somewhat suppressed. $H$ can be discovered by the LHC in the $ZZ \to
4 l$ mode, since the signal normalized to the SM value, is $0.65$. We note that here both $m_{h}$ and $m_{H}$ are near the region where the $WW$ channel opens up leading to a suppression in the sensitivity of the $ZZ$ search mode at the LHC. We recall that such a heavy SM-like $H$ is not a feature of the MSSM, being a unique characteristic of the BMSSM Higgs sector. The CP-odd $A$ decays almost entirely to bottom and tau pairs, while the charged Higgs has sizable decays into both the $\tau \nu_{\tau}$ and $t \bar{b}$ channels.
The last subset of the Tevatron covered points corresponds to those than can be probed by the $H \to WW$ search, for which the $gg \to H
\to WW$ signal goes between $0.4 - 4$ times the SM value. In such scenarios, $h$ and $A$ decay mostly into bottoms and taus. In some cases the $h \to \gamma \gamma$ or $q q H \to q q \tau \tau$ signal might be observable at the LHC. The charged Higgs is relatively light (always below $200~{\rm GeV} $) and will decay almost $100 \% $ of the time into $\tau \nu_{\tau}$ for masses below $160~{\rm GeV}$, and in $t \bar{b}$ for the remaining points. The $H^{\pm} \to h W^{\pm}$ decay mode is closed for kinematical reasons, as we already know from Fig. \[fig:Ctb2\]. In addition, when $m_A$ is light, the $h \to AA$ and $H^{\pm} \to A W^{\pm}$ channels might become important. Since this also happens with the Tevatron uncovered points, we will defer further comments and the study of a suitable benchmark point for the next subsection.
### LHC searches {#sec:LHCPoints}
Regarding the Tevatron uncovered points, we can also split them into two disjoint subsets, corresponding to each of the blue stripes in Figs. \[fig:mlvsma\] or \[fig:mhvsml\]: we will refer to them as low mass ($m_h \lesssim 140~{\rm GeV}$) and high mass (above $190~{\rm GeV}$) regions. In the high $m_h$ case, one can make a further distinction according to whether $m_A$ is below or above $160~{\rm GeV}$. Again, we illustrate the possibilities with a few benchmark points.
*Point D: Two peaks in the $ZZ \to 4l$ signal*
**POINT D**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
-------------------- --------------------- ------------------------- -----------------------------
184 204 234 203
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.3 0.7 1.39 0.36
channel BMSSM (SM) channel BMSSM (SM)
$h \to WW $ 0.73 (0.72) $ h \to ZZ$ 0.25 (0.27)
$ H \to WW $ 0.70 (0.71) $ H \to ZZ $ 0.29 (0.29)
$A \to b \bar{b}$ 0.87 $ H^{+} \to t \bar{b} $ 0.99
: [*Masses and branching fractions in the BMSSM (and in the SM for $h$ and $H$), for point D.* ]{}[]{data-label="tab:pointD"}
General features of the high $m_{h}$, high $m_{A}$ case are: increased $gg \to h$ cross section with respect to the SM, and negligible (below 2%) changes in the $h \to WW/ZZ$ decay modes. Regarding $H$, one has that the signal in the $gg \to H \to WW/ZZ$ channel is always suppressed with respect to the SM. As an example, we show point D [^13] in Table \[tab:pointD\]. Here, the $1/M^2$ operators contribute about $40~{\rm GeV}$ to $m_{h}$. Given the features of this point, it again makes sense to compare both CP-even Higgs bosons with the SM.
The rise in the mass of $h$ automatically closes the $H \to hh$, $A
\to h Z$ and $H^{\pm} \to h W^{\pm}$ decay modes, which could be important in the MSSM case. This picture can suffer some alterations if $h$ stays around $200~{\rm GeV}$, while the rest of the Higgs bosons attain values around $400~{\rm GeV}$, since this will open not only the previously mentioned channels, but possibly also decays into sparticles. In such a case, one would run into a sort of *MSSM decoupling limit*, but with a mass for the lightest Higgs which is unattainable within the MSSM. Concentrating on point D, we emphasize that both $h$ and $H$ couple in a sizable way to the electroweak gauge bosons, and thus the measurement of both couplings will permit a detailed study of the EWSB mechanism, as it arises from a 2HDM. One possibility to discover these CP-even Higgs bosons would be to search for two isolated peaks in the $ZZ \to 4l$ golden mode. Notice that the branching fractions of both $h$ and $H$ are very close to their SM counterparts, while there is a difference in the gluon fusion production cross section. Since the $gg \to H \to ZZ$ signal can be sizably suppressed with respect to the SM, the direct detection of $H$ in this channel might not be feasible. Nevertheless, a large number of models similar to point D would in fact present two clear peaks in the di-lepton invariant mass distribution. We also note that these points have a $gg \to h \to WW/ZZ$ signal that can be 20-70 % larger than the SM value.
*Point E: Non-SM-like Higgs with a clear di-boson signal*
**POINT E**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
------------------------- --------------------- ------------------------------ -----------------------------
134 181 205 165
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.03 0.95 0.79 0.99
channel BMSSM (SM) channel BMSSM (SM)
$h \to b \bar{b}$ 0.23 (0.005) $ h \to \tau \bar{\tau} $ 0.03 (0.0005)
$h \to WW $ 0.68 (0.92) $ h \to ZZ$ 0.04 (0.07)
$ H \to WW $ 0.72 (0.73) $ H \to ZZ $ 0.27 (0.27)
$A \to b \bar{b}$ 0.89 $A \to \tau \bar{\tau}$ 0.10
$ H^{+} \to t \bar{b} $ 0.57 $ H^{+} \to \tau \nu_{\tau}$ $0.40$
: [*Masses and branching fractions (and in the SM for $h$ and $H$) for point E.* ]{}[]{data-label="tab:pointE"}
One interesting example of a point where $m_h$ is still high, but $m_A$ is below $160~{\rm GeV}$ is given in point E [^14] , shown in Table \[tab:pointE\]. We see that this point has an unusual Higgs hierarchy, since here $h$ is heavier than both $A$ and $H^{\pm}$ (in this respect similar to point C). The signal for $h \to WW $, normalized to the SM, is $0.54$. The CP-even $H$ will be discovered first and will appear to be the SM Higgs, since the signal is very close to the SM one. Soon after, $h$ will be found in both the $ZZ$ and $WW$ channels, thus providing a clear evidence of new physics. Notice that here the coupling of $h$ to gauge bosons is extremely small, but due to kinematics it still decays preferentially into gauge bosons. We stress again that this is a unique characteristic of the BMSSM Higgs sector in the low $\tan
\beta$ regime, since in the MSSM a similar behavior can only occur for $H$. For this point, the $1/M^2$ operators contribute about $30~{\rm
GeV}$ to $m_{h}$.
*Point F: Multi-Higgs decay chains*
**POINT F**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
------------------------------ --------------------- ----------------------------- -----------------------------
64 135 155 125
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.002 0.991 0.65 1.17
channel BMSSM channel BMSSM
$h \to b \bar{b} $ 0.15 $ h \to AA$ 0.84
$ H \to WW $ 0.12 $ H \to AA $ 0.84
$H \to b \bar{b} $ 0.02 $ A \to b \bar{b} $ 0.92
$ H^{+} \to \tau \nu_{\tau}$ 0.56 $H^{\pm} \to W^{\pm} + A $ $0.40$
: [*Masses and branching fractions in the BMSSM for point F.* ]{}[]{data-label="tab:pointF"}
For the points outside the Tevatron reach with $m_h$ below $140~{\rm
GeV}$, the most remarkable feature is the possibility of having the channels $h \to AA$ and $H \to AA$ kinematically open. For the points where these channels are closed, the situation is not as interesting, so we will focus on the first scenario. As an example, we show point $F$ [^15] in Table \[tab:pointF\], where these channels are the dominant decay modes of both $h$ and $H$. For this point, the $1/M^2$ operators contribute about $10~{\rm GeV}$ to $m_{h}$. Focusing on the $AA$ channel, the possible final states for $h$ and $H$ are $b \bar{b} b
\bar{b}$, $b \bar{b} \tau \bar{\tau}$ and $\tau \bar{\tau} \tau
\bar{\tau}$ [@Carena:2007jk; @Dermisek:2008uu; @Cheung:2007sva]. The first one is very challenging due to the enormous QCD background, while the third one suffers from a reduced signal \[$BR(A \to \tau
\bar{\tau}) \sim 10 \%$\]. This leaves the $b \bar{b} \tau \bar{\tau}$ channel as the most promising one. For the case of $H$, one may also look at the $gg \to H \to WW$ channel, whose signal is 1/5 of the SM value, and could be discovered with about 100 fb$^{-1}$ [@atlasphystdr]. For the charged Higgs, the dominant decay mode is $\tau \nu_{\tau }$. Notice also that $H^{\pm}\to
AW^{\pm}$ can have a sizable branching fraction, offering the possibility to discover both $A$ and $H^{\pm}$ simultaneously in this decay mode.
Large $\tan\beta$ searches: general features {#sec:largetb}
--------------------------------------------
In this subsection we present our analysis for the large $\tan \beta$ regime, fixing $\tan \beta=20$. As shown in Section \[sec:spectra\], the changes in the spectrum with respect to the MSSM are less important than in the low $\tan \beta$ case.
We use Eq. (\[eq:ggheff\]) to estimate the gluon fusion production cross section at NLO in $\alpha_{s}$. Although the impact of the bottom loop in the $K$-factor is more important for larger $\tan\beta$, the NLO K-factor in our model is still expected to be within $20\%$ of the NLO SM K-factor, as discussed at the beginning of Section \[sec:lowtb\]. Furthermore, as shown in [@Spira:1997dg], the effects on the K factor due to a light sparticle spectrum like the one we are considering are negligible at large $\tan\beta$. Hence, we conclude that simply computing the right-hand side of Eq. (\[eq:ggheff\]) allows us to obtain the NLO gluon fusion production cross section within $20\%$ accuracy even at large $\tan\beta$. In this regime, production in association with a $b\bar{b}$ pair can become important, and can be obtained in our model from existing results by a simple rescaling with the effective coupling $g^{2}_{\phi bb}$, where $\phi = h, H, A$ \[see Eq. (\[geff\])\].
{width="7.9cm"} {width="7.9cm"}
We show in Fig. \[fig:lbblwwvsggl\] the effective coupling of $h$ to down-type fermions, $g_{hb\bar{b}}^2$, for both large (left panel) and small (right panel) $\tan\beta$. At large $\tan\beta$, one sees that the currently allowed models (blue and red points) have a gluon fusion production cross section which ranges from $0.7 - 5$ times the SM value. The most striking feature is that the coupling to bottom pairs can be strongly suppressed. For large $\tan \beta$ this happens for a value of $\sigma (gg \to h) /SM$ of around $1.4$. These models have $110~{\rm GeV} \lesssim m_{h} \lesssim 150~{\rm GeV}$, where the decays into $b\bar{b}$ of the SM Higgs are important. The suppression in $g_{hb\bar{b}}^2$ can also be observed at low $\tan\beta$ (right panel). In this case, the associated values of $m_{h}$ are in the somewhat higher range from $120~{\rm GeV}$ to $250~{\rm GeV}$, with the strongest suppressions occurring for $m_{h} > 150~{\rm GeV}$.
The suppression in the coupling to down-type fermions is somewhat reminiscent of the small $\alpha_{\rm eff}$ scenario [@Carena:2002qg; @Carena:2005ek], but there are important differences. In the small $\alpha_{\rm eff}$ scenario the $g_{h b
\bar{b}}$ coupling is suppressed as a result of a cancellation between the tree-level and one-loop contributions. This can happen at large $\tan\beta$, where the radiative effect is enhanced at the same time that the tree-level contribution is somewhat suppressed, thus allowing for a cancellation. Besides large $\tan\beta$, sizable values of $\mu
A_t/M_{SUSY}^2$ are necessary, and the cancellation is found to happen only for certain values of $m_{A}$ (below or of order $200~{\rm GeV}$) that are highly correlated with $\tan\beta$ [@Carena:2002qg]. In contrast, the suppression we find occurs as a result of a cancellation between the tree-level MSSM contribution and those due to the higher-dimension operators (we have checked that the picture remains unchanged by turning off all loop effects). Most importantly, the fact that the suppression occurs at tree-level implies that the couplings to bottom and tau pairs are simultaneously (and strongly) suppressed. This does not tend to happen in the small $\alpha_{\rm
eff}$ scenario, since the radiative enhancements for bottoms and taus happen in different regions of parameter space. Also, in spite of the large number of parameters, there is a clear correlation between the $g_{hb\bar{b}}^2$ suppression and the $\sigma(gg \to h)$ enhancement. The increase in the gluon fusion cross section is due to the destructive interference of the bottom loop in the gluon fusion cross section, and also to light SUSY particles running in the loop.
{width="7.9cm"}
The suppression of the down-type fermion channels implies a general enhancement in the branching fractions of the remaining channels. The most interesting enhancements are those in the gauge boson channels: $WW$, $ZZ$ and $\gamma\gamma$. In the left panel of Fig. \[fig:htoggtb20\] we show the branching fraction into WW at large $\tan\beta$, again as a function of the gluon fusion production cross section normalized to the SM value, which can be compared to the left panel of Fig. \[fig:lbblwwvsggl\]. We see that the region where the $hb\bar{b}$ coupling is suppressed is exactly where the WW branching fraction is greatly enhanced, and leads to an interesting Tevatron sensitivity in the $W$ channel over a wide range of $m_{h}$. The left panel of Fig. \[fig:htoggtb20\] clearly exhibits how the Tevatron covered (red) points arise. The upper red region corresponds to those models within Tevatron reach in the $h \to WW$ search, while the two lower red regions contain only points that can be probed in the $h \to b \bar{b}$ channel. These latter models always have a branching fraction into WW below about 20%.
In the right panel of Fig. \[fig:htoggtb20\] we show the $gg \to h
\to \gamma \gamma$ cross section, normalized to the SM. We see that the di-photon signal can be increased with respect to the SM one by up to a factor of 10. This strong enhancement is a direct result of the decreased branching fraction into $b \bar{b}$, together with the enhancement in the gluon production cross section discussed above. The points with enhanced signal in the diphoton channel correspond to values of $m_h$ between $110~{\rm GeV}$ and $130~{\rm GeV}$. It is interesting to compare to the latest available diphoton analysis from CDF [@:cdftwogammas] and D0 [@:d0twogammas]. The CDF analysis, performed with 5.4 fb$^{-1} $ of data, quotes an observed limit of 18.7-25.9 for the di-photon cross section normalized to the SM. The D0 analysis, with 4.2 fb$^{-1} $, gives a corresponding limit of 11.9-28.3 . [^16] As a result, the enhancement in the di-photon signal we find can be interesting at the Tevatron, and of course it would be spectacular at the LHC. One should also notice that for models with enhanced $BR(h \to b \bar{b})$, the signal into photons can be reduced by up to a factor of 10.
The $gg \to h \to WW$ signal (not shown here) presents the same behavior as the $\gamma \gamma$ one. This can be easily understood as follows. In the SM, the $h \to \gamma \gamma$ decay mode proceeds via $W$ and top loops, the former giving the dominant effect. In our currently allowed (blue and red) points, the coupling of $h$ to tops and W’s is very close to the SM value (the differences are below 2%). Although $g_{hb\bar{b}}$ can be enhanced by a factor of $10$, the bottom loop is still a small contribution to the $h \to \gamma \gamma$ process. Therefore, the partial widths $\Gamma (h \to WW)$ and $\Gamma (h \to \gamma \gamma)$ in our model are very close to the SM ones, and the changes in the branching ratios of each channel are common and strictly due to the variation of $BR(h \to b \bar{b})$ with respect to the SM. Therefore, enhancements in the $WW/ZZ$ channels can also be interestingly large.
With respect to the remaining Higgs bosons, the situation resembles the large $\tan \beta$ regime of the MSSM. Both $H$ and $A$ decay mainly into bottoms and taus, while the charged Higgs goes to either $\tau \nu_{\tau}$ or $t \bar{b}$ depending on its mass. It is also possible for a heavy Higgs to decay into the lightest one: ${\rm BR}(H
\to h h)$ can reach 30%, while both ${\rm BR}(A \to h Z)$ and ${\rm
BR}(H^{\pm} \to h W^{\pm})$ can reach 10%, provided the decaying Higgs boson mass is above $200~{\rm GeV}$. For this mass range, the decay mode into sparticles can also be important, if kinematically allowed.
Large $\tan\beta$ searches: benchmark points {#sec:largetbbench}
--------------------------------------------
Having described the main differences of the large tangent beta regime with respect to the MSSM, we show a selected sample of benchmark points.
### Scenarios within Tevatron reach
We start with the points covered in the near future by the Tevatron via the $h \to b \bar{b}$ search. From Fig. \[fig:htoggtb20\] one sees that for these points (lowest red region in the left panel), the signal into $\gamma \gamma$ (and $WW$) can be enhanced by at most a factor of $2$. Since such enhancement factors can also be obtained within the MSSM (for instance with sparticle masses around $500~{\rm
GeV}$), we will not show a benchmark point here, but will briefly comment on the main characteristics of these type of models. The Tevatron could claim a hint in the $b \bar{b}$ channel, while at the LHC the signals into $\gamma \gamma$ and $\tau \bar{\tau}$ are enhanced by up to a factor of $2$ with respect to the SM, thus allowing for a discovery using these decays modes. Regarding the remaining Higgs bosons, one sees that both $H$ and $A$ decay mainly into bottoms and taus. The gluon fusion production cross section for $H$ and $A$ is around 80% of the SM-value, while the $bbh$ production becomes an important mechanism due to the large $\tan\beta$ enhancement (the CP-even $H$ has highly suppressed couplings to W’s and Z’s). Thus a discovery in the $H/A \, \tau\bar{\tau}$ search may be feasible [@atlasphystdr]. We note also that $A$ and $H$ can be very close in mass, so that the two states cannot be disentangled at the LHC, but rather the signals have to be added up. Because of a sizable branching fraction into the $\tau \nu_{\tau}$ channel, the charged Higgs can be within LHC reach, even for $m_{H^{\pm}} > m_{t}$ [@Aad:2009wy]. The way to distinguish such a situation from the MSSM will be through the observation of relatively light superparticles.
As we mentioned before, an interesting possibility is to have sizable branching fractions for the decay modes $H \to h h$, $A \to h Z$ and $H^{\pm} \to h W^{\pm}$. This requires heavy Higgs bosons with a mass above $250 ~{\rm GeV}$. Depending on the details of the SUSY spectrum, also decays into sparticles may be open. We have found that the branching fractions in these multi-Higgs channels are below 10% in most cases, and that one would still have both $A$ and $H$ decaying sizably into down-type fermions, with $H^{+}$ decaying preferably into $t \bar{b}$ but with a non-negligible branching fraction into $\tau\nu_{\tau}$ due to the large $\tan\beta$ enhancement. Provided that $m_{H} > 300~{\rm GeV}$, the $H \to h h$ branching fraction can reach values of up to $20 - 30\%$, which is interesting since it allows for the potential observation of several Higgs states.
*Point G: SM-like Higgs heavier than the MSSM upper bound*
**POINT G**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
--------------------------------- --------------------- ------------------------------- -----------------------------
267 148.6 297 283
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.97 0.03 1.64 0.14
channel BMSSM (SM) channel BMSSM (SM)
$h \to b \bar{b} $ 0.43 (0.20) $ h \to \tau \bar{\tau} $ 0.07 (0.02)
$h \to ZZ $ 0.08 (0.05) $ h \to WW $ 0.41 (0.66)
$ H \to b \bar{b} $ 0.75 $ H \to \tau \bar{\tau} $ 0.13
$A \to b \bar{b} $ 0.84 $ A \to \tau \bar{\tau} $ 0.14
$ H^{\pm} \to \tau \nu_{\tau} $ 0.21 $H^{\pm} \to t \bar{b} $ 0.75
: [*Masses and branching fractions in the BMSSM (and in the SM for $h$) for point G.* ]{}[]{data-label="tab:pointG"}
We turn now to the models covered at the Tevatron via the $h \to WW$ search (upper red region in Figs. \[fig:htoggtb20\]). We show in Table \[tab:pointG\] [^17] an example where $m_h$ is above the maximum attainable value in the $m_h$ max scenario of the MSSM, with sparticles at the ${\rm TeV}$ scale. The $1/M^2$ operators contribute about $45~{\rm GeV}$ to $m_{h}$. We notice that in this point the enhancement in the gluon fusion production cross section \[$1.64 \times \sigma_{SM}(gg \to h)$\] is compensated by the reduction in the $WW$ branching fraction with respect to the SM ($0.41/0.66$), thus resulting in a $gg \to h \to WW$ signal close to the SM one. As can be seen in the right panel of Fig. \[fig:htoggtb20\] (interpreted for the $WW$ channel) this is a general feature of the Tevatron covered points in the higher range of $m_{h}$. The nonstandard neutral Higgs bosons, $H$ and $A$, can be detected in the $\tau \bar{\tau}$ (or $\mu \bar{\mu}$) channels, as is well known for the large $\tan\beta$ region of the MSSM. The charged Higgs can be searched for in the $\tau\nu_{\tau}$ channel. However, we emphasize again that the observation of light SUSY signals would give compelling evidence for BMSSM physics. There are also Tevatron covered (red) points at smaller $m_{h}$ values, around $110~{\rm GeV}$ with an enhanced di-photon signal. We discuss these type of scenarios in the next section, together with the Tevatron uncovered (blue points) in the same region.
LHC searches {#lhc-searches}
------------
Referring to Fig. \[fig:htoggtb20\] we split the Tevatron uncovered (blue) points according to whether their signal into photons (and W’s) is enhanced or suppressed. For the latter case, one has that $h$ decays mainly into bottom and tau pairs. In these scenarios, $h$ can be within the reach of the LHC in the $\tau \bar{\tau}$ channel, and if the suppression of the ZZ coupling is not extreme, maybe also in the $ZZ \to 4l$ channel. Higgs decay chains, such as $H \to hh \to b
\bar{b} \tau \bar{\tau}$, can also give rise to interesting (if challenging) signatures. We do not show a benchmark point here since the branching fractions of the relevant Higgs decay chain modes will depend on the details of the sparticle spectrum.
*Point H: SM-like Higgs with enhanced di-photon signal*
**POINT H**
$m_A ~({\rm GeV})$ $m_h ~({\rm GeV}) $ $m_H ~({\rm GeV}) $ $m_{H^{\pm}} ~({\rm GeV}) $
--------------------------------- --------------------- ---------------------------------- -----------------------------
210 111.3 215 225
$g_{hWW}^2$ $g_{HWW}^2$ $g_{hgg}^2$ $g_{Hgg}^2$
0.98 0.02 1.39 0.84
channel BMSSM (SM) channel BMSSM (SM)
$h \to b \bar{b} $ 0.03 (0.79) $ h \to \gamma \gamma / 10^{-3}$ 12.1 (2.1)
$h \to $ jets 0.56 (0.07) $ h \to WW $ 0.36 (0.05)
$ H \to b \bar{b} $ 0.86 $ H \to \tau \bar{\tau} $ 0.14
$A \to b \bar{b} $ 0.86 $ A \to \tau \bar{\tau} $ 0.14
$ H^{\pm} \to \tau \nu_{\tau} $ 0.35 $H^{\pm} \to t \bar{b} $ 0.64
: [*Masses and branching fractions in the BMSSM (and in the SM for $h$) for point H.* ]{}[]{data-label="tab:pointH"}
We illustrate the features of models with a strong enhancement of the di-photon signal with point H [^18] (shown in Table \[tab:pointH\]). We see that $h$ is rather light (the $1/M^2$ operators contribute about $10~{\rm GeV}$ to $m_{h}$), but escaped detection at LEP due to the strong suppression of the $b\bar{b}$ channel. The $gg \to h \to \gamma \gamma$ signal is larger than the SM one by a factor of $8$, thus allowing for a very nice and clean detection of $h$ at the LHC. As was discussed in the context of Fig. \[fig:htoggtb20\], the same enhacement also occurs for the $WW$ and $ZZ$ channels. Therefore, and in spite of such a light Higgs mass, the $gg \to h \to ZZ \to 4l$ channel would be at the reach of the LHC.
For the remaining neutral Higgs bosons ($H$ and $A$), one will have to consider the $\tau \bar{\tau}$ search. The charged Higgs may be detected at the LHC in the $\tau \nu_{\tau}$ channel.
Note that the benchmark point H has nonstandard Higgs bosons that are too light to allow decays into $hh$. However, given that $h$ is rather light in the region with suppressed $b\bar{b}$ couplings, it is possible that such *exotic* channels might be open, while still having an interesting di-photon signal. As mentioned before, in such cases it is possible that other channels involving SUSY particles are also open.
Conclusions {#sec:conclu}
===========
We studied the Higgs collider phenomenology of BMSSM scenarios, i.e. supersymmetric extensions of the MSSM within an EFT framework where the effects of the BMSSM degrees of freedom enter through higher-dimension operators. As emphasized in [@Carena:2009gx] the first two orders in the $1/M$ expansion can be phenomenologically significant, and should be included. In the present work, we have performed a model-independent study to highlight the variety of collider signals that become available in such scenarios.
The coupling of the lightest CP-even Higgs to bottom pairs can be suppressed due to cancellations between the MSSM contribution and those from the higher-dimension operators. It does not seem to require a special tuning of parameters and occurs in both the low and large $\tan\beta$ regimes. As a result, the signals in clean channels, such as the di-photon or $WW$ ones, can be greatly enhanced. This suppression in the $h b \bar{b}$ induces an enhancement in the gluon fusion production cross section, beyond the one arising from light sparticles in the loop.
To emphasize the interplay between the Tevatron and the LHC, we have analyzed projections for the Tevatron assuming a total integrated luminosity of $10~{\rm fb}^{-1}$ per experiment and a 50% efficiency improvement in the $WW$ and $b\bar{b}$ search channels with respect to present results. We find that the current Tevatron data already probes a large class of SUSY models, especially in the $WW$ channel. The future projections indicate that the $b \bar{b}$ channel can become effective for a SM-like Higgs search. Moreover, a combination of the $b\bar{b}$ and $WW$ search channels, together with the $\tau\bar{\tau}$ decay mode in the large $\tan \beta$ region, would further enlarge the set of BMSSM models that can be probed at the Tevatron. However, our main interest in this work was to survey the types of signals that might be expected in SUSY scenarios, many of which are not realized in the MSSM limit. Improving the analysis by combining channels and/or moderately increasing the luminosity will not significantly change our conclusions. Lightest CP-even Higgs bosons with masses above $180~{\rm GeV}$, that can not be probed by the Tevatron, will be at the reach of the LHC.
Most of the changes in the expected Higgs signals, compared to the MSSM, can be understood in large part from the altered Higgs spectrum. We have surveyed a wide range of possibilities by scanning over the parameter space of the higher-dimension operators. Motivated by naturalness arguments, we have chosen the SUSY-breaking scale close to the EW scale, with the BMSSM physics at the ${\rm TeV}$ scale. In this case, the contributions from the SUSY particles to the Higgs spectrum are subleading compared to the ones coming from the BMSSM physics. In the case of a heavier SUSY spectrum, and for a scale $M$ such that the effective field theory approach remains valid, the qualitative features of the Higgs phenomenology triggered by the BMSSM physics will be similar. However, a detailed study should be performed for each specific choice of the heavy scale $M$, the scale of SUSY-breaking $m_S$, and the $\mu$-term to address the quantitative features of the Higgs sector.
We have defined a number of “benchmark points” in order to discuss the correlations between different Higgs signals. Interestingly, we find that there can be significant mixing in the CP-even Higgs sector, allowing non-negligible couplings of both CP-even Higgs eigenstates to the EW gauge bosons. In addition, they can both be in the right mass range to decay predominantly into $W$’s or $Z$’s, thus enabling a detailed and direct study of the physics of EWSB. For these benchmark points, the $1/M^2$ effects add a few tens of GeV to $m_{h}$, and have a rather relevant impact on the collider phenomenology (but we remind the reader that the $1/M^2$ operators can easily give a much larger contribution to $m_{h}$; see Fig. 1 of Ref. [@Carena:2009gx]). Furthermore, we have found viable examples where the nonstandard CP-odd Higgs can be produced in charged Higgs decays. Moreover, unusual decay chains such as $h \to AA$ or $H \to AA$ are also possible, without $A$ being ultralight. These channels are most interesting in the low $\tan\beta$ region where the $\tan\beta$-enhanced production of the nonstandard Higgs bosons is not available. These Higgs decay chains open the possibility of fully reconstructing the Higgs content of a 2HDM in such supersymmetric scenarios. We also find scenarios where observing the Higgs sector is more challenging, and would require dedicated studies that go beyond the scope of this work.
In conclusion, we find that Higgs signals in supersymmetric scenarios can be markedly different from those in the MSSM paradigm. If all third generation squarks turn out to be light ($m_S \leq 300~{\rm
GeV}$), given the LEP Higgs mass bounds, this will imply a clear case for BMSSM physics. The heavier degrees of freedom could be at the kinematic reach of the LHC, but depending on their nature the direct discovery might be elusive. In either case, supersymmetric Higgs searches can provide evidence of physics beyond the MSSM.
We would like to thank Oliver Brein and Karina Williams for making an unofficial version of the HiggsBounds code available to us, and for the help provided. J.Z would like to thank the Theory Division of Fermilab for hospitality during the final stages of this work. Fermilab is operated by the Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy. E.P. is supported by DOE Grant No. DE-FG02-92ER40699. The work of J.Z is supported by the Swiss National Science Foundation (SNF) under Contract No. 200020-126691.
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<http://www-d0.fnal.gov/Run2Physics/WWW/results/prelim/HIGGS/H66/H66.pdf>
[^1]: The present Tevatron bounds on supersymmetric charged Higgs bosons are beyond the parameter space that we study in this work.
[^2]: We have checked that these results are consistent, in the appropriate limits, with those of Ref. [@Antoniadis:2009rn], that appeared soon after Ref. [@Carena:2009gx]. We emphasize, however, that one has to treat near degenerate cases in the CP-even sector with care, as explained in [@Carena:2009gx].
[^3]: We thank the authors of [@Bechtle:2008jh; @Bechtle:2009ic] for providing us with a modified version of the code that includes the LEP 2 jet analysis.
[^4]: Here $m_{S}$ gives the scale of SUSY breaking in the heavy sector. The detailed differences between the various SUSY-breaking operators are parametrized via ${\cal O}(1)$ dimensionless parameters over which we scan. See [@Carena:2009gx] for complete details.
[^5]: We evaluate the scale inside the logarithms associated with SUSY loops at $\sqrt{M^2_{SUSY} + m^2_{t}} \approx
347~{\rm GeV}$.
[^6]: We have not included the future projection of this channel in our analysis. We expect that the increase in luminosity has a minor incidence in the additional number of points excluded.
[^7]: The difference between $g^2_{h/H WW}$ and $g^2_{h/HZZ}$ arises only from the custodially-violating higher-dimension operators, and was shown in [@Carena:2009gx] to be numerically negligible.
[^8]: Here the inversion between $h$ and $A$ also takes place, but the $h \to A Z$ decay mode is typically suppressed by a factor of 10 or more with respect to the dominant decay mode $h \to AA$, when kinematically allowed.
[^9]: Notice, however, that we have not combined the $h
\to WW$ and $h \to b \bar{b}$ channels.
[^10]: Following the notation of [@Carena:2009gx], the coefficients of the effective operators for this point are: , , , , , , , , , , , , , , , , , , , , , .
[^11]: The Lagrangian parameters for this point are: , , , , , , , , , , , , , , , , , , , , , .
[^12]: The Lagrangian parameters for this point are: , , , , , , , , , , , , , , , , , , , , , .
[^13]: The Lagrangian parameters for this point are: , , , , , , , , , , , , , , , , , , , , , .
[^14]: The Lagrangian parameters for this point are: , , , , , , , , , , , , , , , , , , , , , .
[^15]: The Lagrangian parameters for this point are: , , , , , , , , , , , , , , , , , , , , , .
[^16]: The factor of almost ten enhancement in the di-photon signal in our model is based on gluon fusion production, which at the Tevatron contributes 73 % - 95 % of the total SM cross section.
[^17]: The Lagrangian parameters for this point are: , , , , , , , , , , , , , , , , , , , , , .
[^18]: The Lagrangian parameters for this point are: , , , , , , , , , , , , , , , , , , , , , .
|
---
author:
- Igal Arav
- and Yaron Oz
title: 'The Sound of Topology in the AdS/CFT Correspondence'
---
Introduction {#sec:intro}
============
The aim of this paper is to analyze the properties of 2-point correlation functions of finite-temperature strongly coupled (deconfined) gauge field theories defined on spacetimes with general spatial topology, and admit a dual black hole description via the AdS/CFT correspondence. Of specific interest is the dependence of these properties on the topology of the space on which the gauge theory is defined, and its temperature. We will see that the topology is indeed encoded in the properties of the correlation function and its poles, corresponding to quasinormal modes in the dual bulk spacetime: the sound of topology. In particular, the asymptotic “slope” of the poles encodes information about the spatial topology and the temperature, but is “universal” otherwise, i.e. it doesn’t depend on the type of operators considered.
One motivation for this work is the fact that important aspects of the strong coupling dynamics of gauge field theories can depend crucially on the topology of space-time. For instance, the confinement-deconfinement phase transition is a property of strongly coupled thermal conformal gauge theories defined on a space-time with spherical spatial topology. The theory is always in the deconfined phase in the planar and hyperbolic cases. Thus, one may hope that insights to the confinement-deconfinement phase transition mechanism can be gained by analyzing the strong coupling dynamics on different spatial topologies. This can be done by using the gravitational dual description. A second motivation is the recently studied correspondence between gravitational dynamics and strongly coupled condensed matter systems. One can naturally envision in and out of equilibrium condensed matter systems realized on different spatial topologies, where our study can be of much relevance. A third motivation, perhaps a more remote one, is the relevance of the interplay between topology and field theory dynamics to astrophysical/cosmological setups.
We distinguish three classes of spatial topology [^1]: The spherical, the flat and the hyperbolic (The relevant thermodynamic quantities and their dependence on the topology are given in Appendix \[app:thermodynamicquant\]). Consider a gauge theory defined on a $(d-1)$-dimensional spacetime with the topology of $ \mathbb{R}\times\Omega_{d-2}^{FT} $, where $ \Omega_{d-2}^{FT} $ is a constant curvature $ (d-2) $-dimensional Riemannian manifold of arbitrary topology[^2]. The two basic scales that characterize the theory are the temperature $ T $ and the scalar curvature of the $ \Omega_{d-2}^{FT} $ manifold $ R_\Omega^{FT} $. In the analysis we will consider two different regimes:
1. The regime of $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \to 0 $: In this case the curvature of the $ \Omega_{d-2}^{FT} $ manifold may essentially be neglected, so we expect all physical quantities to behave as they do in the case of a flat ($ R_\Omega^{FT} = 0 $) space. Therefore, in this limit one does not distinguish between different topologies (aside from the known changes of the Laplace operator spectra between different topologies). This limit includes the hydrodynamic limit[^3]: $ \frac{\omega}{T},\frac{L_s^{FT}}{T} \to 0 $, where $ \omega $ denotes the frequency and $ L_s^{FT} $ is the spatial Laplace operator eigenvalue.
2. The regime of $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} = \text{fixed} $: In this case it will be possible to distinguish between the different topologies. In particular we will study the limit of $ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $, where analytical asymptotic expressions will be derived for the quasinormal modes and the 2-point correlation functions.
Note that the $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \gg 1 $ regime does not exist in the black hole description in the case of spherical topology (since the temperature has a finite minimal value) - it corresponds to a different (confining) phase on the field theory side. Nor does it exist in the case of flat topology (for which $ R_\Omega^{FT} = 0 $). While this regime does exist for the hyperbolic case, there is nothing separating it qualitatively from the $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} = \text{fixed} $ regime in the context of this paper. For these reasons, we won’t consider this regime separately here.
The main new results of this work are (see Appendix \[app:notations\] for a list of notations and definitions):
- In the asymptotic limit, where $ \frac{L_s^{FT}}{T} \sim \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $, the following holds:
- For all types of topologies and all the temperatures , the quasinormal frequencies (and therefore the poles of the field theory correlators) are of the form $$\lambda_n = \lambda_0 + n\Delta\lambda \ ,$$ where $ \Delta\lambda $ depends on the topology and temperature.
- In this limit we derive asymptotic expressions for the field theory 2-point retarded correlation functions of the operators dual to the different perturbation types. These are given in Equations \[eq:asymptcftcorrelatorscalarevend\] and \[eq:asymptcftcorrelatorscalaroddd\] for the scalar perturbation modes, in Equations \[eq:asymptcftcorrelatorlongvectorevend\] and \[eq:asymptcftcorrelatorlongvectoroddd\] for the longitudinal vector perturbation modes, and in Equations \[eq:asymptcftcorrelatortransvectorevend\] and \[eq:asymptcftcorrelatortransvectoroddd\] for the transverse vector perturbation modes.
- In the hyperbolic case there is a specific temperature $ T_c $, where the bulk “black hole” solution has no singularity, and is isometric to AdS. In this special case the following holds:
- At $ T=T_c $ we derive exact expressions for the field theory 2-point retarded correlation functions of the operators dual to the different perturbation types. These are given in Equations \[eq:scalarexactcorrelatorintceven\] and \[eq:scalarexactcorrelatorintcodd\] for the scalar perturbation modes, in Equations \[eq:longvectorexactcorrelatorintceven\] and \[eq:longvectorexactcorrelatorintcodd\] for the longitudinal vector perturbation modes and in Equations \[eq:transvectorexactcorrelatorintceven\] and \[eq:transvectorexactcorrelatorintcodd\] for the transverse vector perturbation modes.
- For $ T>T_c $, the asymptotic $ \Delta\lambda $ has a real component $ \operatorname{Re}(\Delta\lambda)>0 $. As $ T\to T_c $, $ \Delta\lambda $ becomes imaginary according to Equation \[eq:asymptparamsaroundtcde4b\] for $ d=4 $ (as previously calculated in [@Koutsoumbas:2006xj]), and according to Equation \[eq:asymptparamsaroundtcdg4b\] for $ d>4 $.
The paper is organized as follows. In Section \[sec:quasinormalmodes\], we will review some known results regarding quasinormal modes (formulated in our notation). We will calculate the QNM equations for massless scalar and vector perturbations, the exact solutions for the “special” case mentioned above (the hyperbolic case with the specific temperature $ T_c $), the approximate quasinormal frequencies in the asymptotic limit ($ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $) and the diffusion mode in the hydrodynamic limit. We will discuss the dependence of the asymptotic quasinormal modes on the topology and the temperature, in particular in the hyperbolic case around the temperature $ T_c $, and demonstrate it using some numerical results. In Section \[sec:cftcorrelators\], we will use AdS/CFT dictionary to calculate the 2-point correlation functions of the dual field theory. We will obtain exact expressions for the hyperbolic case with $ T=T_c $ and approximate expressions in the asymptotic limit for general topology and temperature. We will compare the asymptotic expressions with exact numerical solutions. Section \[sec:discussion\] is devoted to a discussion. Details of calculations are outlined in the appendices.
Quasi-Normal Modes {#sec:quasinormalmodes}
==================
QNM Definition {#subsec:qnmdefinition}
--------------
Quasinormal modes of black holes or black branes are defined as the late-time oscillation modes of the black hole metric and the fields coupled to it, satisfying certain boundary conditions. Put differently, these are the eigenmodes of the linearized equations of motion over the black hole background. The boundary conditions are specified at the black hole horizon and at spatial infinity. The boundary condition at the horizon is chosen so that the solution corresponds to a wave ingoing into the horizon, while the boundary condition choice at spatial infinity depends on the asymptotic nature of the spacetime.
Choosing the incoming wave solution gives the boundary condition: $$\left.\psi\right|_{z=0} \sim z^{-\frac{i\omega}{C}} \ ,$$ where $ z=0 $ corresponds to the BH horizon and $ z=1 $ corresponds to spatial infinity. As for the boundary condition at spatial infinity, different boundary conditions have been investigated for different asymptotic spacetime geometries. In the context of AdS/CFT, the relevant boundary condition is the one that corresponds to the poles of the retarded correlator of the CFT operators dual to the investigated field. For the fields discussed here, this condition amounts to the Dirichlet boundary condition (see [@Son:2002sd], [@Berti:2009kk] and [@Nunez:2003eq]): $$\left.\psi\right|_{z=1}=0 \ .$$
QNM Equations {#subsec:qnmequations}
-------------
We consider the d-dimensional black hole metric given by a metric of the form: $$\label{eq:rmetric}
{\,\mathrm{d}}s_{bulk}^2 = -f(r){\,\mathrm{d}}t^2 + \frac{1}{f(r)} {\,\mathrm{d}}r^2 + r^2 {\,\mathrm{d}}\Omega_{d-2}^2 \ ,$$ where $
{\,\mathrm{d}}\Omega_{d-2}^2 = \sum_{i,j} (g_\Omega)_{ij} {\,\mathrm{d}}x^i {\,\mathrm{d}}x^j
$ is the inner metric of a constant curvature, (d-2)-dimensions Riemannian manifold with of same topology as the $ \Omega_{d-2}^{FT} $ manifold.
From the vacuum Einstein equation with a cosmological constant one can deduce the form of $ f(r) $ to be $$f(r) = k + \frac{r^2}{R^2} - \frac{r_0^{d-3}}{r^{d-3}} \ ,$$ where $ R $ is related to the cosmological constant by $
R^2 = -\frac{(d-2)(d-1)}{2\Lambda}
$ and $ k $ is related to the scalar curvature $ R_\Omega $ of the manifold $ \Omega_{d-2} $ by (see, for example, [@Birmingham:1998nr]) $
k = \frac{R_\Omega}{(d-2)(d-3)}
$. In terms of these coordinates, the horizon of the black hole is located at $ r=r_+ $ where $ f(r_+)=0 $, while the boundary is located at $ r\to\infty $.
After making the transformation to the coordinate $ z = 1-\frac{r_+}{r} $, the metric takes the form: $$\label{eq:zmetric}
{\,\mathrm{d}}s^2 = -\frac{\rho^2 \tilde{g}(z)}{(1-z)^2} {\,\mathrm{d}}t^2 + \frac{r_+^2}{\rho^2\tilde{g}(z)(1-z)^2}{\,\mathrm{d}}z^2 + \frac{r_+^2}{(1-z)^2}{\,\mathrm{d}}\Omega_{d-2}^2 \ ,$$ where $ \rho \equiv \frac{r_+}{R} $, $ K \equiv \frac{k}{\rho^2} $ and $$\label{eq:gzexp}
\tilde{g}(z) \equiv 1 + K(1-z)^2 - (1+K)(1-z)^{d-1} \ .$$ In terms of the new coordinate $ z $, the horizon is located at $ z=0 $ while the boundary is located at $ z=1 $.
Let us assume that the spectrum of the scalar and vector Laplace operator $ \Delta_\Omega $[^4] defined on the manifold $ \Omega_{d-2} $ is given by $$\begin{aligned}
\Delta_\Omega H_{L^2}(x) & = L_s^2 H_{L^2}(x) \\
\Delta_\Omega \boldsymbol{A}_{L^2}(x) & = L_v^2 \boldsymbol{A}_{L^2}(x)\end{aligned}$$ respectively. In order for a QNM corresponding to the frequency $ \omega $ and the above Laplace operator eigenvalue to exist, the corresponding boundary conditions problem should have a non-trivial solution. Introducing the dimensionless parameters $
\lambda \equiv \frac{\omega r_+}{\rho^2}
$, $
q_s \equiv \frac{L_s}{\rho}
$, $
q_v \equiv \frac{L_v}{\rho}
$ and $
C \equiv \tilde{g}'(0)
$, we find the following equations (see Appendix \[app:qnmequationsderivation\] for a detailed derivation):
- For a (massless, minimally coupled) scalar QNM, the equation is $$\label{eq:scalareq}
(1-z)^{d-2}\partial_z\left[ \frac{\tilde{g}(z)}{(1-z)^{d-2}} \partial_z \psi \right] +
\left[ \frac{\lambda^2}{\tilde{g}(z)} - q_s^2 \right] \psi = 0 \ ,$$ along with the boundary conditions: $$\label{eq:scalarbc}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1} = 0 \ .$$
- For a “longitudinal” vector QNM, the equation is $$\label{eq:longvectoreq}
\partial_z \left[ \tilde{g}(z)(1-z)^{d-4} \partial_z \left( \frac{1}{(1-z)^{d-4}} \psi \right) \right] + \left[ \frac{\lambda^2}{\tilde{g}(z)} - q_s^2 \right] \psi = 0 \ ,$$ along with the boundary conditions: $$\begin{gathered}
\label{eq:longvectorbc}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}\\
\left. (1-z)^{d-4}\partial_z\left[ \frac{1}{(1-z)^{d-4}} \psi \right] \right|_{z=1} = \left.\left( \partial_z \psi + \frac{d-4}{1-z}\psi \right)\right|_{z=1} = 0 \ ,\end{gathered}$$ or *equivalently* the equation $$\begin{gathered}
\label{eq:longvectoreq2}
(1-z)^{d-4}\partial_z\left[ \frac{\tilde{g}(z)}{(1-z)^{d-4}} \partial_z \psi \right]\\
+ \frac{q_s^2}{\lambda^2-q_s^2 \tilde{g}(z)} \tilde{g}(z)\partial_z\tilde{g}(z) \partial_z\psi + \left[ \frac{\lambda^2}{\tilde{g}(z)} - q_s^2 \right] \psi =0 \ ,\end{gathered}$$ along with the boundary conditions: $$\label{eq:longvectorbc2}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1} = 0 \ .$$
- For a “transverse” vector QNM, the equation is $$\label{eq:transvectoreq}
(1-z)^{d-4}\partial_z\left[ \frac{\tilde{g}(z)}{(1-z)^{d-4}} \partial_z \psi \right] +
\left[ \frac{\lambda^2}{\tilde{g}(z)} - q_v^2 \right] \psi = 0 \ ,$$ along with the boundary conditions: $$\label{eq:transvectorbc}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1} = 0 \ .$$
It can be easily seen from the above equations that if $ \psi(z) $ is as a solution for any of the equations (satisfying the corresponding boundary conditions) with frequency $ \lambda $, then $ \psi^*(z) $ is a solution for the same equation (with the same boundary conditions) with frequency $ -\lambda^* $. This means that the QNM frequencies are always symmetric with respect to the imaginary axis.
A detailed derivation of the QNM Equations can be found in Appendix \[app:qnmequationsderivation\].
Exact solutions for the K=-1 case {#subsec:qnmexactsolutionsfortc}
---------------------------------
In the case where the $ \Omega_{d-2} $ manifold is hyperbolic, we consider the special state corresponding to the temperature: $$T_C \equiv \frac{\sqrt{|k|}}{2\pi R} = \frac{1}{2\pi}\sqrt{\frac{|R_\Omega^{FT}|}{(d-2)(d-3)}} \ ,$$ for which $ K=-1 $, and the horizon radius is given by $ \rho_C = \sqrt{|k|} $. In this case, the QNM equations (Equations \[eq:scalareq\], \[eq:longvectoreq\] and \[eq:transvectoreq\]) can be solved analytically. This has been done in [@Birmingham:2006zx], [@Aros:2002te] and [@LopezOrtega:2007vu]. Here we summarize these solutions in our notation.
Making the transformation: $$\begin{aligned}
w &\equiv{\tilde{g}(z)}= 1-(1-z)^2 \\
\psi &\equiv w^\gamma (1-w)^\delta \phi\end{aligned}$$ (where $ \gamma $ and $ \delta $ are defined in Appendix \[app:exactsolutionsfortcderivation\] for each perturbation type), one obtains the hypergeometric equation. The solution to the equation satisfying the incoming-wave boundary condition at the horizon is $\sb{2}F_1\left(a,b;c;w\right)$, where $ a $, $ b $ and $ c $ are given in Appendix \[app:exactsolutionsfortcderivation\] for each case.
Applying the QNM boundary condition at the AdS boundary we get for the scalar and transverse vector cases: $$\sb{2}F_1\left(a,b;c;1\right)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}=0 \ ,$$ and for the longitudinal vector case (and using known formula fot the hypergeometric function, taking into account $ c-a-b = \frac{d-5}{2} $) we get: $$\begin{gathered}
\left.(1-w)^{\frac{d-3}{2}}\partial_w\left[\frac{1}{(1-w)^\frac{d-5}
{2}}\sb{2}F_1(a,b;c;w)\right]\right|_{w=1} \\
= \frac{(c-a)(c-b)}{c}\sb{2}F_1(a,b;c+1;1) \\
= \frac{(c-a)(c-b)}{c} \frac{\Gamma(c+1)\Gamma(c-a-b+1)}{\Gamma(c-a+1)\Gamma(c-b+1)}
= 0 \ .\end{gathered}$$ The solution for which is given by $c-a=-n$ or $c-b=-n$ where $n\in\mathbb{Z}$ and $n\geq0$, so that the quasi-normal frequencies are:
- For the massless scalar case: $$\lambda_n = -\frac{i}{2}(d+1) \pm \sqrt{q^2-\frac{1}{4}(d-3)^2} -2ni \ .$$
- For the longitudinal vector case: $$\lambda_n=-\frac{i}{2}(d-3)\pm\sqrt{q^2-\frac{1}{4}(d-3)^2}-2ni \ .$$
- For the transverse vector case: $$\lambda_n = -\frac{i}{2}(d-1) \pm \sqrt{q^2-\frac{1}{4}(d-5)^2} -2ni \ .$$
QNM Asymptotics (for K>-1) {#subsec:qnmasymoptotics}
-----------------------------
For the case $ K>-1 $, one may find an approximate analytical expression for the $n$-th QNM frequency for $ n\to\infty $ - that is, in the asymptotic limit of $ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $. This has been done in [@Natario:2004jd] using the “monodromy” method. Here we summarize this calculation using our notations. The general stages of the calculation are as follows:
1. We bring each of the QNM equations to a Schrödinger form, with some effective potentials, using appropriate coordinate and function transformations.
2. We find the poles of these effective potentials on the complex plane, and write approximate forms for the equations in the vicinity of the poles.
3. We solve the approximate equations around each pole using Bessel functions.
4. Using the asymptotic limit of $ |\lambda| \gg 1 $, we replace the Bessel functions with their asymptotic forms.
5. We match the solutions on each region of the complex plane, and apply the appropriate boundary conditions, to get equations for values of $ \lambda $ that allow for non-trivial solutions.
6. We solve the equations analytically and obtain an asymptotic expression of the form: $$\lambda_n = \lambda_0 + n\Delta\lambda \ .$$
### Calculation of Asymptotic QNMs
Starting from the QNM Equations, we perform the following transformation: $$\psi(z) = (1-z)^{\frac{\alpha}{2}} \phi(z)$$ $${\tilde{z}}\equiv \int \frac{1}{{\tilde{g}(z)}} {\,\mathrm{d}}z = \sum_{k=1}^{d-1} \gamma_k \ln(z-z_k) + {\tilde{z}}_0 \ ,$$ where: $$\alpha=
\begin{cases}
d-2 & \text{for scalar},\\
d-4 & \text{for vector}
\end{cases} \ ,$$ $ z_k $ ($k=1,\ldots, d-1 $) are the zeros of the polynomial $ {\tilde{g}(z)}$ and[^5] $$\gamma_k = \lim_{z\to z_k} \frac{z-z_k}{{\tilde{g}(z)}} = \frac{1}{\tilde{g}'(z_k)}$$ $${\tilde{z}}_0 = - \sum_{k=1}^{d-1} \gamma_k \ln(1-z_k)$$ (so that $ {\tilde{z}}(z=1) = 0 $). After this transformation the equations take the Schrödinger form: $$\partial_{{\tilde{z}}}^2\phi + \left[ \lambda^2-V({\tilde{z}}) \right]\phi = 0 \ ,$$ where the effective potential $ V(z) $ is given for each case in Appendix \[app:effectivepotentials\].
Next we turn to the calculation of the asymptotic QNMs using the monodromy method. We first note that the asymptotic QNM frequencies satisfy (see [@Cardoso:2004up] and [@Natario:2004jd]) $$\operatorname{Im}(\lambda{\tilde{z}}_0)=0 \ .$$
Looking at the anti-Stokes lines, defined by $$\operatorname{Im}(\lambda{\tilde{z}}) = 0 \ ,$$ we choose two of them to be the contour for the calculation[^6]. We then proceed by developing the equation, and finding its solutions, around points along the chosen contour (the boundary and $ z\to\infty $ ) and “matching” the solutions. The chosen contour is shown in Figure \[fig:contour\][^7].
We first look at the equation at $ z\to\infty $ or $ {\tilde{z}}\to{\tilde{z}}_0 $. As shown in Appendix \[app:effectivepotentials\], the equation at $ {\tilde{z}}\to{\tilde{z}}_0 $ is approximately $${\partial_{\tilde{z}}}^2\phi + \left[\lambda^2-\frac{j_\infty^2-1}{4({\tilde{z}}-{\tilde{z}}_0)^2}\right] = 0 \ .$$ The solution to this equation can be written as: $$\phi \approx A_+ P_+({\tilde{z}}-{\tilde{z}}_0) + A_- P_-({\tilde{z}}-{\tilde{z}}_0) \ .$$ In the non-scalar case where $ j_\infty \neq 0 $ (and more generally $ j_\infty $ isn’t an even integer), $$P_\pm({\tilde{z}}-{\tilde{z}}_0) \equiv \sqrt{2\pi\lambda({\tilde{z}}-{\tilde{z}}_0)}J_{\pm\frac{j_\infty}{2}}\left(\lambda({\tilde{z}}-{\tilde{z}}_0)\right) \ ,$$ where $ J_{\pm\frac{j_\infty}{2}} $ are the Bessel functions. The fact that asymptotically[^8] $ |\lambda| \gg 1 $ allows us to replace the Bessel functions with their corresponding asymptotic forms. For direction (1) (as appears in Figure \[fig:contour\]), we have $ \lambda({\tilde{z}}-{\tilde{z}}_0) > 0 $, so that $$\begin{gathered}
\label{eq:monoasympinftyplus}
\phi \approx
\left[A_+ {\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_+)} + A_- {\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_-)}\right] {\mathrm{e}}^{i\lambda{\tilde{z}}}\\
+ \left[A_+ {\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_+)} + A_- {\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_-)} \right] {\mathrm{e}}^{-i\lambda{\tilde{z}}}
\ ,\end{gathered}$$ where $$\alpha_\pm = \frac{\pi}{4}(1 \pm j_\infty) \ .$$ For direction (2) we have $ \lambda({\tilde{z}}-{\tilde{z}}_0) < 0 $, so that $$\begin{gathered}
\label{eq:monoasympinftymin}
\phi \approx
\left[A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_+)} + A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_-)}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}}\\
+\left[A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_+)} + A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_-)}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ .\end{gathered}$$
Next we look at the equation at the boundary. As shown in Appendix \[app:effectivepotentials\], the equation at $ {\tilde{z}}\to 0 $ is approximately $${\partial_{\tilde{z}}}^2\phi + \left[\lambda^2-\frac{j_1^2-1}{4({\tilde{z}}-{\tilde{z}}_0)^2}\right] = 0 \ .$$ The solution is therefore: $$\phi \approx B_+ P_+({\tilde{z}}) + B_- P_-({\tilde{z}})\\
= B_+\sqrt{2\pi\lambda{\tilde{z}}} J_{\frac{j_1}{2}}(\lambda{\tilde{z}})
+ B_-\sqrt{2\pi\lambda{\tilde{z}}} J_{-\frac{j_1}{2}}(\lambda{\tilde{z}}) \ .$$ We now apply the boundary conditions at the boundary ($ z=1 $, $ {\tilde{z}}=0 $) to this solution. In all cases (the scalar, transverse vector and longitudinal vector) we end up with the condition $$B_- = 0 \ .$$ Next we again assume $ |\lambda| \gg 1 $ so that the Bessel function can be replaced with its asymptotic form. For direction (2), we have $ \lambda{\tilde{z}}> 0 $, so that $$\label{eq:monoasympboundary}
\phi \approx 2B_+\cos(\lambda{\tilde{z}}-\beta_+)
= \left[B_+ {\mathrm{e}}^{-i\beta_+}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}} + \left[B_+ {\mathrm{e}}^{i\beta_+}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ ,$$ where $$\beta_+ = \frac{\pi}{4}(1+j_1) \ .$$
Finally from the boundary conditions at the horizon ($ z=0 $, $ {\tilde{z}}\to -\infty $) we have (for line (1) as it appears in Figure \[fig:contour\]) $$\label{eq:monoasymphorizon}
\phi \sim {\mathrm{e}}^{-i\lambda{\tilde{z}}} \ .$$
Now we “match” the solutions on lines (1) and (2). In the non-scalar case, in which $ j_\infty \ne 0 $ so that $ \alpha_+ \ne \alpha_- $. In this case we have for line (1) (from Equations \[eq:monoasympinftyplus\] and \[eq:monoasymphorizon\]) $$\label{eq:monoeq1}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_+)} + A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_-)} = 0 \ .$$ For line (2) we have (from Equations \[eq:monoasympinftymin\] and \[eq:monoasympboundary\]) $$\begin{aligned}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_+)} + A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_-)} &= B_+{\mathrm{e}}^{-i\beta_+}\\
A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_+)} + A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_-)} &= B_+{\mathrm{e}}^{i\beta_+} \ .\end{aligned}$$
The condition for having a non-trivial solution for this system of equations is: $$\tan(\lambda{\tilde{z}}_0-\beta_+)
= \frac{{\mathrm{e}}^{i3\alpha_+}\sin(\alpha_+) - {\mathrm{e}}^{i3\alpha_-}\sin(\alpha_-)}{{\mathrm{e}}^{i3\alpha_+}\cos(\alpha_+) - {\mathrm{e}}^{i3\alpha_-}\cos(\alpha_-)} \ .$$ Finally, we obtain the result[^9]: $$\label{eq:asymptoticlambda}
\lambda_n = \lambda_0 + n\Delta\lambda \ ,$$ where $$\lambda_0 = \frac{\beta_+ + \arctan\left[\frac{{\mathrm{e}}^{i3\alpha_+}\sin(\alpha_+) - {\mathrm{e}}^{i3\alpha_-}\sin(\alpha_-)}{{\mathrm{e}}^{i3\alpha_+}\cos(\alpha_+) - {\mathrm{e}}^{i3\alpha_-}\cos(\alpha_-)}\right]}{{\tilde{z}}_0}
\qquad
\Delta\lambda = \frac{\pi}{{\tilde{z}}_0} \ .$$
In the scalar case, in which $ j_\infty = 0 $, we have: $$\begin{aligned}
P_+({\tilde{z}}-{\tilde{z}}_0) &\equiv \sqrt{2\pi\lambda({\tilde{z}}-{\tilde{z}}_0)}J_0\left(\lambda({\tilde{z}}-{\tilde{z}}_0)\right)\\
P_-({\tilde{z}}-{\tilde{z}}_0) &\equiv \sqrt{2\pi\lambda({\tilde{z}}-{\tilde{z}}_0)}Y_0\left(\lambda({\tilde{z}}-{\tilde{z}}_0)\right) \ .\end{aligned}$$ In a calculation similar to the non-scalar case, we again obtain the asymptotic form in Equation \[eq:asymptoticlambda\], where $$\lambda_0 = \frac{\beta_+ + \frac{\pi}{4}-\frac{i}{2}\ln(2)}{{\tilde{z}}_0}
\qquad
\Delta\lambda = \frac{\pi}{{\tilde{z}}_0} \ .$$
### Asymptotic Dependence on Topology and Temperature
From the results of the calculation in this section we see that the asymptotic parameters of the QNM frequencies (namely $ \lambda_0 $ and $ \Delta\lambda $) of each perturbation type depend on the $ \Omega_{d-2} $ manifold topology and the temperature through the parameter $ {\tilde{z}}_0 $, which depends only on $ d $ and $ K $ (where $ K $ contains the dependency on both the topology and the temperature). In fact, the asymptotic slope of the frequencies ($ \Delta\lambda $) depends only on the spatial topology and the temperature - it is the same for all types of perturbations (even for types that aren’t discussed here, such as tensorial perturbations): $$\Delta\lambda = \frac{\pi}{{\tilde{z}}_0} \ .$$ The dependence of $ {\tilde{z}}_0 $ on $ K $ is plotted in Figure \[fig:zt0valuesd4\] for $ d=4 $ and in Figure \[fig:zt0valuesd5\] for $ d=5 $. In the case of spherical ($ K>0 $) and flat ($ K=0 $) topologies, the parameter $ {\tilde{z}}_0 $, and therefore the slope $ \Delta\lambda $, always has an imaginary part and a real part (for all temperature values). However in the hyperbolic case, at the temperature $ T_c $ (as defined in Section \[sec:intro\]) where $ K=-1 $, the slope becomes completely imaginary[^10] (as seen in the exact results given in Subsection \[subsec:qnmexactsolutionsfortc\] for this case). Moreover, it can be clearly seen in the Figures \[fig:zt0valuesd4\] and \[fig:zt0valuesd5\] that $ {\tilde{z}}_0 (K) $ (and therefore $ \Delta\lambda(T) $) is not smooth at this temperature.
![The dependence of the parameter $ {\tilde{z}}_0 $ on the parameter $ K $ on the $ {\tilde{z}}_0 $ complex plane, in $ d=4 $ dimensions.[]{data-label="fig:zt0valuesd4"}](Zt0Valuesd4.pdf){width="10cm"}
![The dependence of the parameter $ {\tilde{z}}_0 $ on the parameter $ K $ on the $ {\tilde{z}}_0 $ complex plane, in $ d=4 $ dimensions.[]{data-label="fig:zt0valuesd5"}](Zt0Valuesd5.pdf){width="10cm"}
In the following we study two specific cases: the case of $ K=0 $, and the case of $ K\to -1$ ($ k<0 $,$ T\to T_c $).
#### The case of $ K=0 $
Let us first review the case of $ K=0 $ (where the $ \Omega_{d-2} $ manifold is “flat”), described for example in [@Natario:2004jd] as the limiting case of large spherical black holes. In this case, the roots of $ {\tilde{g}(z)}$ are analytically known and we can get an accurate expression for the asymptotic QNM frequencies (and most importantly the gap $ \Delta\lambda $).
Assuming that $ K=0 $, we get for $ {\tilde{g}(z)}$: $${\tilde{g}(z)}= 1-(1-z)^{d-1} \ .$$ Its roots are given by: $$z_k = 1 - {\mathrm{e}}^{i2\pi\frac{k-1}{d-1}} \qquad k=1,\ldots,(d-1) \ .$$
After calculating the residues and using the definition of $ {\tilde{z}}_0 $, we obtain the results: $${\tilde{z}}_0 = -\frac{\pi}{d-1}{\mathrm{e}}^{-i\frac{\pi}{d-1}}\frac{1}{\sin\left(\frac{\pi}{d-1}\right)}$$ $$\Delta\lambda = \frac{\pi}{{\tilde{z}}_0} = -(d-1){\mathrm{e}}^{i\frac{\pi}{d-1}}\sin\left(\frac{\pi}{d-1}\right) \ .$$ For example, for $ d=4 $ we get $ \Delta\lambda = -\frac{3\sqrt{3}}{4}-i\frac{9}{4} $, and for $ d=5 $ we get $ \Delta\lambda = -2(1+i) $.
#### The case of $ K\to -1 $
Here we consider the case of $ K\to -1 $ ($ T\to T_c $) from above (meaning $ K>-1 $). Define: $$K\equiv -1+\epsilon \qquad \epsilon>0 \ ,$$ so that $${\tilde{g}(z)}= 1-(1-\epsilon)(1-z)^2-\epsilon(1-z)^{d-1} \ .$$ Notice also that from the discussion in Section \[sec:intro\] and Appendix \[app:thermodynamicquant\] we can relate $ \epsilon $ to the temperature difference by: $$\epsilon = \frac{2}{d-2} \frac{\Delta T}{T_c} \ .$$
We first find an approximation for the roots of $ {\tilde{g}(z)}$. For $ K=-1 $ ($ \epsilon=0 $) we have: $${\tilde{g}(z)}= 1-(1-z)^2 = 0
\qquad\Rightarrow\qquad
z_1=0, z_2=2 \ .$$ In this case there are only *two* roots. When $ K>-1 $, there are $ d-1 $ roots. Two of them will be close to the two roots of the $ K=-1 $ case, while the others go to $ \infty $ as $ K\to -1 $.
Developing the roots and residues around $ z=0 $ and $ z=2 $ to first order in $ \epsilon $, we get: $$\begin{aligned}
z_1 & =0 \\
\gamma_1 &\approx \frac{1}{2+(d-3)\epsilon} \approx \frac{1}{2}-\frac{d-3}{4}\epsilon \\
z_2 &\approx 2+\frac{1+(-1)^d}{2}\epsilon \\
\gamma_2 &\approx -\frac{1}{2}-\frac{(-1)^d(d-2)+1}{4}\epsilon \ .\end{aligned}$$
Approximating the other roots we get: $$\begin{aligned}
1-z_k &\approx \epsilon^{-\frac{1}{d-3}} {\mathrm{e}}^{i\left(\frac{\pi}{d-3}+\frac{2\pi}{d-3}j\right)} \\
\gamma_k &\approx -\frac{1}{d-3}\epsilon^{\frac{1}{d-3}}{\mathrm{e}}^{-i\left(\frac{\pi}{d-3}+\frac{2\pi}{d-3}j\right)} \ ,\end{aligned}$$ where $ j \equiv d-k-1 = 0,\ldots,d-4 $.
We put these values into the definition for $ {\tilde{z}}_0 $, and after some calculations we obtain the following results:\
For the case $ d=4 $:
[align]{} \[eq:asymptparamsaroundtcde4a\] \_0 & +\
\[eq:asymptparamsaroundtcde4b\] &= -2i + ,
And for the case $ d>4 $:
[align]{} \[eq:asymptparamsaroundtcdg4a\] \_0 & - \^\
\[eq:asymptparamsaroundtcdg4b\] &= -2i - \^ .
For example, for $ d=5 $ we get $$\Delta\lambda \approx -2i-2\sqrt{\epsilon} \ .$$
Hydrodynamic Approximation
--------------------------
Taking the hydrodynamic limit of $ \lambda,q \to 0 $ ($ \frac{\omega}{T},\frac{L_s^{FT}}{T} \to 0 $, $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \to 0 $ is required) in the appropriate QNM equations, one may calculate the hydrodynamic constants of the dual gauge theory (see [@Son:2007vk], [@Kovtun:2005ev]). Since the longitudinal vector mode is coupled to charge density fluctuations in the gauge theory, its QNM spectrum in the hydrodynamic limit has to contain a mode corresponding to the charge diffusion mode in the gauge theory. Here we review the derivation of the diffusion mode from the QNM Equation in our notation.
Starting from Equation \[eq:longvectoreq\] for a longitudinal vector QNM, let us define a new function $ \psi' = z^{-\frac{i\lambda}{C}}\psi $, and get a new “shifted” equation: $$\begin{gathered}
\label{eq:longvectoreqshifted}
\left( {\partial_z}-\frac{i\lambda}{Cz} \right)
\left[ {\tilde{g}(z)}(1-z)^{d-4} \left({\partial_z}-\frac{i\lambda}{Cz}\right) \left( \frac{1}{(1-z)^{d-4}}\psi \right) \right]\\
+ \left[ \frac{\lambda^2}{{\tilde{g}(z)}}-q^2 \right] \psi = 0 \ ,\end{gathered}$$ with boundary conditions $$\left. \psi \right|_{z=0} \sim \text{const.}
\qquad
\left. (1-z)^{d-4} \left({\partial_z}-\frac{i\lambda}{Cz}\right) \left(\frac{1}{(1-z)^{d-4}}\psi\right) \right|_{z=1} = 0 \ .$$
For the hydrodynamic approximation we shall assume $ \lambda \sim q^2 \ll 1 $, and develop $ \psi $ in orders of $ \lambda $ : $$\psi = \psi^{(0)} + \psi^{(1)} + \psi^{(2)} + \ldots \ .$$ We put this into Equation \[eq:longvectoreqshifted\], equate each order to 0 and solve the equation with the corresponding boundary conditions in terms of the higher order solutions.
For the 0-order we get the general solution: $$\psi^{(0)}
\sim C_0\left[ \frac{1}{C}\ln z + \text{const.} \right] + D_0 (1-z)^{d-4} \ ,$$ since $ \left. {\tilde{g}(z)}\right|_{z=0} \sim Cz $. From the boundary condition at $ z=0 $ we get $$C_0 = 0 \ ,$$ so that $$\psi^{(0)} = D_0 (1-z)^{d-4} \ ,$$ while the boundary conditions at $ z=1 $ are automatically fulfilled.
Next we turn to the $ \lambda^1 $ order, for which we get the solution: $$\begin{gathered}
\psi^{(1)} = \frac{i\lambda D_0}{C}(1-z)^{d-4} \ln z
-(1-z)^{d-4} \frac{q^2 D_0}{d-3} \int \frac{1-z}{{\tilde{g}(z)}} {\,\mathrm{d}}z\\
+C_1 (1-z)^{d-4} \int \frac{1}{{\tilde{g}(z)}(1-z)^{d-4}} {\,\mathrm{d}}z
+D_1 (1-z)^{d-4} \ .\end{gathered}$$ Applying the boundary condition at $ z=0 $ we get $$\left.\psi^{(1)}\right|_{z=0} \sim \frac{i\lambda D_0}{C}\ln z
-\frac{q^2 D_0}{(d-3)C}\ln z
+\frac{C_1}{C}\ln z + D_1 \sim \text{const.} \ ,$$ from which we get the condition $$\label{eq:hydrocond1}
\left( i\lambda - \frac{q^2}{d-3} \right)D_0 + C_1 = 0 \ .$$ From the boundary condition at $ z=1 $ we have $$\begin{gathered}
\left. (1-z)^{d-4}{\partial_z}\left( \frac{1}{(1-z)^{d-4}}\psi^{(1)} \right) \right|_{z=1}
-\frac{i\lambda}{C}\left. \psi^{(0)} \right|_{z=1} \\
= \frac{i\lambda D_0}{C}\delta_{d,4} + C_1 - \frac{i\lambda D_0}{C}\delta_{d,4}
= C_1 = 0 \ .\end{gathered}$$ Putting this into Equation \[eq:hydrocond1\] we get to the conclusion $$\lambda = -\frac{i}{d-3} q^2 \ ,$$ which is a (normalized) diffusion relation, with the diffusion constant equal to $ \frac{1}{d-3} $, or, in terms of $ \omega $ and $ L_\mathbf{s}^{FT} $: $$\omega \approx -iD \left(L_\mathbf{s}^{FT}\right)^2
\qquad
D = \frac{d-1}{d-3} \frac{1}{4\pi T} \ .$$ This is true for any dimension $ d \ge 4 $ and any $ \Omega_{d-2} $ topology and temperature for whom the hydrodynamic condition can be fulfilled. As expected, the leading term in this approximation doesn’t depend on the topology of $ \Omega_{d-2} $.
Numerical Calculation of QNMs
-----------------------------
In the following we present some results of exact numerical calculations of the QNM spectra in several cases in order to illustrate the dependence on topology and temperature. The spectra have been calculated using the method outlined in Appendix \[app:numericalmethods\].
Figures \[fig:qnmresultsforseveralkscalard4q0K10m05\] and \[fig:qnmresultsforseveralklongvectord5q141K10m05\] contain the exact QNM spectra for the cases of spherical ($ K=1 $), flat ($ K=0 $) and hyperbolic ($ K=-0.5 $) topologies, for the scalar and longitudinal vector cases. The dependence of the asymptotic slope of the spectrum on the parameter $ K $ can be easily seen in these figures (as well as its independence of other parameters such as the spatial mode or perturbation type). The (pure imaginary) hydrodynamic mode can also be seen in the spectra of the longitudinal vector perturbations.
![Numerically calculated QNM frequencies for the case of longitudinal vector perturbations with $ d=5 $,$ K = 1 , 0 , -0.5 $ and $ q_\mathbf{s} = \sqrt{2} $.[]{data-label="fig:qnmresultsforseveralklongvectord5q141K10m05"}](QNMResultsForSeveralKScalard4q0K10m05.pdf){width="\textwidth"}
![Numerically calculated QNM frequencies for the case of longitudinal vector perturbations with $ d=5 $,$ K = 1 , 0 , -0.5 $ and $ q_\mathbf{s} = \sqrt{2} $.[]{data-label="fig:qnmresultsforseveralklongvectord5q141K10m05"}](QNMResultsForSeveralKLongVectord5q141K10m05.pdf){width="\textwidth"}
The numerical results for $ -1.1 < K < -0.9 $, as illustrated in Figures \[fig:qnmresultsforseveralkscalard4q1\] and \[fig:qnmresultsforseveralklongvectord5q1p5\], demonstrate the phenomenon discussed in Sections \[subsec:qnmexactsolutionsfortc\] and \[subsec:qnmasymoptotics\] for an hyperbolic $ \Omega_{d-2} $ manifold: At $ T=T_c $ ($ K=-1 $) the asymptotic QNM frequency gap ($ \Delta\lambda $) becomes imaginary, and for $ d=4,5 $ it remains imaginary for $ T<T_c $ ($ K<-1 $).
![Numerically calculated QNM frequencies for the case of longitudinal vector perturbations with $ d=5 $,$ -1.1 < K < -0.9 $ and $ q_\mathbf{s} = 1.5 $.[]{data-label="fig:qnmresultsforseveralklongvectord5q1p5"}](QNMResultsForSeveralKScalard4q1.pdf){width="\textwidth"}
![Numerically calculated QNM frequencies for the case of longitudinal vector perturbations with $ d=5 $,$ -1.1 < K < -0.9 $ and $ q_\mathbf{s} = 1.5 $.[]{data-label="fig:qnmresultsforseveralklongvectord5q1p5"}](QNMResultsForSeveralKLongVectord5q1p5.pdf){width="\textwidth"}
CFT Correlators {#sec:cftcorrelators}
===============
CFT Correlators From Holographic Principle {#subsec:cftcorrelatorsfromhol}
------------------------------------------
Via the AdS/CFT corresondence, a black hole (or black brane) background in the bulk spacetime, such as the ones discussed here, corresponds to a dual CFT at finite temperature in the deconfined phase. The correspondence allows one to calculate the 2-point correlation function of CFT operators by calculating the on-shell action of the bulk field dual to that operator.
The general prescription for calculating the dual CFT correlators from the AdS/CFT correspondence is discussed in [@Maldacena:1997re], [@Witten:1998qj], [@Gubser:1998bc], and [@Aharony:1999ti]. In the case of Minkowski spacetime, the calculation of the Minkowski correlators from AdS/CFT requires some additional subtleties, as explained in [@Son:2002sd]. In particular, the retarded green function defined as: $$G^R\left(t,x;t',x'\right) = -i\theta(t-t')\left\langle\left[O(t,x),O(t',x')\right]\right\rangle$$ can be calculated by choosing the incoming-wave boundary condition at the black hole horizon for the solution of the classical EOM in the bulk spacetime (see Subsection \[subsec:qnmdefinition\]).
One consequence of the above prescription is the fact that the QNM frequency spectrum of the bulk field (as defined in Subsection \[subsec:qnmdefinition\]) comprises the poles of the retarded correlator of the dual CFT operator. Therefore all of the results of Section \[sec:quasinormalmodes\] serve to teach about the poles of the retarded correlators of the theories dual to the discussed black holes. The purpose of this section is the expansion of those results from knowledge of the poles to expressions for the correlators themselves.
General Formulae For Correlators {#subsec:generalformulaeforcorrelators}
--------------------------------
The background spacetime metric on which the field theory dual to the bulk black hole is defined will be given by: $$\label{eq:cftmetricnoR}
{\,\mathrm{d}}s_{FT}^2 = -{\,\mathrm{d}}t^2 + {\,\mathrm{d}}\Omega_{FT,d-2}^{2} \ ,$$ where $
{\,\mathrm{d}}\Omega_{FT,d-2}^2 = \sum_{i,j}(g_\Omega^{FT})_{ij}{\,\mathrm{d}}x_{FT}^i {\,\mathrm{d}}x_{FT}^j
$ is the inner metric of the $ \Omega_{d-2}^{FT} $ manifold. This metric is conformally related to the bulk background metric at its boundary, with the conformal factor chosen to cancel the divergence in the bulk metric. The relation between the metrics is then given by (see [@Emparan:1999gf]) $$g_{\mu\nu}^{FT} = \lim_{r\to\infty}\left(\frac{R^2}{r^2}g_{\mu\nu}^{bulk}\right) \ ,$$ or: $$\label{eq:cftmetric}
{\,\mathrm{d}}s_{FT}^2 = -{\,\mathrm{d}}t^2 + R^2 {\,\mathrm{d}}\Omega_{d-2}^2 \ .$$ Comparing this with Equation (\[eq:cftmetricnoR\]) we get the relation between the field theory manifold and the bulk manifold $ {\,\mathrm{d}}\Omega_{FT,d-2}^2 = R^2{\,\mathrm{d}}\Omega_{d-2}^2 $, and therefore the relation between the corresponding curvatures - $
R_\Omega^{FT} = \frac{R_\Omega}{R^2}
$ and $
k_{FT} \equiv \frac{R_\Omega^{FT}}{(d-2)(d-3)} = \frac{k}{R^2}
$, and the relation between the corresponding Laplace operator eigenvalues $
L_s^{FT} = \frac{L_s}{R}
$.
Using the AdS/CFT prescription for a massless scalar bulk field and a gauge vector bulk field, we may find expressions for the retarded correlation functions of the dual field theory operators. This is done by integrating by parts the expressions for the action of these fields, using the EOMs they satisfy to get an expression that depends only on the boundary values of these fields and taking the derivative of the expression with respect to the the boundary values. We find the following expressions:
- For a massless scalar field, the correlation function is given by: $$\label{eq:scalarcftcorrelator}
G^R(\omega,\mathbf{s})
= \left.-2C_s\frac{r_+^{d-1}}{R^d}\frac{1}{(1-z)^{d-2}}
{\partial_z}\widehat{\psi}_{\omega,\mathbf{s}}(z)\right|_{z\to 1} \ ,$$ where $ \widehat{\psi}_{\omega,\mathbf{s}} $ satisfies the scalar QNM equation (Equation \[eq:scalareq\]) along with the boundary conditions: $$\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z\to 1} = 1 \ .$$ $ C_s $ is the appropriate normalization constant for the bulk scalar field (see Appendix C).
- For the longitudinal component of a vector field, the correlation function is given by: $$\label{eq:longvectorcftcorrelator}
G_{tt}^R(\omega,\mathbf{s})
= \left.4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z\to 1}$$ $$\begin{aligned}
\label{eq:longvectorcftcorrelator2}
G_{t\|}^R(\omega,\mathbf{s}) &= i\frac{\lambda}{q_\mathbf{s}}G_{tt}^R(\omega,\mathbf{s}) \\
\label{eq:longvectorcftcorrelator3}
G_{\|\|}^R(\omega,\mathbf{s}) &= \frac{\lambda^2}{q_\mathbf{s}^2}G_{tt}^R(\omega,\mathbf{s}) \ ,\end{aligned}$$ where $ \widehat{\psi}_{\omega,\mathbf{s}} $ satisfies the longitudinal vector QNM equation (Equation \[eq:longvectoreq\]) along with the boundary conditions: $$\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\quad
\left.\widehat{\chi}_{\omega,\mathbf{s}}\right|_{z\to 1}
= \left.\frac{1}{q_\mathbf{s}^2}{\tilde{g}(z)}(1-z)^{d-4}{\partial_z}\left[\frac{1}{(1-z)^{d-4}}\widehat{\psi}_{\omega,\mathbf{s}}\right]\right|_{z\to 1} = 1 \ .$$ $ C_v $ is the appropriate normalization constant for the bulk vector field (see Appendix C).
- For the transverse component of a vector field, the correlation function is given by: $$\label{eq:transvectorcftcorrelator}
G_{\bot\bot}^R(\omega,\mathbf{v})\\
= \left.-4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
{\partial_z}\widehat{\psi}_{\omega,\mathbf{v}}(z)\right|_{z\to 1} \ ,$$ where $ \widehat{\psi}_{\omega,\mathbf{v}} $ satisfies the transverse vector QNM equation (Equation \[eq:transvectoreq\]) along with the boundary conditions: $$\left.\widehat{\psi}_{\omega,\mathbf{v}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\widehat{\psi}_{\omega,\mathbf{v}}\right|_{z\to 1} = 1 \ .$$
Several notes about taking the limit $ z\to 1 $:
1. The expression $ {\partial_z}\widehat{\psi}_{\omega,\mathbf{s}}(z) $ in the limit $ z\to 1 $ is calculated by normalizing $ \psi $ at $ z=1-\epsilon $, differentiating it and then taking $ \epsilon\to 1 $, so that: $$\left.{\partial_z}\widehat{\psi}_{\omega,\mathbf{s}}(z)\right|_{z\to 1}
= \left.\frac{{\partial_z}\psi_{\omega,\mathbf{s}}(z)}{\psi_{\omega,\mathbf{s}}(z)}\right|_{z\to 1} \ .$$
2. When taking the limit $ z\to 1 $ , one must drop the contact terms - the terms that are polynomials in $ \omega $ and $ L_\mathbf{s} $ and diverge as $ z\to 1 $ (These terms are removed by process of renormalization). This can be done by developing $ {\partial_z}\widehat{\psi}_{\omega,\mathbf{s}}(z) $ in orders of $ 1-z $, and then dropping all terms up to the lowest non-contact-term order.
Note also that the spacetime coordinates correlators for the vector perturbations are related to the above longitudinal and transverse components by the following relations: $$\begin{aligned}
G_{tt}^R(t,x;t',x') &=& \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s}
G_{tt}^R(\omega,\mathbf{s})H_\mathbf{s}^{FT*}(x)H_\mathbf{s}^{FT}(x'){\mathrm{e}}^{-i\omega(t'-t)}\\
G_{ti}^R(t,x;t',x') &=& \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s}
G_{t\|}^R(\omega,\mathbf{s})H_\mathbf{s}^{FT*}(x)\frac{\partial_i H_\mathbf{s}^{FT}(x')}
{L_\mathbf{s}^{FT}}{\mathrm{e}}^{-i\omega(t'-t)} \\
G_{ij}^R(t,x;t',x') &=& \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s}
G_{\|\|}^R(\omega,\mathbf{s})\frac{\partial_i H_\mathbf{s}^{FT*}(x)}
{L_\mathbf{s}^{FT}}\frac{\partial_j H_\mathbf{s}^{FT}(x')}
{L_\mathbf{s}^{FT}}{\mathrm{e}}^{-i\omega(t'-t)} \nonumber\\
&& + \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{v}
G_{\bot\bot}^R(\omega,\mathbf{v})\tilde{A}_{\mathbf{v},i}^{FT*}(x)
\tilde{A}_{\mathbf{v},j}^{FT}(x'){\mathrm{e}}^{-i\omega(t'-t)} \ .\end{aligned}$$
A detailed derivation of these expressions can be found in Appendix \[app:corrformulaederivation\].
Exact Correlators for the K=-1 Case {#subsec:cftexactcorrelatorsfortc}
-----------------------------------
As in Subsection \[subsec:qnmexactsolutionsfortc\], in the $K=-1$ case (where the $\Omega_{d-2}$ manifold is hyperbolic and $T=T_c$), analytical expressions for the CFT correlators may be found. This section is dedicated to the calculation of the correlators in this case.
### Scalar Correlator
Continuing from the transformation defined in Appendix \[app:exactsolutionsfortcderivation\], the expression in Equation \[eq:scalarcftcorrelator\] can be written in terms of $ \phi $ and $ w $: $$\label{eq:scalarcftcorrelatorinw}
G^R(\omega,\mathbf{s})
= \left.-4C_s\frac{r_+^{d-1}}{R^d}\frac{1}{(1-w)^\frac{d-3}{2}}
\frac{\partial_w\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{w\to 1}$$ (where a contact term has been dropped).
As explained in Appendix \[app:exactsolutionsfortcderivation\], the solution to the EOM with an incoming-wave boundary condition at the horizon is $ \phi_{\omega,\mathbf{s}} = {}_2F_1\left(a,b;c;w\right) $, where $ a,b $ and $ c $ are given by Equations \[eq:scalartransexactsolutionfortcparams1\] and \[eq:scalartransexactsolutionfortcparams2\]. Define: $$\Delta \equiv c-a-b = \frac{d-1}{2} \ .$$
There are now two possible cases:
1. $ d $ is even, so that $ \Delta $ is non-integer. In this case, the hypergeometric function satisfies the following connection formula: $$\begin{gathered}
\label{eq:hypergeometricevenconnectionformula}
\phi_{\omega,\mathbf{s}}(w) = {}_2F_1\left(a,b;c;w\right) \\
= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(c-a)\Gamma(c-b)}\,{}_2F_1\left(a,b;1-\Delta;1-w\right)\\
+ \frac{\Gamma(c)\Gamma(-\Delta)}{\Gamma(a)\Gamma(b)}(1-w)^\Delta
\,{}_2F_1\left(c-a,c-b;\Delta+1;1-w\right) \ .\end{gathered}$$ Defining the coefficients: $$\begin{aligned}
A(\omega,\mathbf{s}) &= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(c-a)\Gamma(c-b)} \\
B(\omega,\mathbf{s}) &= \frac{\Gamma(c)\Gamma(-\Delta)}{\Gamma(a)\Gamma(b)} \ ,\end{aligned}$$ we get: $$\left.\frac{\partial_w\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{w=1-\epsilon}
\approx -\Delta\frac{B(\omega,\mathbf{s})}{A(\omega,
\mathbf{s})}\epsilon^{\Delta-1}\\
= -\Delta \frac{\Gamma(-\Delta)}{\Gamma(\Delta)} \frac{\Gamma(c-a)\Gamma(c-b))}
{\Gamma(a)\Gamma(b)} \epsilon^{\Delta-1} \ .$$ Putting this into Equation \[eq:scalarcftcorrelatorinw\] we get: $$\label{eq:scalarexactcorrelatorintceven}
\boxed{
G^R(\omega,\mathbf{s})
= 2(d-1)C_s \frac{r_+^{d-1}}{R^d}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}
\frac{\Gamma(a+\Delta)}{\Gamma(a)}\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ .
}$$
2. $ d $ is odd, so that $ \Delta $ is an integer. In this case, the hypergeometric function satisfies[^11]: $$\begin{gathered}
\label{eq:hypergeometricoddconnectionformula}
\phi_{\omega,\mathbf{s}}(w) = {}_2F_1\left(a,b;c;w\right) \\
= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(a+\Delta)\Gamma(b+\Delta)}
\sum_{n=0}^{\Delta-1}\frac{(a)_n (b)_n}{n!(1-\Delta)_n}(1-w)^n \\
- \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}(w-1)^\Delta\sum_{n=0}^{\infty}
\frac{(a+\Delta)_n(b+\Delta)_n}{n!(n+\Delta)!}(1-w)^n [\ln(1-w)\\
+\psi(a+\Delta+n)+\psi(b+\Delta+n)-\psi(n+1)-\psi(n+\Delta+1)] \ ,\end{gathered}$$ so that[^12]: $$\begin{aligned}
A(\omega,\mathbf{s}) &= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(a+\Delta)\Gamma(b+\Delta)} \\
B(\omega,\mathbf{s}) &= \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\frac{(-1)^{\Delta+1}}{(\Delta)!}
\left[\psi(a+\Delta)+\psi(b+\Delta)\right]\end{aligned}$$
$$\left.\frac{\partial_w\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{w=1-\epsilon}
\approx -\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}(a)_\Delta(b)_\Delta
\left[\psi(a+\Delta)+\psi(b+\Delta)\right]\epsilon^{\Delta-1} \ .$$
Putting this into Equation \[eq:scalarcftcorrelatorinw\] we get: $$\label{eq:scalarexactcorrelatorintcodd}
\boxed{
G^R(\omega,\mathbf{s})
= 4C_s\frac{r_+^{d-1}}{R^d}\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}(a)_\Delta(b)_\Delta
\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}$$
### Vector Correlators
The calculation of the exact vector correlators in the $ K=-1 $ case is similar to the scalar case: We write the expressions for the correlators in Equations \[eq:longvectorcftcorrelator\] and \[eq:transvectorcftcorrelator\] in terms of the variables defined in Appendix \[app:exactsolutionsfortcderivation\] and calculate the limits using the exact solutions. A detailed derivation is given in Appendix \[app:exactvectorcorrelatorsderivationfortc\], and here we quote the final results.
For the longitudinal vector mode, we define: $$\Delta \equiv \frac{d-5}{2} \ .$$ For the case of even $ d $ ($ \Delta $ is non-integer), we obtain the expression: $$\label{eq:longvectorexactcorrelatorintceven}
\boxed{
G_{tt}^R(\omega,\mathbf{s})
= \frac{4C_v}{d-5}\frac{r_+^{d-3}}{R^{d-2}}q_\mathbf{s}^2
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\frac{\Gamma(a+\Delta)}{\Gamma(a)}
\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ .
}$$ For the case of odd $ d $ ($ \Delta $ is an integer), we obtain the expression: $$\label{eq:longvectorexactcorrelatorintcodd}
\boxed{
G_{tt}^R(\omega,\mathbf{s})
= 2C_v\frac{r_+^{d-3}}{R^{d-2}}q_\mathbf{s}^2\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta+1)}
(a)_\Delta(b)_\Delta\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}$$ The other components of the longitudinal mode correlator can be calculated from Equations \[eq:longvectorcftcorrelator2\] and \[eq:longvectorcftcorrelator3\].
For the transverse vector mode, we define: $$\Delta \equiv \frac{d-3}{2} \ .$$ The caluclation gives for even $ d $: $$\label{eq:transvectorexactcorrelatorintceven}
\boxed{
G_{\bot\bot}^R(\omega,\mathbf{v})
= 4(d-3)C_v \frac{r_+^{d-3}}{R^{d-2}}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}
\frac{\Gamma(a+\Delta)}{\Gamma(a)}\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ ,
}$$ and for odd $ d $: $$\label{eq:transvectorexactcorrelatorintcodd}
\boxed{
G_{\bot\bot}^R(\omega,\mathbf{v})
= 8C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}(a)_\Delta(b)_\Delta
\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}$$
Asymptotic Correlator Expressions (for K>-1 ) {#subsec:cftasymptoticcorrelatorexpressions}
------------------------------------------------
Continuing Subsection \[subsec:qnmasymoptotics\], the goal of this subsection is to find an approximate expression for the correlators of the CFT operators coupled to the scalar and vector fields in the bulk spacetime for large values of the frequency, where $ |\lambda| \gg q $ (corresponding to $ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $) and $ \operatorname{Im}\lambda < 0 $. The general stages of the calculation are as follows:
1. We start from the effective potentials and their poles calculated in Subsection \[subsec:qnmasymoptotics\].
2. We solve the approximate QNM equations around each pole using Bessel functions.
3. Using the asymptotic limit of $ |\lambda| \gg 1 $, we replace the Bessel functions with their asymptotic forms.
4. We match the solutions on each region of the complex plane, and apply the boundary condition at the horizon, to get an expression for the solution near the boundary up to a multiplicative constant.
5. We use the approximate solution near the boundary and the expressions for the retarded correlators from Subsection \[subsec:generalformulaeforcorrelators\] to obtain an asymptotic expression for the correlators.
### Calculation of Asymptotic Correlators for the Scalar Case {#subsubsec:cftasymptoticcorrelatorexpressionsscalar}
Proceeding from the definitions in Subsection \[subsec:qnmasymoptotics\], the expression in Equation \[eq:scalarcftcorrelator\] can be written in terms of $ \phi $ and $ {\tilde{z}}$ (after dropping the contact term and using the fact that $ {\tilde{z}}\approx z-1 $ for $ z\to 1 $): $$\label{eq:scalarcftcorrelatorinzt}
G^R(\omega,\mathbf{s}) = \left.-2C_s\frac{r_+^{d-1}}{R^d}\frac{(-1)^d}{{\tilde{z}}^{d-2}}
\frac{{\partial_{\tilde{z}}}\phi_{\omega,\mathbf{s}}({\tilde{z}})}{\phi_{\omega,\mathbf{s}}({\tilde{z}})}\right|_{{\tilde{z}}\to 0} \ .$$
We continue in a method similar to the one applied in Subsection \[subsec:qnmasymoptotics\]: find asymptotic solutions around the poles of the effective potential and “match” the solutions[^13]. As before, we assume that $ |\lambda|\gg 1 $ and $ |\lambda|\gg q $. We also assume that $ \lambda $ is in the 3rd quadrant of the complex plane ($ \operatorname{Re}\lambda<0 $, $ \operatorname{Im}\lambda<0 $). The solutions around $ {\tilde{z}}\to{\tilde{z}}_0 $ ($ z\to\infty $) are given by: $$\begin{aligned}
P_+({\tilde{z}}-{\tilde{z}}_0) &\equiv \sqrt{2\pi\lambda({\tilde{z}}-{\tilde{z}}_0)}J_0\left(\lambda({\tilde{z}}-{\tilde{z}}_0)\right)\\
P_-({\tilde{z}}-{\tilde{z}}_0) &\equiv \sqrt{2\pi\lambda({\tilde{z}}-{\tilde{z}}_0)}Y_0\left(\lambda({\tilde{z}}-{\tilde{z}}_0)\right) \ .\end{aligned}$$ Replacing the Bessel functions with their asymptotic forms gives for direction (1): $$\label{eq:monoasympinftyplus2}
\phi \approx
\left[A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\frac{\pi}{4})}+A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\frac{3\pi}{4})}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}}\\
+\left[A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\frac{\pi}{4})}-A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0-\frac{\pi}{4})}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ ,$$ and for direction (2): $$\label{eq:monoasympinftymin2}
\phi \approx
\left[A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+\frac{3\pi}{4})}-3A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+\frac{\pi}{4})}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}}\\
+\left[A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\frac{\pi}{4})}-A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0-\frac{\pi}{4})}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ .$$ From the boundary condition at the horizon ($ z=0 $, $ {\tilde{z}}\to -\infty $) we have: $$\phi \approx {\mathrm{e}}^{-i\lambda{\tilde{z}}} \ .$$
There are now two possible cases:
1. $ d $ is even. In this case the solution around $ {\tilde{z}}\to 0 $ is approximately: $$\phi \approx B_+ P_+({\tilde{z}}) + B_- P_-({\tilde{z}})\\
= B_+\sqrt{2\pi\lambda{\tilde{z}}} J_{\frac{j_1}{2}}(\lambda{\tilde{z}})
+ B_-\sqrt{2\pi\lambda{\tilde{z}}} J_{-\frac{j_1}{2}}(\lambda{\tilde{z}}) \ ,$$ where $ j_1 = d-1 $ (so that $ \frac{j_1}{2} $ is non-integer). Replacing the Bessel functions with their asymptotic form, we have: $$\phi
\approx \left[B_+{\mathrm{e}}^{-i\beta_+} + B_-{\mathrm{e}}^{-i\beta_-}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}} +
\left[B_+{\mathrm{e}}^{i\beta_+} + B_-{\mathrm{e}}^{i\beta_-}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ ,$$ where $ \beta_\pm \equiv \frac{\pi}{4}(1 \pm j_1)$. We now match the solutions on lines (1) and (2). We have for line (1): $$\label{eq:cftmonoeq3}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\frac{\pi}{4})}+A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\frac{3\pi}{4})} = 0
\qquad\Rightarrow\qquad
A_- = -iA_+ \ .$$ For the section (2) we have: $$\begin{aligned}
\label{eq:cftmonoeq4}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+\frac{3\pi}{4})}-3A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+\frac{\pi}{4})}
&= B_+{\mathrm{e}}^{-i\beta_+} + B_-{\mathrm{e}}^{-i\beta_-}\\
\label{eq:cftmonoeq5}
A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\frac{\pi}{4})}-A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0-\frac{\pi}{4})}
&= B_+{\mathrm{e}}^{i\beta_+} + B_-{\mathrm{e}}^{i\beta_-} \ .\end{aligned}$$ Putting Equation \[eq:cftmonoeq3\] into Equations \[eq:cftmonoeq4\] and \[eq:cftmonoeq5\] we get: $$\begin{aligned}
B_+{\mathrm{e}}^{-i\theta_+} + B_-{\mathrm{e}}^{-i\theta_-} &= 2iA_+ \\
B_+{\mathrm{e}}^{i\theta_+} + B_-{\mathrm{e}}^{i\theta_-} &= 4iA_+ \ ,\end{aligned}$$ where $ \theta_\pm \equiv \lambda{\tilde{z}}_0 - \frac{\pi}{4} - \beta_\pm $. Solving for $ B_\pm $ we can get an expression for $ \frac{B_+}{B_-} $: $$\frac{B_+}{B_-}
= \frac{\left|\begin{array}{cc}
2 & {\mathrm{e}}^{-i\theta_-} \\
4 & {\mathrm{e}}^{i\theta_-}
\end{array}\right|}
{\left|\begin{array}{cc}
{\mathrm{e}}^{-i\theta_+} & 2 \\
{\mathrm{e}}^{i\theta_+} & 4
\end{array}\right|}
= -i^{d-1}\frac{{\mathrm{e}}^{2i\theta_-}-2}{{\mathrm{e}}^{2i\theta_-}+2}$$ (here we used the fact that $ d $ is even).
Next we evaluate the correlator. Developing $ \phi $ around $ {\tilde{z}}=0 $ we have: $$\begin{gathered}
\label{eq:asymptoticcorrelatorsphiaround0even}
\phi
\approx B_+\sqrt{2\pi\lambda{\tilde{z}}} J_{\frac{j_1}{2}}(\lambda{\tilde{z}})
+ B_-\sqrt{2\pi\lambda{\tilde{z}}} J_{-\frac{j_1}{2}}(\lambda{\tilde{z}}) \\
\approx B_-\sqrt{2\pi\lambda{\tilde{z}}}(\frac{1}{2}\lambda{\tilde{z}})^{-\frac{j_1}{2}}
\sum_{k=0}^\infty \frac{(-\frac{1}{4}\lambda^2{\tilde{z}}^2)^k}{k!\Gamma(-\frac{j_1}{2}+k+1)} \\
+ B_+\sqrt{2\pi\lambda{\tilde{z}}}(\frac{1}{2}\lambda{\tilde{z}})^{\frac{j_1}{2}}
\sum_{k=0}^\infty \frac{(-\frac{1}{4}\lambda^2{\tilde{z}}^2)^k}{k!\Gamma(\frac{j_1}{2}+k+1)} \ .\end{gathered}$$ Defining the coefficients: $$\begin{aligned}
A(\omega,\mathbf{s}) &= \frac{\sqrt{2\pi}\lambda^{-\Delta+\frac{1}{2}}}
{2^{-\Delta}\Gamma(-\Delta+1)} B_- \\
B(\omega,\mathbf{s}) &= \frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}
{2^{\Delta}\Gamma(\Delta+1)} B_+\end{aligned}$$ (where $ \Delta\equiv \frac{j_1}{2} = \frac{d-1}{2} $), we get: $$\left.\frac{{\partial_{\tilde{z}}}\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{{\tilde{z}}=\epsilon}
= (2\Delta)\frac{B(\omega,\mathbf{s})}{A(\omega,\mathbf{s})}\epsilon^{2\Delta-1}
\approx (d-1)\left(\frac{i\lambda}{2}\right)^{d-1}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}
\frac{{\mathrm{e}}^{2i\theta_-}-2}{{\mathrm{e}}^{2i\theta_-}+2}\epsilon^{d-2} \ .$$ And finally putting this into Equation \[eq:scalarcftcorrelatorinzt\]: $$G^R(\omega,\mathbf{s}) \approx -2(d-1)C_s\frac{r_+^{d-1}}{R^d}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\lambda}{2}\right)^{d-1}
\frac{{\mathrm{e}}^{2i\theta_-}-2}{{\mathrm{e}}^{2i\theta_-}+2}$$ (Note that this expression is true up to the addition of contact terms). This asymptotic formula is only true for $ \operatorname{Re}\lambda<0 $. In order to get an expression for $ \operatorname{Re}\lambda>0 $, we can make use of the symmetry properties of the correlator[^14]: $$G^R(\omega,\mathbf{s}) = G^{R*}(-\omega^*,\mathbf{s}) \ .$$ We get for $ \operatorname{Re}\lambda>0 $: $$G^R(\omega,\mathbf{s}) \\
= -2(d-1)C_s\frac{r_+^{d-1}}{R^d}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\lambda}{2}\right)^{d-1}
\frac{{\mathrm{e}}^{2i\overline{\theta_-}}-2}{{\mathrm{e}}^{2i\overline{\theta_-}}+2} \ ,$$ where we define: $$\overline{\theta_-}(\lambda) \equiv -\theta_-^*(-\lambda*)
= \lambda{\tilde{z}}_0^*-\frac{\pi}{4}(d-3) \ .$$ Looking at the asymptotic behaviour of the expressions for $ \operatorname{Re}\lambda<0 $ and $ \operatorname{Re}\lambda>0 $, we may unify them into one asymptotic expression: $$G^R(\omega,\mathbf{s}) \approx 2(d-1)C_s\frac{r_+^{d-1}}{R^d}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\lambda}{2}\right)^{d-1}
\frac{{\mathrm{e}}^{2i\theta_-}-2}{{\mathrm{e}}^{2i\theta_-}+2}\,
\frac{{\mathrm{e}}^{2i\overline{\theta_-}}-2}{{\mathrm{e}}^{2i\overline{\theta_-}}+2} \ .$$ Finally, using $ \lambda=\frac{\omega R^2}{r_+} $, the asymptotic expression may be written: $$\label{eq:asymptcftcorrelatorscalarevend}
\boxed{
G^R(\omega,\mathbf{s}) \approx 2(d-1)C_s R^{d-2}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\omega}{2}\right)^{d-1}
\frac{{\mathrm{e}}^{2i\theta_-}-2}{{\mathrm{e}}^{2i\theta_-}+2}\,
\frac{{\mathrm{e}}^{2i\overline{\theta_-}}-2}{{\mathrm{e}}^{2i\overline{\theta_-}}+2} \ .
}$$
2. $ d $ is odd. In this case the solution around $ {\tilde{z}}\to 0 $ is approximately: $$\phi \approx B_+ P_+({\tilde{z}}) + B_- P_-({\tilde{z}})\\
= B_+\sqrt{2\pi\lambda{\tilde{z}}} J_{\frac{j_1}{2}}(\lambda{\tilde{z}})
+ B_-\sqrt{2\pi\lambda{\tilde{z}}} Y_{\frac{j_1}{2}}(\lambda{\tilde{z}}) \ ,$$ where $ j_1 = d-1 $ (so that $ \frac{j_1}{2} $ is an integer). Replacing the Bessel functions with their asymptotic form, we have: $$\phi
\approx \left[(B_+-iB_-){\mathrm{e}}^{-i\beta_+}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}} +
\left[(B_++iB_-){\mathrm{e}}^{i\beta_+}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ ,$$ where $ \beta_+ = \frac{\pi}{4}(1+j_1) = \frac{\pi}{4}d $. We again match the solutions on lines (1) and (2). For (1) we again have: $$\label{eq:cftmonoeq6}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\frac{\pi}{4})}+A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\frac{3\pi}{4})} = 0 \ .$$ For (2): $$\begin{aligned}
\label{eq:cftmonoeq7}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+\frac{3\pi}{4})}-3A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+\frac{\pi}{4})}
&= (B_+-iB_-){\mathrm{e}}^{-i\beta_+}\\
\label{eq:cftmonoeq8}
A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\frac{\pi}{4})}-A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0-\frac{\pi}{4})}
&= (B_++iB_-){\mathrm{e}}^{i\beta_+} \ .\end{aligned}$$ Putting Equation \[eq:cftmonoeq6\] into Equations \[eq:cftmonoeq7\] and \[eq:cftmonoeq8\] we get (defining $ \theta_+ \equiv \lambda{\tilde{z}}_0 - \frac{\pi}{4} - \beta_+ $): $$\begin{aligned}
(B_+-iB_-){\mathrm{e}}^{i\theta_+} &= 4iA_+ \\
(B_++iB_-){\mathrm{e}}^{-i\theta_+} &= 2iA_+ \ .\end{aligned}$$ Solving for $ B_\pm $ we can get an expression for $ \frac{B_+}{B_-} $: $$\frac{B_+}{B_-}
= \frac{\left|\begin{array}{cc}
4 & -i{\mathrm{e}}^{i\theta_+} \\
2 & i{\mathrm{e}}^{-i\theta_+}
\end{array}\right|}
{\left|\begin{array}{cc}
{\mathrm{e}}^{i\theta_+} & 4 \\
{\mathrm{e}}^{-i\theta_+} & 2
\end{array}\right|}
= 2i\frac{{\mathrm{e}}^{2i\theta_+}}{{\mathrm{e}}^{2i\theta_+}-2}-i \ .$$
Next we evaluate the correlator. Developing $ \phi $ around $ {\tilde{z}}=0 $ we have (we again define $ \Delta\equiv \frac{j_1}{2} = \frac{d-1}{2} $): $$\label{eq:asymptoticcorrelatorsphiaround0odd}
\phi
\approx B_+\sqrt{2\pi\lambda{\tilde{z}}} J_{\Delta}(\lambda{\tilde{z}})
+ B_-\sqrt{2\pi\lambda{\tilde{z}}} Y_{\Delta}(\lambda{\tilde{z}}) \\$$ Defining the coefficients: $$A(\omega,\mathbf{s}) = -\frac{\sqrt{2\pi}\lambda^{-\Delta+\frac{1}{2}}}{\pi 2^{-\Delta}}
(\Delta-1)!\,B_-$$ $$\begin{gathered}
B(\omega,\mathbf{s}) = \frac{2\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}
{\pi 2^\Delta\Gamma(\Delta+1)}\ln\left(-\frac{\lambda}{2}\right)\,B_- \\
-\frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}{\pi 2^\Delta(\Delta)!}
\left[\psi(1)+\psi(\Delta+1)\right]\,B_-
+\frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}{2^\Delta \Gamma(\Delta+1)}\,B_+ \ ,\end{gathered}$$ we again have: $$\left.\frac{{\partial_{\tilde{z}}}\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{{\tilde{z}}=\epsilon}
\approx (2\Delta)\frac{B(\omega,\mathbf{s})}{A(\omega,\mathbf{s})}\epsilon^{2\Delta-1} \ .$$ And finally after putting this into Equation \[eq:scalarcftcorrelatorinzt\], and again making use of the symmetry properties of the correlator we obtain:
[multline]{} \[eq:asymptcftcorrelatorscalaroddd\] G\^R(,) 4C\_s R\^[d-2]{} ()\^[d-1]{}\
,
where: $$\overline{\theta_+}(\lambda) \equiv -\theta_+^*(-\lambda^*) = \lambda{\tilde{z}}_0^*
+\frac{\pi}{4}(d+1) \ .$$
### Calculation of Asymptotic Correlators for the Vector Case {#subsubsec:cftasymptoticcorrelatorexpressionsvector}
The calculation of asymptotic expressions for the vector correlators is similar to the scalar case. A detailed derivation is given in Appendix \[app:cftasymptoticcorrelatorexpressionsvector\], and here we quote the final results.
For the longitudinal vector mode, we define: $$\Delta \equiv \frac{d-5}{2} \ .$$ We distinguish between two possible cases[^15]:
1. $ d $ is even. In this case we obtain the asymptotic expression:
[multline]{} \[eq:asymptcftcorrelatorlongvectorevend\] G\_[tt]{}\^R(,) R\^[d-4]{} (L\_\^[FT]{})\^2()\^[d-5]{}\
,
where: $$\begin{aligned}
\theta_- &\equiv \lambda{\tilde{z}}_0 + \frac{\pi}{4}(d-7) \\
\overline{\theta_-}(\lambda) &\equiv -\theta_-^*(-\lambda^*)
= \lambda{\tilde{z}}_0^*-\frac{\pi}{4}(d-7) \ .\end{aligned}$$
2. $ d $ is odd. In this case we obtain the asymptotic expression:
[multline]{} \[eq:asymptcftcorrelatorlongvectoroddd\] G\_[tt]{}\^R(,) 2C\_v R\^[d-4]{}(L\_\^[FT]{})\^2 ()\^[d-5]{}\
,
where: $$\begin{aligned}
\theta_+ &\equiv \lambda{\tilde{z}}_0 - \frac{\pi}{4}(d-3) \\
\overline{\theta_+}(\lambda) &\equiv -\theta_+^*(-\lambda*)
= \lambda{\tilde{z}}_0^*+\frac{\pi}{4}(d-3) \ .\end{aligned}$$
The other components of the longitudinal mode correlator can be calculated from Equations \[eq:longvectorcftcorrelator2\] and \[eq:longvectorcftcorrelator3\].
For the transverse vector mode, we define: $$\Delta \equiv \frac{d-3}{2} \ .$$ We again distinguish between two possible cases:
1. $ d $ is even. In this case we obtain the asymptotic expression:
[multline]{} \[eq:asymptcftcorrelatortransvectorevend\] G\_\^R(,) 4(d-3)C\_v R\^[d-4]{} ()\^[d-3]{}\
,
where: $$\begin{aligned}
\theta_- &\equiv \lambda{\tilde{z}}_0 + \frac{\pi}{4}(d-5) \\
\overline{\theta_-}(\lambda) &\equiv -\theta_-^*(-\lambda*)
= \lambda{\tilde{z}}_0^*-\frac{\pi}{4}(d-5) \ .\end{aligned}$$
2. $ d $ is odd. In this case we obtain the asymptotic expression:
[multline]{} \[eq:asymptcftcorrelatortransvectoroddd\] G\_\^R(,) 8C\_v R\^[d-4]{} ()\^[d-3]{}\
,
where: $$\begin{aligned}
\theta_+ &\equiv \lambda{\tilde{z}}_0 - \frac{\pi}{4}(d-1) \\
\overline{\theta_+}(\lambda) &\equiv -\theta_+^*(-\lambda*)
= \lambda{\tilde{z}}_0^*+\frac{\pi}{4}(d-1) \ .\end{aligned}$$
Notice that, taking into account the differences in the definition of $ \Delta $ and $ \theta_\pm $ between the longitudinal and the transverse perturbation modes: $$\begin{aligned}
\Delta^{\text{long}} &= \Delta^{\text{trans}} - 1 \\
\theta_\pm^{\text{long}} &= \theta_\pm^{\text{trans}} \pm \frac{\pi}{2} \ ,\end{aligned}$$ the asymptotic expressions for $ G_{\bot\bot}^R(\omega,\mathbf{v}) $ and $ G_{\|\|}^R(\omega,\mathbf{s}) $ are identical. This is expected and serves as a check for our calculations, since in this limit we have neglected the spatial mode $ L_{\mathbf{s}/\mathbf{v}}^{FT} $, so that the differences between the longitudinal and the transverse modes are negligible.
Numerical Calculation of Correlators
------------------------------------
In the following we present some results of exact numerical calculations of the gauge theory retarded correlation functions for several cases, and compare them to the asymptotic analytical expressions (as discussed in Subsection \[subsec:cftasymptoticcorrelatorexpressions\]). The correlation functions have been calculated using the method outlined in Appendix \[app:numericalmethods\].
The following calculations were made for a degree of $ N=200 $ and $ z_0 = 0.6 $ (see Appendix \[app:numericalmethods\] for details):
- The case of scalar perturbation in $ d=4 $ bulk dimensions, with $ K=1 $ (spherical boundary topology) and $ q_\mathbf{s} = 0 $ (s-wave). The results are given in Figures \[fig:correlationfunctionresultspolesscalard4K1q0\], \[fig:correlationfunctionresultscirclescalard4K1q0R20\], \[fig:correlationfunctionresultscontourscalard4K1q0\] and \[fig:correlationfunctionresultsspectralscalard4K1q0\].
\
- The case of longitudinal vector perturbations in $ d=5 $ bulk dimensions, with $ K=0 $ (flat boundary topology) and $ q_\mathbf{s} = 0.6 $. The results are given in Figures \[fig:correlationfunctionresultspoleslongvectord5K0q06\], \[fig:correlationfunctionresultscirclelongvectord5K0q06R20\], \[fig:correlationfunctionresultscontourlongvectord5K0q06\] and \[fig:correlationfunctionresultsspectrallongvectord5K0q06\].
\
- The case of transverse vector perturbations in $ d=5 $ bulk dimensions, with $ K=-0.5 $ (hyperbolic boundary topology) and $ q_\mathbf{v} = 1 $. The results are given in Figures \[fig:correlationfunctionresultspolestransvectord5Km05q1\], \[fig:correlationfunctionresultscircletransvectord5Km05q1R20\], \[fig:correlationfunctionresultscontourtransvectord5Km05q1\] and \[fig:correlationfunctionresultsspectraltransvectord5Km05q1\].
\
For the sake of better visual representation of the results, we define the “normalized” retarded correlators $ I(\lambda) $ by “normalizing out” the power and logarithmic behaviour of the correlators, keeping only the part of the correlators that behaves as $ O(1) $ for $ |\lambda| \to \infty $[^16]. These “normalized” correlators have poles at the QNM frequencies, and approach a constant on each side of the QNM frequencies line for frequencies far enough away from that line.
Figures \[fig:correlationfunctionresultspolesscalard4K1q0\], \[fig:correlationfunctionresultspoleslongvectord5K0q06\] and \[fig:correlationfunctionresultspolestransvectord5Km05q1\] show the value of $ |I(\lambda_0 + s\Delta\lambda)| $ as a function of $ s $, where $ \lambda_0 $ and $ \Delta\lambda $ are the analytical asymptotic parameters of the QNM spectrum as calculated in Subsection \[subsec:qnmasymoptotics\] and $ s $ is a positive, real parameter. As expected, we find the poles of the correlation function approximately at integer values of $ s $.
Figures \[fig:correlationfunctionresultscirclescalard4K1q0R20\], \[fig:correlationfunctionresultscirclelongvectord5K0q06R20\] and \[fig:correlationfunctionresultscircletransvectord5Km05q1R20\] show the value of $ \operatorname{Re}I\left(\lambda_R{\mathrm{e}}^{-i\theta}\right) $ as a function of $ \theta $, where $ \lambda_R $ is some constant radius in the frequency complex plane (in this case $ \lambda_R = 20 $). As expected, the numerically calculated values are close to the ones of the analytical asymptotic expressions calculated in Subsection \[subsec:cftasymptoticcorrelatorexpressions\].
Figures \[fig:correlationfunctionresultscontourscalard4K1q0\], \[fig:correlationfunctionresultscontourlongvectord5K0q06\] and \[fig:correlationfunctionresultscontourtransvectord5Km05q1\] show a contour plot of $ \operatorname{Re}I(\lambda) $ in the 4-th quadrant of the complex frequency plane. Both the poles at the QNM frequencies and the step-function-like behaviour of the “normalized” correlation function can be clearly seen in the plots.
Figures \[fig:correlationfunctionresultsspectralscalard4K1q0\], \[fig:correlationfunctionresultsspectrallongvectord5K0q06\] and \[fig:correlationfunctionresultsspectraltransvectord5Km05q1\] show the values of the appropriate spectral functions on the real frequency axis ($ \rho(\lambda) $ in the case of scalar perturbations, $ \rho_{tt}(\lambda) $ in the case of longitudinal vector perturbations and $ \rho_{\bot\bot}(\lambda) $ in the case of transverse vector perturbations). At large frequency values, these functions approach the analytical asymptotic expressions and are proportional to $ \lambda^s $ (where $ s $ is the appropriate exponent for the perturbation type), as expected from conformal symmetry considerations. In the longitudinal vector case (Figure \[fig:correlationfunctionresultsspectrallongvectord5K0q06\]), the spectral function oscillates at lower frequencies, due to the existence of the hydrodynamic diffusion pole.
Discussion {#sec:discussion}
==========
In this work we studied some of the properties of the retarded correlation function of composite operators of finite temperature gauge field theories with dual gravity descriptions in the large coupling limit. Of specific interest was the dependence of these properties on the topology of the space on which the gauge theory is defined, and its temperature. This goal was achieved by performing exact analytical calculations for specific cases where it was possible (the hyperbolic case with $ T=T_c $), by performing numerical calculations and by obtaining approximate expressions for the large frequency limit ($ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $).
In the large frequency limit we saw that in all cases (all topologies, temperatures and perturbation types) the correlation functions exhibit the same kind of asymptotic pole structure, with a constant “gap” between two consecutive poles. Moreover, we saw that asymptotically these poles are produced by Bose-Einstein-like factors in the correlation functions, with some complex parameter multiplying the frequency.
We have also seen that the spatial topology on which the gauge theory is defined is indeed noticeable in the properties of the correlation function and its poles (corresponding to quasinormal modes in the dual bulk spacetime), in the case where $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} = \text{fixed} $. In particular, the asymptotic “slope” of the correlator poles encodes information about the spatial topology and the temperature, but is “universal” otherwise - it doesn’t depend on the type of operators considered. In the spherical (positive curvature) and flat (zero curvature) cases, this slope always has a real part and an imaginary part. In the spherical case, it depends on the relation between the spatial curvature and the temperature, while in the flat case it’s constant (as expected, since in this case the temperature is the only scale in the theory, and the correlator poles are necessarily proportional to it). In the hyperbolic (negative curvature) case, the slope has a real part for temperatures above $ T_c $, which goes to $ 0 $ in a non-smooth way for $ T\to T_c $ (In $ d = 4,5 $ dimensions the real part is also $ 0 $ for temperatures below $ T_c $). At $ T=T_c $ the slope is completely imaginary, and exact analytical expressions can be found for the correlation functions and their poles. Interestingly, the poles “gap” in this case, and indeed the general form of the correlation functions, is exactly what one would expect to find for the retarded correlation function of composite operators in a free thermal field theory (the poles “gap” in this case corresponds to the Matsubara frequency “gap” - see [@Hartnoll:2005ju]).
Several open issues are left for further investigation:
- This work has dealt with uncharged bulk black hole solutions, but we can also expand the scope and discuss charged black holes, that correspond to gauge theories with non-zero chemical potentials (see [@Son:2006em]). The background of the perturbations in such solutions contains both a black hole metric and a non-zero gauge field. We can then ask - how does the charge change the asymptotic properties of the gauge theory correlation functions and their dependence on the spatial topology?
- As noted in Subsubsection \[subsubsec:cftasymptoticcorrelatorexpressionsvector\], the method we have employed for calculating the asymptotic expressions for the correlation functions doesn’t work as is for the vector perturbation modes in $ d=4 $ bulk dimensions. This case requires a different treatment using complex WKB analysis, although numerical calculations show that our conclusions regarding the asymptotic “slope” of the correlator poles remain true in this case as well.
- While the asymptotic structure of the correlators (and their poles) has a geometrical meaning in the gravity (bulk) side, it is unclear if it has an interpretation in the strongly coupled dual field theory. Weak coupling calculations don’t seem to exhibit such a structure (see [@Hartnoll:2005ju]). The asymptotic poles “gap” in the large coupling limit depends only on the spatial topology and the temperature, suggesting it may have an interpretation involving them, but it can’t be trivially deduced from the symmetries of the theory alone.
- Considering the properties of the correlation functions in the hyperbolic case for $ T\to T_c $, does this temperature have any importance in the dual gauge theory? In [@Shen:2007xk], it is suggested that in $ d=4 $, the behaviour of the QNM frequencies slope near this temperature is related to the known phase transition between the topological (hyperbolic) black hole and the MTZ solution (a hyperbolic black hole with scalar hair - see [@Martinez:2004nb], [@Myung:2008ze]). It is not clear, however, what is the interpretation of such a phase transition in the gauge theory side, and whether the argument extends to higher dimensions.
Acknowledgements {#acknowledgements .unnumbered}
================
The work is supported in part by the Israeli Science Foundation center of excellence.
List of Notations and Definitions {#app:notations}
=================================
Here we list for convenience some of the main notations used throughout this paper.
- $ d $ is the number of *bulk* dimensions.
- $ R $ is the AdS radius.
- $ R_\Omega \equiv R^2 R_\Omega^{FT} $ is the scalar curvature of the $ \Omega_{d-2} $ manifold.
- $ r_+ $ is the radius of the bulk black hole horizon.
- $ k \equiv k_{FT}R^2 \equiv \frac{R_\Omega}{(d-2)(d-3)} $.
- $ \rho \equiv \frac{r_+}{R} $.
- $ K \equiv \frac{k}{\rho^2} $.
- $ \omega $ is the frequency of a perturbation mode, $ L_\mathbf{s/v}^{FT} \equiv \frac{L_\mathbf{s/v}}{R} $ is the Laplace operator eigenvalue of the perturbation mode on the dual field theory side, for scalar or vector respectively.
- $
\lambda \equiv \frac{\omega r_+}{\rho^2} = \frac{\omega R}{\rho}
\qquad
q_\mathbf{s/v} \equiv \frac{L_\mathbf{s/v}}{\rho} = \frac{L_\mathbf{s/v}^{FT}R}{\rho}
$.
- $
T_C \equiv \frac{\sqrt{|k|}}{2\pi R} = \frac{1}{2\pi}\sqrt{\frac{|R_\Omega^{FT}|}{(d-2)(d-3)}}
$ (for the case of $ k<0 $).
- $
\tilde{g}(z) \equiv 1 + K(1-z)^2 - (1+K)(1-z)^{d-1}
$.
- $
{\tilde{z}}_0 = - \sum_{k=1}^{d-1} \gamma_k \ln(1-z_k)
$, where $
\tilde{g}(z_k)=0
$ and $
\gamma_k = \frac{1}{\tilde{g}'(z_k)}
$ for $ k = 1, \ldots ,d-1 $.
- $ C_{s/v} $ is the normalization constant for the bulk action of the scalar/vector field respectively.
- $
\Delta \equiv
\begin{cases}
\frac{d-1}{2} & \text{scalar perturbation,} \\
\frac{d-3}{2} & \text{transverse vector perturbation,} \\
\frac{d-5}{2} & \text{longitudinal vector perturbation.}
\end{cases}
$
- $
a,b(\lambda) \equiv
\begin{cases}
-\frac{1}{4}(d-3)-\frac{i\lambda}{2}\pm\frac{i}{2}\sqrt{q_\mathbf{s}^2 - \frac{1}{4}(d-3)^2} &
\text{scalar perturbation,} \\
-\frac{1}{4}(d-5)-\frac{i\lambda}{2}\pm\frac{i}{2}\sqrt{q_\mathbf{v}^2 - \frac{1}{4}(d-5)^2} &
\text{transverse vector perturbation,} \\
-\frac{1}{4}(d-7)-\frac{i\lambda}{2}\pm\frac{i}{2}\sqrt{q_\mathbf{s}^2 - \frac{1}{4}(d-3)^2} &
\text{longitudinal vector perturbation.}
\end{cases}
$\
$
c(\lambda) \equiv 1-i\lambda
$.
- $
\theta_-(\lambda) \equiv
\begin{cases}
\lambda{\tilde{z}}_0 + \frac{\pi}{4}(d-3) & \text{scalar perturbation,} \\
\lambda{\tilde{z}}_0 + \frac{\pi}{4}(d-5) & \text{transverse vector perturbation,} \\
\lambda{\tilde{z}}_0 + \frac{\pi}{4}(d-7) & \text{longitudinal vector perturbation.}
\end{cases}
$\
$
\overline{\theta_-}(\lambda) \equiv -\theta_-^*(-\lambda^*) \equiv
\begin{cases}
\lambda{\tilde{z}}_0^* - \frac{\pi}{4}(d-3) & \text{scalar perturbation,} \\
\lambda{\tilde{z}}_0^* - \frac{\pi}{4}(d-5) & \text{transverse vector perturbation,} \\
\lambda{\tilde{z}}_0^* - \frac{\pi}{4}(d-7) & \text{longitudinal vector perturbation.}
\end{cases}
$
- $
\theta_+(\lambda) \equiv
\begin{cases}
\lambda{\tilde{z}}_0 - \frac{\pi}{4}(d+1) & \text{scalar perturbation,} \\
\lambda{\tilde{z}}_0 - \frac{\pi}{4}(d-1) & \text{transverse vector perturbation,} \\
\lambda{\tilde{z}}_0 - \frac{\pi}{4}(d-3) & \text{longitudinal vector perturbation.}
\end{cases}
$\
$
\overline{\theta_+}(\lambda) \equiv -\theta_+^*(-\lambda^*) \equiv
\begin{cases}
\lambda{\tilde{z}}_0^* + \frac{\pi}{4}(d+1) & \text{scalar perturbation,} \\
\lambda{\tilde{z}}_0^* + \frac{\pi}{4}(d-1) & \text{transverse vector perturbation,} \\
\lambda{\tilde{z}}_0^* + \frac{\pi}{4}(d-3) & \text{longitudinal vector perturbation.}
\end{cases}
$
Thermodynamic Quantities {#app:thermodynamicquant}
========================
Given the black hole metric, one may calculate the usual thermodynamic quantities related to the black hole. In the context of AdS/CFT, these quantities are related to the respective quantities in the field theory side. Defining: $
\rho \equiv \frac{r_+}{R}
$ and $
K\equiv\frac{k}{\rho^2}
$, we have for the temperature: $$T = T_{FT} = \frac{f'(r_+)}{4\pi} = \frac{\rho}{4\pi R}\left[2+(K+1)(d-3)\right] \ .$$ For the total energy we have: $$E = E_{FT}
= \frac{(d-2)Vol(\Omega^{FT}_{d-2})}{16\pi GR}\rho^{d-1}(K+1)+E_0 \ ,$$ where $ Vol(\Omega_{d-2}^{FT}) $ is the total volume of the $ \Omega_{d-2}^{FT} $ manifold, $ G $ is the bulk gravitational constant and $ E_0 $ is independent of the black hole parameters. Finally the total entropy is given by: $$S = S_{FT} = \frac{Vol(\Omega_{d-2})}{4G}r_+^{d-2} = \frac{Vol(\Omega_{d-2}^{FT})}{4G}\rho^{d-2} \ .$$ For the sake of discussing the field theory correlators and the quasinormal modes, we shall also define the following dimensionless quantities, for an oscillation mode of frequency $ \omega $ and Laplace operator eigenvalue $ L_s $: $$\begin{aligned}
\lambda &\equiv \frac{\omega r_+}{\rho^2} = \frac{\omega R}{\rho}
= \frac{d-1}{2\pi} \frac{\omega}{T}
\frac{1}{1+\sqrt{1-\frac{d-1}{d-2}\frac{R_\Omega^{FT}}{(2\pi T)^2}}} \\
q_s &\equiv \frac{L_s}{\rho} = \frac{L_s^{FT}R}{\rho}
= \frac{d-1}{2\pi} \frac{L_s^{FT}}{T}
\frac{1}{1+\sqrt{1-\frac{d-1}{d-2}\frac{R_\Omega^{FT}}{(2\pi T)^2}}} \ .\end{aligned}$$ Table \[tab:bulkcftquantities\] contains a summary of the physical quantities defined in the bulk spacetime, the corresponding field theory quantities and their dimensions.
![$ TR $ as a function of $ \rho $ for the spherical ($ k=1 $), flat ($ k=0 $) and hyperbolic($ k=-1 $) cases in $ d=5 $ bulk dimensions.[]{data-label="fig:thermodynamicquantitiestrforrho"}](ThermodynamicQuantitiesTRforRho.pdf){width="12cm"}
QNM Equations Derivation {#app:qnmequationsderivation}
========================
QNM Scalar Equation {#appsubsec:qnmscalarequationderivation}
-------------------
The EOM for a minimally coupled, massless scalar field $ \Phi $ are given by $$\nabla_\mu \nabla^\mu \Phi = 0 \ ,$$ or $$\label{eq:scalareom}
\frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g}g^{\mu\nu} \partial_{\nu}\Phi \right) = 0$$ (where $ g = \det(g_{\mu\nu}) $ ).
To find a QNM, we look for solutions of the form $$\Phi(t,z,\mathbf{x}) = \psi(z) {\mathrm{e}}^{-i\omega t} H_{L^2}(\mathbf{x}) \ ,$$ where $ H_{L^2}(\mathbf{x}) $ is a (scalar) eigenfunction of the Laplace operator $ \Delta_\Omega $ defined on the manifold $ \Omega_{d-2} $ with corresponding eigenvalue of $ -L_s^2 $. Putting this in, and noticing that $$\label{eq:scalarlaplaciandef}
\Delta_\Omega H_{L^2}(\mathbf{x}) \equiv {\,\mathrm{\delta}}{\,\mathrm{d}}H_{L^2}(\mathbf{x})\\
\equiv -\sum_{i,j}\frac{1}{\sqrt{g_\Omega}} \partial_i\left( \sqrt{g_\Omega} g_\Omega^{ij} \partial_j H_{L^2}(\mathbf{x}) \right)
= L_s^2 H_{L^2}(\mathbf{x})$$ (where $ {\,\mathrm{d}}$ and $ {\,\mathrm{\delta}}$ are the exterior derivative and codifferential defined on $ \Omega_{d-2} $, respectively), we obtain Equation \[eq:scalareq\] (scalar QNM equation). The QNMs in asymptotically AdS spacetimes are defined as the solutions to the equations of motion obeying the incoming wave boundary condition at the horizon and the Dirichlet boundary condition at the AdS boundary (at least in the context of AdS/CFT - see Subsection \[subsec:qnmdefinition\] and also [@Son:2002sd], [@Berti:2009kk], [@Nunez:2003eq]). In terms of the defined variables and coordinates, these translate to $$\left. \psi \right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left. \psi \right|_{z=1} = 0 \ ,$$ as in Equation \[eq:scalarbc\].
QNM Vector Equations {#appsubsec:qnmvectorequationsderivation}
--------------------
The EOM for a vector (gauge) field $ \mathbf{A} $ are given by $$\nabla_\mu F^{\mu\nu} = 0 \ ,$$ or $$\label{eq:vectoreom}
\frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g}g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta} \right) = 0 \ .$$ Putting the metric from Equation \[eq:zmetric\] Equation \[eq:vectoreom\] we get for the $ \beta $ component of the equation: $$\begin{gathered}
\label{eq:vectoreom2}
(1-z)^{d-2}\partial_z\left( \frac{{\tilde{g}(z)}}{(1-z)^{d-2}}g^{\beta\gamma}F_{z\gamma} \right)
-\frac{r_+^2}{\rho^4}\frac{1}{{\tilde{g}(z)}}g^{\beta\gamma}\partial_t F_{t\gamma}\\
+\frac{1}{\rho^2}\sum_{i,j} \frac{1}{\sqrt{g_\Omega}}\partial_i \left( \sqrt{g_\Omega}g_\Omega^{ij}g^{\beta\gamma}F_{j\gamma} \right) = 0 \ .\end{gathered}$$
To find a QNM, we look for solutions proportional to eigenfunctions of the Laplace operator. The $ t $ and $ z $ components would be proportional to the eigenfunctions of the scalar Laplace operator: $
A_t, A_z \propto H_{L^2}(\mathbf{x})
$, where $ H_{L^2} $ is defined as in Equation \[eq:scalarlaplaciandef\]. The other components would be proportional to the corresponding components of the eigenfunction of the vector Laplace operator: $
A_i \propto A_{L^2,i}(\mathbf{x})
$, where $ A_{L^2} $ is defined as follows: $$\label{eq:vectoreigenfunctionsdefex}
\Delta_\Omega \mathbf{A}_{L^2}(\mathbf{x})
\equiv {\,\mathrm{\delta}}{\,\mathrm{d}}\mathbf{A}_{L^2}(\mathbf{x})\\
= \sum_j\frac{1}{\sqrt{g_\Omega}}\partial_j\left( \sqrt{g_\Omega}F_{L^2}^{ij} \right)
= L_v^2 A_{L^2}^i (\mathbf{x}) \ .$$
There are now two possibilities that correspond to two different modes:
1. The “longitudinal” mode. This is the case where $
L_v = 0 \ ,
$ or $
{\,\mathrm{\delta}}{\,\mathrm{d}}\mathbf{A}_{0} = 0
$, from which we can deduce $$\left< {\,\mathrm{d}}\mathbf{A}_{0}, {\,\mathrm{d}}\mathbf{A}_{0} \right>
= \left< \mathbf{A}_{0} , {\,\mathrm{\delta}}{\,\mathrm{d}}\mathbf{A}_{0} \right> = 0
\qquad \Rightarrow \qquad
\mathbf{F}_{0} = {\,\mathrm{d}}\mathbf{A}_{0} = 0 \ .$$ $ \mathbf{A}_0 $ is, therefore, a closed form and can be written as $
\mathbf{A}_{0,i} = \partial_i\phi
$, where $ \phi $ is some (scalar) function[^17]. To get the equation for a QNM we choose $ \phi $ to be an eigenfunction of the scalar Laplace operator $ \phi=H_{L^2} $. The solution will then be of the form $$\begin{pmatrix}
A_t(t,z,\mathbf{x})\\
A_z(t,z,\mathbf{x})\\
A_1(t,z,\mathbf{x})\\
\vdots\\
A_{d-2}(t,z,\mathbf{x})
\end{pmatrix}
=
\begin{pmatrix}
f(t,z)H_{L^2}(\mathbf{x})\\
h(t,z)H_{L^2}(\mathbf{x})\\
k(t,z)\partial_1 H_{L^2}(\mathbf{x})\\
\vdots\\
k(t,z)\partial_{d-2} H_{L^2}(\mathbf{x})
\end{pmatrix} \ .$$ Putting this into Equation \[eq:vectoreom2\] we get the following equations:\
For the $ t $ component- $$\label{eq:longvectortcomp}
I \equiv {\tilde{g}(z)}(1-z)^{d-4} {\partial_z}\left[ \frac{1}{(1-z)^{d-4}}\left( \partial_t h - \partial_z f \right) \right]
- q_s^2\left( \partial_t k - f \right) = 0 \ .$$ For the $ z $ component- $$\label{eq:longvectorzcomp}
II \equiv -\frac{r_+^2}{\rho^4}\frac{1}{{\tilde{g}(z)}}\partial_t\left( \partial_t h - \partial_z f \right)
-q_s^2 \left( h-\partial_z k \right) = 0 \ .$$
From Equations \[eq:longvectortcomp\] and \[eq:longvectorzcomp\] we can take $
\partial_z(I) + \partial_t(II) = 0
$, and defining\
$
\left( \partial_t h - \partial_z f \right) \equiv \psi(z) {\mathrm{e}}^{-i\omega t}
$, we finally obtain Equation \[eq:longvectoreq\] (longitudinal vector QNM equation). As for the boundary conditions, the incoming wave boundary condition at the horizon implies that $$\left. \psi \right|_{z=0} \sim z^{-\frac{i\lambda}{C}} \ ,$$ while the Dirichlet boundary condition at the boundary implies $$\left. f,h,k \right|_{z=1} = 0 \ .$$ Setting $ z=1 $ in Equation \[eq:longvectortcomp\] and applying the boundary condition we get for $ \psi $ the condition: $$\left. (1-z)^{d-4}\partial_z\left[ \frac{1}{(1-z)^{d-4}} \psi \right] \right|_{z=1} = 0 \ .$$
2. The “transverse” mode. In this case $
L_v \ne 0
$. Applying the codifferential $ \delta $ to Equation \[eq:vectoreigenfunctionsdefex\] we get $$0={\,\mathrm{\delta}}{\,\mathrm{\delta}}{\,\mathrm{d}}\mathbf{A}_{L^2} = L_v^2 {\,\mathrm{\delta}}\mathbf{A}_{L^2} \ ,$$ and since $ L_v \ne 0 $ we see that $$\label{eq:transvectorzerodiv}
{\,\mathrm{\delta}}\mathbf{A}_{L^2} = -\sum_i \frac{1}{\sqrt{g_\Omega}}\partial_i\left( \sqrt{g_\Omega} A_{L^2}^i \right) = 0 \ .$$ The solution in this case will be of the form $$\begin{pmatrix}
A_t(t,z,\mathbf{x})\\
A_z(t,z,\mathbf{x})\\
A_1(t,z,\mathbf{x})\\
\vdots\\
A_{d-2}(t,z,\mathbf{x})
\end{pmatrix}
=
\begin{pmatrix}
0\\
0\\
a(t,z)A_{L^2,1}(\mathbf{x})\\
\vdots\\
a(t,z)A_{L^2,d-2}(\mathbf{x})
\end{pmatrix} \ .$$ Putting this into Equation \[eq:vectoreom2\], the equations for the $ t $ and $ z $ components are automatically satisfied accroding to Equation \[eq:transvectorzerodiv\].\
Taking into account Equation \[eq:vectoreigenfunctionsdefex\], we get for the i component $$(1-z)^{d-4} {\partial_z}\left( \frac{{\tilde{g}(z)}}{(1-z)^{d-4}}{\partial_z}a \right)
-\frac{r_+^2}{\rho^4}\frac{1}{{\tilde{g}(z)}}\partial_t^2 a
-\frac{L_v^2}{\rho^2}a = 0 \ .$$ Setting $ a(t,z) = {\mathrm{e}}^{-i\omega t} \psi(z) $ we finally obtain Equation \[eq:transvectoreq\] (transverse vector QNM equation). The boundary conditions are again given by the incoming wave boundary condition at the horizon and the Dirichlet boundary condition at the boundary, so that: $$\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1} = 0 \ .$$
Derivation of General Formulae For Correlators {#app:corrformulaederivation}
==============================================
Scalar Correlators
------------------
Consider a massless, minimally coupled scalar field in the bulk spacetime (where the bulk metric is given by Equation \[eq:zmetric\]). The action of the scalar field is given by: $$S[\phi] = C_s\int_0^1{\,\mathrm{d}}z\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int_{\Omega_{d-2}}{\,\mathrm{d}}x^1\ldots{\,\mathrm{d}}x^{d-2}
\sqrt{-g}\nabla_\mu\phi\nabla^\mu\phi \ ,$$ where $ C_s $ is the appropriate normalization constant for the bulk scalar field. Since $ \phi $ is on-shell, it satisfies the EOM: $$\nabla_\mu\nabla^\mu\phi = 0 \ ,$$ so that, using integration by parts: $$\begin{gathered}
S = C_s\int_0^1{\,\mathrm{d}}z\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int_{\Omega_{d-2}}{\,\mathrm{d}}x^1\ldots{\,\mathrm{d}}x^{d-2}
\sqrt{-g}\left(\nabla_\mu\left(\phi\nabla^\mu\phi\right)-\phi\nabla_\mu\nabla^\mu\phi\right) \\
= C_s\left.\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int_{\Omega_{d-2}}{\,\mathrm{d}}x^1\ldots{\,\mathrm{d}}x^{d-2}
\sqrt{-g}g^{zz}\phi{\partial_z}\phi\right|^{z=1}_{z=0} \ .\end{gathered}$$ Using the metric from Equation \[eq:zmetric\] gives: $$\sqrt{-g} = \frac{r_+^{d-1}}{(1-z)^d}\sqrt{g_\Omega} \ ,$$ so that $$S[\phi] = C_s\frac{r_+^{d-1}}{(1-z)^d}g^{zz}\left.\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int{\,\mathrm{d}}\Omega_{d-2}
\phi{\partial_z}\phi\right|_{z=0}^{z=1}$$ (where $ {\,\mathrm{d}}\Omega_{d-2} \equiv \sqrt{g_\Omega}{\,\mathrm{d}}x^1\ldots{\,\mathrm{d}}x^{d-2} $).
Let $ {H_\mathbf{s}(x)} $ be a complete basis of eigenfunctions of the Laplace operator defined on the $ \Omega_{d-2} $ manifold, where $ \mathbf{s} $ here stands for all of the indices (discrete or continuous) required to uniquely identify the eigenfunctions. The normalization of these functions will be chosen so that: $$\int{\,\mathrm{d}}\Omega_{d-2} H_{\mathbf{s'}}^*(x)H_{\mathbf{s}}(x) = \delta_{\mathbf{ss'}}\ ,$$ where $ \delta_{\mathbf{ss'}} $ here stands for the Kronecker delta in case of a discrete spectrum and a Dirac delta in case of a continuous spectrum. One may then expand the field $ \phi $ in terms of these functions and the Fourier components so that: $$\label{eq:scalarexpansion}
\phi(z,t,x) = \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s} {\mathrm{e}}^{-i\omega t}H_{\mathbf{s}}(x)
\psi_{\omega,\mathbf{s}}(z) \ ,$$ where $ \sum_\mathbf{s} $ stands here for either a sum in case of a discrete spectrum or an integral (with the appropriate measure) in case of a continuous spectrum. Since $ \phi $ is a solution for the EOM, as stated in Subsection \[subsec:cftcorrelatorsfromhol\], $ \psi_{\omega,\mathbf{s}} $ would be a solution of the scalar QNM Equation (Equation \[eq:scalareq\]), with the incoming-wave boundary condition at the horizon: $$\left.\psi_{\omega,\mathbf{s}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}} \ .$$ As for the boundary, notice that the metric of CFT spacetime is actually given by Equation \[eq:cftmetric\], so that in the CFT spacetime the normalized eigenfunctions $ H_{\mathbf{s}}^{FT}(x) $ are given by $$\begin{gathered}
\int{\,\mathrm{d}}\Omega_{d-2}^{FT} H_{\mathbf{s'}}^{FT*}(x)H_{\mathbf{s}}^{FT}(x)
= R^{d-2}\int{\,\mathrm{d}}\Omega_{d-2} H_{\mathbf{s'}}^{FT*}(x)H_{\mathbf{s}}^{FT}(x)
=\delta_{\mathbf{ss'}} \\
\Rightarrow \qquad H_{\mathbf{s}}^{FT}(x) = R^{-\frac{d-2}{2}}H_{\mathbf{s}}(x) \ .\end{gathered}$$ As a consequence, on the boundary the field $ \phi $ may be written as: $$\phi(1,t,x) = \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s} {\mathrm{e}}^{-i\omega t}
R^{-\frac{d-2}{2}}H_{\mathbf{s}}(x)\phi_0(\omega,\mathbf{s})$$ (where $ \phi_0(\omega,\mathbf{s}) $ are the components of $ \phi $ in the CFT spacetime), which means the boundary condition on $ \psi_{\omega,\mathbf{s}} $ at the AdS boundary is: $$\left.\psi_{\omega,\mathbf{s}}\right|_{z\to 1} = R^{-\frac{d-2}{2}}\phi_0(\omega,\mathbf{s}) \ .$$ Defining a normalized $ \widehat{\psi}_{\omega,\mathbf{s}} $ by $ \psi_{\omega,\mathbf{s}} \equiv R^{-\frac{d-2}{2}}\phi_0(\omega,
\mathbf{s})\widehat{\psi}_{\omega,\mathbf{s}} $, the normalized function satisfies the boundary conditions: $$\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z\to 1} = 1 \ .$$
Putting Equation \[eq:scalarexpansion\] into the expression for the action, we have: $$\begin{gathered}
S[\phi] = C_s\frac{r_+^{d-1}}{(1-z)^d}g^{zz}\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int{\,\mathrm{d}}\Omega_{d-2}
\int\frac{{\,\mathrm{d}}\omega}{2\pi}\int\frac{{\,\mathrm{d}}\omega'}{2\pi}\sum_{\mathbf{s,s'}}\\
\left.{\mathrm{e}}^{i(\omega'-\omega)t}H_{\mathbf{s'}}^*(x)H_{\mathbf{s}}(x)\psi_{\omega',\mathbf{s'}}^*(z)
{\partial_z}\psi_{\omega,\mathbf{s}}(z)\right|_{z=0}^{z=1} \ .\end{gathered}$$ Using $$\int{\,\mathrm{d}}t \,{\mathrm{e}}^{i(\omega'-\omega)t} = 2\pi\delta(\omega'-\omega)$$ and the orthogonality of $ H_{\mathbf{s}}(x) $, the action becomes: $$\begin{gathered}
S[\phi] = \left.C_s\frac{r_+^{d-1}}{(1-z)^d}g^{zz}\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}
\psi_{\omega,\mathbf{s}}^*(z){\partial_z}\psi_{\omega,\mathbf{s}}(z)\right|_{z=0}^{z=1} \\
= C_s\frac{r_+^{d-1}}{(1-z)^d}\frac{{\tilde{g}(z)}(1-z)^2}{R^2}\frac{1}{R^{d-2}}
\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}\\
\left.\phi_0^*(\omega,\mathbf{s})
\widehat{\psi}_{\omega,\mathbf{s}}^*(z){\partial_z}\widehat{\psi}_{\omega,\mathbf{s}}(z)
\phi_0(\omega,\mathbf{s})\right|_{z=0}^{z=1} \ ,\end{gathered}$$ so that $$S[\phi] = \left.\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}\phi_0^*(\omega,\mathbf{s})
\mathcal{F}(\omega,\mathbf{s},z)\phi_0(\omega,\mathbf{s})\right|_{z=0}^{z=1} \ ,$$ where $$\mathcal{F}(\omega,\mathbf{s},z) = C_s\frac{r_+^{d-1}}{R^d}\frac{{\tilde{g}(z)}}{(1-z)^{d-2}}
\widehat{\psi}_{\omega,\mathbf{s}}^*(z){\partial_z}\widehat{\psi}_{\omega,\mathbf{s}}(z) \ .$$ Using the prescription described in [@Son:2002sd], the retarded correlator is then given by: $$G^R(\omega,\mathbf{s}) = \left.-2\mathcal{F}(\omega,\mathbf{s},z)\right|_{z\to 1}
= \left.-2C_s\frac{r_+^{d-1}}{R^d}\frac{1}{(1-z)^{d-2}}
{\partial_z}\widehat{\psi}_{\omega,\mathbf{s}}(z)\right|_{z\to 1} \ .$$
Vector Correlators
------------------
Next consider a vector gauge field in the bulk spacetime. The action of the gauge field is given by: $$S[A] = C_v\int_0^1{\,\mathrm{d}}z\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int_{\Omega_{d-2}}
{\,\mathrm{d}}x^1\ldots{\,\mathrm{d}}x^{d-2}\sqrt{-g}F_{\mu\nu}F^{\mu\nu} \ ,$$ where $ C_v $ is the appropriate normalization constant for the bulk vector field. Since $ A_\mu $ is on-shell, it satisfies the EOM: $$\nabla_\mu F^{\mu\nu} = 0 \ ,$$ so that $$F_{\mu\nu} F^{\mu\nu} = 2\nabla_\mu\left(A_\nu F^{\mu\nu}\right) \ ,$$ and using integration by parts: $$S = \left.2C_v\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int_{\Omega_{d-2}}{\,\mathrm{d}}x^1\ldots{\,\mathrm{d}}x^{d-2}
\sqrt{-g}A_\nu F^{z\nu} \right|_{z=0}^{z=1} \ .$$ Using the metric from Equation \[eq:zmetric\] gives: $$S = 2C_v\frac{r_+^{d-1}}{(1-z)^d}g^{zz}\left.\int_{-\infty}^{\infty}{\,\mathrm{d}}t\int{\,\mathrm{d}}\Omega_{d-2}
\left[g^{tt}A_t F_{zt} + g^{ij}A_i F_{zj}\right]\right|_{z=0}^{z=1}$$ (where $ {\,\mathrm{d}}\Omega_{d-2} \equiv \sqrt{g_\Omega}{\,\mathrm{d}}x^1\ldots{\,\mathrm{d}}x^{d-2} $).
Let $ H_\mathbf{s}(x) $ again be a complete basis of eigenfunctions of the Laplace operator on the $ \Omega_{d-2} $ manifold, normalized so that: $$\label{eq:orthogonality1}
\int{\,\mathrm{d}}\Omega_{d-2} H_{\mathbf{s'}}^*(x) H_\mathbf{s}(x) = \delta_{\mathbf{ss'}} \ .$$ Let us further define $ \mathbf{\tilde{A}_v}(x) $ as a set of vector eigenfunctions of the Laplace operator, such that: $${\,\mathrm{\delta}}{\,\mathrm{d}}\mathbf{\tilde{A}_v} = L_\mathbf{v}^2 \mathbf{\tilde{A_v}} \qquad L_\mathbf{v} \ne 0 \ ,$$ and $ {\,\mathrm{\delta}}\mathbf{\tilde{A}_v} = 0 $. Here $ \mathbf{v} $ stands for all of the indices required to uniquely identify the eigenfunctions. The normalization of these functions will be chosen so that: $$\label{eq:orthogonality2}
\int{\,\mathrm{d}}\Omega_{d-2}\, g_\Omega^{ij} \tilde{A}_{\mathbf{v},i}^*(x)\tilde{A}_{\mathbf{v'},j}(x) =
\delta_{\mathbf{vv'}} \ .$$ Note that $ \mathbf{\tilde{A}}_v $ along with $ \frac{\partial_i H_{\mathbf{s}}(x)}{L_s} $ form a complete orthonormal basis. One may expand the field $ \mathbf{A} $ in terms of these functions and the Fourier components: $$\begin{aligned}
A_t(z,t,x) &= \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}{\mathrm{e}}^{-i\omega t}H_{\mathbf{s}}(x)
f_{\omega,\mathbf{s}}(z) \\
A_z(z,t,x) &= \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}{\mathrm{e}}^{-i\omega t}H_{\mathbf{s}}(x)
h_{\omega,\mathbf{s}}(z) \\
A_i(z,t,x) &= \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}{\mathrm{e}}^{-i\omega t}\partial_i H_{\mathbf{s}}(x)
k_{\omega,\mathbf{s}}(z) +
\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{v}}{\mathrm{e}}^{-i\omega t}\tilde{A}_{\mathbf{v},i}(x)
a_{\omega,\mathbf{v}}(z) \ .\end{aligned}$$ Since $ \mathbf{A} $ is a solution for the EOM, as stated in Subsection \[subsec:cftcorrelatorsfromhol\], $ f_{\omega,\mathbf{s}} $, $ h_{\omega,\mathbf{s}} $, $ k_{\omega,\mathbf{s}} $ and $ a_{\omega,\mathbf{v}} $ would be solutions of the vector QNM Equations (see Appendix \[app:qnmequationsderivation\]) with incoming-wave boundary conditions at the horizon. As for the boundary, since the field theory metric is given by Equation \[eq:cftmetric\], in the field theory side the normalized eigenfunctions are given by: $$\begin{aligned}
H_\mathbf{s}^{FT}(x) &= R^{-\frac{d-2}{2}}H_\mathbf{s}(x) \\
\mathbf{\tilde{A}_v}^{FT}(x) &= R^{-\frac{d-4}{2}}\mathbf{\tilde{A}_v} \ .\end{aligned}$$ As a consequence, on the boundary the field $ \mathbf{A} $ may be written as: $$\begin{aligned}
A_t(1,t,x) &=& \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}{\mathrm{e}}^{-i\omega t}R^{-\frac{d-2}{2}}
H_\mathbf{s}(x)A_t^0(\omega,\mathbf{s}) \\
A_i(1,t,x) &=& \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}{\mathrm{e}}^{-i\omega t}R^{-\frac{d-4}{2}}
\frac{\partial_i H_\mathbf{s}(x)}{L_\mathbf{s}} A_\|^0(\omega,\mathbf{s}) \nonumber\\
&& + \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{v}}{\mathrm{e}}^{-i\omega t}R^{-\frac{d-4}{2}}
\tilde{A}_{\mathbf{v},i}(x) A_\bot^0(\omega,\mathbf{v})\end{aligned}$$ (where $ A_t^0(\omega,\mathbf{s}) $, $ A_\|^0(\omega,\mathbf{s}) $ and $ A_\bot^0(\omega,\mathbf{v}) $ are the components of $ \mathbf{A} $ in the field theory spacetime). The boundary conditions at the AdS boundary are therefore: $$\begin{aligned}
\label{eq:fboundarycondition}
\left.f_{\omega,\mathbf{s}}\right|_{z\to 1} &= R^{-\frac{d-2}{2}}A_t^0(\omega,\mathbf{s}) \\
\label{eq:kboundarycondition}
\left.k_{\omega,\mathbf{s}}\right|_{z\to 1} &= R^{-\frac{d-4}{2}}L_\mathbf{s}^{-1}
A_\|^0(\omega,\mathbf{s}) \\
\label{eq:aboundarycondition}
\left.a_{\omega,\mathbf{v}}\right|_{z\to 1} &= R^{-\frac{d-4}{2}}A_\bot^0(\omega,\mathbf{v}) \ .\end{aligned}$$
Next, one may derive expansions for the $ F_{\mu\nu} $ tensor: $$F_{zt} = -\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}{\mathrm{e}}^{-i\omega t}H_{\mathbf{s}}(x)
\left[-i\omega h_{\omega,\mathbf{s}}-{\partial_z}f_{\omega,\mathbf{s}}\right]$$ $$F_{zj} = \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{s}}{\mathrm{e}}^{-i\omega t}\partial_j H_\mathbf{s}(x)
\left[{\partial_z}k_{\omega,\mathbf{s}}-h_{\omega,\mathbf{s}}\right] \\
+ \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_{\mathbf{v}}{\mathrm{e}}^{-i\omega t}\tilde{A}_{\mathbf{v},j}(x)
{\partial_z}a_{\omega,\mathbf{v}} \ .$$ We proceed by inserting these expressions into the expression for the action. Using the orthogonality properties of Equations \[eq:orthogonality1\] and \[eq:orthogonality2\], and also the following derived properties: $$\int{\,\mathrm{d}}\Omega_{d-2}\, g_\Omega^{ij}\partial_i H_\mathbf{s'}^*(x) \partial_j H_\mathbf{s}(x)
= \left\langle{\,\mathrm{d}}H_\mathbf{s'},{\,\mathrm{d}}H_\mathbf{s}\right\rangle = \left\langle H_\mathbf{s'},{\,\mathrm{\delta}}{\,\mathrm{d}}H_\mathbf{s}\right\rangle = L_\mathbf{s}^2 \delta_\mathbf{ss'}$$ $$\int{\,\mathrm{d}}\Omega_{d-2}\, g_\Omega^{ij}\partial_i H_\mathbf{s'}^*(x) \tilde{A}_{\mathbf{v},j}(x)
= \left\langle {\,\mathrm{d}}H_\mathbf{s'} , \mathbf{\tilde{A}_v} \right\rangle
= \left \langle H_\mathbf{s'} , {\,\mathrm{\delta}}\mathbf{\tilde{A}_v} \right\rangle = 0 \ .$$ We see that the action decomposes into a longitudinal part $ S_\| $, that involves $ f_{\omega,\mathbf{s}} $, $ h_{\omega,\mathbf{s}} $ and $ k_{\omega,\mathbf{s}} $, and a transverse part $ S_\bot $ that involves only $ a_{\omega,\mathbf{v}} $, so that: $$S = S_\| + S_\bot \ .$$
For the longitudinal part of the action, we get the following expression: $$\begin{gathered}
S_\| = 2C_v\frac{r_+^{d-1}}{(1-z)^d}g^{zz} \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s}
\left[\frac{(1-z)^2}{\rho^2{\tilde{g}(z)}} f_{\omega,\mathbf{s}}^*
\left(-i\omega h_{\omega,\mathbf{s}}-{\partial_z}f_{\omega,\mathbf{s}}\right) \right.\\
\left.\left.+\frac{(1-z)^2}{R^2}q_\mathbf{s}^2 k_{\omega,\mathbf{s}}^*
\left({\partial_z}k_{\omega,\mathbf{s}}-h_{\omega,\mathbf{s}}\right)\right]\right|_{z=0}^{z=1}\end{gathered}$$ (where the definition $ q_\mathbf{s}\equiv \frac{L_\mathbf{s}}{\rho} $ was used). Since $ f_{\omega,\mathbf{s}} $, $ h_{\omega,\mathbf{s}} $ and $ k_{\omega,\mathbf{s}} $ satisfy the longitudinal vector QNM Equations, we may use Equation \[eq:longvectorzcomp\] to get the relation: $$q_\mathbf{s}^2\left({\partial_z}k_{\omega,\mathbf{s}} - h_{\omega,\mathbf{s}}\right)
= -\frac{r_+^2}{\rho^4{\tilde{g}(z)}}i\omega\left(-i\omega h_{\omega,\mathbf{s}}
-{\partial_z}f_{\omega,\mathbf{s}}\right) \ .$$ Using this relation, along with the definitions: $$\begin{aligned}
\psi_{\omega,\mathbf{s}} &\equiv -i\omega h_{\omega,\mathbf{s}} -{\partial_z}f_{\omega,\mathbf{s}} \\
\chi_{\omega,\mathbf{s}} &\equiv -i\omega k_{\omega,\mathbf{s}} -f_{\omega,\mathbf{s}} \ ,\end{aligned}$$ we obtain the expression: $$S_\| =-2C_v \frac{r_+^{d-3}}{(1-z)^{d-4}} \left.\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s}
\chi_{\omega,\mathbf{s}}^*(z)\psi_{\omega,\mathbf{s}}(z)\right|_{z=0}^{z=1} \ .$$ Looking at Equations \[eq:longvectortcomp\] and \[eq:longvectoreq\], we see that $ \chi_{\omega,\mathbf{s}} $ and $ \psi_{\omega,\mathbf{s}} $ satisfy the following system of differential equations: $$\begin{aligned}
\label{eq:longvectorchipsiequation1}
\chi_{\omega,\mathbf{s}} &= \frac{1}{q_\mathbf{s}^2}{\tilde{g}(z)}(1-z)^{d-4}{\partial_z}\left[\frac{1}{(1-z)^{d-4}}\psi_{\omega,\mathbf{s}}\right] \\
\label{eq:longvectorchipsiequation2}
\psi_{\omega,\mathbf{s}} &= -\frac{q_\mathbf{s}^2{\tilde{g}(z)}}{\lambda^2-q_\mathbf{s}^2{\tilde{g}(z)}}{\partial_z}\chi_{\omega,\mathbf{s}} \ .\end{aligned}$$ From Equations \[eq:kboundarycondition\] and \[eq:fboundarycondition\] we can derive the boundary condition for $ \chi_{\omega,\mathbf{s}} $: $$\left.\chi_{\omega,\mathbf{s}}\right|_{z\to 1} = -i\omega R^{-\frac{d-4}{2}}L_\mathbf{s}^{-1}
A_\|^0(\omega,\mathbf{s}) - R^{-\frac{d-2}{2}}A_t^0(\omega,\mathbf{s}) \ .$$ Next, define the normalized $ \widehat{\chi}_{\omega,\mathbf{s}} $ and $ \widehat{\psi}_{\omega,\mathbf{s}} $ by: $$\begin{aligned}
\chi_{\omega,\mathbf{s}} &\equiv \left[-i\omega R^{-\frac{d-4}{2}}L_\mathbf{s}^{-1}
A_\|^0(\omega,\mathbf{s}) - R^{-\frac{d-2}{2}}A_t^0(\omega,\mathbf{s})\right]
\widehat{\chi}_{\omega,\mathbf{s}}\\
\psi_{\omega,\mathbf{s}} &\equiv \left[-i\omega R^{-\frac{d-4}{2}}L_\mathbf{s}^{-1}
A_\|^0(\omega,\mathbf{s}) - R^{-\frac{d-2}{2}}A_t^0(\omega,\mathbf{s})\right]
\widehat{\psi}_{\omega,\mathbf{s}} \ ,\end{aligned}$$ so that $ \widehat{\chi}_{\omega,\mathbf{s}} $ and $ \widehat{\psi}_{\omega,\mathbf{s}} $ also satisfy Equations \[eq:longvectorchipsiequation1\] and \[eq:longvectorchipsiequation2\], with the boundary condition: $$\left.\widehat{\chi}_{\omega,\mathbf{s}}\right|_{z\to 1} = 1 \ .$$ Using these definitions in the expression for the action gives: $$\begin{gathered}
S_\| = \int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{s} \\
\left.\left[i\frac{\lambda}{q_\mathbf{s}}A_\|^0(\omega,\mathbf{s})+A_t^0(\omega,\mathbf{s})\right]^*
\mathcal{F}_\|(\omega,\mathbf{s},z)
\left[i\frac{\lambda}{q_\mathbf{s}}A_\|^0(\omega,\mathbf{s})+A_t^0(\omega,\mathbf{s})\right]
\right|_{z=0}^{z=1} \ ,\end{gathered}$$ where: $$\mathcal{F}_\|(\omega,\mathbf{s},z) = -2C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\widehat{\chi}_{\omega,\mathbf{s}}^*\widehat{\psi}_{\omega,\mathbf{s}} \ .$$ Using the prescription described in [@Son:2002sd], we can then get the components of the retarded correlators that correspond to the longitudinal mode: $$\begin{gathered}
G_{tt}^R(\omega,\mathbf{s}) = \left.-2\mathcal{F}_\|(\omega,\mathbf{s},z)\right|_{z\to 1}
= \left.4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\widehat{\chi}_{\omega,\mathbf{s}}^*\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z\to 1}\\
= \left.4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{z\to 1}\end{gathered}$$ $$\begin{aligned}
G_{t\|}^R(\omega,\mathbf{s}) &= i\frac{\lambda}{q_\mathbf{s}}G_{tt}^R(\omega,\mathbf{s}) \\
G_{\|\|}^R(\omega,\mathbf{s}) &= \frac{\lambda^2}{q_\mathbf{s}^2}G_{tt}^R(\omega,\mathbf{s}) \ .\end{aligned}$$ Note that from Equation \[eq:longvectorchipsiequation1\] we have: $$\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{z\to 1}
= \left.q_\mathbf{s}^2\frac{\psi_{\omega,\mathbf{s}}}
{(1-z)^{d-4}{\partial_z}\left[\frac{1}{(1-z)^{d-4}}\psi_{\omega,\mathbf{s}}\right]}\right|_{z\to 1} \ .$$
Turning to the transverse part of the action, we get the following expression: $$S_\bot = \left.2C_v\frac{r_+^{d-3}}{(1-z)^{d-4}}\frac{{\tilde{g}(z)}}{R^2}
\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{v} a_{\omega,\mathbf{v}}^*(z){\partial_z}a_{\omega,\mathbf{v}}(z)
\right|_{z=0}^{z=1} \ .$$ Considering the boundary condition in Equation \[eq:aboundarycondition\] we define the normalized $ \widehat{a}_{\omega,\mathbf{v}} $ by: $$a_{\omega,\mathbf{v}} \equiv R^{-\frac{d-4}{2}}A_\bot^0(\omega,\mathbf{v})
\widehat{a}_{\omega,\mathbf{v}} \ ,$$ so that: $$\left.\widehat{a}_{\omega,\mathbf{v}}\right|_{z\to 1} = 1 \ .$$ Inserting this definition into the expression for the action, we get: $$S_\bot = \left.\int\frac{{\,\mathrm{d}}\omega}{2\pi}\sum_\mathbf{v}
A_\bot^{0*}(\omega,\mathbf{v})\mathcal{F}_\bot(\omega,\mathbf{v},z)A_\bot^{0}(\omega,\mathbf{v})
\right|_{z=0}^{z=1} \ ,$$ where: $$\mathcal{F}_\bot(\omega,\mathbf{v},z)
= 2C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{{\tilde{g}(z)}}{(1-z)^{d-4}}
\widehat{a}_{\omega,\mathbf{v}}^*(z){\partial_z}\widehat{a}_{\omega,\mathbf{v}}(z) \ .$$ The components of the retarded correlators corresponding to the transverse mode are then: $$\begin{gathered}
G_{\bot\bot}^R(\omega,\mathbf{v})
= \left.-2\mathcal{F}_\bot(\omega,\mathbf{v},z)\right|_{z\to 1}\\
= \left.-4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\widehat{a}_{\omega,\mathbf{v}}^*(z){\partial_z}\widehat{a}_{\omega,\mathbf{v}}(z)\right|_{z\to 1}\\
= \left.-4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\frac{{\partial_z}a_{\omega,\mathbf{v}}(z)}{a_{\omega,\mathbf{v}}(z)}\right|_{z\to 1} \ .\end{gathered}$$
Derivation of Exact Solutions for the K=-1 Case {#app:exactsolutionsfortcderivation}
===============================================
In the $K=-1$ case, both the massless scalar and the transverse vector equations can be written in the form: $$(1-z)^\alpha{\partial_z}\left[\frac{{\tilde{g}(z)}}{(1-z)^\alpha}{\partial_z}\psi\right]+\left[\frac{\lambda^2}{{\tilde{g}(z)}}-q^2\right]\psi = 0 \ ,$$ or $${\tilde{g}(z)}\partial_z^2\psi+\left[\tilde{g}'(z)+\frac{\alpha{\tilde{g}(z)}}{1-z}\right]{\partial_z}\psi
+\left[\frac{\lambda^2}{{\tilde{g}(z)}}-q^2\right]\psi = 0 \ ,$$ where $$\alpha=
\begin{cases}
d-2 & \text{for scalar},\\
d-4 & \text{for transverse vector}
\end{cases} \ ,$$ with the boundary conditions: $$\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1}=0 \ .$$ In the case $K=-1$ we have $${\tilde{g}(z)}=1-(1-z)^2 \ .$$ In order to solve the equation, first make the transformation: $$w\equiv{\tilde{g}(z)}=1-(1-z)^2=z(2-z) \ ,$$ so that $${\partial_z}= \frac{\partial w}{\partial z}\partial_w = \tilde{g}'(z)\partial_w = 2(1-z)\partial_w
= 2\sqrt{1-w}\partial_w \ .$$ The equation then becomes: $$w(1-w)\partial_w^2\psi
+\left[1+\frac{1}{2}(\alpha-3)w\right]\partial_w\psi
+\left[\frac{\lambda^2}{4w}-\frac{1}{4}q^2\right]\psi = 0 \ ,$$ with the boundary conditions: $$\left.\psi\right|_{w=0} \sim w^{-\frac{i\lambda}{2}}
\qquad
\left.\psi\right|_{w=1} = 0 \ .$$ Next, define $$\psi \equiv w^\gamma \phi \ ,$$ where $\gamma=-i\frac{\lambda}{2}$, and get an equation for $\phi$: $$\begin{gathered}
w(1-w)\partial_w^2\phi
+\left[(1+2\gamma)+\left(\frac{1}{2}(\alpha-3)-2\gamma\right)w\right]\partial_w\phi\\
+\left[\gamma-\gamma^2+\frac{1}{2}\gamma(\alpha-3)-\frac{1}{4}q^2\right]\phi = 0 \ ,\end{gathered}$$ with the boundary conditions: $$\label{eq:hypergeometriceq1}
\left.\phi\right|_{w=0}=1
\qquad
\left.\phi\right|_{w=1}=0 \ .$$ Equation \[eq:hypergeometriceq1\] is known as the hypergeometric equation. The solution to this equation that also satisfies the boundary condition at $w=0$ is $\sb{2}F_1\left(a,b;c;w\right)$, where: $$\begin{aligned}
\label{eq:scalartransexactsolutionfortcparams1}
c &= 1-i\lambda \\
\label{eq:scalartransexactsolutionfortcparams2}
a,b &= -\frac{1}{4}(\alpha-1)-i\frac{\lambda}{2} \pm \frac{1}{2}\sqrt{\frac{1}{4}(\alpha-1)^2-q^2} \ .\end{aligned}$$
The longitudinal vector equation can be written in the form: $${\partial_z}\left[{\tilde{g}(z)}(1-z)^\alpha{\partial_z}\left(\frac{1}{(1-z)^\alpha}\psi\right)\right]
+\left[\frac{\lambda^2}{{\tilde{g}(z)}}-q^2\right]\psi = 0 \ ,$$ or $${\tilde{g}(z)}\partial_z^2\psi
+\left[\tilde{g}'(z)+\frac{\alpha{\tilde{g}(z)}}{1-z}\right]{\partial_z}\psi
+\left[{\partial_z}\left(\frac{\alpha{\tilde{g}(z)}}{1-z}\right)+\frac{\lambda^2}{{\tilde{g}(z)}}-q^2\right]\psi = 0 \ ,$$ where $$\alpha=d-4 \ ,$$ with the boundary conditions: $$\left.\psi.\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.(1-z)^\alpha{\partial_z}\left[\frac{1}{(1-z)^\alpha}\psi\right]\right|_{z=1} = 0 \ .$$ Defining again: $$w \equiv {\tilde{g}(z)}= 1-(1-z)^2 \ ,$$ the equation becomes: $$\begin{gathered}
w(1-w)\partial_w^2\psi
+\left[1+\frac{1}{2}(\alpha-3)w\right]\partial_w\psi\\
+\left[\frac{\lambda^2}{4w}-\frac{1}{4}q^2+\frac{1}{2}\alpha+\frac{\alpha w}{4(1-w)}\right]\psi = 0 \ .\end{gathered}$$ Next, define $$\psi \equiv w^\gamma (1-w)^\delta \phi \ ,$$ where $ \gamma = -i\frac{\lambda}{2} $ and $ \delta=\frac{1}{2} $, and get an equation for $ \phi $: $$\begin{gathered}
w(1-w)\partial_w^2\phi
+\left[(1+2\gamma)+\left(\frac{1}{2}(\alpha-3)-2\gamma-2\delta\right)w\right]\partial_w\phi\\
+\left[\gamma-\gamma^2+\frac{1}{2}\gamma(\alpha-3)-\frac{1}{4}q^2
+\frac{1}{4}\alpha-\delta(\delta-1)+\left(\frac{1}{2}(\alpha-3)-2\gamma\right)\delta\right]\phi = 0 \ .\end{gathered}$$ As for the boundary conditions, at $ w=0 $ we get: $$\left.\phi\right|_{z=0}=0 \ ,$$ while at $ w=1 $ we have [^18] $$\begin{gathered}
\left.(1-z)^\alpha{\partial_z}\left[\frac{1}{(1-z)^\alpha}\psi\right]\right|_{z=1}
= \left.2(1-w)^{\frac{\alpha+1}{2}}\partial_w\left[\frac{w^\gamma}{(1-w)^\frac{\alpha-1}
{2}}\phi\right]\right|_{w=1} \\
= \left.2w^\gamma(1-w)^{\frac{\alpha+1}{2}}\partial_w\left[\frac{1}{(1-w)^\frac{\alpha-1}
{2}}\phi\right] + 2\gamma(1-w)w^{\gamma-1}\phi\right|_{w=1} \\
\sim \left.2(1-w)^{\frac{\alpha+1}{2}}\partial_w\left[\frac{1}{(1-w)^\frac{\alpha-1}
{2}}\phi\right]\right|_{w=1} = 0 \ .\end{gathered}$$
The solution is once again $ \sb{2}F_1(a,b;c;w) $, where: $$\begin{aligned}
\label{eq:longvectorexactsolutionfortcparams1}
c &= 1-i\lambda\\
\label{eq:longvectorexactsolutionfortcparams2}
a,b &= -\frac{1}{4}(\alpha-3)-i\frac{\lambda}{2}\pm\frac{1}{2}\sqrt{\frac{1}{4}(\alpha+1)^2-q^2} \ .\end{aligned}$$
Derivation of Vector Correlators for the K=-1 Case {#app:exactvectorcorrelatorsderivationfortc}
==================================================
For the longitudinal vector mode, we continue from the transformation defined in Appendix \[app:exactsolutionsfortcderivation\]. The expression in Equation \[eq:longvectorcftcorrelator\] may be written in terms of $ \phi $ and $ w $: $$\label{eq:longvectorcorrelatorinw}
G_{tt}^R(\omega,\mathbf{s})
=\left.4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-w)^\frac{\alpha}{2}}\frac{q_\mathbf{s}^2}{2}
\frac{(1-w)^\frac{1}{2}\phi_{\omega,\mathbf{s}}}
{(1-w)^\frac{\alpha+1}{2}\partial_w\left[\frac{1}{(1-w)^\frac{\alpha-1}{2}}\phi_{\omega,\mathbf{s}}\right]
+\gamma\frac{1-w}{w}\phi_{\omega,\mathbf{s}}}\right|_{w\to 1} \ .$$
As explained in Appendix \[app:exactsolutionsfortcderivation\], the solution to the EOM with an incoming-wave boundary condition at the horizon is $ \phi_{\omega,\mathbf{s}} = {}_2F_1\left(a,b;c;w\right) $, where $ a $, $ b $ and $ c $ are given by Equations \[eq:longvectorexactsolutionfortcparams1\] and \[eq:longvectorexactsolutionfortcparams2\]. Define: $$\Delta \equiv c-a-b = \frac{\alpha-1}{2} = \frac{d-5}{2} \ .$$
There are now several possible cases:
1. $ d $ and is even ($ \Delta $ is non-integer). In this case, the connection formula in Equation \[eq:hypergeometricevenconnectionformula\] holds, so that: $$\begin{gathered}
\phi_{\omega,\mathbf{s}} = {}_2F_1\left(a,b;c;w\right)
= A \,{}_2F_1\left(a,b;-\Delta+1;1-w\right) \\
+ B (1-w)^\Delta\,{}_2F_1\left(c-a,c-b;\Delta+1;1-w\right)\end{gathered}$$ $$\begin{gathered}
(1-w)^{\frac{\alpha+1}{2}}\partial_w\left[\frac{1}{(1-w)^\frac{\alpha-1}
{2}}\phi_{\omega,\mathbf{s}}\right]
= \frac{(c-a)(c-b)}{c}{}_2F_1(a,b;c+1;w) \\
= \tilde{A} \,{}_2F_1\left(a,b;-\Delta;1-w\right) \\
+ \tilde{B} (1-w)^{\Delta+1}\,{}_2F_1\left(c-a+1,c-b+1;\Delta+2;1-w\right) \ ,\end{gathered}$$ where: $$\begin{aligned}
A &= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(c-a)\Gamma(c-b)} \\
B &= \frac{\Gamma(c)\Gamma(-\Delta)}{\Gamma(a)\Gamma(b)} \\
\tilde{A} &= \frac{(c-a)(c-b)}{c}\frac{\Gamma(c+1)\Gamma(\Delta+1)}{\Gamma(c-a+1)\Gamma(c-b+1)}
= \Delta A \\
\tilde{B} &= \frac{(c-a)(c-b)}{c}\frac{\Gamma(c+1)\Gamma(-\Delta-1)}{\Gamma(a)\Gamma(b)} \ .\end{aligned}$$ We proceed by calculating the limit: $$\label{eq:longvectorexactcorrelatorlimitcalculation}
\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{w=1-\epsilon}
\approx \frac{q_\mathbf{s}^2}{2\Delta}\frac{B}{A}\epsilon^\frac{\alpha}{2} \ ,$$ where contact terms have been dropped. Putting this into Equation \[eq:longvectorcorrelatorinw\] we get: $$\boxed{
G_{tt}^R(\omega,\mathbf{s})
= \frac{4C_v}{d-5}\frac{r_+^{d-3}}{R^{d-2}}q_\mathbf{s}^2
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\frac{\Gamma(a+\Delta)}{\Gamma(a)}
\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ .
}$$ The other components of the longitudinal mode correlator can be calculated from Equations \[eq:longvectorcftcorrelator2\] and \[eq:longvectorcftcorrelator3\].
2. $ d>5 $ and is odd ($ \Delta>0 $ is an integer). In this case, the connection formula in Equation \[eq:hypergeometricoddconnectionformula\] holds, so that: $$\left.\phi_{\omega,\mathbf{s}}\right|_{w\to 1} = \left.{}_2F_1\left(a,b;c;w\right)\right|_{w\to 1}
= A+\ldots+B(1-w)^\Delta+\ldots$$ $$\begin{gathered}
\left.(1-w)^{\frac{\alpha+1}{2}}\partial_w\left[\frac{1}{(1-w)^\frac{\alpha-1}
{2}}\phi_{\omega,\mathbf{s}}\right]\right|_{w\to 1}\\
= \left.\frac{(c-a)(c-b)}{c}{}_2F_1(a,b;c+1;w)\right|_{w\to 1} \\
= \tilde{A}+\ldots+\tilde{B}(1-w)^{\Delta+1}+\ldots \ ,\end{gathered}$$ where: $$\begin{aligned}
A &= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(c-a)\Gamma(c-b)} \\
B &= \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\frac{(-1)^{\Delta+1}}{(\Delta)!}
\left[\psi(a+\Delta)+\psi(b+\Delta)\right]\\
\tilde{A} &= \frac{(c-a)(c-b)}{c}\frac{\Gamma(c+1)\Gamma(\Delta+1)}
{\Gamma(c-a+1)\Gamma(c-b+1)} = \Delta A \\
\tilde{B} &= \frac{(c-a)(c-b)}{c}\frac{\Gamma(c+1)}{\Gamma(a)\Gamma(b)}
\frac{(-1)^{\Delta+2}}{(\Delta+1)!}
\left[\psi(a+\Delta+1)+\psi(b+\Delta+1)\right] \ .\end{aligned}$$
Proceeding in the same manner as in Equation \[eq:longvectorexactcorrelatorlimitcalculation\] above, we have: $$\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{w=1-\epsilon}
\approx \frac{q_\mathbf{s}^2}{2\Delta}\frac{B}{A}\epsilon^\frac{\alpha}{2} \ ,$$ so that: $$\boxed{
G_{tt}^R(\omega,\mathbf{s})
= 2C_v\frac{r_+^{d-3}}{R^{d-2}}q_\mathbf{s}^2\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta+1)}
(a)_\Delta(b)_\Delta\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}$$ The other components of the longitudinal mode correlator can be calculated from Equations \[eq:longvectorcftcorrelator2\] and \[eq:longvectorcftcorrelator3\].
3. $ d=5 $ ($ \Delta=0 $). In this case the following connection formula holds: $$\begin{gathered}
\phi_{\omega,\mathbf{s}} = {}_2F_1(a,b;c;w)
= -\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(n!)^2}(1-w)^n \ln(1-w)\\
-\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(n!)^2}
\left[\psi(a+n)+\psi(b+n)-2\psi(n+1)\right](1-w)^n\end{gathered}$$ (while for $ {}_2F_1(a,b;c+1;w) $ the formula in Equation \[eq:hypergeometricoddconnectionformula\] still holds), so that: $$\left.\phi_{\omega,\mathbf{s}}\right|_{w\to 1} = \left.{}_2F_1\left(a,b;c;w\right)\right|_{w\to 1}
= A\ln(1-w)+\ldots+B+\ldots$$ $$\begin{gathered}
\left.(1-w)^{\frac{\alpha+1}{2}}\partial_w\left[\frac{1}{(1-w)^\frac{\alpha-1}
{2}}\phi_{\omega,\mathbf{s}}\right]\right|_{w\to 1}\\
= \left.\frac{(c-a)(c-b)}{c}{}_2F_1(a,b;c+1;w)\right|_{w\to 1} \\
= \tilde{A}+\ldots+\tilde{B}(1-w)+\ldots \ ,\end{gathered}$$ where: $$\begin{aligned}
A &= -\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \\
B &= -\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\left[\psi(a)+\psi(b)\right]\\
\tilde{A} &= \frac{(c-a)(c-b)}{c}\frac{\Gamma(c+1)}
{\Gamma(c-a+1)\Gamma(c-b+1)} = -A \\
\tilde{B} &= \frac{(c-a)(c-b)}{c}\frac{\Gamma(c+1)}{\Gamma(a)\Gamma(b)}
\left[\psi(a+1)+\psi(b+1)\right] \ .\end{aligned}$$
Calculating the limit, we get: $$\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{w=1-\epsilon}
\approx -\frac{q_\mathbf{s}^2}{2}\frac{B}{A}\epsilon^\frac{1}{2} \ ,$$ where contact terms have been dropped. Putting this into Equation \[eq:longvectorcorrelatorinw\] we get: $$G_{tt}^R(\omega,\mathbf{s})
= -2C_v\frac{r_+^{2}}{R^{3}}q_\mathbf{s}^2 \left[\psi(a)+\psi(b)\right] \ .$$
This result also matches the general result for $ d>5 $ in Equation \[eq:longvectorexactcorrelatorintcodd\], so that one may extend that result to $ d=5 $ as well. The other components of the longitudinal mode correlator can be calculated from Equations \[eq:longvectorcftcorrelator2\] and \[eq:longvectorcftcorrelator3\].
As for the transverse vector mode, the calculations are very similar to the scalar case. Writing the expression in Equation \[eq:transvectorcftcorrelator\] in terms of $ \phi $ and $ w $ we get: $$G_{\bot\bot}^R(\omega,\mathbf{v})
=\left.-8C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-w)^\frac{d-5}{2}}
\frac{\partial_w\phi_{\omega,\mathbf{v}}}{\phi_{\omega,\mathbf{v}}}\right|_{w\to 1} \ .$$ The solution to the EOM with an incoming-wave boundary condition at the horizon is again $ \phi_{\omega,\mathbf{s}} = {}_2F_1\left(a,b;c;w\right) $, where $ a,b $ and $ c $ are given by Equations \[eq:scalartransexactsolutionfortcparams1\] and \[eq:scalartransexactsolutionfortcparams2\]. The rest of the calculations are identical to the scalar case, except with: $$\Delta \equiv c-a-b = \frac{\alpha+1}{2} = \frac{d-3}{2} \ ,$$ so that we get for even $ d $: $$\boxed{
G_{\bot\bot}^R(\omega,\mathbf{v})
= 4(d-3)C_v \frac{r_+^{d-3}}{R^{d-2}}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}
\frac{\Gamma(a+\Delta)}{\Gamma(a)}\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ ,
}$$ and for odd $ d $: $$\boxed{
G_{\bot\bot}^R(\omega,\mathbf{v})
= 8C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}(a)_\Delta(b)_\Delta
\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}$$
Effective Potentials and Their Asymptotics {#app:effectivepotentials}
==========================================
In order to obtain an asymptotic expression for the QNM frequencies and corresponding correlation functions, the Equations \[eq:scalareq\], \[eq:longvectoreq\] and \[eq:transvectoreq\] must first be put into the Schrödinger form: $$\label{eq:schrodingerform}
\partial_{{\tilde{z}}}^2\phi + \left[ \lambda^2-V({\tilde{z}}) \right]\phi = 0 \ .$$
Both the scalar equation and the “transverse” vector equation are of the form $$\label{eq:scalartransvectoreq}
(1-z)^\alpha\partial_z\left[ \frac{\tilde{g}(z)}{(1-z)^\alpha} \partial_z \psi \right] +
\left[ \frac{\lambda^2}{\tilde{g}(z)} - q^2 \right] \psi = 0 \ ,$$ where $$\alpha=
\begin{cases}
d-2 & \text{for scalar},\\
d-4 & \text{for transverse vector}
\end{cases} \ .$$ In these cases we can bring the equation to the form in Equation \[eq:schrodingerform\] by defining $$\psi(z) = (1-z)^{\frac{\alpha}{2}} \phi(z) \ .$$ After substituting $ \psi $ in Equation \[eq:scalartransvectoreq\] and some algebra we get the equation $${\tilde{g}(z)}{\partial_z}\left[{\tilde{g}(z)}{\partial_z}\phi\right] + \left[\lambda^2-V_1(z)\right] = 0 \ ,$$ where $$\begin{gathered}
V_1(z) = \frac{\alpha}{2}{\tilde{g}(z)}(1-z)^{\frac{\alpha}{2}}{\partial_z}\left(\frac{{\tilde{g}(z)}}{(1-z)^{\frac{\alpha}{2}+1}} \right) + q^2{\tilde{g}(z)}\\
= \frac{\alpha}{2}\frac{{\tilde{g}(z)}{\partial_z}{\tilde{g}(z)}}{1-z} + \frac{\alpha}{2}\left(\frac{\alpha}{2}+1\right)\frac{\tilde{g}^2(z)}{(1-z)^2} + q^2{\tilde{g}(z)}\ .\end{gathered}$$ The boundary conditions are accordingly $$\left.\phi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.(1-z)^{\frac{\alpha}{2}}\phi\right|_{z=1} = 0 \ .$$
Next we define a new coordinate as follows: $${\tilde{z}}\equiv \int \frac{1}{{\tilde{g}(z)}} {\,\mathrm{d}}z \ ,$$ so that $ {\tilde{g}(z)}{\partial_z}= {\partial_{\tilde{z}}}$. Changing to this new coordinate we obtain the form of Equation \[eq:schrodingerform\] with $ V_1\left(z({\tilde{z}})\right) $ as the effective potential.
The “longitudinal” vector equation is: $$\label{eq:longvectoreq_withalpha}
\partial_z \left[ \tilde{g}(z)(1-z)^{\alpha} \partial_z \left( \frac{1}{(1-z)^{\alpha}} \psi \right) \right] + \left[ \frac{\lambda^2}{\tilde{g}(z)} - q_s^2 \right] \psi = 0 \ ,$$ where $$\alpha = d-4 \ .$$ Define again $$\psi(z) = (1-z)^{\frac{\alpha}{2}} \phi(z) \ .$$ Substituting into Equation \[eq:longvectoreq\_withalpha\] we again get $${\tilde{g}(z)}{\partial_z}\left[{\tilde{g}(z)}{\partial_z}\phi\right] + \left[\lambda^2-V_2(z)\right]\phi = 0 \ ,$$ only with the effective potential $$\begin{gathered}
V_2(z) = -\frac{\alpha}{2}{\tilde{g}(z)}(1-z)^{-\frac{\alpha}{2}}{\partial_z}\left((1-z)^{\frac{\alpha}{2}-1}{\tilde{g}(z)}\right)\\
= -\frac{\alpha}{2}\frac{{\tilde{g}(z)}{\partial_z}{\tilde{g}(z)}}{1-z}+\frac{\alpha}{2}\left(\frac{\alpha}{2}-1\right)\frac{\tilde{g}^2(z)}{(1-z)^2} + q^2 {\tilde{g}(z)}\ .\end{gathered}$$ The boundary conditions are accordingly $$\left.\phi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.(1-z)^{\alpha}{\partial_z}\left[(1-z)^{-\frac{\alpha}{2}}\phi\right]\right|_{z=1} = 0 \ .$$
Transforming again to $ {\tilde{z}}$ we obtain the form of Equation \[eq:schrodingerform\] with $ V_2\left(z({\tilde{z}})\right) $ as the effective potential.
Looking at the effective potentials, we can see their behaviour near the horizon, near the boundary and at $ z\to\infty $ (or $ {\tilde{z}}={\tilde{z}}_0 $).
At the horizon ($ z\to 0 $, $ {\tilde{z}}\to -\infty $) we have: $$\begin{aligned}
V_1(z) &\to 0\\
V_2(z) &\to 0 \ .\end{aligned}$$
At the boundary ($ z\to 1 $, $ {\tilde{z}}\to 0 $) we have: $$\begin{aligned}
V_1(z) &\approx \frac{\alpha}{2}\left(\frac{\alpha}{2}+1\right) \frac{1}{(1-z)^2} \approx \frac{\alpha}{2}\left(\frac{\alpha}{2}+1\right)\frac{1}{{\tilde{z}}^2} = \frac{j_1^2-1}{4{\tilde{z}}^2}\\
V_2(z) &\approx \frac{\alpha}{2}\left(\frac{\alpha}{2}-1\right) \frac{1}{(1-z)^2} \approx \frac{\alpha}{2}\left(\frac{\alpha}{2}-1\right)\frac{1}{{\tilde{z}}^2} = \frac{j_1^2-1}{4{\tilde{z}}^2} \ ,\end{aligned}$$ so that, generally $$V(z) \approx \frac{j_1^2-1}{4{\tilde{z}}^2}
\qquad
j_1 =
\begin{cases}
d-1 & \text{for scalar},\\
d-3 & \text{for transverse vector},\\
|d-5| & \text{for longitudinal vector}
\end{cases} \ .$$
At the limit $ z\to\infty $ ($ {\tilde{z}}\to{\tilde{z}}_0 $) we have: $$V_1(z)\approx\frac{\alpha}{2}(1+K)^2\left[\frac{\alpha}{2}-d+2\right]z^{2d-4}\\
= \frac{1}{(d-2)^2}\frac{\alpha}{2}\left[\frac{\alpha}{2}-d+2\right]\frac{1}{({\tilde{z}}-{\tilde{z}}_0)^2}$$ $$V_2(z) \approx \frac{\alpha}{2}(1+K)^2\left[\frac{\alpha}{2}+d-2\right]z^{2d-4}\\
= \frac{1}{(d-2)^2}\frac{\alpha}{2}\left[\frac{\alpha}{2}+d-2\right]\frac{1}{({\tilde{z}}-{\tilde{z}}_0)^2} \ ,$$ so that, generally $$V(z) \approx \frac{j_{\infty}^2-1}{4({\tilde{z}}-{\tilde{z}}_0)^2}
\qquad
j_\infty =
\begin{cases}
0 & \text{for scalar,}\\
\frac{2}{d-2} & \text{for transverse vector,}\\
2-\frac{2}{d-2} & \text{for longitudinal vector}
\end{cases} \ .$$
Asymptotic Correlators for the Vector Case {#app:cftasymptoticcorrelatorexpressionsvector}
==========================================
Proceeding from the definitions in Subsection \[subsec:qnmasymoptotics\], the expressions in Equation \[eq:longvectorcftcorrelator\] and \[eq:transvectorcftcorrelator\] can be written in terms of $ \phi $ and $ {\tilde{z}}$: $$\label{eq:longvectorcftcorrelatorinzt}
G_{tt}^R(\omega,\mathbf{s})
= \left.4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{(-1)^d}{{\tilde{z}}^{d-4}}q_\mathbf{s}^2
\frac{\phi_{\omega,\mathbf{s}}}{{\tilde{z}}^\frac{\alpha}{2}{\partial_{\tilde{z}}}\left[\frac{1}{{\tilde{z}}^\frac{\alpha}{2}}
\phi_{\omega,\mathbf{s}}\right]}\right|_{{\tilde{z}}\to 0}$$ $$\label{eq:transvectorcftcorrelatorinzt}
G_{\bot\bot}^R(\omega,\mathbf{v})
= \left.-4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{(-1)^d}{{\tilde{z}}^{d-4}}
\frac{{\partial_{\tilde{z}}}\phi_{\omega,\mathbf{v}}}{\phi_{\omega,\mathbf{v}}}\right|_{{\tilde{z}}\to 0} \ .$$
We again continue with the method applied in Subsection \[subsec:qnmasymoptotics\] and Subsubsection \[subsubsec:cftasymptoticcorrelatorexpressionsscalar\]. In this case, the solution around $ {\tilde{z}}\to{\tilde{z}}_0 $ ($ z\to\infty $) is given by Equation \[eq:monoasympinftyplus\] in direction (1) and Equation \[eq:monoasympinftymin\] in direction (2). From the boundary condition at the horizon ($ z=0 $, $ {\tilde{z}}\to -\infty $) we have: $$\phi \approx {\mathrm{e}}^{-i\lambda{\tilde{z}}} \ .$$
We again distinguish between two possible cases:
1. $ d $ is even. In this case the solution around $ {\tilde{z}}\to 0 $ is approximately: $$\phi \approx B_+ P_+({\tilde{z}}) + B_- P_-({\tilde{z}})\\
= B_+\sqrt{2\pi\lambda{\tilde{z}}} J_{\frac{j_1}{2}}(\lambda{\tilde{z}})
+ B_-\sqrt{2\pi\lambda{\tilde{z}}} J_{-\frac{j_1}{2}}(\lambda{\tilde{z}}) \ ,$$ where: $$j_1 =
\begin{cases}
d-3 & \text{for transverse vector,}\\
|d-5| & \text{for longitudinal vector}
\end{cases}$$ (so that $ \frac{j_1}{2} > 0 $ is non-integer). Replacing the Bessel functions with their asymptotic form, we have: $$\phi
\approx \left[B_+{\mathrm{e}}^{-i\beta_+} + B_-{\mathrm{e}}^{-i\beta_-}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}} +
\left[B_+{\mathrm{e}}^{i\beta_+} + B_-{\mathrm{e}}^{i\beta_-}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ ,$$ where $ \beta_\pm \equiv \frac{\pi}{4}(1 \pm j_1)$. We now match the solutions on lines (1) and (2). We have for line (1): $$\label{eq:cftmonoeq9}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_+)} + A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_-)} = 0 \ .$$ For the section (2) we have: $$\begin{aligned}
\label{eq:cftmonoeq10}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_+)} + A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_-)}
&= B_+{\mathrm{e}}^{-i\beta_+} + B_-{\mathrm{e}}^{-i\beta_-} \\
\label{eq:cftmonoeq11}
A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_+)} + A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_-)}
&= B_+{\mathrm{e}}^{i\beta_+} + B_-{\mathrm{e}}^{i\beta_-} \ .\end{aligned}$$ Equations \[eq:cftmonoeq9\], \[eq:cftmonoeq10\] and \[eq:cftmonoeq11\] form a linear system of equations. Solving for $ B_\pm $, $ A_- $ we can get an expression for $ \frac{B_+}{B_-} $: $$\frac{B_+}{B_-}
= -i^{j_1}\frac{{\mathrm{e}}^{2i\theta_-}-2\cos\left(\frac{\pi}{2}j_\infty\right)}
{{\mathrm{e}}^{2i\theta_-}+2\cos\left(\frac{\pi}{2}j_\infty\right)} \ ,$$ where $ \theta_- \equiv \lambda{\tilde{z}}_0 + \frac{\pi}{4}(j_1 -2) $.
Next we evaluate the correlators. Developing $ \phi $ around $ {\tilde{z}}=0 $, we again have: $$\phi = A{\tilde{z}}^{-\Delta+\frac{1}{2}}+\ldots+B{\tilde{z}}^{\Delta+\frac{1}{2}}+\ldots \ ,$$ with: $$\begin{aligned}
A &= \frac{\sqrt{2\pi}\lambda^{-\Delta+\frac{1}{2}}}
{2^{-\Delta}\Gamma(-\Delta+1)} B_- \\
B &= \frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}
{2^{\Delta}\Gamma(\Delta+1)} B_+\end{aligned}$$ (where $ \Delta\equiv \frac{j_1}{2} $). For the longitudinal vector mode (for which $j_1 = d-5 = \alpha-1 = 2\Delta $), we have: $$\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{{\tilde{z}}=\epsilon}
\approx -\frac{q_\mathbf{s}^2}{j_1} \frac{B}{A}\epsilon^\alpha \ .$$ Putting this into Equation \[eq:longvectorcftcorrelatorinzt\], and using the fact that for the longitudinal vector mode $ j_\infty = 2-\frac{2}{d-2} $ and the symmetry properties of the retarded correlator (as in Subsubsection \[subsubsec:cftasymptoticcorrelatorexpressionsscalar\]), we obtain the correlator:
[multline]{} G\_[tt]{}\^R(,) R\^[d-4]{} (L\_\^[FT]{})\^2()\^[d-5]{}\
,
where: $$\overline{\theta_-}(\lambda) \equiv -\theta_-^*(-\lambda^*)
= \lambda{\tilde{z}}_0^*-\frac{\pi}{4}(d-7) \ .$$ The other components of the longitudinal mode correlator can be calculated from Equations \[eq:longvectorcftcorrelator2\] and \[eq:longvectorcftcorrelator3\].
For the transverse vector mode (for which $ j_1 = d-3 = \alpha+1 = 2\Delta $), we have: $$\left.\frac{{\partial_{\tilde{z}}}\phi_{\omega,\mathbf{v}}}{\phi_{\omega,\mathbf{v}}}\right|_{{\tilde{z}}=\epsilon}
= j_1 \frac{B}{A}\epsilon^\alpha \ .$$ Putting this into Equation \[eq:transvectorcftcorrelatorinzt\], and using the fact that for the transverse vector mode $ j_\infty = \frac{2}{d-2} $ and the symmetry properties of the retarded correlator , we obtain the result:
[multline]{} G\_\^R(,) 4(d-3)C\_v R\^[d-4]{} ()\^[d-3]{}\
,
where: $$\overline{\theta_-}(\lambda) \equiv -\theta_-^*(-\lambda*)
= \lambda{\tilde{z}}_0^*-\frac{\pi}{4}(d-5) \ .$$
2. $ d $ is odd (so that $ \frac{j_1}{2} \geq 0 $ is an integer). In this case the solution around $ {\tilde{z}}\to 0 $ is approximately: $$\phi \approx B_+ P_+({\tilde{z}}) + B_- P_-({\tilde{z}})\\
= B_+\sqrt{2\pi\lambda{\tilde{z}}} J_{\frac{j_1}{2}}(\lambda{\tilde{z}})
+ B_-\sqrt{2\pi\lambda{\tilde{z}}} Y_{\frac{j_1}{2}}(\lambda{\tilde{z}}) \ .$$
Replacing the Bessel functions with their asymptotic form, we have: $$\phi
\approx \left[(B_+-iB_-){\mathrm{e}}^{-i\beta_+}\right]{\mathrm{e}}^{i\lambda{\tilde{z}}} +
\left[(B_++iB_-){\mathrm{e}}^{i\beta_+}\right]{\mathrm{e}}^{-i\lambda{\tilde{z}}} \ ,$$ where $ \beta_+ = \frac{\pi}{4}(1+j_1) $. We now match the solutions on lines (1) and (2). We have for line (1): $$\label{eq:cftmonoeq12}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_+)} + A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0-\alpha_-)} = 0 \ .$$ For the section (2) we have: $$\begin{aligned}
\label{eq:cftmonoeq13}
A_+{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_+)} + A_-{\mathrm{e}}^{i(-\lambda{\tilde{z}}_0+3\alpha_-)}
&= (B_+ -iB_-){\mathrm{e}}^{-i\beta_+} \\
\label{eq:cftmonoeq14}
A_+{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_+)} + A_-{\mathrm{e}}^{i(\lambda{\tilde{z}}_0+\alpha_-)}
&= (B_+ +iB_-){\mathrm{e}}^{i\beta_+} \ .\end{aligned}$$ Equations \[eq:cftmonoeq12\], \[eq:cftmonoeq13\] and \[eq:cftmonoeq14\] form a linear system of equations. Solving for $ B_\pm $, $ A_- $ we get an expression for $ \frac{B_+}{B_-} $: $$\frac{B_+}{B_-}
= 2i\frac{{\mathrm{e}}^{2i\theta_+}}{{\mathrm{e}}^{2i\theta_+} - 2\cos\left(\frac{\pi}{2}j_\infty\right)} - i \ ,$$ where $ \theta_+ \equiv \lambda{\tilde{z}}_0 - \frac{\pi}{4}(j_1 +2) $.
Next we evaluate the correlators. We first assume that $ j_1>0 $ (this is satisfied for the transverse mode with $ d\geq 5 $, or the longitudinal mode with $ d>5 $). Developing $ \phi $ around $ {\tilde{z}}=0 $, we again have: $$\phi = A{\tilde{z}}^{-\Delta+\frac{1}{2}}+\ldots+B{\tilde{z}}^{\Delta+\frac{1}{2}}+\ldots \ ,$$ with: $$A = -\frac{\sqrt{2\pi}\lambda^{-\Delta+\frac{1}{2}}}{\pi 2^{-\Delta}}
(\Delta-1)!\,B_-$$ $$\begin{gathered}
B = \frac{2\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}
{\pi 2^\Delta\Gamma(\Delta+1)}\ln\left(-\frac{\lambda}{2}\right)\,B_- \\
-\frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}{\pi 2^\Delta(\Delta)!}
\left[\psi(1)+\psi(\Delta+1)\right]\,B_-
+\frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}{2^\Delta \Gamma(\Delta+1)}\,B_+\end{gathered}$$ (where $ \Delta\equiv \frac{j_1}{2} $). For the longitudinal vector mode (for which $j_1 = d-5 = \alpha-1 = 2\Delta $), we again have: $$\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{{\tilde{z}}=\epsilon}
\approx -\frac{q_\mathbf{s}^2}{j_1} \frac{B}{A}\epsilon^\alpha\ .$$ Putting this into Equation \[eq:longvectorcftcorrelatorinzt\], and using the fact that for the longitudinal vector mode $ j_\infty = 2-\frac{2}{d-2} $ and the symmetry properties of the correlator, we obtain the correlator:
[multline]{} G\_[tt]{}\^R(,) 2C\_v R\^[d-4]{}(L\_\^[FT]{})\^2 ()\^[d-5]{}\
,
where: $$\overline{\theta_-}(\lambda) \equiv -\theta_-^*(-\lambda*)
= \lambda{\tilde{z}}_0^*+\frac{\pi}{4}(d-3) \ .$$ The other components of the longitudinal mode correlator can be calculated from Equations \[eq:longvectorcftcorrelator2\] and \[eq:longvectorcftcorrelator3\].
Similarly, for the transverse vector mode (for which $ j_1 = d-3 = \alpha+1 = 2\Delta $), we get:
[multline]{} G\_\^R(,) 8C\_v R\^[d-4]{} ()\^[d-3]{}\
,
where: $$\overline{\theta_-}(\lambda) \equiv -\theta_-^*(-\lambda*)
= \lambda{\tilde{z}}_0^*+\frac{\pi}{4}(d-1) \ .$$
Next we turn to the case of $ j_1=0 $, which is satisfied for the longitudinal mode with $ d=5 $. In this case, Equation \[eq:asymptoticcorrelatorsphiaround0odd\] is still valid with $ \Delta = \frac{j_1}{2} = 0 $, so that: $$\phi = A{\tilde{z}}^\frac{1}{2}\ln(-{\tilde{z}})+\ldots+B{\tilde{z}}^\frac{1}{2}+\ldots \ ,$$ with: $$A = \frac{2\sqrt{2\pi}\lambda^\frac{1}{2}}{\pi}B_-$$ $$B
= \frac{2\sqrt{2\pi}\lambda^\frac{1}{2}}{\pi}\left[\ln\left(-\frac{\lambda}{2}\right)-\psi(1)\right]B_-
+ \sqrt{2\pi}\lambda^\frac{1}{2}B_+ \ .$$ This case is restricted for the longitudinal case, for which we have: $$\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{{\tilde{z}}=\epsilon}
= \left.q_\mathbf{s}^2 \frac{\phi_{\omega,\mathbf{s}}}{{\tilde{z}}^\frac{1}{2}{\partial_{\tilde{z}}}\left[\frac{1}
{{\tilde{z}}^\frac{1}{2}}\phi_{\omega,\mathbf{s}}\right]}\right|_{{\tilde{z}}=\epsilon}\\
= q_\mathbf{s}^2\left[\epsilon\ln(-\epsilon)+\ldots+\frac{B}{A}\epsilon+\ldots\right]
\approx q_\mathbf{s}^2 \frac{B}{A}\epsilon \ ,$$ where contact terms were dropped in the last equality. Putting $ \frac{B}{A} $ into this expression we get: $$\left.\frac{\psi_{\omega,\mathbf{s}}}{\chi_{\omega,\mathbf{s}}}\right|_{{\tilde{z}}=\epsilon}
= \frac{q_\mathbf{s}^2}{2}\left[\pi\frac{B_+}{B_-}+2\ln\left(-\frac{\lambda}{2}\right)\right] \epsilon
\ .$$ This result is consistent with the corresponding expression for $ d>5 $, so that we may extend the rest of the results for $ d>5 $ to $ d=5 $ as well.
Numerical Methods {#app:numericalmethods}
=================
Here we outline the numerical methods used for the exact calculation of the QNM spectrum and the retarded correlation functions associated with the massless scalar and vector perturbation modes.
QNM spectrum calculation {#appsubsec:qnmnumericalmethod}
------------------------
For each of the QNM Equations \[eq:scalareq\], \[eq:longvectoreq\] and \[eq:transvectoreq\] and their corresponding boundary conditions, given the values of the parameters $ d $, $ K $ and $ q $, one may use the Frobenius method to calculate the exact values of $ \lambda $ for which the eqautions have non-trivial solutions. The stages of the calculation are as follows:
1. We write the QNM equation in the form: $$\label{eq:qnmequationpolynomform}
P_1(z){\partial_z}^2\psi + P_2(z){\partial_z}\psi + P_3(z)\psi = 0 \ ,$$ where $ P_1(z) $, $ P_2(z) $ and $ P_3(z) $ are polynomials in $ z $ that depend on the values of $ d $, $ K $, $ q $ and $ \lambda $. We write the boundary condition at $ z=1 $ (the AdS boundary) in a similar form: $$\left.Q_1(z){\partial_z}\psi + Q_2(z)\psi\right|_{z\to 1} = 0 \ ,$$ where $ Q_1(z) $ and $ Q_2(z) $ are polynomials in $ z $[^19]. For example, for the scalar QNM perturbation we have: $$\begin{aligned}
P_1(z) &= \tilde{g}^2(z)(1-z) \\
P_2(z) &= {\tilde{g}(z)}\left[\tilde{g}'(z)(1-z)+(d-2){\tilde{g}(z)}\right] \\
P_3(z) &= (1-z)\left[\lambda^2-q^2{\tilde{g}(z)}\right] \\
Q_1(z) &= 0 \\
Q_2(z) &= 1 \ .\end{aligned}$$
2. We calculate the exponents of the QNM equation at $ z=0 $ and $ z=1 $ via its indicial equation. At $ z=0 $ we have: $$\left.\psi\right|_{z=0} \sim z^{\pm\frac{i\lambda}{C}} \ ,$$ where $ C=\tilde{g}'(0) $. In accordance with the incoming-wave boundary condition at the horizon, we always choose the exponent $ \gamma = -\frac{i\lambda}{C}$. At $ z=1 $ we get the real exponents $ \beta_1 $ and $ \beta_2 $, and we define: $$\beta \equiv \min(\beta_1,\beta_2) \ .$$
3. We make the transformation: $$\label{eq:qnmnumericaltransform}
\psi = z^\gamma(1-z)^\beta\phi \ ,$$ so that $ \phi $ is regular and finite at $ z=0 $. Re-writing the equation in terms of $ \phi $ we get: $$\label{eq:numericalnormalizedqnmequation}
\widetilde{P}_1(z){\partial_z}^2\phi + \widetilde{P}_2(z){\partial_z}\phi + \widetilde{P}_3(z)\phi = 0 \ ,$$ with the boundary condition at the horizon: $$\left.\widetilde{Q}_1(z){\partial_z}\phi + \widetilde{Q}_2(z)\phi\right|_{z\to 1} = 0 \ ,$$ where $ \widetilde{P}_1(z) $, $ \widetilde{P}_2(z) $, $ \widetilde{P}_3(z) $, $ \widetilde{Q}_1(z) $ and $ \widetilde{Q}_2(z) $ are polynomials in $ z $. We assume that $ \widetilde{P}_1 $, $ \widetilde{P}_2 $ and $ \widetilde{P}_3 $ are given by: $$\begin{aligned}
\widetilde{P}_1(z) &= \sum_{k=0}^{n_1} c_k^1 z^k \\
\widetilde{P}_2(z) &= \sum_{k=0}^{n_2} c_k^2 z^k \\
\widetilde{P}_3(z) &= \sum_{k=0}^{n_3} c_k^3 z^k \ .\end{aligned}$$
4. In order to increase the numerical stability of the calculation and decrease the time required for calculation we minimize the degree of the polynomials in the equation by dividing $ \widetilde{P}_1 $, $ \widetilde{P}_2 $ and $ \widetilde{P}_3 $ by their greatest common divisor (as polynomials in $ z $).
5. We solve the equation using the Frobenius method: We develop $ \phi $ as a power series around $ z=0 $ up to the $ N $-th degree: $$\label{eq:numericalexpansion}
\phi(z) = \sum_{k=1}^{N+1} a_{k-1}z^{k-1}$$ We then put the series into the Equation \[eq:numericalnormalizedqnmequation\] and obtain an equation for each power of $ z $ up to the $ (N-1) $-th degree. The homogeneous system of equations reads: $$S\mathbf{a} = 0 \ ,$$ where $$\mathbf{a} =
\begin{pmatrix}
a_0 \\
a_1 \\
\vdots \\
a_N
\end{pmatrix} \ ,$$ and $$\label{eq:numericallinearequations}
S_{ij} = c_{i-j+2}^1 (j-1)(j-2) + c_{i-j+1}^2 (j-1) + c_{i-j}^3$$ (for $ 1 \leq i \leq N $, $ 1 \leq i \leq N+1 $ and $ -2 \leq i-j \leq \max(n_1-2,n_2-1,n_3) $). The boundary condition at $ z=1 $ gives another equation: In the case of $ \beta_1 = \beta_2 $, the series in Equation \[eq:numericalexpansion\] doesn’t converge at $ z=1 $ for $ N\to\infty $, since one of the solutions to Equation \[eq:numericalnormalizedqnmequation\] goes like $ \log(1-z) $ near $ z=1 $, while the other is finite at $ z=1 $. In order to choose the finite solution (as dictated by the boundary condition in this case), we choose the following condition that is necessary for convergence: $$a_N = 0 \ .$$ In the case of $ \beta_1 \neq \beta_2 $, we have the boundary condition: $$b_1{\partial_z}\phi + b_2\phi = 0 \ ,$$ where: $$b_1 = \left.\frac{\widetilde{Q}_1(z)}{(1-z)^s}\right|_{z\to 1}
\qquad
b_2 = \left.\frac{\widetilde{Q}_2(z)}{(1-z)^s}\right|_{z\to 1} \ ,$$ and $ s $ is the minimal exponent such that $ b_1 \neq 0 $ or $ b_2 \neq 0 $. In terms of the series expansion this gives the equation: $$\sum_{j=1}^{N+1} S_{N+1,j}a_{j-1} = 0 \ ,$$ where $$S_{N+1,j} = b_1(j-1) + b_2 \ .$$ We end up with a linear system of $ N+1 $ equations, where the coefficients are polynomials in $ \lambda $: $$S_{ij}(\lambda,d,K,q) = \sum_{l=0}^M S_{ij}^{(l)}(d,K,q)\lambda^l \ .$$ Finally we find all the values of $ \lambda $ for which a non-trivial solution exists to the system of equations, by solving the generalized eigenvalue problem: $$\det(S) = \det\left(\sum_{l=0}^M S_{ij}^{(l)}\lambda^l\right) = 0 \ .$$
6. We filter out all of the values of $ \lambda $ that don’t converge as we take $ N\to\infty $, by performing the above calculation for the $ N $-th degree and for the $ (N+1) $-th degree, and including only values of $ \lambda $ that satisfy the condition: $$\left| \lambda_k^{(N+1)} - \lambda_k^{(N)} \right| < \epsilon \ ,$$ where $ \epsilon $ is some convergence threshold.
Correlators calculation
-----------------------
For each type of perturbation discussed in this chapter, and given the values of the parameters $ d $, $ K $, $ q $ and $ \lambda $, we outline a numerical method based on the Frobenius solution to calculate the exact value of the retarded correlation function of the dual gauge theory operators. The calculation of the correlators require the evaluation of the expressions in Equations \[eq:scalarcftcorrelator\], \[eq:longvectorcftcorrelator\] and \[eq:transvectorcftcorrelator\]. The stages of the calculation are as follows:
1. As in Subsection \[appsubsec:qnmnumericalmethod\], we write the relevant QNM equation in the form of Equation \[eq:qnmequationpolynomform\].
2. We again calculate the exponents of the QNM equation at $ z=0 $ and $ z=1 $ via the indicial equation. At the horizon we choose the the exponent $ \gamma = -\frac{i\lambda}{C} $ corresponding to the incoming-wave condition (and the retarded correlator in the gauge theory). At $ z=1 $ we get the real exponents $ \beta_1 $ and $ \beta_2 $, and we define: $$\beta \equiv \min(\beta_1,\beta_2)
\qquad
\beta' \equiv \max(\beta_1,\beta_2) - \min(\beta_1,\beta_2) \ .$$
3. We again perform the transformation in Equation \[eq:qnmnumericaltransform\], and get the Equation \[eq:numericalnormalizedqnmequation\].
4. We divide the polynomials $ \widetilde{P}_1 $, $ \widetilde{P}_2 $ and $ \widetilde{P}_3 $ by their greatest common divisor.
5. We solve the equation around $ z=0 $ using the Frobenius method: We develop $ \phi $ as a power series around $ z=0 $ up to the N-th degree: $$\phi_1(z) = \sum_{k=1}^{N+1} a_{k-1}^{(1)} z^{k-1} \ .$$ We put the expansion into the equation \[eq:numericalnormalizedqnmequation\] and obtain an equation for each power of $ z $ up to the $ (N-1) $-th degree. We end up with the system of equations: $$S\mathbf{a} = 0 \ ,$$ where $ S $ is given by Equation \[eq:numericallinearequations\]. To get a single solution a normalization equation needs to be added, and we choose: $$a_0^{(1)} = 1 \ .$$ Solving this system of equations for specific values of $ d $, $ K $, $ q $ and $ \lambda $, we obtain the solution $ \phi _1 $ (which is regular at $ z=0 $).
6. We find the two independent solutions of the equation, $ \phi_2 $ and $ \phi_3 $ around $ z=1 $ using the Frobenius method: First, we assume that: $$\left.\phi_2\right|_{z=1} \approx (1-z)^{\beta'} \ .$$ We define: $$\label{eq:qnmnumericaltransform2}
\phi_2 = (1-z)^{\beta'}\widetilde{\phi}_2 \ ,$$ and again re-write the equation in terms of $ \widetilde{\phi}_2 $. We develop $ \widetilde{\phi}_2 $ as a power series around $ z=1 $ up to the $ N $-th degree: $$\widetilde{\phi}_2(z) = \sum_{k=1}^{N+1} a_{k-1}^{(2)} (z-1)^{k-1} \ .$$ Putting the expansion into the equation, we again end up with a system of $ N $ linear equations for the coefficients $ a_k^{(2)} $ (similar to Equations \[eq:numericallinearequations\], but here the parameters $ c_k^i $ are the coefficients of the polynomials in the QNM equation after the transformation given in Equation \[eq:qnmnumericaltransform2\], developed around $ z=1 $). Adding the normalization equation: $$a_0^{(2)} = 1 \ ,$$ we solve the system of equations to get the solution $ \widetilde{\phi}_2 $ and then transform back to $ \phi_2 $. The other solution around $ z=1 $ takes the form: $$\phi_3 = \widetilde{\phi}_3 + c\ln(1-z)\phi_2 \ ,$$ where: $$\left.\widetilde{\phi}_3\right|_{z=1} \approx 1 \ .$$ We develop $ .\widetilde{\phi}_3 $ as a power series around $ z=1 $ up to the $ N $-th degree: $$\widetilde{\phi}_3(z) = \sum_{k=1}^{N+1} a_{k-1}^{(3)} (z-1)^{k-1} \ ,$$ and put $ \phi_3 $ into the equation. We end up with a system of $ N $ linear equations for the coefficients $ a_k^{(3)} $ and the coefficient $ c $. Two equations must now be added: A normalization equation, and another equation to fix the extra degree of freedom of adding to $ \phi_3 $ a function that is proportional to $ \phi_2 $. In the case where $ \beta' > 0 $, we choose the following two equations: $$\begin{aligned}
a_0^{(3)} &= 1 \\
a_{\beta'}^{(3)} &= 0 \ .\end{aligned}$$ If $ \beta' = 0 $, we choose: $$\begin{aligned}
c &= 1 \\
a_0^{(3)} &= 0 \ .\end{aligned}$$ Solving this system of $ N+2 $ equations we get the solution $ \phi_3 $.
7. We extract the connection coefficients $ A,B $ linearly relating the solution $ \phi_1 $ to the solutions $ \phi_2 $, $ \phi_3 $: $$\label{eq:numericalconnectionrelation}
\phi_1(z) = A\phi_3(z) + B\phi_2(z) \ .$$ This can be accomplished by choosing a value $ 0 < z_0 < 1 $[^20], and setting $ z=z_0 $ both in Equation \[eq:numericalconnectionrelation\] and its derivative with respect to $ z_0 $, thereby obtaining 2 linear equations for $ A,B $: $$\begin{aligned}
\phi_1(z_0) &= A\phi_3(z_0) + B\phi_2(z_0) \\
\phi_1'(z_0) &= A\phi_3'(z_0) + B\phi_2'(z_0) \ .\end{aligned}$$ Solving these equations, we get the values of $ A,B $, and may then calculate the relation $ \frac{B}{A} $ that enters into the expressions for the retarded correlators (as explained in Subsection \[subsec:generalformulaeforcorrelators\]).
8. For the sake of better visual representation of the results, we define and calculate the “normalized” retarded correlators $ I(\lambda) $ in the following way: In all cases we may write the retarded correlator in the form: $$G^R(\lambda) = N(\lambda) \left[I(\lambda) + M(\lambda)\right] \ ,$$ where $ |I(\lambda)| = O(1) $ for $ |\lambda|\to\infty $, $ N(\lambda) $ is proportional to $ \lambda^s $ ($ s $ being the appropriate exponent for perturbation type) and $ M(\lambda) $ is proportional to $ \ln(a\lambda) $. For example, for the scalar perturbation mode (see Subsubsection \[subsubsec:cftasymptoticcorrelatorexpressionsscalar\]): If $ d $ is even- $$N(\lambda) = -2(d-1)C_s\frac{r_+^{d-1}}{R^d}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\lambda}
{2}\right)^{d-1}
\qquad
M(\lambda) = 0 \ .$$ If $ d $ is odd- $$N(\lambda) = 8\pi iC_s\frac{r_+^{d-1}}{R^d} \frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}\left(\frac{i\lambda}
{2}\right)^{d-1}
\qquad
M(\lambda) = -\frac{i}{\pi}\ln\left(\frac{i\lambda}{2}\right) \ .$$ Using the numerically calculated value of $ \frac{B}{A} $ and the values of $ N $ and $ M $ appropriate for each perturbation type we may extract a numerical value for $ I(\lambda) $.
Acknowledgements {#acknowledgements-1 .unnumbered}
================
The work is supported in part by the Israeli Science Foundation center of excellence.
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[^1]: In the spherical case, a Hawking-Page phase transition exists, which is interpreted in terms of the AdS/CFT correspondence as a transition between a higher temperature deconfined phase and a lower temperature confined phase (see [@Witten:1998qj], [@Witten:1998zw]). Phase transitions might exist also for the flat and hyperbolic cases, e.g. the transition between the black hole solution at a higher temperature and the Horowitz-Myers “AdS soliton” at a lower temperature (see [@Surya:2001vj]).
[^2]: We assume that $ d\ge 4 $.
[^3]: The hydrodynamic condition $ \frac{L_s^{FT}}{T} \to 0$ requires that $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \to 0 $ since the minimal eigenvalue $ L_{s,min}^{FT} $ is of the same order as $ \sqrt{|R_\Omega^{FT}|} $.
[^4]: In the context of this work we shall define the Laplace operator as: $ \Delta \equiv {\,\mathrm{\delta}}{\,\mathrm{d}}$ where $ {\,\mathrm{d}}$ and $ {\,\mathrm{\delta}}$ are the exterior derivative and codifferential operators respectively.
[^5]: The branch cuts are chosen here so that $ 0\leq\arg(z)<2\pi $.
[^6]: The choice of branch cuts so that $ 0\leq\arg(z)<2\pi $ makes sure that the contours don’t pass through a branch cut. Choosing $ 0<\arg(z)\leq 2\pi $ instead would give the complex conjugates of these contours.
[^7]: The assumed shape of the anti-Stokes lines is only true in the case of $ K>-1 $.
[^8]: Another assumption is that $ |\lambda| \gg |q|$, so that the effect of the effective potential can be neglected in the regions between the the “special” points ($ z\rightarrow0 $, $ z\rightarrow1 $, $ z\rightarrow\infty $)
[^9]: Obviously, $ -\lambda_n^* $ would also be a solution. Choosing the branch cuts so that $ 0<\arg(z)\leq 2\pi $ would give this solution instead.
[^10]: In dimensions $ d=4 $ and $ d=5 $, $ \operatorname{Re}{\tilde{z}}_0 \ge 0 $ below the temperature $ T_c $, so that the asymptotic slope of the QNM frequencies remains imaginary below this temperature as well.
[^11]: Here and in related formulae, $ \psi(z) $ represents the Digamma function, defined by : $ \psi(z) \equiv \frac{\Gamma'(z)}{\Gamma(z)}$.
[^12]: The terms $ \psi(1) $ and $ \psi(\Delta+1) $ have been dropped from $ B $ because after dividing by A they amount to a contact term.
[^13]: While in the general case, where $ \operatorname{Im}(\lambda{\tilde{z}}_0)\neq 0 $, the anti-Stokes lines emanating from $ {\tilde{z}}=0 $ and $ {\tilde{z}}_0 $ don’t align, the “matching” of the solutions in the region (2) is still possible since there are no Stokes lines between these two anti-Stokes lines in this region, so that the asymptotic solution remains the same between them.
[^14]: This property can also be deduced from the QNM equations themselves.
[^15]: Note that for the case of vector gauge field perturbations with $ d=4 $, the effective potential for both modes is $ V(z) = q^2 {\tilde{g}(z)}$, and has a pole of degree $ -\frac{3}{2} $ at $ {\tilde{z}}\to{\tilde{z}}_0 $. In this case the method applied here can’t be used since the equation is not of the Bessel type around the pole $ {\tilde{z}}_0 $, and since $ q $ obviously can’t be neglected (neglecting it results in an an expression with no poles). This case therefore requires a separate and more complete treatment, which shall not be investigated here. This section will therefore focus on the case of $ d\geq 5 $.
[^16]: See Appendix \[app:numericalmethods\] for a more precise definition.
[^17]: More precisely, $ \mathbf{A}_0 $ can be written as a sum of an exact form $ {\,\mathrm{d}}\phi $ and some harmonic form $ \mathbf{B} $. The harmonic component satisfies $ {\,\mathrm{\delta}}\mathbf{B} = 0 $, and can therefore be treated as a “transverse” mode.
[^18]: Since the exponents of the equation at $ w=1 $ are $0$ and $\frac{\alpha-1}{2}$, and since $ d\geq 4 $, near $w=1$ the solution $ \phi=o((1-w)^{-1}) $.
[^19]: This boundary condition should be interpreted at the first order of $ 1-z $ that doesn’t trivially vanish.
[^20]: $ z_0 $ should be chosen so that all three series solutions are well converged in the vicinity of $ z_0 $.
|
---
abstract: |
Let $m,n\ge 2$ be positive integers. Denote by $M_m$ the set of $m\times m$ complex matrices and by $w(X)$ the numerical radius of a square matrix $X$. Motivated by the study of operations on bipartite systems of quantum states, we show that a linear map $\phi: M_{mn}\rightarrow M_{mn}$ satisfies $$w(\phi(A\otimes B))=w(A\otimes B)~~{\rm for~ all}~ A\in M_{m} ~{\rm and}~B\in M_{n}$$ if and only if there is a unitary matrix $U \in M_{mn}$ and a complex unit $\xi$ such that $$\phi(A\otimes B) = \xi U(\varphi_1(A) \otimes \varphi_2(B))U^* \quad \hbox{ for all }
A\in M_{m} ~\hbox {\rm and}~B\in M_{n},$$ where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,2$, and the maps $\varphi_1$ and $\varphi_2$ will be of the same type if $m,n \ge 3$. In particular, if $m,n \ge 3$, the map corresponds to an evolution of a closed quantum system (under a fixed unitary operator), possibly followed by a transposition. The results are extended to multipartite systems.
address:
- 'Ajda Fošner, Faculty of Management, University of Primorska, Cankarjeva 5, SI-6104 Koper, Slovenia'
- 'Zejun Huang, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong'
- 'Chi-Kwong Li, Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA; Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong'
- ' Nung-Sing Sze, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong'
author:
- Ajda Fošner
- Zejun Huang
- 'Chi-Kwong Li'
- 'Nung-Sing Sze'
title: |
Linear maps preserving numerical radius\
of tensor products of matrices
---
1.1
[*2010 Math. Subj. Class.*]{}: 15A69, 15A86, 15A60, 47A12.
[*Key words*]{}: Complex matrix, linear preserver, numerical range, numerical radius, tensor product.
Introduction and preliminaries
==============================
Let $M_n$ be the set of $n\times n$ complex matrices for any positive integer $n$. For $A \in M_n$, define (and denote) its numerical range and numerical radius by $$W(A)=\bigg\{u^*Au : u\in {\mathbb C}^n, u^*u = 1\bigg\}
\quad \hbox{ and } \quad
w(A)=\sup \{|\mu|: \mu\in W(A)\},$$ respectively. The study of numerical range and numerical radius has a long history and is still under active research. Moreover, there are many generalizations motivated by pure and applied topics; see [@G; @Ha; @HJ].
By the convexity of the numerical range, $$W(A) = \{\tr(Auu^*): u \in \IC^n, u^*u = 1\} = \{\tr(AX): X \in D_n\},$$ where $D_n$ is the set of density matrices (positive semidefinite matrices with trace one) in $M_n$. In particular, in the study of quantum physics, if $A \in M_n$ is Hermitian corresponding to an observable and if quantum states are represented as density matrices, then $W(A)$ is the set of all possible measurements under the observables and $w(A)$ is a bound for the measurement. If $A = A_1+iA_2$, where $A_1, A_2 \in M_n$ are Hermitian, then $W(A)$ is the set of the joint measurement of quantum states under the two observables corresponding to $A_1$ and $A_2$.
Suppose $m,n\ge 2$ are positive integers. Denote by $ A\otimes B$ the tensor (Kronecker) product of the matrices $A\in M_m$ and $B\in M_n$. If $A$ and $B$ are observables of two quantum systems, then $A\otimes B$ is an observable of the composite bipartite system. Of course, a general observable on the composite system corresponds to $C \in M_{mn}$, and observable of the form $A\otimes B$ with $A\in M_m$, $B\in M_n$ is a very small (measure zero) set. Nevertheless, one may be able to extract useful information about the bipartite system by focusing on the set of tensor product matrices. In particular, in the study of linear operators $\phi: M_{mn}\rightarrow M_{mn}$ on bipartite systems, the structure of $\phi$ can be determined by studying $\phi(A\otimes B)$ with $A\in M_m$, $B\in M_n$; see [@FLPS; @FHLS; @FHLS2; @J] and their references.
In this paper, we determine the structure of linear maps $\phi: M_{mn} \rightarrow M_{mn}$ satisfying $w(A\otimes B) = w(\phi(A\otimes B))$ for all $A\in M_m$ and $B\in M_n$. We show that for such a map there is a unitary matrix $U \in M_{mn}$ and a complex unit $\xi$ such that $$\phi(A\otimes B) = \xi U(\varphi_1(A) \otimes \varphi_2(B))U^* \quad \hbox{ for all }
A\in M_{m} \ \hbox { and } B\in M_{n},$$ where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,2$, and the maps $\varphi_1$ and $\varphi_2$ will be of the same type if $m,n \ge 3$. In particular, if $m,n \ge 3$, the map corresponds to an evolution of a closed quantum system (under a fixed unitary operator), possibly followed by a transposition.
The study of linear maps on matrices or operators with some special properties are known as preserver problems; for example, see [@LP] and its references. In connection to preserver problems on bipartite quantum systems, it is quite common that if one considers a linear map $\phi:M_{mn} \rightarrow M_{mn}$ and imposes conditions on $\phi(A\otimes B)$ for $A\in M_m$, $B\in M_n$, then the partial transpose maps $A \otimes B \mapsto A \otimes B^t$ and $A\otimes B \mapsto A^t\otimes B$ are admissible preservers. It is interesting to note that for numerical radius preservers and numerical range preservers in our study, if $m,n \ge 3$, then the partial transpose maps are not allowed and that the (linear) numerical radius preserver $\phi$ on $M_{mn}$ will be of the standard form $$X \mapsto \xi V^*XV \qquad \hbox{ or } \qquad X \mapsto \xi V^*X^tV$$ for some complex unit $\xi$ and unitary $V \in M_{mn}$. This is the first example of such results in this line of study. It would be interesting to explore more matrix invariant or quantum properties that the structure of preservers on $M_{mn}$ can be completely determined by the behavior of the map on the small class of matrices of the form $A\otimes B \in M_{mn}$ with $A\in M_m$, $B\in M_n$.
In our study, we also determine the linear map $\phi: M_{mn} \rightarrow M_{mn}$ such that $$W(A\otimes B) = W(\phi(A\otimes B))\quad \hbox{ for all }
(A,B) \in M_m \times M_n.$$
We will denote by $X^t$ the transpose of a matrix $X\in M_n$ and $X^*$ the conjugate transpose of a matrix $X\in M_n$. The $n\times n$ identity matrix will be denoted by $I_n$. Let $E_{ij}^{(n)} \in M_n$ be the matrix whose $(i,j)$-entry is equal to one and all the others are equal to zero. We simply write $E_{ij} = E_{ij}^{(n)}$ if the size of the matrix is clear.
We will prove our main result on bipartite systems in Section 2 and extend the results to multi-partite systems in Section 3.
In [@FHLS], we considered linear maps $\phi$ preserving the spectrum $\sigma(A\otimes B)$ and spectral radius $r(A\otimes B)$ of Hermitian matrices $A \in M_m$ and $B \in M_n$, which are of the form $$\phi(A\otimes B) = U(\varphi_1(A) \otimes \varphi_2(B))U^*~{\rm for~ all}~ A \in H_m~{\rm and}~ H \in M_n,$$ where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,2$ and $U \in M_{mn}$ is unitary. In [@FHLS2], we characterized the linear maps $\phi$ on $M_{mn}$ preserving Ky Fan norms and Schatten norms of all matrices $A\otimes B$ with $A\in M_{m}$ and $B\in M_{n}$, which are of the form $$\phi(A\otimes B) = U(\varphi_1(A) \otimes \varphi_2(B))V~{\rm for~ all}~ A \in M_m~{\rm and}~ B \in M_n,$$ where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,2$ and $U,V\in M_{mn}$ are unitary.
In this paper, we characterize the linear map $\phi: M_{mn} \rightarrow M_{mn}$ with the property $$w(\phi(A\otimes B)) = w(A\otimes B)~~{\rm for~ all}~ A \in M_{m }~{\rm and}~B\in M_{n}.$$ It is shown that $\phi$ is of the form $$\phi(A\otimes B) = \xi U(\varphi_1(A) \otimes \varphi_2(B))U^* ~~{\rm for~ all}~ A \in M_{m }~{\rm and}~B\in M_{n}$$ where $U\in M_{mn}$ is unitary, $\xi$ is complex unit, $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,2$, and the maps $\varphi_1$ and $\varphi_2$ will be of the same type when $m,n \ge 3$. We also extend the result to multipartite systems. Note that it has been shown in [@L] that the linear map $ \psi: M_n\rightarrow M_n$ preserving the numerical radius of all matrices in $M_n$ is of the form $$\psi(A) = \xi UAU^* ~~{\rm or}~~\psi(A) = \xi UA^tU^*$$ with $U\in M_{n}$ being unitary and $\xi$ being complex unit.
Bipartite systems
=================
The following example is useful in our discussion.
*\[Ex1\] Suppose $m, n \ge 3$. Let $A = X \oplus O_{m-3}$ and $B = X \oplus O_{n-3}$ with $X=\begin{pmatrix}0 & 2 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr\end{pmatrix}$. Then $A \otimes B$ is unitarily similar to $$\begin{pmatrix}0 & 1 \cr 0 & 0 \end{pmatrix}
\oplus \begin{pmatrix}0 & 1 \cr 0 & 0 \end{pmatrix}
\oplus \begin{pmatrix}0 & 2 & 0 \cr 0 & 0 & 1/2 \cr 0 & 0 & 0 \cr\end{pmatrix}
\oplus O_{mn-7},$$ and $A \otimes B^t$ is unitarily similar to $$\begin{pmatrix}0 & 1/2 \cr 0 & 0 \end{pmatrix}
\oplus \begin{pmatrix}0 & 2 \cr 0 & 0 \end{pmatrix}
\oplus \begin{pmatrix}0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr\end{pmatrix}
\oplus O_{mn-7}.$$ One readily checks (see also [@LT]) that $W(A\otimes B)$ and $W(A\otimes B^t) = W(A^t\otimes B)$ are circular disks centered at the origin with radii $w(A\otimes B)$ and $w(A\otimes B^t)$, respectively. Moreover, we have $$\begin{aligned}
w(A\otimes B) &=&
\lambda_{\max} (A\otimes B + (A\otimes B)^t)/2 = \sqrt{4.25} \\
&>& 2.0000 = \lambda_{\max} (A\otimes B^t + (A\otimes B^t)^t)/2 = w(A\otimes B^t).\end{aligned}$$ -.2in*
In the following, we first determine the structure of linear preservers of numerical range using the above example and the results in [@FHLS].
In this section we will use the trace function $\tr:M_k\to \mathbb C$, i.e., $\tr (X)$ is the sum of the of a matrix $X$. Note that $\tr$ is a similarity-invariant linear functional.
\[T2\] The following are equivalent for a linear map $\phi: M_{mn} \rightarrow M_{mn}$.
1. $W(\phi(A\otimes B)) = W(A\otimes B)$ for any $A \in M_m$ and $B \in M_n$.
2. There is a unitary matrix $U \in M_{mn}$ such that $$\phi(A\otimes B) = U(\varphi_1(A) \otimes \varphi_2(B))U^*~ for~all~ A\in M_m ~and~ B\in M_n,$$ where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,2$, and the maps $\varphi_1$ and $\varphi_2$ will be of the same type if $m,n \ge 3$.
Suppose (b) holds. If $m, n \ge 3$, then the map has the form $C \mapsto UCU^*$ or $C \mapsto UC^tU^*$. Thus, the condition (a) holds. If $m = 2$, then $A^t$ and $A$ are unitarily similar for every $A \in M_2$. So, $W(A \otimes B) = W(A^t \otimes B)$ for any $B\in M_n$. Hence, the condition (a) holds. Similarly, if $n = 2$, then (a) holds.
Conversely, suppose that $W(\phi(A\otimes B)) = W(A\otimes B)$ for all $A \in M_m$ and $B \in M_n$. Assume for the moment that $A \otimes B\in M_{mn}$ is a Hermitian matrix. Then $$W(\phi(A\otimes B)) = W(A\otimes B) \subseteq \IR.$$ This yields that $\phi(A\otimes B)$ is a Hermitian matrix, as well. Thus, $\phi$ maps Hermitian matrices to Hermitian matrices and preserves numerical radius, which is equivalent to spectral radius for Hermitian matrices. By Theorem 3.3 in [@FHLS], we conclude that $\phi$ has the asserted form on Hermitian matrices and, hence, on all matrices in $M_{mn}$. However, if $m, n \ge 3$, then, by Example \[Ex1\], neither the map $A\otimes B \mapsto A\otimes B^t$ nor the map $A \otimes B \mapsto A^t\otimes B$ will preserve the numerical range. So, the last statement about $\varphi_1$ and $\varphi_2$ holds.
Next, we turn to linear preservers of the numerical radius. We need the following (well-known) lemma to prove our result. We include a short proof for the sake of the completeness.
\[le3\] Let $A\in M_n$ with $w(A)=|x^*Ax|=1$ for some unit $x\in \mathbf{C}^n$. Then for any unitary $U\in M_n$ with $x$ being its first column, there exists some $y\in \mathbf{C}^{n-1}$ such that $$\label{eq1}
U^*AU=x^*Ax\begin{pmatrix}1&y^*\\-y&* \end{pmatrix}.$$
Write $(x^*Ax)^{-1} A=G+iH$ with $G$ and $H$ Hermitian. Then the largest eigenvalue of $ G$ is 1 with $x$ as its corresponding eigenvector and $U^* GU =[1]\oplus G_1$ for some $G_1\in M_{n-1}$. Moreover, the (1,1)-entry of $iU^* HU$ is 0 since $w(A)=1$. Since $U^* HU$ is a Hermitian matrix we have $iU^* HU =\begin{pmatrix}0 & y^* \cr -y & * \cr\end{pmatrix}$. Thus, $U^*AU$ has the claimed form.
\[T3\] The following are equivalent for a linear map $\phi: M_{mn} \rightarrow M_{mn}$.
1. $w(\phi(A\otimes B)) = w(A\otimes B)$ for any $A \in M_m$ and $B \in M_n$.
2. There is a unitary matrix $U \in M_{mn}$ and a complex unit $\xi$ such that $$\phi(A\otimes B) = \xi U(\varphi_1(A) \otimes \varphi_2(B))U^*~ for~all~ A\in M_m ~and~ B\in M_n,$$ where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,2$, and the maps $\varphi_1$ and $\varphi_2$ will be of the same type if $m,n \ge 3$.
The implication (b) $\Rightarrow$ (a) can be verified readily. Now, suppose (a) holds and let $B_{ij} = \phi(E_{ii}\otimes E_{jj})$ for $1 \le i \le m, 1 \le j \le n$. According to the assumptions, for all $1 \le i \le m, 1 \le j \le n$, there is a unit vector $u_{ij}\in \mathbf{C}^{mn}$ and a complex unit $\xi_{ij}$ such that $u_{ij}^*B_{ij}u_{ij}=\xi_{ij}$. We will first show that there exists a unitary matrix $U\in M_{mn}$ and complex units $\xi_{ij}$ such that $$B_{ij}= \xi_{ij} U(E_{ii}\otimes E_{jj})U^*, \qquad 1 \le i \le m, 1 \le j \le n.$$ We divide the proof into several claims.
[**Claim 1.**]{} Suppose $(i,j) \ne (r,s)$. If $u \in \IC^{mn}$ is a unit vector such that $|u^*B_{ij}u| = 1$, then $B_{rs}u = 0$.
*Proof. Suppose $u$ is a unit vector such that $ u^*B_{ij}u = e^{i\theta}$. Let $\xi = u^*B_{rs}u$. We first consider the case if $i = r$. For any $\mu \in \IC$ with $|\mu| \le 1$, $$\label{ck-1}
1 = w(E_{ii} \otimes (E_{jj} + \mu E_{ss})) = w(B_{ij} + \mu B_{rs})
\ge |u^*(B_{ij} + \mu B_{rs})u|
= |e^{i\theta} + \mu \xi|.$$ Then we must have $\xi = 0$. Otherwise, $|e^{i\theta} + \mu \xi |
= |e^{i\theta} + e^{i\theta} |\xi|| > 1$ if one chooses $\mu = e^{i\theta} \bar \xi / |\xi|$. Suppose $\xi = 0$. Then the inequality in (\[ck-1\]) become equality and $w(B_{ij} + \mu B_{rs}) = |u^*(B_{ij} +\mu B_{rs})u| = 1$. By Lemma \[le3\], there is $y_\mu \in \IC^{mn-1}$ such that $$U^*B_{ij}U + \mu U^*B_{rs} U = U^*(B_{ij} + \mu B_{rs})U = e^{i\theta}
\begin{pmatrix} 1 & y_\mu^* \cr -y_\mu & * \end{pmatrix},$$ where $U$ is a unitary matrix with $u$ as its first column. Since the above equation holds for any $\mu \in \IC$ with $|\mu| \le 1$, the matrix $U^*B_{rs}U$ must have the form $\begin{pmatrix} 0 & 0 \cr 0 & * \end{pmatrix}$. So, $U^*B_{rs}U$ is a matrix with zeros in its first column and row, or equivalently, $B_{rs} u = 0$, as desired. Similarly, we can prove the case if $j = s$.*
Now we consider the case when $i\ne r$ and $j \ne s$. By the previous argument, we have $B_{is} u = B_{rj} u = 0$. Then for any $\mu \in \IC$ with $|\mu| \le 1$, $$\begin{aligned}
%\label{ck-2}
1 &=& w( (E_{ii} + E_{rr}) \otimes (E_{jj} + \mu E_{ss})) \cr
&=& w(B_{ij} + B_{rj} + \mu (B_{is} + B_{rs})) \cr
&\ge& |u^*(B_{ij} + B_{rj} + \mu (B_{is} + B_{rs}))u| \cr
&=& |u^*B_{ij}u + \mu u^*B_{rs}u|\cr
&=& |e^{i\theta} + \mu \xi|.\end{aligned}$$ It follows that $\xi = u^*B_{rs} u = 0$ and hence $w(B_{ij} + B_{rj} + \mu (B_{is} + B_{rs})) = |u^*B_{ij} u| = 1$. By Lemma \[le3\], we conclude that $(B_{is} + B_{rs})u = 0$ and thus, $B_{rs} u = 0$.
[**Claim 2.**]{} Suppose $(i,j) \ne (r,s)$. If $u_{ij}, u_{rs} \in \IC^{mn}$ are two unit vectors such that $|u_{ij}^*B_{ij} u_{ij}| = |u_{rs}^*B_{rs} u_{rs}| = 1$, then $u_{ij}^* u_{rs} = 0$.
*Proof. Suppose $u_{rs} = \alpha u_{ij} + \beta v$ with $\alpha = u_{ij}^* u_{rs}$ and $\beta = v^*u_{rs}$, where $v$ is a unit vector orthogonal to $u_{ij}$. Notice that $|\alpha|^2 + |\beta|^2 = 1$. By Claim 1, $u_{ij}^*B_{rs} = 0$ and $B_{rs} u_{ij} = 0$ and so $$1 = |u_{rs}^*B_{rs} u_{rs} | = |\beta|^2 |v^* B_{rs} v| \le |\beta|^2 w(B_{rs}) = |\beta|^2 \le 1.$$ Thus, $|\beta| = 1$ and hence $u_{ij}^* u_{rs} = \alpha = 0$.*
[**Claim 3.**]{} Let $U=[u_{11}~\cdots~u_{1n}~u_{21}\cdots~u_{2n}~\cdots~u_{m1}~\cdots~u_{mn}].$ Then $U^*U = I_{mn}$ and $U^*B_{ij}U = \xi_{ij} (E_{ii}\otimes E_{jj})$ for all $1\le i\le m$ and $1\le j\le n$.
*Proof. By Claim 2, $\{u_{ij}: 1\le i\le m, 1\le j\le n\}$ forms an orthonormal basis. Thus, $U^*U = I_{mn}$. Next by Claim 1, $u_{rs}^* B_{ij} u_{k\ell} = 0$ for all $(r,s)$ and $(k,\ell)$, except the case when $(r,s) = (k,\ell) = (i,j)$. Therefore, the result follows.*
[**Claim 1.**]{} Suppose $(i,j) \ne (r,s)$. If $u \in \IC^{mn}$ is a unit vector such that $|u^*B_{ij}u| = 1$, then $u^*B_{rs}u = 0$.
*Proof. Suppose $u$ is a unit vector such that $ u^*B_{ij}u = e^{i\theta}$. Let $\xi = u^*B_{rs}u$. Assume it is not zero, and assume that $\mu$ is a complex unit such that $\mu \xi = |\xi|$.*
First, suppose that $i=r$ or $j=s$. Without loss of generality, assume $(i,j) = (1,1)$ and $(r,s) = (1,2)$. Then $$|u^*(e^{-i\theta}B_{11} + \mu B_{12})u| = 1 + |\xi| > 1 = w(E_{11}\otimes (e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}B_{11} + \mu B_{12}),$$ which is a contradiction.
Now, suppose $i\ne r$ and $j\ne s$, say, $(i,j) = (1,1)$ and $(r,s) = (2,2)$. Then by the previous case, $u^*B_{12}u = 0 = u^*B_{21}u$, and hence $$|u^*(e^{-i\theta}(B_{11} + B_{21}) + \mu(B_{12} + B_{22}))u|
= 1 + |\xi| > 1$$ $$= w((E_{11}+E_{22})\otimes (e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}(B_{11} + B_{21}) + \mu(B_{12} + B_{22})),$$ which is a contradiction.
[**Claim 2.**]{} Suppose $(i,j) \ne (r,s)$.
[(a)]{} If $i=r$ or $j=s$, then $V = [u_{ij}~ u_{rs}]$ satisfies $V^*V = I_2$ and $$V^*B_{ij}V = \begin{pmatrix}\xi_{ij} & 0 \cr 0 & 0 \cr\end{pmatrix} \quad \hbox{and }
\quad V^*B_{rs}V = \begin{pmatrix}0 & 0 \cr 0 & \xi_{rs} \cr\end{pmatrix}.$$
[(b)]{} If $i\ne r$ and $j\ne s$, then $V = [u_{ij}~ u_{is}~ u_{rj} ~u_{rs}]$ satisfies $V^*V = I_4$ and $V^*B_{ij}V = \xi_{ij} E_{11} \otimes E_{11}, \
V^*B_{is}V = \xi_{is} E_{11} \otimes E_{22}, \
V^*B_{rj}V = \xi_{rj} E_{22} \otimes E_{11}, \
V^*B_{rs}V = \xi_{rs} E_{22} \otimes E_{22},$ where $E_{11}, E_{22} \in M_2$.
*Proof. We may assume that $(i,j) = (1,1)$.*
[(a)]{} Since $i=r$ or $j=s$, we may assume that $(r,s) = (1,2)$ or $(r,s) = (2,1)$. In the first case, let $V = [u_{ij}~ \tilde u_{rs}]$ be such that $\span\{u_{ij},\tilde u_{rs}\}=\span\{u_{ij}, u_{rs}\}$, $V^*V = I_2$ and $V^*B_{11}V = F$, $V^*B_{12}V = G$. Then $1 = w(F) = w(G)$ and $$\label{ck-1}
1 = w(B_{11} + \mu B_{12}) \ge w(F+\mu G) \ge |(F+\mu G)_{11}| = 1.$$ By Lemma \[le3\], $G_{11} = G_{12} = G_{21} = 0$. Since $w(G) = 1$, we have $|G_{22}| = 1$, and we may assume that $\tilde u_{rs} = u_{rs}$. Now, by (\[ck-1\]), we see that $0 = F_{12} = F_{21} = F_{22}$. The proof goes in the same way if $(r,s) = (2,1)$.
[(b)]{} Suppose $(r,s) = (2,2)$. Let $V = [u_{11}~ u_{12}~ \tilde u_{21}~ \tilde u_{22}]$, where $$\span\{u_{11},u_{12}, \tilde u_{21}\} = \span\{u_{11}, u_{12}, u_{21}\} \ \hbox{
and } \ \span\{u_{11},u_{12}, \tilde u_{21}, \tilde u_{22}\}
= \span\{u_{11}, u_{12}, u_{21}, u_{22}\}.$$ Set $V^*B_{11}V = F_{11}$, $V^*B_{12}V = F_{12}$, $V^*B_{21}V = F_{21}$, $V^*B_{22}V = F_{22}$. By Case 1, we see that the leading $2\times 2$ principal submatrix of $F_{11}$ is $\xi_{11} E_{11} \in M_2$ and that of $F_{12}$ is $\xi_{12} E_{22} \in M_2$.
Denote the $(p,q)$ entry of $F_{ij}$ by $F_{ij}(p,q)$ for $i,j\in\{1,2\}$ and $p,q\in \{1,2,3,4\}.$ By claim 1, $F_{ij}(p,p)\ne 0$ only if $p=2(i-1)+j$. Since $$1=w(B_{11}+\mu B_{21})\geq w(F_{11}+\mu F_{21})\geq |(F_{11}+\mu F_{21})_{11}|=1$$ for any unit complex number $\mu$, applying Lemma \[le3\], we get $F_{21}(1,q)=F_{21}(q,1)=0$ for $q=2,3,4$. Similarly, considering $B_{21}+\mu B_{22}$ we obtain $F_{22}(2,q)=F_{22}(q,2)=0$ for $q=2,3,4$. Next, considering $B_{11}+B_{12}+\mu(B_{21}+B_{22})$ we get $F_{21}=0_2\oplus G_{21}$ and $F_{22}=0_2\oplus G_{22}$. Be the fact $\span\{u_{11}, u_{12},\tilde u_{21}\} = \span\{u_{11}, u_{12}, u_{21}\}$, we have $G_{21}(1,1)=F_{21}(3,3)=1$. As in the proof of Claim 2, considering $w(\mu B_{21}+\xi B_{22})$ for all unit complex numbers $\mu$ and $\xi$, applying Lemma \[le3\] on $\mu G_{21}+\xi G_{22}$ we can get $F_{21}=\xi_{21}E_{22}\otimes E_{11}$ and $F_{22}=\xi_{22}E_{22}\otimes E_{22}$. On the other hand, considering $B_{11}+\mu B_{12}$ and $\mu (B_{11}+B_{12})+B_{21}+B_{22}$ we have $F_{11}=\xi_{11}E_{11}\otimes E_{11}$ and $F_{12}=\xi_{12}E_{11}\otimes E_{22}$. Therefore, $\tilde u_{21}, \tilde u_{22}$ may be assumed to be $u_{21}, u_{22}$, and $F_{ij} = \xi_{ij} E_{ii}\otimes E_{jj} \in M_2 \otimes M_2$ for $i,j \in \{ 1,2\}$.
[**Claim 3.**]{} Let $U=[u_{11}~\cdots~u_{1n}~u_{21}\cdots~u_{2n}~\cdots~u_{m1}~\cdots~u_{mn}].$ Then $U^*B_{ij}U = \xi_{ij} (E_{ii}\otimes E_{jj})$ for all $1\le i\le m$ and $1\le j\le n$.
*Proof. Consider $U^*B_{11}U$. Label the rows and columns of $X \in M_{mn}$ by $(p,q)$ with $1 \le p \le m$ and $1 \le q \le n$, i.e., we denote the row (column) index $(p-1)n+q$ by $(p,q)$. By Claim 2 (b), for any $(r,s)$ such that $r,s\ne 1$, the $4\times 4$ submatrix of $U^*B_{11}U$ with rows and column indices $(1,1), (1,s), (r,1), (r,s)$ has the form $\xi_{11} E_{11} \otimes E_{11} \in M_2 \otimes M_2$. One sees that $U^*B_{11}U = \xi_{11} E_{11} \otimes E_{11} \in M_m \otimes M_n$. Applying the same argument to $U^*B_{ij}U$ for any $(i,j)$ pair, the result follows. According to our assumptions and by Claims $1,2,3$, we see that up to some unitary similarity $$\phi(E_{ii}\otimes E_{jj})= \xi_{ij} (E_{ii}\otimes E_{jj})$$ for $1 \le i \le m, 1 \le j \le n$ and some complex units $\xi_{ij}$. Now, for any unitary $X \in M_m$, using the same arguments as above, there exists some unitary $U_X$ and some complex units $\mu_{ij}$ such that $$\phi(XE_{ii}X^* \otimes E _{jj}) = \mu_{ij} U_X(E_{ii} \otimes E _{jj})U_X^*$$ for all $1\le i \le m$ and $1\le j\le n$. So, $\phi(XE_{ii}X^*\otimes E_{jj})$ is a unit multiple of rank one Hermitian matrix with numerical radius one. Thus, $\phi(XE_{ii}X^*\otimes E_{jj}) = \mu_{ij} xx^*$ for some unit vector $x\in \IC^{mn}$. Note also that $\phi(I_m \otimes E_{jj}) = D \otimes E_{jj}$ for some diagonal unitary matrix $D$. If $\gamma > 0$, then $$w(\phi((XE_{ii}X^*+\gamma I_m)\otimes E_{jj})) = 1+\gamma.$$ Furthermore, there exists a unit vector $u\in \IC^{mn}$ such that $|u^*x| = 1$ and $|u^* (D\otimes E_{jj}) u| = 1$. From the second equality, $u$ must have the from $u = \hat u \otimes e_j$ for some unit vector $\hat u\in \IC^m$. Therefore, $|u^*x| = 1$ implies $x = \hat x \otimes e_j$ for some unit vector $\hat x \in \IC^m$. Thus, $\phi(XE_{ii}X^*\otimes E_{jj})$ has the form $R_{i,X} \otimes E_{jj}$ for some $R_{i,X} \in M_m$. Since this is true for any $1\le i \le m$ and unitary $X\in M_m$, we have $$\phi(A\otimes E_{jj}) = \varphi_j(A)\otimes E_{jj}$$ for all matrices $A\in M_m$ and some linear map $\varphi_j$. Clearly, $\varphi_j$ preserves numerical radius and, hence, has the form $$A \mapsto \xi_j W_jAW_j^* \quad \hbox{ or } \quad A \mapsto \xi_j W_jA^tW_j^*$$ for some complex unit $\xi_j$ and unitary $W_j \in M_m$. In particular, $\varphi_j(I_m)=\xi_j I_m$ and $\phi(I_{mn}) = I_m\otimes D$ for some diagonal matrix $D\in M_n$. Using the same arguments as above, we can show that $$\phi(E_{ii}\otimes B) = E_{ii}\otimes \varphi_i(B)$$ for all matrices $B\in M_n$ and some linear map $\varphi_i$ of the form $$B \mapsto \tilde{\xi_i} \tilde{W_i}B\tilde{W_i}^* \quad \hbox{ or } \quad B \mapsto \tilde{\xi_i}\tilde{W_i}B^t\tilde{W_i}^*,$$ where $\tilde{\xi_i}$ is a complex unit and $\tilde{W_i} \in M_n$ a unitary matrix. Therefore, we have $\varphi_i(I_n)=\tilde{\xi_i} I_n$ and $\phi(I_{mn}) = \tilde{D}\otimes I_n$ for some diagonal matrix $\tilde{D}\in M_m$. Since $\phi(I_{mn}) = \tilde D\otimes I_n =I_m\otimes D$, we conclude that $\phi(I_{mn})=\xi I_{mn}$ for some complex unit $\xi$. For the sake of the simplicity, let us assume that $\phi(I_{mn})= I_{mn}$. Then $\phi(E_{ii} \otimes E_{jj}) = E_{ii} \otimes E_{jj}$ for all $1\le i \le m$, $1\le j \le n$.*
For any Hermitian matrices $A\in M_m$ and $B\in M_n$, suppose their spectral decompositions are $A=XD_1X^*$ and $B=YD_2Y^*$. Repeating the above argument and using the assumption $\phi(I_{mn})= I_{mn}$, one sees that there exists a unitary matrix $U_{X,Y}$ such that $$\phi(XE_{ii}X\otimes YE_{jj}Y^*)=U_{X,Y}(XE_{ii}X^*\otimes YE_{jj}Y^*)U_{X,Y}^*,\qquad 1\le i \le m, 1\le j \le n,$$ and, hence, $\phi(A\otimes B)=U_{X,Y}(A\otimes B)U_{X,Y}^*$. So, $\phi$ maps Hermitian matrices to Hermitian matrices and preserves numerical range on the tensor product of Hermitian matrices. Thus, by the same argument as in the proof of Theorem \[T2\], $\phi$ has the asserted form on Hermitian matrices and, hence, on all matrices in $M_{mn}$. If $m, n \ge 3$, we can use Example \[Ex1\] to conclude that $\varphi_1$ and $\varphi_2$ should both be the identity map, or both be the transpose map. The proof is completed.
[**Old proof.**]{}
Consider first the matrix $\phi(E_{11}\otimes E_{11})$. According to the assumptions, there is a unit vector $x\in \mathbf{C}^{mn}$ and a real number $\theta_1\in [0,2\pi)$ such that $x^*\phi(E_{11}\otimes E_{11})x=e^{i\theta_1}$. Let $U_1$ be a unitary matrix with $x$ being its first column. By Lemma \[le3\], $$U_1\phi(E_{11}\otimes E_{11})U_1^* = e^{i\theta_1}\begin{pmatrix}1 & x_1^* \cr - x_1 & * \cr\end{pmatrix}.$$ Now, for any scalar $\gamma$ with $|\gamma| \le 1$, we have $$w(\phi(E_{11}\otimes E_{11}) + \gamma \phi(E_{11} \otimes E_{22})) = 1.$$ It follows that $x^*\phi(E_{11}\otimes E_{22})x=0$ and, by Lemma \[le3\], we have $U_1 \phi(E_{11}\otimes E_{22})U_1^* = 0 \oplus B$ with $B\in M_{mn-1}$ and $w(B)=1$. Again, there is a unitary $U_2\in M_{mn-1}$ and a complex number $e^{i\theta_2}$ such that $$U_2BU_2^* = e^{i\theta_2}\begin{pmatrix}1 & x_2^* \cr - x_2 & * \cr\end{pmatrix}.$$ Moreover, if we replace $U_1$ with $(1\oplus U_2)U_1$, the (1,2)-entry and (2,1)-entry of $U_1\phi(E_{11}\otimes E_{11})U_1^*$ are equal to zero. Repeating the same argument for $\phi(E_{11}\otimes E_{jj})$ with $j = 2, \dots, n$, we see that there exists some unitary $U_1\in M_{mn}$ such that $$U_1 \phi(E_{11} \otimes E_{jj})U_1^* = \begin{pmatrix} e^{i\theta_j} E_{jj} & * \cr * & * \cr \end{pmatrix}$$ with $E_{jj}\in M_n.$ Replacing $\phi$ by the map $A\otimes B \mapsto U_1^*\phi(A\otimes B)U_1$, if necessary, we may assume that $U_1 = I_{mn}$.
For any diagonal unitary $D\in M_n$ and any scalar $\gamma$ with $|\gamma| \le 1$ we have $$w(\phi(E_{11} \otimes D) + \gamma \phi(E_{22} \otimes D)) = 1.$$ It follows that $\phi(E_{22} \otimes D)= 0_n \oplus Y$ for some $Y\in M_{mn-n}$, where $0_n$ denotes the $n\times n$ zero matrix. Furthermore, using the previous arguments, there is unitary $U_2 \in M_{mn-n}$ and a complex number $\mu_{2j}$ such that $$(I_n \oplus U_2) \phi(E_{22} \otimes E_{jj})(I_n\oplus U_2)^*= 0_n \oplus \begin{pmatrix} \mu_{2j} E_{jj} & * \cr * & * \cr
\end{pmatrix}$$ with $j=1,\ldots n$. Moreover, if $(I_n \oplus U_2) \phi(E_{11} \otimes E_{jj})(I_n\oplus U_2)^* = (Y_{rs})_{1 \le r,s \le m}$ such that $Y_{rs} \in M_n$, then $Y_{12} = 0_n = Y_{21}$. Again, we can modify $\phi$ and assume that $U_2 = I_{mn-n}$. Continuing with the same procedure, we see that, up to some unitary similarity, we have $$\phi(E_{ii} \otimes E_{jj}) = \xi_{ij} (E_{ii} \otimes E_{jj})$$ for all $1\le i\le m$, $1\le j\le n$, and some complex units $\xi_{ij}$.
For any unitary $X \in M_m$, using the same arguments as above we see that $\phi(XE_{ii}X^*\otimes E_{jj})$ is a rank one matrix with numerical radius one. If $\gamma > 0$, then $$w(\phi((XE_{ii}X^*+\gamma I_m)\otimes E_{jj})) = 1+\gamma.$$ Since this is true for any unitary $X\in M_m$, we have $$\phi(A\otimes E_{jj}) = \varphi_j(A)\otimes E_{jj}$$ for all Hermitian matrices $A\in M_m$ and some linear map $\varphi_j$. Clearly, $\varphi_j$ preserves numerical radius and, hence, has the form $$A \mapsto \xi_j W_jAW_j^* \quad \hbox{ or } \quad A \mapsto \xi_j W_jA^tW_j^*$$ for some complex unit $\xi_j$ and unitary $W_j \in M_m$. In particular, $\varphi_j(I_m)=\xi_j I_m$ and $\phi(I_{mn}) = I_m\otimes D$ for some diagonal matrix $D\in M_n$. Using the same arguments as above, we can show that $$\phi(E_{ii}\otimes B) = E_{ii}\otimes \varphi_i(B)$$ for all Hermitian matrices $B\in M_n$ and some linear map $\varphi_i$ of the form $$B \mapsto \tilde{\xi_i} \tilde{W_i}B\tilde{W_i}^* \quad \hbox{ or } \quad B \mapsto \tilde{\xi_i}\tilde{W_i}B^t\tilde{W_i}^*,$$ where $\tilde{\xi_i}$ is a complex unit and $\tilde{W_i} \in M_n$ a unitary matrix. Therefore, we have $\varphi_i(I_n)=\tilde{\xi_i} I_n$ and $\phi(I_{mn}) = \tilde{D}\otimes I_n$ for some diagonal matrix $\tilde{D}\in M_m$. Since $\phi(I_{mn}) = \tilde D\otimes I_n =I_m\otimes D$, we conclude that $\phi(I_{mn})=\xi I_{mn}$ for some complex unit $\xi$. For the sake of the simplicity, let us assume that $\phi(I_{mn})= I_{mn}$. Then $\phi(E_{ii} \otimes E_{jj}) = E_{ii} \otimes E_{jj}$ for all $1\le i \le m$, $1\le j \le n$.
For any Hermitian matrices $A\in M_m$ and $B\in M_n$, suppose their spectral decompositions are $A=XD_1X^*$ and $B=YD_2Y^*$. Repeating the above argument and using the assumption $\phi(I_{mn})= I_{mn}$, we can conclude that there exists a unitary matrix $U_{X,Y}$ such that $$\phi(XE_{ii}X\otimes YE_{jj}Y^*)=U_{X,Y}(XE_{ii}X^*\otimes YE_{jj}Y^*)U_{X,Y}^*\quad {\rm for~all}~1\le i \le m, 1\le j \le n,$$ and hence $\phi(A\otimes B)=U_{X,Y}(A\otimes B)U_{X,Y}^*$. So, $\phi$ maps Hermitian matrices to Hermitian matrices and preserves numerical range on the tensor product of Hermitian matrices. Thus, by Theorem \[T2\], it has the asserted form on Hermitian matrices and, hence, on all matrices in $M_{mn}$. If $m, n \ge 3$, we can use Example \[Ex1\] to conclude that $\varphi_1$ and $\varphi_2$ should both be the identity map, or both be the transpose map.
The converse can be verified readily. The proof is completed.
Multipartite systems
====================
In this section we extend Theorem \[T2\] and Theorem \[T3\] to multipartite systems $\M$, $m\ge 2$.
\[T5\] Let $n_1, \dots, n_m \ge 2$ be positive integers and $N = \prod_{j=1}^m n_j$. The following are equivalent for a linear map $\phi: M_N \rightarrow M_N$.
1. $W(\phi(\A)) = W(\A)$ for any $(A_1, \dots , A_m) \in M_{n_1} \times \cdots \times M_{n_m}$.
2. There is a unitary matrix $U \in M_{N}$ such that $$\label{eq2}
\phi(\A) = U(\varphi_1(A_1)\otimes \cdots\otimes\varphi_m(A_m))U^*$$ for all $(A_1, \dots , A_m) \in M_{n_1} \times \cdots \times M_{n_m}$, where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,\ldots,m$, and the maps $\varphi_j$ are of the same type for those $j$’s such that $n_j\geq 3$.
The sufficient part is clear. For the converse, as in the proof of Theorem \[T2\], consider Hermitian matrix $A = A_1 \otimes \cdots \otimes A_m $ with $A_j \in H_{n_j}$ for $j = 1, \dots, m$. By [@FHLS Theorem 3.4], $\phi$ has the asserted form on Hermitian matrices and, hence, on all matrices in $\M$.
However, if $n_i, n_j \ge 3$ with $i<j$, let $A_i = X \oplus O_{n_i-3}$ and $A_j = X \oplus O_{n_j-3}$, where $X$ is defined as in Example \[Ex1\], and $A_k=E_{11}\in M_{n_k}$ for $k\ne i,j$. Then $w(\A) = \sqrt{4.25}$ and $w(A_1\otimes\cdots\otimes A_{j-1}\otimes A_j^t\otimes A_{j+1}\otimes\cdots\otimes A_m)
= 2.0000$. Thus, $$W(\A) \ne W(A_1\otimes\cdots\otimes A_{j-1}\otimes A_j^t\otimes A_{j+1}\otimes\cdots\otimes A_m),$$ and we see that the last statement about $\varphi_k$ holds.
\[T6\] Let $n_1, \dots, n_m \ge 2$ be positive integers and $N = \prod_{j=1}^m n_j$. The following are equivalent for a linear map $\phi: M_N \rightarrow M_N$.
1. $w(\phi(\A)) = w(\A)$ for any $(A_1, \dots , A_m) \in M_{n_1} \times \cdots \times M_{n_m}$.
2. There is a unitary matrix $U \in M_N$ and a complex unit $\xi$ such that $$\label{eq2b}
\phi(\A) =\xi U(\varphi_1(A_1)\otimes \cdots\otimes\varphi_m(A_m))U^*$$ for all $(A_1, \dots , A_m) \in M_{n_1} \times \cdots \times M_{n_m}$, where $\varphi_k$ is the identity map or the transposition map $X \mapsto X^t$ for $k=1,\ldots,m$, and the maps $\varphi_j$ are of the same type for those $j$’s such that $n_j\geq 3$.
The sufficiency part is clear. We verify the necessity. First of all, one can use similar arguments as in the proof of Theorem \[T3\] (see Claim 1, Claim 2, and Claim 3) to show that up to some unitary similarity, $$\label{E}
\phi(E_{i_1i_1} \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m}) =
\xi_{i_1i_2\cdots i_m} (E_{i_1i_1} \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m})$$ for all $1\le i_k\le n_k$ with $1\le k\le m$ and some complex units $\xi_{i_1i_2\cdots i_m}$. Below we give the details of the proof for the case $M_{n_1} \otimes M_{n_2} \otimes M_{n_3}$. One readily extends the arguments to the general case.
Denote $B_{ijk} = \phi(E_{ii} \otimes E_{jj} \otimes E_{kk})$ for $1 \le i \le n_1, 1 \le j \le n_2, 1 \le k \le n_3$ and let $N=n_1n_2n_3$. According to the assumptions, for any $1 \le i \le n_1, 1 \le j \le n_2, 1 \le k \le n_3$, there is a unit vector $u_{ijk}\in \mathbf{C}^N$ and a complex unit $\xi_{ijk}$ such that $u_{ijk}^*B_{ijk}u_{ijk}=\xi_{ijk}$.
[**Claim 4.**]{} Suppose $(i,j,k) \ne (r,s,t)$. If $u \in \IC^N$ is a unit vector such that $|u^*B_{ijk}u| = 1$, then $B_{rst}u = 0$.
*Proof. Suppose $u$ is a unit vector such that $ u^*B_{ijk}u = e^{i\theta}$. Let $\xi = u^*B_{rst}u$. First, assume that $\delta_{ir}+\delta_{js}+\delta_{kt}=2$, where $\delta_{ab}$ equals to 1 when $a=b$ and zero otherwise. In other words, we assume that exactly two of the three sets $\{i,r\}, \{j,s\}, \{k,t\}$ are singletons. Without loss of generality, assume that $i = r$, $j = s$, and $k \ne t$. For any $\mu \in \IC$ with $|\mu| \le 1$, $$1 = w(B_{ijk} + \mu B_{rst})
\ge |u^*(B_{ijk} + \mu B_{rst})u|
= |e^{i\theta} + \mu \xi|.$$ Then we must have $\xi = 0$. Furthermore, $w(B_{ijk} + \mu B_{rst}) = |u^*(B_{ijk} +\mu B_{rst})u| = 1$. By Lemma \[le3\], one conclude that $U^*B_{rst}U$ has the form $\begin{pmatrix} 0& 0 \cr 0 & * \end{pmatrix}$, where $U$ is a unitary matrix with $u$ as its first column, and hence $B_{rst} u = 0$.*
Next, suppose $\delta_{ir}+\delta_{js}+\delta_{kt}=1$, say $i = r$. By the previous case, $B_{isk} u = B_{ijt} u = 0$. Then for any $\mu \in \IC$ with $|\mu| \le 1$, $$\begin{aligned}
%\label{ck-2}
1 &=& w(B_{ijk} + B_{isk} + \mu (B_{ijt} + B_{ist})) \cr
&\ge& |u^*(B_{ijk} + B_{isk} + \mu (B_{ijt} + B_{ist}))u| \cr
&=& |u^*B_{ijk}u + \mu u^*B_{ist}u|
= |e^{i\theta} + \mu \xi|.\end{aligned}$$ It follows that $\xi = u^*B_{ist} u = 0$ and hence $w(B_{ijk} + B_{isk} + \mu (B_{ijt} + B_{ist})) = |u^*B_{ijk} u| = 1$. By Lemma \[le3\], we conclude that $(B_{ijt} + B_{ist})u = 0$ and thus, $B_{ist} u = 0$.
Finally, suppose $\delta_{ir}+\delta_{js}+\delta_{kt}=0$. By the previous cases, $B_{isk} u = B_{rjk} u = B_{rsk} u = B_{ijt} u = B_{ist} u = B_{rjt} u = 0$. Then for any $\mu \in \IC$ with $|\mu| \le 1$, $$\begin{aligned}
%\label{ck-2}
1 &=& w((B_{ii}+B_{rr})\otimes (B_{jj}+B_{ss})\otimes (B_{kk}+\mu B_{tt})) \cr
&\ge& |u^*((B_{ii}+B_{rr})\otimes (B_{jj}+B_{ss})\otimes (B_{kk}+\mu B_{tt}))u| \cr
&=& |u^*B_{ijk}u + \mu u^*B_{rst}u|
= |e^{i\theta} + \mu \xi|.\end{aligned}$$ Then by a similar argument, one conclude that $\xi = 0$ and $B_{rst} u = 0$.
[**Claim 5.**]{} Suppose $(i,j,k) \ne (r,s,t)$. If $u_{ijk}, u_{rst} \in \IC^{mn}$ are two unit vectors such that $|u_{ijk}^*B_{ijk} u_{ijk}| = |u_{rst}^*B_{rst} u_{rst}| = 1$, then $u_{ijk}^* u_{rst} = 0$.
*Proof. Suppose $u_{rst} = \alpha u_{ijk} + \beta v$ with $\alpha = u_{ijk}^* u_{rst}$ and $\beta = v^*u_{rst}$, where $v$ is a unit vector orthogonal to $u_{ijk}$. Note that $|\alpha|^2 + |\beta|^2 = 1$. By Claim 1, $u_{ijk}^*B_{rst} = 0$ and $B_{rst} u_{ijk} = 0$ and so $$1 = |u_{rst}^*B_{rst} u_{rst} |
= |\beta|^2 |v^* B_{rst} v| \le |\beta|^2 w(B_{rst}) = |\beta|^2 \le 1.$$ Thus, $|\beta| = 1$ and hence $u_{ijk}^* u_{rst} = \alpha = 0$.*
[**Claim 6.**]{} Let $$U=[u_{111}~\cdots~u_{11n_3}~u_{121}~\cdots~u_{12n_3}~\cdots~u_{1n_2n_3}
~\cdots~u_{211}~\cdots~u_{2n_2n_3}~\cdots~u_{n_111}~\cdots~u_{n_1n_2n_3}].$$ Then $U^*U = I_N$ and $U^*B_{ijk}U = \xi_{ijk} (E_{ii}\otimes E_{jj}\otimes E_{kk})$ for all $1\le i\le n_1, 1\le j\le n_2$, $1\le k\le n_3$.
*Proof. By Claim 2, $\{u_{ijk}: 1\le i\le n_1, 1\le j\le n_2, 1\le k \le n_3\}$ forms an orthonormal basis and thus $U^*U = I_{N}$. Next by Claim 1, $u_{rst}^* B_{ijk} u_{ \ell pq} = 0$ for all $(r,s,t)$ and $( \ell, p,q)$, except the case when $(r,s,t) = ( \ell, p,q) = (i,j,k)$. Therefore, the result follows.*
[**Claim 4.**]{} If $(i,j,k) \ne (r,s,t)$ and if $u \in \IC^N$ is a unit vector such that $|u^*B_{ijk}u| = 1$, then $u^*B_{rst}u = 0$.
*Proof. Let $\theta$ be a real number such that $ u^*B_{ijk}u = e^{i\theta}$ and let $\xi = u^*B_{rst}u$. Suppose that $\xi\ne 0$ and let $\mu$ be a complex unit such that $\mu \xi = |\xi|$.*
First, assume that $\delta_{ir}+\delta_{js}+\delta_{kt}=2$, where $\delta_{uv}$ equals to 1 when $u=v$ and zero otherwise. Without loss of generality, assume $(i,j,k) = (1,1,1)$ and $(r,s,t) = (1,1,2)$. Then $$|u^*(e^{-i\theta}B_{111} + \mu B_{112})u| = 1 + |\xi| > 1 = w(E_{11}\otimes E_{11}\otimes(e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}B_{111} + \mu B_{112}),$$ which is a contradiction.
Next, suppose $\delta_{ir}+\delta_{js}+\delta_{kt} = 1$, say, $(i,j,k) = (1,1,1)$ and $(r,s,t) = (1,2,2)$. Then by the previous case $u^*B_{112}u = 0 = u^*B_{121}u$, and, hence, $$|u^*(e^{-i\theta}(B_{111} + B_{121}) + \mu(B_{112} + B_{122}))u|
= 1 + |\xi| > 1$$ $$= w(E_{11}\otimes(E_{11}+E_{22})\otimes (e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}(B_{111} + B_{121}) + \mu(B_{112} + B_{112})),$$ which is a contradiction.
Finally, suppose $\delta_{ir}+\delta_{js}+\delta_{kt}= 0$, say, $(i,j,k) = (1,1,1)$ and $(r,s,t) = (2,2,2)$. By the previous cases $u^*B_{121}u = u^*B_{211}u= u^*B_{221}u= u^*B_{112}u = u^*B_{122}u= u^*B_{212}u=0$, and, hence, $$\begin{aligned}
&&|u^*(e^{-i\theta}(B_{111} +B_{121}+B_{211}+ B_{221}) + \mu(B_{112} + B_{122}+B_{212}+B_{222}))u|\\
&=& 1 + |\xi| \\
&>& 1 = w((E_{11}+E_{22})\otimes(E_{11}+E_{22})\otimes (e^{-i\theta}E_{11} + \mu E_{22}))\\
&=& w(e^{-i\theta}(B_{111} +B_{121}+B_{211}+ B_{221}) + \mu(B_{111} +B_{121}+B_{211}+ B_{221})),\end{aligned}$$ which is a contradiction.
[**Claim 5.**]{} Suppose $1\leq r\ne i\leq n_1$, $1\le s\ne j\le n_2$, $1\le t\ne k\le n_3$. Then $V = [u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~u_{rjk}~ u_{rjt}~ u_{rsk} ~u_{rst}]$ satisfies $V^*V = I_8$ and $$\begin{aligned}
\label{eq111}
\begin{array}{c}
V^*B_{ijk}V = \mu_{ijk} E_{11} \otimes E_{11}\otimes E_{11}, \quad
V^*B_{ijt}V = \mu_{ijt} E_{11} \otimes E_{11}\otimes E_{22}, \\
V^*B_{isk}V = \mu_{isk} E_{11} \otimes E_{22}\otimes E_{11}, \quad
V^*B_{ist}V = \mu_{ist} E_{11} \otimes E_{22}\otimes E_{22},\\
V^*B_{rjk}V = \mu_{rjk} E_{22} \otimes E_{11}\otimes E_{11}, \quad
V^*B_{rjt}V = \mu_{rjt} E_{22} \otimes E_{11}\otimes E_{22}, \\
V^*B_{rsk}V = \mu_{rsk} E_{22} \otimes E_{22}\otimes E_{11}, \quad
V^*B_{rst}V = \mu_{rst} E_{22} \otimes E_{22}\otimes E_{22},
\end{array}\end{aligned}$$ where $E_{11}, E_{22} \in M_2$.
*Proof. We may assume $(i,j,k) = (1,1,1)$ and $(r,s,t)=(2,2,2)$. Let $$U=[u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt}~ \tilde u_{rsk} ~\tilde u_{rst}]$$ with $$\begin{aligned}
&&\span\{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}\}=\span\{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}\},\label{eq121}\\
&&\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt} \}=\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}~ u_{rjt} \},\label{eq122}\\
&&\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt}~ \tilde u_{rsk} \}=\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}~ u_{rjt}~ u_{rsk} \},\label{eq123}\\
&&\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt}~ \tilde u_{rsk} ~\tilde u_{rst}\}=\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}~ u_{rjt}~ u_{rsk} ~ u_{rst}\}\label{eq124}.\end{aligned}$$*
First, applying Claim 4 and using the same arguments as in the proof of Claim 2 with replacing the role of $B_{ij}$ by $B_{1ij}$, we can get that the leading $4\times 4$ principal submatrix of $U^*B_{1jk}U$ is $\xi_{1jk} E_{jj}\otimes E_{kk}$ for $j,k\in\{1,2\}$.
Next, applying Lemma \[le3\] on $U^*(B_{1jk}+\mu B_{2jk})U$, $U^*((B_{1j1}+B_{1j2})+\mu (B_{2j1}+B_{2j2}))U$ with $j,k\in\{1,2\}$ and $U^*((B_{111}+B_{112}+B_{121}+B_{122})+\mu (B_{111}+B_{112}+B_{121}+B_{122}))U$ successively, we obtain that $U^*B_{2jk}U=0_4\oplus X_{jk}$ for all $j,k\in\{1,2\}$. As in the proof of Claim 2, by (\[eq121\]) and $w(B_{211})=1$, we have $|X_{11}(1,1)|=1$. Now repeating the same arguments as in the proof of Claim 2 with replacing the role of $B_{ij}$ by $B_{2ij}$, we get $X_{jk}=\mu_{jk}E_{jj}\otimes E_{kk}$ with $|\mu_{jk}|=1$ for all $j,k\in\{1,2\}$.
Finally, applying Lemma \[le3\] on $U^*(\mu B_{1jk}+ B_{2jk})U$, $U^*(\mu (B_{1j1}+B_{1j2})+ (B_{2j1}+B_{2j2}))U$ with $j,k\in\{1,2\}$ and $U^*(\mu (B_{111}+B_{112}+B_{121}+B_{122})+(B_{111}+B_{112}+B_{121}+B_{122}))U$ successively, we obtain that $U^*B_{1jk}U=\xi_{1jk}E_{11}\otimes E_{jj}\otimes E_{kk}$ for all $j,k\in\{1,2\}$. By (\[eq121\])-(\[eq124\]), we can assume $U=V$ and (\[eq111\]) holds.
[**Claim 6.**]{} Let $$U=[u_{111}~\cdots~u_{11n_3}~u_{121}~\cdots~u_{12n_3}~\cdots~u_{1n_2n_3}
~\cdots~u_{211}~\cdots~u_{2n_2n_3}~\cdots~u_{n_111}~\cdots~u_{n_1n_2n_3}].$$ Then $U^*B_{ijk}U = \xi_{ijk} (E_{ii}\otimes E_{jj}\otimes E_{kk})$ for all $1\le i\le n_1, 1\le j\le n_2$, $1\le k\le n_3$.
*Proof. Consider $U^*B_{111}U$. Denote the row (column) index $(r-1)n_2n_3+(s-1)n_3+t$ by $(r,s,t)$, where $1 \le r \le n_1$, $1 \le s \le n_2$ and $1\le t\le n_3$. By Claim 5, for any $(r,s,t)$ such that $r\ne 1,s\ne 1,t\ne 1$, the $8\times 8$ submatrix of $U^*B_{111}U$ with rows and column indices $$\{(1,1,1), (1,1,t), (1,s,1), (1,s,t), (r,1,1),(r,1,t),(r,s,1),(r,s,t)\},$$ has the form $\xi_{111} (E_{11} \otimes E_{11}\otimes E_{11}) \in M_2 \otimes M_2\otimes M_2$. One sees that $U^*B_{111}U = \xi_{111} (E_{11}\otimes E_{11} \otimes E_{11}) \in M_{n_1} \otimes M_{n_2}\otimes M_{n_3}$. Applying the same argument to $U^*B_{ijk}U$ for any triple $(i,j,k)$, we get the conclusion.*
Similarly, for any unitary $X\in M_{n_1}$, there exists some unitary $U_X$ and some complex units $\mu_{i_1i_2\cdots i_m}$ such that $$\phi(XE_{i_1i_1}X^* \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m}) = \mu_{i_1i_2\cdots i_m} U_X(XE_{i_1i_1}X^* \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m})U_X^*$$ for all $1\le i_k\le n_k$ with $1\le k\le m$. We see that $\phi(XE_{i_1i_1}X^* \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m})$ is a rank one matrix with numerical radius one. If $\gamma > 0$, then $$w(\phi((XE_{i_1i_1}X^*+\gamma I_{n_1})\otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m})) = 1+\gamma.$$ Thus, $\phi(XE_{i_1i_1}X^* \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m})$ has the form $R \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m})$ for some $R \in M_{n_1}$. Since this is true for any unitary $X\in M_{n_1}$, we have $$\phi(A\otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m}) = \varphi_{ i_2\cdots i_m}(A)\otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m}$$ for all Hermitian matrices $A\in M_{n_1}$ and some linear map $\varphi_{ i_2\cdots i_m}$. Clearly, $ \varphi_{ i_2\cdots i_m}$ preserves numerical radius and, hence, has the form $$A \mapsto \xi_{ i_2\cdots i_m} W_{ i_2\cdots i_m}AW_{ i_2\cdots i_m}^* \quad \hbox{ or } \quad A \mapsto \xi_{ i_2\cdots i_m} W_{ i_2\cdots i_m}A^tW_{ i_2\cdots i_m}^*$$ for some complex unit $\xi_{ i_2\cdots i_m}$ and unitary $W_{ i_2\cdots i_m} \in M_{n_1}$. In particular, $\varphi_{ i_2\cdots i_m}(I_{n_1})=\xi_ { i_2\cdots i_m} I_{n_1}$ and $\phi(I_N) = I_{n_1}\otimes D_1$ for some diagonal matrix $D_1\in M_{n_2\cdots n_m}$.
Given $2\leq k\leq m$ and using the same arguments as above, one can show that $\phi(I_N) = D_{k1}\otimes I_{n_k}\otimes D_{k2}$ for some diagonal matrix $D_{k1}\in M_{n_1\cdots n_{k-1}}$ and $D_{k2}\in M_{n_{k+1}\cdots n_m}$. Since $$\phi(I_N) = I_{n_1}\otimes D_1 =D_{k1}\otimes I_{n_k}\otimes D_{k2}~{\rm for}~k=2,\ldots,m,$$ we conclude that $\phi(I_N)=\xi I_N$ for some complex unit $\xi$. For the sake of the simplicity, let us assume that $\phi(I_N)= I_N$. Then $$\phi(E_{i_1i_1} \otimes E _{i_2i_2}\otimes \cdots \otimes E_{i_mi_m}) = E_{i_1i_1} \otimes E_{i_2i_2}\otimes \cdots \otimes E_{i_mi_m}$$ for all $1\le i_k\le n_k$ with $1\le k\le m$.
For any Hermitian matrix $\A\in M_N$, suppose its spectral decomposition is $$\A=X_1D_1X_1^*\otimes \cdots \otimes X_mD_mX_m^*.$$ Repeating the above argument and using the assumption $\phi(I_{N})= I_{N}$, we can conclude that there exists a unitary matrix $U_{X_1,\ldots,X_m}$ such that $$\phi(X_1E_{i_1i_1}X_1^*\otimes \cdots\otimes X_mE_{i_mi_m}X_m^*)=U_{X_1,\ldots,X_m}(X_1E_{i_1i_1}X_1^*\otimes \cdots\otimes X_mE_{i_mi_m}X_m^*)U_{X_1,\ldots,X_m}^*$$ for all $1\le i_k\le n_k$ with $1\le k\le m$. By linearity, we have $$\phi(\A)=U_{X_1,\ldots,X_m}(\A)U_{X_1,\ldots,X_m}^*.$$ So, $\phi$ maps Hermitian matrices to Hermitian matrices and preserves numerical range on the tensor product of Hermitian matrices. By Theorem \[T5\], it has the asserted form on Hermitian matrices and, hence, on all matrices in $M_{n_1\cdots n_m}$. If $n_i, n_j \ge 3$, we observe the matrices $A_i = X \oplus 0_{n_i-3}$ and $A_j = X \oplus 0_{n_j-3}$, where $X$ is defined as in Example \[Ex1\], and $A_k=E_{11}\in M_{n_k}$ for $k\ne i,j$, to conclude that $\varphi_i$ and $\varphi_j$ should both be the identity map, or both be the transpose map. The proof is completed.
[**Assertion 1.**]{} *There is a unitary $U \in M_n$ and a diagonal unitary $V_1 \in M_{n_1}$ such that $$U^*\phi(D\otimes E_{11} \otimes E_{11})U=\begin{pmatrix} V_1D & * \cr * & * \cr\end{pmatrix}$$ for any diagonal matrix $D\in M_{n_1}$.*
Proof. Consider $A_j = \phi(E_{jj} \otimes E_{11} \otimes E_{11})$ for $1\le j\le n_1$. According to the assumptions, there is a unit vector $x\in \mathbf{C}^N$ and a complex unit $\alpha_1$ such that $x^*A_1x=\alpha_1$. Let $U_1\in M_N$ be a unitary matrix with $x$ being its first column. By Lemma \[le3\], $$U_1A_1U_1^* = \alpha_1\begin{pmatrix}1 & x_1^* \cr - x_1 & * \cr\end{pmatrix}.$$ Now, for any scalar $\gamma$ with $|\gamma| \le 1$, we have $w(A_1)=1=w(A_1 + \gamma A_2)$. It follows that $x^*A_2x=0$ and, by Lemma \[le3\], we have $U_1 A_2U_1^* = 0 \oplus X$ with $X\in M_{N-1}$ and $w(X)=1$. Again, there is a unitary $U_2\in M_{N-1}$ and a complex unit $\alpha_2$ such that $$U_2XU_2^* = \alpha_2\begin{pmatrix}1 & x_2^* \cr - x_2 & * \cr\end{pmatrix}.$$ Moreover, if we replace $U_1$ with $(1\oplus U_2)U_1$, then the (1,2)-entry and (2,1)-entry of $U_1A_1U_1^*$ are equal to zero as $1 = w(\gamma A_1 + A_2)$ for all $\gamma \in \IC$ with $|\gamma| = 1$. Repeating the same argument for $A_3$, we see that there exist a unitary $U_3\in M_{N-2}$ and a complex unit $\alpha_3$ such that $U_1 A_3U_1^* = 0_2 \oplus Y$, where $$U_3YU_3^* = \alpha_3\begin{pmatrix}1 & x_3^* \cr - x_3 & * \cr\end{pmatrix}.$$ Furthermore, if we replace $U_1$ with $(I_2 \oplus U_3)U_1$, we see that the $(k,3)$-entry and $(3,k)$-entry of $U_1A_1U_1^*$ are equal to zero for $k = 1,2$. Repeating the same procedure for all $j\le n_1$, we conclude that there is a unitary $U\in M_{N}$ and complex units $\alpha_j$ such that $$UA_jU^* = \begin{pmatrix} \alpha_j E_{jj} & * \cr * & * \cr \end{pmatrix},
\qquad j=1, \ldots, n_1.$$
[**Assertion 2.**]{} *There is a unitary $U\in M_N$ and diagonal unitary matrices $V_{1}, \dots, V_{n_2} \in M_{n_1}$ such that $$U^*\phi(D \otimes E_{jj} \otimes E_{11})U=\begin{pmatrix}(V_jD) \otimes E_{jj} & * \cr * & * \cr\end{pmatrix}, \quad j = 1, \dots, n_2,$$ for any diagonal matrix $D \in M_{n_1}$.*
Proof. Replacing $\phi$ by the map $A\otimes B\otimes C \mapsto U^*\phi(A\otimes B\otimes C)U$, if necessary, we may assume that $U = I_{N}$ in the conclusion of Assertion 1. For any diagonal unitary $D\in M_{n_1}$ and any scalar $\gamma$ with $|\gamma|\le 1$, we have $$\label{w}
w(\phi(D \otimes E_{jj} \otimes E_{11})) = 1 = w(\phi(D \otimes E_{jj} \otimes E_{11}) + \gamma \phi(D \otimes E_{ii} \otimes E_{11})),$$ where $1\le i ,j\le n_2$ with $i\ne j$. Taking $i = 1$ and using the fact that $\phi(D\otimes E_{11} \otimes E_{11}) = \begin{pmatrix} V_1 D
& * \cr * & * \cr\end{pmatrix}$, we see that $\phi(D\otimes E_{22} \otimes E_{11}) = 0_{n_1} \oplus Z_D$ with $Z_D\in M_{N-n_1}$. Moreover, by linearity, $\phi(D\otimes E_{22} \otimes E_{11}) = 0_{n_1} \oplus Z_D$ for any diagonal matrix $D\in M_{n_1}$. Thus, for each $k = 1, \dots, n_1$, we have $\phi(E_{kk}\otimes E_{22} \otimes E_{11}) = 0_{n_1} \oplus Z_k$ with $Z_{k}\in M_{N-n_1}$. Because $w(Z_{k_1}) = 1 =
w(Z_{k_1} + \gamma Z_{k_2})$ for any $k_1 \ne k_2$, there is a unitary $V$ and complex units $\beta_{k}$ such that $$VZ_k V^* = \begin{pmatrix}
\beta_k E_{kk} & * \cr * & * \cr\end{pmatrix}, \quad k = 1, \dots, n_1.$$ Therefore, there exists a diagonal unitary $V_2\in M_{n_1}$ such that $$\phi(D \otimes E_{22} \otimes E_{11})=0_{n_1} \oplus \begin{pmatrix} V_2D & * \cr * & * \cr \end{pmatrix}$$ for any diagonal $D \in M_{n_1}$. Now, taking $i=1,2$ in (\[w\]), we see that $\phi(D\otimes E_{33} \otimes E_{11}) = 0_{n_1}\oplus 0_{n_1} \oplus W_D$ with $W_D\in M_{N-2n_1}$. Continuing with the same procedure as above, we see that $\phi(E_{kk}\otimes E_{33} \otimes E_{11}) = 0_{n_1} \oplus 0_{n_1}\oplus W_k$ with $$VW_k V^* = \begin{pmatrix} \delta_k E_{kk} & * \cr * & * \cr\end{pmatrix}, \quad k = 1, \dots, n_1.$$ Thus, there exists a diagonal unitary $V_3\in M_{n_1}$ such that $$\phi(D \otimes E_{33} \otimes E_{11})=0_{n_1} \oplus 0_{n_1} \oplus \begin{pmatrix} V_3D & * \cr * & * \cr \end{pmatrix}$$ for any diagonal $D \in M_{n_1}$. Using the same arguments, one can see that there are diagonal unitary matrices $V_{1}, \dots, V_{n_2} \in M_{n_1}$ such that $$\phi(D \otimes E_{jj} \otimes E_{11})=\begin{pmatrix}(V_jD) \otimes E_{jj} & * \cr * & * \cr\end{pmatrix}, \quad j = 1, \dots, n_2,$$ for any diagonal matrix $D \in M_{n_1}$.
[**Assertion 3.**]{} *There is a unitary $U \in M_N$ and a diagonal unitary $V \in M_n$ such that $$U^*\phi(D)U = VD$$ for any diagonal matrix $D \in M_N$.*
Proof. In the following, we assume that $U = I_{N}$ in the conclusion of Assertion 2. Recall that Assertion 2 implies that there exists a diagonal unitary $W_1\in M_{n_1n_2}$ such that $$\phi(D \otimes E_{jj} \otimes E_{11})=\begin{pmatrix} W_1(D \otimes E_{jj}) & * \cr * & * \cr\end{pmatrix},
\quad j = 1, \dots, n_2,$$ for any diagonal matrix $D \in M_{n_1}$. Thus, for any diagonal $D_1\in M_{n_1}$, $D_2\in M_{n_2}$, we have $$\phi(D_1 \otimes D_2 \otimes E_{11})=\begin{pmatrix} W_1(D_1 \otimes D_2) & * \cr * & * \cr\end{pmatrix}.$$ Furthermore, for any diagonal unitary $D_1\in M_{n_1}$, $D_2\in M_{n_2}$, and any scalar $\gamma$ with $|\gamma|\le 1$, we have $$w(\phi(D_1 \otimes D_2 \otimes E_{jj})) = 1 = w(\phi(D_1
\otimes D_2\otimes E_{jj}) + \gamma \phi(D_1 \otimes D_2 \otimes E_{ii})),$$ where $1\le i ,j\le n_3$ with $i\ne j$. Taking $i = 1$, we see that $\phi(D_1
\otimes D_2\otimes E_{22}) = 0_{n_1n_2} \oplus R_D$ with $R_D\in M_{N-n_1n_2}$. Thus, for each $k=1,\ldots, n_1$, $\ell=1, \ldots, n_2$, we have $\phi(E_{kk}\otimes
E_{\ell\ell} \otimes E_{22}) = 0_{n_1n_2} \oplus R_{k\ell}$ with $R_{k\ell}\in M_{N-n_1n_2}$. Since $w(R_{k_1\ell_1}) = 1 = w(R_{k_1\ell_1} + \gamma R_{k_2\ell_2})$, where either $k_1\ne k_2$ or $\ell_1 \ne \ell_2$, there is a unitary $W$ and complex units $\delta_{k\ell}$ such that $$WR_{k\ell} W^* = \begin{pmatrix} \delta_{k\ell}
(E_{kk}\otimes E_{\ell\ell}) & * \cr * & * \cr \end{pmatrix}, \quad k=1,\ldots, n_1, \ell=1, \ldots, n_2.$$ It follows that there exists a diagonal unitary $W_2\in
M_{n_1n_2}$ such that $$\phi(D_ \otimes D_2 \otimes E_{22})=0_{n_1n_2} \oplus \begin{pmatrix} W_2(D_1\otimes D_2) & * \cr * & * \cr \end{pmatrix}$$ for any diagonal $D_1\in M_{n_1}$, $D_2\in M_{n_2}$. Using the same arguments, we can show that there are diagonal unitary matrices $W_{1}, \dots, W_{n_3} \in M_{n_1}$ such that $$\phi
(D_1 \otimes D_2 \otimes E_{jj})=W_j(D_1 \otimes D_2) \otimes E_{jj}, \quad j = 1, \dots, n_3,$$ for any diagonal $D_1\in M_{n_1}$, $D_2\in M_{n_2}$. Hence, there exists a diagonal unitary $W \in M_n$ such that $$\phi(D_1 \otimes D_2 \otimes D_3)=W(D_1 \otimes D_2\otimes D_3)$$ for any diagonal $D_1\in M_{n_1}$, $D_2\in M_{n_2}$, $D_3\in M_{n_3}$.
Reorganization of the proofs
============================
For bipartite system $M_m\otimes M_n$, let $B_{ij} = \phi(E_{ii}\otimes E_{jj})$ for $1 \le i \le m, 1 \le j \le n$.
[**Claim 1.**]{} If $(i,j) \ne (r,s)$ and if $u \in \IC^{mn}$ is a unit vectors such that $|u^*B_{ij}u| = 1$, then $u^*B_{rs}u = 0$.
*Proof. Suppose $u$ is a unit vector such that $ u^*B_{ij}u = e^{i\theta}$. Let $\xi = u^*B_{rs}u$. Assume it is not zero, and assume that $\mu$ is a complex unit such that $\mu \xi = |\xi|$.*
First, suppose that $\{i,j\} \cap \{r,s\}\ne\emptyset$. Without loss of generality, assume $(i,j) = (1,1)$ and $(r,s) = (1,2)$. Then $$|u^*(e^{-i\theta}B_{11} + \mu B_{12})u| = 1 + |\xi| > 1 = w(E_{11}\otimes (e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}B_{11} + \mu B_{12}),$$ which is a contradiction.
Now, suppose $\{i,j\} \cap \{r,s\} = \emptyset$, say, $(i,j) = (1,1)$ and $(r,s) = (2,2)$. Then by the previous case, $u^*B_{12}u = 0 = u^*B_{21}u$, and hence $$|u^*(e^{-i\theta}(B_{11} + B_{21}) + \mu(B_{12} + B_{22}))u|
= 1 + |\xi| > 1$$ $$= w((E_{11}+E_{22})\otimes (e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}(B_{11} + B_{21}) + \mu(B_{12} + B_{22})),$$ which is a contradiction.
[**Claim 2.**]{} Let $u_{ij} \in \IC^{mn}$ be unit vector such that $\mu_{ij}=u_{ij}^* B_{ij} u_{ij}$ is unit for $1 \le i \le m, 1 \le j \le n$. Suppose $(i,j) \ne (r,s)$.
\(a) Suppose $|\{i,j\} \cap \{r,s\}| = 1.$ Then $V = [u_{ij}~ u_{rs}]$ satisfies $V^*V = I_2$ and $$V^*B_{ij}V = \begin{pmatrix}\mu_{ij} & 0 \cr 0 & 0 \cr\end{pmatrix} \quad \hbox{
and } \quad V^*B_{rs}V = \begin{pmatrix}0 & 0 \cr 0 & \mu_{rs} \cr\end{pmatrix}.$$
\(b) Suppose $|\{i,j\} \cap \{r,s\}| = 0.$ Then $V = [u_{ij}~ u_{is}~ u_{rj} ~u_{rs}]$ satisfies $V^*V = I_4$ and $$V^*B_{ij}V = \mu_{ij} E_{11} \otimes E_{11}, \
V^*B_{is}V = \mu_{is} E_{11} \otimes E_{22}, \
V^*B_{rj}V = \mu_{rj} E_{22} \otimes E_{11}, \
V^*B_{rs}V = \mu_{rs} E_{22} \otimes E_{22},$$ where $E_{11}, E_{22} \in M_2$.
*Proof. We may assume that $(i,j) = (1,1)$.*
\(a) Since $|\{i,j\} \cap \{r,s\}| = 1$, we may take $(r,s) = (1,2)$ or $(r,s) = (2,1)$. In the first case, let $V = [u_{ij}~ \tilde u_{rs}]$ be such that $\span\{u_{ij},\tilde u_{rs}\}=\span\{u_{ij}, u_{rs}\}$, $V^*V = I_2$ and $V^*B_{11}V = F$, $V^*B_{12}V = G$. Then $1 = w(F) = w(G)$ and $$1 = w(B_{11} + \mu B_{12}) \ge w(F+\mu G) \ge |(F+\mu G)_{11}| = 1.$$ By Lemma \[le3\], $G_{11} = G_{12} = G_{21} = 0$. Since $w(G) = 1$, we have $|G_{22}| = 1$, and we may assume that $\tilde u_{rs} = u_{rs}$. Now, by (\[ck-1\]), we see that $0 = F_{12} = F_{21} = F_{22}$. The proof goes in the same way if $(r,s) = (2,1)$.
\(b) Suppose $(r,s) = (2,2)$. Let $V = [u_{11}~ u_{12}~ \tilde u_{21}~ \tilde u_{22}]$, where $\span\{u_{11},u_{12}, \tilde u_{21}\} = \span\{u_{11}, u_{12}, u_{21}\}$ and $\span\{u_{11},u_{12}, \tilde u_{21}, \tilde u_{22}\}
= \span\{u_{11}, u_{12}, u_{21}, u_{22}\}$. Set $V^*B_{11}V = F_{11}$, $V^*B_{12}V = F_{12}$, $V^*B_{21}V = F_{21}$, $V^*B_{22}V = F_{22}$. By Case 1, we see that the leading $2\times 2$ principal submatrix of $F_{11}$ is $\mu_{11} E_{11} \in M_2$ and that of $F_{12}$ is $\mu_{12} E_{22} \in M_2$.
Denote the $(p,q)$ entry of $F_{ij}$ by $F_{ij}(p,q)$ for $i,j\in\{1,2\}$ and $p,q\in \{1,2,3,4\}.$ By claim 1, $F_{ij}(p,p)\ne 0$ only if $p=2(i-1)+j$. Since $$1=w(B_{11}+\mu B_{21})\geq w(F_{11}+\mu F_{21})\geq |(F_{11}+\mu F_{21})_{11}|=1$$ for any unit complex number $\mu$, applying Lemma \[le3\], we get $F_{21}(1,q)=F_{21}(q,1)=0$ for $q=2,3,4$. Similarly, considering $B_{21}+\mu B_{22}$ we can obtain $F_{22}(2,q)=F_{22}(q,2)=0$ for $q=2,3,4$. Next, considering $B_{11}+B_{12}+\mu(B_{21}+B_{22})$ we get $F_{21}=0_2\oplus G_{21}$ and $F_{22}=0_2\oplus G_{22}$. Now using the fact $\span\{u_{11}, u_{12},\tilde u_{21}\} = \span\{u_{11}, u_{12}, u_{21}\}$ we can get $G_{21}(1,1)=F_{21}(3,3)=1$. As in the proof of Claim 2, considering $w(\mu B_{21}+\xi B_{22})$ for all unit complex numbers $\mu$ and $\xi$, applying Lemma \[le3\] on $\mu G_{21}+\xi G_{22}$ we can get $F_{21}=\mu_{21}E_{22}\otimes E_{11}$ and $F_{22}=\mu_{22}E_{22}\otimes E_{22}$ for some unit numbers $\mu_{21},\mu_{22}$. On the other hand, considering $B_{11}+\mu B_{12}$ and $\mu (B_{11}+B_{12})+B_{21}+B_{22}$ we will have $F_{11}=\mu_{11}E_{11}\otimes E_{11}$ and $F_{12}=\mu_{12}E_{11}\otimes E_{22}$. Therefore, $\tilde u_{21}, \tilde u_{22}$ may be assumed to be $u_{21}, u_{22}$, and $F_{ij} = \mu_{ij} E_{ii}\otimes E_{jj} \in M_2 \otimes M_2$ for $i,j \in \{ 1,2\}$.
[**Claim 3.**]{} Using the orthonormal family $\{u_{ij}: 1 \le i \le m, 1 \le j \le n \}$ to form a unitary matrix $U$, we have $U^*B_{ij}U = \xi_{ij} E_{ii}\otimes E_{jj}$ for all $(i,j)$ pair.
*Proof. Consider $U^*B_{11}U$. Label the rows and columns of $X \in M_{mn}$ by $(p,q)$ with $1 \le p \le m$ and $1 \le q \le n$. By Claim 2 (b), for any $(r,s)$ such that $|\{1,1\} \cap \{r,s\}| = 0$, the $4\times 4$ submatrix of $U^*B_{11}U$ with rows and column indices $(1,1), (1,s), (r,1), (r,s)$ has the form $\xi_{11} E_{11} \otimes E_{11} \in M_2 \otimes M_2$. One sees that $U^*B_{11}U = \xi_{11} E_{11} \otimes E_{11} \in M_m \otimes M_n$. The same argument applies to $U^*B_{ij}U$ for any $(i,j)$ pair. The result follows.*
------------------------------------------------------------------------
[**Tripartite case**]{}
Let $X_{ijk} = \phi(E_{ii}\otimes E_{jj}\otimes E_{kk})$ for $1 \le i \le n_1, 1 \le j \le n_2, 1\le k\le n_3$.
[**Claim 4**]{} If $(i,j,k) \ne (r,s,t)$ and if $u \in \IC^{n_1n_2n_3}$ is a unit vectors such that $|u^*X_{ijk}u| = 1$, then $u^*B_{rst}u = 0$.
*Proof. Suppose $u$ is a unit vector such that $ u^*X_{ijk}u = e^{i\theta}$. Let $\xi = u^*X_{rst}u$. Assume it is not zero, and $\mu$ is a complex unit such that $\mu \xi = |\xi|$.*
First, assume that $\delta_{ir}+\delta_{js}+\delta_{kt}=2$, where $\delta_{uv}$ equals to 1 when $u=v$ and zero otherwise. Without loss of generality, assume $(i,j,k) = (1,1,1)$ and $(r,s,t) = (1,1,2)$. Then $$|u^*(e^{-i\theta}X_{111} + \mu X_{112})u| = 1 + |\xi| > 1 = w(E_{11}\otimes E_{11}\otimes(e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}X_{111} + \mu X_{112}),$$ which is a contradiction.
Next, suppose $\delta_{ir}+\delta_{js}+\delta_{kt} = 1$, say, $(i,j,k) = (1,1,1)$ and $(r,s,t) = (1,2,2)$. Then by the previous case $u^*X_{112}u = 0 = u^*X_{121}u$, and hence $$|u^*(e^{-i\theta}(X_{111} + X_{121}) + \mu(X_{112} + X_{122})u|
= 1 + |\xi| > 1$$ $$= w(E_{11}\otimes(E_{11}+E_{22})\otimes (e^{-i\theta}E_{11} + \mu E_{22}))
= w(e^{-i\theta}(X_{111} + X_{121}) + \mu(X_{112} + X_{112})),$$ which is a contradiction.
Now, suppose $\delta_{ir}+\delta_{js}+\delta_{kt}= 0$, say, $(i,j,k) = (1,1,1)$ and $(r,s,t) = (2,2,2)$. By the previous cases $u^*X_{121}u = u^*X_{211}u= u^*X_{221}u= u^*X_{112}u = u^*X_{122}u= u^*X_{212}u=0$, and hence $$\begin{aligned}
&&|u^*(e^{-i\theta}(X_{111} +X_{121}+X_{211}+ X_{221}) + \mu(X_{112} + X_{122}+X_{212}+X_{222})u|\\
&=& 1 + |\xi| \\
&>& 1 = w((E_{11}+E_{22})\otimes(E_{11}+E_{22})\otimes (e^{-i\theta}E_{11} + \mu E_{22}))\\
&=& w(e^{-i\theta}(X_{111} +X_{121}+X_{211}+ X_{221}) + \mu(X_{111} +X_{121}+X_{211}+ X_{221})),\end{aligned}$$ which is a contradiction.
[**Claim 5**]{} Let $u_{ijk} \in \IC^{n_1n_2n_3}$ be unit vector such that $ u_{ijk}^* X_{ijk} u_{ijk} = \xi_{ijk}$ is unit for $1 \le i \le n_1, 1 \le j \le n_2, 1\le k\le n_3$. Suppose $1\leq r\ne i\leq n_1$, $1\le s\ne j\le n_2$, $1\le t\ne k\le n_3$. Then $V = [u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~u_{rjk}~ u_{rjt}~ u_{rsk} ~u_{rst}]$ satisfies $V^*V = I_8$ and $$\begin{aligned}
\label{eq111}
\begin{array}{c}
V^*X_{ijk}V = \mu_{ijk} E_{11} \otimes E_{11}\otimes E_{11}, \quad
V^*X_{ijt}V = \mu_{ijt} E_{11} \otimes E_{11}\otimes E_{22}, \\
V^*X_{isk}V = \mu_{isk} E_{11} \otimes E_{22}\otimes E_{11}, \quad
V^*X_{ist}V = \mu_{ist} E_{11} \otimes E_{22}\otimes E_{22},\\
V^*X_{rjk}V = \mu_{rjk} E_{22} \otimes E_{11}\otimes E_{11}, \quad
V^*X_{rjt}V = \mu_{rjt} E_{22} \otimes E_{11}\otimes E_{22}, \\
V^*X_{rsk}V = \mu_{rsk} E_{22} \otimes E_{22}\otimes E_{11}, \quad
V^*X_{rst}V = \mu_{rst} E_{22} \otimes E_{22}\otimes E_{22},
\end{array}\end{aligned}$$ where $E_{11}, E_{22} \in M_2$.
*Proof. We may assume $(i,j,k) = (1,1,1)$ and $(r,s,t)=(2,2,2)$. Let $$U=[u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt}~ \tilde u_{rsk} ~\tilde u_{rst}]$$ with $$\begin{aligned}
&&\span\{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}\}=\span\{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}\},\label{eq121}\\
&&\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt} \}=\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}~ u_{rjt} \},\label{eq122}\\
&&\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt}~ \tilde u_{rsk} \}=\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}~ u_{rjt}~ u_{rsk} \},\label{eq123}\\
&&\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~\tilde u_{rjk}~ \tilde u_{rjt}~ \tilde u_{rsk} ~\tilde u_{rst}\}=\span \{u_{ijk}~ u_{ijt}~ u_{isk} ~u_{ist}~ u_{rjk}~ u_{rjt}~ u_{rsk} ~ u_{rst}\}\label{eq124}.\end{aligned}$$ Firstly, applying Claim 4 and the same argument as in the proof of Claim 2 with replacing the role of $B_{ij}$ by $X_{1ij}$, we can get that the leading $4\times 4$ principal submatrix of $U^*X_{1jk}U$ is $\xi_{1jk} E_{jj}\otimes E_{kk}$ for $j,k\in\{1,2\}$.*
Next, applying Lemma \[le3\] on $U^*(X_{1jk}+\mu X_{2jk})U$, $U^*((X_{1j1}+X_{1j2})+\mu (X_{2j1}+X_{2j2}))U$ with $j,k\in\{1,2\}$ and $U^*((X_{111}+X_{112}+X_{121}+X_{122})+\mu (X_{111}+X_{112}+X_{121}+X_{122}))U$ successively we can obtain that $U^*X_{2jk}U=0_4\oplus F_{jk}$ for all $j,k\in\{1,2\}$. As in the proof of Claim 2, by (\[eq121\]) and $w(X_{211})=1$ we have $|F_{11}(1,1)|=1$.
Now repeating the argument as in the proof of Claim 2 again with replacing the role of $B_{ij}$ by $X_{2ij}$, we get $F_{jk}=\mu_{jk}E_{jj}\otimes E_{kk}$ with $|\mu_{jk}|=1$ for all $j,k\in\{1,2\}$.
Finally, applying Lemma \[le3\] on $U^*(\mu X_{1jk}+ X_{2jk})U$, $U^*(\mu (X_{1j1}+X_{1j2})+ (X_{2j1}+X_{2j2}))U$ with $j,k\in\{1,2\}$ and $U^*(\mu (X_{111}+X_{112}+X_{121}+X_{122})+(X_{111}+X_{112}+X_{121}+X_{122}))U$ successively we can obtain that $U^*X_{1jk}U=\xi_{1jk}E_{11}\otimes E_{jj}\otimes E_{kk}$ for all $j,k\in\{1,2\}$. By (\[eq121\])-(\[eq124\]), we can assume $U=V$ and (\[eq111\]) holds.
[**Claim 6**]{} Using the orthonormal family $\{u_{ijk}: 1 \le i \le n_1, 1 \le j \le n_1,1\le k\le n_3 \}$ to form a unitary matrix $U$, we have $U^*X_{ijk}U = \xi_{ijk} E_{ii}\otimes E_{jj}\otimes E_{kk}$ for all $1\le i\le n_1, 1\le j\le n_2$ and $1\le k\le n_3$.
*Proof. Consider $U^*X_{111}U$. Denote by $(r,s,t)=(r-1)n_2n_3+(s-1)n_3+t)$ with $1 \le r \le n_1$, $1 \le s \le n_2$ and $1\le t\le n_3$. By Claim 5, for any $(r,s,t)$ such that $r\ne 1,s\ne 1,t\ne 1$, the $8\times 8$ submatrix of $U^*X_{111}U$ with rows and column indices $$\{(1,1,1), (1,1,t), (1,s,1), (1,s,t), (r,1,1),(r,1,t),(r,s,1),(r,s,t)\}$$ has the form $\xi_{11} E_{11} \otimes E_{11}\otimes E_{11} \in M_2 \otimes M_2\otimes M_2$. One sees that $U^*X_{111}U = \xi_{111} E_{11}\otimes E_{11} \otimes E_{11} \in M_{n_1} \otimes M_{n_2}\otimes M_{n_3}$. The same argument applies to $U^*X_{ijk}U$ for any $(i,j,k)$. The result follows.*
------------------------------------------------------------------------
------------------------------------------------------------------------
Since $w(B_{ij}) = 1$, there is a unit vector $u_{ij}$ such that $|u_{ij}^* B_{ij} u_{ij}| = 1$ for each $(i,j)$ pair. By the above claim, $\{u_{ij}: 1 \le i \le m, 1 \le j \le n\}$ form an orthonormal basis for $\IC^{mn}$. By a suitable unitary similarity transform, we may assume that for each $(i,j)$ pair, the diagonal of $B_{ij}$ is the same as that of $\xi_{ij} E_{ii} \otimes E_{jj}$.
Next, we show that all the off-diagonal entries of $B_{ij}$ are zero so that $B_{ij}$ is indeed equal to $\xi_{ij} E_{ii} \otimes E_{jj}$. Note that for every diagonal unitary $D \in M_m$ we can show that [**Details!**]{} for any diagonal unitary $D \in M_m$, if $\phi(D\otimes E_{jj}) = (Y_{rs})$ with $Y_{rs} \in M_n$ then (1) $Y_{jj} = \tilde D$ for some diagonal unitary $\tilde D$ and $Y_{rr} = 0$ for $r \ne j$. After that, we can show that $Y_{rs} = 0$ for all $r \ne s$.
[**Now for multipartite system**]{}, it suffices to demonstrate the proof for $M_{n_1}\otimes M_{n_2} \otimes M_{n_3}$.
We let $B_{ijk} = \phi(E_{ii} \otimes E_{jj} \otimes E_{kk})$. Again, we claim that if $u \in \IC^N$ with $N = n_1n_2n_3$ is a unit vector such that $|u^*B_{ijk}u| = 1$, then $u^*B_{pqr}u = 0$ if $(p,q,r) \ne (i,j,k)$. Assume the contrary that $u^*B_{pqr}u = \xi \ne 0$. Let $\mu$ be a complex unit such that $\mu\xi = |\xi|$.
Case 1. If $|\{i,j,k\} \cap \{p,q,r\}| = 2$, then we may assume that $(i,j,k) = (1,1,1)$ and $(p,q,r) = (1,1,2)$. Then $$|u^*(B_{111} + \mu B_{112})u| = 1 + |\xi| > 1
= w(E_{11}\otimes E_{11} \otimes (E_{11} + \mu E_{22}))
= w(B_{111} + \mu B_{112}),$$ which is a contradiction.
Case 2. If $|\{i,j,k\} \cap \{p,q,r\}| = 1$, then we may assume that $(i,j,k) = (1,1,1)$ and $(p,q,r) = (1,2,2)$. Then by the previous case $u^*B_{112}u = 0 = u^*B_{121}u$, and hence $$|u^*(B_{111} + B_{121} + \mu(B_{121} + B_{121}))u|
= 1 + |\xi| > 1$$ $$= w(E_{11} \otimes ((E_{11}+E_{22})\otimes (E_{11} + \mu E_{22}))
= w(B_{111} + B_{121} + \mu(B_{121} + B_{121})),$$ which is a contradiction.
Case 3. If $\{i,j,k\} \cap \{p,q,r\} = 0$, then we may assume that $(i,j,k) = (1,1,1)$ and $(p,q,r) = (2,2,2)$. Then by Cases 1 and 2, $u^*B_{abc}u = 0$ if $\{a,b,c\} = \{1,2\}$. we can then consider $u^*Bu$ for $B = (E_{11} + E_{22}) \otimes (E_{11}+E_{22}) \otimes (E_{11} + \mu E_{22})$, and derive a contradiction if $\xi \ne 0$.
Now, by a unitary similarity, we may assume that the diagonal of $B_{ijk}$ is the same as that of $\xi_{ijk} E_{ii} \otimes E_{jj} \otimes E_{kk}$. Next, we show that $B_{ijk}$ actually equal $\xi_{ijk} E_{ii} \otimes E_{jj} \otimes E_{kk}$. ([**Lot of details**]{}.)
[**Acknowledgment**]{}
This research was supported by a Hong Kong GRC grant PolyU 502411 with Sze as the PI. The grant also supported the post-doctoral fellowship of Huang and the visit of Fošner to the Hong Kong Polytechnic University in the summer of 2012. She gratefully acknowledged the support and kind hospitality from the host university. Fošner was supported by the bilateral research program between Slovenia and US (Grant No. BI-US/12-13-023). Li was supported by a GRC grant and a USA NSF grant; this research was done when he was a visiting professor of the University of Hong Kong in the spring of 2012; furthermore, he is an honorary professor of Taiyuan University of Technology (100 Talent Program scholar), and an honorary professor of the Shanghai University.
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|
---
author:
- 'Gary Steigman, James P. Kneller, and Andrew Zentner'
title: 'CMB (And Other) Challenges To BBN'
---
Introduction {#sec:intro}
============
Even though diamonds may not be forever, experimental constraints on proton stability are very strong ($\tau_{\rm N} > 10^{25}$ yr) and baryon (nucleon) number should be preserved during virtually the entire evolution of the universe. If so, then in the standard theories of particle physics and cosmology the baryon density at very early epochs is simply related to the baryon density throughout the later evolution of the universe. In particular, the nucleon-to-photon ratio ($\eta \equiv n_{\rm N}/n_{\gamma}$) during primordial nucleosynthesis when the universe is only minutes old should be identical to $\eta$ measured when the universe is several hundred thousand years old and the cosmic microwave background (CMB) photons last scattered, as well as to $\eta$ in the present universe billions of years after the “bang". Probing $\eta$ at such widely separated epochs in the evolution of the universe is a key test of the consistency of the standard models of particle physics and cosmology.
The current status of this confrontation between theory and observations is reviewed here and our key results appear in Figure 1 where estimates of the universal baryon abundance at widely separated epochs are compared. In § \[sec:bbn\] the predicted BBN abundance of deuterium is compared with the primordial value inferred from observational data to derive the early-universe value of $\eta$. After testing for the internal consistency of the standard model of Big Bang Nucleosynthesis (SBBN) by comparing the BBN-predicted and observed abundances of the other light elements (he, li), an independent estimate of $\eta$ in the present (recent) universe is derived in § \[sec:sn1a\] utilizing observations of clusters of galaxies and of type Ia supernovae (SNIa). These [*independent*]{} estimates of $\eta$ are compared to each other and, in § \[sec:cmb\] to that from observations of the CMB anisotropy spectrum, a probe of $\eta$ in the several hundred thousand year old universe. Having established that some “tension" exists between $\eta_{\rm BBN}$ and $\eta_{\rm CMB}$, in § \[sec:nsbbn\] a modification of SBBN involving “degenerate" neutrinos is introduced and its consequences for the CMB anisotropies is explored. In § \[sec:disc\] we summarize our conclusions. The material presented here is extracted from our recent work [@swz; @kssw] where further details and more extensive references may be found.
An alternate measure of the baryon abundance is the baryon density parameter, $\Omega_{\rm B}$, the ratio of the baryon mass density to the critical mass density. In terms of the present value of the Hubble parameter $h$ (H$_{0} \equiv 100h$ kms$^{-1}$Mpc$^{-1}$), and for a present CMB temperature of 2.725 K [@mather], $\eta_{10} \equiv 10^{10}\eta = 274\Omega_{\rm B}h^{2}$.
![The likelihood distributions, normalized to unit maximum, for the baryon-to-photon ratio $\eta_{10} = 274\Omega_{\rm B}h^{2}$. The solid curve is the early universe value for low-D BBN while the dashed curve is for high-D BBN; the dotted curve (SNIa) is the present universe estimate; the dot-dashed curve shows the CMB inferred range.[]{data-label="fig:"}](eta.eps){width="\columnwidth"}
The Early Universe Baryon Density {#sec:bbn}
=================================
During its early evolution the universe is too hot to allow the presence of astrophysically interesting abundances of bound nuclei and primordial nucleosynthesis doesn’t begin in earnest until the temperature drops below $\approx 80$ keV, when the universe is a few minutes old (for a recent review and further references see ). Prior to this time neutrons and protons have been interconverting, at first rapidly, but more slowly after the first few seconds, driven by such “charged-current" weak interactions as: $p + e^{-} \leftrightarrow n + \nu_{e}$, $n + e^{+} \leftrightarrow
p + \bar{\nu}_{e}$, and $n \leftrightarrow p + e^{-} + \bar{\nu}_{e}$ ($\beta$-decay). Once BBN begins neutrons and protons quickly combine to form deuterium which, in turn, is rapidly burned to , he, and he. There is a gap at mass-5 which, in the expanding, cooling universe, is difficult to bridge. As a result, most neutrons available when BBN began are incorporated in the most tightly bound light nuclide, he. For this reason, the he abundance (by mass, Y) is largely independent of the nuclear reaction rates but does depend on the neutron abundance at BBN which is determined by the competition between the weak interaction rates and the universal expansion rate (the early universe Hubble parameter, H). In contrast, the abundances of D and he ( is unstable, decaying to he) depend on the competition between the expansion rate and the nuclear reaction rates and, hence, on the baryon abundance $\eta$. As a result, while D (and to a lesser extent, he) can provide a baryometer, he offers a test of the internal consistency of SBBN. Although the gap at mass-5 is a barrier to the synthesis of heavier nuclides in the early universe, there is some production of mass-7 nuclei (li and ), albeit at a much suppressed level. The second mass gap at mass-8 eliminates (within SBBN) the synthesis of any astrophysically interesting abundances of heavier nuclides. The abundance of lithium (after BBN, when the universe is sufficiently cool, will capture an electron and decay to li) is rate driven and can serve as a complementary baryometer to deuterium.
SBBN is overdetermined in the sense that for one adjustable parameter $\eta$, the abundances of four light nuclides (D, he, he, li) are predicted. Here we concentrate on D and he. Deuterium is an ideal baryometer candidate [@rafs] since it is only [**destroyed**]{} (by processing in stars) in the post-BBN universe [@els]. Deuterium is observed in absorption in the spectra of distant QSOs and its abundance in these high-redshift (relatively early in the star-forming history of the universe), low-metallicity (confirming that very little stellar processing has occurred) systems should represent the primordial value. For three, high-z, low-Z QSO absorption-line systems a “low" value of the deuterium abundance is found [@bt; @omeara], from which derive: D/H $= 3.0 \pm 0.4 \times 10^{-5}$. Given the steep dependence of (D/H)$_{\rm BBN}$ on $\eta$ ($\propto \eta^{-1.6}$), this leads to a reasonably precise prediction for the baryon abundance at BBN: $\eta_{10} = 5.6 \pm 0.5$ ($\Omega_{\rm B}h^{2} = 0.020 \pm
0.002$). The likelihood distribution for this BBN-determined baryon density is shown in Figure 1 by the curve labelled “BBN(Low-D)".
![Comparison of the SBBN-predicted relation between the primordial abundances of helium-4 (mass fraction, Y) and deuterium (ratio by number to hydrogen, D/H) and four sets of observationally inferred abundances. The SBBN prediction, including uncertainties, is shown by the solid band. The “low-D" deuterium abundance is from O’Meara (2000); the “high-D" value is from Webb (1997). The solid ellipses reflect the Izotov & Thuan (1998) helium abundance, while the dashed ellipses use the Olive, Steigman & Walker (2000) value.[]{data-label="fig:bbn"}](hevsd.eps){width="\columnwidth"}
Although any deuterium, observed anywhere (and at any time) in the universe, provides a [**lower**]{} limit to its primordial abundance, not all absorption identified with deuterium need actually be due to deuterium. The absorption spectra of hydrogen and deuterium are identical, save for the wavelength/velocity shift (81 kms$^{-1}$) due to the very slightly different reduced masses. Thus, any “observed" deuterium can only provide an [**upper**]{} bound to the true deuterium abundance. It is dismaying that such crucial implications for cosmology rely at present on only three pieces of observational data. Indeed, the most recently determined deuterium abundance [@omeara] is somewhat more than 3$\sigma$ lower than the previous primordial value based on the first two systems. In fact, there is a fourth absorption-line system for which it has been claimed that deuterium is observed [@webb]. The deuterium abundance derived for this system is very high (“high-D"), nearly an order of magnitude larger than the low-D value, leading to a considerably smaller baryon abundance estimate. This determination suffers from a lack of sufficient data on the velocity structure of the absorbing cloud(s) and is a likely candidate for confusion with a hydrogen interloper masquerading as deuterium. Nonetheless, for completeness, this estimate of the baryon density is included in Figure 1 by the curve labelled “BBN(High-D)". We believe that the low-D value provides a better estimate of the true primordial abundance and, use it in the following for our “preferred" estimate of the SBBN baryon density.
In Figure \[fig:bbn\] the band extending from upper left to lower right shows the relation between the SBBN-predicted abundances of D and he; the width of the band represents the (1$\sigma$) uncertainties in the predictions due to uncertainties in the nuclear and weak interaction rates. Note that while D/H changes by an order of magnitude, Y hardly changes at all ($\approx 0.015$). Figure \[fig:bbn\] exposes the first observational challenge to SBBN. For the observed (low-D) deuterium abundance (including its uncertainty), the SBBN-predicted helium abundance is Y $= 0.248 \pm 0.001$. This is in disagreement with several determinations of the primordial helium abundance derived from observations of low-metallicity, extragalactic regions. From their survey of the literature find $= 0.234
\pm 0.003$ (see also and ), while from their own, independent data set derive $= 0.244 \pm 0.002$. It is clear that these results are in conflict and it is likely that unaccounted for systematic errors dominate the error budget. For this reason a “compromise" was advocated in : $= 0.238 \pm
0.005$. Recently, in an attempt to either uncover or avoid some potential systematic errors, studied the nearby, albeit relatively metal-rich, region NGC 346 in the SMC. They found Y $= 0.2405 \pm 0.0018$ and, correcting for the evolution of Y with metallicity, derived $= 0.235 \pm 0.003$. It is clear (see Fig. \[fig:bbn\]) that [**none**]{} of these observational estimates is in agreement with the predictions of SBBN (low-D), although the gravity of the disagreement may be in the eye of the beholder. The observationally inferred primordial helium abundance is “too small" for the observationally determined deuterium abundance. Either one (or both) of the abundance determinations is inaccurate at the level claimed, or some interesting physics (and/or cosmology) is missing from SBBN. Notice that if the high-D abundance is the true primordial value there is no conflict between SBBN and the helium abundance, while the abundance is now too high. Before addressing the role of possible non-standard BBN in relieving the tension between D and he, other, non-BBN, bounds on the baryon abundance are considered and compared to $\Omega_{\rm BBN}$.
The Present Universe Baryon Density {#sec:sn1a}
===================================
It is notoriously difficult to inventory baryons in the present universe. have attempted to find the density of those baryons which reveal themselves by shining (or absorbing!) in some observationally accessible part of the electromagnetic spectrum: “luminous baryons". It is clear from that most baryons in the present universe are “dark" since they find $\Omega_{\rm LUM} \approx 0.0022 +
0.0006h^{-1.3} \ll \Omega_{\rm BBN}$. At the very least this lower bound to $\Omega_{\rm B}$ is good news for SBBN, demonstrating that the baryons present during BBN [*may*]{} still be here today. In a more recent inventory which includes some estimates of dark baryons, find a larger range ($0.007 \la \Omega_{\rm B} \la 0.041$) that has considerable overlap with $\Omega_{\rm BBN}$.
A complementary approach to the present universe baryon density is to combine an estimate of the [*total*]{} mass density, baryonic plus “cold dark matter" (CDM), $\Omega_{\rm M}$, with an independent estimate of the universal baryon [*fraction*]{} $f_{\rm B}$ to find $\Omega_{\rm B} = f_{\rm B}\Omega_{\rm M}$. Recently, we [@swz] imposed the assumption of a “flat" universe and used the SNIa magnitude-redshift data [@sn1a] to find $\Omega_{\rm M}$ ($0.28^{+0.08}_{-0.07}$), which was combined with a baryon fraction estimate ($f_{\rm B}h^{2} = 0.065^{+0.016}_{-0.015}$) based on X-ray observations of rich clusters of galaxies [@xray] and the HST Key Project determination of the Hubble parameter ($h = 0.71 \pm 0.06$; ) to derive $\eta_{10} = 4.8^{+1.9}_{-1.5}$ ($\Omega_{\rm
B}h^{2} = 0.018^{+0.007}_{-0.005}$). Subsequently , utilizing observations of the Sunyaev-Zeldovich effect in X-ray clusters, have reported a very similar value for the cluster hot gas fraction to that adopted in . For the value for $f_{\rm B}$, which may be less vulnerable to systematics, the present universe baryon density is, $\eta_{10} = 5.1^{+1.8}_{-1.4}$ ($\Omega_{\rm B}h^{2} = 0.019^{+0.007}_{-0.005}$). This distribution is shown in Figure 1 by the curve labelled SNIa. Although the uncertainties in this estimate at $z \approx 0$ are large, the excellent overlap lends support to the low-D SBBN baryon abundance. The poor overlap with the high-D SBBN baryon density argues against the high D/H being representative of the primordial deuterium abundance.
The Baryon Density At $z \sim 1000$ {#sec:cmb}
===================================
At redshift $z \sim 1000$, when the universe is several hundred thousand years old, the temperature of the CMB radiation has cooled sufficiently for neutral hydrogen (and helium) to form. The CMB photons are now freed from the tyranny of electron scattering and they propagate freely carrying the imprint of cosmic perturbations as well as encoding the parameters of the cosmological model, in particular the baryon density. Observations of the CMB anisotropies therefore provide a probe of at a time in the evolution of the universe intermediate between BBN and the present epoch.
Recent observations of the CMB fluctuations by the BOOMERANG [@boom] and MAXIMA [@max] experiments have provided a means for constraining the baryon density at $z \sim 1000$. The relative height of the first two “acoustic peaks" in the CMB anisotropy spectrum is sensitive to the baryon density. Although the precise value of $\Omega_B h^2$ depends on the choice of “priors" for the other cosmological parameters which must be included in the analysis, the CMB-inferred baryon density exceeds that derived from BBN (low-D) by $\sim 50\%$, $\Omega_B h^2 \sim 0.03$ ($\eta_{10} \sim 8$). The baryon density likelihood distribution shown in Figure 1 is based on the combined Boomerang and Maxima analysis of who find $\Omega_B h^2 = 0.032 \pm 0.005$ ($\eta_{10} = 8.8 \pm 1.4$).
It is clear from Figure 1 that while there is excellent overlap between the low-D SBBN and SNIa baryon density estimates, the high-D SBBN value is discordant. Furthermore, there is a hint that the CMB value may be too large. Note that the apparent “agreement" (or, minimal apparent disagreement) in Figure 1 is an artifact of normalizing each likelihood function to unit maximum. In fact, the CMB data excludes the central value of low-D SBBN at greater than 98% confidence. Although it may well be premature to take this “threat" to SBBN seriously, this potential discrepancy has led to the suggestion that new physics may need to be invoked to reconcile the BBN and CMB predictions for $\Omega_B h^2$. This possibility is discussed next.
Beyond SBBN {#sec:nsbbn}
===========
![Iso-abundance contours for deuterium (D/H), lithium (Li/H) and helium (mass fraction, Y) in the [$\Delta N_\nu\;$]{}– $\eta_{10}$ plane for four choices of $\nu_{e}$ degeneracy ($\xi_{e}$). The shaded areas highlight the range of parameters consistent with the adopted abundance ranges.[]{data-label="fig:iso"}](iso.eps){width="\columnwidth"}
Observations of deuterium and helium (and, perhaps lithium) offer the first challenge to SBBN and the baryon density derived from it (see § \[sec:bbn\]). Setting aside the very real possibility of errors in the observationally derived abundances, how might SBBN be modified to account for a helium abundance which is (predicted to be) too large? Not surprisingly, the options are manifold. One possibility is to modify the expansion rate of the early universe. If for some reason the universe were to expand more slowly than in the standard model, there would be more time for neutrons to convert to protons, resulting in a lower primordial helium abundance. In addition, a slower expansion would leave more time for deuterium to be destroyed resulting in a lower D-abundance. To compensate for this, the BBN baryon density would need to be [**reduced**]{}. This has the further beneficial effect of reducing the predicted lithium abundance, as well as reducing (very slightly) the predicted helium abundance. Thus, a [**slower**]{} expansion rate in the early universe can reconcile the predicted and observed deuterium and helium abundances (c.f. ). But, since this “solution" requires a [**lower**]{} baryon density, it exacerbates the tension between BBN and the CMB.
Although a [**speed up**]{} in the expansion rate offers the possibility of reconciling the observed deuterium abundance with the high baryon density favored by the CMB, it greatly exacerbates the helium abundance discrepancy and increases the tension between the predicted and observed lithium abundances. To reconcile the BBN and CMB estimates of the baryon density, while maintaining (or, establishing!) consistency between the predicted and observed primordial abundances, additional “new physics" needs to be invoked.
The simplest possibility for reducing the BBN-predicted helium abundance is a non-zero chemical potential for the electron neutrinos. An excess of $\nu_{e}$ over $\bar{\nu}_{e}$ can drive down the neutron-proton ratio, leading to reduced production of helium-4. Thus, one path to reconciling BBN with a high baryon density is to “arrange" for a faster than standard expansion rate ($S \equiv H'/H >
1$) [**and**]{} for degenerate $\nu_{e}$. Although these two effects need not be related, neutrino degeneracy can, in fact, provide an economic mechanism for both since the energy density contributed by degenerate neutrinos exceeds that from non-degenerate neutrinos, leading to an enhanced expansion rate during radiation-dominated epochs ($H'/H =
(\rho '/\rho)^{1/2} > 1$). Thus, one approach to non-standard BBN is to introduce two new parameters, the speed up factor ${\bf S}$ and the electron neutrino degeneracy parameter ${\bf \xi_e}$, where $\xi_e = \mu_e/T_\nu$ is the ratio of the electron neutrino chemical potential $\mu_e$ to the neutrino temperature $T_\nu$. For degenerate neutrinos the energy density ($\rho_{\nu}(\xi)$) exceeds the non-degenerate energy density ($\rho_{\nu}^{0}$)
$$\Delta\rho_{\nu}/\rho_{\nu}^{0} = 15/7[(\xi/\pi)^4 + 2(\xi/\pi)^2].$$
Thus, neutrino degeneracy has the same effect (on the early universe expansion rate) as would additional species of light, non-degenerate neutrinos. In terms of the equivalent number of “extra", non-degenerate, two-component neutrinos [**$\Delta N_\nu$**]{}, the speed up factor is
$$S = (1 + 7\Delta N_\nu/43)^{1/2}.$$
To facilitate comparison with the published literature, [$\Delta N_\nu\;$]{}is used in place of $S$. Since $\Delta N_\nu = \Delta\rho_{\nu}/
\rho_{\nu}^{0}$, [$\Delta N_\nu\;$]{}accounts for the additional energy density contributed by all the degenerate neutrinos (see eq. 1) [*as well as any other energy density not accounted for in the standard model of particle physics*]{} (e.g., additional relativistic particles) expressed in terms of the equivalent number of extra, non-degenerate, two-component neutrinos. However, our results are independent of whether [$\Delta N_\nu\;$]{}(or the corresponding value of $S$) arises from neutrino degeneracy, from “new” particles, or from some other source. Note that a non-zero value of $\xi_e$ implies a non-zero contribution to $\Delta N_\nu$ from the electron neutrinos alone. This contribution has been included in our calculations. However, for the range of $\xi_e$ which proves to be of interest ($\xi_e \la 0.5$; see Fig. 3), the degenerate electron neutrinos contribute only a small fraction of an additional neutrino species to the energy density ($\Delta N_\nu \la 0.1$). As and have shown, the observed primordial abundances of the light nuclides can be reconciled with very large baryon densities provided that $\xi_{e} > 0$ and [$\Delta N_\nu\;$]{}is sufficiently large.
The parameter space investigated is three-dimensional: $\eta$, $\xi_{e}$, and $\Delta N_\nu$. Generous ranges for the primordial abundances were chosen which are large enough to encompass systematic errors in the observations, as well as to account for the BBN uncertainties due to imprecisely known nuclear and/or weak reaction rates: $0.23 \le {\rm Y}_P \le 0.25$, $2 \times
10^{-5} \le {\rm D/H} \le 5 \times 10^{-5}$, $1 \times 10^{-10}
\le {\rm ^7Li/H} \le 4 \times 10^{-10}$. Since we wish to compare to the predictions of the CMB, which are sensitive to $\eta$ and $\Delta N_\nu$, but independent of $\xi_e$, the allowed BBN region is projected onto the $\eta - \Delta N_\nu$ plane. The BBN results are shown in the four panels of Figure 3 where, for four choices of $\xi_e$ the iso-abundance contours for Y$_{P}$, D/H and Li/H are shown. The shaded areas highlight the acceptable regions in our parameter space. As $\xi_e$ increases, the allowed region moves to higher values of $\eta$ and $\Delta N_\nu$, tracing out a BBN-consistent band in the $\eta - \Delta N_\nu$ plane. This band is shown in Figure 4 where the CMB constraints on the same parameters (under the assumption of a flat universe; for details and other cases, see ) are shown. The trends are easy to understand: as the baryon density increases the universal expansion rate (measured by $\Delta N_\nu$) increases to keep the deuterium and lithium unchanged, while the $\nu_{e}$ degeneracy ($\xi_{e}$) increases to maintain the helium abundance at its (correct!) BBN value.
![The BBN (dashed) and CMB (solid) contours (flat, $\Lambda$CDM model) in the [$\Delta N_\nu\;$]{}– $\eta_{10}$ plane. The corresponding best fit iso-age contours are shown for 11, 12, and 13 Gyr. The shaded region delineates the parameters consistent with BBN, CMB, and t$_{0} >
11$ Gyr.[]{data-label="fig:comp"}](cmb.eps){width="\columnwidth"}
The CMB anisotropy spectrum depends on the baryon density and on the universal expansion rate (through the relativistic energy density as measured by $\Delta N_{\nu}$) as well as on many other cosmological parameters which play no role in BBN. But, in fitting the CMB data, choices must be made (“priors") of the values or ranges of these other parameters. In several cosmological models and several choices for the “priors" were explored. Figure 4 shows the BBN/CMB comparison for the “flat, $\Lambda$CDM" model (Case C of ). The significant overlap between the BBN-allowed band and the CMB contours, confirms that if we allow for “new physics" ($\xi_{e} > 0$ and [$\Delta N_\nu\;$]{}$> 0$), the tension between BBN and the CMB can be relieved.
Since the points in the $\eta$ – [$\Delta N_\nu\;$]{}plane are projections from a multi-dimensional parameter space, the relevant values of the “hidden" parameters may not always be consistent with other, independent observational data which could provide additional constraints. As an illustration, three iso-age contours (11, 12, and 13 Gyr), are shown in Figure 4. The iso-age trend is easy to understand since as [$\Delta N_\nu\;$]{}increases, so too do the corresponding values of the matter density ($\Omega_{M}$) and the Hubble parameter (H$_{0}$) which minimize $\chi^{2}$. Furthermore, since $\Omega_{M} + \Omega_{\Lambda} =
1$, $\Omega_{\Lambda}$ decreases. All of these lead to younger ages for larger values of $\Delta N_\nu$. Note that if an age constraint is imposed (that the universe today is [**at least**]{} 11 Gyr old [@chaboyer]), then the BBN and CMB overlap is considerably restricted (to the shaded region in Figure 4). Even with this constraint it is clear that for modest “new physics” ([$\Delta N_\nu\;$]{}$\la 4$; $\xi_{e} \la 0.3$) there is a small range of baryon density ($0.020 \la \Omega_{B}h^{2} \la 0.026$) which is concordant with both the BBN and CMB constraints, as well as the present universe baryon density.
Summary and Conclusions {#sec:disc}
=======================
According to the standard models of cosmology and particle physics, as the universe evolves from the first few minutes to the present, the ratio of baryons (nucleons) to photons, $\eta$, should be unchanged. The abundance of deuterium, a relic from the earliest epochs, identifies a nucleon abundance $\eta_{10} \sim 5.6$. The CMB photons, relics from a later, but still distant epoch in the evolution of the universe suggest a somewhat higher value, $\eta_{10}
\sim 8.8$. Although most baryons in the present universe are dark and the path to the current nucleon-to-photon ratio is indirect, our estimates suggest $\eta_{10} \sim 5.1$. That these determinations are all so close to one another is a resounding success of the standard model. The possible differences may either reflect the growing pains of a maturing field whose predictions and observations are increasingly precise, or perhaps, be pointing the way to new physics. Exciting times indeed!
We are pleased to acknowledge Bob Scherrer and Terry Walker for their contributions to the work reviewed here. Financial support for this research at OSU has been provided by the DOE (DE-FG02-91ER-40690).
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---
abstract: 'Advances in Data Science permeate every field of Transportation Science and Engineering, making it straightforward to imagine that developments in the transportation sector will be data-driven. Nowadays, Intelligent Transportation Systems (ITS) could be arguably approached as a “story” intensively producing and consuming large amounts of data. A diversity of sensing devices densely spread over the infrastructure, vehicles or the travelers’ personal devices act as sources of data flows that are eventually fed to software running on automatic devices, actuators or control systems producing, in turn, complex information flows between users, traffic managers, data analysts, traffic modeling scientists, etc. These information flows provide enormous opportunities to improve model development and decision-making. This work aims to describe how data, coming from diverse ITS sources, can be used to learn and adapt data-driven models for efficiently operating ITS assets, systems and processes; in other words, for data-based models to fully become *actionable*. Grounded on this described data modeling pipeline for ITS, we define the characteristics, engineering requisites and challenges intrinsic to its three compounding stages, namely, data fusion, adaptive learning and model evaluation. We deliberately generalize model learning to be adaptive, since, in the core of our paper is the firm conviction that most learners will have to adapt to the ever-changing phenomenon scenario underlying the majority of ITS applications. Finally, we provide a prospect of current research lines within Data Science that can bring notable advances to data-based ITS modeling, which will eventually bridge the gap towards the practicality and actionability of such models.'
author:
- |
Ibai Laña, Javier J. Sanchez-Medina, , Eleni I. Vlahogianni, ,\
and Javier Del Ser, [^1] [^2] [^3]
bibliography:
- 'biblio.bib'
title: |
From Data to Actions in Intelligent Transportation Systems: a Prescription of Functional Requirements\
for Model Actionability
---
Data Science, Machine Learning, model actionability, functional requirements, data fusion, model evaluation.
Introduction
============
In the last years Intelligent Transportation Systems (ITS) have experienced an unparalleled expansion for many reasons. The availability of cost-effective sensor networks, pervasive computation in assorted flavors (distributed/edge/fog computing) and the so-called Internet of Things are all accelerating the evolution of ITS [@zhu2018big]. On top of them, Smart Cities cannot be understood anyhow without Smart Mobility and ITS as technological pillars sustaining their operation [@albino2015smart]. Smartness springs from connectivity and intelligence, which implies that massive flows of information are acquired, processed, modeled and used to enable faster and informed decisions.
For the last couple of decades, ITS have grown enough to cross pollinate with previously distant areas such as Machine Learning and its superset in the Artificial Intelligence taxonomy: Data Science. These days Data Science is placed at the methodological core of works ranging from traffic and safety analysis, modeling and simulation, to transit network optimization, autonomous and connected driving and shared mobility. Since the early 90’s most ITS systems exclusively relied on traditional statistics, econometric methods, Kalman filters, Bayesian regression, auto-regressive models for time series and Neural Networks, to mention a few [@zhang2011data; @karlaftis2011statistical]. What has changed dramatically over the years is the abundance of available data in ITS application scenarios as a result of new forms of sensing (e.g. crowd sensing) with unprecedented levels of heterogeneity and velocity. Zhang et al. [@zhang2011data] have defined this new form of data-driven ITS as the systems that have vision, multisource, and learning algorithms driven to optimize its performance and augment its privacy-aware people-centric character.
The exploitation of this upsurge of data has been enabled by advances in computational structures for data storage, retrieval and analysis, which have rendered it feasible to train and maintain extremely complex data-based models. These baseline technologies have laid a solid substrate for the proliferation of studies dealing with powerful modeling approaches such as Deep Learning or bio-inspired computation [@del2019bioinspired], which currently protrude in the literature as the *de facto* modeling choice for a myriad of data-intensive applications.
However, significant consideration must be placed to the systematic and myopic selection of complex data-based solutions over well-established modeling choices. The current research mainstream seems to be misleadingly focusing on performance-biased studies, in a fast-paced race towards incorporating sophisticated data-based models to manifold research area, leaving aside or completely disregarding the operational aspects for the applicability of such models in ITS environment. The scope of this work is to review existing literature on data-driven modeling and ITS, and identify the functional elements and specific requirements of engineering solutions, which are the ultimate enablers for data-based models to lead towards efficient means to operate ITS assets, systems and processes; in other words, for data-based models to fully become *actionable*.
Bearing the above rationale in mind, this work underscores the need for formulating the requirements to be met by forthcoming research contributions around data-based modeling in ITS systems.To this end, we focus mainly on system-level on-line operations that hinge on data-based pipelines. However, ITS is a wide research field, encompassing operations held at longer time scales (e.g. long-term and mid-term planning) that may not demand some of the functional requirements discussed throughout our work. Furthermore, our discussions target system-level operations rather than user-level or vehicle-level applications, since in the latter the information flow from and to the system is scarce. Nevertheless, some of the described functional requirements for system-level real-time decisions can be extrapolated to other levels and time scales seamlessly.
Bearing the above observation in mind, our ultimate goal is to prescribe – or at least, set forth – the main guidelines for the design of models that rely heavily on the collection, analysis and exploitation of data. To this end, we delve into a series of contributions that are summarized below:
- In the first place, we identify the gap between the data-driven research reported so far, and the practical requirements that ITS experts demand in operation. We capitalize on this gap to define what we herein refer to as *actionable data-based modeling workflow*, which comprises all data processing stages that should be considered by any actionable data-based ITS model. Although diverse data-based modeling workflows can be found in literature with different purposes, most of them count on recognized stages, that are presented in this work from an actionability perspective, i.e., what to take into account from the operational point of view when designing the workflow, how to capture and preprocess data, how to develop a model and how to prescribe its output. These guidelines are proposed and argued within an ITS application context. However, they can be useful for any other discipline in which data-based modeling is performed.
- Next, functional requirements to be satisfied by the aforementioned workflow are described and framed in the context of ITS systems and processes, with examples exposing their relevance and consequences if they are not fulfilled.The contributions of this section are twofold: on one hand, we identify and define the holistically actionable ITS model along with its main features; on the other hand, we enumerate requirements for each feature to be considered actionable, as well as a review of the latest literature dealing with these features and requisites.
- Finally, on a prospective note we elaborate on current research areas of Data Science that should progressively enter the ITS arena to bridge the identified gap to actionability. Once the challenges of modeling and ITS requirements have been stated, we review emerging research areas in Artificial Intelligence and Data Science that can contribute to the fulfilment of such requirements. We expect that our reflexive analysis serves as a guiding material for the community to steer efforts towards modeling aspects of more impact for the field than the performance of the model itself.
The above contribution is organized throughout the rest of the paper as follows: Section \[pipeline\] delves into the *actionable data-based modeling workflow*, i.e. the canonical data processing pipeline that should be considered by a fully actionable ITS system with data-based models in use. Section \[requirements\] follows by elaborating on the functional features that an ITS system should comply with so as to be regarded as *actionable*. Once these requirements are listed and argued in detail, Section \[challenges\] analyzes research paths under vibrant activity in areas related to Data Science that could bring profitable insights in regards to the actionability of data-based models for the ITS field, such as explainable AI, the inference of causality from data, online learning and adaptation to non-stationary data flows. Finally, Section \[conclusion\] concludes the paper with summarizing remarks drawn from our prospect.
From Data to Actions: An Actionable Data-based Modeling Workflow {#pipeline}
================================================================
ITS applications with data driven modeling problems underneath range from the characterization of driving behavioral patterns, the inference of typical routes or traffic flow forecasting, among others. Data driven modeling can be considered to include the family of problems where a computational model or system must be characterized or learned from a set of inputs and their expected outputs [@eiben2003introduction]. In the context of this definition, actionability complements the data-driven model by prescribing the actions (in the form of rules, optimized variable values or any other representation alike) that build upon the output knowledge enabled by the model.
In general, a design workflow for data-based modeling consists of 4 sequential stages: 1) data acquisition (*sensing*), which usually considers different sources; 2) data preprocessing, which aims at building consistent, complete, statistically robust datasets; 3) data modeling, where a model is learned for different purposes; and 4) model exploitation, which includes the definition of actions to be taken with respect to the insights provided by models in real life application scenarios. These 4 stages can be regarded as the core of off-line data-driven modeling; however, when time dimension joins the game, a fifth stage – adaptation – must be considered as an iterative stage of this data pipeline, aimed at maintaining learned models updated and adapted to eventual changes in the data distribution. This adaptation is crucial for real-life scenarios, where changes can happen in all stages, from variations of the input data sources, to interpretation adjustments and other sources of non-stationarity imprinting the so-called *concept drift* in the underlying phenomenon to be modelled [@ditzler2015learning]. We now delve into these five data processing stages in the context of their implementation in ITS applications, following the diagram in Figure \[fig:1\].
{width="1.8\columnwidth"}
Data Acquisition (Sensing) {#sensing}
--------------------------
The path towards concrete data-based actions departs from the capture of available ITS information, which in this specific sector is plentiful and highly diverse. The advent of data science for ITS has come along with the unfolding of copious data sources related to transportation. Indeed, ITS are pouring volumes of sensed data, from the environment perception layer of intelligent and connected vehicles, to human behaviour detection/estimation (drivers, passengers, pedestrians) and the multiple technologies deployed to sense traffic flow and behaviour. Concurrently, many other non-traditional sources that were useful to infer behavioral needs and expectations of people that use transportation, such as social media, have started to become increasingly available and exploited augmenting the more conventional sensing sources towards more efficient mobility solutions. Some of these data sources are currently used in almost any domain of ITS, from operational perspectives such as the estimation of future transportation demands, adaptive signaling or the discovery of mobility patterns, to the provision and of practical solutions, such as the development of autonomous vehicles.
Five main categories can be established to describe the spectrum of ITS data sources:
1. *Roadside sensing*, which brings together tools and mechanisms that directly capture and convey data measurements from the road, obtaining valuable metrics such as speed, occupation, flow or even which vehicles are traversing a given road segment. These are the most commonly used sensors in ITS, most frequently based on computer vision and radar, as they directly provide traffic information close to the point where it originates. However, this information is tethered to the exact points where the sensors are placed, thus the actionability of a system built upon these data is subject to the geographical area where such sensing devices are deployed and their range.
2. *Structured/static data*, which refers to data sources that provide information of elements that have a direct impact on transportation, such as public transportation lines and timetables, or municipal bike rental services. Due to their inherently structured nature, data provided by these sources are often arranged in a fixed format, making it easier to incorporate to subsequent data-based modeling stages. These information sources must be considered for an intelligent transportation system to be actionable, being a particularly essential piece of urban and interurban mobility.
3. *Cooperative sensing*, which denotes the general family of data collection strategies that regards the information provided by different users of the ITS ecosystem as a whole, thus being grouped and jointly processed forward. This inner perspective of traffic and transportation can be obtained through many mechanisms, and, although it is more specific and scarce, it is also more complete than the one obtained from roadside sensing. These data open the door to mobility profiling and anomaly detection, enriching the outlook of a transportation model by means of the fusion of different data-based *views* of an ITS scenario. This includes all forms of mobile sensing data, from call detail record data that can be used to obtain users trajectories [@kujala2016estimation], to GPS data [@sun2015trajectory]. These sources are the foundation of abundant research [@rodrigues2011mobile; @lana2018road], but in most cases the data fusion part is obviated. Crowdsourced and Social Media sensing can be analogously considered in this category. These data sources can also contribute to data-based ITS models by means of sentiment analysis and geolocation. The use of crowd-sourced data is well established among technology-based companies (Google, Uber etc), yet not very often available to research community and private and public authorities in the transport operations management. The limited information that becomes available is deprived from the necessary statistical representativeness and truthfulness in order to be easily integrated to legacy management systems.
4. *External data sources*, which include all data that are not directly related to traffic of demand, but have an impact on it, such as weather, calendar, or planned events, social and economic indicators, demographic characteristics etc. These data are usually easy to obtain, and their incorporation to ITS models augments in general the quality of their produced insights and ultimately, the actionability of the actions yielded therefrom. It is also true that this data source is typically unstructured, which can pose a challenge regarding its automatic integration.
Data Preprocessing
------------------
The variety of the above mentioned sensing sources comes with promises and perils. These data come in various forms and formats, various time resolutions, synchronously or asynchronously and different rates of accumulation. To leverage the full spectrum of knowledge these data can bring to the sake of informed decision making, the more the sensing opportunities the larger the needs for powerful preprocessing and skills are before reaching the stage of modeling.
A principled data-driven modeling workflow requires more than just applying off-the-shelf tools. In this regard, preprocessing raw data is undoubtedly an elementary step of the modeling process [@garcia2015data], but still persists nowadays as a step frequently overlooked by researchers in the ITS field [@lopes2010traffic].
To begin with, when a model is to be built on real ITS data, an important fact to be taken into account is the proneness of real environments to present missing or corrupted data due to many uncertain events that can affect the whole collection, transformation, transmission and storage process [@vlahogianni2004short]. This issue needs to be assessed, controlled and suitably tackled before proceeding further with next stages of the processing pipeline. Otherwise, missing and/or corrupted instances within the captured data may distort severely the outcome of data-based models, hindering their practical utility [@chen2001study]. A wide extent of missing data imputation strategies can be found in literature [@qu2009ppca; @tan2013tensor], as well as methods to identify, correct or discriminate abnormal data inputs [@li2014missing]. However, they are often loosely coupled to the rest of the modeling pipeline [@ran2016tensor]. An actionable data preprocessing should focus not only on improving the quality of the captured data in terms of completeness and regularity, but also on providing valuable insights about the underlying phenomena yielding missing, corrupted and/or outlying data, along with their implications on modeling [@lana2018imputation].
Next, the cleansed dataset can be engineered further to lie an enriched data substrate for the subsequent modeling [@krempl2014open; @etemad2018predicting]. A number of operations can be applied to improve the way in which data are further processed along the chain. For instance, data transformation methods can be applied for different purposes related to the representation and distribution of data (e.g. dimensionality reduction, standardization, normalization, discretization or binarization). Although these transformations are not mandatory in all cases, a deep knowledge of what input data represent and how they contribute to modeling is a key aspect to be considered in this preprocessing stage.
Furthermore, data enrichment can be held from two different perspectives that can be adopted depending on the characteristics of the dataset at this point. As such, feature selection/engineering refers to the implementation of methods to either discard irrelevant features for the modeling problem at hand, or to produce more valuable data descriptors by combining the original ones through different operations. Likewise, instance selection/generation implies a transformation of the original data in terms of the examples. Removing instances can be a straight solution for corrupted data and/or outliers, whereas the addition of synthetic instances can help train and validate models for which scarce real data instances are available. Besides, these approaches are among the most predominant techniques to cope with class imbalance [@zheng2013using], a very frequent problem in predictive modeling with real data. Whether each of these operations is required or not depends entirely on the input data, their quality, abundance and the relations among them. This entails a deep understanding of both data and domain, which is not always a common ground among the ITS field practitioners [@smith2004investigation].
Finally, data fusion embodies one of the most promising research fields for data-driven ITS [@zhang2011data; @el2011data], yet remains marginally studied with respect to other modeling stages despite its potential to boost the actionability of the overall data-based model. Indeed, an ITS model can hardly be actionable if it does not exploit interactions among different data sources. Upon their availability, ITS models can be enriched by fusing diverse data sources. A recent review on different operational aspects of data-driven ITS developments states that these models rarely count on more than one source of data [@lana2018road]. This fact clearly unveils a niche of research when taking into account the increasing availability of data provided by the growing amount of sensors, devices and other data capturing mechanisms that are deployed in transportation networks, in all sorts of vehicles, or even in personal devices held by the infrastructure users. Despite the relative scarcity of contributions dealing with this part of the data-based modeling workflow, the combination of multiple sources of information has been proven to enrich the model output along different axis, from accuracy to interpretability [@choi2002data; @chang2010intelligent; @han2010radar; @treiber2011reconstructing].
Modeling
--------
Once data are obtained, fused, preprocessed and curated, the modeling phase implies the extraction of knowledge by constructing a model to characterize the distribution of such data or their evolution in time. The distillation of such knowledge can be performed for different purposes: to represent unsupervised data in a more valuable manner (as in e.g. clustering or manifold learning), for instance, to insight patterns relating the input data to a set of supervised outputs (correspondingly, classification/regression) aiming to automatically label unseen data observations, to predict future values based on the previous values (time series forecasting), or to inspect the output produced by a model when processing input data (simulation). To do so, in data-based modeling machine learning algorithms are often put to use, which allow automating the modeling process itself.
The above purposes can serve as a discrimination criterion for different algorithmic approaches for data-based modeling. However, when the goal is to model data interactions within complex systems such as transportation networks, it is often the case that the modeling choice resorts to ensembles of different learner types. For instance, when applying regression models for road traffic forecasting, a first clustering stage is often advisable to unveil typicalities in the historical traffic profiles and to feed them as priors for the subsequent predictive modeling [@vlahogianni2009enhancing; @lana2019; @liu2019mining]. However, when it comes to model actionability, a key feature of this stage is the *generalization* of the developed model to unseen data. This characteristic implies making a model useful beyond the data on which it is trained, which implies that the model design efforts should not only be put on making the model achieve a marginally superior performance, but also to be useful in other spatial or temporal circumstances. Achieving good generalization properties for the developed can be tackled by diverse means, which often depend on the modeling purpose at hand (e.g. cross-validation, regularization, or the use of ensembles in predictive modeling). Essentially, the design goal is to find the trade-off between performance (through representing much of the intrinsic variance of data) and generalization (staying away from an overfitted model to a particular training set). This aspect becomes utterly relevant when data modeling is done on time-varying data produced by dynamic phenomena. ITS are, in point of fact, complex scenarios subject to strong sources of non-stationarity, thereby calling for an utmost focus on this aspect.
The complexity met in traffic and transportation operations is usually treated with heterogeneous modeling approaches that aim to complement each other to improve accuracy [@moretti2015urban; @cong2016traffic; @kim2015urban]. This can be done either by comparing different models and selecting the most appropriate one every time, or by combining different models to produce the final outcome. Additionally, in some fields of ITS, such as traffic modeling, physical (namely, theory- or simulation-based) models have been available for decades. Their integration into data-based modeling workflows, considering the knowledge they can provide, can become crucial for a manifold of purposes, e.g. to enforce traffic theory awareness in models learned from ITS data. Indeed, the hybridization of physical and data-based models has a yet to be developed potential that has only been timidly explored in some recent works [@fusco2015short; @montanino2015trajectory; @chaulwar2016hybrid].
Interestingly, complex data driven modeling solutions to transportation phenomena have been numerous and resourceful ranging from modular structures, to model combinations, surrogate modeling [@vlahogianni2015optimization] and so on. Regardless of the approach, literature emphasizes on the critical issue of model hyperparameter optimization using for example nature inspired algorithms, namely Evolutionary Computation or Swarm Intelligence [@cong2016traffic; @teodorovic2008swarm]. Assuming that there is a feasible and acceptable solution to the problem of selecting the proposed parameters for a data drive model, when dealing with complex modeling structure this task should be conducted automatically by optimizing the hyperparameter space usually based on the models’ predictive error. It is to note that, the greater the number of models involved the more difficult the optimization task becomes. Moreover, relying on nature inspired stochastic approaches, full determinism in the solution and convergence stability can not be formally guaranteed [@del2019bioinspired].
Prescription
------------
Once the modeling phase itself has been completed, the resulting model faces its application to a real ITS environment. It is at this stage when actuations deriving from the data insights are defined/learned/decided, and when the actionability of the model can be best assessed. Actions taken as a result of the outcome of a data-based model can serve for strategic, tactical or operational decision making, for which the output of the data-based model supports decisions made by transportation networks managers. Such an output can be consumed directly without any need for further modeling, or exploited by means of a secondary modeling process aimed at optimizing the decision making process. This latter case can be exemplified, for instance, by the formulation of the decision making process as an optimization problem, in which actions are represented by the variables compounding a solution to the problem, and the output of the previous data-based modeling phase can be used to quantitatively estimate the quality or fitness of the solution. One of the most prominent examples of this prescription mode deals with routing problems, since they often use simulation tools or predictive models to assess the travel time, pollutant emissions or any other optimization objective characterizing the fitness of the tested routes [@kumar2012survey; @osaba2016improved]. Other examples of prescription based on data emerge in tactical and strategic planning, such as the modification of public transportation lines [@mendes2015validating], the establishment of special lanes (e.g. taxi, bike) [@szeto2015sustainable], the improvement of road features [@van2016automatic], the adaptive control of traffic signaling [@mannion2016experimental], the identification of optimal delivery (or pickup) routes for different kinds of transportation services [@osaba2017discrete], the incident detection and management [@imprialou2013methods], learning for automated driving [@yu2019distributed], or the design of urban sustainable mobility plans based on the current and future demand or the drivers’ behavior [@lecue2014star; @gindele2015learning].
In any of the above presented ITS cases, a data-based model should be equipped with a certain set of features that guarantee its actionability. For instance, if a traffic manager is not able to interpret a model or understand its outcome in terms of confidence, it can be hardly applied for practical decision making. When the model is used for adaptive control purposes (as in automated traffic light scheduling), the adaptability of the model to contextual changes is a key requirement for prescribed actions to be matched to the current traffic status [@kammoun2014adapt]. Interestingly, some control techniques with a long history in the field (e.g. Stochastic Model Predictive Control, SMPC, [@mesbah2016stochastic]) serve as a good example of the triple-play between application requirements, decision making and data-based models. When dealing with the design of control methods in ITS, SMPC has been proven to perform efficiently in highly-complex systems subject to the probabilistic occurrence of uncertainties [@hrovat2012development]. Specifically, SMPC leverages at its core data-based prediction modeling and low-complexity chance-constrained optimization to deal with control problems that impose that the method to be used must operate in real time. In this case, and in most actionable data-based workflows where decision making is formulated as an optimization problem, we note a clear entanglement between application requirements (e.g. real-time processing), decision making (low-complexity, dynamic optimization techniques) and data-based models (predictive modeling for system dynamics forecasting).
Adaptation {#sec:adapt}
----------
Finally, the proposed actionable data processing workflow considers model adaptation as a processing layer that can be applied over different modeling stages along the pipeline. When models are based on data, they are subject to many kinds of uncertainties and non-stationarities that can affect all stages of the process. Streaming data initially used to build the model can experience long-term drifts (for instance, an increase of the average number of vehicles), sudden changes (a newly available road), or unexpected events (for example, a public transportation strike)[@buchanan2015traffic; @pan2013crowd; @davison2006bus]. A closed lane, a new tram line, the opening of a tunnel or simply the opening of a new commercial center, may change completely the way in which network users behave, and thus, affect the data-based models that are intended to reflect such a mobility. Therefore, data-based modeling cannot be conceived as a static design process. This critical adaptation should be considered in all parts of the workflow, and constantly updated with new data:
1. In the preprocessing stage, adaptation could be understood from many perspectives: the incorporation of new sources of data, the partial or total failure of data capturing sensors, which lead to an increased need for data fusion, imputation, engineering or augmentation.
2. In the modeling stage, adaptations could range from model retraining, adaptation to new data or alternative model switching, to the change of the learning algorithm due to a change in the requested system requirements (for instance, in terms of processing latency any other performance indicator).
3. In the prescription stage, adaptation is intended to dynamically support decisions accounting for changes in data that propagate to the output of preceding modeling stages. Data-based modeling can deal with such changes and adapt their output accordingly, yet they are effective to a point. For instance, online learning strategies devised to overcome from concept drift in data streams can speed up the learning process after the drift occurs (by e.g. diversity induction in ensembles or active learning mechanisms). Unfortunately, even when model adaptation is considered the performance of the adapted model degrades at different levels after the drift. Extending adaptation to the prescription stage provides an additional capacity of the overall workflow to adapt to changes, leveraging techniques from prescriptive analysis such as dynamic or stochastic optimization.
Adaptations within the above stages can be observed from two perspectives: automatic adaptations that the system is prepared to do when certain circumstances occur, or adaptations that are derived from changes that are introduced by the user. Thus, the adaptation layer is strongly linked to actionability: an ITS model will be more actionable if adaptations, either needed or imposed, are accessible to its final users. For instance, a system could be required to introduce a new set of data, and its impact on all the stages should be controlled by the transportation network manager, or if a drift is detected, the system should consider if it is relevant to inform the user.
Functional Requirements for Model Actionability {#requirements}
===============================================
Any data-based modeling process should embrace actionability as its most desirable feature for the engineered model to yield insights of practical value, so that field stakeholders can harness them in their decision making processes. This is certainly the case of ITS, in which managers, transportation users and policy makers rely on models and research results to make better and more informed decisions. Thus, once the main stages of data-driven modeling have been outlined, this section places the spotlight on the main functional features that should be mandatory to produce fully-actionable ITS data-based models. These functional requirements, which are shown in Figure \[fig:2\], should not be understood as a compulsory list of features, but rather as an enumeration of possibilities to make a model actionable. Not all ITS scenarios requiring actionable data-based models should impose all these requirements, nor can actionability be thought to be a Boolean property. Different loosely defined degrees of actionability may hold depending on the practicality of decisions stemming from the model.
![Functional requirements for actionable data-based models in ITS.[]{data-label="fig:2"}](PIPELINE2col.pdf){width="1\columnwidth"}
Usability
---------
The way in which humans interact with information systems has been thoroughly studied in last decades and formalized under the general *usability* term [@nielsen1994usability]. Although usability is a feature that can be associated to any system in which there is some kind of interaction with the user, most of its definitions to date gravitate around the design of software systems [@nielsen1994usability2; @brooke1996sus; @nielsen199510], which is not necessarily the case of ITS research. Usable designs imply defining a clear purpose for a system, and helping users making use of it to reach their objectives [@nielsen2003usability]. Within ITS, there are domains where this definitions apply directly [@noy1997human], such as vehicle user interfaces [@green1999estimating; @green1999navigation; @burns2010importance], the development of navigation systems [@burnett2000turn], road signalization [@dos2017proposal], or even the way in which public transportation systems information is shown to users [@avelar2006design; @roberts2016radi].
The aforementioned domains of application, and mostly any system lying at the core of Advanced Traveler Information Systems (ATIS), have an explicit interaction component. On the other hand, models developed for Advanced Transportation Management Systems (ATMS) are less related to user interaction (beyond the interface design of decision making tools), hence this canonical definition of usability seems to be less applicable. However, the general concept of usability can also accommodate the notion of *utility* as the quality of a system of being useful for its purpose, or the concept of *effectiveness*, in regards to how effective is the information provided by them [@lyons2001advanced]. Since ITS are systems developed as tools designed to help the different stakeholders that take part in transportation activities, the actionability of data-based models used for this regard depends stringently in this general idea of usability [@barfield2014human]. Models’ usability is a feature largely disregarded in literature. A clear example of this situation is traffic forecasting, a preeminent subfield of ATMS, in which the link between the high end deep learning models with the requirements by the road operators in forecasts to support the decision making is very weak [@Vlahogianni2014]. Usability may relate to the person that is going to operate the model, and to the type and complexity of the model, which relate to specific skills. Achieving usable ITS models does not entail the same efforts for all ITS subdomains. Thus, while for research contributions related to ATIS there is a clear interest in this matter [@horan2006assessing], for ATMS developments some extra considerations need to be made. Usability in ITS has, therefore, a facet oriented towards user interface, where interfaces reflect at least one of the outputs of an ITS data-based model, and another facet towards creating models that are more aware of the way their outputs are going to be consumed afterwards by the decision maker.
### User Interface
For the first of these facets, Spyridakis et al. [@barfield2014human] propose general software usability measuring tools and scales such as System Usability Scale (SUS) [@brooke1996sus], ethnographic field studies, or even questionnaires. These basic techniques are also proposed in [@ross2001evaluating] in order to evaluate navigation systems interfaces. There are also many other evaluation measures that are more specific to the field, such as [@fischer2002human], or those defined by public authorities [@dingus1996development]. Some of the main techniques to appraise ITS interface usability are:
- *Usability techniques:* if the output of the developed model is consumed through the use of an interface, common techniques like asking directly the users about their experience can be adopted [@ross2001evaluating]. Among them SUS surveys are the standard to provide interpretable metrics that can be used for the evaluation of passenger information systems [@beul2014usability] or any other kind of automated traveler information system [@horan2006assessing].
- *Quality of the provided information:* in [@lyons2001advanced], another perspective is proposed, based on estimating the quality of the information provided by the model. Characteristics such as the means to access the information, the reliability of the information provider, or the awareness of the information availability can be measured for assessing the model’s usability.
- *Transportation-aware strategies:* an alternative way to measure usability is to take into account the transportation context and how the use of the model impacts the system. As many of these systems are used during the course of transportation, the environment must be considered in order to provide an adequate and pertinent output [@dingus1996development]. This particular aspect is regarded below in section \[sec:appcon\].
- *Public transportation guidelines:* when ITS developments are intended for the public domain, inclusion of disadvantaged collectives in the usability evaluation is a must [@fischer2002human]. The extent in which these concerns are addressed by the ITS solution should not be disregarded.
### Consumption of the Model’s Output
For this second usability facet, there are no scales or measurements in literature that provide an objective (or even subjective) usability assessments, but we propose some angles that should be considered when designing this kind of models:
- *Confidence-based outputs:* data-driven models are often subject to stochasticity as a result of their learning procedure or the uncertainty/randomness of their input data (as specially occurs in crowdsourced and Social Media data). This randomness imprints a certain degree of uncertainty in their outputs, which can be estimated values, predicted categories, solutions to an optimization problem or any other alike. Such outcomes are often assessed in terms of their similarity to a ground truth in order to quantitatively assess the performance of the data-based model. Thus, a practitioner aiming to make decisions based on the model’s output is informed with a nominal performance score (which has been computed over test data), and the predicted output for a given input. However, when one of such data-based models is intended to work in a real environment, there is no ground truth to evaluate the quality of the result they are providing towards making a decision.For instance, a predictive model could score high on average as per the errors made during the testing phase. However, predictions produced by the model could be less reliable during peak hours than during the night, being less trustworthy in the first case as per the variability of the data from which it was learned, and/or the model’s learning algorithm itself. For this reason, the estimation of the confidence of outputs from a data-based model must be analyzed for the sake of its usability. For example, a public transportation model that provides outlooks of future demand could be more usable if, besides the estimation itself, some kind of confidence metric was provided. Elaborating on this aspect is not very frequent in academic research, mainly due to the fact that confidence is not always that easy to obtain and the estimation procedure is, in most cases, model-specific,requiring a previous statistical analysis of input data to properly understand their variability and characteristics. Unfortunately, such a confidence analysis is usually left out of the scope of research contributions, which rather focus on finding the best scoring model for a particular problem. Exceptions to this scarcity of related works are [@mazloumi2011prediction], in which the uncertainty inherent to artificial neural networks is analyzed in a real ITS context; [@van2009bayesian], in which a committee of different models provides intervals of confidence to predictions;or the more recent contribution in [@liu2019dynamic], which departs from previous findings in [@tsekeris2009short; @khosravi2011prediction] to estimate the uncertainty of traffic demand. This uncertainty estimation is then used as an input to assess the confidence of traffic demand predictions. These few references exemplify good practices that should be universally considered in contributions to appear.
- *Interpretability:* a stream of work has been lately concentrated around the noted need for *explaining* how complex models process input data and produce decisions/actions therefrom. Under the so-called XAI (eXplainable Artificial Intelligence) term, a torrent of techniques have been reported lately to explain the rationale behind traditional black-box models, mainly built for prediction purposes [@gunning2017explainable; @arrieta2020explainable]. Nowadays, Deep Learning is arguably the family of data-driven models mostly targeted by XAI-related studies [@samek2017explainable; @ras2018explanation].
The interest of transport researchers to interpretable data-driven models is not new; intuitively, any decision in transportation and traffic operations should be based on a solid understanding of the mechanism by which different factors interact and influence transportation phenomena [@vlahogianni2012modeling]. In the transportation context explainability is closely related to integrability, when it comes to traffic managers, as ensuring that data-based models can be understood by non-AI expert can make them appropriately trust and favor the inclusion of data-based models in their decisional processes. When framed within ITS systems and processes, the need for explainable data-based models can help decision makers understand how information is processed along the data modeling pipeline, including the quantification of insightful managerial aspects such as the relationship and sensibility of a predicted output with respect to their inputs.
- *Trade-off between accuracy and usability:* when ITS data-based models aim at superior performances, they often work in ideal scenarios where the real context of application is disregarded; should that context apply in practice, the claimed suitability of the developed model for its particular purpose could be compromised. For instance, the goodness of an ITS model devised to detect users’ typical trajectories can be measured with regard to the exactitude of the detected trajectories. If the pursuit of a superb performance relies on a constant stream of data (hence, eventually depleting the user’s phone battery), it could be a pointless achievement when put to practice. This particular example has been already considered by plenty of researchers [@thiagarajan2009vtrack; @thiagarajan2011probabilistic]. However, there is a long way to go in this aspect, as most ITS research developments consider only ideal circumstances without regarding the implications that an accurate design could have on its final usability.
Self-Sustainability {#sec:self-sust}
-------------------
In general, self-sustainability of a model refers to its ability to survive – hence, to continue to be useful – in a dynamic environment. ITS models and developments are usually intended to operate during long periods of time. However, it is widely accepted that traffic and transportation phenomena are strongly dynamic in nature, meaning that these phenomena exhibit long term trends, evolve in space and time, but also, at the occurrence of an unexpected event, they are susceptible to abrupt changes and exhibit long term memory effects. For instance, a trip information system based on traffic forecasts on a certain part of the network trained with historical data coming from recurrent traffic conditions may not be easily transferable to other road networks or not efficient in case of a severe disruption in traffic operation (accident). What is more, if the specific system does not undergo constant training with new data over time, eventually it will fail to correctly operate even for the network location it was originally designed to operate due to contextually induced non-stationarities. Thus, an intelligent transportation system developed based on data-based approaches should at least follows a set of minimum self-sustainability requirements during the design workflow.
To better understand the importance of self-sustainability as a significant aspect of model’s actionability, one should bring to mind the case of cooperative ITS systems (e.g. advanced vehicle control systems) and the automated driving. To this end, a self-sustainable data-based model should bridge the gap between the development of a model prototype and its deployment in a real, potentially non-stationary environment.
When an ITS system or model is deployed to operate in changing conditions, self-sustainability involves dealing with the effects of such changes in the learned knowledge. To this end, different strategies and design approaches could be required depending on the nature of the change and its effects on the model. We next delve into several attributes that can be desirable to deploy data-driven systems or models in changing environments, rendering them actionable:
1. *Adaptable*: Data-driven models for ITS applications created in controlled conditions, with static, self-contained datasets, can provide great performance metrics, but could also fail if data evolve along time [@geisler2012evaluation]. Adaptation is the reaction of a system, model or process to new circumstances intended to reduce its performance deterioration in comparison to the one expected before the change in the environment happened. If data change over time, their evolution is not detected by the model and it does not adapt to it whatsoever, then the developed model will eventually provide an obsolete output. When these contextual variations occur over data streams and models are learned on-line therefrom (for e.g. on-line clustering or classification), such variations can imprint changes in the statistical distribution of input and/or output data, making it necessary to update such models to reflect this change in their learned knowledge. This phenomenon is known as *concept drift* [@gama2014survey], and has been identified as an active research challenge for most of fields connected to machine learning in non-stationary environment [@vzliobaite2016overview]. Many of those fields are already studying this topic, from spam detection [@delany2005case; @mendez2006tracking] to medicine [@stiglic2011interpretability].
There are two main lines related to concept drift: how to detect drift, and how to adapt to it. Both lines should be scheduled in the research agenda of data-driven ITS, as they have obvious implications when analyzing traffic [@moreira2014improving]. Situations like road works can modify completely traffic profiles over a certain area during a period of time, after which the situation goes back to normal. A similar casuistry happens with road design changes (i.e. new lanes, transformation of types of lanes, new accesses, roundabouts, etc), although in those cases there is a new stable traffic profile largely after the change. Even without man-crafted changes, traffic profiles may change for social-economical reasons [@lana2016role]. Besides, analysis of drift can be used to detect anomalies in the normal operation of roads [@moreira2015drift3flow], or to analyze patterns in maritime traffic flow data [@osekowska2017maritime]. However, the adaptability of ITS models to evolving data is scarcely found in literature, and certainly, in many cases concept drift management is the scope of the work, and not a circumstance that is considered to achieve a greater goal [@moreira2015drift3flow; @wibisono2016traffic]. There are though some online approaches to typical ITS problems that consider the effects of drift in data [@lana2019; @wu2012online; @procopio2009learning], and we consider this kind of initiatives should lead the way for an actionable ITS research.
2. *Robust*: When an ITS system is deployed in a real-life environment, diverse kinds of setbacks can affect its normal operation, from power failures that preclude its functioning to the interruption of the input data flow. Robustness is a self-sustainability trait that prevents a system to stop working when external disruptions occur. Although in most research-level designs this is not a relevant feature, it is essential for actionable, self-sustainable designs. Robustness, defined as the ability to recover from failures, would have, however, different requirements depending on the criticality of the ITS system. Thus, in a traffic flow forecasting system robustness could only imply that the system does not crack when input data fail [@zhang2008short], and it continues to operate; on the other hand, for critical systems such as air traffic management, robustness would require additional measurements to contain damage [@isaacson2010concept; @chen2017air]. All in all, robust data-based workflows should be able to accommodate unseen operational circumstances, such as data distribution shifts or unprecedented levels of information uncertainty, which particularly prevail in crowdsourced and Social Media data [@wechsler2019pervasive; @adar2007managing].
3. *Stable and resilient*: Actionable systems require a certain output stability in order to be understandable by their users. This notion is apparently opposed to adaptability, but while the latter is the ability to adapt the output to environment or data changes, stability pursues maintaining the output statistically bounded even when contextual changes occur, through e.g. model adaptation techniques. When adaptation is not perfect and the model violates a given level of statistical stability, stability requires another kind of adaptation, namely *resilience*, to make the model return to its normal operation and thus, minimize the impact of external changes on the quality of its output [@de2017mathematical]. This entails, in essence, going one step further in the knowledge of the environment and taking into account those circumstances that can affect the system, and it could be linked to transferable models, which would be addressed below. For instance, a traffic volume characterization model would be adaptable if it considers the changes inherent to traffic volume (an increase over time due to economical factors), and it would be stable if a change in the weather conditions does not deteriorate its performance, or in other words, it has considered this essential circumstance. These kind of considerations are almost nonexistent in literature [@Vlahogianni2014], but however crucial for a model to be self-sustainable.
4. *Scalable*: In the research environment, tests are run in a delimited scale, constrained to the size of data, and useful for the experiments, in contrast with large, multi-variate real environments. Scaling up is not, of course, a matter of ITS research, but an engineering problem. However, models should be designed to be scalable since their conception.
Leaving aside calibration and training phases, classic transportation theories tend in general to be computationally more affordable than data-driven models. However, the unprecedented amount of computing power available nowadays discards any real pragmatic limitation due to the computational complexity of learning algorithms in data-based modeling. An exception occurs with models falling within the Deep Learning category which, depending on their architecture and size of training data, may require specialized computing hardware such as GPU or multi-core equipment. Nevertheless, the rising trend in terms of scalability is to make data-based models incremental and adaptable [@zhang2011data], which finds its rationale not only in the environmental sustainability of data centers (lower energy consumption and thereby, carbon footprint), but also in the deployment of scalable model architectures on edge devices, usually with significantly less computing resources than data centers.
Although some ITS problems are easier to scale and this feature would not be troublesome, there are some fields that can be very sensitive to scalability. For instance, route planners frequently consist of shortest-path problem and travel-salesman problem implementations that increase in complexity when the number of nodes grow [@colpaert2016impact]. This is a good example where artificial intelligence and optimization tools provide solutions that are actionable in terms of scalability, and where cases are found effortlessly [@basu2016genetic; @schmitt2018experimental]. Caring about aspects like the easiness to introduce new variables when needed, the complexity of tuning if applies, or the execution time, would make a model more actionable, by increasing its self-sustainability. This need for scalability is not just a matter related to the computational complexity of modeling elements along the pipeline, but also links to the feasibility of migrating the designed models from a lab setup to a, e.g., Big Data computing architecture. Unfortunately, scarce publications reflect nowadays on whether their proposed data-based workflows can be deployed and run on legacy ITS systems, thereby avoiding costly upgrade investments in computing equipment.
Traffic Theory Awareness
------------------------
Theoretical representations of traffic attempt to construct (mostly simple) models with causal aspects. These models are usually of a closed form and are frequently dictated by simplifying assumptions, which leads to limited performance when modeling complex spatio-temporal dynamics in the microscopic analysis context. In these models, data are instrumental to estimate how well they fit real world conditions. On the other hand, and since their upsurge in the 80s, data driven models rely exclusively on the data to extract the dynamics that govern the phenomena. This, at least theoretically, makes them more adaptable and more efficient in complex conditions when compared to theory based models. But, they can hardly claim applicability in large scale scenarios (city level traffic management) due to significant computational resources requirements. Such data-driven traffic models have been systematically implemented as proof of concepts and are now dominant in Traffic Engineering literature [@lana2018road], incorporating most well-known advanced techniques, and, in many cases, ignoring the elementary knowledge of traffic and focusing blindly on performance.
Owing to the above, researchers in traffic modeling have diversified the way in which their models are developed and evaluated, fitting them to the technology that is introduced, as opposed to fitting the model to the well established knowledge described in well established theories of traffic flow. This results in models that are hardly actionable for traffic engineers, in terms of integration to legacy traffic control and management systems and relevance to the decision making process of road network operators. Besides, there is a lack of standards in what regards to data and scenarios used to assess the performance, usually due to the availability of real data for each researcher. This was already identified in [@Vlahogianni2014], where test-beds were proposed, either generating them or using some of the existent as standards. This would help compare models, understand them better, as they can be evaluated in a known environment, and obtain their insights concerning traffic theory. Besides, as we anticipated in Section \[sensing\] there is a industrial trend towards the consideration of different data sources when modeling traffic dynamics. In many cases, these data sources do not have any straightforward relationship to traffic itself. The integration of these sources of data, the models learned from them and theoretical representations of transportation scenarios remains an open challenge that has started to be addressed in literature [@zhang2012exploring; @zhang2016exploratory; @zhang2017understanding].
In this line of reasoning, linking data-driven to theory based models in transportation may resort to efficient and physically consistent representations of transportation phenomena. In fields like traffic modeling and forecasting, this hybrid approach permits to consider theoretical aspects of traffic, such as the relationship among speed, flow and density, the three phases of traffic [@kerner2004three], or the Breakdown Minimization Principle [@kerner1999physics] when modeling bottlenecks. The consideration of these theoretical concepts takes effect mainly in the preprocessing, modeling and prescription phases of the modeling workflow. In preprocessing, domain knowledge can be crucial for feature engineering, by describing how available features are related to each other, estimating collinearities in advance, deleting irrelevant predictors, or obtaining feature combinations with improved modeling power [@Vlahogianni2014]. Applying traffic theories and principles can also be useful for data augmentation and missing data imputation, by simulating or generating data that are more akin to what the context can provide [@lana2019]. In the modeling phase, previously defined mathematical frameworks can help define the constraints, operation ranges and correct the output of data-based models, which do not take into account the compliance of their output with respect to well-established theories. Lastly, in the prescription phase, model outputs can be linked to traffic theory knowledge to improve the way in which they are applied: a predicted flow value can be more useful if the travel time or the bottleneck probability can be computed afterwards. Furthermore, in the case of predictive models, they can reach a point in which the provided predictions ultimately affect the future behavior of the models themselves, if they are trained only with observed past data. For instance, a model that assists traffic management decisions, like closing a lane, might lead to a situation that has not been observed by the model before, thus making the knowledge captured by the traffic model obsolete and useless until the data captured from the environment is exploited for retraining. Physical models can be highly useful to anticipate scenarios and complement data-based models, providing additional information of what theories or simulations determine that the behavior of the scenario should be.
This emergent modeling paradigm is known as Theory Guided Data Science, and aims to enhance data driven models by integrating scientific knowledge [@karpatne2017theory]. The main objective of this approach is to enable an insightful learning using theoretic foundations of a specific discipline to tackle the problem of data representativeness, spurious patterns found in datasets, as well as providing physically inconsistent solutions. From the algorithmic point of view, this induction of domain knowledge can be done in assorted means, such as the use of specially devised regularization terms in predictive models (e.g. in the loss function of Deep Learning models), data cleansing strategies that account for known data correlations, or memetic solvers that incorporate local search methods embedding problem-specific heuristics. In transportation, there has been several example of theory enhanced models departing from traffic conditions identification and characterization [@vlahogianni2007spatio; @ramezani2012estimation], to data driven and agent based traffic simulation models for control and management [@zhang2011data; @chen2010review; @montanino2015trajectory; @shahrbabaki2018data], or cooperative intelligent driving services [@mintsis2017evaluation].
Awareness with domain-specific knowledge can be also enforced at the end of the workflow. When decision making is formulated as an optimization problem, the family of optimization strategies known as Memetic Computing [@gupta2018memetic; @neri2012memetic] has been used for years to incorporate local search strategies compounded by global search techniques and low-level local search heuristics. These heuristics can be driven by intuition when tackling the optimization problem at hand or, more suitably for actionability purposes, by a priori knowledge about the decision making process gained as a result of human experience or prevailing theories. For instance, traffic management under incidences in the road network can largely benefit from the human knowledge acquired for years by the manager in charge, since this knowledge may embed features of the traffic dynamics that are not easily observable from historical data. This knowledge can be inserted in an optimization algorithm devised to decide e.g. which lanes should be rerouted in an accident.
Application Context Awareness {#sec:appcon}
-----------------------------
Transportation is exceptionally diverse around the world, with notable differences in modes, preferences and availability due to social, economic and cultural disparity. Moreover, Intelligent Transportation Systems with different purposes have also characteristic requirements that can also be very divergent with respect to space and time. To address this landscape of complex and some times conflicting goals, policies and decision making should span from few seconds (traffic management and control) to years (planning and designing of new systems). It is strongly argued that data driven framework are able to cope with context aware datasets, due to their inherent capabilities of learning patterns hindering in resourceful data and reconstruct - in a sense - the context of the application. Typical examples of such context aware systems are the extraction of Origin-Destination matrices from cellphone based data [@gonzalez2008understanding], the mobility applications that aim to improve the the mobility footprint of users [@chatterjee2018type2motion], as well as the smartphone based driving insurance systems [@tselentis2017innovative]. Although these approaches seem to be appropriate to complement the user or system’s experience on a problem, significant uncertainty lies in their transferability and accuracy, owing to the lack of context-aware knowledge.
A certain degree of awareness of the context should be a matter of concern when developing ITS models that intend to be actionable. Context aware information is usually introduced in the modeling, for example accounting fro the demographic characteristics of the application area, the type of the road or network, the mode, the travel purpose etc. However, what is usually disregarded is a much broader consideration of the operational and system’s characteristics, such as how models can be introduced to the operations at hand, what the privacy concerns are with respect to data and information flows, what is the regulatory framework and policy level restrictions and goals to be reached.
First, within the operation, the deployment context where a developed model is intended to be implemented can enforce a series of operative constraints. Creating and proposing an ITS model without observing these requirements is an exercise of futility, for its lack of actionability. From this operation perspective, the context covers from deployment and operation costs – is the system cost-efficient considering its potential service? – to functioning modes – has the model the expected response times? can it operate in reduced computational power environments? As an illustration, a system designed to detect and identify pedestrians can be very effective in terms of performance, but if it does not operate at an appropriate speed, or it needs more demanding computations that cannot be embarked in a vehicle, it is useless for an autonomous driving context [@andreasson2015autonomous]. A similar reasoning holds if by *operation cost* one thinks about the energy consumption of the model at hand. Questions such as whether the energy consumed by the model compliant with the system should be kept in mind at design time, but also from the academic perspective, where efforts should be directed to the development of models that are consequent with the actual operative circumscription.
Second, regulations constitute a hard and highly contextual constraint in the implementation of ITS. Besides the wide regulatory differences that can be found across regions, there are transport frameworks where regulations are specially rigid. A typical example is the case of airports [@kulik2016intelligent], and where there is a broad field for specialized ITS. Another example is the constantly rising use of drone systems to monitor traffic [@barmpounakis2016unmanned]. Models that fail to relate to the application’s regulatory environment are not actionable.
Third, data privacy and sovereignty constitute a growing concern in a connected world where, after a decade of handing over data with complacency, an awareness about personal information sharing is springing. A recent example is the introduction of the EU General Data Protection Regulation (GDPR) framework, that severely disputed the manner data were introduced to models, as well as data availability. ITS models that are based on personal data are common nowadays, for instance in floating car data based developments [@lin2014mining]. However, there are fields where this aspect is becoming crucial (autonomous driving connectivity [@khodaei2015key], security in public transport environments [@menouar2017uav]), and research is steering to privacy-preserving approaches [@sucasas2016autonomous], spheres where technologies such as Blockchain can have a major dominance [@yuan2016towards; @lei2017blockchain].
Fourth, social aspects of the application play a major role in modeling. Social transportation is the subfield in ITS where the “social” information coming from mobile devices, wearable devices and social media is used for a number of ITS management related applications [@zheng2015big]. The outcomes from social transportation may be, to name a few, traffic analysis and forecasting [@he2013improving; @ni2016forecasting], transportation based social media [@evans2012microparticipation], transportation knowledge automation in the form of recommending systems and decision support systems [@kuflik2017automating], and services for the collection of further signal to be used later for the already mentioned purposes or others. However, cultural differences can have a relevant impact in how these systems operate, as social data are most commonly strongly linked to geographical information. This is a key aspect for their actionability.
Fifth, transportation is currently a large source of greenhouse-gas emissions [@woodcock2009public]. These concerns are gaining momentum in a wide range of ITS applications, such as the discovery of parking spots [@chen2013development], multimodality applications that grant travelers the chance of using collective transportation systems efficiently and conveniently [@kramers2014designing], the improvement of logistics operations [@zhang2015swarm], shared mobility applications, which help reducing the number of one-passenger vehicles in the road network [@feigon2016shared], or driving analytics to improve safety and ecological footprint [@vlahogianni2017driving; @huang2018eco; @adamidis2019impacts].
Of course, research goes beyond the application context and does not need to be always connected to a certain application scenario. A prototype can be far from the practical requirements of its eventual deployment; still, knowing the essential application common grounds is key to converge to actionable models. Unfortunately, this is a matter frequently disregarded in ITS research.
Transferability
---------------
Within the research context, it is common to employ test data to assess the models. Regardless if these data are obtained from real sources or synthetically generated, the resulting models have been built around them, and can be heavily linked to that experimentation context. Would these models work in other context or with other input data? Transferability could be defined as the quality of a data-driven model to be applied in other environment with other data, and it is directly linked with actionability: the application of a model should be generalizable to different datasets and transportation settings. This definition stems from the concept of *Transfer Learning* [@pan2010survey], which is more general and can entail that models that have been trained in a certain domain are applied to other domains, and the previous knowledge obtained from the first makes them perform better in the latter than models without it.
Depending on the subcategory of ITS, this requirement can be easily met or arduous to achieve, as some subcategories are more oriented towards the application and rely less on the environment than others; the key is defining what is *environment*. For example, a travel time forecasting model developed with data of a certain location could be transferable to another location without great complications, if it is built considering this feature [@bajwa2005performance]. In fact, many ITS models that are spatial-sensitive are developed using real data, but within the experimentation context, they are evaluated only in certain locations. Transferability for these scenarios would imply that the obtained results are reproducible (with certain degree of tolerance) in other locations.This could entail from plainly extrapolating the model to other locations [@getachew2007simplified], to implementing of techniques such as *soft-sensing*, aimed at modeling situations where no sensor is available [@habibzadeh2018soft], and the environment information is enough to obtain these models. A similar case in terms of spatial contexts, but with more parameter complexity, requires plenty of information about the environment. As an illustration, the case of crash risk estimation implies a higher calibration and adjustment needs due to the higher number of parameters that take part in this type of estimations. In these circumstances, works such as [@shew2013transferability] or [@xu2014using] work with posterior probability models and give more relevance to models that behave with a certain performance in many contexts than to models that perform better in a particular location. On the other extreme, for cases like autonomous driving, the change of environment is connatural to the domain (a moving vehicle constantly changing its location), and the parameters of these models are abundant and highly variable. Thus, these applications need transferable solutions, transferability that is specifically sought by researchers, for instance in LIDAR based localization [@ibisch2013towards] or pedestrian motion estimation [@shen2018transferable]. In any case, and regardless the domain, ITS research is in an incipient stage (probably with the exception of autonomous driving) of developing transferable models, and evaluating this feature, and some machine learning paradigms can help improve this characteristic.
Emerging AI Areas towards Actionable ITS {#challenges}
========================================
We have hitherto elaborated on the requisites that a model should meet towards leading to actionable data-based insights in ITS applications and processes. Some of these requirements can be fulfilled by properly designing the data-based workflow (e.g. interpretability can be straightforward for certain prediction models, whereas adaptability can be enforced by periodically scheduling the learning algorithm under use and feeding it with new data). However, several research areas have stemmed in the last years from the wide fields of Data Science and Artificial Intelligence that may serve to catalyze the compliance of data-based ITS workflows with the prescribed requisites, and thereby attain the sought actionability of their produced insights.
{width="1.8\columnwidth"}
\[fig:3\]
The main AI areas that have been identified as potentially appropriate for addressing the requirements can be summarized briefly as follows:
- Real-time data processing and online learning, which are not brand new research avenues in ITS, as we can find advanced developments in the literature. However, as we will later show, emerging fields with great potential such as dynamic data fusion and dynamic optimization can expedite and proliferate the widespread adoption of incremental data-based models in more ITS-related applications.
- Transfer learning and domain adaptation, that could allow to develop models for certain contexts and export them to others, linking directly to the transferability requirement, but also to the integration of transportation theories and physical models to data-based models.
- Gray-box modeling, a paradigm halfway between white-box (physical) and black-box (data-based) models. Gray-box modeling represents a promising area to bring awareness to traffic theory and other physical modeling when developing data-based models, with the potential to increase the performance, usability and comprehensibility of the latter.
- Green AI, a trend in Artificial Intelligence research that connects directly with energy and cost efficiency. Developing efficient models has a relevant impact in their sustainability and context awareness.
- Fairness, Accountability, Transparency and Ethics: Data-based models – specially those learning from large amounts of diverse data from many sources – are fragile to biases, and can compromise aspects such as the fairness of decisions or the differential privacy of data. In this context of growing sources of data, including those gathered from people, and increasingly opaque data-based models, it has become essential to understand what models have learned from data, and to analyze them beyond their predictive performance to consider ethical, societal and legal aspects. These aspects have been scarcely considered in ITS research.
- Other Artificial Intelligence areas such as imbalanced learning, reinforcement learning, adversarial machine learning are later highlighted for their noted relevance in ITS.
We next discuss on the research opportunities spurred by the above research lines, their connections with the requirements presented in Section \[requirements\] (shown in Figure \[fig:3\]), as well as the challenges that stem from the consideration of these AI areas in the context of ITS.
Online Learning and Dynamic Data Fusion/Optimization
----------------------------------------------------
Previously sketched in Section \[sec:self-sust\], by online learning we refer to the capability of the learning model and in general, of the entire workflow, to learn from fastly arriving data possibly produced by non-stationary phenomena that enforces a need for adapting the knowledge captured by the model along time. Changes over data streams can make the data pipeline obsolete, thus demanding active or passive techniques to update it with the characteristics of the stream [@ditzler2015learning; @gama2014survey].
Although activity around online learning has mostly revolved on certain clustering and classification paradigms (the latter giving rise to the so-called concept drift term to refer to pattern changes), it is important to note that adaptation can be also needed in other stages of the actionable data-based workflow, from data fusion to the prescription of actions. This being said, research areas such as dynamic optimization and dynamic multi-sensor data fusion should be also investigated deeply in future studies related to actionable data-based models, specially when the scenario under analysis can produce information with non-stationary statistical characteristics. When merging different data sources, fusion strategies at different levels can be designed and implemented, from traditional means (data-level fusion, knowledge-level fusion) to modern methods (corr. model-based fusion, federated learning or multiview learning) [@smirnov2019knowledge; @wang2019data]. Fusion of correlated data sources can compensate for missing entries or noisy instances in static environments. However, when data evolve over time as a result of their non-stationarity, new challenges may arise in regards to the inconsistency among multiple information sources, including measurement discrepancy, inconsistent spatial and temporal resolutions, or the timeliness/obsolescence of the data flows to be merged, among other issues. For this reason, close attention should be paid to advances reported around adaptive fusion methods capable of detecting, counteracting and correcting misalignments between data flows that occur and evolve over time. This branch of dynamic data fusion schemes aims at combining together information flows produced by non-stationary sources, synthesizing a representation of the recent history of each of the flows to be merged into a set of more coherent, useful data inputs to the rest of the data-based pipeline [@khaleghi2013multisensor; @ramachandran2006dynamic]. On the other hand, dynamic optimization techniques can efficiently deliver optimized actionable policies when the objectives and/or constraints of the underlying optimization problem varies [@nguyen2012evolutionary; @mavrovouniotis2017survey]. We energetically advocate for a widespread embrace of advances in these fields by the ITS community, emphasizing on those scenarios whose dynamic nature can make the obtained actionable insights eventually obsolete. This is the case, for instance, of traffic related modeling problems (e.g. traffic flow forecasting and optimal routing) or driver characterization for consumption minimization, among many others.
Other requirements for actionability can also benefit from the adoption of the above models in dynamic ITS contexts. For instance, cost efficiency in terms of energy consumption can largely harness the incrementality that often features an online learning model. The use of dynamic data fusion can also yield a drastically less usage of communication resources in wireless V2V links, such as those established in cooperative driving scenarios. All in all, the recent literature poses no question around the relevance of adaptation in data-based modeling exercises noted in this work, with an increasing volume of contributions dealing with the extrapolation of adaptation mechanisms to ITS problems [@chang2009online; @saadallah2018bright; @moreira2013predicting].
Transfer Learning and Domain Adaptation {#sec:tfada}
---------------------------------------
In close semantics to its related actionability requirement (*transferability*), transfer learning aims at deriving novel means to export the knowledge captured by a data-based model for a given task to another task with different inputs and/or outputs [@pan2009survey]. Depending on the amount of alikeness between the origin task and the destination task, we may be also referring to *domain adaptation*, by which we adapt the model built to perform a certain task to make it generalize better when processing new unseen inputs that do not follow the same distribution as their original counterparts (only the distribution changes [@sun2015survey]). Techniques such as subspace mapping, representation learning, of feature weighting arise as those methods most used to allow knowledge to be transferred between data-based models used for prediction.
In essence, transfer learning can provide higher prediction accuracy for models whose number of parameters to be learned (e.g. weights in a Neural Network) demands higher amounts of labeled data than those available in practice. However, data augmentation is not the only goal targeted by transfer learning. Domain adaptation may yield a better performance when used between ITS models that can become severely affected by a lack of calibration, different configurations or diverging specifications. An immediate example illustrating this hypothesis is the use of camera sensors for vehicular perception. Models trained to detect and identify objects in the surroundings of the vehicle can fail if the images provided as their inputs are produced by image sensors with new specs. The same holds for car engine prognosis: replaced components can make a data-based characterization of the normal operation of the engine be of no practical use unless a domain adaptation mechanism is applied. Personalization of ITS services can be another problem where domain adaptation can help refine a model trained with data from many sources: a clear example springs from naturalistic driving, where a behavioral characterization model built at first instance from driving data produced by many individuals (source domain) can be progressively specialized to the particular driver of the car where it is deployed [@ou2018transfer; @xing2019driver; @xing2018end].
In regards to actionability, several functional requisites can be approached by using elements from Transfer Learning over the data-based pipeline. To begin with, it should be clear that the transferability of learned models for their deployment in different locations and contexts could be vastly improved by Transfer Learning, as the purpose of this AI branch is indeed to meet this requirement in data-based learning models. In fact, this approach is currently under study and wide adoption within the ITS community working on vehicular perception: when the capability of the vehicle to sense and identify its surrounding hinges on learning models (e.g. Deep Learning for image segmentation with cameras), a plethora of contributions depart from pretrained models, which are later particularized for the problem/scenario at hand [@ye2018machine]. This exemplified use case supports our advocacy for further efforts to incorporate transfer learning methods in other ITS applications, specially those where data collection and supervision are not straightforward to achieve in practice. Another functional requirement where Transfer Learning can pose a difference in ITS developments to come is cost efficiency. The knowledge transferred between models learned from different contexts can improve their performance, thereby reducing the need for supervising data instances and ultimately, the time, costs and resources required to perform the data annotation.
Finally, the more recent paradigm coined as Federated Learning refers to the privacy-preserving exchange of captured knowledge among models deployed in different contexts [@konevcny2016federated; @mcmahan2017communication]. Although the main motivation for the initial inception of Federated Learning targeted the mobile sector, techniques supporting the federation of distributed data-based models can be of utmost importance in the future of ITS, specially for V2V communications among autonomous vehicles and in-vehicle ATIS systems. Definitely the enrichment of models with global knowledge about the data-based task(s) at hand will pose a differential breakthrough in vehicular safety and driving experience. For instance, federated models can collectively identify, assess and countermeasure the risk of more complex vehicular scenarios than each of them in isolation [@ferdowsi2019deep]. Likewise, ATIS systems can learn from the preferences and habits of other users to better anticipate the preferences of the driver and act accordingly [@vogel2018emotion]. In a few words: an enhanced and more effective actionability of the data-based workflows built to undertake such tasks.
Gray-box Modeling
-----------------
Gray-box modelling refers to the design of models that combine theoretical developments and structures related to the problem, with data that serve as a complement for such theories to make the overall model match better the scenario under analysis [@kroll2000grey; @oussar2001gray]. Gray-box models lie in between white-box models, for which the learned structure is deterministic and grounded on theoretical concepts; and ii) black-box models, whose internal structure lacks physical significance and is learned from data. An example of white-box model in ITS systems is the use of computational fluid dynamics for macroscopic traffic flow modeling, whereas Deep Learning models for traffic forecasting can exemplify black-box modeling in this domain. Gray-box models have been lately embraced by the ITS community in a number of modeling scenarios, such as those combining biological concepts and data-based models for driver characterization [@inga2015gray; @flad2017cooperative].
Gray-box modeling can contribute to the actionability of data-based workflows for ITS applications in two different albeit interconnected directions. To begin with, the incorporation of theoretical models to data-based pipelines can narrow the gap between engineers and practitioners more acquainted with traditional tools to analyze ITS systems and processes. Indeed, hybrid modeling can tie both worlds together not only without questioning the validity of prevalent theoretical developments, but also evincing the complementarity and synergy of both approaches. On the other hand, using validated theoretical models can help data-based modeling overcome difficult learning contexts such as class imbalance, outlier characterization or the partial interpretability of data clusters, among others.
Green Artificial Intelligence
-----------------------------
A profitable strand of literature has recently stressed on the energy efficiency of data-based models, highlighting the need for redesigning their learning algorithms to minimize their energy consumption and thereby, make them implementable and usable in practice [@mittal2016survey; @alwadi2015energy; @han2013approximate]. While this issue is particularly relevant for resource-constrained devices (e.g. mobile hand-helds), the concern with energy efficiency goes beyond usability towards environmental friendliness. For this reason many recent contributions are striving for computationally lightweight variants of machine learning models that sacrifice performance for a notable reduction of their energy demand. This is not only the case of predictive models capable of incrementally learning from data, but also of specific Deep Learning architectures tailored for their deployment on embedded devices [@lane2015can].
Based on the above rationale, cost efficiency is arguably the most evident functional requirement around which energy-aware model designs can pose a breakthrough towards improving the actionability of the overall data-based workflow. In addition, other aspects can be made more actionable by using energy-aware model designs, such as usability [@faisal2015towards]. Despite achieving unprecedented levels of predictive accuracy, a data-based workflow may become useless should it deplete the battery of the system on which it is deployed for operation. Therefore, energy efficiency should be under the target of future research efforts, specially when dealing with ITS applications running on battery-powered devices, inspecting interesting paths rooted thereon such as the trade-off between performance and energy consumption, or the adaptation of the model’s operation regime depending on the remaining battery life, among others [@zliobaite2015towards].
Fairness, Accountability, Transparency and Ethics
-------------------------------------------------
To end with, the prescription of actions based on the insights provided by a data-based pipeline must be buttressed by a thorough understanding of the mechanisms behind its provided decisions [@1910.10045]. Extended information about the model must be presented to the end user for several reasons:
- To gauge as many consequences of the actions as possible, identifying situations where decision making based on the outputs of the data-based workflow gives rise to socially unfair scenarios due to the propagation of inadvertently encoding bias to the automated decisions of the model.
- To ensure him/her that the output of the model is reliable and invariant under the same data stimuli, maintaining a record of the intermediate decisions made along the pipeline, allowing for the post-mortem, potentially correcting analysis of bad decision paths, and thereby maximizing the trust and certainty of the user when embracing its output.
- To make the user understand why the developed model produces its prescriptive output when fed with a set of data inputs, shedding light on which inputs correlate more significantly with the prescribed actions, tracing back causal relations between intermediate data inputs, and discriminating extreme cases where decisions can change radically under slight modifications of the model inputs.
- To supervise the ethics of data-based workflows, identifying potentially illegal uses of unlawful data given the prevailing legislation, guaranteeing the privacy and governance of personal data by third-party data-based ITS applications and processes, and certifying that the output of the model’s output does not favor inequalities in terms of gender, religion, race or any other aspect alike.
The above requirements have been lately collectively compiled under the FATE (Fairness, Accountability, Transparency and Ethics) concept, which refers to the design of actionable data-based pipelines whose internal operations can be explained, accounted and critically examined in regards to the consequences of their eventual bias in privacy, fairness and ethical issues [@martin2018ethical; @veale2017fairer; @stoyanovich2017fides]. This recent concern with the operation of machine learning models spawns from the proliferation of real cases where practical model installments have unveiled deficiencies of different kind, from differential privacy breaks (data revealing the identity of the persons to whom they belong) to unnoticed output bias that caused racist discriminatory issues [@whittaker2018ai]. For instance, data-based models for vehicular perception, obstacle detection and avoidance must be also endowed with ethics and legal design factors to make the overall decision not just drifted by the data themselves. Another clear domain where FATE can be crucial is modeling with crowd-sourced Big Data, where aspects like privacy preservation [@victor2016privacy] and bias avoidance [@rashidi2017exploring] are arguably more critical [@boyd2012critical; @chen2017traffic]. The construction of the data-based modeling workflow must i) ensure that protected features remain as such once the workflow has been built, without any chance for reverse engineering (via e.g. XAI techniques [@arrieta2020explainable]) that could compromise the differential privacy of data; and ii) that learning algorithms along the workflow counteract hidden bias in data that could eventually lead to discriminatory decisions (due to skewed samples, tainted annotation, limited data sizes or imbalanced data). From our perspective, these are among the most concerning challenges in the exploitation of Big Data in ITS, and the main source of motivation for a number of recent studies in areas related to data-driven transportation systems such as pedestrian detection [@wilson2019predictive], autonomous vehicles [@lim2019algorithmic; @bigman2020life] or urban computing [@fu2019batman]. Bias-related issues can be identified by a proper analysis of the decisions made by the workflow, which in turn requires models to be accountable and transparent enough to thoroughly characterize their sensitivity to bias, and how inputs and outputs (decisions) correlate in regards to protected features. It is also remarkable to note that several proposals have been made to quantify fairness in machine learning pipelines, yielding useful metrics that account for the parity of models when processing groups of inputs [@leben2020normative; @verma2018fairness]. Without these aspects being considered jointly with performance measures, data-based ITS developments in years to come are at the risk of being restricted to the academia playground [@zook2017ten].
Other AI Research Areas connected to Actionability
--------------------------------------------------
The above areas have been highlighted as the main propellers for model actionability in ITS systems. However, it is worthwhile to mention other research areas from the AI realm that can also help completing the chain from data to actions:
- Few-shot learning [@fei2006one], which aims at overcoming the lack of reliably annotated data and the practical difficulty of performing annotation in certain application scenarios. For instance, accident prevention models cannot be enriched with positive samples unless a fatality occurs and the data captured in place is fed to the model. Few shots learning and related subareas (zero-shot, one-shot) deriving solutions that can automatically learn from very small amounts of training data, incorporating mechanisms (e.g. generative models, regularization techniques, guided simulation) to prevent the overall model from overfitting [@1904.05046]. In regards to actionability, this family of learning techniques can be helpful to make data-based ITS models deployable in situations lacking data supervision, specially when such a data annotation cannot be guaranteed to be achievable over time.
- Imbalanced and cost-sensitive learning [@krawczyk2016learning; @branco2016survey], which link to the need for avoiding model bias, not only to ensure the generalization of its output, but also to reduce the likeliness of the workflow to cause discriminatory issues as the ones exemplified above. The history of these AI areas in the ITS community has been going for years now [@zhang2011data]. However, we here emphasize the crucial role of these techniques beyond performance boosting: the techniques originally aimed to counteract the effects of class imbalance in the output of data-based models could be also leveraged to reflect legal impositions that not necessarily relate to the model’s performance nor can they be inferred easily for the attributes within the data themselves. The lack of compliance of the model with fairness and ethics standards does not necessarily render a performance degradation observed at its output, nor can it be inferred easily from the available data.
- Hybrid models encompassing linguistic rules and data-based learning techniques, capable of supporting the transition from the traditional way of doing to the new data-based modeling era in the management of ITS systems. We foresee that the community will witness a renaissance of data mining methods incorporating methods such as fuzzy logic not only to implement human knowledge to decision workflows, but also to explain and describe the internal structure of learned models, as it is currently under investigation in many contributions under the XAI umbrella [@fernandez2019evolutionary; @mencar2018paving].
- New prescriptive data-based techniques such as Deep Reinforcement Learning [@1701.07274] and Algorithmic Game Theory [@nisan2007algorithmic] will also grasp interest in the near future for their close connection to actionable data science. The interaction of data-based workflows with humans will require techniques capable of learning actions from experience, and eventually orchestrating the interaction and negotiation among users when their actions are governed by interrelated yet conflicting objectives. In fact such new prescriptive elements are progressively entering the literature in certain ITS applications that target machine autonomy (e.g. autonomous vehicle [@sallab2017deep; @ruch2019value] or automated signaling [@mannion2016experimental]), but it is our vision that they will gain momentum in many other ITS setups.
- Privacy-preserving Data Mining [@aldeen2015comprehensive; @agrawal2000privacy], which has garnered a great interest in the last year with major breakthroughs reported in the intersection between machine learning, cryptography, homomorphic encryption, secure enclaves and blockchains [@mendes2017privacy]. The use of personal data and the stringent pressure placed by governments and agencies on differential privacy preservation has spurred a flurry of research to prevent models from revealing sensitive data from their training instances [@ding2019survey; @victor2016privacy]. Within the ITS domain, it is possible to find many areas in which privacy preservation has recently been a subject of intense research: from origin-destination flow estimation [@zhou2013privacy] to route planners [@florian2014privacy; @rabieh2015privacy], or pattern mining [@kim2008privacy], a glance at recent literature reveals the momentum this topic has acquired lately. In any of these examples data are available as a result of the sensing pervasiveness (specially in the case of VANETs) and the capture of user data. While previous works explored how to used these data in a proper way with respect to privacy matters, it is straightforward to think that the natural evolution of this research line arrives at how protected data is preserved through the modeling workflow.
- Furthermore, the proven vulnerability of data-based models against adversarial attacks has also motivated the community to lay the foundations of an entirely new research area – Adversarial Machine Learning –, committed to the design of robust models against attacks crafted to confuse their outputs [@huang2011adversarial; @1312.6199]. Interestingly, one of the most widely exemplified scenarios in this research area relates to ITS: automated traffic signal classification models were proven to be vulnerable to adversarial attacks by placing a simple, intelligently designed sticker on the traffic sign itself [@akhtar2018threat]. Likewise, the rationale behind Federated Learning (discussed in Section \[sec:tfada\]) also spans beyond the efficient distribution of locally captured knowledge among models: since no raw data instances are involved in the information transfer, privacy of local data is consequently preserved. In short: security also matters in actionable data-based pipelines.
- Finally, the ever-growing scales of ITS scenarios demand more research invested in scaling up learning algorithms in a computationally efficient manner [@nguyen2019machine]. Automated traffic, smart cities, mobility as a service constitute ITS scenarios where a plethora of information sources interact with each other. Definitely more efforts must be invested in aggregation strategies for data-based models learned from different interrelated data ecosystems, either in a distributed fashion (e.g. federated learning) or in a centralized system (correspondingly, Map-Reduce implementations of data-based models, cloud-based architectures, etc). Computational aspects of large-scale implementations should be also under study due to their implications in terms of actionability, such as the latency of the system when prescribing decisions from data. This latter aspect can be a key for real-time ITS applications for which the gap from data to actions must be shortened to its minimum.
Concluding Notes and Outlook {#conclusion}
============================
This work has built upon the overabundance of contributions within the ITS community dealing with performance-based comparisons among data-based models. Our claim is that, as in any other domain of application, data-based modeling should bridge the gap between data and actions, providing further value to the ITS application at hand than superior model performance statistics. It is our firm belief that the research community should embrace actionability as the primary design motto, with negligible performance improvements being left behind in favor of relevant aspects such as adaptability, usability, resiliency, scalability or efficiency.
To provide a solid rationale for our postulations, we have first presented a reference model for actionable data-based workflows, placing emphasis on the different phases that should be undertaken to translate data into actions of added value for the decision maker. Adaptation has been highlighted as a necessary albeit often neglected processing step in data-based modeling, which allows models to be effective when deployed on dynamic ITS environments with time-varying data sources. Next, our study has listed the main functional requirements that models along the reference model should meet to guarantee their actionability, followed by an overview of incipient research areas in Data Science and Artificial Intelligence that should progressively enter the ITS arena. Indeed, advances in XAI, Online Learning, Gray-box Modeling and Transfer Learning are currently investigated mostly from an application-agnostic perspective. Their undoubted connection to actionability makes them the core of a promising future for data-based modeling in ITS systems, processes and applications.
Data-based modeling has brought a deep transformation to ITS. A vast amount of research works in the field are produced by data-based modeling specialists attracted by the profusion of available data, and with limited knowledge of transportation. Data-based models are getting progressively more complex, increasing the gap between research and practice. This situation calls for a change of paradigm, to a one in which actionability requirements of models is desired by researchers, and practitioners are aware of the technologies available to provide it. Model actionability is a great whole that can act as an incentive to perform smaller steps towards its realization. It is probably unthinkable to develop, in a research environment, a data-based model that meets all proposed requirements. However, addressing some of the postulated requirements while developing a competing data-based ITS model will make it closer to actionability. There is, therefore, a long road to be travelled in ITS model actionability, with interesting avenues around the thorough understanding of models, and the adoption of emerging AI technologies to endow data-based workflows with the requirements needed to make them actionable in practice. As exposed in our study, there is a germinal interest in these research topics. Nevertheless, we foresee vast opportunities for future work when model actionability is set as a design priority.
On a closing note, we advocate for a new dawn of Data Science in the ITS domain, where advances in modeling performance concurrently emerge along with histories and reports about how such models have helped decision making in practical scenarios. Data mining has limited merit without actions prescribed from its outputs, always in compliance and close match with the specificities of its context.
Acknowledgements {#acknowledgements .unnumbered}
================
Ibai Laña and Javier Del Ser would like to thank the Basque Government for its funding support through the EMAITEK program. Javier Del Ser receives funding support from the Consolidated Research Group MATHMODE (IT1294-19) granted by the Department of Education of the Basque Government.
[Ibai Laña]{} received his B.Sc. degree in Computer Engineering from Deusto University, Spain, in 2006, the M.Sc. degree in Advanced Artificial Intelligence from UNED, Spain, in 2014, and the PhD in Artificial Intelligence from the University of the Basque Country in 2018. He is currently a senior researcher at TECNALIA (Spain). His research interests fall within the intersection of Intelligent Transportation Systems (ITS), machine learning, traffic data analysis and data science. He has dealt with urban traffic forecasting problems, where he has applied machine learning models and evolutionary algorithms to obtain longer term and more accurate predictions. He also has interest in other traffic related challenges, such as origin-destination matrix estimation or point of interest and trajectory detection.
[Javier J. Sanchez-Medina]{} (M’11-SM’17) is an associate professor at ULPGC, in Spain. His research interests include mainly the application of Data Mining, Machine Learning and Parallel Computing to Intelligent Transportation Systems. He has a wide experience on the development of traffic models and simulation platforms. Javier Sanchez-Medina is also a very active volunteer in the ITS community, serving regularly at IEEE conferences in their organizing committee, and as an IEEE ITS Society’s officer and BoG member. He has published more than 60 publications in these areas. He has also been a keynote speaker at ANT2017 and SOLI2017, and a distinguished lecturer at the IEEE ITSS Tunisian Chapter in 2016.
[Eleni I. Vlahogianni]{} (M’07) received the Diploma degree in civil engineering and the Ph.D. degree from National Technical University of Athens, Athens, Greece, specializing in traffic operations. She is currently an Associate Professor with the Department of Transportation Planning and Engineering, National Technical University of Athens. Her professional and research experience includes projects and consultancies, in national and European levels, focusing on urban traffic flow management, public transport, and traffic safety. Moreover, she has authored over 80 publications in journals and conference proceedings. Her primary research field is traffic flow analysis and forecasting. Her other research fields are nonlinear dynamics, statistical modeling, and advanced data mining techniques.
[Javier Del Ser]{} (M’07-SM’12) received his first PhD in Telecommunication Engineering (Cum Laude) from the University of Navarra, Spain, in 2006, and a second PhD in Computational Intelligence (Summa Cum Laude, Extraordinary PhD Prize) from the University of Alcala, Spain, in 2013. He is currently a Research Professor in data analytics and optimization at TECNALIA (Spain), a visiting fellow at the Basque Center for Applied Mathematics (Spain) and an adjunct professor at the University of the Basque Country (UPV/EHU). He also serves as a senior AI advisor at the technological startup Sherpa. He has published more than 300 journal articles, book chapters and conference contributions, co-supervised 10 Ph.D. Theses, edited 6 books and co-invented 7 patents in the broad topics of Artificial Intelligence, Data Science and Optimization. He is associate editor of several journals related to areas of Artificial Intelligence, including Swarm and Evolutionary Computation, Information Fusion, Cognitive Computation and IEEE Transactions on ITS.
[^1]: I. Laña and J. Del Ser are with TECNALIA, 48170 Zamudio, Spain J. Del Ser is also with the University of the Basque Country (UPV/EHU) and the Basque Center for Applied Mathematics (BCAM), 48009 Bilbao, Spain. (e-mails: [ibai.lana,javier.delser]{}@tecnalia.com).
[^2]: J. J. Sanchez-Medina is with CICEI, at the University of Las Palmas de Gran Canaria, 35001 Las Palmas, Spain (e-mail: [email protected])
[^3]: E. I. Vlahogianni is with the Department of Transportation Planning and Engineering of the National Technical University of Athens, 15780 Zografou, Greece (e-mail: [email protected])
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abstract: 'We study the elasto-plastic behavior of dense attractive emulsions under mechanical perturbation. The attraction is introduced through non-specific depletion interactions between the droplets and is controlled by changing the concentration of surfactant micelles in the continuous phase. We find that such attractive forces are not sufficient to induce any measurable modification on the scalings between the local packing fraction and the deformation of the droplets. However, when the emulsions are flown through 2D microfluidic constrictions, we uncover a measurable effect of attraction on their elasto-plastic response. Indeed, we measure higher levels of deformation inside the constriction for attractive droplets. In addition, we show that these measurements correlate with droplet rearrangements that are spatially delayed in the constriction for higher attraction forces.'
author:
- 'I. Golovkova'
- 'L. Montel'
- 'E. Wandersman'
- 'T. Bertrand'
- 'A. M. Prevost'
- 'L.-L. Pontani'
title: Depletion attraction favors the elastic response of emulsions flowing in a constriction
---
INTRODUCTION
============
The flow of particulate systems is a problem of great importance both theoretically and practically, with direct applications to the industry. It is relevant for a wide range of soft materials, from granular packings to foams and emulsions. While these materials present obvious differences, they share universal features, e.g. they generically undergo what is known as a jamming transition [@Liu2010; @Vanhecke2009]. As the particle or droplet volume fraction $\phi$ increases, this rigidity transition between liquid and amorphous-solid states controls the phase behavior of these disordered solids. At a critical volume fraction $\phi_c$ (random close packing), the system jams and develops a yield stress [@Cates1998; @Ohern2003; @Olsson2007; @Goyon2008]. The mechanical and rheological properties, such as the elastic modulus or the local pressure, of these systems are known to display a power law dependence with the distance to the jamming onset $(\phi-\phi_c)$ [@Ohern2002; @Ohern2003; @Olsson2007; @Mason1995; @Lacasse1996; @Ellenbroek2006; @Majmudar2007; @Jorjadze2013].\
Jammed solids are characterised by a spatially heterogeneous network of interparticle contacts, with a broad distribution of contact forces exhibiting an exponential tail [@Jaeger1996; @Ohern2003; @Majmudar2007; @Brujic2003] in which only a small subset of the particles sustain most of the mechanical load [@Liu1995; @Cates1999; @Majmudar2005; @Zhou2006]. Below the yield stress, these systems responds elastically, while above it, they deform and flow plastically [@Chen2010]. In these soft glassy flows, it was shown that stress and strain rates are coupled nonlocally [@Goyon2008; @Kamrin2012; @Bocquet2009]. In two-dimensional materials, the flow properties can easily be probed both at the microscopic and macroscopic scales [@Desmond2015; @Chen2015; @Hartley2003; @Lauridsen2004; @Kabla2007; @Dollet2007; @Utter2008; @Keim2014; @Keim2015; @Bares2017]. As a consequence, previous experimental studies examined the microscopic rearrangements in a variety of two-dimensional model systems under stress [@Lauridsen2002; @Gai2016]. This plastic flow is generically governed by local structural rearrangements which relieve stresses and dissipate energy [@Argon1979; @Goyon2008; @Desmond2015; @Chen2015]. Local plastic rearrangements have been connected to the fluctuating macroscopic flow in both simulations [@Durian1995; @Kabla2003; @Maloney2004; @Maloney2006; @Mansard2013] and theoretical studies [@Falk1998; @Picard2004; @Goyon2008; @Bocquet2009; @Kamrin2012] of model systems. Nevertheless, the intimate link between the microscopic dynamics of an amorphous material and its macroscopic elasto-plastic response is still an open question for a broad class of more realistic materials.\
In emulsions, the use of surfactants prevents the coalescence of the droplets and leads to short-range purely repulsive droplet-droplet interactions [@Desmond2013; @Desmond2015; @Chen2015]. As such, dense stable emulsions are examples of jammed solids. In the last decades, a number of experimental works studied the structural, mechanical and rheological properties of purely repulsive emulsions [@Desmond2013; @Hebraud1997; @Coussot2002; @Becu2006; @Jop2012; @Jorjadze2013; @Lin2016]. In particular, as in other soft materials [@Lundberg2008; @Graner2008; @Cohen-Addad2013; @Marmottant2008; @Keim2014; @Bi2015; @Bares2017], recent studies in quasi-2D flowing emulsions have also highlighted the importance of T1 events for local rearrangements and stress redistribution [@Chen2015; @Desmond2015]. Monodisperse emulsions allow one to study material properties such as grain boundaries, dislocations and plasticity [@Bragg1947; @Schall2004; @Schall2007; @Arciniaga2011; @Arif2012]; in particular, a recent study showed the existence of a spatiotemporal periodicity in the dislocation dynamics of these emulsions [@Gai2016]. However, none of these studies have so far adressed the question of how interdroplet attractive forces modify the flow response of these emulsions.\
Indeed, in a variety of natural settings and industrial applications, emulsion droplets do display additional attractive interactions that have been shown to change the nature of the jamming transition [@Trappe2001; @Lois2008; @Jorjadze2011]. In contrast with the purely repulsive case, droplets in attractive emulsions can form bonds and thus a soft gel-like elastic structure which can sustain shear stresses below isostaticity [@Bibette1993; @Poulin1999; @Becu2006; @Datta2011]. However, the microscopic dynamics of the material, i.e. at the scale of the particles, was not explored. As a consequence, it is of particular importance to ask how the response to stress and in particular, the structural and mechanical properties of emulsions are modified by the presence of attractive interactions. Despite their broad applicability, our understanding of the influence of particle-particle interactions on the macroscopic properties of soft matter systems with attractive interactions is currently hindered by a crucial lack of controlled experimental settings.\
In this article, we propose a first step towards completing our understanding of the microscopic origin for the macroscopic properties of adhesive emulsions. In particular, we study emulsions in which droplets interact through depletion attraction. First, we find that the static structure of 2D polydisperse emulsions remains unchanged by the introduction of depletion forces. However, the response of 2D monodisperse emulsions under mechanical constraint is impacted by the presence of depletion forces. Indeed, we flow the droplets through a microfluidic constriction in which they have to undergo elasto-plastic remodelling in order to go from a wide channel to a narrow one. In particular, we find that attractive droplets deform more inside the constriction, which we correlate to a shift in the expected position of rearrangements. These findings show that depletion attraction forces are sufficient to modify the elasto-plastic response of dense emulsions under a mechanical perturbation. This attraction, even though it is not evidenced in static conditions, impairs rearrangements and in turn promotes an enhanced elastic response under flow.
MATERIALS AND METHODS
=====================
Emulsion preparation
--------------------
For static experiments, polydisperse emulsions are prepared using a pressure emulsifier (Internal Pressure Type, SPG Technology co.). Silicon oil (viscosity $50 \mathrm{mPa.s}$, Sigma Aldrich) is pushed through a cylindrical Shirasu Porous Glass membrane decorated with $10~\mu \mathrm{m}$ pores, direcly into a 10mM SDS solution that is maintained under vigorous agitation. The resulting droplets display an average diameter of $42~\mu \mathrm{m}$ (polydispersity $21\%$). In order to prepare the emulsion with both SDS concentrations, we use the same droplets and only replace their continuous phase. To do so, the emulsion is washed in a separating funnel in order to replace the continuous phase by solutions of 10 or 45mM SDS in a water/glycerol mixture (60:40 in volume). This enhances the optical quality of the oil/water interface visualization through bright field and confocal microscopy.\
For experiments in the constriction, we use monodisperse emulsions with an average droplet diameter of $45~\mu\mathrm{m}$ (polydispersity $3.9\%$). These emulsions are obtained with a custom made flow-focusing microfluidic set-up (channel size $60~\mu \mathrm{m} \times 60~\mu \mathrm{m}$, width at the flow-focusing junction $30~\mu \mathrm{m}$). We use the same oil and continuous phases for polydisperse and monodisperse emulsions.
Observation and image analysis of 2D static packings
----------------------------------------------------
When studying 2D static packings, we consider polydisperse emulsions that are fluorescently labelled with Nile Red (Sigma Aldrich). To label the emulsion, we incubate it overnight in a SDS buffer (with \[SDS\] = 10 or 45 mM) saturated in Nile Red allowing the dye to partition between the oil and water phases over time. A $10~\mu\mathrm{L}$ drop of emulsion is placed between a microscope glass slide (76 x 26 mm, Objekttrager) and a cover slip (24 x 60mm, Knittel Glaser) separated by 50 $\mu$m spacers (polymethylmethacrilate -PMMA- film, Goodfellow). Droplets are imaged through confocal microscopy (Spinning Disc Xlight V2, Gataca systems) using a 20x objective.\
To study the local structure of these static packings, we use a custom Matlab (MathWorks) routine that works as follows. We first threshold the images and perform a watershed tessellation, we then measure the perimeter $p$ and area $a$ of each droplet as well as the area $a_{c}$ of the associated watershed tesselation cell (see Fig. \[Fig1\]D). Following @Boromand2019, we study the relation between the deformation of the droplets and their local packing fraction. To do so, we compute their shape factor $\mathcal{A} = p^2/4 \pi a$ and determine the local packing fraction $\phi_l = a/a_{c}$. Note that we only consider droplets in the center of the packing, i.e. we exclude those that are partially cut by the edge of the image frame. The shape parameter $\mathcal{A}$ equals 1 for circular disks and is greater than 1 for all nonspherical particles [@Boromand2018].
{height="6.2cm"}
Experimental set-up for emulsion flow
-------------------------------------
We designed the constriction in a microfluidic channel composed of three main sections (Fig. \[Fig1\]): at the entrance, the channel is 50 $\mu$m deep and 200 $\mu$m wide over a 5 mm length, then at the constriction the width is reduced from 200 to 50 $\mu$m over a length of 200 $\mu$m, finally the channel remains 50 $\mu$m wide over a final 5 mm length. The channel is made in polydimethylsiloxane using a negative cast micromachined in a block of PMMA ($50\times50\times5~\mathrm{mm}^{3}$) using a desktop CNC Mini-Mill machine (Minitech Machinary Corp., USA). After passivating the channel with casein 0.05 mg/ml ($\beta$-casein from bovine milk, Sigma Aldrich) for 20 minutes, the emulsion is flown in the device using a pressure pump (MFCS-8C Fluigent, P = 30 mbar). After droplets fill the constriction area, the pressure is decreased to stop the emulsion flow, and droplets are left to cream in the supply tube overnight, thus compressing the droplets in the microfluidic device in order to reach high values of packing fraction. After this passive compression phase, the emulsion is flown again in the channel at a constant pressure. The flow of the droplets at the constriction is imaged in bright field microscopy with a 10x objective at a frequency of 20 frames per second (fps).
Image analysis of the emulsions flowing in the constriction
-----------------------------------------------------------
To analyse the videos of flowing emulsions, we first threshold the images to subsequently determine the center and perimeter of each droplet in the channel using a custom made Matlab routine. When studying droplet deformation, we only consider the droplets located in the constriction region. We define this area along the channel as a window that includes the 200 $\mu$m of the constriction itself, plus 50 $\mu$m before and after the constriction (Fig. \[Fig1\]). To quantify the deformation of each droplet, we use the approach proposed by @Chen2012. The perimeter of the droplet is discretized at evenly spaced 1024 angles $\theta$ and the deformation $d$ is calculated as a standard deviation of the radii $r(\theta)$ for each of these angles divided by the mean value of $r$:
$$d = \frac{\sqrt{\langle r^2\rangle+\langle r \rangle^2}}{\langle r\rangle}$$
We also determine the global packing fraction of the emulsion in each video frame. To this end, we calculate the ratio between the sum of all droplets area and the area of the channel within the window of $200\times200~\mu$m located before the constriction region. Finally, frames are sorted according to the emulsion packing fraction, and the distributions of droplet deformations for each packing fraction are computed.
For rearrangement and flow analysis, the droplets were tracked using a custom Python routine. All droplets are sorted according to the lane they belong to in the channel ahead of the constriction. In our experiments, they are thus sorted into four lanes. The instantaneous velocity of the droplets was computed as the distance travelled between two consecutive frames acquired at a fixed frame rate. The localization of the minima in the instantaneous velocity are then measured for each droplet trajectory and sorted as a function of the original lane the droplet belonged to.
RESULTS
=======
Analysis of static packings
---------------------------
We first study 2D static packings of polydisperse emulsions with two distinct depletion interactions. Using silicon oil droplets stabilized with two different concentrations of SDS (10mM and 45mM) allows us to change the depletion forces between the droplets. In our experiment, the continuous aqueous phase is supplemented in glycerol (40 % in volume of glycerol). Note that in addition to allowing for a better imaging of the droplets, it also shifts the critical micellar concentration (CMC) of SDS. However, the CMC is only raised from 8mM (in pure water) to about 9mM in our experimental conditions [@Ruiz2008; @Khan2019], which ensures that the system is still above the CMC under both SDS concentrations and that the surface tension remains the same when the concentration of SDS is increased from 10 to 45 mM. Above the CMC, depletion attraction forces increase linearly with the concentration of micelles [@Asakura1958], which itself grows with increasing concentrations of SDS. Considering the aggregation number of SDS at both SDS concentrations (i.e. the number of SDS molecule per micelle at a given concentration), we estimate that there is approximately 30 times more micelles at 45mM SDS than at 10mM SDS. Depletion forces at 45mM SDS are thus expected to be 30 times larger than at 10mM SDS.
![[Analysis of static 2D packings]{} — [(A)]{} $\phi$ versus $\mathcal{A}-1$ for 10mM SDS emulsion. The total number of droplets is N = 1193. The experimental data (red open diamonds) are plotted together with the DP model with the exponent fixed to 1/3 and $\phi_c=0.842$ (black dashed line). The pink dashed line is the best power law fit with the prefactor fixed. [(B)]{} Log-log plot of $\phi_l-\phi_c$ versus $\mathcal{A}-1$ for 10mM (red open diamonds) and 45mM SDS (blue open squares) emulsions. The data points for the 10mM SDS emulsion are the same as in (A). The total number of droplets for the 45mM SDS emulsion is N = 1735. The DP model is plotted as a black dashed line.[]{data-label="Local-contour"}](fig2_panel.pdf){width="45.00000%"}
To study the impact of depletion forces on static 2D packings, we first quantify the deformation of the droplets as a function of their local packing fraction. Recent studies[@Boromand2018; @Boromand2019] developed a new numerical model to study the structural and mechanical properties of 2D bubbles and emulsions, including at high compressions. In the so-called deformable particle (DP) model, particles deform in response to mechanical constraints to minimize their perimeter while keeping their area fixed. This leads to a model of deformable disks with potential energies that includes an energy term associated to the line tension and a penalization energy term quadratic in the change of area of the droplets, thus associated to their compressibility. Further, the deformable particles interact via a purely repulsive potential energy. Within the framework of this DP model and in our range of deformations, it was predicted that the distance to jamming onset $\phi_l - \phi_c$ scales with asphericity $\mathcal{A}-1$ as $\phi_l - \phi_c \sim (\mathcal{A}-1)^\omega$ with $\omega \approx 0.3$.
Thus, we measure the asphericity and local volume fraction of each droplet in several images of 2D packings for both 10 and 45mM SDS concentrations (see Fig. \[Fig1\]C-D and Materials and Methods). In Fig. \[Local-contour\]A, we first plot $\phi_l$ *vs* $\mathcal{A}-1$ for 10mM SDS emulsions in order to compare our data with the theoretical predictions. We then fit our data with the equation $\phi_l = \alpha (\mathcal{A}-1)^{1/3} + \phi_c $ with a fixed $\phi_c=0.842$ as numerically computed in @Boromand2019. We find a good agreement between theory and experiments with a prefactor $\alpha=0.26$. Conversely, when we only fix the prefactor $\alpha=0.26$, and keep both $\omega$ and $\phi_c$ as free fitting parameters, we recover $\omega = 0.36\pm0.1$ and $\phi_c=0.848\pm0.02$, which are also very close to the numerically computed ones. For the 45mM SDS emulsion, we also find that $\omega=0.37\pm0.07$ and $\phi_c=0.852\pm0.02$. As shown in Fig. \[Local-contour\]B, both SDS concentrations cannot be distinguished. Indeed, the data points corresponding to both depletion forces overlap and are captured by the same scaling that was predicted by the DP model for repulsive discs. This indicates that depletion attraction does not induce any measurable modification in the static packings of droplets. In other words, the interaction energy term that could be included in the DP model does not affect significantly the scaling of $\phi_l - \phi_c$ *versus* $\mathcal{A}-1$ for depletion induced attractive interactions.
![[Analyzing the droplet deformation in the constriction]{} — [(A)]{} Still snapshot of the image analysis in the channel at a given instant for an attractive emulsion (\[SDS\]=45mM). The color of the droplets codes for their deformation $d$ calculated for their detected contours displayed on the image. [(B-C)]{} Average deformation of the droplets along the x-axis of the channel for different packing fractions in [(B)]{} the low attraction case (\[SDS\]=10mM) and [(C)]{} high attraction case (\[SDS\]=45mM). The deformation is averaged in bins that are 45 $\mu$m wide along the x-axis, corresponding to about a droplet diameter. The average deformation peaks inside the area of the constriction for both conditions. The error bars correspond to the standard deviation of the distributions of $d$ calculated for all droplets in all experiments in each bin. The total number of droplets, combining all packing fractions, is N = 27219 for 10mM SDS and N = 91391 for 45 mM SDS.[]{data-label="Xspace"}](fig3_panel.pdf){width="45.00000%"}
Despite the fact that static packings cannot be distinguished as a function of depletion forces, we reveal in what follows that significantly distinct behaviors can be evidenced in the context of a dynamic flow.
Emulsion flow in a constriction
-------------------------------
In order to study their response under mechanical perturbations, monodisperse emulsions are flown in microfluidic channels exhibiting a single physical constriction (Fig. \[Fig1\]). In particular, we use monodisperse droplets whose diameter is comparable to the channel height, constraining the system to a 2D monolayer of droplets. We focus our analysis on the area of the constriction in which droplets have to rearrange and deform in order to go from a large channel into a narrower one. The width of the narrow channel is chosen such that it only allows for the passage of one droplet diameter (Fig. \[Fig1\]) in order to maximize the number of rearrangements.
A typical experiment is carried out in two phases. The channel is first filled with the emulsion using a pressure pump. After a waiting time (see Materials and Methods), the pressure is increased again so that this packed emulsion can flow in the channel. We usually require a typical pressure of the order of 30 mbar to establish a continuous flow. For each experiment, we image the droplets upstream, in order to evaluate their packing fraction, as well as inside the constriction to measure their deformation. We choose to quantify the deformation $d$ of each droplet in the channel through the standard deviation of droplet radii as previously done [@Chen2015] (see Materials and Methods).
Deformation along the channel
-----------------------------
We first study the deformation of the droplets inside the channel. To do so, we measure the volume fraction of the emulsion in a window located upstream of the constriction (on the left of the image) and that encompasses 200$\mu$m of the channel length (Fig. \[Xspace\]A). We show in Fig. \[Xspace\] the average deformation $\left<d\right>$ along the channel for both SDS concentrations. The shape of the obtained curves differ for the two SDS concentrations both in the constriction region and in the thinner channel. For 10mM SDS (Fig. \[Xspace\]B) the deformation builds up in the constriction to a first maximum average deformation until it is released to a lower value of $\left<d\right>$ at $x \approx 420\mu m$. Then the deformation builds up again to a second maximum and is decreased to a lower deformation. Qualitatively, this behavior can be explained as the signature of a local stress release after a rearrangement. Indeed, @Chen2015 showed that in compressed emulsions, T1 events were immediately followed by a local decrease of deformation inside compressed emulsions. Here the observed peaks are separated by about 40$\mu$m, corresponding to a droplet diameter. This indicates that droplet rearrangements indeed occur at positions that are set by the topology of the packing in the channel [@Gai2016]. However, at 45mM SDS (Fig. \[Xspace\]C) droplets progressively deform and reach a peak deformation value until they escape it. In that framework, the higher attractive depletion forces could impair the order of rearrangements, explaining why we do not observe a localized deformation release compared to the low attraction case. The plastic response of the attractive emulsion is thus less ordered spatially, which in turn leads to a higher elastic contribution.
The other difference between the two conditions can be observed in the thinner channel region, after the constriction, where droplets enter one by one and release their deformation. In the case of low depletion forces (\[SDS\]=10mM), droplets relax to a deformation value that is close to the initial one at the entry of the channel $\left(\left<d\right>_{out}-\left<d\right>_{in} \approx 0.0025\right)$. However, with high depletion forces (\[SDS\]=45mM), droplets relax to a plateau at higher values of deformation than at the entry $\left(\left<d\right>_{out}-\left<d\right>_{in} \approx 0.01\right)$. This impaired relaxation could be a signature of long range effects that could also explain why droplets enter the constriction with a slightly higher value of deformation in the high attraction case.
![[Statistics of deformation under flow]{} — (A) Probability density function of the deformation $d$ calculated in the constriction for different packing fractions in the case of low attraction forces (\[SDS\] = 10mM, open circles) and high attraction forces (\[SDS\] = 45mM, stars). [(B)]{} Cumulative distributions of the deformation $d$ in the constriction for low attraction forces (open circles) and high depletion forces (stars) for different packing fractions. []{data-label="distributions"}](fig4_panel.pdf){width="45.00000%"}
Deformation as a function of packing fraction
---------------------------------------------
To further confirm these observations, we study the distribution of deformation of all droplets at all positions inside the constriction (taken in a window whose length spans 50$\mu$m before and after the constriction – see Materials and Methods). Since the global volume fraction can evolve over the course of one experiment, we separate each experiment into stacks according to their upstream volume fraction. We then pool together the image sequences corresponding to the same volume fraction throughout all performed experiments, for each concentration. Note that we also checked that the deformation in the constriction does not depend on the instantaneous droplet velocity within the investigated range (from 120 to 360 $\mu m/s$, see ESI$^\dag$).
We compare the distributions of the deformations observed for different packing fractions and for each SDS concentration (Fig. \[distributions\]). The distributions peak at smaller values of deformation in the low attraction case than in the case of strongly attractive droplets (Fig. \[distributions\]A). This shift can also be clearly evidenced when plotting the cumulative distributions for each condition at various volume fractions (see Fig. \[distributions\]B). As expected, for low depletion forces (10mM SDS) we find that the distributions exhibit lower values of deformation in all conditions. When attraction is introduced between droplets, all curves are shifted to higher values of deformation. In the previous section we showed that depletion alone was not sufficient to induce significant additional deformations in static packings of droplets. The shift observed in these deformation distributions must thus originate from differences in the local topological changes of the emulsions. Hence, we next examine the spatial localization of rearrangements in the constriction as a function of SDS concentration.
{height="4.5cm"}
Rearrangements and velocity distributions in the constriction
-------------------------------------------------------------
We here test the hypothesis that rearrangements are impaired by the attraction between the droplets, which would in turn force them to deform more to overcome the constriction. To study the rearrangements in the constriction, we simply track the position of the droplets in separate lines of the channel for both conditions (Fig. \[Rearrangements\]A). Indeed, since the size of the channel as well as the diameter of the droplets are fixed, there are always four lines of droplets flowing in the channel, ahead of the constriction. In this framework the droplets will exchange neighbors to do the necessary rearrangements in given zones of the channel that are defined by geometry. At a point of rearrangement, droplets should thus transiently decrease their speed and subsequently accelerate once the rearrangement is achieved. Typical instant velocity profiles are plotted in Fig. \[Rearrangements\]B where 2 contiguous zones of rearrangements are highlighted. These zones were chosen as areas where the velocity of the droplets hits a local minimum. In this figure, one can see that the minimum values in zone 1 and 2 seem to be reached further down into the constriction for 45mM SDS than for 10mM SDS (see orange and purple arrows respectively). In order to quantify this observation, we extracted the position on the x-axis of the minimum velocity for each droplet in zones 1 and 2 and plotted their cumulative distributions in Fig. \[Rearrangements\]C. The distributions for attractive droplets are both shifted by about 5$\mu$m (measured shift at 50$\%$), indicating that rearrangements are indeed delayed in the channel compared to the low depletion case.
DISCUSSION
==========
Attractive interactions between particles is expected to affect their packing topology as well as their rheological and mechanical response to local mechanical perturbations. Below the jamming transition, previous work showed that attraction induced by depletion forces tuned significantly the structure of 3D packings and could mechanically stabilize them below the isostatic limit [@Jorjadze2011]. Above the jamming transition, one expects adhesive forces in packings of deformable spheres to change how droplet deformation and coordination numbers scale with the packing fraction [@Boromand2018; @Boromand2019]. To the best of our knowledge, this issue has been addressed neither in theoretical models nor in experimental systems.
In our experimental study, we provide a first step towards the understanding of the mechanical response of adhesive emulsions by introducing attractive interactions induced by depletion between oil droplets. We first evidence that such attraction forces are too low to induce any measurable effect in 2D static packings of droplets. Indeed, for both attraction forces, we recover the scaling laws predicted by @Boromand2019 for purely repulsive packings, with a critical packing fraction $\phi_c \approx 0.842$. However, using monodisperse emulsions, we uncovered distinct changes in their elasto-plastic response when the droplets are flown through a 2D physical constriction. The first manifestation of attraction is an increase of the average deformation of the droplets in the constriction. The second one is the delay of topological rearrangements inside the constriction as attraction forces are increased. Depletion forces thus appear adequate to change the elasto-plastic response of emulsions in our system.
Such findings could be relevant for biological tissues in which adhesion controls to a large extent remodelling events that occur on timescales that are beyond those of cytoskeletal activity. In order to isolate the role of adhesion in biological processes, cellular tissues can indeed be mimicked with droplet assemblies connected by specific binders [@Pontani2012; @Feng2013; @Pontani2016]. Within that framework, emulsions have been shown to exhibit similar mechanical properties and have for this reason been used to measure cellular forces both *in vitro* [@Molino2016] and *in vivo* [@Campas2014; @Mongera2018]. This reductionist approach could thus shed light on behavioral transitions in developping tissues upon adhesion modulation and will be the focus of future investigations.\
The authors thank Yannick Rondelez for providing flow focusing devices for the production of the emulsions, as well as Jacques Fattaccioli for allowing us to use his pressure emulsifier, they also thank Georges Debregeas and Zorana Zeravcic for fruitful discussions and acknowledge financial support from Agence Nationale de la Recherche (BOAT, ANR-17- CE30-0001).
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|
---
abstract: 'We introduce a simple, experimentally realisable, entanglement manipulation protocol for exploring mixed state entanglement. We show that for both non-maximally entangled pure, and mixed polarisation-entangled two qubit states, an increase in the degree of entanglement and purity, which we define as concentration, is achievable.'
address: |
Special Research Centre for Quantum Computer Technology,\
University of Queensland, Brisbane, Australia
author:
- 'R. T. Thew[@RTT] and W. J. Munro[@WJM]'
title: Entanglement Manipulation and Concentration
---
\[INTRO\]
The increasing interest in quantum information and computing as well as other quantum mechanical dependent operations such as teleportation [@Bennett2:96] and cryptography [@Deutsch:96] have as their cornerstone a reliance on entanglement. There has been a great deal of discussion of measures and manipulation of entanglement in recent years with respect to purification [@Bennett1:96], concentration [@Bennett3:96], and distillable entanglement [@Rains1:99; @Vedral:97] especially concerning states subject to environmental noise. It is this noise that takes the initially pure maximally entangled resource and leaves us with, at best, a non maximally entangled state, or at worst a mixed state, both less pure and less entangled. We introduce a simple, experimentally realisable [@Kwiat:00], protocol to manipulate and explore both pure and mixed-state entanglement. While the scheme will have limitations, in part due to its simplicity, it will allow experimental investigation of the large Hilbert space associated with mixed states.
The motivation for this scheme comes from focusing ideas and proposals of several groups from the past few years into a simple realisation of mixed state entanglement manipulation. It was proposed that quantum correlations on mixed states could be enhanced by positive operator valued measurements [@Popescu1:95]. A more specific example by Gisin[@Gisin1:96] considered the manipulation of a $ 2\times 2 $ system using local filters. The scheme we propose here combines these ideas and uses an arrangement similar to the original Procrustean method [@Bennett3:96] which dealt solely with pure states. The primary motivation here is in proposing a scheme that can be easily realised experimentally. With the recent advances in the preparation of nonmaximally entangled pure[@White:99] and mixed[@White:00] polarisation-entangled quantum states we now have a source for which there is a high degree of control over the degree of entanglement and purity of the state. This allows us to consider a wide variety of states and examine what operations can be performed so as to make the state more useful in the context of an entanglement resource.
For the purposes of describing the possible manipulation of a state we will define the following three concepts of distillation, purification and concentration (illustrated schematically in Figure (\[fig:ent-pur\])) as follows,
- *Distillation: Increasing the entanglement of a state.*
- *Purification: Increasing the purity of a state (decreasing its entropy). This is not purification with respect to some particular state, for example obtaining a singlet state from a mixed state.*
- *Concentration: Increasing both the entanglement and the purity of a mixed state.*
These concepts have been used almost interchangeable in the literature but we will follow our primitive definitions to avoid potential confusion. In this letter it is the concentration of a state that is the main aim for the maintenance or recovery of an entanglement resource.
Let us now specify the measures which we will be using to characterise the degree of entanglement and purity of a state. The entanglement and purity of a state can be determined using distinct measures. Here we will restrict our attention to $2 \times 2$ systems and hence will use analytic expressions for The Entanglement of Formation and Entropy as our respective measures. The Entanglement of formation as introduced by Wootters [@Wooters1:98] is found by considering that for a general two qubit state, $ \rho $, the “spin-flipped state” $\tilde{\rho}$ is given by $$\begin{aligned}
\tilde{\rho} = (\sigma _y \otimes \sigma _y) \rho^{*} (\sigma _y \otimes
\sigma _y)\end{aligned}$$ where $\sigma_y$ is the Pauli operator in the computational basis. We calculate the square root of the eigenvalues $ \tilde{\lambda}_{i} $ of $\rho \tilde \rho$, in descending order, to determine the “Concurrence”, $$\begin{aligned}
C(\rho) = max\{\tilde{\lambda}_1 - \tilde{\lambda}_2 -\tilde{\lambda}_3
-\tilde{\lambda}_4,0 \} \end{aligned}$$ The Entanglement of Formation (EOF) is then given by $$\begin{aligned}
E(C(\rho)) = h\left(\frac{1 + \sqrt{1 - C(\rho)^2}}{2} \right)\end{aligned}$$ where $ h $ is the binary entropy function $$\begin{aligned}
h(x) = -x\log(x) - (1 - x)\log(1 - x)\end{aligned}$$ The entropy of the density matrix $ \rho $ (our purity measure) is given by $$\begin{aligned}
S = -\sum_{i=1}^{4} \lambda_i \log_{4} \lambda_i\end{aligned}$$ where $ \lambda_i $ are the eigenvalues of $ \rho $.
We will now describe our entanglement manipulation protocol and emphasise its simplicity. The experimental arrangement for our protocol is described by the schematic in figure(\[fig:exp2\]). The aim of our protocol is to manipulate mixed states and enhance their degree of entanglement. Let us consider an initial state composed of two subsystems, A and B, each represented by a general $2 \times 2$ matrix. We will describe the joint state of the system, $AB$, in the polarisation basis, $\{|VV\rangle,|VH\rangle, |HV\rangle,
|HH\rangle\}$, as $$\begin{aligned}
\hat{\rho}_{ABin} &=& \left( \begin{array}{cccc}
\rho_{11} & \rho_{12} & \rho_{13} & \rho_{14} \\
\rho_{12}^* & \rho_{22} & \rho_{23} & \rho_{24} \\
\rho_{13}^* & \rho_{23}^* & \rho_{33} & \rho_{34} \\
\rho_{14}^* & \rho_{24}^* & \rho_{34}^* & \rho_{44} \\ \end{array} \right)\end{aligned}$$ with the $\hat{\rho}_{ij}$ satisfying the requirements for a legitimate density matrix. From our source (see figure(\[fig:exp2\])) we have four polarisation modes (two for A and two for B). These polarisation modes are spatially separated and input onto beam splitters (BS), with independent and variable transmission coefficients. The second input port of each of these beam splitters are assumed to be vacuums. With perfectly efficient photodetectors it would be possible to monitor the second output mode of each of these beamsplitter and use the results to conditionally select the concentrated state we wish to produce. We know that if the detection of a photon is made in any of the second output ports then the preparation process is considered to have failed. Non-detection (with perfectly efficient detectors) at all the second output ports is required to prepare our state and here is the problem with current single photon detection efficiencies. Photon detectors have a finite efficiency and it possible that a photon present at these second output ports will not be detected. Hence we will not get the conditioned state we desire. Instead we will examine the transmitted modes of the beamsplitter and consider the situations where joint coincidences are registered at the photodetectors of the two subsystems A and B, or Alice and Bob if you prefer. While this is a post selective process it has the advantage that poor detection efficiency only decreases the coincidence count rate. As we discard any information present at the second output of the beamsplitters, the protocol we describe is not unitary.
\
If we consider that having each mode incident on a BS has the effect of expanding the Hilbert space of the system, then in the expanded Hilbert space we can manipulate the state and then project it back onto the polarisation coincidence basis. The BSs transform each mode in the following way $$\begin{aligned}
|V,H \rangle |0 \rangle \rightarrow \eta_{v,h}|V,H \rangle |0 \rangle +
\sqrt{(1-\eta_{v,h}^2)}|0\rangle |1\rangle\end{aligned}$$ and hence we obtain an output density matrix for this reduced system of the form $$\begin{aligned}
\hat{\rho}_{ABout} &=&{\cal N} \left( \begin{array}{cccc}
\rho_{11}\eta_{va}^2\eta_{vb}^2&\rho_{12}\eta_{va}^2\eta_{vb}\eta_{hb}& \rho_{13}\eta_{va}\eta_{ha}\eta_{vb}^2 & \rho_{14}\eta \\
\rho_{12}^*\eta_{va}^2\eta_{vb}\eta_{hb}&\rho_{22}\eta_{va}^2\eta_{hb}^2&\rho_{23}\eta&\rho_{24}\eta_{va}\eta_{ha}\eta_{hb}^2 \\
\rho_{13}^*\eta_{va}\eta_{ha}&\rho_{23}^*\eta&\rho_{33}\eta_{ha}^2\eta_{vb}^2&\rho_{34}\eta_{ha}^2\eta_{vb}\eta_{hb} \\
\rho_{14}^*\eta&\rho_{24}^*\eta_{va}\eta_{ha}\eta_{hb}^2&\rho_{34}^*\eta_{ha}^2\eta_{vb}\eta_{hb} &\rho_{44}\eta_{ha}^2\eta_{hb}^2 \\
\end{array}
\right)\end{aligned}$$ where $\eta = \eta_{va}\eta_{ha}\eta_{vb}\eta_{hb}$ and $\eta_{v,h|a,b} $ are the vertical and horizontal polarisation transmission coefficients for subsystems A and B. The normalisation is given by $$\begin{aligned}
{\cal N} = [\rho_{11}\eta_{va}^2\eta_{vb}^2 + \rho_{22}\eta_{va}^2\eta_{hb}^2 + \rho_{33}\eta_{ha}^2\eta_{vb}^2 + \rho_{44}\eta_{ha}^2\eta_{hb}^2]^{-1}\end{aligned}$$ and the probability of obtaining the desired output state is determined from the trace of the unnormalized BS-transformed density matrix, ${\cal
N}^{-1}$, and thus is dependent on the transmission coefficients. This is the probability of obtaining the output state once the BS parameters have been determined.
This scheme is more easily understood by considering the behaviour of pure states under the protocol. As such we now illustrate the distillation process with a specific example. We will examine a non-maximally entangled pure state and show how to recover a maximally entangled state via our protocol. Consider an initial state produced by our source of the form $$\begin{aligned}
\label{eq:nmeps1}
|\varphi_{in} \rangle_{ab} = {\cal N}_{1}[\epsilon_1|VV\rangle_{ab} + \epsilon_2e^{i\phi}|HH\rangle_{ab}]\end{aligned}$$ or alternatively $$\begin{aligned}
\label{eq:nmeps2}
|\varphi_{in} \rangle_{ab} = {\cal N}_{1}[\epsilon_1|VH\rangle_{ab} + \epsilon_2e^{i\phi}|HV\rangle_{ab}]\end{aligned}$$ where $$\begin{aligned}
{\cal N}_{1}^2=[|\epsilon_1|^2 + |\epsilon_2|^2]^{-1}\end{aligned}$$ Assuming the polarisation modes are all spatially separated we input them onto separate BSs (see figure (\[fig:exp2\])). We can choose to manipulate the BSs at A and B independently to find the optimal output for a given state. For convenience we consider a state of the form of (\[eq:nmeps1\]) which allows us to simplify the analysis. With this in mind we can set $\eta_{va} = \eta_{vb} = \eta_v$ and $\eta_{ha} = \eta_{hb} = \eta_b$.
The state of our system after the BSs (assuming vacuum inputs to the second BS ports) is $$\begin{aligned}
|\varphi_{total} \rangle_{AB} &=&{\cal N}_{1} \left[
\epsilon_1\eta_v^2|VV\rangle_{AB}|00\rangle +
\epsilon_2e^{i\phi}\eta_h^2|HH\rangle_{AB} |00\rangle \right. \nonumber \\
&\;&\;\;\;+\epsilon_1\eta_v \sqrt{(1-\eta_{v}^2)}\left\{
|V0\rangle_{AB}|01\rangle+
|0V\rangle_{AB}|10\rangle \right\} \nonumber \\
&\;&\;\;\;+\epsilon_2e^{i\phi}\eta_h \sqrt{(1-\eta_{h}^2)}\left\{
|H0\rangle_{AB}|01\rangle+
|0H\rangle_{AB}|10\rangle \right\} \nonumber \\
&\;&\;\;\; + \epsilon_1 \left(1-\eta_v^2\right)|00\rangle_{AB}|11\rangle
\nonumber \\
&\;&\;\;\;\left. + \epsilon_2e^{i\phi}\left(1-\eta_h^2\right)|00\rangle_{AB}
|11\rangle\right]\end{aligned}$$ The outcomes we are interested in are in the joint coincidence basis of A,B and hence the vacuum state components are removed from consideration leaving an effective output state of the form $$\begin{aligned}
|\varphi_{out} \rangle_{AB} = {\cal
N}_{2}[\epsilon_1\eta_v^2|VV\rangle_{AB} +
\epsilon_2e^{i\phi}\eta_h^2|HH\rangle_{AB}]\end{aligned}$$ where the normalisation in this coincidence basis is $$\begin{aligned}
{\cal N}_{2}^2 = [|\epsilon_1|^2\eta_v^4 + |\epsilon_2|^2\eta_h^4]^{-1}\end{aligned}$$ For maximal entanglement we have the following simple relationship $$\begin{aligned}
|\epsilon_1| \eta_v^2 = |\epsilon_2| \eta_h^2\end{aligned}$$ We observe that the entanglement of the output state is dependent on the transmission coefficients of the BSs. Further, this protocol can always take a non-maximally entangled state and obtain a pure maximally entangled one. This protocol can also incorporate a phase adjuster at either A or B to tune any relative phase difference for the state. If we had considered states of the form of (\[eq:nmeps2\]) then we would need to consider the tuning parameters independently such that the requirement for a pure maximally entangled state is then $$\begin{aligned}
|\epsilon_1| \eta_{va}\eta_{hb} = |\epsilon_2| \eta_{vb}\eta_{ha}\end{aligned}$$ This is where the protocol differs from the Procrustean method of Bennett *et.al [@Bennett3:96]. We have introduced individual depolarising channels, thus obtaining more degrees of freedom, and so allowing the protocol to be extended to mixed states. It is important to mention again that with perfect single photon detection it is possible to monitor the discarded ports for each of the modes, thus preparing the desired state by conditioned measurements.*
Let us now turn our attention to the concentration of mixed states. As an extension to the distillation process we take the density matrix $\hat{\rho}_{ABin}$ to be a mixture of the density matrices of two of the Bell-type states, (\[eq:nmeps1\]) and (\[eq:nmeps2\]), one of which, say (\[eq:nmeps1\]), is maximally entangled, $\epsilon_1 = \epsilon_2 = 1$. The mixing can be controlled by the parameter $\gamma$, that is, $$\begin{aligned}
\label{eq:rhoineg}
\hat{\rho}_{ABin} &=& \gamma {\cal N}_{1}^{2} \left( \begin{array}{cccc}
|\epsilon_1|^2 & 0 & 0 & \epsilon_1^{*}\epsilon_2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\epsilon_1\epsilon_2^{*} & 0 & 0 & |\epsilon_2|^2 \\ \end{array} \right)
+ \frac{1-\gamma}{2}\left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 \\ \end{array} \right)\end{aligned}$$ This state is one of many of the range of mixed states that can be concentrated and has been chosen to easily show the protocols extension from pure to mixed states, from distillation to concentration. Using the BS protocol illustrated in figure(\[fig:exp2\]) the output state for (\[eq:rhoineg\]) in the coincidence basis, $AB$, can be represented as $$\begin{aligned}
\hat{\rho}_{ABout}&=&{\cal N}_{3}^{2} \left( \begin{array}{cccc}
\gamma|\epsilon_1|^2\eta_{va}^2\eta_{vb}^2&0&0&\gamma \epsilon_1\epsilon_2^{*}\eta \\
0 &\Gamma \eta_{va}^2\eta_{hb}^2 &\Gamma \eta & 0 \\
0 & \Gamma \eta & \Gamma \eta_{ha}^2\eta_{vb}^2 & 0 \\
\gamma \epsilon_1^{*}\epsilon_2\eta & 0 & 0 &\gamma |\epsilon_2|^2\eta_{ha}^2\eta_{hb}^2 \\ \end{array} \right)\end{aligned}$$ with $\Gamma = \frac{(1-\gamma)}{2}$ and the normalisation ${\cal N}_{3}$ given by $$\begin{aligned}
{\cal N}_{3}^{2} =
[\gamma(|\epsilon_1|^2\eta_{va}^2\eta_{vb}^2+|\epsilon_2^2|\eta_{ha}^2\eta_{hb}^2)+\Gamma(\eta_{va}^2\eta_{h_{b}}^2 + \eta_{ha}^2\eta_{vb}^2)]^{-1}\end{aligned}$$
In figure (\[fig:conc\]) we display the effect of our protocol for a range of $\gamma$ values with $\epsilon_1=1$ and $\epsilon_2=0.1$ (the $\gamma$ values are labeled at the peak of each curve). The initial points for the fixed $\gamma$, $\epsilon_1$ and $\epsilon_2$ are displayed as solid dots. These curves represent the behaviour of the Entropy and EOF of the states as the BSs are tuned to optimise both. We see how this class of state can be improved is dependent on the amount mixing. The behaviour of the state is similarly dependent on the degree of entanglement in the pure state components of the mixed state of (\[eq:rhoineg\]), variations in $\epsilon_{1,2}$, though this is not explicitly shown here.
The curves in figure (\[fig:conc\]) represent the range of (S,EOF) values for the output states from our protocol. We take the specific case of $\gamma = 0.1$ and observe the variation of (S,EOF) as we tune $\eta_{va} = \eta_{vb} = \eta_v $. From the initial state marked with a black circle at (S,EOF) = (0.23,0.84) with $\eta_v=1$ we then adjust the BSs, moving up the curve, to a state with (S,EOF) = (0.075,0.94) for $\eta_v= 0.32$. This constitutes a turning point on the plane and if we continue decreasing $\eta_v$ we follow the curve back to our initial point in the plane after which the entanglement-entropy properties of the state deteriorate from the original values.
What does the state look like? We observe that with ($\epsilon_1,\epsilon_2,\gamma) = (1.00,0.10,0.30)$ and allowing all the light through the horizontal BS (an optimal setting provided $|\epsilon_1| > |\epsilon_2|$ to maximise the output), and tuning the vertical beam splitters transmission to $\eta_{v} = 0.32$ we can take an initial state $$\begin{aligned}
\label{eqn:rhoinsp}
\hat{\rho}_{ABin} = \left( \begin{array}{cccc}
0.297 & 0 & 0 & 0.030 \\
0 & 0.350 & 0.350 & 0 \\
0 & 0.350 & 0.350 & 0 \\
0.030 & 0 & 0 & 0.003 \\ \end{array} \right)\end{aligned}$$ to an output state $$\begin{aligned}
\hat{\rho}_{ABout} = \left( \begin{array}{cccc}
0.039 & 0 & 0 & 0.039 \\
0 & 0.461 & 0.461 & 0 \\
0 & 0.461 & 0.461 & 0 \\
0.039 & 0 & 0 & 0.039 \\ \end{array} \right)\end{aligned}$$ This output state has an increase in the Entanglement of Formation from EOF = 0.52 to EOF = 0.78, while the entropy of the system has decreased from S = 0.30 to S = 0.20, this result is achieved with a finite probability P = 7.6%.
There exists a critical point with respect to concentration at $\gamma = 0.5$ which corresponds to the case where the two pure states of (\[eq:rhoineg\]) are evenly mixed. For those states with the mixing parameter $\gamma \le 0.5$ concentration is possible whilst for those states above this value the entanglement can be increased but this is at the cost of purity. All of these states can be concentrated if we choose to tune another BS, thus highlighting the need for all four BSs. Similarly if we considered a mixture of the pure states of (\[eq:nmeps1\]) and (\[eq:nmeps2\]), where both had $\epsilon_{1,2} \ne 1$, then we find that concentration is still achievable.
Now let us consider the incoherent sum of a pure state and a mixed state and take as an example of this the Werner state, a mixture of the identity and some fraction of a pure state. If the pure state fraction of the Werner state is a non-maximally entangled pure state, then it is possible to increase the entanglement of the state. However this entanglement increase comes at the cost of purity and is bound by the amount of entanglement that would be inherent in a Werner state using a maximally entangled pure state.
\[CONCLUSION\]
In conclusion, we have proposed an entanglement concentration protocol that is experimentally realisable and can produce a finite concentration of Bell pairs from some initially mixed states. The key point here is that whilst this is achievable we are more interested in the entanglement properties then the final form of the state. Indeed with such a simple protocol the range of possible tests with respect to quantum information and entanglement are quite diverse, and whilst this protocol does require some knowledge of the state in determining the tuning parameters and is a non-unitary operation, we believe it should provide a most useful tool in the exploration of mixed state entanglement.\
The authors would like to thank A.G. White and P.G. Kwiat for useful discussions with respect to the practicality of the experimental implementation of this scheme. WJM would like to acknowledge the support of the Australian Research Council.
Electronic address: [email protected] Electronic address: [email protected]
C.H. Bennett, G.Brassard, S. Popescu, and B. Schumacher, J.A. Smolin, and W.K. Wooters, Phys. Rev. Lett. [**76**]{}, 722 (1996). D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. [**77**]{}, 2818 (1996). C.H. Bennett,D.P. Vincenzo, J.A. Smolin and W.K. Wootters, Phys. Rev. A [**54**]{}, 3824 (1996). C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A [**53**]{}, 2046 (1996). E.M. Rains, Phys. Rev. A [**60**]{}, 173 (1999). V. Vedral, M.B. Plenio, M.A. Rippen, and P.L. Knight, Phys.Rev. Lett. [**78**]{}, 2275 (1997). P.G. Kwiat, private communication. S. Popescu, Phys. Rev. Lett. [**74**]{}, 2619 (1995). N. Gisin, Phys. Lett. A [**210**]{}, 151 (1996). A.G. White, D.F.V. James, P.H. Eberhard, and P.G. Kwiat, Phys.Rev. Lett. [**83**]{}, 3103 (1999). A.G. White, D.F.V. James, W.J.Munro, and P.G. Kwiat, In preparation. W.K. Wooters, Phys. Rev. Lett. [**80**]{}, 2245 (1998).
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abstract: |
Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the derivative and real roots of graph polynomials, which have applications in chemistry, control theory and computer science. Related to the derivatives of graph polynomials, polynomial reconstruction of the matching polynomial is also introduced.\
\[2mm\] Keywords: graph polynomial; derivatives; real roots; polynomial reconstruction\
\[2mm\] AMS Subject Classification (2010): 05C31, 05C90, 05C35, 05C50
author:
- |
Xueliang Li and Yongtang Shi\
Center for Combinatorics and LPMC\
Nankai University, Tianjin 300071, China\
Email: [email protected], [email protected]
date:
title: Derivatives and Real Roots of Graph Polynomials
---
Introduction
============
Many kinds of graph polynomials have been introduced and extensively studied, such as characteristic polynomial, chromatic polynomial, Tutte polynomial, matching polynomial, independence polynomial, clique polynomial, etc.
Let $G$ be a simple graph with $n$ vertices and $m$ edges, whose vertex set and edge set are $V(G)$ and $E(G)$, respectively. The [*complement*]{} $\overline{G}$ of $G$ is the simple graph whose vertex set is $V(G)$ and whose edges are the pairs of nonadjacent vertices of $G$. For terminology and notation not defined here, we refer to [@BondyMurty].
Denote by $A(G)$ the adjacency matrix of $G$. The [*characteristic polynomial*]{} of $G$ is defined as $$\phi(G,x)=det(\lambda I-A(G))=\sum_{i=0}^{n}a_{i}^{n-i}.$$ The roots $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ of $\phi(G,x)=0$ are called the [*eigenvalues*]{} of $G$. For more results on $\phi(G,x)$, we refer to [@CDGT; @CDS].
Denote by $m(G,k)$ the [*$k$-th matching number*]{} of $G$ for $k \geq 0$. We assume that $m(G,0)=1$. For $k\geq 1$, $m(G,k)$ is defined as the number of ways in which $k$ pairwise independent edges can be selected in $G$. The [*matching polynomial*]{} is defined as $$\alpha(G,x)=\sum_{k\geq 0}(-1)^km(G,k)x^{n-2k}.$$ There is also an auxiliary polynomial $\alpha(G,x,y)$, which is defined as $$\alpha(G,x,y)=\sum_{k\geq 0}(-1)^km(G,k)x^{n-2k}y^k.$$ Note that $\alpha(G,x,y)=y^{n/2}\alpha(G,xy^{-1/2})$. In view of this fact, we may define an auxiliary polynomial of $\phi(G,x,y)$: $$\phi(G,x,y)=y^{n/2}\phi(G,xy^{-1/2})=\sum_{k\geq 0}a_kx^{n-k}y^{k/2}.$$ Note that $\phi(G,x,y)$ is a polynomial in $y$ if and only if $G$ is bipartite.
Denote by $n(G,k)$ the [*$k$-th independence number*]{} of $G$ for $k \geq 0$. We assume that $n(G,0)=1$. For $k\geq 1$, $n(G,k)$ is defined as the number of ways in which $k$ pairwise independent vertices can be selected in $G$. The [*independence polynomial*]{} is defined as $$\omega(G,x)=\sum_{k\geq 0}(-1)^kn(G,k)x^{n-k},$$ which is also called independent set polynomial in [@HoedeLi] and stable set polynomial in [@Stanley]. For more results on the independence polynomials, we refer the surveys [@LevitMadrescu; @Trinks].
Denote by $c(G,k)$ the $k$-th [*clique number*]{} of $G$ for $k \geq
0$. We assume that $c(G,0)=1$. For $k\geq 1$, $c(G,k)$ is defined as the number of ways in which $k$ pairwise adjacent vertices can be selected in $G$. Note that $c(G,1)=n$ and $c(G,2)=m$. The [*clique polynomial*]{} is defined as $$c(G,x)=\sum_{k\geq 0}(-1)^kc(G,k)x^{n-k}.$$ Note that the clique polynomial of a graph $G$ is exactly the independence polynomial of the complement $\overline{G}$ of $G$, i.e., $c(G,x)=\alpha(\overline{G},x)$. Obviously, we also have $$c(G,x)+c(\overline{G},x)=\alpha(G,x)+\alpha(\overline{G},x).$$ The following results are easily obtained.
Let $G_1$ and $G_1$ be two vertex-disjoint graphs. Then we have $$c(G_1\cup G_2,x)=c(G_1,x)+c(G_2,x)-1,\qquad \alpha(G_1\cup G_2,x)=\alpha(G_1,x)\cdot \alpha(G_2,x);$$ $$c(G_1+ G_2,x)=c(G_1,x)\cdot c(G_2,x),\qquad \alpha(G_1+ G_2,x)=\alpha(G_1,x)+ \alpha(G_2,x)-1.$$
In [@HoedeLi], the authors obtained the following similar result.
Let $G_1$ and $G_2$ be two vertex-disjoint graphs with $n_1$ and $n_2$ vertices, respectively. Then $$c(G_1\times G_2, x)=n_2\cdot c(G_1,x)+n_1\cdot c(G_2,x)-(n_1+n_2+n_1n_2x)+1.$$
For more properties on $c(G,x)$ and $\alpha(G,x)$, we refer to [@HoedeLi].
Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the derivative and real roots of graph polynomials, which have applications in chemistry, control theory and computer science. Related to the derivatives of graph polynomials, polynomial reconstruction of the matching polynomial is also introduced.
Derivatives of graph polynomials
================================
The derivatives of the characteristic polynomial were examined by Clarke [@Clarke] and the following result was showed.
Let $G$ be a simple graphs and $\phi(G,x)$ be the characteristic polynomial of $G$. Then $$\mathrm{d} \phi(G,x)/ \mathrm{d}x=\sum_{v\in V(G)}\phi(G-v,x).$$
Gutman and Hosoya [@GutmanHosoya] got a similar result for the matching polynomial.
Let $G$ be a simple graphs and $\alpha(G,x)$ be the matching polynomial of $G$. Then $$\mathrm{d} \alpha(G,x)/ \mathrm{d}x=\sum_{v\in V(G)}\alpha(G-v,x).$$
One can get the first derivative of the independence polynomial and clique polynomial, which have similar expressions as the matching polynomial and characteristic polynomial. That is, $$\mathrm{d} \omega(G,x)/ \mathrm{d}x=\sum_{v\in V(G)}\omega(G-v,x),\ \ \mathrm{d}
c(G,x)/ \mathrm{d}x=\sum_{v\in V(G)}c(G-v,x).$$ Although the four first derivatives obey fully analogous expressions, their proofs existing in the literatures, are quite dissimilar. Li and Gutman [@LiGutman1995] provided a unified approach to all these formulas by introducing a general graph polynomial.
Let $f$ be a complex-valued function defined on the set of graphs $\mathcal{G}$ such that $G_1\cong G_2$ implies $f(G_1)=f(G_2)$. Let $G$ be a graph on $n$ vertices and $S(G)$ be the set of all subgraphs of $G$. Define $$S_k(G)=\{H: H\in S(G)\ and
\ |V(H)|=k\}, \ \ \ p(G,k)=\sum_{H\in S_k(G)}f(H).$$ Then, the general graph polynomial of $G$ is defined as $$P(G,x)=\sum_{k=0}^np(G,k)x^{n-k}.$$ Actually, let $$f(H)=\left\{
\begin{array}{ll}
(-1)^{|V(H)|/2} &{\text{if $H$ is $1$-regular};}\\
0 &{\text{otherwise.}}
\end{array}
\right.$$ Then the resulting polynomial is the matching polynomial. Let $$f(H)=\left\{
\begin{array}{ll}
(-1)^{|V(H)|} &{\text{if $H$ is no edges};}\\
0 &{\text{otherwise.}}
\end{array}
\right.$$ Then the resulting polynomial is the independence polynomial. Let $$f(H)=\left\{
\begin{array}{ll}
(-1)^{r(H)}\cdot 2^{c(H)} &{\text{if all components of $H$ are 1- or 2-regular};}\\
0 &{\text{otherwise,}}
\end{array}
\right.$$ where $r(H)$ is the number of components in $H$ and $c(H)$ is the number of cycles in $H$. Then the resulting polynomial is the characteristic polynomial. Let $$f(H)=\left\{
\begin{array}{ll}
(-1)^{|V(H)|} &{\text{if $H$ is a complete graph};}\\
0 &{\text{otherwise.}}
\end{array}
\right.$$ Then the resulting polynomial is the clique polynomial.
The following theorem was obtained by Li and Gutman in [@LiGutman1995].
For the graph polynomial $P(G,x)$ of $G$, we have $$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}x} (P(G,x)) =\sum_{v\in
V(G)}P(G-v,x).\end{aligned}$$
Furthermore, Gutman [@gutman1992; @Gutman2] got the first derivative formula for $\alpha(G,x,y)$: $$\partial \alpha(G,x,y)/ \partial y=-\sum_{uv\in E(G)}\alpha(G-u-v,x,y).$$ To find an expression for $\partial \phi(G,x,y)/ \partial y$ of a bipartite graph was posed by Gutman as a problem in [@Gutman0]. A solution of this problem was offered by Li and Zhang [@LiZhang].
For a bipartite graph $G$, $$\begin{aligned}
\label{eq1}
\partial \phi(G,x,y)/
\partial y=-\sum_{uv\in E(G)}\phi(G-u-v,x,y)-\sum_{C\subseteq G}
n(C)y^{n(C)/2-1}\phi(G-C,x,y),\end{aligned}$$ where $C$ is a cycle, possessing $n(C)$ vertices.
The above theorem was proved by using Sachs Theorem for the coefficients of the characteristic polynomial and by verifying the equality of the respective coefficients of the polynomials on the left- and right-hand sides of Eq. . In [@GutmanLiZhang], the authors put forward another route to Eq. , from which it become evident that Eq. holds for an arbitrary graph.
Moreover, if we define $$P(G,x,y)=\sum_{i+j=n}p(G,k)x^iy^j,$$ then we can obtain $$\frac{\partial P(G,x,y)}{\partial y}=ny^{-1}P(G,x,y)-xy^{-1}\sum_{v\in V(G)}P(G-v,x,y).$$
Derivatives of other graph polynomials have also been studied, such as the cube polynomial [@BresarKlavzarSkrekovski], the Tutte polynomial [@Ellis-MonaghanMerino], the Wiener polynomial [@KonstantinovaDiudea], etc.
Polynomial reconstruction of the matching polynomial
====================================================
The derivative of a graph polynomial is related the problem of polynomial reconstruction. This section aims to prove that graphs with pendant edges are polynomial reconstructible and, on the other hand, to display some evidence that arbitrary graphs are not, which is given in [@LiShiTrinks].
The famous (and still unsolved) reconstruction conjecture of Kelly [@kelly1957] and Ulam [@ulam1960] states that every graph $G$ with at least three vertices can be reconstructed from (the isomorphism classes of) its vertex-deleted subgraphs.
With respect to a graph polynomial $P(G)$, this question may be adapted as follows: Can $P(G)$ of a graph $G = (V, E)$ be reconstructed from the graph polynomials of the vertex deleted-subgraphs, that is from the collection $P(G_{-v})$ for $v
\in V$ ? Here, this problem is considered for the matching polynomial of a graph. For results about the polynomial reconstruction of other graph polynomials, see the article by Brešar, Imrich, and Klavžar [@bresar2005 Section 1] and the references therein. For additional results, see [@tittmann2011 Section 7] [@trinks2012c Subsection 4.7.3].
The matching polynomial we considered here is the generating function of the number of its matchings with respect to their cardinality, denoted by $M(G, x, y)$, which is different from the above $\alpha(G, x)$ and $\alpha(G, x, y)$. Let $G = (V, E)$ be a graph. A [*matching*]{} in $G$ is an edge subset $A \subseteq E$, such that no two edges in $A$ have a common vertex. The [*matching polynomial*]{} $M(G, x, y)$ is defined as $$M(G, x, y) = \sum_{A \subseteq E \text{ is a matching}}
x^{\text{def}(G, A)} y^{|A|},$$ where $\text{def}(G, A) = |V| - |\bigcup_{e \in A}{e}|$ is the number of vertices not included in any of the edges of $A$. A matching $A$ is a [*perfect matching*]{}, if its edges include all vertices, that means if $def(G, A) = 0$. A [*near-perfect matching*]{} $A$ is a matching that includes all vertices except one, that means $def(G, A) = 1$. For more information about matchings and the matching polynomial, see [@farrell1979b; @gutman1977; @lovasz1986].
For a graph $G = (V, E)$ with a vertex $v \in V$, $G_{-v}$ is the graph arising from the [*deletion*]{} of $v$, i.e., arising by the removal of $v$ and all the edges incident with $v$. The multiset of (the isomorphism classes of) the vertex-deleted subgraphs $G_{-v}$ for $v \in V$ is the [*deck*]{} of $G$. The [*polynomial deck*]{} $\mathcal{D}_P(G)$ with respect to a graph polynomial $P(G)$ is the multiset of $P(G_{-v})$ for $v \in V$. A graph polynomial $P(G)$ is [*polynomial reconstructible*]{}, if $P(G)$ can be determined from $\mathcal{D}_P(G)$.
By arguments analogous to those used in Kelly’s Lemma [@kelly1957], the derivative of the matching polynomial of a graph $G = (V, E)$ equals the sum of the polynomials in the corresponding polynomial deck.
Let $G = (V, E)$ be a graph. The matching polynomial $M(G, x, y)$ satisfies $$\begin{aligned}
\frac{\delta}{\delta x} M(G, x, y) = \sum_{v \in V}{M(G_{-v}, x,
y)}.\end{aligned}$$
In other words, all coefficients of the matching polynomial except the one corresponding to the number of perfect matchings can be determined from the polynomial deck and thus also from the deck: $$\begin{aligned}
m_{i, j}(G) = \frac{1}{i} \sum_{v \in V}{m_{i, j}(G_{-v})} \qquad
\forall i \geq 1,\end{aligned}$$ where $m_{i, j}(G)$ is the coefficient of the monomial $x^i y^j$ in $M(G,x,y)$.
Consequently, the (polynomial) reconstruction of the matching polynomial reduces to the determination of the number of perfect matchings.
\[prop:polynomial\_reconstruction\] The matching polynomial $M(G, x, y)$ of a graph $G$ can be determined from its polynomial deck $\mathcal{D}_M(G)$ and its number of perfect matchings. In particular, the matching polynomial $M(G, x, y)$ of a graph with an odd number of vertices is polynomial reconstructible.
Tutte [@tutte1979 Statement 6.9] showed that the number of perfect matchings of a simple graph can be determined from its deck of vertex-deleted subgraphs and therefore gave an affirmative answer on the reconstruction problem for the matching polynomial.
The matching polynomial of a simple graph can also be reconstructed from the deck of edge-extracted and edge-deleted subgraphs [@farrell1987 Theorem 4 and 6] and from the polynomial deck of the edge-extracted graphs [@gutman1992 Corollary 2.3]. For a simple graph $G$ on $n$ vertices, the matching polynomial is reconstructible from the collection of induced subgraphs of $G$ with $\lfloor{\frac{n}{2}}\rfloor + 1$ vertices [@godsil1981b Theorem 4.1].
The following result is from [@LiShiTrinks] for simple graphs with pendant edges.
\[theo:pendant\_perfect\_matching\] Let $G = (V, E)$ be a simple graph with a vertex of degree $1$. Then, $G$ has a perfect matching if and only if each vertex-deleted subgraph $G_{-v}$ for $v \in V$ has a near-perfect matching.
As proved recently by Huang and Lih [@huang2014], this statement can be generalized to arbitrary simple graphs.
\[coro:forest\_perfect\_matching\] Let $G = (V, E)$ be a forest. Then $G$ has a perfect matching if and only if each vertex-deleted subgraph $G_{-v}$ for $v \in V$ has a near-perfect matching.
Forests have either none or one perfect matching, because every pendant edge must be in a perfect matching (in order to cover the vertices of degree $1$) and the same holds recursively for the subforest arising by deleting all the vertices of the pendant edges. Therefore, from Proposition \[prop:polynomial\_reconstruction\] and Corollary \[coro:forest\_perfect\_matching\] the polynomial reconstructibility of the matching polynomial follows.
The matching polynomial $M(G, x, y)$ of a forest is polynomial reconstructible.
On the other hand, arbitrary graphs with pendant edges can have more than one perfect matching. However, Corollary \[coro:forest\_perfect\_matching\] can be extended to obtain the number of perfect matchings. For a graph $G = (V, E)$, the number of perfect matchings and of near-perfect matchings of $G$ is denoted by $p(G)$ and $np(G)$, respectively.
\[theo:pendant\_number\_perfect\_matching\] Let $G = (V, E)$ be a simple graph with a pendant edge $e = \{u,
w\}$ where $w$ is a vertex of degree $1$. Then we have $$\begin{aligned}
&p(G) = np(G_{-u}) \leq np(G_{-v}) \qquad \forall v \in V \text{ and particularly} \\
&p(G) = \min{\{np(G_{-v}) \mid v \in V\}}.\end{aligned}$$
By applying this theorem, the number of perfect matchings of a simple graph with pendant edges can be determined from its polynomial deck and the following result is obtained as a corollary.
The matching polynomial $M(G, x, y)$ of a simple graph with a pendant edge is polynomial reconstructible.
While it is true that the matching polynomials of graphs with an odd number of vertices or with a pendant edge are polynomial reconstructible, it does not hold for arbitrary graphs.
There are graphs which have the same polynomial deck and yet their matching polynomials are different. Although there are already counterexamples with as little as six vertices, it seems that nothing has been published before in connection with the question addressed here.
The matching polynomial $M(G, x, y)$ of an arbitrary graph is not polynomial reconstructible. The minimal counterexample for simple graphs (with respect to the number of vertices and edges) are constructed in [@LiShiTrinks].
The question arises here: whether or not there are such counterexamples consisting of graphs with an arbitrary even number of vertices. In [@LiShiTrinks], we gave an affirmative answer to this question.
Roots of beta-polynomials and independence polynomials
======================================================
Polynomials whose all zeros are real-valued numbers are said to be [*real*]{}. Several graph polynomials have been known to be real; among them the matching polynomial $\alpha(G,x)$ plays a distinguished role [@godsilgutman; @HeilmannLieb].
Polynomials with only real roots arise in various applications in control theory and computer science [@Visontai], but also admit interesting mathematical properties on their own. Newton noted that the sequence of coefficients of such polynomials form a log-concave (and hence unimodal) sequence. These polynomials also have strong connections to totally positive matrices.
The fact that for all graphs, all zeros of the matching polynomial are real-valued was first established by Heilmann and Lieb [@HeilmannLieb].
Let $C$ be a circuit contained in a graph $G$. If $C$ is a Hamiltonian cycle, then $\alpha(G-C, x)\equiv 1$. In certain considerations in theoretical chemistry [@Aihara; @LepovicGutmanPetrovicMizoguchi; @Mizoguchi1; @Mizoguchi2], graph polynomials $\beta(G,C,x)$ are encountered, defined as $$\begin{aligned}
\label{eq2}
\beta(G,C,x)=\alpha(G,x)-2\alpha(G-C,x)\end{aligned}$$ and $$\begin{aligned}
\label{eq3}
\beta(G,C,x)=\alpha(G,x)+2\alpha(G-C,x)\end{aligned}$$ Formula is used in the case of so-called Hückel-type circuits, whereas formula for the so-called Möbius-type circuits. For details, see [@Mizoguchi1]. These polynomials are also called [*circuit characteristic polynomials*]{} [@Aihara].
Already in the first paper devoted to this topic [@Aihara], Aihara mentioned that the zeros of the $\beta$-polynomials are real-valued, but gave no argument to support his claim. In the meantime, for a number of classes of graphs it was shown that $\beta(G,C, x)$ is indeed a real polynomial [@Gutman3; @Gutman4; @GutmanMizoguchi; @LepovicGutmanPetrovicMizoguchi; @LiZhaoGutman; @Mizoguchi2; @LepovicGutmanPetrovic]. In addition to this, by means of extensive computer searches not a single graph with non-real $\beta$-polynomial could be detected. The following conjecture has been put forward by Gutman and Mizoguchi in [@Gutman3; @Gutman4; @GutmanMizoguchi].
For any circuit $C$ contained in any graph $G$, the $\beta$-polynomials $\beta(G,C,x)$, Eqs. and , are real.
Many results have been obtained. In particular, $\beta(G,C,x)$ has been shown to be real for unicyclic graphs [@GutmanMizoguchi], bicyclic graphs [@Mizoguchi2], graphs in which no edge belongs to more than one circuit [@Mizoguchi2], graphs without 3-matchings (i.e., $m(G,3)=0$) [@LepovicGutmanPetrovicMizoguchi], several (but not all) classes of graphs without 4-matchings (i.e., $m(G,4)=0$) [@LepovicGutmanPetrovic].
In [@LiGutmanMilovanovic], Li et al. showed that the conjecture is true for complete graphs. Actually, they proved a stronger result for complete graphs.
For any circuit $C$ in the complete graph $K_n$, the polynomial $$\beta (K_n, C, t; x)=
\alpha (K_n, x)+t\alpha (K_n-C, x)$$ is real for any real $t$ such that $|t|\leq n-1$.
The proof offered in [@LiGutmanMilovanovic] relies on an earlier published theorem by Turán. In [@LiGutman2000], Li and Gutman presented an elementary self-contained proof for complete graphs. Finally, in [@LiZhaoWang], Li et al. showed that the conjecture is true for all graphs, and therefore completely solved this conjecture.
For any circuit $C$ contained in any graph $G$, all roots of the polynomial $\beta(G,C, x)$ are real.
Chudnovsky and Seymour [@ChudnovskySeymour] proved the following result for independence polynomial.
\[thm1\] If $G$ is clawfree, then all roots of its independence polynomial are real.
Theorem \[thm1\] extends a theorem of [@HeilmannLieb], answering a question posed by Hamidoune [@Hamidoune] and Stanley [@Stanley]. Since all line graphs are clawfree, this extends the result of [@HeilmannLieb]. Later, Levit and Mandrescu studied the roots of independence polynomials of almost all very well-covered graphs [@LevitMadrescu1]. In [@Mandrescu], Mandrescu showed that starting from a graph $G$ whose independence polynomial has only real roots, one can build an infinite family of graphs, whose independence polynomials have only real roots.
Real roots of other graph polynomials have also been extensively studied, such as edge-cover polynomial [@AkbariOboudi], the expected independence polynomial [@BrownDilcherManna], domination polynomial [@BrownTufts], sigma-polynomial [@ZhaoLiZhangLiu], chromatic polynomial [@DongKoh; @Jackson; @Woodall], Wiener polynomial [@DehmerIlic], flow polynomial [@Jackson], Tutte polynomial [@EM; @Verg], etc. For more results on the roots of graph polynomials, we refer to [@DehmerShiMowshowitz; @Haglund; @HaglundOnoWagner; @MakowskyRavveBlanchard; @Nijenhuis; @SavageVisontai; @Visontai].
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors are supported by NSFC and the “973" program.
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|
---
author:
- '[^1]'
bibliography:
- 'IEEEabrv.bib'
- 'Literatur.bib'
title: |
\
Cooperative Protocols for Random Access Networks
---
[^1]: This work has been supported by the UMIC Research Centre, RWTH Aachen University
|
---
abstract: 'To a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo Laplacians on the Weyl chamber. We develop the time-dependent scattering theory for these discrete pseudo Laplacians and determine the corresponding wave operators and scattering operators in closed form. As an application, we describe the scattering behavior of certain hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser models associated with the Macdonald polynomials.'
address: ' Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile'
author:
- 'J.F. van Diejen'
title: 'Scattering Theory of Discrete (Pseudo) Laplacians on a Weyl Chamber'
---
[^1]
Introduction {#sec1}
============
A fundamental property of the solitonic solutions of integrable nonlinear wave equations is that their multi-particle scattering process decomposes into pairwise two-particle interactions [@sco-chu-mcl:soliton; @abl-seg:solitons; @nmpz:theory; @new:solitons; @fad-tak:hamiltonian]. This phenomenon is preserved at the quantum level: the corresponding solitonic quantum field theories are characterized by an $N$-particle scattering matrix that factorizes in terms of two-particle scattering matrices [@mat-lie:many-body; @kor-bog-ize:quantum]. As it turns out, this type of factorization can be understood heuristically as being a consequence of the integrability of the models in question [@kul:factorization; @rui-sch:new].
An archetype example of an integrable system with factorized scattering is the celebrated nonlinear Schrödinger equation (NLS). The quantum version of this model boils down to a bosonic $N$-particle system with a pairwise interaction via delta-functional potentials. The factorization of the scattering manifests itself through the asymptotics of the wave function, which is characterized by (products of) two-particle scattering matrices (or $c$-functions) [@mat-lie:many-body; @gau:fonction; @oxf:hamiltonian; @kor-bog-ize:quantum].
In recent work, Ruijsenaars constructed a remarkably large class of quantum integrable lattice models of $N$-particles exhibiting factorized scattering [@rui:factorized]. The discrete systems in question arise by interpreting recurrence relations (or Pieri formulas) for symmetric multivariate orthogonal polynomials as quantum eigenvalue equations. Here the polynomial variable plays the role of the spectral parameter and the index (i.e. partition) labelling the polynomials is thought of as the discrete spatial variable. By analyzing the asymptotics of the polynomials as the degree tends to infinity, Ruijsenaars demonstrated that—for factorized orthogonality measures subject to certain technical conditions ensuring that the particle interaction is short-range and the spectrum is absolutely continuous—the corresponding discrete models are governed by a scattering matrix that factorizes into two-particle scattering matrices. An interesting particular case is that of the Macdonald polynomials [@mac:symmetric]. The corresponding $N$-particle model can be identified as a hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser system [@rui:finite-dimensional; @rui:systems]. At the level of classical Hamiltonian mechanics, the scattering of the corresponding integrable system was studied in great detail in Ref. [@rui:action-angle].
It is known that the root systems of simple Lie algebras form a fruitful context for understanding Calogero-Moser systems and particle models with delta-functional potentials [@ols-per:quantum; @gut:integrable; @hec-sch:harmonic; @hec-opd:yang; @opd:lecture]. From this perspective, it is natural to ask for a generalization of Ruijsenaars’ construction to the case of arbitrary root systems. The purpose of the present paper is to provide such a construction.
More specifically, we associate to a crystallographic root system a system of Weyl-group invariant multivariate orthogonal polynomials on the Weyl alcove, characterized by a weight function that factorizes over the roots (of the root system) in terms of one-dimensional $c$-functions. The orthogonality implies that the polynomials satisfy a system of recurrence relations (Pieri formulas). These recurrence relations are interpreted as eigenvalue equations for an integrable system of discrete pseudo Laplacians on the Weyl chamber. We develop the time-dependent scattering theory for these discrete pseudo Laplacians and determine the corresponding wave operators and scattering operators in closed form. For a specific choice of the weight function, our polynomials amount to the Macdonald polynomials associated with root systems [@mac:orthogonal; @mac:affine]. Again the corresponding integrable lattice model then permits identification as a discrete hyperbolic Ruijsenaars-Schneider type Calogero-Moser system [@rui:finite-dimensional; @rui:systems; @die:integrability]. For the type $A$ root systems the Weyl group is the symmetric group and we reproduce the results of Ruijsenaars [@rui:factorized].
The wave- and scattering operators computed in this paper compare the dynamics generated by the discrete pseudo Laplacian to that of a free discrete Laplacian (corresponding to the case that the $c$-functions reduce to constant functions). Our study of the scattering consists of two parts. In the first (time-independent) part it is shown that the wave function of the discrete pseudo Laplacian has plane wave asymptotics, provided that the $c$-functions determining the orthogonality measure of the polynomials satisy certain analyticity requirements (guaranteeing that the spectrum of the discrete pseudo Laplacian is absolutely continuous). This part of the discussion hinges on previous results describing the large-degree asymptotics of the class of multivariate orthogonal polynomials under consideration [@die:asymptotic; @die:asymptotics]. The second (time-dependent) part consists of a stationary phase analysis that permits proving the existence and unitarity of the wave operators and scattering operators given the plane wave asymptotics of the wave functions. Key ingredient of this part of the discussion is a stationary phase estimate from [@ree-sim:methods p. 38-39] that controls the decay for $t\to\pm\infty$ of certain oscillatory integrals describing the difference between interacting and freely evolving wave packets.
The paper is organized as follows. Section \[sec2\] describes the construction of orthogonal polynomials related to root systems. In Section \[sec3\] we introduce a commuting system of discrete pseudo Laplacians on the Weyl chamber diagonalized by the orthogonal polynomials in question. The wave operators and scattering operators for our discrete pseudo Laplacians are determined in Section \[sec4\]. The stationary phase analysis that lies at the basis of the computation of these wave– and scattering operators is relegated to Section \[sec5\]. Finally, in Section \[sec6\] we specialize to the case of Macdonald polynomials and detail the scattering theory of the associated hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser models. Some key properties of the Macdonald polynomials invoked in Section \[sec6\] have been collected in Appendix \[appA\] at the end of the paper. For the reader’s convenience, we have also included an index of notations in Appendix \[appB\].
Let us conclude this introduction by providing a brief description of what the main results amount to in the elementary (classical) situation of a root system of rank 1. Let $\hat{c}(z)$ be a zero-free analytic function on the disc $|z|\leq \varrho$, with $\varrho >1$, that is real-valued for $z$ real and normalized such that $\hat{c}(0)=1$. We associate to $\hat{c}(z)$ an orthonormal basis of trigonometric polynomials $P_0(\xi), P_1(\xi),P_2(\xi),\ldots$ for the Hilbert space $L^2((0,\pi),\frac{2\sin^2(\xi)\text{d}x}
{\pi\hat{c}(e^{i\xi})\hat{c}(e^{-i\xi})})$ that is obtained by applying the Gram-Schmidt process to the Fourier-cosine basis $1,\cos (\xi),\cos (2\xi),\ldots$. It is an immediate consequence of the three-term recurrence relation for the orthonormal polynomials $P_\ell (\xi)$ that the wave function $$\label{wave}
\Psi_\ell (\xi)=
\frac{2\sin(\xi)P_\ell (\xi)}{\sqrt{\hat{c}(e^{i\xi})\hat{c}(e^{-i\xi})}},
\quad \xi\in (0,\pi),\; \ell\in\mathbb{N},$$ satisfies an eigenvalue equation of the form $L\Psi=2\cos(\xi)\Psi$, where $L$ represents a discrete (self-adjoint) Laplacian acting on lattice functions $\phi\in
\ell^2 (\mathbb{N})$ as $$\label{lap}
L\phi_\ell = a_\ell
\phi_{\ell+1}+b_\ell\phi_\ell+a_{\ell-1}\phi_{\ell-1} \qquad
(\phi_{-1}\equiv 0),$$ with $a_\ell, b_\ell $ denoting the coefficients of the three-term recurrence relation. For $\hat{c}(z)=1$, the polynomials $P_\ell(\xi)$ amount to the Chebyshev polynomials of the second kind $U_\ell (\cos \xi)=\sin (\ell+1)\xi/\sin\xi$, whence the wave function in Eq. reduces in this case to the Fourier-sine kernel $\Psi_\ell^{(0)}(\xi)=2\sin (\ell +1)\xi$. The Laplacian $L$ then amounts to a free Laplacian $L^{(0)}$ whose action on lattice functions is given by $L^{(0)}\phi_\ell=\phi_{\ell +1}+\phi_{\ell-1}$.
Theorem \[plane:thm\] (below) now states that for $\ell\to\infty
$ the wave function $\Psi_\ell (\xi)$ converges exponentially fast in $L^2((0,\pi),(2\pi)^{-1}\text{d}\xi )$ to the anti-symmetric combination of plane waves $$\label{pw-exp}
\Psi_l^{\infty}(\xi)=\hat{s}^{1/2}(\xi) e^{i(\ell+1)\xi}-\hat{s}^{-1/2}(\xi)
e^{-i(\ell+1)\xi},$$ with $\hat{s}(\xi )=\hat{c}(e^{-i\xi})/\hat{c}(e^{i\xi} )$.
Furthermore, let us denote by $\mathcal{F}:l^2(\mathbb{N})\mapsto
L^2((0,\pi),(2\pi)^{-1}\text{d}\xi)$ and $\mathcal{F}^{(0)}:l^2(\mathbb{N})\mapsto
L^2((0,\pi),(2\pi)^{-1}\text{d}\xi)$ the Fourier pairings with kernel $\Psi_\ell(\xi)$ and $\Psi_\ell^{(0)}(\xi)$, respectively: $$\begin{cases}
{\displaystyle \hat{\phi}(\xi) = \sum_{\ell\in\mathbb{N}}
\phi_\ell\Psi_\ell(\xi)} \\[2ex]
{\displaystyle \phi_\ell = \frac{1}{2\pi} \int_0^\pi
\hat{\phi}(\xi)\Psi_\ell(\xi)\text{d}\xi }
\end{cases} ,
\qquad
\begin{cases}
{\displaystyle \hat{\phi}(\xi) = \sum_{\ell\in\mathbb{N}}
\phi_\ell\Psi^{(0)}_\ell(\xi)} \\[2ex]
{\displaystyle \phi_\ell = \frac{1}{2\pi} \int_0^\pi
\hat{\phi}(\xi)\Psi_\ell^{(0)}(\xi)\text{d}\xi }
\end{cases}.$$ Then Theorem \[wave:thm\] and Corollary \[scattering:cor\] (below) state that the wave operators $\Omega_\pm=s-\lim_{t\to
\pm\infty} e^{itL} e^{-itL^{(0)}}$ and the scattering operator $\mathcal{S}=\Omega_+^{-1}\Omega_-$ exist in $l^2(\mathbb{N})$ and are given explicitly by the unitary operators $\Omega_\pm =
\mathcal{F}^{-1}\circ \hat{\mathcal{S}}^{\mp 1/2}\circ
\mathcal{F}^{(0)}$ and $\mathcal{S} =
(\mathcal{F}^{(0)})^{-1}\circ \hat{\mathcal{S}}\circ
\mathcal{F}^{(0)}$, where $\hat{\mathcal{S}}$ denotes a unitary scattering matrix that is characterized by its multiplicative action on a wave packet $\hat{\phi}\in L^2((0,\pi),(2\pi)^{-1}\text{d}\xi)$ of the form $(\hat{\mathcal{S}} \hat{\phi})(\xi)=
\hat{s}(-\xi)\hat{\phi}(\xi) $ for $0 <\xi <\pi$ (with $\hat{s}(\xi)$ as defined just below Eq. ).
Thanks are due to S.N.M. Ruijsenaars for several helpful discussions and to the referees for suggesting some improvements in the presentation.
Orthogonal Polynomials Related to Root Systems {#sec2}
==============================================
In this section we introduce a class of multivariate orthogonal polynomials related to root systems. For basic facts on root systems we refer to the standard works [@bou:groupes; @hum:introduction].
Polynomials on the Weyl Alcove {#sec2.1}
------------------------------
Let $\mathbf{E}$, $\langle\cdot ,\cdot\rangle$ be a real $N$-dimensional Euclidean vector space and let $\boldsymbol{R}\subset \mathbf{E}$ denote an irreducible crystallographic root system spanning $\mathbf{E}$. We write $\mathcal{Q}$ and $\mathcal{Q}^+$ for the root lattice and its nonnegative semigroup generated by the positive roots $\boldsymbol{R}^+$ $$\mathcal{Q}= \text{Span}_\mathbb{Z}
(\boldsymbol{R}),\;\;\;\mathcal{Q}^+= \text{Span}_\mathbb{N}
(\boldsymbol{R}^+) ,$$ and we write $\mathcal{P}$ and $\mathcal{P}^+$ for the weight lattice its nonnegative cone of dominant weights
$$\begin{aligned}
\mathcal{P} &=& \{ \lambda\in \mathbf{E} \mid \langle \lambda
,\alpha^\vee
\rangle \in\mathbb{Z},\; \forall \alpha\in \boldsymbol{R} \} ,\\
\mathcal{P}^+ &=&
\{ \lambda\in \mathbf{E} \mid \langle \lambda ,\alpha^\vee
\rangle \in\mathbb{N},\; \forall \alpha\in \boldsymbol{R}^+ \} ,\end{aligned}$$
where we have introduced the coroot $\alpha^\vee\equiv 2\alpha
/\langle \alpha ,\alpha \rangle$. The algebra of (trigonometric) polynomials on the Weyl alcove $$\mathbf{A}= \{ \xi\in \mathbf{E} \mid 0 <\langle \xi ,\alpha
\rangle < 2\pi, \; \forall \alpha\in \boldsymbol{R}^+ \}$$ is spanned by the basis of the monomial symmetric functions $$m_\lambda (\xi) = \frac{1}{| W_\lambda |} \sum_{w\in W} e^{i
\langle \lambda , \xi_w \rangle },\qquad \lambda\in \mathcal{P}^+,$$ where $W\subset \text{GL}(\mathbf{E})$ denotes the Weyl group of the root system $\boldsymbol{R}$, $\xi_w\equiv w(\xi)$, and $|W_\lambda|$ stands for the order of the stabilizer subgroup $W_\lambda =\{ w\in W \mid w(\lambda) =\lambda \}$.
Factorized Weight Functions {#sec2.2}
---------------------------
We will now introduce a class of smooth weight functions on the Weyl alcove $\mathbf{A}$ that factorize over the root system $\boldsymbol{R}$. To this end we write $\boldsymbol{R}_0=\{ \alpha \in \boldsymbol{R} \mid
2\alpha \not\in \boldsymbol{R}\}$ and $\boldsymbol{R}_1=\{ \alpha
\in \boldsymbol{R} \mid \frac{\alpha}{2} \not\in
\boldsymbol{R}\}$. (So for a reduced root system one has that $\boldsymbol{R}_0=\boldsymbol{R}_1=\boldsymbol{R}$ and for the nonreduced root system $\boldsymbol{R}=BC_N$ one has that $\boldsymbol{R}_0=C_N$ and $\boldsymbol{R}_1=B_N$.) The weight functions under consideration are of the form
$$\label{p-m}
\hat{\Delta} (\xi)=\frac{1}{\hat{\mathcal{C}} (\xi)
\hat{\mathcal{C}}(-\xi)},$$
with $$\label{c-f}
\hat{\mathcal{C}} (\xi) =
\prod_{\alpha \in
\boldsymbol{R}_1^+} \hat{c}_{|\alpha|}(e^{-i\langle \alpha ,\xi
\rangle} ) ,$$
where it assumed that the $c$-functions $\hat{c}_{|\alpha|}(z)$ building $\hat{\mathcal{C}}(\xi)$ depend only on the length of the root $\alpha$ (so $\hat{c}_{|\alpha|}(z)=\hat{c}_{|\beta|}(z)$ if $\alpha$ and $\beta$ lie on the same Weyl-orbit). For technical reasons, we will furthermore assume that these $c$-functions $
\hat{c}_{|\alpha|}(z)$ are [*(i)*]{} analytic and zero-free on a closed disc $\mathbb{D}_\varrho =\{ z\in\mathbb{C}\mid |z|\leq
\varrho \}$ of radius $\varrho
>1$, [*(ii)*]{} normalized such that $ \hat{c}_{|\alpha|}(0)=1$, and [*(iii )*]{} real-valued for $z\in\mathbb{R}$ (so $\hat{\mathcal{C}}(-\xi)=\overline{\hat{\mathcal{C}}(\xi)}$).
Gram-Schmidt Orthogonalization {#sec2.3}
------------------------------
The technical conditions on the $c$-functions ensure that $\hat{\Delta} (\xi)$ , defines a smooth positive weight function on $\mathbf{A}$ (which extends analytically to a Weyl-group invariant function on $\mathbf{E}$). We employ this weight function to endow the space of trigonometric polynomials on the Weyl alcove with an inner product structure via embedding in the Hilbert space $L^2(\mathbf{A},\hat{\Delta}|\delta|^2\, \text{d}\xi)$: $$( f , g)_{\hat{\Delta}} =
\frac{1}{|W|\, \text{Vol}(\mathbf{A}) }\,
\int_{\mathbf{A}} f(\xi) \overline{g(\xi)}\,\hat{\Delta}
(\xi)\,|\delta(\xi) |^2 \text{d} \xi ,\qquad \forall f,g\in
L^2(\mathbf{A},\hat{\Delta}|\delta|^2\,\text{d}\xi),$$ where $\overline{g(\xi)}$ stands for the complex conjugate of $g(\xi)$, $\text{Vol}(\mathbf{A})=\int_{\mathbf{A}}\text{d}\xi$, and $\delta (\xi)$ denotes the Weyl denominator $$\label{delta} \delta(\xi)= \prod_{\alpha \in \boldsymbol{R}_0^+} (e^{i
\langle \alpha ,\xi \rangle /2}-e^{-i \langle \alpha ,\xi \rangle
/2}) .$$
Let $\succeq$ be a (partial) order of the dominant weights $\mathcal{P}^+$ refining the [*dominance partial order*]{} $$\label{po}
\lambda \geqslant \mu \;\Longleftrightarrow \;\lambda -\mu \in
\mathcal{Q}^+$$ such that the highest-weight spaces $\text{Span} \{ m_\mu
\}_{\mu\in\mathcal{P}^+,\mu\preceq\lambda}$ remain finite-dimensional for all $\lambda \in\mathcal{P}^+$. By applying the Gram-Schmidt process to the partially ordered monomial basis $\{ m_\lambda \}_{\lambda\in \mathcal{P}^+}$, we construct a normalized basis $\{ P_\lambda \}_{\lambda\in \mathcal{P}^+}$ of $L^2(\mathbf{A},\hat{\Delta}|\delta|^2\,\text{d}\xi)$ given by trigonometric polynomials of the form
$$\label{op1}
P_\lambda (\xi) = \sum_{\mu\in\mathcal{P}^+,\, \mu \preceq
\lambda} a_{\lambda\mu} m_\mu (\xi),\qquad
\lambda\in\mathcal{P}^+,$$
with coefficients $a_{\lambda\mu}\in\mathbb{C}$ such that $$\label{op2}
( P_\lambda , P_\mu )_{\hat{\Delta}} =
\begin{cases}
0 & \text{if}\; \mu \prec\lambda , \\
1 &\text{if}\; \mu =\lambda
\end{cases}$$
(where $ a_{\lambda \lambda}>0$ by convention). The Gram-Schmidt process guarantees that the polynomials $P_\lambda$, $\lambda\in\mathcal{P}^+$ are orthogonal when comparable in the (partial) order $\succeq$ (i.e. $( P_\lambda , P_\mu
)_{\hat{\Delta}} = 0$ when $\lambda \succ \mu$ or $\lambda\prec\mu$). Hence, a sufficient condition to ensure that our polynomials form an orthonormal basis of the Hilbert space $L^2(\mathbf{A}, \hat{\Delta} |\delta|^2\,\text{d}x)$ is to require the refinement $\succeq$ of the dominance order $\geqslant$ to be a [*linear*]{} ordering of $\mathcal{P}^+$. (The fact that the polynomials in Eqs. , form a complete set in $L^2(\mathbf{A}, \hat{\Delta} |\delta|^2\,\text{d}x)$ is a consequence of the Stone-Weierstrass theorem.) In general, the orthonormal basis in question depends on the choice of such linear refinement. It will turn out below, however, that for our principal applications the $c$-functions are such that the orthogonality is already guaranteed when taking for $\succeq$ simply the dominance ordering $\geqslant$ itself (in other words, in such case the construction results to be independent of the choice of the linear refinement). From now on we will always assume that we have fixed a sufficiently fine (partial) ordering $\succeq$ so as to guarantee that the basis $\{
P_\lambda \}_{\lambda\in \mathcal{P}^+}$ be [*orthogonal*]{} (i.e. $( P_\lambda , P_\mu )_{\hat{\Delta}} = 0$ when $\lambda \neq
\mu$).
Weyl Characters {#weyl:sec}
---------------
\[sec2.4\] The simplest example of the above construction is the special case with unit $c$-functions, i.e., with $\hat{c}_{|\alpha|} (z)=1$, $\forall \alpha \in \boldsymbol{R}_1^+$. The weight function then becomes of the form $\hat{\Delta} (\xi)=1$ and the Gram-Schmidt process turns out to be independent of the choice of the refinement $\succeq$ of $\geqslant$ (i.e. in this case we may take $\succeq$ to be equal to $\geqslant$ without restriction). The corresponding orthonormal polynomials $P_\lambda (\xi)$ amount to the celebrated Weyl characters [@mac:orthogonal; @mac:affine] $$\label{weylcars}
P_\lambda (\xi)= \chi_\lambda (\xi)\equiv \delta^{-1} (\xi)
\sum_{w\in W} (-1)^w\, e^{i\langle \rho+\lambda , \xi_w\rangle }
,\qquad \lambda\in\mathcal{P}^+,$$ where $(-1)^w\equiv\det (w)$ and $\rho
\equiv\frac{1}{2}\sum_{\alpha\in \boldsymbol{R}_0^+}\alpha$.
For later use, it will actually be convenient to extend the definition of the Weyl characters $\chi_\lambda (\xi )$ in Eq. to the case of nondominant weights $\lambda$. It is immediate from this definition that for $\lambda\in\mathcal{P}\setminus\mathcal{P}^+$ $$\label{weylcars2}
\chi_\lambda (\xi ) =
\begin{cases}
(-1)^{w_{\rho+\lambda}} \chi_{w_{\rho+\lambda}
(\rho+\lambda)-\rho}(\xi)
&\text{if}\;\; |W_{\rho+\lambda}| =1, \\
0& \text{if}\;\; |W_{\rho+\lambda}| >1,
\end{cases}$$ where, for $\mu\in\mathcal{P}$ regular, $w_\mu\in W$ denotes the unique Weyl group element such that $w_\mu (\mu
)\in\mathcal{P}^+$.
Discrete (Pseudo) Laplacians on the Weyl Chamber {#sec3}
================================================
In this section we associate a commuting system of discrete pseudo Laplacians on $\mathcal{P}^+$ to our orthonormal polynomials $P_\lambda (\xi)$.
Fourier Transform {#sec3.1}
-----------------
Let $\mathcal{H}$ be the Hilbert space $ l^2(\mathcal{P}^+)$ of square-summable functions over the dominant cone $\mathcal{P}^+$ equipped with the standard inner product
$$(f,g)_{\mathcal{H}}=\sum_{\lambda\in\mathcal{P}^+} f_\lambda
\overline{g_\lambda}\qquad (f,g\in l^2 (\mathcal{P}^+)),$$
and let $\hat{\mathcal{H}}$ be the Hilbert space $L^2
(\mathbf{A},\text{d}\xi)$ of square-integrable functions over the Weyl alcove equipped with the normalized inner product $$(\hat{f},\hat{g})_{\hat{\mathcal{H}}}=\frac{1}{|W|\,\text{Vol}(\mathbf{A})}\int_{\mathbf{A}}
\hat{f}(\xi) \overline{\hat{g}(\xi)} \text{d}\xi \qquad
(\hat{f},\hat{g}\in L^2 (\mathbf{A},\text{d}\xi)).$$
By construction, the functions $$\label{wave-f}
\Psi_\lambda (\xi) = \hat{\Delta}^{1/2} (\xi) \delta (\xi)
P_\lambda (\xi) , \qquad \lambda\in\mathcal{P}^+$$ form an orthonormal basis of $\hat{\mathcal{H}}$. As a result, the mapping $\mathcal{F}:\mathcal{H}\mapsto\hat{\mathcal{H}}$ given by $\phi_\lambda
\stackrel{\mathcal{F}}{\longrightarrow}\hat{\phi}(\xi)$ with
$$\begin{aligned}
\label{F1}
\hat{\phi} (\xi) &=& (\phi ,\Psi (\xi))_{\mathcal{H}} \\
&=& \sum_{\lambda\in\mathcal{P}^+} \phi_\lambda
\overline{\Psi_\lambda (\xi )} \nonumber\end{aligned}$$
constitutes a unitary Hilbert space isomorphism between $\mathcal{H}$ and $\hat{\mathcal{H}}$. The inverse mapping $\mathcal{F}^{-1}:\hat{\mathcal{H}}\mapsto\mathcal{H}$ takes the form $\hat{\phi}(\xi)\stackrel{\mathcal{F}^{-1}}{\longrightarrow}
\phi_\lambda$ with $$\begin{aligned}
\label{F2}
\phi_\lambda &=& (\hat{\phi}
,\overline{\Psi}_\lambda)_{\hat{\mathcal{H}}} \\
&=& \frac{1}{|W|\,\text{Vol}(\mathbf{A})}\int_{\mathbf{A}}
\hat{\phi}(\xi) \Psi_\lambda(\xi) \text{d}\xi. \nonumber\end{aligned}$$
We will refer to $\mathcal{F}$ as the [*Fourier transform*]{} associated to the polynomials $P_\lambda (\xi)$. In the simplest case with unit $c$-functions, the wave functions amount to plane waves (cf. Section \[weyl:sec\]) $$\label{pwaves}
\Psi_\lambda^{(0)} (\xi) = \sum_{w\in W} (-1)^w\, e^{i\langle
\rho+\lambda , \xi_w\rangle } .$$ The corresponding Fourier transform reduces to the conventional Fourier transform $\mathcal{F}^{(0)}:\mathcal{H}\mapsto\hat{\mathcal{H}}$ of the form $\phi_\lambda
\stackrel{\mathcal{F}^{(0)}}{\longrightarrow}\hat{\phi} (\xi)$ with
$$\label{F01}
\hat{\phi} (\xi) = \sum_{w\in W} (-1)^w\,
\sum_{\lambda\in\mathcal{P^+}}
\phi_\lambda e^{-i\langle \rho+\lambda , \xi_w\rangle } .$$
The inverse transform $(\mathcal{F}^{(0)})^{-1}:\hat{\mathcal{H}}\mapsto\mathcal{H}$ is then given by $\hat{\phi} (\xi)
\stackrel{(\mathcal{F}^{(0)})^{-1}}{\longrightarrow} \phi_\lambda$ with $$\label{F02}
\phi_\lambda = \frac{1}{|W|\,\text{Vol} (\mathbf{A})} \sum_{w\in
W} (-1)^w\, \int_{\mathbf{A}} \hat{\phi} (\xi) e^{i\langle
\rho+\lambda , \xi_w\rangle } \text{d}\xi .$$
Pseudo Laplacians {#sec3.2}
-----------------
To the basis of fundamental weights $\omega_1,\ldots ,\omega_N$ generating $\mathcal{P}^+$, we associate bounded multiplication operators $\hat{E}_1,\ldots ,\hat{E}_N$ in $\hat{\mathcal{H}}$ of the form $$\hat{E}_r(\xi) = \sum_{\nu\in W(\omega_r)} \exp
( i \langle \nu , \xi \rangle ), \qquad r=1,\ldots ,N,$$ where the sum is over all weights in the Weyl orbit of $\omega_r$. The pullbacks of $\hat{E}_1,\ldots ,\hat{E}_N$ with respect to the Fourier transform $\mathcal{F}$ define an integrable system of bounded commuting operators in $\mathcal{H}$ $$\label{pL}
L_r = \mathcal{F}^{-1}\circ \hat{E}_r \circ \mathcal{F}, \qquad
r=1,\ldots ,N.$$ We will refer to the commutative algebra $\mathbb{R}[L_1,\ldots
,L_N]$ generated by these operators as the (algebra of) [*discrete pseudo Laplacians*]{} associated to the polynomials $P_\lambda (\xi)$. It is immediate from its construction as the pullback of a multiplication operator in $\hat{\mathcal{H}}$ (cf. Eq. ) that the pseudo Laplacian $L_r$ has a purely absolutely continuous spectrum in $\mathcal{H}$ given by the compact set $\sigma (L_r)=\{ \hat{E}_r(\xi) \mid
\xi\in\overline{\mathbf{A}}\}\subset\mathbb{C}$. By acting with both sides of the operator equality $L_r \mathcal{F}^{-1}= \mathcal{F}^{-1} \hat{E}_r$ on (the complex conjugate of) an arbitrary element $\hat{\phi}\in\hat{\mathcal{H}}$, we get
$$\label{ev-eqr}
L_r(\Psi_\lambda ,\hat{\phi})_{\hat{\mathcal{H}}}=
(\hat{E}_r\Psi_\lambda,\hat{\phi})_{\hat{\mathcal{H}}},
\qquad\forall\hat{\phi}\in\hat{\mathcal{H}}.$$
In other words, the functions $\Psi_\lambda (\xi)$ form a complete (as $\mathcal{F}:\mathcal{H}\rightarrow\hat{\mathcal{H}}$ is a Hilbert space isomorphism) set of generalized joint eigenfunctions of our pseudo Laplacians, i.e. formally $$\label{ev-eq}
L_r \Psi_\lambda (\xi ) = \hat{E}_r (\xi) \Psi_\lambda (\xi) .$$
Here $\xi\in\overline{\mathbf{A}}$ plays the role of the spectral parameter and the weight $\lambda\in\mathcal{P}^+$ is interpreted as the discrete geometric variable (i.e. the position variable).
[*i.*]{} Below we will sometimes write formal equalities of the form in Eq. that admit a rigorous interpretation of the form in Eq. upon taking the inner product (smearing) with an arbitrary (stationary) wave packet $\hat{\phi}\in\hat{\mathcal{H}}$.
[*ii.*]{} In general the Laplacian $L_r$ is not self-adjoint. Indeed, the adjoint $L_r^*$ is given by $L_s$ with $\omega_s=-w_0(\omega_r)$, where $w_0$ denotes the longest element of the Weyl group $W$ (i.e., the unique Weyl group element $w_0$ such that $w_0(\mathbf{A})=-\mathbf{A}$). Thus $L_r$ is self-adjoint if and only if $w_0(\omega_r)=-\omega_r$.
Localization {#sec3.3}
------------
Let $\phi:\mathcal{P}^+\rightarrow\mathbb{C}$ be square-summable a lattice function. The action of $L_r$ on $\phi$ is of the form $$L_r \phi_\lambda = \sum_{\mu\in\mathcal{P}^+} a_{\lambda\mu ;r}
\phi_\mu,$$ for certain coefficients $a_{\lambda\mu ;r}\in \mathbb{C}$. We will now show that in fact only a finite number of these coefficients is nonzero.
\[localization:prp\] The action of the pseudo Laplacian $L_r$ on $\phi\in\mathcal{H}$ is of the form
$$L_r \phi_\lambda = \sum_{\mu\in \mathcal{P}^+_{\lambda ;r}}
a_{\lambda\mu ;r} \phi_\mu ,
\qquad a_{\lambda\mu ;r}\in \mathbb{C},$$
where $$\label{locset}
\mathcal{P}^+_{\lambda ;r} =\{ \mu\in\mathcal{P}^+ \mid \mu\preceq
\lambda +\omega_r \; \text{and}\; \mu-w_0(\omega_r)\succeq \lambda
\} .$$
From the triangularity of the monomial expansion of $P_\lambda
(\xi)$ it is immediate that $$m_{\omega_r}(\xi) P_\lambda (\xi) = \sum_{\begin{subarray}{c}
\mu\in\mathcal{P}^+\\ \mu\preceq \lambda+\omega_r\end{subarray}}
a_{\lambda\mu
;r } P_\mu (\xi ) ,\qquad a_{\lambda\mu ;r}\in \mathbb{C}.$$ The orthonormality furthermore implies that $$a_{\lambda\mu ;r }=(m_{\omega_r} P_\lambda ,
P_\mu)_{\hat{\Delta}}= ( P_\lambda ,
m_{\omega_s}P_\mu)_{\hat{\Delta}}=\overline{a_{\mu\lambda ;s
}} ,$$ with $\omega_s=-w_0(\omega_r)$ (cf. Note [*ii*]{}. above). Hence $$a_{\lambda\mu ;r }\neq 0\Rightarrow \mu\in\mathcal{P}^+_{\lambda
;r} .$$ Since $\hat{E}_r(\xi)=m_{\omega_r}(\xi)$ and $ \Psi_\lambda (\xi )=\hat{\Delta}^{1/2}(\xi)\delta(\xi)P_\lambda(\xi)$, we conclude that $$\hat{E}_r (\xi) \Psi_\lambda (\xi) = \sum_{\mu\in
\mathcal{P}^+_{\lambda ;r} }
a_{\lambda\mu ;r}
\Psi_\mu (\xi ) .$$ Taking the innerproduct with an arbitrary wave packet $\hat{\phi}\in\hat{\mathcal{H}}$ and comparison with the eigenvalue equation in Eq. entails that $$L_r (\Psi_\lambda ,\hat{\phi})_{\hat{\mathcal{H}}}=
\sum_{\mu\in \mathcal{P}^+_{\lambda ;r}}
a_{\lambda\mu ;r} (\Psi_\mu ,\hat{\phi})_{\hat{\mathcal{H}}} ,
\qquad \forall\hat{\phi}\in\hat{\mathcal{H}},$$ i.e. formally (cf. Note [*i.*]{} above) $$L_r \Psi_\lambda (\xi )= \sum_{\mu\in \mathcal{P}^+_{\lambda ;r}}
a_{\lambda\mu ;r} \Psi_\mu (\xi ) .$$ The proposition then follows by the completeness of the generalized eigenfunctions $\Psi_\lambda (\xi )$, $\xi \in \overline{\mathbf{A}}$ in the Hilbert space $\mathcal{H}$ (i.e. by the fact that the Fourier transform $\mathcal{F}$ , constitutes a unitary Hilbert space isomorphism between $\mathcal{H}$ and $\hat{\mathcal{H}}$).
A priori the cardinality of the set $\mathcal{P}^+_{\lambda
;r}$ may be unbounded as a function of $\lambda\in\mathcal{P}^+$. Hence, in general our pseudo Laplacians need [*not*]{} be difference operators. If the ordering of the dominant weights $\succeq$ coincides with the dominance order $\geqslant$, however, then it follows from the definition in Eq. that the size of the set $\mathcal{P}^+_{\lambda ;r}$ is bounded by the number of weights in the interval $\{ \nu \in\mathcal{P} \mid w_0(\omega_r)
\leqslant \nu \leqslant \omega_r \}$. Consequently, in this situation our pseudo Laplacian $L_r$ is actually a difference operator in $\mathcal{H}$. (We will refer in such case to $L_r$ as a [*discrete Laplacian*]{} as opposed to merely a pseudo Laplacian.)
\[dl:prp\] When our ordering $\succeq$ coincides with the dominance ordering $\geqslant$ , then the pseudo Laplacians in $\mathbb{R}[L_1,\ldots ,L_N]$ are discrete difference operators in $\mathcal{H}$.
In the case of unit $c$-functions (cf. Section \[weyl:sec\]), our discrete Laplacians $L_1,\ldots ,L_N$ amount to conventional free Laplacians $L_1^{(0)},\ldots ,L_N^{(0)}$ over the dominant cone $\mathcal{P}^+$.
\[fl:prp\] If $\hat{c}_{|\alpha |}(z)=1$, $\forall \alpha\in
\boldsymbol{R}_1^+$, then our discrete Laplacians $L_r$ reduce to the free Laplacians $$L_r^{(0)} \phi_\lambda = \sum_{\nu\in W(\omega_r)}
\phi_{\lambda +\nu} , \qquad r=1,\ldots ,N,$$ with the boundary condition that for $\mu\in\mathcal{P}\setminus\mathcal{P}^+$ $$\phi_\mu =
\begin{cases}
(-1)^{w_{\rho+\mu}} \phi_{w_{\rho+\mu} (\rho+\mu)-\rho}
&\text{if}\;\; |W_{\rho+\mu}| =1, \\
0& \text{if}\;\; |W_{\rho+\mu}| >1
\end{cases}$$ (where $w_{\rho+\mu}$ denotes the Weyl permutation taking the regular weight $\rho+\mu$ to the dominant cone).
As pointed out in Section \[weyl:sec\], the case of unit $c$-functions corresponds to orthonormal polynomials $P_\lambda
(\xi)$ given by the Weyl characters $ \chi_\lambda (\xi)$. It is immediate from the explicit expression for $\chi_\lambda$ in Eq. that the Weyl characters satisfy the well-known recurrence relations $$m_{\omega_r}(\xi) \chi_\lambda (\xi) = \sum_{\nu\in W(\omega_r)}
\chi_{\lambda+\nu} (\xi ) .$$ Starting from these recurrence relations, the proposition readily follows by repeating the arguments in the proof of Proposition \[localization:prp\]. The boundary condition stems from the property of the Weyl characters.
By Proposition \[localization:prp\], the eigenvalue equations in Eq. take the form
$$\sum_{\mu\in \mathcal{P}^+_{\lambda
;r} } a_{\lambda\mu ;r}
\Psi_\mu (\xi ) = \hat{E}_r (\xi) \Psi_\lambda (\xi), \qquad 1,\ldots
,N,$$
or equivalently $$\sum_{\mu\in \mathcal{P}^+_{\lambda
;r} } a_{\lambda\mu ;r}
P_\mu (\xi ) = \hat{E}_r (\xi) P_\lambda (\xi), \qquad 1,\ldots
,N$$
(upon dividing out the trivial overall normalization factor $\delta (\xi)/
\sqrt{\hat{\mathcal{C}}(\xi)\hat{\mathcal{C}}(-\xi)}$ on both sides). The latter equations admit an alternative interpretation as a system of recurrence relations (or Pieri formulas) for the polynomials $P_\lambda (\xi)$.
The above construction of the discrete (pseudo) Laplacians has its origin in the works of Macdonald [@mac:spherical; @mac:orthogonal; @mac:affine]. Specifically, for $c_{|\alpha |}(z)=(1-t_{|\alpha|}z)$ with $-1<t_{|\alpha |}<1$ the polynomials $P_\lambda (\xi)$ , amount to (the parameter deformations of) Macdonald’s zonal spherical functions on $p$-adic Lie groups [@mac:spherical; @mac:orthogonal]. The algebra of discrete Laplacians $\mathbb{R}[L_1,\ldots ,L_N]$ corresponds in this case to the $K$-spherical Hecke algebra of the $p$-adic Lie group. When $c_{|\alpha |}(z)$ is given by a $q$-shifted factorial (cf. Eq. below), then the polynomials $P_\lambda (\xi)$ specialize to the Macdonald polynomials [@mac:symmetric; @mac:orthogonal; @mac:affine]. The discrete Laplacians appear in this context in Cherednik’s double affine Hecke algebra as “coordinate multiplication operators” dual to Macdonald’s difference operators [@che:macdonalds; @mac:affine].
Time-Dependent Scattering Theory {#sec4}
================================
In this section we determine the wave operators and scattering operator associated to our discrete pseudo Laplacians. For background literature on scattering theory the reader is referred to e.g. Refs. [@ree-sim:methods; @pea:quantum; @thi:course].
Plane Wave Asymptotics {#sec4.1}
----------------------
The dominant Weyl chamber is given by the open convex cone $$\label{dwc}
\mathbf{C}^+=\{ \mathbf{x}\in \mathbf{E}\mid \langle \mathbf{x},
\alpha \rangle
> 0,\; \forall \alpha\in \boldsymbol{R}^+ \} .$$ We will now describe the asymptotics of the wave function $\Psi_\lambda (\xi)$ diagonalizing the pseudo Laplacians $L_1,\ldots
,L_N$ for $\lambda$ deep in the Weyl chamber, i.e., for $\lambda$ growing to infinity in such a way that $\langle \lambda ,\alpha^\vee\rangle \to +\infty$ for all positive roots $\alpha\in\boldsymbol{R}^+$.
To this end we define for $\lambda\in\mathcal{P}^+$ $$m(\lambda) \equiv \min_{\alpha\in \boldsymbol{R}^+} \langle
\lambda ,\alpha^\vee\rangle .$$ In previous work, it was shown that the strong $L^2$-asymptotics of the polynomials $P_\lambda (\xi)$ for $m(\lambda)\to\infty$ is given by [@rui:factorized; @die:asymptotic; @die:asymptotics] $$P_\lambda^\infty (\xi) = \delta^{-1}(\xi) \sum_{w\in W} (-1)^w
\hat{\mathcal{C}}(\xi_w) e^{i\langle \rho +\lambda , \xi_w
\rangle}.$$ More precisely, one has that $$\label{pol-as}
\| P_\lambda -P_\lambda^\infty \|_{\hat{\Delta}} =
O(e^{-\epsilon\, m(\lambda) })\quad \text{as}\;\;
m(\lambda)\longrightarrow \infty ,$$ where $\| \cdot \|_{\hat{\Delta}} \equiv (\cdot ,\cdot
)_{\hat{\Delta}}^{1/2}$ and $\epsilon>0$ denotes a decay rate that depends on the radius $\varrho>1$ of the analyticity disc $\mathbb{D}_\varrho$ of the $c$-functions $\hat{c}_{|\alpha |}(z)$ (see the technical assumptions in Section \[sec2\]).
The idea of the proof in [@rui:factorized; @die:asymptotic; @die:asymptotics] of this exponential convergence goes along the following lines. Firstly, a direct (constant term) computation reveals that $$\label{prf1}
\langle P_\lambda^{\infty} ,m_\mu\rangle_{\hat{\Delta}} =
\begin{cases}
0& \text{if}\; \mu\prec\lambda , \\
1& \text{if}\: \mu = \lambda .
\end{cases}$$ Next, we denote by $P_\lambda^{(m(\lambda ))}(\xi) $ the polynomial approximation of the asymptotic function $P_\lambda^\infty (\xi)$ obtained by replacing the overall $c$-function $\hat{\mathcal{C}}(\xi)$ by its Taylor polynomial of degree $m(\lambda)$. Then a combinatorial analysis shows that this polynomial approximation expands triangularly on the basis of monomial symmetric functions $$\label{prf2}
P_\lambda^{(m(\lambda ))}(\xi)=m_\lambda
(\xi)+\sum_{\mu\in\mathcal{P}^+,\mu\prec\lambda} b_{\lambda\mu}
m_\mu (\xi)$$ (for certain coefficients $b_{\lambda\mu}\in\mathbb{C}$). Moreover, the analyticity requirements on the $c$-functions $\hat{c}_{|\alpha |}(z)$ guarantee that $$\label{prf3}
P_\lambda^\infty (\xi) = P_\lambda^{m(\lambda )}(\xi) +
O(e^{-\epsilon m(\lambda )})$$ (this is because the technical conditions ensure that the Taylor coefficients of the $c$-function $\hat{c}_{|\alpha |}(z)$ decay exponentially fast).
From Eqs. - one concludes that—up to an $O(e^{-\epsilon m(\lambda )})$ error term—the asymptotic function $P_\lambda^\infty(\xi)$ amounts to a monic polynomial obtained by performing the Gram-Schmidt process on the monomial symmetric basis with respect to the inner product $(\cdot ,\cdot
)_{\hat{\Delta}}$. In other words, the asymptotic functions coincide up to exponentially decaying error terms with the monic versions of the polynomials $P_\lambda (\xi)$ defined in Eqs. , . The convergence in Eq. now follows from the fact that the orthonormalized polynomials $P_\lambda(\xi)$ are asymptotically monic: $a_{\lambda\lambda}=1+O(e^{-\epsilon
m(\lambda)})$. (This estimate for the leading coefficient in the monomial expansion of $P_\lambda (\xi)$ follows starting from the equality $a_{\lambda\lambda} =\langle P_\lambda
,P_\lambda^\infty\rangle_{\hat{\Delta}}$, upon substituting and expanding the polynomial part $P_\lambda^{m(\lambda)}(\xi)$ in terms of the orthonormalized polynomials $P_\mu(\xi)$, $\mu\preceq\lambda$, taking into account the orthogonality .)
The asymptotic estimate in Eq. for the polynomials $P_\lambda (\xi)$ immediately gives rise to the following plane wave asymptotics for the wave functions $\Psi_\lambda (\xi )$ :
$$\begin{aligned}
\label{aswave}
\Psi_\lambda^\infty (\xi) &= & \hat{\Delta}^{1/2}(\xi) \delta
(\xi) P_\lambda^\infty
(\xi) \\
&=& \sum_{w\in W} (-1)^w \hat{S}_w^{1/2}(\xi) e^{i\langle \rho
+\lambda , \xi_w \rangle} ,\end{aligned}$$
where $$\begin{aligned}
\hat{S}_w (\xi) &=& \frac{\hat{\mathcal{C}}(\xi_w)}{\hat{\mathcal{C}}(-\xi_w)} \\
&=& \prod_{\alpha\in \boldsymbol{R}^+_1\cap
w^{-1}(\boldsymbol{R}^+_1)} \hat{s}_{|\alpha|}(\langle \alpha ,
\xi\rangle) \prod_{\alpha\in \boldsymbol{R}^+_1\cap
w^{-1}(-\boldsymbol{R}^+_1)} \overline{\hat{s}_{|\alpha|}(\langle
\alpha , \xi\rangle)} , \label{Sw}\end{aligned}$$ with $$\label{smat}
\hat{s}_{|\alpha|}(\langle \alpha , \xi\rangle)=
\frac{\hat{c}_{|\alpha|}(e^{-i\langle \alpha ,
\xi\rangle})}{\hat{c}_{|\alpha|}(e^{i\langle \alpha ,
\xi\rangle})}$$
(so $\overline{\hat{s}_{|\alpha|}(\langle \alpha ,
\xi\rangle)}=\hat{s}_{|\alpha|}(-\langle \alpha ,
\xi\rangle)=\hat{s}_{|\alpha|}^{-1}(\langle \alpha , \xi\rangle)$ and $|\hat{s}_{|\alpha|}(\langle \alpha , \xi\rangle)|=1$ ).
\[plane:thm\] The wave function $\Psi_\lambda$ tends to the plane waves $\Psi_\lambda^\infty$ for $\lambda$ deep in the Weyl chamber: $$\| \Psi_\lambda -\Psi_\lambda^\infty \|_{\hat{\mathcal{H}}} =
O(e^{-\epsilon\, m(\lambda) })\quad \text{as}\;\;
m(\lambda)\longrightarrow \infty ,$$ where $\| \cdot \|_{\hat{\mathcal{H}}}\equiv (\cdot
,\cdot)_{\hat{\mathcal{H}}}^{1/2}$.
We see from Theorem \[plane:thm\] that the asymptotics of the wave functions $\Psi_\lambda (\xi)$ for $\lambda$ deep in the Weyl chamber is given by an anti-symmetric combination of plane waves $e^{i\langle \lambda ,\xi \rangle}$ with phase-factors that factorize over the root system in terms of one-dimensional $c$-functions.
Scattering and Wave Operators {#sec4.2}
-----------------------------
For any [*real*]{} multiplication operator $\hat{E}(\xi) \subset
\mathbb{R}[\hat{E}_1 (\xi),\ldots ,\hat{E}_N(\xi)]$, let $L=
\mathcal{F}^{-1}\circ \hat{E} \circ\mathcal{F}$ and let $L^{(0)}=
(\mathcal{F}^{(0)})^{-1}\circ \hat{E} \circ\mathcal{F}^{(0)}$. In other words, the operators $L\subset \mathbb{R}[L_1,\ldots ,L_N]$ and $L^{(0)}\subset \mathbb{R}[L_1^{(0)},\ldots ,L_N^{(0)}]$ are [*self-adjoint*]{} (pseudo) Laplacians in $\mathcal{H}$ such that (formally) $$L \Psi_\lambda (\xi ) = \hat{E} (\xi) \Psi_\lambda (\xi) \quad
\text{and}\quad L^{(0)} \Psi_\lambda^{(0)} (\xi ) = \hat{E} (\xi)
\Psi_\lambda^{(0)} (\xi) .$$ (So the spectrum of $L$ and $L^{(0)}$ in $\mathcal{H}$ is absolutely continuous and given by the compact interval $\sigma
(L)=\sigma (L^{(0)})=\{ \hat{E}(\xi) \mid
\xi\in\overline{\mathbf{A}}\}$.) We will now describe the scattering of the interacting dynamics generated by the discrete pseudo Laplacian $L$ with respect to the free dynamics generated by the discrete Laplacian $L^{(0)}$. Let us to this end define the regular sector of the Weyl alcove as $$\mathbf{A}_{\text{reg}}=\{ \xi\in \mathbf{A}\mid \langle \nabla
\hat{E}, \alpha \rangle \neq 0,\; \forall \alpha\in
\boldsymbol{R}^+ \} .$$ Due to the analyticity of $\hat{E}(\xi)$, the regular sector $\mathbf{A}_{\text{reg}}$ is an open dense subset of the Weyl alcove $\mathbf{A}$. For every $\xi \in\mathbf{A}_{\text{reg}}$, there exists now a unique Weyl group element $\hat{w}_\xi\in W$ such that $\hat{w}_\xi (\nabla \hat{E})$ lies in the dominant Weyl chamber $ \mathbf{C}^+$ . Clearly, the Weyl-group valued function $\xi\rightarrow \hat{w}_\xi$ is constant on the connected components of $\mathbf{A}_{\text{reg}}$ (by continuity). We are now in the position to define the unitary multiplication operator $\hat{\mathcal{S}}_L:\hat{\mathcal{H}}\mapsto\hat{\mathcal{H}}$ (the so-called [*scattering matrix*]{}) via its restriction to the dense subspace of (say) smooth complex test functions with compact support in $\mathbf{A}_{\text{reg}}$: $$\label{Sm}
(\hat{\mathcal{S}}_L\hat{\phi} ) (\xi)=
\hat{S}_{\hat{w}_\xi}(\xi)\hat{\phi} (\xi) \qquad ( \hat{\phi}\in
C_0^\infty (\mathbf{A}_{\text{reg}}) ),$$ where $\hat{S}_{w}(\xi)$ is given by Eq. .
The main result of this paper is the following explicit formula for the wave operators and the scattering operator in terms of the scattering matrix $\hat{\mathcal{S}}_L$ and the Fourier transforms $\mathcal{F}$ , and $\mathcal{F}^{(0)}$ , , thus relating the long-time asymptotics of [*interacting dynamics*]{} $e^{itL}$ to that of the [*free dynamics*]{} $e^{itL^{(0)}}$. The proof, which is relegated to Section \[sec5\] below, consists of a stationary phase analysis based on the asymptotic formula for the wave functions in Theorem \[plane:thm\].
\[wave:thm\] The operator limits $$\Omega_\pm = s-\lim_{t\to\pm\infty} e^{itL}e^{-itL^{(0)}}$$ converge in the strong $\|\cdot \|_{\mathcal{H}}$-norm topology (where $\|\cdot \|_{\mathcal{H}}=(\cdot,\cdot)_{\mathcal{H}}^{1/2}$), and the corresponding wave operators $\Omega_\pm:\mathcal{H}\mapsto\mathcal{H}$ are given by the unitary operators $$\begin{aligned}
\Omega_+ &=& \mathcal{F}^{-1}\circ\hat{\mathcal{S}}_L^{-1/2}\circ
\mathcal{F}^{(0)} , \\
\Omega_- &=& \mathcal{F}^{-1}\circ\hat{\mathcal{S}}_L^{1/2}\circ
\mathcal{F}^{(0)} .\end{aligned}$$
\[scattering:cor\] The scattering operator $\mathcal{S}_L:\mathcal{H}\mapsto\mathcal{H}$ for the self-adjoint discrete pseudo Laplacian $L\in \mathbb{R}[L_1,\ldots ,L_N]$ is given by the unitary operator $$\mathcal{S}_L
\equiv\Omega_+^{-1}\Omega_-=(\mathcal{F}^{(0)})^{-1}\circ\hat{\mathcal{S}}_L\circ
\mathcal{F}^{(0)} .$$
We see from Corollary \[scattering:cor\] and Eqs. , that the scattering matrix for the self-adjoint discrete pseudo Laplacian $L$ factorizes over the root system $\boldsymbol{R}$. For the type $A$ root systems, Theorem \[wave:thm\] and Corollary \[scattering:cor\] reproduce the results of Ruijsenaars in Ref. [@rui:factorized].
Stationary Phase Analysis {#sec5}
=========================
In this section the fundamental formulas for the wave operators stated in Theorem \[wave:thm\] are proven. To this end we employ a stationary phase method that generalizes Ruijsenaars’ approach in Ref. [@rui:factorized] from the type $A$ root systems to the case of arbitrary crystallographic root systems. Throughout this section the notational conventions of Sections \[sec3\] and \[sec4\] are adopted.
Asymptotics of Wave Packets {#sec5.1}
---------------------------
Let us introduce the [*free wave packet*]{} $\phi^{(0)}(t) $ and the [*interacting wave packets*]{} $\phi_{\pm }(t)$ of the form
$$\begin{aligned}
\phi^{(0)}(t) &=& (\mathcal{F}^{(0)})^{-1}\, e^{-i
t\hat{E}}\,\hat{\phi} , \\
\phi_{\pm }(t) &=& \mathcal{F}^{-1}\, e^{-i
t\hat{E}}\,\hat{\mathcal{S}}_L^{\mp 1/2}\, \hat{\phi},\end{aligned}$$
or more explicitly
$$\begin{aligned}
\phi^{(0)}_\lambda (t) &=&
\frac{1}{|W|\,\text{Vol} (\mathbf{A})} \sum_{w\in W} (-1)^w\,
\int_{\mathbf{A}} e^{i\langle \rho+\lambda ,
\xi_w\rangle-it\hat{E}(\xi) } \hat{\phi} (\xi) \text{d}\xi , \\
\phi_{\pm,\lambda} (t)&=& \frac{1}{|W|\text{Vol}(\mathbf{A})}
\int_{\mathbf{A}} \Psi_\lambda (\xi) e^{-it\hat{E}(\xi)}
\hat{\mathcal{S}}_L^{\mp 1/2} (\xi) \hat{\phi}(\xi) \text{d}\xi ,\end{aligned}$$
with $\hat{\phi}\in C_0^\infty (\mathbf{A}_{\text{reg}})$. The following lemma states that the long-time asymptotics of the interacting wave packets $\phi_{+ }(t)$ and $\phi_{- }(t)$ for $t\to +\infty$ and $t\to -\infty$, respectively, coincides with the corresponding asymptotics of the free wave packet $\phi^{(0)}(t)$.
\[asf:prp\] For $t\to\pm \infty$, the difference between the interacting wave packet $\phi_{\pm }(t)$ and the free wave packet $\phi^{(0)}(t)$ tends to zero: $$\forall \kappa >0:\qquad \| \phi_{\pm }(t)- \phi^{(0)}(t)
\|_{\mathcal{H}} =O(1/|t|^\kappa)\quad \text{as}\;t\to \pm\infty .$$
Before proving Proposition \[asf:prp\] (cf. below), let us first infer that Theorem \[wave:thm\] arises as an immediate consequence. Indeed, since the space of test functions $C_0^{\infty}(\mathbf{A}_{\text{reg}})$ is dense in $\hat{\mathcal{H}}$ and the operators in question are unitary, to validate Theorem \[wave:thm\] it is sufficient to demonstrate that for $\phi = (\mathcal{F}^{(0)})^{-1}\hat{\phi}$ with $\hat{\phi}\in C_0^{\infty}(\mathbf{A}_{\text{reg}})$ $$\lim_{t\to\pm\infty} \| e^{itL}e^{-itL^{(0)}}\phi-\Omega_{\pm}
\phi \|_{\mathcal{H}} =0,$$ where $\Omega_{\pm}\equiv\mathcal{F}^{-1}\circ
\hat{\mathcal{S}}_L^{\mp 1/2}\circ \mathcal{F}^{(0)}$. From the unitarity of $e^{itL}$ and the intertwining relations $$e^{-itL^{(0)}} \circ (\mathcal{F}^{(0)})^{-1} =
(\mathcal{F}^{(0)})^{-1}\circ e^{-it\hat{E}}\quad\text{and} \quad
e^{-itL} \circ\mathcal{F}^{-1} = \mathcal{F}^{-1}\circ
e^{-it\hat{E}},$$ it is clear that $$\begin{aligned}
\| e^{itL}e^{-itL^{(0)}}\phi-\Omega_{\pm} \phi \|_{\mathcal{H}}
&=& \| e^{-itL^{(0)}}(\mathcal{F}^{(0)})^{-1}\hat{\phi}-
e^{-itL}\Omega_{\pm} (\mathcal{F}^{(0)})^{-1}\hat{\phi} \|_{\mathcal{H}}\\
&=& \| \phi^{(0)}(t)- \phi_{\pm }(t) \|_{\mathcal{H}},\end{aligned}$$ whence the theorem follows from Proposition \[asf:prp\].
The Classical Wave Packet {#sec5.2}
-------------------------
To prove Proposition \[asf:prp\], we may assume without restricting generality that the test function $\hat{\phi}$ has in fact compact support inside a connected component of $\mathbf{A}_{\text{reg}}$. In this situation there thus exists a unique Weyl group element $\hat{w}\in W$ such that $\hat{w}(\nabla\hat{E})$ lies inside the dominant Weyl chamber $\mathbf{C}^+$ for all $\xi$ in the support of $\hat{\phi}$. Let $\mathbf{V}_{\text{clas}}\subset\mathbf{E}$ be an open bounded neighborhood of the compact range of classical wave-packet velocities $\text{Ran}_{\hat{\phi}} (\nabla \hat{E}) \equiv\{
\nabla\hat{E}(\xi) \mid \xi \in \text{Supp}(\hat{\phi})\}$ staying away from the walls of the Weyl chamber $\hat{w}^{-1}(\mathbf{C}^+)$ in the sense that there exists a lower-bound $\varepsilon>0$ such that $\langle \zeta_{\hat{w}}
,\alpha^\vee \rangle
>\varepsilon$ for all $\zeta\in\mathbf{V}_{\text{clas}}$ and all $\alpha\in \boldsymbol{R}^+$. We will now introduce a [*classical wave packet*]{} that is finitely supported on the following $t$-dependent region of the cone of dominant weights $\mathcal{P}^+$ $$\mathcal{P}^{+}_{\text{clas}}(t)=
\begin{cases}
\{ \lambda\in \mathcal{P}^+ \mid \rho+\lambda\in
t\hat{w}(\mathbf{V}_{\text{clas}}) \} &\text{for}\; t> 0 ,\\
\{ \lambda\in \mathcal{P}^+ \mid \rho+\lambda\in
tw_0\hat{w}(\mathbf{V}_{\text{clas}}) \} &\text{for}\; t< 0 .
\end{cases}$$ Because of dimensional considerations, it is clear that the cardinality of the support $\mathcal{P}^{+}_{\text{clas}}(t)$ grows at most polynomially in $t$ $$|\mathcal{P}^{+}_{\text{clas}}(t)|=O(t^N)\qquad\text{for}\;\;
|t|\rightarrow\infty .$$ The classical wave packet is defined as $$\begin{aligned}
\label{clas:wp}
\lefteqn{\phi^{(\text{clas})}_{\lambda} (t) =} && \\
&& \begin{cases} {\displaystyle
\frac{(-1)^{\hat{w}}}{|W|\text{Vol}(\mathbf{A})} \int_{\mathbf{A}}
e^{i\langle \rho+\lambda
,\xi_{\hat{w}}\rangle-it\hat{E}(\xi)}\hat{\phi}(\xi) \text{d}\xi }
&\text{for}\;\lambda\in
\mathcal{P}^{+}_{\text{clas}}(t)\;\text{and}\; t>0 ,
\\[3ex]
{\displaystyle \frac{(-1)^{w_0\hat{w}}}{|W|\text{Vol}(\mathbf{A})}
\int_{\mathbf{A}} e^{i\langle \rho+\lambda
,\xi_{w_0\hat{w}}\rangle-it\hat{E}(\xi)}\hat{\phi}(\xi)
\text{d}\xi } &\text{for}\;\lambda\in
\mathcal{P}^{+}_{\text{clas}}(t)\;\text{and}\; t<0 ,\\[0.5ex]
\makebox[6em]{} 0 & \text{otherwise}.
\end{cases} \nonumber\end{aligned}$$
The next lemma compares the long-time asymptotics of the free wave packet $\phi^{(0)}(t)$ with that of the classical wave packet $\phi^{(\text{clas})}(t)$.
\[clas:lem\] For $t\to\pm \infty$, the difference between the free wave packet $\phi^{(0 )}(t)$ and the classical wave packet $\phi^{(\text{clas})}(t)$ tends to zero: $$\forall\kappa >0:\qquad \|
\phi^{(0)}(t)-\phi^{(\text{clas})}(t)\|_{\mathcal{H}} =O(1/|t|^\kappa)
\quad\text{as}\;t\to\pm\infty .$$
It is immediate from the definitions of the wave packets under consideration that
$$\label{qwavea}
\phi^{(0)}_\lambda(t)-\phi^{(\text{clas})}_\lambda(t) =
\frac{1}{|W|\,\text{Vol} (\mathbf{A})} \sum_{w\in \hat{W}}
(-1)^w\, \int_{\mathbf{A}} e^{i\langle \rho+\lambda ,
\xi_w\rangle-it\hat{E}(\xi) } \hat{\phi} (\xi) \text{d}\xi ,$$
with $$\label{qwaveb}
\hat{W}\equiv
\begin{cases}
W\setminus \{ \hat{w}\} & \text{if}\;
\lambda\in\mathcal{P}^+_{\text{clas}}(t)\;\text{and}\;t>0 ,\\
W\setminus \{ w_0\hat{w}\} & \text{if}\;
\lambda\in\mathcal{P}^+_{\text{clas}}(t)\;\text{and}\;t<0 ,\\
W & \text{otherwise}.
\end{cases}$$
The proof of the lemma now hinges on a stationary phase estimate extracted from the Corollary of Theorem XI.14 in Ref. [@ree-sim:methods p. 38-39], which states that for any $k>0$ there exists a (positive) constant $c_k$ such that
$$\label{spa}
\left| \int_{\mathbf{A}} e^{i\langle \mathbf{x},\xi\rangle -it
\hat{E}(\xi)} \hat{\phi}(\xi) \text{d}\xi \right|\leq
\frac{c_k}{(1+|\mathbf{x}|+|t|)^k}$$
for all $\mathbf{x}\in\mathbf{E}$ and $t\in\mathbb{R}$ such that $$\label{spb}
\mathbf{x}\not\in t \mathbf{V}_{\text{clas}}.$$
Indeed, invoking of the stationary phase estimate in Eqs. , with $k> N/2$ and $\mathbf{x}=w^{-1}
(\rho+\lambda)$, reveals that the norm of the difference between the wave packets given by Eqs. , in the Hilbert space $\mathcal{H}$ is $O(1/|t|^{k-N/2})$ as $t\to\pm\infty$. (Notice in this connection that $w^{-1} (\rho+\lambda)\in
t\mathbf{V}_{\text{clas}}$ if and only if $\lambda\in\mathcal{P}^+_{\text{clas}}(t)$ and $w\in
W\setminus\hat{W}$.)
The Asymptotic Wave Packet {#sec5.3}
--------------------------
Let us define [*asymptotic wave packets*]{} $\phi^{(\infty)}_{\pm}(t)$ that are obtained from the interacting wave packets $\phi_{\pm}(t)$ upon replacing the Fourier kernel $\Psi_\lambda (\xi)$ by its plane wave asymptotics $\Psi_\lambda^{(\infty )} (\xi)$
$$\label{as:wp}
\phi^{(\infty)}_{\pm,\lambda }(t) =
\frac{1}{|W|\text{Vol}(\mathbf{A})} \int_{\mathbf{A}}
\Psi_\lambda^{(\infty )} (\xi) e^{-it\hat{E}(\xi)}
\hat{\mathcal{S}}_L^{\mp 1/2} (\xi) \hat{\phi}(\xi) \text{d}\xi ,$$
or more explicitly $$\begin{aligned}
\phi^{(\infty)}_{+,\lambda }(t)\!\! &=&
\!\!\frac{1}{|W|\,\text{Vol} (\mathbf{A})} \sum_{w\in W} (-1)^w\,
\int_{\mathbf{A}} e^{i\langle \rho+\lambda ,
\xi_w\rangle-it\hat{E}(\xi) }
\frac{\hat{\mathcal{C}}(\xi_w)}{\hat{\mathcal{C}}(\xi_{\hat{w}})}
\hat{\phi} (\xi)\text{d}\xi , \\
\phi^{(\infty)}_{-,\lambda }(t)\!\! &=&\!\!
\frac{1}{|W|\,\text{Vol} (\mathbf{A})} \sum_{w\in W} (-1)^w\,
\int_{\mathbf{A}} e^{i\langle \rho+\lambda ,
\xi_w\rangle-it\hat{E}(\xi) }
\frac{\hat{\mathcal{C}}(\xi_w)}{\hat{\mathcal{C}}(\xi_{w_0\hat{w}})}
\hat{\phi} (\xi)\text{d}\xi .\end{aligned}$$
The next lemma states that the long-time behavior of the asymptotic wave packets is governed by the classical wave packet $\phi^{(\text{clas})}(t)$ .
\[as1:lem\] For $t\to\pm \infty$, the difference between the asymptotic wave packet $\phi^{(\infty )}_\pm (t)$ and the classical wave packet $\phi^{(\text{clas})}(t)$ tends to zero: $$\forall \kappa >0:\qquad \| \phi^{(\infty)}_\pm
(t)-\phi^{(\text{clas})}(t)\|_{\mathcal{H}} =O(1/|t|^\kappa)
\quad\text{as}\;t\to\pm\infty .$$
The proof of Lemma \[clas:lem\] applies verbatim, upon replacing $\phi^{(0)}(t)$ by $\phi^{(\infty )}_\pm (t)$ and the introduction of minor modifications in the formulas so as to incorporate the additional (smooth) factors $\hat{\mathcal{C}}(\xi_w)/\hat{\mathcal{C}}(\xi_{\hat{w}})$ and $\hat{\mathcal{C}}(\xi_w)/\hat{\mathcal{C}}(\xi_{w_0\hat{w}})$, respectively.
Let $P^{(\text{clas})}_t:\mathcal{H}\mapsto\mathcal{H}$ denote the orthogonal projection onto the finite-dimensional subspace $l^2(\mathcal{P}^+_{\text{clas}}(t))\subset l^2(\mathcal{P}^+)$: $$(P^{(\text{clas})}_t\phi)_\lambda =
\begin{cases}
\phi_\lambda & \text{if}\; \lambda\in\mathcal{P}^+_{\text{clas}}(t) , \\
0 & \text{if}\;
\lambda\in\mathcal{P}^+\setminus\mathcal{P}^+_{\text{clas}}(t).
\end{cases}$$ It is clear from the definition of the classical wave packet that $P^{(\text{clas})}_t(\phi^{(\text{clas})}(t))=\phi^{(\text{clas})}(t)$. As a consequence, we get from from Lemma \[as1:lem\] upon projection onto $l^2(\mathcal{P}^+_{\text{clas}}(t))$ that $$\label{as1:eq}
\forall \kappa >0:\qquad \| P^{(\text{clas})}_t\phi^{(\infty)}_\pm
(t)-\phi^{(\text{clas})}(t)\|_{\mathcal{H}} = O(1/|t|^\kappa)\quad
\text{as}\; t\to\pm\infty.$$
Our final lemma states that the long-time asymptotics of the interacting wave packet $\phi_{\pm}(t)$ coincides with that of the asymptotic wave packet $\phi^{(\infty)}_\pm (t)$.
\[as2:lem\] For $t\to\pm \infty$, the difference between the interacting wave packet $\phi_{\pm }(t)$ and the asymptotic wave packet $\phi^{(\infty )}_\pm (t)$ tends to zero: $$\forall \kappa >0:\qquad \| \phi_{\pm}(t)-\phi^{(\infty )}_\pm
(t)\|_{\mathcal{H}} = O(1/|t|^\kappa)\quad\text{as}\ t\to\pm\infty
.$$
From the definitions it is immediate that $$\phi_{\pm,\lambda}(t)-\phi^{(\infty )}_{\pm,\lambda}(t) =
(e^{-it\hat{E}}\hat{\mathcal{S}}_L^{\mp 1/2} \hat{\phi},
\overline{\Psi}_\lambda-
\overline{\Psi}_\lambda^{(\infty)})_{\hat{\mathcal{H}}}.$$ Hence $$\begin{aligned}
\| P_t^{(\text{clas})}\left( \phi_{\pm}(t)-\phi^{(\infty )}_\pm
(t)\right) \|_{\mathcal{H}}^2&=&
\sum_{\lambda\in\mathcal{P}^+_{\text{clas}}(t)} |
(e^{-it\hat{E}}\hat{\mathcal{S}}_L^{\mp 1/2} \hat{\phi},
\overline{\Psi}_\lambda-\overline{\Psi}_\lambda^{(\infty)})_{\hat{\mathcal{H}}}|^2
\\
&\leq & \| \hat{\phi}\|_{\hat{\mathcal{H}}}^2
\sum_{\lambda\in\mathcal{P}^+_{\text{clas}}(t)}
\|\Psi_\lambda-\Psi_\lambda^{(\infty)}\|_{\hat{\mathcal{H}}}^2\end{aligned}$$ (by the Cauchy-Schwarz inequality). Now, since $|
\mathcal{P}^+_{\text{clas}}(t)|=O(t^N)$ and $m(\lambda)
>|t|\varepsilon -1$ for $\lambda\in \mathcal{P}^+_{\text{clas}}(t)$, we conclude from this estimate combined with Theorem \[plane:thm\] that $\| P^{(\text{clas})}_t\left(
\phi_{\pm}(t)-\phi^{(\infty )}_\pm (t)\right) \|_{\mathcal{H}}$ converges to zero exponentially fast for $t\to\pm\infty$, so in particular $$\label{as2:eq}
\forall\kappa >0:\qquad \| P^{(\text{clas})}_t\left(
\phi_{\pm}(t)-\phi^{(\infty )}_\pm (t)\right) \|_{\mathcal{H}} =
O(1/|t|^\kappa)\quad\text{as}\; t\to\pm\infty.$$ The lemma now follows from the vanishing of the tails $(\text{Id}-P^{(\text{clas})}_t)\phi^{(\infty )}_\pm (t)$ and $(\text{Id}-P^{(\text{clas})}_t) \phi_{\pm }(t)$ for $t\to\pm\infty$:
$$\begin{aligned}
&& \|
(\text{Id}-P^{(\text{clas})}_t)\phi^{(\infty
)}_\pm (t) \|_{\mathcal{H}} =O(1/|t|^\kappa)\quad\text{as}\;t\to\pm\infty
\label{tail1} ,\\
&& \| (\text{Id}-P^{(\text{clas})}_t)
\phi_{\pm }(t) \|_{\mathcal{H}} =O(1/|t|^\kappa)\quad\text{as}\; t\to\pm\infty .
\label{tail2}\end{aligned}$$
Notice in this connection that the limiting relation in Eq. is immediate from Lemma \[as1:lem\] and Eq. , and that the limiting relation in Eq. follows by compairing the norm equality $$\| \phi_{\pm}(t)\|_{\mathcal{H}}=\| \hat{\phi}
\|_{\hat{\mathcal{H}}}$$ with the norm estimate for $t\to\pm\infty$ $$\begin{aligned}
\| P^{(\text{clas})}_t\phi_{\pm }(t)
\|_{\mathcal{H}} &\stackrel{\text{Eq.}~\eqref{as2:eq}}{=}&
\|
P^{(\text{clas})}_t\phi^{(\infty )}_\pm (t) \|_{\mathcal{H}}
+ O(1/|t|^\kappa) \\
&\stackrel{\text{Eq.}~\eqref{as1:eq}}{=}& \| \phi^{(clas )}_t \|_{\mathcal{H}}
+ O(1/|t|^\kappa) \\
&\stackrel{\text{Lemma}~\ref{clas:lem}}{=}&
\|
\phi^{(0 )}(t) \|_{\mathcal{H}}+ O(1/|t|^\kappa) \\
&=& \| \hat{\phi} \|_{\hat{\mathcal{H}}}+ O(1/|t|^\kappa) ,\end{aligned}$$ which entails that $$\| (\text{Id}-P^{(\text{clas})}_t)
\phi_{\pm }(t) \|_{\mathcal{H}}=
\sqrt{\| \phi_{\pm}(t)\|_{\mathcal{H}}^2-\| P^{(\text{clas})}_t\phi_{\pm }(t)
\|_{\mathcal{H}}^2}=O(1/|t|^{\kappa /2}).$$
Proof of Proposition \[asf:prp\] {#sec5.4}
--------------------------------
After these preparations, the proof of Proposition \[asf:prp\] reduces to the telescope $$\begin{aligned}
\lefteqn{\| \phi_{\pm}(t)-\phi^{(0)}(t)\|_{\mathcal{H}}} &&
\\&& \leq \| \phi_{\pm}(t)-\phi^{(\infty )}_\pm (t)\|_{\mathcal{H}} + \|
\phi^{(\infty )}_\pm (t)-\phi^{(clas )}(t)\|_{\mathcal{H}} + \|
\phi^{(clas)}(t)-\phi^{(0 )}(t)\|_{\mathcal{H}}, \nonumber\end{aligned}$$ and the application of Lemmas \[clas:lem\], \[as1:lem\] and \[as2:lem\].
Application: Scattering of Hyperbolic Lattice Calogero-Moser Models {#sec6}
===================================================================
In this section we specialize our $c$-functions so as to describe the scattering of the hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser models associated with the Macdonald polynomials. Initially, viz. in the first three subsections, it will be assumed that our root system $\boldsymbol{R}$ be [*reduced*]{} (so $\boldsymbol{R}_0=\boldsymbol{R}_1=\boldsymbol{R}$) except when explicitly stated otherwise. In the fourth subsection, we then indicate how the results extend to the case of a [*nonreduced*]{} root system (so $\boldsymbol{R}=BC_N$, $\boldsymbol{R}_0=C_N$ and $\boldsymbol{R}_1=B_N$). We end our study of the lattice Calogero-Moser models in the fifth subsection by providing some illuminating additional details describing what the results boil down to in the simplest situation of a root system of rank [*one*]{}.
Macdonald Wave Function {#sub51}
-----------------------
\[sec6.1\] For $c$-functions of the form $$\label{m-c}
\hat{c}_{|\alpha|}(z) =
\frac{(q^{g_{|\alpha|}}z;q)_\infty}{(qz;q)_\infty},\qquad
q=e^{-s}, \quad g_{|\alpha|},s >0,$$ with $(z;q)_\infty\equiv\prod_{n=0}^\infty (1-zq^n)$, the weight function $\hat{\Delta} (\xi)$ – becomes $$\label{m-weight}
\hat{\Delta} (\xi)= \prod_{\alpha\in \boldsymbol{R}}
\frac{(qe^{i\langle \alpha ,\xi\rangle};q)_\infty}
{(q^{g_{|\alpha|}} e^{i\langle \alpha
,\xi\rangle};q)_\infty}.$$ (The positivity restrictions on the parameters $g_{|\alpha|}$ and $s$ guarantee that the $c$-function meets the technical requirements stated in Section \[sec2\].) Our polynomials $P_\lambda (\xi)$, $\lambda\in\mathcal{P}^+$ now amount to the orthonormalized Macdonald polynomials [@mac:symmetric; @mac:orthogonal; @mac:affine]
$$\label{omp}
P_\lambda (\xi) =\frac{1}{\mathcal{N}_0^{1/2}}
\Delta^{1/2}(\lambda) \mathbf{P}_\lambda (\xi) ,$$
where $$\label{hm}
\Delta (\lambda)= \frac{\mathcal{C}^+
(\rho_g)\mathcal{C}^- (\rho_g)}{\mathcal{C}^+
(\rho_g+\lambda)\mathcal{C}^- (\rho_g+\lambda)},\qquad
\mathcal{N}_0 =\frac{\mathcal{C}^- (\rho_g)}{\mathcal{C}^+
(\rho_g)},$$ with $\rho_{g} \equiv \frac{1}{2}\sum_{\alpha\in \boldsymbol{R}^+}
g_{|\alpha |}\,\alpha$ and $$\begin{aligned}
\label{cm-f} \mathcal{C}^\pm
(\mathbf{x})&=&\prod_{\alpha\in \boldsymbol{R}^+} c^\pm_{|\alpha
|}(\langle\mathbf{x},\alpha^\vee \rangle ) , \\
c^+_{|\alpha |}(x) &=& q^{g_{|\alpha|} x /2}
\frac{(q^{g_{|\alpha|}+x};q)_\infty}
{(q^{x};q)_\infty} , \label{cm-f1} \\
c^-_{|\alpha |}(x) &=& q^{g_{|\alpha|} x/2} \frac{(
q^{1+x};q)_\infty}
{(q^{1-g_{|\alpha|}+x};q)_\infty}. \label{cm-f2}\end{aligned}$$
Here $\mathbf{P}_\lambda (\xi )$ denotes the Macdonald polynomial
$$\label{mp}
\mathbf{P}_\lambda (\xi )= c_\lambda p_\lambda (\xi),\qquad
c_\lambda = \frac{\mathcal{C}^+(\rho_g+\lambda
)}{\mathcal{C}^+(\rho_g )} ,$$
where $$\label{mp1}
p_\lambda (\xi) = m_\lambda (\xi) + \sum_{\mu\in\mathcal{P}^+,\,
\mu \prec \lambda} c_{\lambda\mu} m_\mu (\xi),\qquad$$ with coefficients $c_{\lambda\mu}\in\mathbb{C}$ such that $$\label{mp2}
( p_\lambda , m_\mu )_{\hat{\Delta}} = 0 \quad \text{for}\; \mu
\prec\lambda .$$
For the Macdonald weight function $\hat{\Delta}(\xi)$ , the coefficients $c_{\lambda\mu}$ turn out to vanish when $\lambda$ and $\mu$ are not comparable in the dominance ordering [@mac:orthogonal]. In other words, in this case one may take the ordering $\succeq$ to be equal to the dominance order $\geqslant$ without restricting generality. Explicit formulas for the expansion coefficients $c_{\lambda\mu}$ when $\mathcal{P}\neq\mathcal{Q}$ (i.e. excluding the root systems $E_8$, $F_4$ and $G_2$) can be found in Ref. [@die-lap-mor:determinantal].
We thus arrive at the following formula for the wave function $\Psi_\lambda (\xi) $ in terms of Macdonald polynomials.
\[mwf:prp\] For $\boldsymbol{R}$ reduced and $c$-functions given by $\hat{c}_{|\alpha|}(z)$ , the wave function $\Psi_\lambda (\xi) $ reads explicitly $$\Psi_\lambda (\xi) = \frac{1}{\mathcal{N}_0^{1/2}}\Delta^{1/2}
(\lambda)\hat{\Delta}^{1/2}(\xi) \delta (\xi) \mathbf{P}_\lambda
(\xi),$$ with $\mathbf{P}_\lambda (\xi)$ denoting the Macdonald polynomial characterized by Eqs. –.
Macdonald-Ruijsenaars Laplacian {#sec6.2}
-------------------------------
Let us recall that a nonzero weight $\pi\in\mathcal{P}^+$ is called [*minuscule*]{} if $\langle \pi,\alpha^\vee\rangle \leq 1$ for all $\alpha\in \boldsymbol{R}^+$ and that it is called [*quasi-minuscule*]{} if $\langle \pi,\alpha^\vee\rangle \leq 1$ for all $\alpha\in \boldsymbol{R}^+\setminus \{ \pi\}$ (and it is not minuscule). The number of minuscule weights is equal to the index of $\mathcal{Q}$ in $\mathcal{P}$ minus $1$ (so for $E_8$, $F_4$ and $G_2$ there are none). As regards to the quasi-minuscule weights: there is always just [*one*]{} and it is given by $\pi=\alpha_0$, where $\alpha_0^\vee$ is the maximal root of the dual root system $\boldsymbol{R}^\vee\equiv \{\alpha^\vee\mid\alpha\in\boldsymbol{R}\}$. For the readers convenience, we have included a list of the (quasi-)minuscule weights for each root system in Table \[qmweight\] (where we have adopted the standard numbering of the fundamental weights in accordance with Refs. [@bou:groupes; @hum:introduction]).
$$\begin{array}{lll}
\boldsymbol{R} & \text{minuscule} & \text{quasi-minuscule} \\[1ex]
A_N :& \omega_1,\ldots ,\omega_{N} & \omega_1+\omega_{N},\\
B_N :& \omega_N & \omega_1 ,\\
C_N :& \omega_1 & \omega_2 ,\\
D_N :& \omega_1,\omega_{N-1},\omega_N & \omega_2, \\ [1ex]
E_6 :& \omega_1,\omega_6 & \omega_2,\\
E_7 :& \omega_7 & \omega_1 ,\\
E_8: & & \omega_8,\\
F_4: & & \omega_4,\\
G_2: & & \omega_1 ,\\[1ex]
BC_N: & & \omega_1 .\\
& &
\end{array}$$
To a (quasi-)minuscule weight $\pi$ we associate the multiplication operator $\hat{E}_\pi
:\hat{\mathcal{H}}\mapsto\hat{\mathcal{H}}$ given by $$\hat{E}_\pi (\xi) =\sum_{\nu\in W (\pi )\cup W(-\pi)} \exp
(i\langle \nu , \xi \rangle ).$$ The [*Macdonald-Ruijsenaars Laplacian*]{} is now defined as the pullback $L_\pi:\mathcal{H}\mapsto\mathcal{H}$ of $\hat{E}_\pi$ with respect to the Fourier transform $\mathcal{F}$ $$L_\pi= \mathcal{F}^{-1}\circ \hat{E}_\pi \circ\mathcal{F} .$$ By Proposition \[dl:prp\], the Macdonald-Ruijsenaars Laplacian $L_\pi\in \mathbb{R}[L_1,\ldots ,L_N]$ constitutes a difference operator in $\mathcal{H}$. The following proposition provides its explicit action on lattice functions over the dominant cone $\mathcal{P}^+$.
\[mrlap:prp\] For $\boldsymbol{R}$ reduced and $\pi$ (quasi-)minuscule, the action of the Macdonald-Ruijsenaars Laplacian $L_\pi$ on a (square-summable) lattice function $\phi:\mathcal{P}^+\rightarrow \mathbb{C}$ is given by $$\begin{aligned}
\lefteqn{L_\pi \phi_\lambda =
E_\pi (\rho_g^\vee)\, \phi_{\lambda} +} && \\
&& \sum_{\begin{subarray}{c} \nu\in W(\pi )\cup W(-\pi)\\
\lambda+\nu\in\mathcal{P}^+\end{subarray}} \Bigl( V_\nu^{1/2}
(\rho_g +\lambda ) V_{-\nu}^{1/2} (\rho_g +\lambda +\nu
) \phi_{\lambda+\nu} - V_\nu (\rho_g +\lambda )\phi_{\lambda} \Bigr) ,\\\end{aligned}$$ where $$\begin{aligned}
V_\nu(\mathbf{x})&=& \prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha^\vee\rangle >0 \end{subarray}}
\frac{(g_{|\alpha |}+\langle \mathbf{x},\alpha^\vee \rangle :
\sinh_s)_{\langle \nu ,\alpha^\vee\rangle}}{(\langle
\mathbf{x},\alpha^\vee \rangle:\sinh_s )_{\langle \nu
,\alpha^\vee\rangle}}
\\
&=&
\prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha^\vee\rangle =1 \end{subarray}}
\frac{\sinh\frac{s}{2}(g_{|\alpha |}+\langle
\mathbf{x},\alpha^\vee \rangle
)}{\sinh\frac{s}{2} (\langle \mathbf{x},\alpha^\vee \rangle)} \times\\
&&
\prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha^\vee\rangle =2 \end{subarray}}
\frac{\sinh\frac{s}{2}(g_{|\alpha |}+\langle
\mathbf{x},\alpha^\vee \rangle )}{\sinh\frac{s}{2} (\langle
\mathbf{x},\alpha^\vee \rangle)} \frac{\sinh\frac{s}{2}(1+
g_{|\alpha |}+\langle \mathbf{x},\alpha^\vee \rangle
)}{\sinh\frac{s}{2} (1+\langle \mathbf{x},\alpha^\vee \rangle)} ,
\\
E_\pi (\mathbf{x}) &=&\sum_{\nu\in W (\pi )\cup W(-\pi)} \exp (
s\langle \nu , \mathbf{x} \rangle ) ,\end{aligned}$$ with $(z:\sinh_s )_m\equiv \prod_{\ell =0}^{m-1}\sinh
\frac{s}{2}(z+\ell )$ and $\rho_g^\vee \equiv
\frac{1}{2}\sum_{\alpha\in \boldsymbol{R}^+} g_{|\alpha |}
\alpha^\vee$.
It is a straightforward consequence of the definitions that the $c$-functions satisfy the difference equations $$\frac{c^+_{|\alpha|}(x+1)}{c^+_{|\alpha|}(x)}= \frac{\sinh
(\frac{sx}{2})}{\sinh \frac{s}{2}(g_{|\alpha |} +x)}, \qquad
\frac{c^-_{|\alpha|}(x+1)}{c^-_{|\alpha|}(x)}= \frac{\sinh
\frac{s}{2}(1+x-g_{|\alpha|})}{\sinh \frac{s}{2}(1 +x)} .$$ With the aid of these difference equations it is not difficult to verify the fundamental functional relation $$\label{ffe}
\Delta (\mathbf{x}+\nu)V_{-\nu}(\rho_g+\mathbf{x} +\nu) =\Delta
(\mathbf{x}) V_\nu(\rho_g+\mathbf{x} ) ,\qquad \nu\in W(\pi).$$ From the recurrence relation for the Macdonald polynomials exhibited in Eq. of the Appendix, it is now readily inferred—upon invoking the functional relation specialized to $\mathbf{x}=\lambda$ with $\lambda$ and $\lambda+\nu$ dominant—that the Macdonald wave function $\Psi_\lambda (\xi )$ – satisfies the eigenvalue equation $$\begin{aligned}
\sum_{\begin{subarray}{c}\nu\in W(\pi )\\
\lambda+\nu\in\mathcal{P}^+\end{subarray}} \Bigl( V_\nu^{1/2}
(\rho_g +\lambda ) V_{-\nu}^{1/2} (\rho_g +\lambda +\nu )
\Psi_{\lambda+\nu} (\xi ) -V_\nu (\rho_g +\lambda )
\Psi_{\lambda}(\xi) \Bigr) && \\
=\sum_{\nu\in W(\pi)} \bigl( e^{i\langle \nu ,\xi\rangle}
-q^{\langle \nu \rho_g^\vee\rangle} \bigr) \Psi_{\lambda}(\xi) .
&&\end{aligned}$$ Combining with the corresponding eigenvalue equation in which $\pi$ is replaced by $-w_0(\pi)$ (where $w_0$ is the longest element of the Weyl group), leads us to the eigenvalue equation for $L_\pi$: $$\begin{aligned}
\sum_{\begin{subarray}{c} \nu\in W(\pi )\cup W(-\pi)\\ \lambda
+\nu\in\mathcal{P}^+
\end{subarray}} \bigl( V_\nu^{1/2} (\rho_g +\lambda ) V_{-\nu}^{1/2}
(\rho_g +\lambda +\nu ) \Psi_{\lambda+\nu} (\xi ) -V_\nu (\rho_g
+\lambda ) \Psi_{\lambda}(\xi) \Bigr) &&\\
=\bigl( \hat{E}_\pi (\xi) -E_\pi (\rho_g^\vee) \bigr) \Psi_{\lambda}(\xi) .
&&\end{aligned}$$ The proposition now follows from the completeness of the Macdonald wave functions $\Psi_\lambda (\xi )$, $\xi\in\mathbf{A}$ in the Hilbert space $\mathcal{H}$.
When $\pi$ is minuscule we have that
$$V_\nu(\mathbf{x}) = \prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha^\vee\rangle =1 \end{subarray}}
\frac{\sinh\frac{s}{2}(g_{|\alpha |}+\langle
\mathbf{x},\alpha^\vee \rangle )}{\sinh\frac{s}{2} (\langle
\mathbf{x},\alpha^\vee \rangle)} ,$$
and that $$\sum_{\nu\in W(\pi )\cup W(-\pi)} V_\nu (\mathbf{x} ) =E_\pi
(\rho_g^\vee)$$ (by the Macdonald identity in Eq. of the Appendix). As a consequence, the action of the Macdonald-Ruijsenaars Laplacian in Proposition \[mrlap:prp\] simplifies in this situation to $$\label{mrlap}
L_\pi \phi_\lambda = \sum_{\begin{subarray}{c}\nu\in W(\pi )\cup
W(-\pi)\\\lambda+\nu\in\mathcal{P}^+\end{subarray}} V_\nu^{1/2}
(\rho_g +\lambda ) V_{-\nu}^{1/2} (\rho_g +\lambda +\nu )
\phi_{\lambda+\nu} .$$
For the root system $A_N$, all fundamental weights are minuscule (cf. Table \[qmweight\]). Hence, in this special case the discrete Laplacians $L_1,\ldots ,L_N$ of Section \[sec3\] are given by the Macdonald-Ruijsenaars Laplacians $L_{\pi}$ with $\pi=\omega_j$, $j=1,\ldots ,N$. The operators in question correspond to the commuting quantum integrals of the hyperbolic relativistic lattice Calogero-Moser model due to Ruijsenaars [@rui:finite-dimensional; @rui:systems]. For the other root systems, only a small part of the discrete Laplacians $L_1,\ldots ,L_N$ can be made explicit by means of the Macdonald-Ruijsenaars Laplacian of Proposition \[mrlap:prp\] and Table \[qmweight\]. In principle, the higher-order commuting Laplacians may be constructed with the aid of the corresponding Dunkl-Cherednik difference-reflection operators [@che:double; @che:macdonalds], however, at present explicit formulas for a set of generators for the algebra of commuting Laplacians $\mathbb{R}[L_1,\ldots ,L_N]$ are available only in the case of the [*classical*]{} root systems [@die:self-dual; @sah:nonsymmetric].
It is of course a consequence of our construction that the algebra of Laplacians $\mathbb{R}[L_1,\ldots ,L_N]$ consists of bounded self-adjoint operators in the Hilbert space $\mathcal{H}$. For the Macdonald-Ruijsenaars Laplacian $L_\pi$, one can also check this fact independently directly from the explicit action in Propostion \[mrlap:prp\].
If $\pi$ is quasi-minuscule then $-\pi\in W(\pi)$ (as $\pi$ is a root). Thus, in this case $W(-\pi)= W(\pi)$. The same simplification also occurs for $\pi$ not necessarily quasi-minuscule when $-\mathbf{1}\in W$ (i.e. when the longest Weyl-group element $w_0$ equals $-\mathbf{1}$). This [*is*]{} the case for the root systems $B_N$, $C_N$, $D_N$ ($N\geq 4$, even), $E_7$, $E_8$, $F_4$, $G_2$ and $BC_N$, but it is [*not*]{} the case for the root systems $A_N$ ($N\geq2$), $D_N$ ($N\geq 3$, odd) and $E_6$.
Scattering Matrix {#sub53}
-----------------
\[sec6.3\] When $g_{|\alpha|}\to 1$, $\forall \alpha\in \boldsymbol{R}$, the Macdonald $c$-functions $\hat{c}_{|\alpha |}(z)$ specialize to the unit $c$-function. The Macdonald-Ruijsenaars Laplacian $L_\pi$ of Proposition \[mrlap:prp\] reduces in this limit to the free Laplacian $L_\pi^{(0)}=
(\mathcal{F}^{(0)})^{-1}\circ \hat{E}_\pi \circ\mathcal{F}^{(0)}$ given by $$\label{fl}
L_\pi^{(0)} \phi_\lambda = \sum_{\nu\in W(\pi)\cup W(-\pi)}
\phi_{\lambda +\nu}$$ with boundary conditions as stipulated in Proposition \[fl:prp\]. The following proposition provides a somewhat more explicit characterization of these boundary conditions (in the case of $\pi$ (quasi-)minuscule).
\[fl-qm:prp\] For $\pi$ (quasi-)minuscule, the action of the free Laplacian $L_\pi^{(0)}$ is of the form $$L_\pi^{(0)} \phi_\lambda = - n_\pi(\lambda) \phi_\lambda+
\sum_{\begin{subarray}{c} \nu\in W(\pi)\cup W(-\pi) \\
\lambda +\nu \in\mathcal{P}^+\end{subarray}} \phi_{\lambda +\nu} ,$$ with $n_\pi(\lambda)=0$ if $\pi$ is minuscule, and with $n_\pi(\lambda)$ denoting the number of short simple roots $\alpha_j$ perpendicular to $\lambda$ if $\pi$ is quasi-minuscule (where, by convention, all roots qualify as short if $\boldsymbol{R}$ is simply laced).
Starting point is the action of $L_\pi^{(0)}$ in Eq. with boundary conditions as detailed in Proposition \[fl:prp\]. If $\lambda +\nu \not\in\mathcal{P}^+$, then there exists a simple root $\alpha_j$ such that $\langle \lambda
+\nu,\alpha_j^\vee\rangle <0$. Hence, since $\lambda\in\mathcal{P}^+$ and $\nu\in W(\pi)\cup W(-\pi)$ with $\pi$ (quasi-)minuscule, it follows that we are in either one of the following three situations:
- $\langle \lambda ,\alpha_j^\vee\rangle =0$ and $\langle
\nu ,\alpha_j^\vee\rangle=-1$,
- $\langle \lambda ,\alpha_j^\vee\rangle =0$ and $\langle
\nu ,\alpha_j^\vee\rangle=-2$,
- $\langle \lambda ,\alpha_j^\vee\rangle =1$ and $\langle
\nu ,\alpha_j^\vee\rangle=-2$.
It is not difficult to see that in the first and last situation the weight $\rho+\lambda+\nu$ is stabilized by the simple reflection $r_{\alpha_j}$. Indeed, we get $$r_{\alpha_j}(\rho+\lambda+\nu)=\rho+\lambda+\nu-\langle\rho+\lambda+\nu,\alpha_j^\vee\rangle\,
\alpha_j=\rho+\lambda+\nu$$ (where we exploited the fact that $\langle \rho
,\alpha^\vee\rangle =1$ for $\alpha$ simple). It thus follows that in these two cases the stabilizer of $\rho+\lambda+\nu$ is nontrivial, whence the corresponding term $\phi_{\lambda +\nu}$ in Eq. vanishes by the boundary condition in Proposition \[fl:prp\]. The second situation occurs only when $\pi$ is quasi-minuscule. Clearly we must then have that $\nu = -\alpha_j$, whence $\langle\rho+\lambda+\nu,\alpha_j^\vee\rangle=-1$ and $\langle\rho+\lambda+\nu,\alpha_k^\vee\rangle=1+\langle\lambda
,\alpha_k^\vee\rangle-\langle \alpha_j,\alpha_k^\vee\rangle >0$ for $k\neq j$ (where we exploited the fact that $\langle \alpha
,\beta^\vee\rangle \leq 0$ for $\alpha,\beta$ simple and distinct). It thus follows that the weight $\rho+\lambda+\nu$ is regular and that the Weyl permutation $w_{\rho+\lambda+\nu}$ taking it to the dominant cone is given by the simple reflection $r_{\alpha_j}$. Indeed, we now get $$r_{\alpha_j}(\rho+\lambda+\nu)=
\rho+\lambda+\nu-\langle\rho+\lambda+\nu,\alpha_j^\vee\rangle\,
\alpha_j=\rho+\lambda .$$ Invoking of the boundary condition in Proposition \[fl:prp\] then reveals that the corresponding term $\phi_{\lambda +\nu}$ in Eq. is equal to $-\phi_\lambda$. Now, every simple root $\alpha_j$ in the Weyl orbit of $\pi$ for which $\langle
\lambda,\alpha_j^\vee\rangle =0$ gives rise to such a contribution $\phi_{\lambda-\alpha_j}=-\phi_\lambda$ in the action on the r.h.s. of Eq. . Furthermore, since a quasi-minuscule weight $\pi$ is a short root of $\boldsymbol{R}$ (as $\pi=\alpha_0$ with $\alpha_0^\vee$ denoting the maximal root of $\boldsymbol{R}^\vee$, whence $\alpha_0^\vee$ is long and $\alpha_0$ is short), it is clear that the nonzero contributions in question occur precisely at all simple short roots perpendicular to $\lambda$.
Our main application of the scattering formalism in Section \[sec4\] is the following explicit formula for the scattering and wave operators for the lattice Calogero-Moser system, relating the long-time asymptotics of the dynamics of the Macdonald-Ruijsenaars Laplacian $L_\pi$ to that of the free Laplacian $L_{\pi}^{(0)}$.
\[lcm:thm\] The wave operators $\Omega_\pm=s-\lim_{t\to \pm\infty} e^{itL_\pi}
e^{-itL_\pi^{(0)}}$ and the scattering operator $\mathcal{S}_{L_\pi}=\Omega_+^{-1}\Omega_-$ for the Macdonald-Ruijsenaars Laplacian $L_\pi$ in relation to the free Laplacian $L_{\pi}^{(0)}$ are of the form stated in Theorem \[wave:thm\] and Corollary \[scattering:cor\], with a unitary scattering matrix $\hat{S}_{L_\pi}(\xi)$ given by Eqs. , and $$\hat{s}_{|\alpha|}(\langle \alpha , \xi\rangle)= \frac{(q
e^{i\langle \alpha ,\xi\rangle};q)_\infty }{(q^{g_{|\alpha|}}
e^{i\langle \alpha ,\xi\rangle};q)_\infty} \frac{(q^{g_{|\alpha|}}
e^{-i\langle \alpha ,\xi\rangle};q)_\infty }{(q e^{-i\langle
\alpha ,\xi\rangle};q)_\infty } .$$
For the type $A$ root systems Theorem \[lcm:thm\] is due to Ruijsenaars [@rui:factorized]. The scattering of the corresponding classical-mechanical system was analyzed previously in Ref. [@rui:action-angle].
The asymptotics of the Macdonald wave function $\Psi_\lambda
(\xi)$ in Proposition \[mwf:prp\] is governed by Theorem \[plane:thm\] (and Eqs. –) with a scattering matrix taken from Theorem \[lcm:thm\].
Extension to Nonreduced Root Systems {#sec6.4}
------------------------------------
We will now indicate how the results of Subsections \[sub51\]–\[sub53\] should be adapted so as to include the case of a [*nonreduced*]{} root system (viz. $\boldsymbol{R}=BC_N$, $\boldsymbol{R}_0= C_N$, $\boldsymbol{R}_1=B_N$ and $W$ amounts to the hyperoctahedral group $S_N\ltimes \mathbb{Z}_2^N$). In short, the bottom line is that all results carry over to the case of nonreduced root systems upon passing from the Macdonald polynomials to the Macdonald-Koornwinder multivariate Askey-Wilson polynomials [@koo:askey-wilson; @die:self-dual; @sah:nonsymmetric]. More specifically, by picking $c$-functions $\hat{c}_{|\alpha|}(z) $, $\alpha \in \boldsymbol{R}_1$ of the form $$\label{mk-c}
\hat{c}_{|\alpha|}(z) =
\begin{cases}
{\displaystyle \frac{(q^{\hat{g}}z;q)_\infty}{(qz;q)_\infty}}
&\text{for}\;\alpha\; \text{long},\\[2ex] {\displaystyle
\frac{(q^{\hat{g}_0}z,-q^{\hat{g}_1}z,q^{\hat{g}_2+1/2}z,-q^{\hat{g}_3+1/2}z;q)_\infty}{(qz^2;q)_\infty}}
&\text{for}\;\alpha\; \text{short},
\end{cases}$$ where $q=e^{-s}$ and $s, \hat{g},\hat{g}_{0},\ldots ,\hat{g}_3 >0$ (and with $(z_1,\ldots ,z_k;q)_\infty\equiv (z_1;q)_\infty\cdots
(z_k;q)_\infty$), we end up with a weight function $\hat{\Delta}
(\xi)$ – given by $$\begin{aligned}
\label{mk-weight}
\lefteqn{\hat{\Delta} (\xi) = \prod_{\begin{subarray}{c}\alpha\in
\boldsymbol{R}_1\\\alpha\;\text{long}\end{subarray}}
\frac{(qe^{i\langle \alpha ,\xi\rangle};q)_\infty}
{(q^{\hat{g}} e^{i\langle \alpha
,\xi\rangle};q)_\infty} } &&\\
&& \times \prod_{\begin{subarray}{c}\alpha\in
\boldsymbol{R}_1\\\alpha\;\text{short}\end{subarray}} \frac{(qe^{2
i\langle \alpha ,\xi\rangle};q)_\infty}
{(q^{\hat{g}_0}e^{i\langle \alpha
,\xi\rangle},-q^{\hat{g}_1}e^{i\langle \alpha
,\xi\rangle},q^{\hat{g}_2+1/2}e^{i\langle \alpha
,\xi\rangle},-q^{\hat{g}_3+1/2}e^{i\langle \alpha
,\xi\rangle};q)_\infty }. \nonumber\end{aligned}$$ The polynomials $P_\lambda (\xi)$, $\lambda\in\mathcal{P}^+$ amount in this case to orthonormalized Macdonald-Koornwinder polynomials [@koo:askey-wilson; @die:self-dual; @sah:nonsymmetric]. The polynomials in question are again of the form in Eqs. , and Eqs. –, but now with
$$\label{cmk-f}
\mathcal{C}^\pm (\mathbf{x})=\prod_{\alpha\in \boldsymbol{R}^+_1}
c^\pm_{|\alpha |}(\langle\mathbf{x},\alpha \rangle ) ,$$
$$\begin{aligned}
\label{cmk-f1}
\lefteqn{c^+_{|\alpha |}(x) =} && \\
&& \begin{cases}{\displaystyle q^{g x/2 } \frac{(q^{g+ x
};q)_\infty}
{(q^{x};q)_\infty} }&\text{for}\;\alpha\;\text{long} ,\\[2ex]
{\displaystyle q^{(g_0+g_1+g_2+g_3) x /2}\times} & \\
{\displaystyle
\frac{(q^{g_0+x},-q^{g_1+x},q^{g_2+1/2+x},-q^{g_3+1/2+x};q)_\infty}
{(q^{2 x};q)_\infty} }&\text{for}\;\alpha\;\text{short},
\end{cases} \nonumber\end{aligned}$$
$$\begin{aligned}
\label{cmk-f2}
\lefteqn{c^-_{|\alpha |}(x) =} && \\
&&\begin{cases} {\displaystyle q^{g x /2} \frac{(
q^{1+x};q)_\infty}
{(q^{1-g+x};q)_\infty}} &\text{for}\;\alpha\;\text{long} ,\\[2ex]
{\displaystyle q^{(g_0+g_1+g_2+g_3)x /2}\times }&
\\{\displaystyle \frac{( q^{1+2x};q)_\infty}
{(q^{1-g_0+x},-q^{1-g_1+x},q^{1/2-g_2+x},-q^{1/2-g_3+x};q)_\infty}}
&\text{for}\;\alpha\;\text{short},
\end{cases} \nonumber\end{aligned}$$
where we have distinguished dual parameters $g$, $g_0,\ldots ,g_3$ that are related to the parameters $\hat{g}$, $\hat{g}_0,\ldots
,\hat{g}_3$ via the linear relations
$$g=\hat{g},\qquad \left(\begin{array}{c} g_0 \\ g_1 \\g_2 \\ g_3
\end{array}\right)
=\frac{1}{2} \left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1
\end{array}\right)
\left(\begin{array}{c} \hat{g}_0 \\ \hat{g}_1 \\\hat{g}_2 \\
\hat{g}_3
\end{array}\right) ,$$
and with the vectors $\rho_g$ and $\rho_g^\vee$ now taken to be $$\begin{aligned}
\rho_g &=& \frac{g}{2}\sum_{\begin{subarray}{c} \alpha\in \boldsymbol{R}_1^+\\
\alpha\;\text{long}
\end{subarray}} \alpha +g_0\sum_{\begin{subarray}{c} \alpha\in \boldsymbol{R}_1^+\\
\alpha\;\text{short}
\end{subarray}} \alpha , \label{rhog1}\\
\rho_g^\vee &=& \frac{\hat{g}}{2}\sum_{\begin{subarray}{c} \alpha\in \boldsymbol{R}_1^+\\
\alpha\;\text{long}
\end{subarray}} \alpha +\hat{g}_0\sum_{\begin{subarray}{c} \alpha\in \boldsymbol{R}_1^+\\
\alpha\;\text{short}
\end{subarray}} \alpha . \label{rhog2}\end{aligned}$$
We thus arrive at the following formula for the wave function $\Psi_\lambda (\xi)$ in terms of Macdonald-Koornwinder polynomials.
\[mkwf:prp\] For $\boldsymbol{R}$ nonreduced and $c$-functions given by $\hat{c}_{|\alpha|}(z)$ , the wave function $\Psi_\lambda (\xi) $ reads explicitly $$\Psi_\lambda (\xi) = \frac{1}{\mathcal{N}_0^{1/2}}\Delta^{1/2}
(\lambda)\hat{\Delta}^{1/2}(\xi) \delta (\xi) \mathbf{P}_\lambda
(\xi),$$ with $\mathbf{P}_\lambda (\xi)$ denoting the Macdonald-Koornwinder polynomial characterized by Eqs. –.
From the second-order recurrence relation for the Macdonald-Koornwinder polynomials [@die:self-dual], we now obtain the following formula for the action of the Macdonald-Koornwinder Laplacian $L_\pi= \mathcal{F}^{-1}\circ
\hat{E}_\pi \circ\mathcal{F}$, associated to the first fundamental weight $\pi=\omega_1$ (which is a quasi-minuscule weight for $\boldsymbol{R}=BC_N$, cf. Table \[qmweight\]).
\[mklap:prp\] For $\boldsymbol{R}$ nonreduced and $\pi=\omega_1$, the action of the Macdonald-Koorwinder Laplacian $L_{\pi}$ on a (square-summable) lattice function $\phi:\mathcal{P}^+\rightarrow \mathbb{C}$ is given by $$\begin{aligned}
\lefteqn{L_\pi \phi_\lambda = E_\pi (\rho_g^\vee )\phi_\lambda+} && \\
&& \sum_{\begin{subarray}{c}\nu\in W(\pi )\\
\lambda +\nu\in\mathcal{P}^+\end{subarray}} \Bigl( V_\nu^{1/2}
(\rho_g +\lambda ) V_{-\nu}^{1/2} (\rho_g +\lambda +\nu )
\phi_{\lambda+\nu} -V_\nu (\rho_g +\lambda ) \phi_{\lambda} \Bigr)
,\end{aligned}$$ where $$\begin{aligned}
V_\nu (\mathbf{x}) &=& \prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R}_1 \\
\alpha \,\text{long},\;\langle \nu ,\alpha\rangle =1
\end{subarray}} \frac{\sinh\frac{s}{2}(g+\langle
\mathbf{x},\alpha \rangle
)}{\sinh\frac{s}{2} (\langle \mathbf{x},\alpha \rangle)} \times\\
&&
\prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R}_1 \\
\alpha \,\text{short},\;\langle \nu ,\alpha\rangle =1
\end{subarray}}
\frac{\sinh\frac{s}{2}(g_0+\langle \mathbf{x},\alpha \rangle
)}{\sinh\frac{s}{2} (\langle \mathbf{x},\alpha \rangle)}
\frac{\cosh\frac{s}{2}(g_1+\langle \mathbf{x},\alpha \rangle
)}{\cosh\frac{s}{2} (\langle \mathbf{x},\alpha \rangle)} \\
&&
\makebox[5em]{}\times\frac{\sinh\frac{s}{2}(g_2+\frac{1}{2}+\langle
\mathbf{x},\alpha \rangle )}{\sinh\frac{s}{2} (\frac{1}{2}+\langle
\mathbf{x},\alpha \rangle)}
\frac{\cosh\frac{s}{2}(g_3+\frac{1}{2}+\langle \mathbf{x},\alpha
\rangle )}{\cosh\frac{s}{2} (\frac{1}{2}+\langle \mathbf{x},\alpha
\rangle)} , \\
E_\pi (\mathbf{x}) &=& \sum_{\nu\in W(\pi)} \exp (s\langle \nu
,\mathbf{x}\rangle ) ,\end{aligned}$$ and with $\rho_g$ and $\rho_g^\vee$ given by Eqs. and , respectively.
For $\hat{g}\longrightarrow 1$ and $ \hat{g}_0,\ldots
,\hat{g}_3\longrightarrow 1/2$ (so $g,g_0\longrightarrow 1$ and $g_1,g_2,g_3\longrightarrow 0$), the $c$-functions $\hat{c}_{|\alpha |}(z)$ tend to $1$ (recall in this connection the duplication formula $(z^2;q)_\infty=(z,-z,q^{1/2}z,-q^{1/2}z;q)_\infty$). The Macdonald-Koornwinder Laplacian $L_\pi$ then reduces to the free Laplacian $$\label{flnr}
L_\pi^{(0)} \phi_\lambda = \sum_{ \nu\in W(\pi),\, \lambda +\nu
\in \mathcal{P}^+} \phi_{\lambda +\nu} .$$
Application of the scattering formalism of Section \[sec4\] now produces the following scattering and wave operators relating the long-time asymptotics of the dynamics of the Macdonald-Koornwinder Laplacian $L_\pi$ to that of the free Laplacian $L_{\pi}^{(0)}$ .
\[lcmNR:thm\] The wave operators $\Omega_\pm=s-\lim_{t\to \pm\infty} e^{itL_\pi}
e^{-itL_\pi^{(0)}}$ and the scattering operator $\mathcal{S}_{L_\pi}=\Omega_+^{-1}\Omega_-$ for the Macdonald-Koornwinder Laplacian $L_\pi$ in relation to the free Laplacian $L_{\pi}^{(0)}$ are of the form stated in Theorem \[wave:thm\] and Corollary \[scattering:cor\], with a unitary scattering matrix $\hat{\mathcal{S}}_{L_\pi}$ given by Eqs. , and $$\begin{aligned}
\lefteqn{\hat{s}_{|\alpha|}(\langle \alpha , \xi\rangle)=} && \\
&& \begin{cases} {\displaystyle \frac{(q e^{i\langle \alpha
,\xi\rangle};q)_\infty }{(q^{\hat{g}} e^{i\langle \alpha
,\xi\rangle};q)_\infty} \frac{(q^{\hat{g}} e^{-i\langle \alpha
,\xi\rangle};q)_\infty}{(q e^{-i\langle \alpha
,\xi\rangle};q)_\infty } }
&\text{for}\;\alpha\;\text{long} , \\[2ex]
{\displaystyle \frac{(qe^{2i\langle \alpha ,\xi\rangle}
;q)_\infty}{(q^{\hat{g}_0}e^{i\langle \alpha
,\xi\rangle},-q^{\hat{g}_1}e^{i\langle \alpha
,\xi\rangle},q^{\hat{g}_2+1/2}e^{i\langle \alpha
,\xi\rangle},-q^{\hat{g}_3+1/2}e^{i\langle \alpha
,\xi\rangle};q)_\infty}}\times & \\[2ex]
{\displaystyle \frac{(q^{\hat{g}_0}e^{-i\langle \alpha
,\xi\rangle},-q^{\hat{g}_1}e^{-i\langle \alpha
,\xi\rangle},q^{\hat{g}_2+1/2}e^{-i\langle \alpha
,\xi\rangle},-q^{\hat{g}_3+1/2}e^{-i\langle \alpha ,\xi\rangle}
;q)_\infty}{(qe^{-2i\langle \alpha ,\xi\rangle};q)_\infty}}
&\text{for}\;\alpha\;\text{short} .
\end{cases}\end{aligned}$$
The asymptotics of the Macdonald-Koornwinder wave function $\Psi_\lambda (\xi)$ in Proposition \[mkwf:prp\] is governed by Theorem \[plane:thm\] (and Eqs. –) with scattering matrices taken from Theorem \[lcmNR:thm\].
Example: The Rank-One Case {#sec6.5}
--------------------------
It is quite instructive to exhibit the results of this section in somewhat further detail for simplest case of a root system of rank [*one*]{}. We will restrict attention the case of a nonreduced root system (i.e. $BC_1$), since the reduced case (viz. $A_1$) can be recovered from it via a specialization of the parameters (corresponding to a standard reduction from the Askey-Wilson polynomials to the $q$-ultraspherical polynomials [@ask-wil:some; @gas-rah:basic]).
In this situation the Macdonald-Koornwinder wave function takes the explicit basic hypergeometric form $$\Psi_l(\xi) = \frac{1}{\mathcal{N}_0^{1/2}} \Delta^{1/2} (\ell )
\hat{\Delta}^{1/2}(\xi ) \delta (\xi)\mathbf{P}_\ell (\xi ),\qquad
\ell\in \mathbb{N}, \quad \xi\in (0,\pi),$$ where
$$\begin{aligned}
\mathcal{N}_0 &=& \frac{c^-(g_0)}{c^+(g_0)} ,\\
\Delta (\ell ) &=& \frac{c^+(g_0)c^-(g_0)}{c^+(g_0+\ell)c^-(g_0+\ell)}, \\
\hat{\Delta}(\xi )&=& \frac{1}{\hat{c}(\xi)\hat{c}(-\xi)} ,\\
\delta (\xi ) &=& 2\sin(\xi) ,\end{aligned}$$
with
$$\begin{aligned}
\hat{c}(\xi) &=&
\frac{(q^{\hat{g}_0}e^{-i\xi},-q^{\hat{g}_1}e^{-i\xi},q^{\hat{g}_2+1/2}e^{-i\xi},-q^{\hat{g}_3+1/2}e^{-i\xi};q)_\infty}
{(qe^{-2i\xi};q)_\infty} ,\\
c^+(x) &=&
q^{(g_0+g_1+g_2+g_3) x /2}\times \\
&&
\frac{(q^{g_0+x},-q^{g_1+x},q^{g_2+1/2+x},-q^{g_3+1/2+x};q)_\infty}
{(q^{2 x};q)_\infty} , \nonumber \\
c^-(x) &=&
q^{(g_0+g_1+g_2+g_3)x /2}\times \\
&& \frac{( q^{1+2x};q)_\infty}
{(q^{1-g_0+x},-q^{1-g_1+x},q^{1/2-g_2+x},-q^{1/2-g_3+x};q)_\infty}, \nonumber\end{aligned}$$
and with $\mathbf{P}_\ell (\xi )$ denoting the Askey-Wilson polynomial [@ask-wil:some; @gas-rah:basic] $$\mathbf{P}_\ell (\xi)=
{}_4\Phi_3 \left(
\begin{array}{c}
q^{-\ell},q^{2g_0+\ell},q^{\hat{g}_0}e^{i\xi},q^{\hat{g}_0}e^{-i\xi} \\
-q^{g_0+g_1},q^{g_0+g_2+1/2},-q^{g_0+g_3+1/2}
\end{array} ;q,q
\right) .$$ Here we have employed standard notation from the theory of basic hypergeometric series [@gas-rah:basic] $${}_{s}\Phi_{s-1} \left( \begin{array}{c} a_1,\ldots ,a_s \\
b_1,\ldots ,b_{s-1}
\end{array} ; q,z \right)\equiv
\sum_{n=0}^\infty \frac{(a_1,\ldots ,a_s ;q)_n }{(b_1,\ldots
,b_{s-1};q)_n } \frac{z^n}{(q;q)_n} ,$$ with $(a;q)_n\equiv\prod_{k=0}^{n-1} (1-aq^k)$ and $(a_1,\ldots
,a_s ;q)_n\equiv(a_1;q)_n\cdots (a_s ;q)_n$.
The asymptotics of the wave function $\Psi_l(\xi)$ for $\ell\longrightarrow\infty$ is given by (cf. also [@ism-wil:asymptotic; @ism:asymptotics])
$$\Psi_l^{\infty}(\xi)=\hat{s}^{1/2}(\xi) e^{i(\ell+1)\xi}-\hat{s}^{-1/2}(\xi)
e^{-i(\ell+1)\xi},$$
with $$\hat{s}(\xi )=\frac{\hat{c}(\xi)}{\hat{c}(-\xi )} .$$
The free plane waves $\Psi_\ell^{(0)} (\xi )$ boil in this case down to the Fourier sine kernel $$\Psi_\ell^{(0)} (\xi )=2\sin (\ell +1)\xi ,\qquad \ell\in
\mathbb{N}, \quad \xi\in (0,\pi).$$
The corresponding Fourier pairings $\mathcal{F}:l^2(\mathbb{N})\mapsto
L^2((0,\pi),(2\pi)^{-1}\text{d}\xi)$ and $\mathcal{F}^{(0)}:l^2(\mathbb{N})\mapsto
L^2((0,\pi),(2\pi)^{-1}\text{d}\xi)$ together with their inversion formulas are given by
$$\begin{cases}
{\displaystyle \hat{\phi}(\xi) = \sum_{\ell\in\mathbb{N}}
\phi_\ell\Psi_\ell(\xi)} , \\[2ex]
{\displaystyle \phi_\ell = \frac{1}{2\pi} \int_0^\pi
\hat{\phi}(\xi)\Psi_\ell(\xi)\text{d}\xi } ,
\end{cases}$$
and $$\begin{cases}
{\displaystyle \hat{\phi}(\xi) = \sum_{\ell\in\mathbb{N}}
\phi_\ell\Psi^{(0)}_\ell(\xi)} , \\[2ex]
{\displaystyle \phi_\ell = \frac{1}{2\pi} \int_0^\pi
\hat{\phi}(\xi)\Psi_\ell^{(0)}(\xi)\text{d}\xi } ,
\end{cases}$$
respectively (where we have omitted the complex conjugations because the relevant kernel functions $\Psi_\ell(\xi)$ and $\Psi_\ell^{(0)}(\xi)$ are real-valued as a consequence of the fact that $-\mathbf{1}\in W\cong\mathbb{Z}_2$).
The Macdonald-Koornwinder Laplacian $L= \mathcal{F}^{-1}\circ
\hat{E} \circ\mathcal{F}$ and the free Laplacian $L^{(0)}=
(\mathcal{F}^{(0)})^{-1}\circ \hat{E} \circ\mathcal{F}^{(0)}$ associated to the multiplication operator $\hat{E}(\xi) =2\cos
(\xi) $ act on on lattice functions $\phi\in l^2 (\mathbb{N})$ respectively as
$$\begin{aligned}
L \phi_\ell &=& V^{1/2}(g_0+\ell) V^{1/2}(-g_0-\ell-1)\phi_{\ell
+1}+ \\
&& V^{1/2}(-g_0-\ell) V^{1/2}(g_0+\ell-1)\phi_{\ell -1} + \nonumber\\
&& \bigl( 2\cosh (s\hat{g}_0)- V(g_0+\ell)
-(1-\delta_{\ell,0})V(-g_0-\ell) \bigr)\phi_\ell ,\nonumber\end{aligned}$$
with $$\begin{aligned}
\lefteqn{V(x)=\frac{\sinh\frac{s}{2}(g_0+x )}{\sinh\frac{s}{2}
(x)} \frac{\cosh\frac{s}{2}(g_1+x )}{\cosh\frac{s}{2} (x)} } \\
&& \makebox[2em]{}\times\frac{\sinh\frac{s}{2}(g_2+\frac{1}{2}+x
)}{\sinh\frac{s}{2} (\frac{1}{2}+x)}
\frac{\cosh\frac{s}{2}(g_3+\frac{1}{2}+x )}{\cosh\frac{s}{2}
(\frac{1}{2}+x)} ,\nonumber\end{aligned}$$
and as $$L^{(0)}\phi_\ell = \phi_{\ell +1}+\phi_{\ell -1} ,$$ with the boundary condition $\phi_{-1}=0$.
The specialization of Theorem \[lcmNR:thm\] to the case $N=1$ now states that the wave operators $\Omega_\pm=s-\lim_{t\to
\pm\infty} e^{itL} e^{-itL^{(0)}}$ and the scattering operator $\mathcal{S}_{L}=\Omega_+^{-1}\Omega_-$ exist in $l^2(\mathbb{N})$, and are moreover of the form $\Omega_\pm =
\mathcal{F}^{-1}\circ \hat{\mathcal{S}}^{\mp 1/2}\circ
\mathcal{F}^{(0)}$ and $\mathcal{S}_{L} =
(\mathcal{F}^{(0)})^{-1}\circ \hat{\mathcal{S}}_L\circ
\mathcal{F}^{(0)}$, respectively, with $\hat{\mathcal{S}}_{L}$ being a unitary scattering matrix whose multiplicative action on a wave packet $\hat{\phi}\in L^2((0,\pi),(2\pi)^{-1}\text{d}\xi)$ is given by $$(\hat{\mathcal{S}}_{L}\hat{\phi})(\xi)=
\frac{\hat{c}(-\xi)}{\hat{c}(\xi
)}\,\hat{\phi}(\xi) \quad\text{for}\; 0 <\xi <\pi .$$
Properties of the Macdonald Polynomials {#appA}
=======================================
This appendix serves to list a number of key properties of the Macdonald polynomials $\mathbf{P}_\lambda (\xi)$, $\lambda\in\mathcal{P}^+$ defined by Eqs. –. We used these properties in Section \[sec6\] to build the Macdonald wave function and to the determine the explicit action of the Macdonald-Ruijsenaars Laplacian. For proofs of the statements below and for further theory concerning the Macdonald polynomials the reader is referred to the seminal works of Macdonald and Cherednik [@mac:symmetric; @mac:orthogonal; @mac:affine; @che:double; @che:macdonalds] (see also [@cha:macdonald] for a different approach). Throughout this appendix it is assumed that our root system $\boldsymbol{R}$ be [*reduced*]{}. For the extension of the statements below to the case of [*nonreduced*]{} root systems the reader is referred to Refs. [@mac:orthogonal; @mac:affine; @koo:askey-wilson; @die:self-dual; @oko:bcn; @sah:nonsymmetric].
The Macdonald polynomials $\mathbf{P}_\lambda (\xi)$ – are normalized such that they satisfy the [*Specialization Formula*]{} $$\label{sp-f}
\mathbf{P}_\lambda (is\rho_g^\vee) =1.$$ In this normalization the [*Orthogonality Relations*]{} read $$\label{ort-r}
( \mathbf{P}_\lambda , \mathbf{P}_\mu )_{\hat{\mathcal{H}}} =
\begin{cases}
0 &\text{if}\; \lambda \neq \mu ,\\
\frac{\mathcal{N}_0}{\Delta (\lambda )} & \text{if}\; \lambda =\mu
.
\end{cases}$$ The specialization formula in Eq. amounts to the special case $\mu=0$ of the more general [*Symmetry Relation*]{} $$\label{sym-r}
\mathbf{P}_\lambda^R (is(\rho_g^\vee +\mu)) =
\mathbf{P}_\mu^{R^\vee} (is(\rho_g +\lambda)),$$ where $\mathbf{P}_\lambda^R(\xi) $ and $\mathbf{P}_\mu^{R^\vee}(\xi)$ refer to the Macdonald polynomials associated to the root system $\boldsymbol{R}$ and the the dual root system $\boldsymbol{R}^\vee$, respectively (so $\lambda$ and $\mu $ are dominant weights of $\boldsymbol{R}$ and $\boldsymbol{R}^\vee$, respectively).
For any (quasi-)minuscule weight $\pi$ of $\boldsymbol{R}^\vee$, we have a corresponding [*Macdonald Difference Equation*]{} given by
$$\begin{aligned}
\label{mdif-eqa}
\sum_{\nu\in W(\pi )} \prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha\rangle >0 \end{subarray}} \frac{(isg_{|\alpha
|}+\langle \xi ,\alpha \rangle : \sin)_{\langle \nu
,\alpha\rangle}}{(\langle \xi,\alpha \rangle:\sin )_{\langle \nu
,\alpha\rangle}} \bigl(\mathbf{P}_{\lambda
}(\xi+is\nu)-\mathbf{P}_\lambda (\xi) \bigr) && \\
=\sum_{\nu\in W(\pi
)} \bigl( q^{\langle \nu ,\lambda+\rho_g\rangle}-q^{\langle \nu
,\rho_g\rangle}\bigr) \mathbf{P}_\lambda (\xi) ,&& \nonumber\end{aligned}$$
where $(z:\sin )_m\equiv \prod_{\ell =0}^{m-1}\sin\frac{1}{2}
(z+is\ell )$. If the weight $\pi$ is minuscule (so $|\langle \pi
,\alpha \rangle | \leq 1$, $ \forall \alpha\in \boldsymbol{R}$), then this difference equation simplifies to $$\label{mdif-eqb}
\sum_{\nu\in W(\pi )} \prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha\rangle =1 \end{subarray}}
\frac{\sin\frac{1}{2} (isg_{|\alpha |}+\langle \xi ,\alpha \rangle
)}{\sin\frac{1}{2} (\langle \xi,\alpha \rangle )}
\mathbf{P}_{\lambda }(\xi+is\nu)
=\sum_{\nu\in W(\pi
)} q^{\langle \nu ,\lambda+\rho_g\rangle} \mathbf{P}_\lambda (\xi)
,$$
because of the [*Macdonald Identity*]{} (for $\pi$ minuscule) $$\label{mac-id}
\sum_{\nu\in W(\pi )}
\prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha\rangle =1 \end{subarray}}
\frac{\sin\frac{1}{2} (isg_{|\alpha |}+\langle \xi ,\alpha \rangle
)}{\sin\frac{1}{2} (\langle \xi,\alpha \rangle )} = \sum_{\nu\in
W(\pi )} q^{\langle \nu ,\rho_g\rangle} .$$
Combination of the Macdonald difference equation in Eq. and the symmetry relation in Eq. leads to the [*Recurrence Relation (or Pieri Formula)*]{}
$$\begin{aligned}
\label{rec-rela}
\lefteqn{\sum_{\nu\in W(\pi )} \bigl( e^{i\langle \nu
,\xi\rangle}-q^{\langle \nu ,\rho_g^\vee\rangle} \bigr)
\mathbf{P}_\lambda (\xi)=} && \\
&& \sum_{\begin{subarray}{c} \nu\in W(\pi )\\ \lambda
+\nu\in\mathcal{P}^+\end{subarray}} \prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha^\vee\rangle >0 \end{subarray}}
\frac{(g_{|\alpha |}+\langle \rho_g+\lambda ,\alpha^\vee \rangle :
\sinh_s )_{\langle \nu ,\alpha^\vee\rangle}}{(\langle
\rho_g+\lambda,\alpha^\vee \rangle:\sinh_s )_{\langle \nu
,\alpha^\vee\rangle}} \bigl(\mathbf{P}_{\lambda
+\nu}(\xi)-\mathbf{P}_\lambda (\xi) \bigr) ,\nonumber\end{aligned}$$
where $\pi$ is now a (quasi-)minuscule weight of $\boldsymbol{R}$ (and $(z:\sinh_s )_m\equiv \prod_{\ell =0}^{m-1}\sinh
(\frac{s}{2}(z+\ell) )$). In the minuscule case this recurrence relation reduces to $$\label{rec-relb}
\sum_{\nu\in W(\pi )} e^{i\langle \nu ,\xi\rangle}
\mathbf{P}_\lambda (\xi)= \sum_{\begin{subarray}{c} \nu\in W(\pi
)\\ \lambda
+\nu\in\mathcal{P}^+\end{subarray}} \prod_{\begin{subarray}{c} \alpha\in \boldsymbol{R} \\
\langle \nu ,\alpha^\vee\rangle =1 \end{subarray}}
\frac{\sinh\frac{s}{2}(g_{|\alpha |}+\langle \rho_g+\lambda
,\alpha^\vee \rangle )}{\sinh\frac{s}{2}(\langle
\rho_g+\lambda,\alpha^\vee \rangle)}\mathbf{P}_{\lambda
+\nu}(\xi).$$
Index of Notations {#appB}
==================
This Appendix provides a list of notations ordered according to the sections in which they were first introduced.
Section \[sec2.1\]: $\mathbf{E}$, $\langle \cdot ,\cdot \rangle$ , $\boldsymbol{R}$, $\boldsymbol{R}^+$, $\mathcal{Q}$, $\mathcal{Q}^+$, $\mathcal{P}$, $\mathcal{P}^+$, $\alpha^\vee$, $\mathbf{A}$, $m_\lambda (\xi)$, $\xi_w$, $W$, $W_\lambda$, $|W_\lambda|$.
Section \[sec2.2\]: $\boldsymbol{R}_0$, $\boldsymbol{R}_1$, $\hat{\Delta}(\xi)$, $\hat{\mathcal{C}}(\xi)$, $\hat{c}_{|\alpha|}(z)$.
Section \[sec2.3\]: $(\cdot ,\cdot)_{\hat{\Delta}}$, $\text{Vol}(\mathbf{A})$, $\delta (\xi)$, $\succeq$, $\geqslant$, $P_\lambda (\xi)$, $a_{\lambda \mu}$.
Section \[sec2.4\]: $\chi_\lambda (\xi)$, $(-1)^w$, $\rho$, $w_\mu$.
Section \[sec3.1\]: $\mathcal{H}$, $(\cdot ,\cdot )_{\mathcal{H}}$, $\hat{\mathcal{H}}$, $(\cdot ,\cdot )_{\hat{\mathcal{H}}}$, $\Psi_\lambda (\xi)$, $\mathcal{F}$, $\Psi_\lambda^{(0)} (\xi)$, $\mathcal{F}^{(0)}$.
Section \[sec3.2\]: $\omega_r$, $\hat{E}_r(\xi)$, $W(\cdot)$, $L_r$, $\sigma(L_r)$, $w_0$.
Section \[sec3.3\]: $a_{\lambda\mu;r}$, $\mathcal{P}^+_{\lambda ;r}$, $L_r^{(0)}$.
Section \[sec4.1\]: $\mathbf{C}^+$, $m(\lambda)$, $P_\lambda^\infty (\xi)$, $\|\cdot\|_{\hat{\Delta}}$, $P_\lambda^{m(\lambda)}(\xi)$, $\Psi_\lambda^\infty(\xi)$, $\hat{S}_w(\xi)$, $\hat{s}_{|\alpha|}(\langle\alpha, \xi\rangle)$, $\|\cdot\|_{\hat{\mathcal{H}}}$.
Section \[sec4.2\]: $\hat{E}(\xi)$, $L$, $L^{(0)}$, $\sigma (L)$, $\mathbf{A}_{\text{reg}}$, $\hat{w}_\xi$, $\hat{\mathcal{S}}_L$, $\|\cdot\|_{\mathcal{H}}$, $\Omega_\pm$, $\mathcal{S}_L$.
Section \[sec5.1\]: $\phi^{(0)}(t)$, $\phi_\pm (t)$.
Section \[sec5.2\]: $\hat{w}$, $\mathbf{V}_{\text{clas}}$, $\mathcal{P}^+_{\text{clas}}(t)$, $\phi^{(\text{clas})}(t)$, $\hat{W}$.
Section \[sec5.3\]: $\phi_\pm^{(\infty)}$, $P_t^{(\text{clas})}$.
Section \[sec6.1\]: $g_{|\alpha|}$, $(z;q)_\infty$, $\Delta (\lambda)$, $\mathcal{N}_0$, $\mathcal{C}^\pm(\mathbf{x})$, $\rho_g$, $c^\pm_{|\alpha|}(x)$, $\mathbf{P}_\lambda(\xi)$.
Section \[sec6.2\]: $\pi$, $\alpha_0^\vee$, $\boldsymbol{R}^\vee$, $\hat{E}_\pi(\xi)$, $L_\pi$, $V_\nu(\mathbf{x})$, $E_\pi(\mathbf{x})$, $(z:\sinh_s)_m$, $\rho_g^\vee$.
Section \[sec6.3\]: $L_\pi^{(0)}$, $n_\pi(\lambda)$, $\alpha_j$, $r_{\alpha_j}$, $\mathcal{S}_{L_\pi}$, $\hat{\mathcal{S}}_{L_\pi}$.
Section \[sec6.4\]: $\hat{g}$, $\hat{g}_r$, $(z_1,\ldots,z_k;q)_\infty$, $g$, $g_r$.
Section \[sec6.5\]: $_{s+1}\Phi_s$, $(a;q)_n$, $(a_1,\ldots,a_s;q)_n$.
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[^1]: Work supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grant \# 1010217 and the Programa Formas Cuadráticas of the Universidad de Talca.
|
---
abstract: |
We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. Let $\Delta_g$ be the Friedrich extension positive Laplacian on $X$, and consider the operator $\mathcal{L}_V=\Delta_g+V$ with $V=V_0r^{-2}$, where $V_0(\theta)\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $\Delta_h+V_0+(n-2)^2/4$ is positive. In the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with $\mathcal{L}_V$. The smallest positive eigenvalue of the operator $\Delta_h+V_0+(n-2)^2/4$ plays an important role in the result.
As an application, for independent of interests, we prove local energy estimates and Keel-Smith-Sogge estimates for the wave equation in this setting.
address:
- 'Department of Mathematics, Beijing Institute of Technology, Beijing 100081;'
- 'Department of Mathematics, Beijing Institute of Technology, Beijing 100081; Department of Mathematics, Cardiff University, UK'
- 'Institute of Applied Physics and Computational Mathematics, Beijing 100088'
author:
- Xiaofen Gao
- Junyong Zhang
- Jiqiang Zheng
title: ' Restriction estimates in a conical singular space: wave equation '
---
[ ]{}\
Introduction
============
Restriction estimate as one of the cores in harmonic analysis is originally proposed by Stein [@Stein79] for sets $S$ having non-vanishing curvature, including hyper-surfaces such as sphere, paraboloid and cone. In this paper, we focus on the wave equation whose characteristic set is a cone, and so we take the cone to illustrate the details of the restriction estimates.
Let $S$ be a smooth compact nonempty subset of the cone $\{(\tau,\xi)\in{\mathbb{R}}\times{\mathbb{R}}^n:\tau=|\xi|\}$ with $n\geq2$. For any Schwartz function $F$ on $S$, the inverse space-time Fourier transform of the measure $F\mathrm{d}\sigma$ is given by $$\begin{aligned}
\label{F-vee}
(F\mathrm{d}\sigma)^\vee(t,x)&=\int_SF(\tau,\xi)e^{2\pi i(x\cdot\xi+t\tau)}\mathrm{d}\sigma(\xi)\\\nonumber
&=\int_{{\mathbb{R}}^n}F(|\xi|,\xi)e^{2\pi i(x\cdot\xi+t|\xi|)}\frac{\mathrm{d}\xi}{|\xi|}.\end{aligned}$$ where the conical measure $\mathrm{d}\sigma$ is the pullback of the measure $\frac{\mathrm{d}\xi}{|\xi|}$ under the projection $(\tau,\xi)\mapsto\xi.$ The restriction problem is to seek the optimal range of $p$ and $q$ satisfying the adjoint restriction estimate $$\label{est:res}
\|(F\mathrm{d}\sigma)^\vee\|_{L^q_{t,x}({\mathbb{R}}\times{\mathbb{R}}^n)}\leq C_{p,q,n,S}\|F\|_{L^p(S,\mathrm{d}\sigma)}.$$ The two necessary conditions such that holds are $$\label{condition}
q>\frac{2n}{n-1}\quad \text{and}\quad \frac{n+1}{q}\leq\frac{n-1}{p'},$$ which come from the decay of $(\mathrm{d}\sigma)^\vee$ and Knapp example, see [@Stein; @Tao].
A famous conjecture is to claim that the two necessary conditions also are sufficient for , see [@Stein; @Tao]. More precisely,
\[conj\] The estimate is true if and only if hold.
This conjecture is a great challenge and has attracted many mathematicians’ attention. This conjecture has been proved to hold true by Barcelo [@Barcelo] for $n=2$ and Wolff [@Wolff] for $n=3$. Very recently, by using the method of polynomial partitioning developed by Guth[@Guth1; @Guth2], Ou and Wang[@OW] solved the cone restriction conjecture for $n=4$ and made some new progress for the conjecture in the higher dimensions. The conjecture is so challenging that it remains open when $n\geq 5$. For recent work, see [@De; @Tao] for more details on process of restriction estimate.
It is known that the restriction problem on cone is closely related to the wave equation, e.g. see Tao [@Tao]. Let $u$ be the solution of the wave equation $$\begin{aligned}
\label{equ:w}
\begin{cases}
&(\partial^2_t-\Delta)u(t,x)=0\quad\quad (t,x)\in{\mathbb{R}}\times {\mathbb{R}}^n\\
&u(0,x)=0, u_t(0,x)=f(x),\quad x\in {\mathbb{R}}^n,
\end{cases}\end{aligned}$$ then the solution $$u(t,x)=\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}f=\sum_{\pm}\int_{{\mathbb{R}}^n}\hat{f}(\xi)e^{2\pi i(x\cdot\xi\pm t|\xi|)}\frac{\mathrm{d}\xi}{|\xi|}.$$ Take $F=\hat{f}$ in , then the inverse space-time Fourier transform of the measure $F\mathrm{d}\sigma$ equals each half-wave $$(F\mathrm{d}\sigma)^\vee(\pm t,x)=\int_{{\mathbb{R}}^n}\hat{f}(\xi)e^{2\pi i(x\cdot\xi\pm t|\xi|)}\frac{\mathrm{d}\xi}{|\xi|}.$$ The restriction problem for the wave equation is to find the optimal range of $p$ and $q$ satisfying the estimate $$\label{est:res-w}
\|u(t,x)\|_{L^q_{t,x}({\mathbb{R}}\times{\mathbb{R}}^n)}\leq C_{p,q,n}\|\hat{f}\|_{L^p_{1/|\xi|}({\mathbb{R}}^n)}.$$ Since the support of $\hat{f}$ may not supported in a compact set, that is the above $S$ is the whole cone instead of a compact subset of the cone, the necessary conditions are strengthened to
$$\label{condition'}
q>\frac{2n}{n-1}\quad \text{and}\quad \frac{n+1}{q}=\frac{n-1}{p'}.$$
The version of Conjecture \[conj\] for wave equation can be stated as
\[conj-w\] The estimate is true if and only if hold.
The above problems and conjectures are proposed to be associated with constant coefficient operator in the Euclidean space. It is natural to ask analogous problems for the wave equation in a curve space or when there is a potential term in the equation. In particular $p=2$, the inequality is known as the Strichartz type estimate for wave equation. There has been a lot of interest in developing Strichartz estimates on manifolds or when there is a potential term in the equation, both for the Schrödinger and wave equations; this is too vast and highly active field to summarize here, but we refer to a very small and incomplete sample of recent results [@BPSS; @MT; @RS; @HTW; @ST]. The restriction theory on manifolds arises in the study of eigenfunctions and the spectral measure of the Laplacian, for example, see Sogge [@sogge] on compact manifold and Guillarmou-Hassell-Sikora[@GHS] on asymptotically conic manifold.
However, for general manifolds and $p\neq 2$, there is little result and the restriction theory is less satisfactory. Due to the geometry of the space and the spectrum of the operator, the results for the variable coefficient operator may be very different from the constant coefficient operator. For instance, one can not expect all the results from the Euclidean theory to carry over to curved space (e.g. see [@MS]). In this paper, we aim to prove a modified adjoint restriction inequality of for the solution of in a conical singular space $X$.
Before stating our main result, we set up our model. Our setting is the metric cone $X$ which is a simple conical singular space as studied in [@CT1; @CT2; @HL; @MSe]. The conical space $(X,g)$ is given by the product space $X=C(Y)=(0,\infty)_r\times Y$ and the metric $g=dr^2+r^2h$ where $(Y,h)$ is a $(n-1)$-dimensional closed Riemannian manifold. A simplest example of a metric cone is the Euclidean space ${\mathbb{R}}^n$ when cross section $Y=\mathbb{S}^{n-1}$ and $h=d\theta^2$ the standard round metric on sphere. We stress that $X$ is more general and different from the Euclidean space. The space $X$ has an isolated conic singularity at cone tip $r=0$ except in the special case of Euclidean space. The space $X$ does not have rotation symmetry and possibly has conjugate points due to the generality of $(Y,h)$ which bring many difficulties in the study of Strichartz estimates in [@ZZ1; @ZZ2].
In this paper, as following the program in [@HL; @Zhang; @ZZ1; @ZZ2], we consider the Schrödinger operator $$\label{oper}
\mathcal{L}_V=\Delta_g+V$$ where $\Delta_g$ is the Friedrichs extension of positive Laplace-Beltrami from the domain $\mathcal{C}_c^\infty(X^\circ),$ compactly supported smooth functions on the interior of the metric cone, and the potential $V=V_0(\theta)r^{-2}$ with $V_0(\theta)\in\mathcal{C}^\infty(Y)$ being such that the operator $\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. There are many works studied this operator from different viewpoints. For example, wave diffraction phenomenon has been extensively studied in [@CT1; @CT2]; Riesz transform and heat kernel has been considered in [@HL; @Li1; @Li2; @Mooer]. In the study of the regularity of wave propagator, Li [@Li3] and Müller-Seeger [@MSe] proved the $L^p$ regularity estimate; the Strichartz estimates were proved by Blair-Ford-Marzuola [@Ford; @BFM] on flat cone $C(\mathbb{S}^1_\rho)$ and then were generalized by the last two authors in [@ZZ1; @ZZ2].
Now, we state our main results. First, we study the adjoint restriction estimate for the solution of the wave equation $$\begin{aligned}
\label{equ:Lw}
\begin{cases}
&(\partial^2_t+{\mathcal{L}}_V)u(t,r,\theta)=0,\quad\quad (t, r,\theta)\in{\mathbb{R}}\times X,\\
&u(0,r,\theta)=0, u_t(0,r,\theta)=f(r,\theta),\quad (r,\theta)\in X.
\end{cases}\end{aligned}$$ More precisely, we prove
\[thm:main\] Let $n\geq2$ and $X$ be a n-dimensional metric cone, and let ${\mathcal{L}}_V=\Delta_g+V$ where $r^2V=:V_0\in\mathcal{C}^\infty(Y)$ such that $\Delta_h+V_0(\theta)+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$ and its smallest eigenvalue is $\nu_0^2$ with $\nu_0>0$. Suppose $f$ to be any Schwartz function and $u$ to be the solution of .
$\bullet$ If $\nu_0\geq \tfrac{n-2}2$, and $(q,p)$ satisfies , then there exists a constant $C$ only depending on $p,q,n$ and $X$ such that $$\label{est:restriction}
\|u(t,r,\theta)\|_{L^q_{t}({\mathbb{R}};L^q_{\mathrm{rad}}(L^2_\mathrm{sph}))}\leq C_{p,q,n,X}\|\rho^{-\frac{1}{p}}\hat{f}(\rho,\omega)\|_{L^p_{\mathrm{rad}}(L^2_\mathrm{sph})},$$ where $L^q_{\mathrm{rad}}(L^2_\mathrm{sph})=L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y))$ and $\hat{f}$ denotes the distorted Fourier transform defined in below.
$\bullet$ If $0<\nu_0< \tfrac{n-2}2$, and $(q,p)$ satisfies and $$\label{add-cond}
q<\frac{2n}{n-2-2\nu_0}$$ then holds true. The additional requirement is necessary.
The admissible pair $(q,p)$ here is almost the same as stated in Conjecture \[conj-w\] when $\nu_0\geq (n-2)/2$. However, from the additional necessary condition , the smallest eigenvalue of $\Delta_h+V_0(\theta)+(n-2)^2/4$ plays an important role.
\[rem:stri\] In particular $p=2$, from , we obtain the Strichartz estimate $$\label{est:stri}
\|u(t,r,\theta)\|_{L^{\frac{2(n+1)}{n-1}}_{t}({\mathbb{R}};L^{\frac{2(n+1)}{n-1}}_{\mathrm{rad}}(L^2_\mathrm{sph}))}\leq C\|f\|_{\dot{H}^{-\frac12}(X)}$$ which is weaker than the Strichartz estimate proved in [@ZZ2]. However, from below, we can prove $$\label{est:stri'}
\|u(t,r,\theta)\|_{L^q_{t}({\mathbb{R}};L^{q}_{\mathrm{rad}}(L^2_\mathrm{sph}))}\leq C M^{\frac{n-1}2-\frac{n+1}q}\|f\|_{\dot{H}^{-\frac12}(X)}, \quad q>\tfrac{2n}{n-1}$$ provided $\text{supp} (\hat{f})\subset\{\rho\sim M\}$, and one needs the restriction $q<\tfrac{2n}{n-2-2\nu_0}$ when $0<\nu_0<\tfrac{n-2}2.$ This includes some new Strichartz estimates since one can choose $q$ to be out of the admissible assumption in [@ZZ2].
Compared with [@Zhang], in which the second author studied the restriction estimate for Schrödinger equation associated with $\mathcal{L}_V$, here we remove the positive assumption on the potential and improve the loss of angular regularity.
The conjugate points do not effect the estimates due to the mixed space $L^q_{\mathrm{rad}}(L^2_\mathrm{sph})$. It would be interesting to investigate the implicit influence of conjugate point by establishing .
We stress that the result does not solve the challenging Conjecture \[conj-w\] in the conical singular space since we use a mixed space $L^q_{\mathrm{rad}}(L^2_\mathrm{sph})$. The modified norm in is motivated to simplify the Conjecture \[conj-w\] from two aspects: the wavelet tubes overlap in the angular direction and the parametrix of the wave propagator.
The modified norm $L^q_{\mathrm{rad}}(L^2_\mathrm{sph}(\mathbb{S}^{n-1}))$ has been used in many famous harmonic analysis problems (such as Fourier restriction estimates, local smoothing conjecture etc.) on Euclidean space ${\mathbb{R}}^n$, we refer the reader to [@Cor; @CG; @CRS; @GS; @Moc; @Shao]. Miao and the last two authors [@MZZ; @MZZ1] proved the restriction estimates for wave and Schrödinger equation when the initial data has additional angular regularity. And later, Córdoba-Latorre [@CL1] revisited some classical conjectures including restriction estimate in harmonic analysis in the mixed space $L^q_{\mathrm{rad}}(L^2_\mathrm{sph})$.
In the conical singular space $X$, the spacetime Fourier transform is no longer so useful as well as in Euclidean space, and so restriction theory is harder to be established; however the spatial Fourier transform can be replaced by the spectral decomposition of the Laplacian, some techniques used in that theory still do apply, for instance the $TT^*$-method which was used in [@GHS] to prove Stein-Tomas type restriction estimates and in [@HZ; @ZZ1; @ZZ2] to prove Strichartz estimates. The $TT^*$-strategy is a key point in those papers to use the approximate microlocalized parametrix for the fundamental solution which is more complicated than the Euclidean’s due to the possibility of appearing conjugate points in the space. But if one aims to establish when $p\neq 2$, the $TT^*$-method breaks down. Instead of using the microlocalized parametrix constructed in [@ZZ1; @ZZ2], we will use the method of Cheeger-Taylor [@CT1; @CT2], even though the method leads to a loss of angular regularity. The method of Cheeger-Taylor has been used by Müller-Seeger [@MSe] to establish local smoothing estimates in the mixed spacetime $L^p_{\mathrm{rad}}(L^2_\mathrm{sph})$ estimates for wave equation in this conical singular space.
Our strategy of proving Theorem \[thm:main\] is to establish the localized estimates for Hankel transform by analyzing Bessel function and using stationary phase argument. As an application of the proof, for independent interest, we will prove local energy estimates and Keel-Smith-Sogge estimates in our setting.
\[thm:KSS\] Let ${\bf R}>0$ be a fixed number and let $u(t,r,\theta)$ be the solution of with initial data $f\in\dot{H}^{-1}$. Then the following results hold:
$\bullet$ Local energy decay estimate: $$\label{est:locendec}
\sup_{{\bf R}>0}{\bf R}^{-1/2}\|u(t,r,\theta)\|_{L^2({\mathbb{R}}; L^2((0,{\bf R}]\times Y))} \lesssim \|f\|_{\dot{H}^{-1}}$$
$\bullet$ Keel-Smith-Sogge estimate: let $\beta>0$ $$\label{kss}
\|\langle r\rangle^{-\beta} u(t,r,\theta)\|_{L^2([0,T];L^2(X))}\leq C_\beta(T)\|f\|_{\dot{H}^{-1}},$$ where $$C_\beta(T)=C\times
\begin{cases}
T^{\frac12-\beta},\quad &\text{if}\;\quad 0\leq \beta< \frac12,\\
(\log((2+T))^{\frac12},\quad &\text{if}\;\quad \beta=\frac12,\\
1 \quad &\text{if}\;\quad \beta>\frac12,
\end{cases}$$ where $C$ is an absolute constant independent of $T$.
$\bullet$ Local smoothing estimate: let $0\leq \beta <\tfrac12$ $$\label{kss2}
\big\| |x|^{-\beta} u(t,r,\theta) \big\|_{L^2([0,T];L^2(X))}\lesssim T^{\frac12-\beta}\|f\|_{\dot{H}^{-1}}.$$
If $\tfrac12<\beta<1+\nu_0$ with $\nu_0$ being given in Theorem \[thm:main\], a global-in-time local smoothing estimate (that is, the constant is independent of $T$), has been proved by the last two authors in [@ZZ2].
The Keel-Smith-Sogge (KSS) estimates were originally developed in [@KSS] to study the lifespan of solution of quasilinear wave equation. We present the KSS estimates here for independent interests of studying the existence theory of the solution of nonlinear wave equation (e.g. Strauss conjecture and Glassey conjecture) in this setting.
This paper is organized as follows: Section 2 gives some preliminaries including the spectral properties, Bessel function and Hankel transform. In Section 3, we prove the key localized estimates of Hankel transform. The proof of Theorem \[thm:main\] is presented in Section 4. Section 5 provides the proof of Theorem \[thm:KSS\].
[**Acknowledgments:**]{}The authors were supported by National Natural Science Foundation of China (11771041, 11831004, 11901041,11671033) and H2020-MSCA-IF-2017(790623).
Preliminaries
=============
In this section, we recall spectral and harmonic analysis results such as orthogonal decomposition of $L^2(Y)$, some basic properties about Hankel transform and Bessel function. In the end of this section, we introduce some notations.
Spectral property of $\Delta_h+V_0(y)+(n-2)^2/4$
------------------------------------------------
To study the operator ${\mathcal{L}}_V$, we recall some spectral result of $\Delta_h+V_0(y)+(n-2)^2/4$, e.g. see [@Wang; @Zhang].
Consider the operator in $$\label{oper'}
{\mathcal{L}}_V=\Delta_g+\frac{V_0(\theta)}{r^2},$$ on the metric cone $X=(0,\infty)_r\times Y.$ In coordinates $(r,\theta)\in{\mathbb{R}}_+\times Y$, $V_0(\theta)$ is a real continuous function and the metric $g$ takes the form $$g=\mathrm{d}r^2+r^2h(\theta,\mathrm{d}\theta),$$ where $h$ is the Riemannian metric on $Y$ independent of $r$. Let $\Delta_h$ be the positive Laplace-Beltrami operator on $(Y,h)$ and let $\nu_0^2$ be the smallest eigenvalue of the operator $\Delta_h+V_0(\theta)+(n-2)^2/4$, that is, for any $f\in L^2(Y)$, it holds [^1] $$\label{nu0}
\big\langle(\Delta_h+V_0(\theta)+(n-2)^2/4)f,f\big\rangle_{L^2(Y)}\geq \nu^2_0\|f\|^2_{L^2(Y)}.$$ Let $\nu^2$ be one eigenvalue of the operator $\Delta_h+V_0(\theta)+(n-2)^2/4$ such that $$(\Delta_h+V_0(\theta)+(n-2)^2/4) Y(\theta)=\nu^2 Y(\theta)$$ where $Y(\theta)$ is an eigenfunction. Since $Y$ is a closed manifold, from the spectral theory, it is known that $\nu^2$ falls in a discrete set, say $\{\nu_j^2\}_{j=0}^\infty$, and moreover $\nu^2_0<\nu^2_1<\cdots<\nu^2_j<\cdots \to \infty$. Let $d(\nu_j)$ be the multiplicity of $\nu_j^2$ and let $\{Y_{\nu_j,\ell}(\theta)\}_{1\leq\ell\leq d(\nu_j)}$ be the corresponding eigenfunctions of $\Delta_h+V_0(\theta)+(n-2)^2/4$, that is $$\label{equ:eig}
\begin{split}
(\Delta_h+V_0(\theta)+(n-2)^2/4)Y_{\nu_j,\ell}(\theta)=\nu_j^2 Y_{\nu_j,\ell}(\theta),\\ \langle Y_{\nu_j,\ell},Y_{\nu_{j'},\ell'}\rangle_{L^2(Y)}=\delta_{j,j'}\delta_{\ell,\ell'}.
\end{split}$$ where $\delta$ is the Kronecker delta function. In particular, when $Y=\mathbb{S}^{n-1}$ and $V_0=0$, $Y_{\nu_j,\ell}$ is spherical harmonics. Define $$\label{Lam}
\Lambda_\infty=\big\{\nu_j>0: \nu_j^2\, \text{is the eigenvalue of }\, \Delta_h+V_0(\theta)+(n-2)^2/4\big\}_{j=0}^\infty.$$ From now on, we drop the superscripts in $\nu_j$ for simple. Define $$\mathcal{H}^\nu=\text{span}\{Y_{\nu,1},\cdots,Y_{\nu,d(\nu)}\},$$ then we have the orthogonal decomposition $$L^2(Y)=\bigoplus_{\nu\in\Lambda_\infty}\mathcal{H}^\nu.$$ Let $\pi_\nu$ denote the orthogonal projection: $$\pi_\nu f=\sum_{\ell=1}^{d(\nu)}Y_{\nu,\ell}(\theta)\int_Yf(r,\omega)Y_{\nu,\ell}(\omega)\mathrm{d}\sigma_h,\quad f\in L^2(X),$$ where $\mathrm{d}\sigma_h$ is the measure on $Y$ under the metric $h$. For any $g\in L^2(X),$ we have the expansion formula $$\label{2.5}
g(r,\theta)=\sum_{\nu\in\Lambda_\infty}\pi_\nu g=\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}a_{\nu,\ell}(r)Y_{\nu,\ell}(\theta)$$ where $a_{\nu,\ell}(r)=\int_Yg(r,\theta)Y_{\nu,\ell}(\theta)\mathrm{d}\sigma_h.$ By orthogonality, it gives $$\label{orth}
\|g(r,\theta)\|^2_{L^2(Y)}=\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|a_{\nu,\ell}(r)|^2.$$
The Bessel Function and Hankel Transform
----------------------------------------
For our purpose, we recall the Bessel function $J_\nu(r)$ of order $\nu$, which is defined by $$J_\nu(r)=\frac{(r/2)^\nu}{\Gamma(\nu+\frac{1}{2})\Gamma(1/2)}\int_{-1}^1e^{isr}(1-s^2)^{(2\nu-1)/2}\mathrm{d}s,$$ where $\nu>-\frac{1}{2}$ and $r>0.$ A simple computation gives the rough estimate $$\label{est:r}
|J_\nu(r)|\leq\frac{Cr^\nu}{2^\nu\Gamma(\nu+\frac{1}{2})\Gamma(\frac12)}\Big(1+\frac{1}{\nu+\frac12}\Big),$$ where $C$ is an absolute constant independent of $r$ and $\nu$. To investigate the behavior of asymptotic on $\nu$ and $r$, we recall Schläfli’s integral representation [@Wolff] of the Bessel function: for $r\in{\mathbb{R}}^+$ and $\nu>-\tfrac12$ $$\begin{aligned}
\label{SIR}
J_\nu(r)&=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ir\sin\theta-i\nu\theta}\mathrm{d}\theta-\frac{\sin(\nu\pi)}{\pi}\int_0^\infty e^{-(r\sinh s+\nu s)}\mathrm{d}s \nonumber \\
&:=\tilde{J}_\nu(r)-E_\nu(r).\end{aligned}$$ We remark that $E_\nu(r)=0$ when $\nu\in\mathbb{Z}^+.$ A simple computation gives that for $r>0$ $$\label{est:E}
|E_\nu(r)|=\Big|\frac{\sin(\nu\pi)}{\pi}\int_0^\infty e^{-(r\sinh s+\nu s)}\mathrm{d}s\Big|\leq C(r+\nu)^{-1}.$$ Next, we recall the properties of Bessel function $J_\nu(r)$ in [@SW; @Watson].
\[lem:bessel\] Assume $\nu\gg1.$ Let $J_\nu(r)$ be the Bessel function of order $\nu$ defined as above. Then there exist a large constant $C$ and a small constant $c$ independent of $\nu$ and $r$ such that:
$\bullet$ when $r\leq\frac{\nu}{2}$ $$\label{2.12}
|J_\nu(r)|\leq Ce^{-c(\nu+r)};$$
$\bullet$ when $\frac{\nu}{2}\leq r\leq2\nu$ $$\label{2.13}
|J_\nu(r)|\leq C\nu^{-\frac{1}{3}}(\nu^{-\frac{1}{3}}|r-\nu|+1)^{-\frac{1}{4}};$$
$\bullet$ when $r\geq2\nu$ $$\label{2.14}
|J_\nu(r)|=r^{-\frac{1}{2}}\sum_{\pm}a_\pm(r,\nu)e^{\pm ir}+E(r,\nu),$$ where $|a_\pm(r,\nu)|\leq C$ and $|E(r,\nu)|\leq Cr^{-1}.$
Let $f\in L^2(X),$ we define the Hankel transform of order $\nu$ by $$\label{Hankel}
(\mathcal{H}_\nu f)(\rho,\theta)=\int_0^\infty(r\rho)^{-\frac{n-2}{2}}J_\nu(r\rho)f(r,\theta)r^{n-1}\mathrm{d}r.$$ In particular, if the function $f$ is independent of $\theta$, then $$\label{Hankel-r}
(\mathcal{H}_{\nu}f)(\rho)=\int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu}(r\rho)f(r)r^{n-1}\mathrm{d}r.$$ We have the following properties of the Hankel transform. We refer the readers to M.Taylor [@Taylor Chapter 9], also see [@BPSS; @PSS].
\[lem:hankel\] Let $\mathcal{H}_\nu$ be the Hankel transform in and $$\label{LV-nu}
A_\nu:=-\partial^2_r-\frac{n-1}{r}\partial_r+\frac{\nu^2-(\frac{n-2}{2})^2}{r^2}.$$ Then
$\mathrm{(1)}$ $\mathcal{H}_\nu=\mathcal{H}^{-1}_\nu,$
$\mathrm{(2)}$ $\mathcal{H}_\nu$ is self-adjoint, i.e.$\quad \mathcal{H}_\nu=\mathcal{H}^{*}_\nu$,
$\mathrm{(3)}$ $\mathcal{H}_\nu$ is an $L^2$ isometry, i.e. $\|\mathcal{H}_\nu f\|_{L^2(X)}=\|f\|_{L^2(X)},$
$\mathrm{(4)}$ $\mathcal{H}_\nu(A_\nu f)(\rho,\theta)=\rho^2(\mathcal{H}_\nu f)(\rho,\theta),$ for $f\in L^2.$
Distorted plan wave and distorted Fourier transform
---------------------------------------------------
In this subsection, we derive the plan wave associated with the operator ${\mathcal{L}}_V$. To this end, we need to find the eigenfunction $\phi(r,\theta;\rho,\omega)$ such that $$\label{equ:eigen}
{\mathcal{L}}_V\phi(r,\theta;\rho,\omega)=\rho^2\phi(r,\theta;\rho,\omega).$$ We claim that $$\label{plan-w}
\phi(r,\theta;\rho,\omega)=(r\rho)^{-\frac{n-2}2}\sum_{\nu\in\Lambda_\infty} J_{\nu}(r\rho)\sum_{\ell=1}^{d(\nu)}Y_{\nu,\ell}(\theta)\overline{Y_{\nu,\ell}(\omega)}$$ where $J_\nu$ is the Bessel function of order $\nu$ and $Y_{\nu,\ell}$ satisfies . To verify this claim, we write ${\mathcal{L}}_V$ in the coordinates $(r,\theta)$ as $$\label{LV-r}
{\mathcal{L}}_V=-\partial^2_r-\frac{n-1}{r}\partial_r+\frac{1}{r^2}(\Delta_h+V_0(\theta)),$$ if it acts on the function in each $\mathcal{H}^\nu$, then it equals to $A_\nu$ as in $$A_\nu:=-\partial^2_r-\frac{n-1}{r}\partial_r+\frac{\nu^2-(\frac{n-2}{2})^2}{r^2}.$$ Therefore it suffices to verify that: for each $\nu$, let $F(r\rho)=(r\rho)^{-\frac{n-2}2}J_{\nu}(r\rho)$, one has $$\label{bess}
\rho^2 F''(r\rho)+\frac{(n-1)\rho}r F'(r\rho)+\Big[\rho^2-\frac{\nu^2-(n-2)^2/4}{r^2}\Big]F(r\rho)=0.$$ Indeed, the Bessel function $J_\nu(\lambda)$ solves $$\label{bessfunc}
G''(\lambda)+\frac{1}\lambda
G'(\lambda)+\Big[1-\frac{\nu^2}{\lambda^2}\Big]G(\lambda)=0,$$ let $\lambda=r\rho$, then $F(\lambda)$ satisfies $$F''(\lambda)+\frac{n-1}\lambda F'(\lambda)+\Big[1-\frac{\nu^2-(n-2)^2/4}{\lambda^2}\Big]F(\lambda)=0$$ which implies . For $$f(r,\theta)=\sum\limits_{\nu\in\Lambda_\infty}\sum\limits_{\ell=1}^{d(\nu)}a_{\nu,\ell}(r)Y_{\nu,\ell}(\theta)\in L^2(X),$$ we define the distorted Fourier transform $$\label{Fourier}
\begin{split}
\hat f(\rho,\omega)&=\int_0^\infty\int_Y f(r,\theta)\overline{\phi(r,\theta;\rho,\omega)} \,r^{n-1}\mathrm{d}r \mathrm{d}h\\&=\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
Y_{\nu,\ell}(\omega)\big(\mathcal{H}_{\nu}a_{\nu,\ell}\big)(\rho).
\end{split}$$
The representation of solution
------------------------------
Based on the above Hankel transform, we write out the explicit expression of solution for wave equation . Recall in coordinates $(r,\theta)$, then the solution $u(t,r,\theta)$ satisfies that $$\label{equ:vt}
\begin{cases}
\pa_{tt}u-\pa_{rr}u-\frac{n-1}{r}\pa_ru+\frac1{r^2}\Delta_hu+\frac{V_0(\theta)}{r^2}u=0,\\
u(0,r,\theta)=0, \quad \partial_tu(0,r,\theta)=f(r,\theta).
\end{cases}$$ We write Schwartz function $f(r,\theta)$ as $$\label{in-f}
f(r,\theta)=\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}a_{\nu,\ell}(r)Y_{\nu,\ell}(\theta).$$ With separation of variables, then we can write $u$ as a superposition $$\label{3.3}
u(t,r,\theta)=\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}u_{\nu,\ell}(t,r)Y_{\nu,\ell}(\theta),$$ where $u_{\nu,\ell}$ satisfies $$\label{equ:vkl}
\begin{cases}
\pa_{tt}u_{\nu,\ell}-\pa_{rr}u_{\nu,\ell}-\frac{n-1}r\pa_ru_{\nu,\ell}+\frac{\nu^2-(\frac{n-2}{2})^2}{r^2}u_{\nu,\ell}=0,\\
u_{\nu,\ell}(0,r)=0,\quad \partial_tu_{\nu,\ell}(0,r)=a_{\nu,\ell}(r)
\end{cases}$$ for each $\nu\in\Lambda_\infty,\ell\in\mathbb{N}$ and $1\leq\ell\leq d(\nu)$. Recall $A_\nu$ in , we consider $$\label{equ:vkln}
\begin{cases}
\pa_{tt}u_{\nu,\ell}+A_{\nu}u_{\nu,\ell}=0,\\
u_{\nu,\ell}(0,r)=0,\quad \partial_tu_{\nu,\ell}(0,r)=a_{\nu,\ell}(r).
\end{cases}$$ Applying the Hankel transform to , by Lemma \[lem:hankel\], we have $$\label{equ:vkln1}
\begin{cases}
\pa_{tt}\tilde{u}_{\nu,\ell}+\rho^2\tilde{u}_{\nu,\ell}=0,\\
\tilde{u}_{\nu,\ell}(0,\rho)=0,\quad \partial_t\tilde{u}_{\nu,\ell}(0,\rho)=b_{\nu,\ell}(\rho),
\end{cases}$$ where $$\label{3.8}
\tilde{u}_{\nu,\ell}(t,\rho)=(\mathcal{H}_{\nu}
u_{\nu,\ell})(t,\rho),\quad
b_{\nu,\ell}(\rho)=(\mathcal{H}_{\nu}a_{\nu,\ell})(\rho).$$ By solving this ODE and using the Hankel transform, we obtain $$u_{\nu,\ell}(t,r)=\mathcal{H}_{\nu}[\rho^{-1}\sin (t\rho) \, b_{\nu,\ell}(\rho)](r)$$ Therefore, by and the definition of Hankel transform, we get $$\begin{aligned}
\label{sol}
u(t,r,\theta)&=\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\mathcal{H}_{\nu}\big[\rho^{-1}\sin (t\rho) b_{\nu,\ell}(\rho)\big](r)Y_{\nu,\ell}(\theta)\nonumber\\
&=\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}Y_{\nu,\ell}(\theta) \int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu}(r\rho)\sin (t\rho) b_{\nu,\ell}(\rho)\rho^{n-2}\;d\rho .\end{aligned}$$
We finally record the Van der Corput lemma for convenience.
\[lem:VdC\] Let $\phi$ be a smooth real-valued function defined on an interval $[a,b]$ and let $|\phi^{(k)}(x)|\geq1$ for all $x\in[a,b].$ Then $$\label{VdC}
|\int_a^be^{i\lambda\phi(x)}\mathrm{d}x|\leq c_k\lambda^{-\frac{1}{k}}$$ holds when:
$\bullet\quad k\geq2$ or
$\bullet\quad k=1$ and $\phi'(x)$ is monotonic.
The bound $c_k$ is independent of $\phi$ and $\lambda.$
Notation
--------
We use $A\lesssim B$ to denote the statement that $A\leq CB$ for some large constant $C$ which may vary from line to line and depend on various parameters, and similarly, we employ $A\backsim B$ to state that $A\lesssim B\lesssim A.$ We also use $A\ll B$ to denote the statement $A\leq C^{-1}B.$ If a constant $C$ depends on a special parameter other than the above, we shall denote it explicitly by subscripts. For instance, $C_\epsilon$ should be understood as a positive constant not only depending on $p,q,n$ and $S,$ but also on $\epsilon.$ Throughout this paper, pairs of conjugate indices are written as $p,p',$ where $\frac{1}{p}+\frac{1}{p'}=1$ with $1\leq p\leq\infty.$ Let $R>0$ be two dyadic numbers, we define $S_R=[R/2,R]$.
Localized estimates of Hankel transform
=======================================
In this section, we utilize the stationary-phase argument to prove the estimates for Hankel transform localized both in frequency and physical spaces. These inequalities are key to prove main theorem in next section.
\[LRE\]
Let $\varphi\in\mathcal{C}_c^\infty({\mathbb{R}})$ be supported in $I:=[1,2]$ and let $R>0$ be a dyadic number and $S_R=[R/2,R]$. Then $$\begin{aligned}
\label{L-2}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\mathcal{H}_{\nu}\big[\rho^{-1}e^{\pm it\rho}\varphi(\rho) b_{\nu,\ell}(\rho)\big](r)\Big|^2\Big)^{1/2}\Big\|_{L^2_tL^2_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)}\nonumber\\
\lesssim& \min\{R^{\nu_0+1},R^{\frac{1}{2}}\}
\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho) \varphi(\rho)|^2\Big)^{1/2}\Big\|_{L^2_\rho(I)},\end{aligned}$$ and $$\begin{aligned}
\label{L-infty1}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\mathcal{H}_{\nu}\big[\rho^{-1}e^{\pm it\rho}\varphi(\rho) b_{\nu,\ell}(\rho)\big](r)\Big|^2\Big)^{1/2}\Big\|_{L^\infty_tL^\infty_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)} \nonumber\\
\lesssim& \min\{R^{\nu_0-\frac{n-2}2},R^{-\frac{n-2}{2}-\frac13}\}
\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^1_\rho(I)},\end{aligned}$$ and $$\begin{aligned}
\label{L-infty}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\mathcal{H}_{\nu}\big[\rho^{-1}e^{\pm it\rho}\varphi(\rho) b_{\nu,\ell}(\rho)\big](r)\Big|^2\Big)^{1/2}\Big\|_{L^\infty_tL^\infty_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)} \nonumber\\
\lesssim& \min\{R^{\nu_0-\frac{n-2}2},R^{-\frac{n-1}{2}}\}
\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^2_\rho(I)}.\end{aligned}$$
To prove this proposition, we divide into two cases $R\lesssim 1$ and $R\gg1$. For $R\lesssim 1$, it suffices to prove, for $q\geq2$ $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty r^{-\frac{n-2}{2}}e^{\pm it\rho}
J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|
_{L^q_tL^q_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)} \nonumber \\
\lesssim& R^{\frac{n}{q}+\nu_0-\frac{n-2}2}
\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho) \varphi(\rho)|^2\Big)^{1/2}\Big\|_{L^2_\rho(I)}.\end{aligned}$$ To this end, since $q\geq2,$ we use the Minkowski inequality and the Hausdorff-Young inequality in $t$ variable to obtain $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty r^{-\frac{n-2}{2}}e^{\pm it\rho}
J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|
_{L^q_tL^q_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)} \nonumber \\
\lesssim&
\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|r^{-\frac{n-2}{2}}J_{\nu}(r\rho)
b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\Big\|^2_{L^{q'}_\rho(I)}\Big)^{1/2}\Big\|_{L^q_{r^{n-1}dr}(S_R)}.\end{aligned}$$ Recall the rough estimate for Bessel function $$|J_\nu(r)|\leq\frac{Cr^\nu}{2^\nu\Gamma(\nu+\frac{1}{2})\Gamma(\frac12)}\Big(1+\frac{1}{\nu+\frac12}\Big),$$ then, by using Stirling’s formula $\Gamma(\nu+1)\sim\sqrt{\nu}(\nu/e)^\nu$, we obtain $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|r^{-\frac{n-2}{2}}J_{\nu}(r\rho)
b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\Big\|^2_{L^{q'}_\rho(I)}\Big)^{1/2}\Big\|_{L^q_{r^{n-1}dr}(S_R)} \nonumber \\
\lesssim&
R^{\frac{n}{q}+\nu_0-\frac{n-2}2}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}
\varphi(\rho)\Big\|_{L^{q'}_\rho(I)},\end{aligned}$$ where we have used Minkowski’s inequality again and $\rho\in I=[1,2]$. Therefore, by choosing $q=2$ and $q=\infty$ respectively, we have proved , and when $R\lesssim 1$.
Next we consider the case $R\gg1$. We first prove . By using the same argument as above (the Minkowski inequality and the Hausdorff-Young inequality in $t$), we have $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty r^{-\frac{n-2}{2}}e^{\pm it\rho}
J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|
_{L^2_tL^2_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)} \nonumber \\
\lesssim&
R^{1/2}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|J_{\nu}(r\rho)
b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\Big\|^2_{L^{2}_\rho(I)}\Big)^{1/2}\Big\|_{L^2_{dr}(S_R)}.\end{aligned}$$ By using Lemma \[lem:bessel\], we can prove $$\int_R^{2R}|J_\nu(r)|^2 \mathrm{d}r\leq C, \quad R\gg1,$$ where the constant $C$ is independent of $R$ and $\nu$. We refer to [@Zhang (3.21)] for details. Thus we prove for $R\gg1$.
Next we prove which is a consequence of $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty J_{\nu}(r\rho)e^{-it\rho}b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho
\Big|^2\Big)^{1/2}\Big\|_{L^\infty_tL^\infty_r({\mathbb{R}}\times S_R)} \nonumber \\
\lesssim&
R^{-1/3}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^1_\rho(I)}.\end{aligned}$$ This is easily proved by using the Minkowski inequality and the Hausdorff-Young inequality in $t$ variable as before due to the uniform estimate $$|J_{\nu}(r)|\leq C r^{-1/3}\qquad r\gg 1,$$ which is implied by Lemma \[lem:bessel\].
Now we prove which will be implied by $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty J_{\nu}(r\rho)e^{-it\rho}b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho
\Big|^2\Big)^{1/2}\Big\|_{L^\infty_tL^\infty_r({\mathbb{R}}\times S_R)} \nonumber \\
\lesssim&
R^{-1/2}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^2_\rho(I)}.\end{aligned}$$
Recall Schläfli’s integral representation , we write $J_{\nu}(r\rho)$ as $$\begin{aligned}
\label{SIR'}
J_{\nu}(r\rho)&=\frac{1}{2\pi}\int_{-\pi}^\pi e^{ir\rho\sin\theta-i\nu\theta}
\mathrm{d}\theta-\frac{\sin(\nu\pi)}{\pi}\int_0^\infty e^{-(r\sinh s+\nu s)}
\mathrm{d}s \nonumber\\
&=\widetilde{J}_{\nu}(r\rho)-E_{\nu}(r\rho),\end{aligned}$$ where $$\label{est:E'}
|E_{\nu}(r\rho)|\leq C(r\rho)^{-1}$$ with $C$ being independent of $k$ and $R.$ There, by the same argument as before and , we easily get $$\begin{aligned}
\label{est:E-term}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty e^{-it\rho}E_{\nu}(r\rho)b_{\nu,\ell}(\rho)
\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|_{L^\infty_t({\mathbb{R}};L^\infty_r(R/2,R))}\nonumber\\
\lesssim& R^{-1}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)\varphi(\rho)|^2\Big)^{1/2}\Big\|_{L^1_\rho(I)}.\end{aligned}$$
It thus only remains to prove $$\begin{aligned}
\label{re:J}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty e^{-it\rho}\widetilde{J}_{\nu}(r\rho)
b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho\Big|^2\Big)^{1/2}
\Big\|_{L^\infty_t({\mathbb{R}};L^\infty_r([R/2,R]))}\nonumber\\
\lesssim& R^{-\frac{1}{2}}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)\varphi(\rho)|^2\Big)^{1/2}\Big\|_{L^2_\rho(I)}.\end{aligned}$$ To use the stationary phase argument, we decompose $[-\pi,\pi]$ into three intervals $$\begin{aligned}
\label{decom}
[-\pi,\pi]=I_1\cup I_3\cup I_2\nonumber\end{aligned}$$ where $$\label{decom'}
\begin{split}
I_1&=[-\delta,\delta],\\ I_2&=[-\pi, -\frac{\pi}{2}-\delta]
\cup[\frac{\pi}{2}+\delta,\pi], \\ I_3&=[-\frac{\pi}{2}-\delta,-\delta]
\cup[\delta,\frac{\pi}{2}+\delta]
\end{split}$$ with $0<\delta\ll1$ being fixed later. Let $$\Phi_{r,\nu}(\theta)=\sin\theta-\frac{\nu}{r}\theta$$ and a simple computation yields $$\Phi'_{r,\nu}(\theta)=\cos\theta-\frac{\nu}{r}, \quad \Phi''_{r,\nu}(\theta)=-\sin\theta.$$ Construct a smooth function $\Lambda_\delta(\theta)$ which is defined by $$\begin{aligned}
\label{4.12}
\Lambda_\delta(\theta)=
\begin{cases}
&1, \quad \theta\in I_1,\\
&0, \quad \theta\notin2I_1.
\end{cases}\end{aligned}$$ Therefore, based on , we write $\widetilde{J}_\nu(r)$ as $$\begin{aligned}
\widetilde{J}_\nu(r)&=\widetilde{J}_\nu^1(r)+\widetilde{J}_\nu^2(r)+\widetilde{J}_\nu^3(r).\end{aligned}$$ where $$\begin{split}
\widetilde{J}_\nu^1(r)
&=\frac{1}{2\pi}\int_{-\pi}^\pi e^{ir\Phi_{r,\nu}(\theta)}\Lambda_\delta(\theta) \mathrm{d}\theta,\\
\widetilde{J}_\nu^2(r)&=\frac{1}{2\pi}\int_{I_2} e^{ir\Phi_{r,\nu}(\theta)}
\mathrm{d}\theta,\\
\widetilde{J}_\nu^3(r)&=
\frac{1}{2\pi}\int_{I_3}e^{ir\Phi_{r,\nu}(\theta)}(1-\Lambda_\delta(\theta))\mathrm{d}\theta.
\end{split}$$ For $\theta\in I_2,$ thus $|\Phi'_{r,\nu}(\theta)|=|\cos\theta-\frac{\nu}{r}|\geq\sin\delta$; When $\theta\in I_3$, one has $|\Phi''_{r,\nu}(\theta)|\geq\sin\delta$. By using Van der Corput lemma \[lem:VdC\], we have $$\label{4.14}
|\widetilde{J}^2_\nu(r)|
\leq C_\delta r^{-1},\quad |\widetilde{J}^3_\nu(r)|
\leq C_\delta r^{-1/2}.$$ Hence, applying the Hausdorff-Young inequality and Minkowski’s inequality, we get $$\begin{aligned}
\label{4.15}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty e^{-it\rho}\big(\widetilde{J}_{\nu}^2(r\rho)+|\widetilde{J}^3_\nu(r\rho)\big)b_{\nu,\ell}(\rho)
\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|_{L^\infty_t({\mathbb{R}};L^\infty_r(R/2,R))}\nonumber\\
\lesssim& R^{-1/2}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^1_\rho(I)}.\end{aligned}$$ So proving is reduced to prove $$\begin{aligned}
\label{4.18}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty e^{-it\rho}\widetilde{J}_{\nu}^1(r\rho)b_{\nu,\ell}(\rho)
\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|_{L^\infty_t({\mathbb{R}};L^\infty_r(R/2,R))}\nonumber\\
\lesssim& R^{-1/2}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^2_\rho(I)}.\end{aligned}$$ For our purpose, we write the Fourier series of $b_{\nu,\ell}(\rho)$ as $$\label{4.19}
b_{\nu,\ell}(\rho)=\sum_jb_{\nu,l}^je^{i\frac{\pi}{2}\rho j}, \quad
b_{\nu,\ell}^j=\frac{1}{4}\int_0^4b_{\nu,l}(\rho)e^{-i\frac{\pi}{2}\rho j} \mathrm{d}\rho.$$
Therefore we have $$\label{4.24}
\|b_{\nu,\ell}(\rho)\|^2_{L^2_\rho(I)}=\sum_j|b_{\nu,\ell}^j|^2$$ and $$\begin{aligned}
\label{4.25}
&\int_0^\infty e^{-it\rho}\widetilde{J}_{\nu}^1(r\rho)
b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho\nonumber\\
=&\frac{1}{2\pi}\int_0^\infty e^{-it\rho}\int_{-\pi}^{\pi}e^{ir\rho\sin\theta-i\nu\theta}
\Lambda_\delta(\theta)\sum_jb_{\nu,\ell}^je^{i\frac{\pi}{2}\rho j}\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho\mathrm{d}\theta\nonumber\\
\lesssim&\sum_jb_{\nu,\ell}^j\int_{{\mathbb{R}}^2}e^{2\pi i\rho(r\sin\theta-(t-\frac{j}{4}))}\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho e^{-i\nu\theta}\Lambda_\delta(\theta)\mathrm{d}\theta.\end{aligned}$$ Then we estimate the term in . Let $m=t-\frac{j}{4}$, we write $$\begin{aligned}
\label{4.26}
\psi_m^\nu(r)
=&\int_{{\mathbb{R}}^2}e^{2\pi i\rho(r\sin\theta-m)}\varphi(\rho)\rho^{\frac{n-2}2}\mathrm{d}\rho e^{-i\nu\theta}\Lambda_\delta(\theta)\mathrm{d}\theta\nonumber\\
=&\int_{{\mathbb{R}}}\check{\varphi}(r\sin\theta-m) e^{-i\nu\theta}\Lambda_\delta(\theta)\mathrm{d}\theta.\end{aligned}$$ It is apparent $\check{\varphi}$ is a Schwartz function, so we have for any $N>0$ $$\label{4.27}
|\check{\varphi}(r\sin\theta-m)|\leq C_N(1+|r\sin\theta-m|)^{-N}.$$ We consider two cases to study the property of function $\psi_m^\nu(r)$.
[**Case 1: $|m|\geq4R.$**]{} Since $r\leq2R\leq|m|$ and $|\theta|\leq2\delta,$ we have $$\label{4.28}
|r\sin\theta-m|\geq|m|-r|\sin\theta|\geq\frac{1}{100}|m|$$ and thus $$\label{4.29}
|\psi_m^\nu(r)|\leq C_{\delta,N}(1+|m|)^{-N}.$$ Using this inequality, we control by $$\label{4.30}
C_{\delta,N}R^{-N}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\sum_{j:4R\leq|t-\frac{j}{4}|}
b_{\nu,\ell}^j\Big(1+\Big|t-\frac{j}{4}\Big|\Big)^{-N}
\Big|^2\Big)^{1/2}\Big\|_{L^\infty_t({\mathbb{R}};L^\infty_r(R/2,R))}.$$ Applying Cauchy-Schwartz’s inequality to the above inequality and then choosing $N$ large enough, we bound by $$\label{4.31}
C_{\delta,N}R^{-N}\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\sum_j|b_{\nu,\ell}^j|^2\Big)^{1/2}
\lesssim R^{-N}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|b_{\nu,\ell}(\rho)\Big|^2\Big)^{1/2}\varphi(\rho)
\Big\|_{L^2_\rho(I)},$$ where we have used and $\text{supp}\varphi\subset I=[1,2].$
[**Case 2: $|m|<4R.$**]{} We get based on and $$\begin{aligned}
|\psi_m^\nu(r)|\leq\frac{C_N}{2\pi}&\Big(\int_{\{\theta: |\theta|<2\delta,|r\sin\theta-m|\leq1\}}\mathrm{d}\theta
\\\nonumber&\quad+\int_{\{\theta: |\theta|<2\delta,|r\sin\theta-m|\geq1\}}
(1+|r\sin\theta-m|)^{-N}\mathrm{d}\theta\Big).\end{aligned}$$ Making the variable change $y=r\sin\theta-m$, we further have $$\begin{aligned}
\label{4.33}
|\psi_m^\nu(r)|\leq\frac{C_N}{2\pi r}\Big(\int_{\{y:|y|\leq1\}}\mathrm{d}y
+\int_{\{y:|y|\geq1\}}(1+|y|)^{-N}\mathrm{d}y\Big)\lesssim r^{-1}.\end{aligned}$$
We put the set $A=\{j\in\mathbb{Z}:|t-\frac{j}{4}|<4R\}$ for fixed $t$ and $R$. Obviously, the cardinality of $A$ is $O(R)$. Then, from and , we obtain $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\sum_{j\in A}b^j_{\nu,\ell}\psi^\nu_m(r)\Big|^2\Big)^{1/2}\Big\|_
{L^\infty_tL^\infty_r({\mathbb{R}}\times S_R)} \\
\leq& C_{\delta,N}R^{-\frac{1}{2}}\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\sum_j|b^j_{\nu,\ell}|^2\Big)^{1/2}\\
=&C_{\delta,N}R^{-\frac{1}{2}}\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\|b_{\nu,\ell}(\rho)\|^2_{L^2_\rho}\Big)^{1/2}\\
\lesssim& R^{-\frac{1}{2}}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\Big\|_{L^2_\rho(I)}.\end{aligned}$$
As a consequence of the interpolation and Proposition \[LRE\], we obtain
\[LRE’\] For $q\geq2$, we have $$\begin{aligned}
\nonumber
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty r^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{-it\rho}b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|_
{L^q_tL^q_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)} \\\label{LRE'1}
\lesssim&
\min\{R^{\frac{n}{q}+\nu_0-\frac{n-2}2},R^{-\frac{3n-4}{6}[1-\frac{2(3n-1)}{(3n-4)q}]}\}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}
\sum_{\ell=1}^{d(\nu)}|
b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^{q'}_\rho(I)},\end{aligned}$$ and $$\begin{aligned}
\nonumber
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty r^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{-it\rho}b_{\nu,\ell}(\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho\Big|^2\Big)^{1/2}\Big\|_
{L^q_tL^q_{r^{n-1}\mathrm{d}r}({\mathbb{R}}\times S_R)} \\\label{LRE'2}
\lesssim&
\min\{R^{\frac{n}{q}+\nu_0-\frac{n-2}2},R^{-\frac{n-1}{2}[1-\frac{2n}{(n-1)q}]}\}\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|
b_{\nu,\ell}(\rho)|^2\Big)^{1/2}\varphi(\rho)\Big\|_{L^2_\rho(I)}.\end{aligned}$$
Proof of Theorem \[thm:main\]
=============================
In this section, we prove Theorem \[thm:main\] based on the estimates of Hankel transform in Proposition \[LRE’\]. In the end of this section, we construct an counterexample to show the necessity of .
From , it suffices to estimate $$\begin{aligned}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\\\nonumber
\lesssim& \sum_{\pm} \Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
\Big|\int_0^\infty r^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{\pm it\rho}
b_{\nu,\ell}(\rho)\rho^{\frac{n-2}{2}}\mathrm{d}\rho\Big|^2\Big)^\frac12\Big\|_{L^q_tL^q_{r^{n-1}\;dr}({\mathbb{R}}\times{\mathbb{R}}_+)}\end{aligned}$$ By the symmetry of $t$, we only need to consider one of signs $\pm$. We only consider the minus sign and apply dyadic decompositions to obtain $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
\Big|\int_0^\infty r^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{-it\rho}
b_{\nu,\ell}(\rho)\rho^{\frac{n-2}{2}} \mathrm{d}\rho\Big|^2\Big)^\frac12\Big\|_{L^q_tL^q_{r^{n-1}\;dr}({\mathbb{R}}\times{\mathbb{R}}_+)}
\\\nonumber
\lesssim& \Big(\sum_R\Big(\sum_M\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty r^{-\frac{n-2}{2}}J_{\nu}(r\rho) e^{-it\rho}
b_{\nu,\ell}(\rho)\varphi(\frac{\rho}{M})\rho^{\frac{n-2}{2}}\;d\rho
\Big|^2\Big)^\frac12\Big\|_{L^q_tL^q_{r^{n-1}\;dr}({\mathbb{R}}\times S_R)}\Big)^q\Big)^{1/q}\end{aligned}$$ where both $R$ and $M$ are dyadic numbers, $\varphi\in C^\infty_c([1,2])$ values in $[0,1]$ such that $\sum\limits_{M\in2^{{\mathbb{Z}}}}\varphi(\rho/M)=1$. Furthermore, by scaling argument, we obtain $$\begin{aligned}
\label{3.12}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\\\nonumber
\lesssim& \Big(\sum_R\Big(\sum_MM^{(n-1)-\frac{n+1}{q}}\Big\|
\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big|\int_0^\infty r^{-\frac{n-2}{2}}J_{\nu}(r\rho)\\\nonumber
&\qquad\qquad \times e^{-it\rho}
b_{\nu,\ell}(M\rho)\varphi(\rho) \rho^{\frac{n-2}{2}}\;d\rho\Big|^2\Big)^\frac12
\Big\|_{L^q_tL^q_{r^{n-1}\;dr}({\mathbb{R}}\times S_{MR})}\Big)^q\Big)^{1/q}.\end{aligned}$$
For our purpose, we divide into two cases.
$\bullet$ [**Case1:**]{} $1\leq p\leq2$. By , we obtain $$\begin{aligned}
\label{3.13}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\\\nonumber
\lesssim& \Big(\sum_R\Big(\sum_MM^{(n-1)-\frac{n+1}{q}}
\min\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},(RM)^{-\frac{3n-4}{6}[1-\frac{2(3n-1)}{(3n-4)q}]}\}
\\\nonumber
&\qquad\qquad\times \Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(M\rho)|^2\Big)^\frac12\varphi(\rho)\Big\|_{L^{p}_\rho(I)}
\Big)^q\Big)^{1/q}\end{aligned}$$ where we use the fact that $$\frac{n+1}q=\frac{n-1}{p'}\Leftrightarrow \frac1{q'}= \frac{1}p+\frac{2}{p'(n+1)}$$ implies $p\geq q'$; On the other hand, since $1\leq p\leq 2$ and $n\geq2$, one has $$\label{p<2}
q\geq \frac{2(n+1)}{n-1}>\frac{2(3n-1)}{3n-4}.$$
If $\nu_0\geq \tfrac{n-2}2$, the fact is enough to guarantee $$\label{S-cond1}
\sup_{R>0}\sum_{M}\min\big\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},
(RM)^{-\frac{3n-4}{6}[1-\frac{2(3n-1)}{(3n-4)q}]}\big\}<\infty,$$ and $$\label{S-cond2}
\sup_{M>0}\sum_{R}\min\big\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},
(RM)^{-\frac{3n-4}{6}[1-\frac{2(3n-1)}{(3n-4)q}]}\big\}<\infty.$$ However, if $0<\nu_0\leq \tfrac{n-2}2$, we need to ensure and to be true. Therefore, by Schur test’s lemma, we show $$\begin{aligned}
\label{3.14}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\nonumber\\
\lesssim& \Big(\sum_M\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
\Big|b_{\nu,\ell}(\rho)\Big|^2\Big)^\frac12\varphi(\frac{\rho}{M})\rho^{\frac{n-2}{p}}
\Big\|^p_{L^p_\rho}\Big)^{1/p}\nonumber\\
\lesssim&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
|b_{\nu,\ell}(\rho)|^2\Big)^\frac12\rho^{-\frac1p}\Big\|_{L^p_{\rho^{n-1}\;d\rho}({\mathbb{R}}_+)}.\end{aligned}$$
$\bullet$ [**Case 2:**]{} $p\geq2$. By , we have $$\begin{aligned}
\label{3.13}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\\\nonumber
\lesssim& \Big(\sum_R\Big(\sum_MM^{(n-1)-\frac{n+1}{q}}
\min\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},(RM)^{-\frac{n-1}{2}[1-\frac{2n}{(n-1)q}]}\}\\\nonumber
&\qquad\qquad\times \Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(M\rho)|^2\Big)^\frac12\varphi(\rho)\Big\|_{L^2_\rho(I)}\Big)^q\Big)^{1/q}.\end{aligned}$$
By noting that $\frac{n+1}q=\frac{n-1}{p'}$ and $p\geq 2$ and using scaling, we have $$\begin{aligned}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\\\nonumber
\lesssim& \Big(\sum_R\Big(\sum_MM^{(n-1)-\frac{n+1}{q}}
\min\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},(RM)^{-\frac{n-1}{2}[1-\frac{2n}{(n-1)q}]}\}\\\nonumber
&\qquad\qquad\times \Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(M\rho)|^2\Big)^\frac12\varphi(\rho)\rho^{\frac{n-2}p}
\Big\|_{L^p_\rho(I)}\Big)^q\Big)^{1/q}
\\\nonumber
\lesssim& \Big(\sum_R\Big(\sum_M\min\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},(RM)^{-\frac{n-1}{2}[1-\frac{2n}{(n-1)q}]}\}\\\nonumber
&\qquad\qquad\times \Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^\frac12\varphi(\frac{\rho}M)\rho^{\frac{n-2}p}
\Big\|_{L^p_\rho({\mathbb{R}})}\Big)^q\Big)^{1/q}.\end{aligned}$$
If $\nu_0\geq \tfrac{n-2}2$, the condition $q>\frac{2n}{n-1}$ in is enough to guarantee $$\label{S-cond1'}
\sup_{R>0}\sum_{M}\min\big\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},
(RM)^{-\frac{n-1}{2}[1-\frac{2n}{q(n-1)}]}\big\}<\infty,$$ and $$\label{S-cond2'}
\sup_{M>0}\sum_{R}\min\big\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},
(RM)^{-\frac{n-1}{2}[1-\frac{2n}{q(n-1)}]}\big\}<\infty.$$ However, if $0<\nu_0\leq \tfrac{n-2}2$, we need again to ensure and to be true. Therefore, by Schur’s lemma and $\ell^{p} \hookrightarrow\ell^q$ since $q>\frac{2n}{n-1}>p$, we show $$\begin{aligned}
\label{3.14}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\nonumber\\
\lesssim& \Big(\sum_M\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
\Big|b_{\nu,\ell}(\rho)\Big|^2\Big)^\frac12\varphi(\frac{\rho}{M})\rho^{\frac{n-2}{p}}
\Big\|^p_{L^p_\rho}\Big)^{1/p}\nonumber\\
\lesssim&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
|b_{\nu,\ell}(\rho)|^2\Big)^\frac12\rho^{-\frac1p}\Big\|_{L^p_{\rho^{n-1}\;d\rho}({\mathbb{R}}_+)}.\end{aligned}$$ In sum, by orthogonality formula , we prove $$\begin{aligned}
\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\lesssim\Big\|\rho^{-\frac1p}\hat{f}(\rho,\omega)\Big\|_{L^p_{\rho^{n-1}\;d\rho}
([0,\infty);L^2_\omega(Y))}.\end{aligned}$$
In the end of this section, we show the necessity of assumption by constructing a counterexample. We will prove
Let $\nu_0$ be in Theorem \[thm:main\] and let $q$ satisfy but $q\geq \frac{2n}{n-2-2\nu_0}$. Then there exists a counterexample such that the inequality $\eqref{est:restriction}$ fails.
We use the argument of [@ZZ2] to construct a counterexample. Choose $\chi(\rho)\in C_c^\infty([1,2])$ to value in $[0,1]$, we take the initial data $f=(\mathcal{H}_{\nu_0}\chi)(r)$, which is independent of the angular variable $\theta$. Then the distorted Fourier transform of $f$ is the Hankel transform. Therefore, we obtain $$\begin{aligned}
\big\|\rho^{-\frac1p}\hat{f}(\rho,\omega)\big\|_{L^p_{\rho^{n-1}\;d\rho}([0,\infty);L^2_\omega(Y))}=\big\|\rho^{-\frac1p}\chi(\rho)
\big\|_{L^p_{\rho^{n-1}\;d\rho}([0,\infty);L^2_\omega(Y))}<\infty.\end{aligned}$$ Since $q\geq \frac{2n}{n-2-2\nu_0}$, one has $\frac{1}{q}\leq\frac{1}{2}-\frac{1+\nu_0}{n}$. To lead a contradiction, we will show $$\label{contradiction}
\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}=\infty,\quad \frac{1}{q}\leq\frac{1}{2}-\frac{1+\nu_0}{n}$$ where $u(t,r,\theta)$ solves , that is, $$\begin{aligned}
u(t,r,\theta)=\int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu_0}(r\rho)\sin (t\rho) \chi(\rho)\rho^{n-2}\;d\rho .\end{aligned}$$ To prove , we recall the behavior of $J_\nu(r)$ as $r\to 0+$. For the complex number $\operatorname*{Re}(\nu)>-1/2$, see [@G Section B.6], then we have that $$\label{Bessel}
J_{\nu}(r)=\frac{r^\nu}{2^\nu\Gamma(\nu+1)}+S_\nu(r)$$ where $$S_{\nu}(r)=\frac{(r/2)^{\nu}}{\Gamma\left(\nu+\frac12\right)\Gamma(1/2)}\int_{-1}^{1}(e^{isr}-1)(1-s^2)^{(2\nu-1)/2}\mathrm{d
}s$$ satisfies $$|S_{\nu}(r)|\leq \frac{2^{-\operatorname*{Re}\nu}r^{\operatorname*{Re}\nu+1}}{(\operatorname*{Re}\nu+1)|\Gamma(\nu+\frac12)|\Gamma(\frac12)}.$$ Now we compute for any $0<\epsilon\ll 1$ $$\begin{aligned}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\\
=&\text{Vol}(Y)^{1/2}\left\|\int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu_0}(r\rho)\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho
\right\|_{L^q({\mathbb{R}};L^q_{r^{n-1}dr}(0,\infty))}\\
\geq &c\left\|\int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu_0}(r\rho)\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho
\right\|_{L^q([\pi/6,\pi/4];L^q_{r^{n-1}dr}[\epsilon,1])}\\
\geq& c\left\|\int_0^\infty(r\rho)^{-\frac{n-2}2}(r\rho)^{\nu_0}\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho
\right\|_{L^q([\pi/6,\pi/4];L^q_{r^{n-1}dr}[\epsilon,1])}\\
&-\left\|\int_0^\infty(r\rho)^{-\frac{n-2}2}S_{\nu_0}(r\rho)\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho
\right\|_{L^q([\pi/6,\pi/4];L^q_{r^{n-1}dr}[\epsilon,1])}.\end{aligned}$$ We first observe that $$\begin{split}
&\left\|\int_0^\infty(r\rho)^{-\frac{n-2}2}S_{\nu_0}(r\rho)\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho
\right\|_{L^q([\pi/6,\pi/4];L^q_{r^{n-1}dr}[\epsilon,1])}\\
\leq& C\left\|\int_0^\infty(r\rho)^{-\frac{n-2}2}(r\rho)^{\nu_0+1}\chi(\rho)\rho^{n-2}\mathrm{d}\rho
\right\|_{L^q([\pi/6,\pi/4];L^q_{r^{n-1}dr}[\epsilon,1])}\\
\leq& C\max\big\{\epsilon^{\nu_0+1-\frac{n-2}2+\frac nq},1\big\}\\
\end{split}$$ Next we estimate the lower boundness $$\begin{aligned}
&\left\|\int_0^\infty(r\rho)^{-\frac{n-2}2}(r\rho)^{\nu_0}\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho
\right\|_{L^q([\pi/6,\pi/4];L^q_{r^{n-1}dr}[\epsilon,1])}\\
=&\left(\int_{\pi/6}^{\pi/4}\int_{\epsilon}^1 \left|\int_0^\infty(r\rho)^{-\frac{n-2}2}(r\rho)^{\nu_0}\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho\right|^q r^{n-1}drdt\right)^{1/q}\\
=&C\left(\int_{\pi/6}^{\pi/4}\left|\int_0^\infty\rho^{-\frac{n-2}2}\rho^{\nu_0}\sin(t\rho)\chi(\rho)\rho^{n-2}
\mathrm{d}\rho\right|^{q}dt\right)^{1/q}\times\begin{cases}\epsilon^{\nu_0-\frac{n-2}2+\frac nq} \quad\text{if}\quad \frac1q<\frac12-\frac{\nu_0+1}{n}\\
\ln\epsilon \quad\text{if}\quad \frac1q=\frac12-\frac{\nu_0+1}{n}
\end{cases}\\
\geq& c\begin{cases}\epsilon^{\nu_0-\frac{n-2}2+\frac nq} \quad\text{if}\quad \frac1q<\frac12-\frac{\nu_0+1}{n}\\
\ln\epsilon \quad\text{if}\quad \frac1q=\frac12-\frac{\nu_0+1}{n}
\end{cases}\end{aligned}$$ where we have used the fact that $\sin(\rho t)\geq 1/2$ for $t\in [\pi/6, \pi/4]$ and $\rho\in [1,2]$, and $$\begin{split}
\left|\int_0^\infty\rho^{-\frac{n-2}2}\rho^{\nu_0}\sin(t\rho)\chi(\rho)\rho^{n-2}\mathrm{d}\rho\right|\geq \frac12\int_0^\infty\rho^{-\frac{n-2}2}\rho^{\nu_0}\chi(\rho)\rho^{n-2}\mathrm{d}\rho\geq c.
\end{split}$$ Hence, we obtain $$\begin{split}
\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}&\geq c\epsilon^{\nu_0-\frac{n-2}2+\frac nq}-C\max\big\{\epsilon^{\nu_0+1-\frac{n-2}2+\frac nq},1\big\}
\\&\geq c\epsilon^{\nu_0-\frac{n-2}2+\frac nq}\to +\infty \quad \text{as}\quad \epsilon\to 0
\end{split}$$ when $\frac1q<\frac12-\frac{\nu_0+1}n.$ And when $\frac1q=\frac12-\frac{\nu_0+1}n$, we get $$\begin{aligned}
\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\geq c\ln\epsilon-C\to +\infty \quad \text{as}\quad \epsilon\to 0.\end{aligned}$$
We conclude this section by proving in Remark \[rem:stri\].
If $\text{supp}~\hat{f}\subset \{\rho\sim M\}$, then by , we have $$\begin{aligned}
\label{est:stri''}
&\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}\\\nonumber
\lesssim& M^{\frac{n-1}2-\frac{n+1}q} \Big(\sum_R\Big(\min\{(RM)^{\frac{n}{q}+\nu_0-\frac{n-2}2},(RM)^{-\frac{n-1}{2}[1-\frac{2n}{(n-1)q}]}\}\\\nonumber
&\qquad\qquad\times \Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^2\Big)^\frac12\varphi(\frac{\rho}M)\rho^{\frac{n-2}2}
\Big\|_{L^2_\rho({\mathbb{R}})}\Big)^q\Big)^{1/q}.\end{aligned}$$ By using the assumption $q>\frac{2n}{n-1}$ when $\nu_0>(n-2)/2$ and $\frac{2n}{n-2-2\nu_0}>q>\frac{2n}{n-1}$ when $0<\nu_0\leq (n-2)/2$, we see the summation in $R$ converges. Thus we obtain $$\begin{aligned}
\|u(t,r,\theta)\|_{L^{q}_{t}({\mathbb{R}}; L^q_{r^{n-1}dr}((0,\infty);L^{2}_\theta(Y)))}
\lesssim M^{\frac{n-1}2-\frac{n+1}q}\|f\|_{\dot H^{-\frac12}},\end{aligned}$$ which is .
The proof of Theorem \[thm:KSS\]
================================
In this section, we prove Theorem \[thm:KSS\] by using the above argument when $q=2$ and a slight modify the original argument for the wave equation in [@KSS].
We first prove . From again, it suffices to estimate $$\begin{aligned}
&\Big\|\Big(\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}
\Big|\int_0^\infty r^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{-it\rho}
b_{\nu,\ell}(\rho)\rho^{\frac{n-2}{2}} \mathrm{d}\rho\Big|^2\Big)^\frac12\Big\|_{L^2_tL^2_{r^{n-1}\;dr}({\mathbb{R}}\times (0,{\bf R}])}\\
\lesssim& \Big(\sum_{R\leq {\bf R}} \sum_M\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|r^{-\frac{n-2}2}J_{\nu}(r\rho)
b_{\nu,\ell}(\rho)\varphi(\frac{\rho}{M})\rho^{\frac{n-2}{2}}
\Big\|^2_{L^2_{d\rho} L^2_{r^{n-1}dr}({\mathbb{R}}\times S_R)}\Big)^{1/2}
\\\nonumber
\lesssim& \Big(\sum_{R\leq {\bf R}} \sum_M M^{n-1}M^{-2}\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|r^{-\frac{n-2}2}J_{\nu}(r\rho)
b_{\nu,\ell}(M\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}
\Big\|^2_{L^2_{d\rho} L^2_{r^{n-1}dr}({\mathbb{R}}\times S_{MR})}\Big)^{1/2}\end{aligned}$$ By the proof of , we have $$\begin{aligned}
\nonumber
& \Big(\sum_{R\leq {\bf R}} \sum_M M^{n-1}M^{-2}\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|r^{-\frac{n-2}2}J_{\nu}(r\rho)
b_{\nu,\ell}(M\rho)\varphi(\rho)\rho^{\frac{n-2}{2}}
\Big\|^2_{L^2_{d\rho} L^2_{r^{n-1}dr}({\mathbb{R}}\times S_{MR})}\Big)^{1/2}\\\nonumber
\lesssim& \Big(\sum_{R\leq {\bf R}} \sum_M M^{-2} \min\big\{(MR)^{2(1+\nu_0)},RM\big\}\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|
b_{\nu,\ell}(\rho)\varphi(\rho/M)\rho^{\frac{n-2}{2}}
\Big\|^2_{L^2_{d\rho}}\Big)^{1/2}\\\nonumber
\lesssim& {\bf R} \Big( \sum_M M^{-2} M\sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|
b_{\nu,\ell}(\rho)\varphi(\rho/M)\rho^{\frac{n-2}{2}}
\Big\|^2_{L^2_{d\rho}}\Big)^{1/2}\\
\lesssim& {\bf R} \Big( \sum_M \sum_{\nu\in\Lambda_\infty}\sum_{\ell=1}^{d(\nu)}\Big\|
b_{\nu,\ell}(\rho)\varphi(\rho/M)\rho^{-1}
\Big\|^2_{L^2_{\rho^{n-1}d\rho}}\Big)^{1/2}\end{aligned}$$ where we have used the fact that $$\begin{aligned}
\sum_{R\leq {\bf R}} \min\{(MR)^{2(1+\nu_0)},RM\}\lesssim {\bf R} M.\end{aligned}$$ Then we prove $$\begin{aligned}
\label{equ:localres}
\sup_{{\bf R}>0}{\bf R}^{-\frac12}\|u(t,r,\theta)\|_{L^{2}_{t}({\mathbb{R}}; L^2_{r^{n-1}dr}((0,{\bf R}];L^{2}_\theta(Y)))}
\lesssim \|f\|_{\dot H^{-1}}.\end{aligned}$$
Next we prove . To this end, we first consider the case $\beta>\tfrac12$: in this range we have, by applying , $$\begin{split}
&\| \langle r\rangle^{-\beta} u(t,r,\theta) \|_{L^2([0,T];L^2(X))}\\
\lesssim& \sum_{j\geq0}2^{-j\beta}\|u(t,r,\theta)\|_{L^2([0,T]; L^2((0,2^{j+1}]\times Y))}\\
\lesssim&
\sum_{j\geq 0}2^{j(1/2-\beta)}\|f\|_{\dot H^{-1}}
\lesssim \|f\|_{\dot H^{-1}}.\end{split}$$
Now we consider the case $0\leq\beta\leq \tfrac12$; we divide into two cases.
[**Case 1: $T\leq 1$.**]{} Here, the estimate is weaker than the energy estimate $$\label{est:energy}
\|u(t,r,\theta) \|_{L^\infty_tL^2(X)}\leq \|f\|_{\dot{H}^{-1}},$$ so that we can immediately write $$\|\langle r\rangle^{-\beta} u(t,r,\theta)\|_{L^2([0,T];L^2(X))}\lesssim T^{1/2}\|u(t,r,\theta)\|_{L^2([0,T];L^2(X))}\leq C_\beta(T)\|f\|_{\dot{H}^{-1}}.$$
[**Case 2: $T\geq1$.**]{} we can use energy estimate to control on the region $\{r:r\geq T\}$ as follows $$\begin{split}
\|\langle r\rangle^{-\beta} u(t,r,\theta)\|_{L^2([0,T]; L^2((T,\infty)\times Y))} &\lesssim T^{-\beta}\|u(t,r,\theta) \|_{L^2([0,T];L^2(X))}\\&
\leq T^{\frac12-\beta}\|f\|_{\dot{H}^{-1}}\leq C_\beta(T)\|f\|_{\dot{H}^{-1}}.
\end{split}$$ While in the region that $\{r:r\leq T\}$, we estimate $$\begin{aligned}
&\| \langle r\rangle^{-\beta} u(t,r,\theta) \|_{L^2([0,T];L^2((0,T]\times Y)}^2\\
\lesssim&\sum_{j=0}^{\ln(T+2)}2^{-2j\beta}\|u(t,r,\theta)\|_{L^2([0,T]; L^2((0,2^{j+1}]\times Y))}^2\\
\lesssim&
\sum_{j=0}^{\ln(T+2)}2^{j(1-2\beta)}\|f\|_{\dot H^{-1}}^2\\
\lesssim& \|f\|_{\dot H^{-1}}^2\times \begin{cases}
T^{1-2\beta}\quad\text{if}\quad 0\leq\beta<\frac12\\
\log(2+T)\quad\text{if}\quad \beta=\frac12
\end{cases}\\
\lesssim& C_\beta(T)^2\|f\|_{\dot H^{-1}}^2\end{aligned}$$ which is accepted.
Finally, we turn to prove . In the region $\{r:\;r\geq T\}$, we utilize the energy estimate to obtain $$\label{equ:outballset}
\big\||r|^{-\beta} u(t,r,\theta)\big\|_{L^2([0,T]; L^2([T,\infty)\times Y))}\\ \lesssim T^{-\beta}\|u(t,r,\theta) \|_{L^2_TL^2_x}
\lesssim T^{\frac12-\beta}\|f\|_{\dot{H}^{-1}}.$$ In the region $\{r:\;r\leq T\}$, by and $0\leq \beta<\tfrac12$, we get $$\begin{aligned}
&\big\||r|^{-\beta} u(t,r,\theta)\big\|_{L^2([0,T];L^2((0,T]\times Y))}\\
\lesssim&\sum_{j=-\infty}^{\log_2 T}2^{-j\beta}\|u(t,r,\theta)\|_{L^2([0,T]; L^2([2^{j},2^{j+1}]\times Y))}\\
\lesssim&
\sum_{j=-\infty}^{\log_2 T}2^{j(\frac12-\beta)}\|f\|_{\dot H^{-1}}\\
\lesssim& T^{\frac12-\beta}\|f\|_{\dot H^{-1}}\end{aligned}$$ which implies .
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[^1]: The assumption here is weaker than the hypothesis in [@Zhang] where one needs $\Delta_h+V_0(\theta)\geq 0$.
|
---
abstract: 'Over the last three decades numerous numerical methods for solving the time-dependent Schrödinger equation within the single-active electron approximation have been developed for studying ionization of atomic targets exposed to an intense laser field. In addition, various numerical techniques for extracting the photoelectron spectra from the time-dependent wave function have emerged. In this paper we compare photoelectron spectra obtained by either projecting the time-dependent wave function at the end of the laser pulse onto the continuum state having proper incoming boundary condition or by using the window-operator method. Our results for three different atomic targets show that the boundary condition imposed onto the continuum states plays a crucial role for obtaining correct spectra accurate enough to resolve fine details of the interference structures of the photoelectron angular distribution.'
author:
- 'B. Feti'' c'
- 'W. Becker'
- 'D. B. Milošević'
title: 'Extracting photoelectron spectra from the time-dependent wave function: Comparison of the projection onto continuum states and window-operator methods'
---
Introduction
============
The pioneering work of K. C. Kulander in the late 1980s [@kulander; @kulander1] has paved the way for the numerical solution of the time-dependent Schrödinger equation (TDSE) to become a very important and powerful tool for studying the laser-atom interaction and related strong-field phenomena. The constant increase in computer power and processor speed of personal computers in the last thirty years has led to the development of numerous numerical methods for solving the TDSE (see, for example, [@tong_gs; @muller; @nurhuda; @bengtsson; @peng; @gordon; @telnov]). Nowadays, many software codes are available for studying processes such as multiphoton ionization, above-threshold ionization, high-order above-threshold ionization, and high-order harmonic generation [@qprop; @altdse; @cltdse; @scid-tdse]. All these methods have one thing in common, namely the TDSE is solved within the single-active-electron (SAE) approximation for a model atom, while the laser-atom interaction is treated in dipole approximation, either using the length or the velocity gauge form of the interaction operator.
Propagation of an initial bound state under the influence of a strong laser field is only one part of the problem. Extraction of the physical observables at the end of the laser pulse poses another challenging task. Modern-day photoionization experiments designed for recording photoelectron spectra (PES) can be used to simultaneously measure the photoelectron kinetic energy and its angular distribution (see, for example, [@holo; @pad_exp1; @pad_exp2]). As the resolution of these experimental techniques increased, the theoretical calculation of highly accurate PES from *ab initio* methods such as numerical solution of the TDSE became essential in order to distinguish different mechanisms that play a role in a photoionization process.
Formal exact PES for a one-electron photoionization process can be calculated by projection of the time-dependent wave function at the end of the laser pulse onto the continuum states of the field-free Hamiltonian. We call this method the PCS (Projection onto Continuum States) method. For long laser pulses at near-infrared wavelengths and moderate intensities the photoelectron can travel very far away from the origin. In order to include the fastest photoelectrons the volume within which the wave function is simulated has to be very large. Another deficiency of the PCS method is that the continuum states, onto which we project the solutions of the TDSE at the end of the laser pulse, are analytically known only for the pure Coulomb potential, while for non-Coulomb potentials they have to be obtained numerically. That is why many approximative methods for extracting PES with no need to calculate the continuum states have emerged in the last three decades. One of the earliest methods used for extracting the PES from the time-dependent wave function is the so-called window-operator (WO) method [@wop]. It has been successfully used in the past for PES calculations for atomic targets exposed to a strong laser field [@wop_app]. Recently, the WO method has also been used for studying high-order above-threshold ionization of the H$_2^+$ molecular ion [@fetic_mhati]. There is also the so-called tSURFF method [@tsurff], which is designed to replace the projection onto continuum states with a time integral of the outer-surface flux, allowing one to use much a smaller simulation volume. An extension of the tSURFF method called iSURF method [@morales] has also been used for calculating PES. Another way of calculating PES without explicit calculation of the continuum states is to propagate the wave function under the influence of the field-free Hamiltonian for some period of time after the laser pulse has been turned off, so that even the slowest parts of the wave function have reached the asymptotic zone [@madsen]. However, for neutral atomic targets this method requires a large spatial grid to include the part of the wave function associated with the fastest photoelectrons.
From a numerical point of view, the above-mentioned approximative methods may be appealing since they are less time consuming than the exact PCS method. However, they can mask some fine details in the PES due to neglecting the nature of the continuum state associated with a photoelectron. Therefore, approximative methods used for extracting PES from the wave function have to be checked for consistency by comparing with the exact method. In this paper we compare the results obtained using the exact PCS method with those obtained with the WO method.
This paper is organized as follows. In Sec. \[sec:num\] we first describe our numerical method for solving the Schrödinger equation. Next, we introduce the method of extracting PES from the time-dependent wave function using the method of projecting onto continuum states and the window-operator method. In Sec. \[sec:results\] we present our results for PES obtained by these two methods. We compare results for three different targets, fluorine negative ions and hydrogen and argon atoms, modeled by different types of the binding potential. Finally, we summarize our results and give conclusions in Sec. \[sec:sum\]. Atomic units (a.u.; $\hbar=1$, $4\pi\varepsilon_0=1$, $e=1$, and $m_e=1$) are used throughout the paper, unless otherwise stated.
Numerical methods {#sec:num}
=================
Method of solving the Schrödinger equation {#subsec:tdse}
------------------------------------------
We start by solving the stationary Schrödinger equation for an arbitrary spherically symmetric binding potential $V({\mathbf{r}})=V(r)$ in spherical coordinates: $$H_0\psi({\mathbf{r}})=E\psi({\mathbf{r}}),\quad H_0=-\frac{1}{2}\nabla^{2}+V(r).$$ We are looking for solutions in the form $$\psi_{n\ell m}({\mathbf{r}})=\frac{u_{n\ell}(r)}{r}Y_\ell^m(\Omega),\quad \Omega\equiv (\theta,\varphi),$$ where the $Y_{\ell}^{m}(\Omega)$ are spherical harmonics. The radial function $u_{n\ell}(r)$ is a solution of the radial Schrödinger equation: $$H_\ell(r)u_{n\ell}(r) = E_{n\ell}u_{n\ell}(r), \label{tdse:rad}$$ $$H_\ell(r)=-\frac{1}{2}\frac{d^{2}}{dr^{2}}+V(r)+\frac{\ell(\ell+1)}{2r^{2}},\label{tdse:rad1}$$ where $n$ is the principal quantum number and $\ell$ is the orbital quantum number. For bound states with the energy $E_{n\ell}<0$ the corresponding radial wave function $u_{n\ell}(r)$ has to obey the boundary conditions $u_{n\ell}(0)=0$ and $u_{n\ell}(r) \to 0$ for $r\to\infty$. The radial equation (\[tdse:rad\]) is solved numerically in the interval $[0,r_{\max}]$ by expanding the radial function into the B-spline basis set as $$u_{n\ell}(r) = \sum_{j=2}^{N-1}c_{j}^{n\ell}B_{j}^{(k_{s})}(r),\label{tdse:bsp}$$ where $N$ represents the number of B-spline functions in the domain $[0,r_{\max}]$ and $k_s$ is the order of the B-spline function. All results presented in this paper have been obtained using the order $k_s=10$ and for simplicity we omit it in further expressions. Since we require that the radial function vanishes at the boundary, we exclude the first and the last B-spline function in the expansion (\[tdse:bsp\]). For more details on the properties of the B-spline basis, see [@Bachau].
Inserting (\[tdse:bsp\]) into (\[tdse:rad\]), multiplying the obtained equation with $B_{i}(r)$, and integrating over the radial coordinate for fixed orbital quantum number $\ell$, we obtain a generalized eigenvalue problem in the form of a matrix equation: $$\mathbf{H}_{0}^{\ell}\mathbf{c}^{n\ell}=E\mathbf{S}\mathbf{c}^{n\ell},\label{tdse:eigen}$$ where $$\left(\mathbf{H}_0^\ell\right)_{ij}=\int_0^{r_{\max}}B_i(r)H_\ell(r)B_j(r)dr,$$ $$(\mathbf{S})_{ij}=\int_{0}^{r_{\max}}B_{i}(r)B_{j}(r)dr.$$ The overlap matrix $\mathbf{S}$ originates from the fact that the B-spline functions do not form an orthogonal basis set. All integrals involving B-spline functions are calculated with the Gauss-Legendre quadrature rule. Using standard diagonalization procedure for solving (\[tdse:eigen\]) we obtain the ground-state energy and the corresponding eigenvector, which is used as an initial state in the TDSE.
In order to describe the laser-atom interaction we numerically solve the time-dependent Schrödinger equation: $$i\frac{\partial\Psi({\mathbf{r}},t)}{\partial t}=\left[H_0+V_I(t)\right]\Psi({\mathbf{r}},t),\label{tdse}$$ where $V_I(t)$ is the interaction operator in the dipole approximation and velocity gauge. We assume that the laser field is linearly polarized along the $z$ axis, so that the interaction operator can be written as $$\begin{aligned}
V_I(t)=-i{\mathbf{A}}(t)\cdot\mathbf{\nabla}=-iA(t)\left(\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta}\right),\end{aligned}$$ where $A(t)=-\int^{t}E(t')dt'$ and $E(t)$ is the electric field given by $$\begin{aligned}
E(t)=E_0\sin^2\left(\frac{\omega t}{2N_c}\right)\cos(\omega t),\quad t\in[0,T_p],\end{aligned}$$ where $\omega=2\pi/T$ is the laser-field frequency and $T_p=N_cT$ is the pulse duration, with $N_c$ the number of optical cycles. The amplitude $E_0$ is related to the intensity $I$ of the laser field by the relation $E_0=\sqrt{I/I_A}$ where $I_A=3.509\times 10^{16}~\text{W}/\text{cm}^{2}$ is the atomic unit of intensity.
The TDSE is solved by expanding the time-dependent wave function in the basis of B-spline functions and spherical harmonics: $$\Psi(r,\Omega, t) = \sum_{j=2}^{N-1}\sum_{\ell=0}^{L-1} c_{j\ell}(t)\frac{B_{j}(r)}{r}Y_{\ell}^{m_0}(\Omega),
\label{tdse:expan}$$ where the expansion coefficients $c_{j\ell}(t)$ are time-dependent. For a linearly polarized laser field, the magnetic quantum number is constant and we set it equal to $m_0=0$. Inserting the expansion (\[tdse:expan\]) into (\[tdse\]), multiplying the obtained result by $B_{i}(r)Y_{\ell'}^{m_{0}*}(\Omega)/r$, and integrating over the spherical coordinates, we obtain the TDSE in the form of the following matrix equation: $$\begin{aligned}
i(\mathbf{S}\otimes\mathbb{1}_{\ell})\frac{d\mathbf{c}(t)}{dt}=\left[\mathbf{H}_{0}^{\ell}\otimes\mathbb{1}_{\ell}
-iA(t)\mathbf{W}_I\right]\mathbf{c}(t),\label{tdse:matrix}\end{aligned}$$ where $\mathbb{1}_{\ell}$ is the identity matrix in $\ell$-space and $$\begin{aligned}
\mathbf{c}(t) &=&
\big[(c_{20},\dots, c_{N-10}),(c_{21},\dots, c_{N-11}),\nonumber\\ &~&\dots,(c_{2L-1},\dots, c_{N-1L-1})\big]^{T},\end{aligned}$$ is a time-dependent vector. The matrices $\mathbf{S}$ and $\mathbf{H}_{0}^{\ell}$ are diagonal in $\ell$-space while the matrix $\mathbf{W}_{I}$ couples the $\ell-1$ and $\ell+1$ $\ell$-block: $$\begin{aligned}
( \mathbf{W}_{I})_{ij}^{\ell'\ell} &=& (\mathbf{Q})_{ij}\left[\ell c_{\ell-1}^{m_0}
\delta_{\ell',\ell-1} -
(\ell+1)c_{\ell}^{m_0}\delta_{\ell',\ell+1}\right]
\nonumber\\&& +(\mathbf{P})_{ij} \left[c_{\ell-1}^{m_0}\delta_{\ell',\ell-1} + c_{\ell}^{m_0}\delta_{\ell',\ell+1}\right],\end{aligned}$$ where $$\begin{aligned}
c_\ell^{m_0}&=&\sqrt{\frac{(\ell+1)^{2}-m_0^2}{(2\ell+1)(2\ell+3)}},\\
(\mathbf{Q})_{ij}&=&\int_{0}^{r_{\max}}\frac{B_{i}(r)B_{j}(r)}{r}dr,\\
(\mathbf{P})_{ij}&=&\int_{0}^{r_{\max}}B_{i}(r) \frac{dB_{j}(r)}{dr}dr.\end{aligned}$$ Since the matrix $\mathbf{W}_I$ couples only the $\ell-1$ and the $\ell+1$ $\ell$-block, it can be decomposed in a sum of mutually commuting matrices $$\begin{aligned}
\mathbf{W}_I = \sum_{\ell=0}^{L-2}\left(\mathbf{P}\otimes\mathbf{L}_{\ell m_0} + \mathbf{Q}\otimes\mathbf{T}_{\ell m_0}\right),\end{aligned}$$ where $$\begin{aligned}
\mathbf{L}_{\ell m_0}&=&c_{\ell}^{m_0}\left(\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right),\\
\mathbf{T}_{\ell m_0} &=&
(\ell+1)c_{\ell}^{m_0}
\left(\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right),\end{aligned}$$ are effectively $2\times2$ matrices acting upon the vector $[\mathbf{c}_{\ell}, \mathbf{c}_{\ell+1}]^{T}=[(c_{2l},\dots, c_{N-1l}),(c_{2\ell+1},\dots, c_{N-1\ell+1})]^{T}$.
The formal solution of the matrix equation (\[tdse:matrix\]) can be written as $$\begin{aligned}
\mathbf{c}(t+\Delta t) &=& \exp\bigg\{-i(\mathbf{S}^{-1} \otimes\mathbb{1}_{\ell})\nonumber\\ &~&\times \int_{t}^{t+\Delta t}
\left[\mathbf{H}_{0}\otimes\mathbb{1}_{\ell} - iA(t')\mathbf{W}_{I}\right]dt'\bigg\} \mathbf{c}(t).\nonumber\\\end{aligned}$$ The evolution of the inital wave function is described by the same numerical recipe as in [@qprop], but without using finite difference expressions. Our final expression for this time evolution is $$\begin{aligned}
\displaystyle\mathbf{c}(t+\Delta t) &=& \prod_{l=L-2}^{0}\Bigg[ \frac{\mathbf{S}\otimes\mathbb{1}_{\ell}-
\frac{\Delta t}{4}A(t+\Delta t)\mathbf{P}\otimes \mathbf{L}_{\ell m_0}}
{\mathbf{S}\otimes\mathbb{1}_{\ell}+\frac{\Delta t}{4}A(t+\Delta t)\mathbf{P}\otimes \mathbf{L}_{\ell m_0}}\nonumber\\
&~&\times \frac{\mathbf{S}\otimes\mathbb{1}_{\ell}- \frac{\Delta t}{4}A(t+\Delta t)\mathbf{Q}\otimes\mathbf{T}_{\ell m_0}}
{\mathbf{S}\otimes\mathbb{1}_{\ell}+\frac{\Delta t}{4}A(t+\Delta t) \mathbf{Q}\otimes \mathbf{T}_{\ell m_0}} \Bigg]
\nonumber\\ &~&\times \prod_{\ell=0}^{L-1} \frac{(\mathbf{S}-i\frac{\Delta t}{2}\mathbf{H}_{0}^{\ell})\otimes\mathbb{1}_{\ell}}
{(\mathbf{S}+i\frac{\Delta t}{2}\mathbf{H}_{0}^{\ell})\otimes\mathbb{1}_{\ell}} \nonumber \\
&~& \times \prod_{\ell=0}^{L-2} \Bigg[ \frac{\mathbf{S}\otimes\mathbb{1}_{\ell}-\frac{\Delta t}{4}A(t)\mathbf{Q}\otimes \mathbf{T}_{\ell m_0}}{\mathbf{S}\otimes\mathbb{1}_{\ell}
+ \frac{\Delta t}{4}A(t)\mathbf{Q}\otimes \mathbf{T}_{\ell m_0}}\nonumber\\&~&\times
\frac{\mathbf{S}\otimes\mathbb{1}_{\ell}-\frac{\Delta t}{4}A(t)\mathbf{P}\otimes \mathbf{L}_{\ell m_0}}
{\mathbf{S}\otimes\mathbb{1}_{\ell}+\frac{\Delta t}{4}A(t)\mathbf{P}\otimes \mathbf{L}_{\ell m_0}}
\Bigg]\mathbf{c}(t).\end{aligned}$$
Extracting the photoelectron spectra from the time-dependent wave function
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The photoelectron spectra can be extracted from the time-dependent wave function $\Psi({\mathbf{r}}, t)$ at the end of the laser pulse by projecting it onto the continuum states having the momentum ${\mathbf{k}}=(k,\Omega_{\mathbf{k}})$, $\Omega_{\mathbf{k}}\equiv(\theta_{\mathbf{k}},\varphi_{\mathbf{k}})$. These continuum states are solutions of the stationary Schrödinger equation for an electron moving in a spherically symmetric potential $V(r)$. There are two linearly independent continuum states labeled $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$ and $ \Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}})$, which satisfy different boundary conditions at large distance from the atomic target: $$\Phi_{{\mathbf{k}}}^{(\pm)}({\mathbf{r}})\xrightarrow{r\to\infty} (2\pi)^{-3/2}\left(e^{i{\mathbf{k}}\cdot {\mathbf{r}}}
+f^{(\pm)}(\theta_{\mathbf{k}})\frac{e^{\pm ikr}}{r}\right),$$ where $f^{(\pm)}(\theta_{\mathbf{k}})$ is the usual scattering amplitude. The solutions $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$ represent continuum states that obey the so-called outgoing boundary condition whereas the solutions $\Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}})$ represent continuum states that obey the so-called incoming boundary condition. The difference between these two continuum states becomes manifest in the time dependence of their corresponding wave packets as shown in [@roman]. Here we only give the main result. Namely, a long time after the interaction with the target, the continuum states $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$ and $\Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}})$ behave as follows: $$\begin{aligned}
\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}},t)&\xrightarrow{t\to\infty}& (2\pi)^{-3/2} e^{i({\mathbf{k}}\cdot {\mathbf{r}}-E_{\mathbf{k}}t)} + \text{a scattering wave},\nonumber \\ \Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}},t)&\xrightarrow{t\to\infty}& (2\pi)^{-3/2} e^{i({\mathbf{k}}\cdot {\mathbf{r}}-E_{\mathbf{k}}t)}.
$$ In an ionization experiment, the electron liberated by ionization winds up in a quantum state having linear momentum ${\mathbf{k}}$. Therefore, the continuum state $\Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}})$ is suitable for describing an ionization experiment while the continuum state $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$ is employed for a collision experiment. For more detailed analysis and discussion, see [@starace]. Both continuum states can be written as partial wave expansions: $$\Phi_{{\mathbf{k}}}^{(\pm)}({\mathbf{r}}) = \sqrt{\frac{2}{\pi}}\frac{1}{k}\sum_{\ell,m} i^{\ell}e^{\pm i\Delta_{\ell}}\frac{u_{\ell}(k,r)}{r}
Y_{\ell}^{m}(\Omega)Y_{\ell}^{m*}(\Omega_{\mathbf{k}}),\label{cont_st}$$ where $\Delta_{\ell}$ is the scattering phase shift of the $\ell$th partial wave. The radial function $u_{\ell}(k,r)$ is a solution of the radial Schrödinger equation (\[tdse:rad\]) for fixed orbital quantum number and kinetic energy $E_{\mathbf{k}}=k^{2}/2$. The continuum states (\[cont\_st\]) are normalized on the momentum scale, i.e., $\langle \Phi_{{\mathbf{k}}'}^{(\pm)}| \Phi_{{\mathbf{k}}}^{(\pm)}\rangle = \delta({\mathbf{k}}'-{\mathbf{k}})$.
For the pure Coulomb potential $V(r) = -Z/r$, the scattering phase shift $\Delta_\ell$ is equal to the Coulomb phase shift $\sigma_\ell=\arg\Gamma(\ell+1 + i \eta)$, with $\eta = -Z/k$ the Sommerfeld parameter. The radial function $u_\ell(k,r)$ is given by the regular Coulomb function $u_\ell(k,r)=F_\ell(\eta,kr)$, which is known in analytical form. Coulomb functions $F_{\ell}(\eta,kr)$ and corresponding phase shifts $\sigma_{\ell}$ are calculated using a subroutine from [@peng1].
For the modified Coulomb potential $$V(r) = -\frac{Z}{r} + V_{s}(r),$$ the scattering phase shift $\Delta_\ell$ is the sum of the Coulomb phase shift $\sigma_\ell$ and the phase shift $\hat{\delta}_\ell$ due to the presence of the short-range potential $V_s(r)$. In this case, the radial equation is solved numerically by the Numerov method in the interval $r\in [0, r_0]$, where $r_0$ is the chosen size of the spherical box, and the phase shift $\hat{\delta}_\ell$ is obtained by matching the numerical solution $u_\ell(k,r)$ to the known asymptotic solution [@joachain]: $$\mathcal{N}u_{\ell}(k, r) = \cos\hat{\delta}_{\ell}F_{\ell}(\eta,kr) + \sin\hat{\delta}_{\ell}G_{\ell}(\eta,kr), \label{matching}$$ where $G_{\ell}(\eta,kr)$ is the irregular Coulomb function and $\mathcal{N}$ is a normalization constant. To avoid having to calculate derivatives, the phase shift $\hat{\delta}_\ell$ is obtained by matching at two different points $r_1$ and $r_2$ close to the boundary $r_0$: $$\tan\hat{\delta}_{\ell} = \frac{\kappa F_{\ell}(\eta,kr_{2}) - F_{\ell}(\eta, kr_{1})} {G_{\ell}(\eta,kr_{1}) -\kappa G_{\ell}(\eta, kr_{2})},\quad
\kappa = \frac{u_{\ell}(k,r_{1})}{u_{\ell}(k, r_{2})}.$$
For a pure short-range potential $V(r) = V_{s}(r)$ ($\eta=0$), the Coulomb functions $F_{\ell}(\eta,kr)$ and $G_{\ell}(\eta,kr)$ must be replaced by the spherical Bessel function $j_{\ell}(kr)$ and the spherical Neumann function $n_{\ell}(kr)$: $$F_{\ell}(0,kr) = krj_{\ell}(kr), \quad G_{\ell}(0,kr) = -krn_{\ell}(kr).$$ The spherical Bessel and Neumann functions and the Coulomb functions are calculated using a subroutine from [@coul90]. After obtaining the phase shift $\hat{\delta}_{\ell}$, the numerical solution $u_{\ell}(k,r)$ is normalized according to (\[matching\]).
The probability of finding the electron at the end of the laser pulse in a continuum state with the momentum ${\mathbf{k}}= (k,\Omega_{{\mathbf{k}}})$ is given by $$P(k, \Omega_{{\mathbf{k}}}) = \frac{d^{3}P}{k^{2} dk d\Omega_{{\mathbf{k}}}} = \left|\langle \Phi_{{\mathbf{k}}}^{(-)} | \Psi(T_{p})\rangle\right|^{2}.\label{pad_1}$$ Inserting (\[cont\_st\]) and (\[tdse:expan\]) into (\[pad\_1\]) we obtain the expression $$P(k, \Omega_{{\mathbf{k}}})= \frac{2}{\pi}\frac{1}{k^{2}}\Big|\sum_{i,\ell}c_{i\ell}(T_{p})(-i)^{\ell}e^{i\Delta_{\ell}}
Y_{\ell}^{m_0}(\Omega_{{\mathbf{k}}})I_{i\ell}(k)\Big|^{2}, \label{prob}$$ where we have introduced the integral $$\begin{aligned}
I_{i\ell}(k)&=& \int_{0}^{r_{0}}u_{\ell}(k,r)B_{i}(r)dr +\int_{r_{0}}^{r_{\max}}\Big[\cos\hat{\delta}_{\ell}F_{\ell}(\eta,kr)
\nonumber\\ &~& + \sin\hat{\delta}_{\ell}G_{\ell}(\eta,kr)\Big] B_{i}(r)dr.\end{aligned}$$ The photoelectron angular distribution (PAD), i.e., the probability $P(E_{\mathbf{k}},\theta_{\mathbf{k}})$ of detecting the electron with kinetic energy $E_{\mathbf{k}}$ emitted in the direction $\theta_{\mathbf{k}}$, is given by replacing $k = \sqrt{2E_{\mathbf{k}}}$ in (\[pad\_1\]) and integrating over $\varphi_{\mathbf{k}}$: $$\begin{aligned}
P(E_{\mathbf{k}},\theta_{\mathbf{k}})&=&\frac{d^2P}{\sin\theta_{\mathbf{k}}dE_{\mathbf{k}}d\theta_{\mathbf{k}}}\nonumber\\ &=&\frac{1}{\pi\sqrt{2E_{\mathbf{k}}}}\Big|\sum_{i,\ell}c_{i\ell}(T_p)(-i)^{\ell}
e^{i\Delta_{\ell}}\nonumber\\&~&\times\sqrt{2l+1}P_{\ell}^{m_0}(\cos\theta_{\mathbf{k}})I_{i\ell}(k)\Big|^{2},\label{pad_2}
$$ where $P_{\ell}^{m_0}(\cos\theta_{\mathbf{k}})$ are associated Legendre polynomials.
Window-operator method
----------------------
Obtaining the photoelectron angular distribution by projecting onto continuum states can be a challenging task since the continuum states are highly oscillatory functions. Therefore, the numerical integration has to be done with high precision and stability to get the photoelectron spectra with an accuracy of a few orders of magnitude. This is especially true for non-Coulomb potentials since in this case the continuum states must be obtained numerically. In this section we present the implementation of the WO method, which can be used for the extraction of the PES without the need to calculate the continuum states. The WO method is based on the projection operator $W_{\gamma}(E_{\mathbf{k}})$ defined by $$W_{\gamma}(E_{\mathbf{k}}) = \frac{\gamma^{2^{n}}}{(H_{0}-E_{\mathbf{k}})^{2^{n}} + \gamma^{2^{n}}},\label{wo}$$ which extracts the component $|\chi_{\gamma}(E_{\mathbf{k}})\rangle$ of the final wave vector $|\Psi(T_{p})\rangle$ that contributes to energies within the bin of the width $2\gamma$, centered at $E_{\mathbf{k}}$: $$|\chi_{\gamma}(E_{\mathbf{k}})\rangle = W_{\gamma}(E_{\mathbf{k}})|\Psi(T_{p})\rangle.\label{wo_eq}$$ We set $n=3$ and expand the wave vector into the basis (\[tdse:expan\]): $$\chi_{\gamma}(E_{\mathbf{k}}, r,\Omega) = \sum_{i=2}^{N-1}\sum_ {\ell=0}^{L-1}b_{i\ell}^{(\gamma)}(E_{\mathbf{k}})\frac{B_{i}(r)}{r}Y_{\ell}^{m_0}(\Omega).$$ To obtain the coefficients $b_{i\ell}^{(\gamma)}(E_{\mathbf{k}})$ we solve Eqn. (\[wo\_eq\]) by factorizing (\[wo\]) [@qprop] and transforming it into a series of matrix equations: $$\begin{aligned}
&~& \mathbb{1}_{\ell}\otimes\left[\mathbf{H}_{0}^{\ell}-\mathbf{S}(E_{\mathbf{k}}-\gamma e^{i\nu_{34}})\right]
\left[\mathbf{H}_0^\ell-\mathbf{S}(E_{\mathbf{k}}+\gamma e^{i\nu_{34}})\right]\mathbf{b}_{1}^{(\gamma)} \nonumber\\
&~& = \gamma^{2^{3}}\mathbb{1}_{\ell}\otimes\mathbf{S}\mathbf{c}(T_{p}),\nonumber\\
&~& \mathbb{1}_{\ell}\otimes\left[\mathbf{H}_{0}^{\ell}-\mathbf{S}(E_{\mathbf{k}}-\gamma e^{i\nu_{33}})\right]
\left[\mathbf{H}_{0}^{\ell}-\mathbf{S}(E_{\mathbf{k}}+\gamma e^{i\nu_{33}})\right]\mathbf{b}_{2}^{(\gamma)} \nonumber\\
&~& = \mathbb{1}_{\ell}\otimes\mathbf{S}\mathbf{b}_{1}^{(\gamma)},\nonumber\\
&~& \mathbb{1}_{\ell}\otimes\left[\mathbf{H}_{0}^{\ell}-\mathbf{S}(E_{\mathbf{k}}-\gamma e^{i\nu_{32}})\right]
\left[\mathbf{H}_{0}^{\ell}-\mathbf{S}(E_{\mathbf{k}}+\gamma e^{i\nu_{32}})\right]\mathbf{b}_{3}^{(\gamma)} \nonumber\\
&~& = \mathbb{1}_{\ell}\otimes\mathbf{S}\mathbf{b}_{2}^{(\gamma)},\nonumber\\
&~& \mathbb{1}_{\ell}\otimes\left[\mathbf{H}_{0}^{\ell}-\mathbf{S}(E_{\mathbf{k}}-\gamma e^{i\nu_{31}})\right]
\left[\mathbf{H}_{0}^{\ell}-\mathbf{S}(E_{\mathbf{k}}+\gamma e^{i\nu_{31}})\right]\mathbf{b}^{(\gamma)} \nonumber\\
&~& = \mathbb{1}_{\ell}\otimes\mathbf{S}\mathbf{b}_{3}^{(\gamma)},\end{aligned}$$ where $\nu_{3j}= (2j-1)\pi/2^{3}$. After obtaining $\mathbf{b}^{(\gamma)}$, the probability of finding the electron with the energy $E_{\mathbf{k}}$ is calculated as $$\begin{aligned}
P_{\gamma}(E_{\mathbf{k}}) &=& \int dV \chi_{\gamma}^{*}(E_{\mathbf{k}},r, \Omega)\chi_{\gamma}(E_{\mathbf{k}},r, \Omega)\nonumber\\
&=&\int d\Omega dr P_{\gamma}(E_{\mathbf{k}}, r,\Omega),\end{aligned}$$ where $$\begin{aligned}
P_{\gamma}(E_{\mathbf{k}}, r,\Omega) =
\Bigg| \sum_{i=2}^{N-1}\sum_{\ell=0}^{L-1}
b_{i\ell}^{(\gamma)}(E_{\mathbf{k}})B_{i}(r)Y_{\ell}^{m_0}(\Omega) \Bigg|^{2}.\end{aligned}$$ Now we make the assumption that the solid-angle element $d\Omega$ in position space is approximately equal to the solid-angle element $d\Omega_{\mathbf{k}}$ in momentum space (for details, see [@deGruyter]). This means that information about the probability distribution in energy and in angle is obtained by integrating $P_{\gamma}(E_{\mathbf{k}}, r,\Omega_{{\mathbf{k}}})\approx P_{\gamma}(E_{\mathbf{k}}, r,\Omega)$ over the radial coordinate. In this case we define the probability $P_{\gamma}(E_{\mathbf{k}},
\Omega_{{\mathbf{k}}}) = P_{\gamma}(E_{\mathbf{k}}, \theta_{\mathbf{k}})/(2\pi)$ which is equal, up to a constant factor, to the PAD, Eq. (\[pad\_2\]).
Results and Discussion {#sec:results}
======================
![The differential detachment probabilities of F$^-$ ions for emission of electrons in the directions $\theta_{\mathbf{k}}=0^{\circ}$, $90^{\circ}$, and $180^{\circ}$, as functions of the photoelectron energy in units of the ponderomotive energy $U_p$, for the following laser-field parameters: $I=1.3\times 10^{13}~\text{W}/\text{cm}^{2}$, $\lambda =1800~\text{nm}$, and $N_c=6$. The results are obtained by projecting the time-dependent wave function $\Psi(T_{p})$ onto the $\Phi_{{\mathbf{k}}}^{(-)}$ states (black solid line) and $\Phi_{{\mathbf{k}}}^{(+)}$ states (green dot-dashed line) and using the WO method with $\gamma = 2\times 10^{-3}$ (red dashed line).[]{data-label="results:f-_pad_vs_wop"}](wop_pm_f-.eps)
![Full PADs for the same parameters as in Fig. \[results:f-\_pad\_vs\_wop\]. The upper panel shows the PAD obtained by projecting onto the continuum states $\Phi_{\mathbf{k}}^{(-)}$ while the lower panel shows the PAD obtained by the WO method. The WO method gives additional structure for angles $\theta_{\mathbf{k}}\in (30^{\circ}, 150^{\circ})$ and energies $E_{\mathbf{k}}>3U_p$. []{data-label="results:f-_pad_full"}](pad_f-.ps)
![The differential ionization probabilities of H atoms for emission of electrons in the directions $\theta_{\mathbf{k}}=0^{\circ}$, $90^{\circ}$, and $180^{\circ}$, as functions of the photoelectron energy in units of the ponderomotive energy $U_p$, for the following laser-field parameters: $I=10^{14}~\text{W}/\text{cm}^{2}$, $\lambda =800~\text{nm}$, and $N_c=6$. The results are obtained by projecting the time-dependent wave function $\Psi(T_p)$ onto the $\Phi_{{\mathbf{k}}}^{(-)}$ states (black solid line) and the $\Phi_{{\mathbf{k}}}^{(+)}$ states (green dot-dashed line) and by using the WO method with $\gamma=6\times 10^{-3}$ (red dashed line).[]{data-label="results:h_pad_vs_wop"}](h_pad_vs_wop.eps)
![Full PADs for the H atom and laser-field parameters as in Fig. \[results:h\_pad\_vs\_wop\]. The upper panel shows the PAD obtained by projecting onto the Coulomb wave for the free particle and the lower panel shows the PAD obtained by the WO method. The WO method gives additional interference structures for angles $\theta_{\mathbf{k}}\in (30^{\circ}, 150^{\circ})$ and $E_{\mathbf{k}}>4U_p$.[]{data-label="results:h_pad_full"}](pad_h.ps)
In this section we present the results for the PES obtained by the methods discussed in the previous section. We begin by comparing the spectra obtained using the PCS and WO methods for a short-range potential. As the target we use the fluorine negative ion $\mathrm{F}^{-}$. Within the SAE approximation we model the corresponding potential by the Green-Sellin-Zachor potential with a polarization correction included [@GSZpot]: $$V(r) = -\frac{Z}{r\left[1+H\left(e^{r/D}-1\right)\right]}-\frac{\alpha}{2\left(r^2 + r_p^2\right)^{3/2}},$$ with $Z=9$, $D=0.6708$, $H=1.6011$, $\alpha=2.002$, and $r_{p}=1.5906$. The $2p$ ground state of F$^-$ has the electron affinity equal to $I_p=3.404~\text{eV}$. In Fig. \[results:f-\_pad\_vs\_wop\] we present the results for PAD in the directions $\theta_{\mathbf{k}}=0^{\circ}$, $90^{\circ}$, and $180^{\circ}$, obtained by projecting the time-dependent wave function $\Psi(T_p)$ onto continuum states satisfying incoming boundary condition (black solid line), outgoing boundary condition (green dot-dashed line), and using the WO method with $\gamma = 2\times 10^{-3}$ (red dashed line) for the laser-field parameters $I=1.3\times10^{13}~\text{W}/\text{cm}^{2}$, $\lambda =1800~\text{nm}$, and $N_c=6$. The photoelectron energy is given in units of the ponderomotive energy $U_p=E_0^2/(4\omega^2)$. The TDSE is solved within a spherical box of the size $r_{\max}=2200~\text{a.u.}$ with the time step $\Delta t = 0.1~\text{a.u.}$ To achieve convergence we used $L=40$ partial waves with $N=5000$ B-spline functions. The convergence was checked with respect to the variation of all these parameters. The continuum states were obtained numerically in a spherical box of the size $r_0=30~\text{a.u.}$ To allow for the best visual comparison, the WO spectra were multiplied by a constant factor so that optimal overlap is achieved with the PAD given by Eq. (\[pad\_2\]). We notice that for $\theta_{\mathbf{k}}=0^{\circ}$ and $\theta_{\mathbf{k}}=180^{\circ}$ these two methods produce almost identical photoelectron spectra, in contrast to the spectrum in the perpendicular direction with respect to the polarization axis, i.e., for $\theta_{\mathbf{k}}=90^{\circ}$, where we notice a significant difference. The WO method gives a large plateau-like annex, which extends approximately up to $9U_p$, whereas the PAD obtained by projection onto the $\Phi_{\mathbf{k}}^{(-)}$ states drops very quickly beyond $2U_p$. The results obtained projecting onto the states $\Phi_{\mathbf{k}}^{(+)}$ exhibit almost the same plateau-like annex. We will discuss this later. We notice here (and will again in the subsequent figures) that the calculated spectra do not observe backward-forward symmetry. This is due to the rather short pulse duration (recall $N_c = 6$); it can nicely be explained in terms of quantum orbits [@fewcyclerapid; @rescTR].
In Fig. \[results:f-\_pad\_full\] we present logarithmically scaled full PADs obtained either by projecting on the states $\Phi_{\mathbf{k}}^{(-)}$ (upper panel) or by the WO method (lower panel). Both spectra have been normalized to unity and the color map covers seven orders of magnitude. As we can see, for small and very large angles, these two methods produce almost identical interference structures in the PADs. However, there is a substantial difference between the two PADs in the angular range $\theta_{\mathbf{k}}\in (25^{\circ}, 150^{\circ})$ for $E_{\mathbf{k}}>3U_p$.
![The differential ionization probabilities of the Ar atom for emission of electrons in the directions $\theta_{\mathbf{k}}=0^{\circ}$, $90^{\circ}$, and $180^{\circ}$, as functions of the photoelectron energy in units of the ponderomotive energy $U_p$, for the following laser-field parameters: $I=8\times 10^{13}~\text{W}/\text{cm}^{2}$, $\lambda =800~\text{nm}$, and $N_c=6$. The results are obtained by projecting the time-dependent wave function $\Psi(T_{p})$ onto the $\Phi_{{\mathbf{k}}}^{(-)}$ states (black solid line) and $\Phi_{{\mathbf{k}}}^{(+)}$ states (green dot-dashed line) and by using the WO method with $\gamma = 6\times 10^{-3}$ (red dashed line).[]{data-label="results:ar_pad_vs_wop"}](wop_pm_ar.eps)
Next we investigate the PAD for the hydrogen atom with its pure Coulomb potential. In Fig. \[results:h\_pad\_vs\_wop\] we show the PES for $I=10^{14}~\text{W}/\text{cm}^{2}$, $\lambda =800~\text{nm}$, and $N_c=6$. The initial state is $1s$ ($I_{p}=13.605~\text{eV}$). The TDSE is solved in a spherical box of the size $r_{\max}=2200~\text{a.u.}$ using $L=40$ partial wave and $N=5000$ B-spline functions. The time step is set to $\Delta t = 0.1~\text{a.u.}$ The spectra obtained using the WO method are calculated with $\gamma = 6\times 10^{-3}$. Again, we see that the WO method as well as PCS on outgoing-boundary-condition states give a plateau-like annex in the perpendicular direction, which is absent from the PAD obtained by projecting onto the Coulomb wave (the state $\Phi_{\mathbf{k}}^{(-)}$). The same conclusion can be obtained by comparing the full PADs, normalized to unity and presented in Fig. \[results:h\_pad\_full\]. In the lower panel the PAD obtained using the WO method clearly shows additional interference structures just as in the case of $\mathrm{F}^{-}$ ions.
As the last example we use modified the Coulomb potential to model the $3p$ state of the argon atom in the SAE approximation. This potential is given by [@tong] $$V(r) = -\frac{1+a_{1}e^{-a_{2}r}+a_{3}re^{-a_{4}r}+ a_{5}e^{-a_{6}r}}{r},\label{TongPot}$$ with $a_{1}=16.039$, $a_{2}=2.007$, $a_{3}=-25.543$, $a_{4}=4.525$, $a_{5}=0.961$, and $a_{6}=0.443$. Using the potential (\[TongPot\]) we calculated the ionization potential of the $3p$ state and obtained $I_{p}=15.774~\text{eV}$. The TDSE is solved within a spherical box of the size $r_{\max}=1800~\text{a.u.}$ with the time step $\Delta t = 0.05~\text{a.u.}$ Convergence is achieved with $L=40$ partial waves with $N=6000$ B-spline functions. The continuum states are calculated within a spherical box of the size $r_{0}=30~\text{a.u.}$ We used the laser-field parameters $I=8\times10^{13}~\text{W}/\text{cm}^{2}$, $\lambda =800~\text{nm}$, and $N_c=6$. The results for $\theta_{\mathbf{k}}=0^{\circ}$, $90^{\circ}$, and $180^{\circ}$ are presented in Fig. \[results:ar\_pad\_vs\_wop\]. For $\theta_{\mathbf{k}}=90^{\circ}$ we again notice a plateau-like structure in the spectrum obtained by the WO method and by projecting on the states $\Phi_{{\mathbf{k}}}^{(+)}$. This is also visible from the full PADs presented in Fig. \[results:ar\_pad\_full\].
![Full PADs for the Ar atom and the same laser-field parameters as in Fig. \[results:h\_pad\_vs\_wop\]. The upper panel shows the PAD obtained by projecting onto the continuum states $\Phi_{\mathbf{k}}^{(-)}$ and the lower panel shows the PAD obtained by the WO method. The WO method gives additional structure for angles $\theta_{\mathbf{k}}\in (30^{\circ}, 150^{\circ})$ and $E_{\mathbf{k}}>4U_p$.[]{data-label="results:ar_pad_full"}](pad_ar.ps)
From all these examples we can conclude that this plateau-like structure observed at large angles is not caused by the nature of the spherical potential $V(r)$ but has a different origin. Let us now explain the discrepancy between the spectra obtained by projection on the states $\Phi_{{\mathbf{k}}}^{(-)}$ on the one hand and by projection on $\Phi_{{\mathbf{k}}}^{(+)}$ or by the WO method on the other, which we noticed in all examples presented above. As we have already discussed, the continuum states have to satisfy the incoming boundary condition in order to properly describe the PES. This boundary condition is automatically included in the continuum state (\[cont\_st\]) by the phase factor $i^{\ell}e^{-i\Delta_{\ell}}$ for each partial wave. For a better understanding of the origin of the artificial plateau-like annex that we see in the spectra obtained using the WO method, in Figs. \[results:f-\_pad\_vs\_wop\], \[results:h\_pad\_vs\_wop\], and \[results:ar\_pad\_vs\_wop\] we have also presented the PADs obtained projecting onto the continuum states $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$. As we can see, the PAD in the direction $\theta_{\mathbf{k}}=90^{\circ}$, calculated using the wrong continuum states $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$, gives the same artificial plateau-like structures as the WO method. Therefore, we conclude that the effect that we see in the PADs obtained by the WO method is caused by the boundary condition satisfied by the continuum states. Since this boundary condition is not included or defined anywhere in the WO method, the energy component $\chi_{\gamma}(E_{\mathbf{k}},r, \Omega)$ extracted from the time-dependent wave function $\Psi({\mathbf{r}},T_{p})$ is a mixture of the contributions from the $\Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}})$ and $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$ continuum states. That is why we see in the spectrum obtained by the WO method a plateau-like structure in the perpendicular direction. Only the continuum states $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$ contribute to this spurious plateau. It is worth noting that another consequence of taking the wrong boundary condition is also visible in the spectrum in the direction $\theta_{\mathbf{k}}=0^{\circ}$ for $\mathrm{Ar}$ (Fig. \[results:ar\_pad\_vs\_wop\]). Namely, the destructive interference at approximately $8.8U_p$ is far less pronounced in the spectrum obtained by the WO method than in the spectrum obtained by projecting onto the states $\Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}})$. The reason is the interplay between the two different contributions, one that comes from the continuum state $\Phi_{{\mathbf{k}}}^{(+)}({\mathbf{r}})$ and the other that comes from the $\Phi_{{\mathbf{k}}}^{(-)}({\mathbf{r}})$ continuum state, which is smaller by a few orders of magnitude. The same feature we see in the spectrum for $\mathrm{F}^{-}$ for $\theta_{\mathbf{k}}=0^{\circ}$ (Fig. \[results:f-\_pad\_vs\_wop\]) at the kinetic energy just above $8U_p$ (it is less pronounced than in the Ar case).
Rescattering plateaus at angles substantially off the polarization direction of the laser field like those calculated for the outgoing boundary conditions or by the WO method and exhibited in Figs. \[results:f-\_pad\_vs\_wop\]–\[results:ar\_pad\_full\] are difficult to understand for physical reasons. All gross features observed so far in angle-dependent above-threshold-ionization spectra have been amenable to explanation in terms of the classical three-step scenario. However, this does not allow for electron energies perpendicularly to the field direction in access of about $2U_p$ [@rescTR; @moeller14]. The reason is that within the three-step model there is no force acting on the electron in the perpendicular direction by the laser field. Hence, the perpendicular momentum has to come either from direct ionization or from rescattering. Direct ionization has a cutoff of about $2U_p$. High-energy rescattering requires that the electron return to its parent atom with high energy, and such an electron will invariably undergo additional longitudinal acceleration after the rescattering, so that its final momentum will not be emitted at right angle to the field.
Summary and conclusions {#sec:sum}
=======================
We presented a method of solving the time-dependent Schrödinger equation (within the SAE and dipole approximations) for an atom (or a negative ion) bound by a spherically symmetric potential and exposed to a strong laser field, by expanding the time-dependent wave function in a basis of B-spline functions and spherical harmonics and propagating it with an appropriate algorithm. The emphasis is on the method of extracting the angle-resolved photoelectron spectra from the time-dependent wave function. This is done by projecting the time-dependent wave function at the end of the laser pulse onto the continuum states $\Phi_{\mathbf{k}}$, which are solutions of the Schrödinger equation in the absence of the laser field (the PCS method). In the context of strong-laser-field ionization, the photoelectrons having the momentum ${\mathbf{k}}$ are observed at large distances ($r\rightarrow\infty$) in the positive time limit ($t\rightarrow +\infty$). Therefore, it is the *incoming (ingoing-wave)* solutions $\Phi_{\mathbf{k}}^{(-)}$ that are relevant. These solutions merge with the plane-wave solutions at the time $t\rightarrow +\infty$: $\Phi_{\mathbf{k}}^{(-)}({\mathbf{r}},t)\rightarrow (2\pi)^{-3/2}e^{i({\mathbf{k}}\cdot{\mathbf{r}}- E_{\mathbf{k}}t)}$.
We have also presented another method of extracting the photoelectron spectra from the TDSE solutions: the window-operator method. The WO method extracts the part of the exact solution of the TDSE at the end of the laser pulse which contributes a small interval of energies near a fixed energy $E_{\mathbf{k}}$. The problem with this method is that it does not single out the contribution of the solution $\Phi_{\mathbf{k}}^{(-)}$, but it includes an unknown linear superposition of the states $\Phi_{\mathbf{k}}^{(-)}$ and $\Phi_{\mathbf{k}}^{(+)}$. Therefore, it may lead and does lead to unphysical results, depending on the considered region of the spectrum. By comparing the results obtained using the exact PCS method with those obtained using the WO method for various potentials $V(r)$ we concluded that the WO method fails for an interval of the electron emission angles around the perpendicular direction (the angle $\theta=90^\circ$ with respect to the polarization axis of the linearly polarized laser field). For $\theta=90^\circ$, the WO method gives a plateau-like structure, which extends up to energies $E_{\mathbf{k}}\sim 9U_p$, while the spectra obtained using the exact PCS method drop very fast beyond $E_{\mathbf{k}}\sim 2-3U_p$. The full PADs show that this unphysical structure in the spectra obtained using the WO method appears for angles $\theta_{\mathbf{k}}\in (30^{\circ}, 150^{\circ})$ and energies $E_{\mathbf{k}}>4U_p$. Furthermore, for values of the angle $\theta_{\mathbf{k}}$ for which the results obtained using the PCS method exhibit interference minima, the WO method smoothes out these minima, due to the spurious contribution of the states $\Phi_{\mathbf{k}}^{(+)}$. We have checked our results using three different type of the potentials $V(r)$: a short-range potential (F$^-$ ion), the pure Coulomb potential (H atom), and a modified Coulomb potential (Ar atom).
Our conclusion is that the WO method is an approximative method that can be used to extract the photoelectron spectrum. It should be used with care since it may produce additional interference structures in the spectrum that have no physical significance. These additional structures are a consequence of the wrong boundary conditions tacitly imposed onto the continuum states by the WO method. That is why every approximative method used for calculating the photoelectron spectra should be tested against the exact method of projecting the time-dependent wave function onto continuum states satisfying incoming boundary condition.
We acknowledge support by the Alexander von Humboldt Foundation and by the Ministry for Education, Science and Youth Canton Sarajevo, Bosnia and Herzegovina.
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|
---
abstract: |
The labyrinth game is a simple yet challenging platform, not only for humans but also for control algorithms and systems. The game is easy to understand but still very hard to master. From a system point of view, the ball behaviour is in general easy to model but close to the obstacles there are severe non-linearities. Additionally, the far from flat surface on which the ball rolls provides for changing dynamics depending on the ball position.
The general dynamics of the system can easliy be handled by traditional automatic control methods. Taking the obstacles and uneaven surface into accout would require very detailed models of the system. A simple deterministic control algorithm is combined with a learning control method. The simple control method provides initial training data. As the learning method is trained, the system can learn from the results of its own actions and the performance improves well beyond the performance of the initial controller.
A vision system and image analysis is used to estimate the ball position while a combination of a PID controller and a learning controller based on LWPR is used to learn to navigate the ball through the maze.
author:
-
bibliography:
- 'bibl.bib'
title: |
Combining Vision, Machine Learning and Automatic Control to Play the\
Labyrinth Game
---
Introduction
============
The <span style="font-variant:small-caps;">brio</span> labyrinth has challenged humans since . The objective is simple: guide the ball through the maze by tilting the plane while avoiding the holes. Most people who have tried it can tell that in practice, the game is really not that simple. By means of computer vision and servo actuators, the challenge can now be handed over to the machines with the same premises as human players.
A platform for evaluation of control algorithms has been created. The controlling system has to determine the correct action solely based on the visual appearance of the game and the knowledge of previous control signals. Building an evaluation system based on the labyrinth game enables humans to easily relate to the performance of the evaluated control strategies.
An overview of the physical system is provided in Fig. \[fig\_systemOverview\]. A short description of the implemented and evaluated control strategies is provided in section \[sec\_sysSetup\]. The evaluation is presented in section \[sec\_eval\] and conclusions in section \[sec\_conclusions\]. A more detailed description of the system is available in [@Ofjall10].
System Setup {#sec_sysSetup}
============
![The system.[]{data-label="fig_systemOverview"}](figures/D2X_4553_oversikt "fig:"){width="1.0\columnwidth"}\
Controllers
-----------
For evaluation purposes, three different control strategies have been implemented. These are designated <span style="font-variant:small-caps;">pid</span>, <span style="font-variant:small-caps;">lwpr</span>-2 and <span style="font-variant:small-caps;">lwpr</span>-4.
All strategies uses the same deterministic path planning, a desired ball position is selected from a fixed path depending on the current ball position.
### <span style="font-variant:small-caps;">Pid</span>
The proportional-integral-derivative controller $$u(t) =
Pe(t) +
D\frac{de(t)}{dt} +
I\int_0^t e(\tau)\ \mathrm{d}\tau$$ is the foundation of classical control theory where $u(t)$ is the control signal and $e(t)$ is the control error. The parameters $P$, $I$ and $D$ are used to adjust the influence of the proportional part, the derivative part and the integrating part respectively. Hand tuned dual <span style="font-variant:small-caps;">pid</span> controllers, one for each maze dimension, is used in the system.
### Learning Controllers
The learning controllers, <span style="font-variant:small-caps;">lwpr</span>-2 and <span style="font-variant:small-caps;">lwpr</span>-4, uses Locally Weighted Projection Regression, <span style="font-variant:small-caps;">lwpr</span> [@Vijayakumar00locallyweighted], to learn the inverse dynamics of the system. <span style="font-variant:small-caps;">Lwpr</span> uses several local linear models weighted together to form the output. The parameters of each local model is adjusted online by a modified partial least squares algorithm. The size and number of local models are also adjusted online depending on the local structure of the function to be learned.
For a time discrete system, the inverse dynamics learning problem can be stated as learning the mapping $$\label{learning:statemapping}
\left(\begin{array}{l}
\mathbf{x} \hfill \\
\mathbf{x}^+ \hfill
\end{array}\right)
\rightarrow
\begin{pmatrix}
\mathbf{u}
\end{pmatrix} \qquad.$$ Consider a system currently in state $\mathbf{x}$, applying a control signal $\mathbf{u}$ will put the system in another state $\mathbf{x}^+$. Learning the inverse dynamics means that given the current state $\mathbf{x}$ and a desired state $\mathbf{x}^+$, the learning system should be able to estimate the required control signal $\mathbf{u}$ bringing the system from $\mathbf{x}$ to $\mathbf{x}^+$.
The desired state of the game is expressed as a desired velocity of the ball in all the conducted experiments involving learning systems. This desired velocity has a constant speed and is directed towards the point selected by the path planner. The learning systems are trained online. The current state and a desired state is fed into the learning system and the control signal is calculated. When the resulting state of this action is known, the triple previous state, applied control signal and the resulting state is used for training. The learning systems are thus able to learn from their own actions.
In the cases where the learning system is unable to make control signal predictions due to lack of training data in the current region, the <span style="font-variant:small-caps;">pid</span> controller is used instead. The state and control signal sequences generated by the <span style="font-variant:small-caps;">pid</span> is used as training data for the learning system. Thus, when starting an untrained system, the <span style="font-variant:small-caps;">pid</span> controller will control the game completely. As the learning system gets trained, control of the game will be handled by the learning system to a greater and greater extent.
In the following expressions, $p$, $v$ and $u$ denote position, velocity and control signal respectively. In Eqs. and a subscript $o$ or $i$ indicates if the aforementioned value correspond to the direction of tilt for the outer or inner gimbal ring of the game.
### <span style="font-variant:small-caps;">Lwpr</span>-2
The <span style="font-variant:small-caps;">lwpr</span>-2 controller tries to learn the mappings $$\label{learning:splitNoPosition1}
\begin{matrix}
\begin{pmatrix}
v_o \hfill \\
v_o^+ \hfill \\
\end{pmatrix}
\rightarrow
\begin{pmatrix}
u_o
\end{pmatrix} ,
%
\qquad
\begin{pmatrix}
v_i \hfill \\
v_i^+ \hfill \\
\end{pmatrix}
\rightarrow
\begin{pmatrix}
u_i
\end{pmatrix} \qquad.
\end{matrix}$$ This setup makes the same assumptions regarding the system as those made for the <span style="font-variant:small-caps;">pid</span> controller. First, the ball can not behave differently in different parts of the maze. Secondly, the outer servo should not affect the ball position in the inner direction and vice versa.
### <span style="font-variant:small-caps;">Lwpr</span>-4
By adding the absolute position to the input vectors, <span style="font-variant:small-caps;">lwpr</span>- is obtained. The mappings are $$\label{learning:splitPosition1}
%\begin{matrix}
\begin{pmatrix}
v_o \hfill \\
p_o \hfill \\
p_i \hfill \\
v_o^+ \hfill \\
\end{pmatrix}
\rightarrow
\begin{pmatrix}
u_o
\end{pmatrix} ,
%
\qquad
\begin{pmatrix}
v_i \hfill \\
p_o \hfill \\
p_i \hfill \\
v_i^+ \hfill \\
\end{pmatrix}
\rightarrow
\begin{pmatrix}
u_i
\end{pmatrix}
%\end{matrix}
\qquad .$$
This learning system should have the possibility to handle different dynamics in different parts of the maze. Still it is assumed that the control signal in one direction has little effect on the ball movement in the other.
Vision and Image Processing
---------------------------
Vision is the only means for feedback available to the controlling system. The controller is dependent on knowing the state of the ball in the maze. The ball position, in a coordinate system fixed in the maze, is estimated by means of a camera system and a chain of image processing steps.
The maze is assumed to be planar and the lens distortion is negliable so the mapping between image coordinates and maze coordinates can be described by a homography, Fig. \[fig\_homographyRect\]. To simplify homography estimation, four colored markers with known positions within the maze are detected and tracked.
![Rectifying homography.[]{data-label="fig_homographyRect"}](figures/homographyRect "fig:"){width="1.0\columnwidth"}\
![Servo installation.[]{data-label="fig_servoInstallation"}](figures/servo_9018_9013_cmyk "fig:"){width="1.0\columnwidth"}\
As the maze is stationary in the rectified images even when the maze or camera is moved, a simple background model and background subtraction can be used to find the position of the ball. An approximate median background model, described in [@ardoPHDbgModels], is used. After background subtraction and removal of large differences originating from the high contrast between the white maze and the black obstacles, the ball position is easily found.
State Estimation
----------------
The ball velocity is needed by the controllers. Direct approximation of the velocity with difference methods provides estimations drowned in noise. A Kalman filter [@kalman] is used to filter the position information as well as to provide an estimate of the ball velocity.
A time discrete Kalman filter is used, based on a linear system model $$\begin{array}{llllllll}
\mathbf{x}_{n+1} & = & \mathbf{A}\mathbf{x}_n & + & \mathbf{B}\mathbf{u}_{n} & + & \mathbf{w}_{n} \\
\mathbf{y}_{n} & = & \mathbf{C}\mathbf{x}_{n} & + & \mathbf{v}_n & & & ,
\end{array}$$ with state vector $\mathbf{x}_n$ at time $n$, output $\mathbf{y}_n$, control signal $\mathbf{u}_n$, process noise $\mathbf{w}_n$, measurement noise $\mathbf{v}$ and system parameters $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$.
### Linear System Model
The servo is modeled as a proportionally controlled motor with a gearbox. The servo motor (<span style="font-variant:small-caps;">dc</span>-motor) and gearbox is modeled as $$\label{hwmodel:dcmotor}
\ddot{\theta} = -a\dot{\theta} + bv$$ where $\theta$ is the output axis angle and $v$ is the input voltage. The internal proportional feedback $v=K(K_2u-\theta)$, where $u$ is the angular reference signal, yields the general second order system $$\label{hwmodel:dcservo}
\ddot{\theta} = -bK\theta -a\dot{\theta} + bKK_2u \qquad.$$
The physical layout of the control linkage provides for an approximate offset linear relation between servo deflection, maze tilt angle and ball acceleration. Thus, the ball motion could be modeled as $$\label{hwmodel:ballmotion}
\ddot{y} = c(\theta + \theta_0) - d\dot{y}$$ as long as the ball avoids any contact with the obstacles.
Using the state vector $\mathbf{x} = \begin{pmatrix} y & \dot{y} & \theta & \dot{\theta} & \theta_0 \end{pmatrix} ^\mathrm{T} $ the combination of equations and can be expressed as the continuous time state space model $$\label{hwmodel:modelcont}
\begin{matrix}
\dot{\mathbf{x}} & = &
\begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & -d & c & 0 & c \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & -bK & -a & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix} \mathbf{x}
& + &
\begin{pmatrix}
0 \\ 0 \\ 0 \\ bKK_2 \\ 0
\end{pmatrix} u
\\
\\
y & = & \begin{pmatrix} 1&0&0&0&0 \end{pmatrix} \mathbf{x} & .
\end{matrix}$$
A time discrete model can be obtained using forward difference approximations of the derivatives $\dot{x} \approx \frac{x^{n+1}-x^n}{T}$ $\Leftrightarrow$ $x^{n+1} \approx x^n + T\dot{x}$ where $T$ is the sampling interval. Using standard methods for system identification, [@mosboken], the unknown parameters can be identified.
Actuators
---------
For controlling the maze, two standard servos for radio controlled models have been installed in the game, see Fig. \[fig\_servoInstallation\].
![Deviation from desired path, scenario 1.[]{data-label="fig_resScen1"}](figures/exp1_raw "fig:"){width="0.8\columnwidth"}\
![Deviation from desired path, smoothed over runs, scenario 2a.[]{data-label="fig_resScen2a"}](figures/exp2a_smoothed "fig:"){width="0.8\columnwidth"}\
Evaluation {#sec_eval}
==========
To facilitate more fine grained performance measurements, a different maze is used for evaluation. The alternative maze is flat and completely free of holes and obstacles. The controllers are evaluated by measuring the deviation from a specified path. <span style="font-variant:small-caps;">Rmsoe</span> is the root mean squared orthogonal deviation of the measured ball positions from the desired path. The <span style="font-variant:small-caps;">rmsoe</span> averaged over runs 171 to 200 for each scenario and controller is shown in Table \[tab\_totalResults\].
Scenario 1
----------
The first scenario is a simple sine shaped path. The deviation from the desired path for the three different controllers are shown in Fig. \[fig\_resScen1\]. The learning controllers are started completely untrained and after some runs they outperform the <span style="font-variant:small-caps;">pid</span> controller used to generate training data initially. As expected, the pure <span style="font-variant:small-caps;">pid</span> controller has a constant performance over the runs.
![Eight runs by the <span style="font-variant:small-caps;">pid</span> controller in scenario 2a. Cyan lines indicate forward runs, blue lines are used for reverse runs. The dashed black line is the desired trajectory.[]{data-label="fig_scen2aPidrun"}](figures/pid_exp2a "fig:"){width="0.8\columnwidth"}\
![Four runs (200 to 203) by the <span style="font-variant:small-caps;">lwpr</span>-4 in scenario 2b. Cyan lines indicate forward runs, blue lines are used for reverse runs. The dashed black line is the desired trajectory.[]{data-label="fig_scen2bLwprRun"}](figures/lwpr4_exp2b_end_200_203 "fig:"){width="1.0\columnwidth"}\
Scenario 2
----------
The desired path for the second scenario is the same as for the first. In the second scenario, the game dynamics are changed depending on the position of the ball. In scenario 2a, a constant offset is added to the outer gimbal servo signal when the ball is in the bottom half of the maze. In scenario 2b, the outer gimbal servo is reversed when the ball is in the bottom half of the maze.
The deviation for scenario 2a is shown in Fig. \[fig\_resScen2a\]. As expected, the position dependent <span style="font-variant:small-caps;">lwpr</span>-4 controller performs best. A few runs by the <span style="font-variant:small-caps;">pid</span> controller in scenario 2a is shown in Fig. \[fig\_scen2aPidrun\]. The effect of the position dependet offset is clear. The integral term need some time to adjust after each change of half planes.
Only the <span style="font-variant:small-caps;">lwpr</span>-4 controller is able to control the ball in scenario 2b, the two other controllers both compensate in the wrong direction. In this scenario, the <span style="font-variant:small-caps;">pid</span> controller can not be used to generate training data. For this experiment, initial training data was generated by controlling the game manually. The position dependent control reversal was hard to learn even for the human subject. A few runs by <span style="font-variant:small-caps;">lwpr</span>-4 is shown in Fig. \[fig\_scen2bLwprRun\].
Scenario 3
----------
The desired path for scenario 3 is the path of the real maze. In this scenario, only <span style="font-variant:small-caps;">lwpr</span>-4 was able to handle the severe nonlinearities close to the edges of the maze. The other two controllers were prone to oscillations with increasing amplitude. Still, the <span style="font-variant:small-caps;">pid</span> controller was useful for generating initial training data as the initial oscillations were dampened when enough training data had been collected. These edge related problems illustrates why only <span style="font-variant:small-caps;">lwpr</span>-4 was able to control the ball in the real maze with obstacles.
Some early runs are shown in Fig. \[fig\_scen3early\], the oscillations from the <span style="font-variant:small-caps;">pid</span> controller can clearly be seen. Some later runs are shown in Fig. \[fig\_scen3late\]. The remaining tendency to cut corners can to some extent be explained by the path planning algorithm.
![Trajectories from early runs by <span style="font-variant:small-caps;">lwpr</span>-4 in scenario 3. Cyan lines indicate forward runs, blue lines are used for reverse runs. The dashed black line is the desired trajectory.[]{data-label="fig_scen3early"}](figures/lwpr4_exp3_beginning "fig:"){width="0.8\columnwidth"}\
Conclusions {#sec_conclusions}
===========
Both <span style="font-variant:small-caps;">lwpr</span> based controlling algorithms outperform the <span style="font-variant:small-caps;">pid</span> in all scenarios. From this, two conclusions may be drawn. First, it should be possible to design a much better traditional controller. Secondly, by learning from their own actions, the learning systems are able to perform better than the controlling algorithm used to provide initial training data.
The <span style="font-variant:small-caps;">lwpr</span>-4 requires more training data than <span style="font-variant:small-caps;">lwpr</span>-2. According to the authors of [@Vijayakumar00locallyweighted], this should not necessarily be the case. However, depending on the initial size of the local models, more local models are needed to fill a higher dimensional input space.
Finally, the combination of a simple deterministic controller and a learning controller has been powerful. Designing a better deterministic controller would require more knowledge of the system to be controlled, which may not be available. A learning controller requires training data before it is useful. Combining a learning controller with a simple deterministic controller, the control performance start at the level of the simple controller and is improved as the system is run by automatic generation of training data.
![Trajectories from late runs by <span style="font-variant:small-caps;">lwpr</span>-4 in scenario 3. Cyan lines indicate forward runs, blue lines are used for reverse runs. The dashed black line is the desired trajectory.[]{data-label="fig_scen3late"}](figures/lwpr4_exp3_end "fig:"){width="0.8\columnwidth"}\
------------- ------ ------- ------ ------- ----- -------
Scenario 1 6.0 (0.8) 3.5 (0.5) 3.7 (0.9)
Scenario 2a 15.5 (1.6) 11.5 (1.8) 6.5 (1.6)
Scenario 2b 5.6 (1.1)
Scenario 3 3.8 (0.6)
------------- ------ ------- ------ ------- ----- -------
: Mean <span style="font-variant:small-caps;">rmsoe</span> for 30 runs in the end of each scenario. The standard deviations are given within parentheses.[]{data-label="tab_totalResults"}
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Fredrik Larsson for inspiration and discussions. This research has received funding from the EC’s 7th Framework Programme (FP7/2007-2013), grant agreement 247947 (GARNICS).
|
---
abstract: |
Many complex ecosystems, such as those formed by multiple microbial taxa, involve intricate interactions amongst various sub-communities. The most basic relationships are frequently modeled as co-occurrence networks in which the nodes represent the various players in the community and the weighted edges encode levels of interaction. In this setting, the composition of a community may be viewed as a probability distribution on the nodes of the network. This paper develops methods for modeling the organization of such data, as well as their Euclidean counterparts, across spatial scales. Using the notion of diffusion distance, we introduce [*diffusion Fréchet functions*]{} and [*diffusion Fréchet vectors*]{} associated with probability distributions on Euclidean space and the vertex set of a weighted network, respectively. We prove that these functional statistics are stable with respect to the Wasserstein distance between probability measures, thus yielding robust descriptors of their shapes.
We apply the methodology to investigate bacterial communities in the human gut, seeking to characterize divergence from intestinal homeostasis in patients with [*Clostridium difficile*]{} infection (CDI) and the effects of fecal microbiota transplantation, a treatment used in CDI patients that has proven to be significantly more effective than traditional treatment with antibiotics. The proposed method proves useful in deriving a biomarker that might help elucidate the mechanisms that drive these processes.
address:
- 'Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510 USA'
- 'Department of Pathology and Molecular Medicine, McMaster University, St Joseph’s Healthcare, 50 Charlton Avenue E, 424 Luke Wing, Hamilton ON L8N4A6 Canada'
- 'Department of Mathematics and Statistics, University of Guelph, Guelph ON N1G 2W1 Canada'
author:
- Diego Hernán Díaz Martínez
- 'Christine H. Lee'
- 'Peter T. Kim'
- Washington Mio
bibliography:
- 'cdi.bib'
title: Probing the Geometry of Data with Diffusion Fréchet Functions
---
Fréchet functions ,diffusion distances ,co-occurrence networks ,[*Clostridium difficile*]{} infection ,fecal microbiota transplantation 62-07 ,92C50
Introduction
============
The state of a complex ecosystem is frequently described by a probability distribution on the nodes of a weighted network. For example, a microbial community may be modeled on a network in which each node represents a taxon and the main interactions between pairs of taxa are represented by weighted edges, the weights reflecting the levels of interaction. In this formulation, bacterial relative abundance in a sample may be viewed as a probability distribution $\xi$ on the vertex set $V$ of the network. Thus, $\xi$ describes the composition of the community whereas the underlying network encodes the expected interactions amongst the various taxa. Identifying sub-communities and their organization at different scales from such data, quantifying variation across samples, and integrating information across scales are core problems. Similar questions arise in other contexts such as probability measures defined on Euclidean spaces or other metric spaces. To address these problems, we develop methods that (i) capture the geometry of data and probability measures across a continuum of spatial scales and (ii) integrate information obtained at all scales. In this paper, we focus on the Euclidean and network cases. We analyze distributions with the aid of a 1-parameter family of diffusion metrics $d_t$, $t>0$, where $t$ is treated as the scale parameter [@coifman]. For (Borel) probability measures $\alpha$ on ${\ensuremath{\mathbb{R}}}^d$, our approach is based on a functional statistic $V_{\alpha, t} \colon {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$ derived from $d_t$ and termed the [*diffusion Fréchet function*]{} of $\alpha$ at scale $t$. Similarly, we define [*diffusion Fréchet vectors*]{} $F_{\xi, t}$ associated with a distribution $\xi$ on the vertex set of a network.
The (classical) Fréchet function of a random variable $y \in {\ensuremath{\mathbb{R}}}^d$ distributed according to a probability measure $\alpha$ with finite second moment is defined as $$\label{E:frechet1}
V_\alpha (x) = {\ensuremath{\mathbb{E}_{\alpha}\!\left[\|y-x\|^2\right]}}
= \int_{{\ensuremath{\mathbb{R}}}^d} \|y-x\|^2 \, d\alpha (y) \,.$$ The Fréchet function quantifies the scatter of $y$ about the point $x$ and depends only on $\alpha$. It is well-known that the [*mean*]{} (or [*expected value*]{}) of $y$ is the unique minimizer of $V_\alpha$. For complex distributions, aiming at more effective descriptors, we introduce diffusion Fréchet functions and show that they encode a wealth of information about the shape of $\alpha$. At scale $t>0$, we define the [*diffusion Fréchet function*]{} $V_{\alpha,t} \colon {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$ by $$\label{E:frechet2}
V_{\alpha,t} (x) = {\ensuremath{\mathbb{E}_{\alpha}\!\left[d_t^2 (y,x)\right]}} = \int_{{\ensuremath{\mathbb{R}}}^d}
d_t^2 (y,x) \, d\alpha (y) \,.$$ This definition simply replaces the Euclidean distance in with the diffusion distance $d_t$. $V_{\alpha,t} (x)$ may be viewed as a localized second moment of $\alpha$ about $x$. In the network setting, Fréchet vectors are defined in a similar manner through the diffusion kernel associated with the graph Laplacian. Unlike $V_\alpha$ that is a quadratic function, diffusion Fréchet functions and vectors typically exhibit rich profiles that reflect the shape of the distribution at different scales. We illustrate this point with simulated data and exploit it in an application to the analysis of microbiome data associated with [*Clostridium difficile*]{} infection (CDI). On the theoretical front, we prove stability theorems for diffusion Fréchet functions and vectors with respect to the Wasserstein distance between probability measures [@villani09]. The stability results ensure that both $V_{\alpha, t}$ and $F_{\xi, t}$ yield robust descriptors, useful for data analysis.
As explained in detail below, $V_{\alpha,t}$ is closely related to the solution of the heat equation $\partial_t u = \Delta u$ with initial condition $\alpha$. In particular, if $p_1, \ldots, p_n \in {\ensuremath{\mathbb{R}}}^d$ are data points sampled from $\alpha$, then the diffusion Fréchet functions for the empirical measure $\alpha_n = \frac{1}{n} \sum_{i=1}^n \delta_{p_i}$ give a reinterpretation of Gaussian density estimators derived from the data as localized second moments of $\alpha_n$. An interesting consequence of this fact is that, for the Gaussian kernel, the scale-space model for data on the real line investigated in [@scalespace] may be recast as a 1-parameter family of diffusion Fréchet functions.
The rest of the paper is organized as follows. Section \[S:distance\] reviews the concept of diffusion distance in the Euclidean and network settings. We define multiscale diffusion Fréchet functions in Section \[S:efrechet\] and prove that they are stable with respect to the Wasserstein distance between probability measures defined in Euclidean spaces. The network counterpart is developed in Section \[S:nfrechet\]. In Section \[S:cdi\] we describe and analyze microbiome data associated with [*C. difficile*]{} infection. We conclude with a summary and some discussion in Section \[S:remarks\].
Diffusion Distances {#S:distance}
===================
Our multiscale approach to probability measures uses a formulation derived from the notion of diffusion distance [@coifman]. In this section, we review diffusion distances associated with the heat kernel on ${\ensuremath{\mathbb{R}}}^d$, followed by a discrete analogue for weighted networks.
For $t>0$, let $G_t \colon {\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$ be the diffusion (heat) kernel $$\label{E:gauss}
G_t (x,y) = \frac{1}{C_d (t)} \exp \left( -\frac{\|y-x\|^2}{4t} \right) \,,$$ where $C_d (t) = (4 \pi t)^{d/2}$. If an initial distribution of mass on ${\ensuremath{\mathbb{R}}}^d$ is described by a Borel probability measure $\alpha$ and its time evolution is governed by the heat equation $\partial_t u = \Delta u$, then the distribution at time $t$ has a smooth density function $\alpha_t (y) = u (y,t)$ given by the convolution $$u (y, t) = \int_{{\ensuremath{\mathbb{R}}}^d} G_t (x,y) \, d\alpha (x) \,.$$ For an initial point mass located at $x \in {\ensuremath{\mathbb{R}}}$, $\alpha$ is Dirac’s delta $\delta_x$ and $u (y,t) = G_t(x, y)$.
The diffusion map $\Psi_t \colon {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{L}_2}}({\ensuremath{\mathbb{R}}}^d)$, given by $\Psi_t (x) = G_t (x, \cdot)$, embeds ${\ensuremath{\mathbb{R}}}^d$ into ${\ensuremath{\mathbb{L}_2}}({\ensuremath{\mathbb{R}}}^d)$. The [*diffusion distance*]{} $d_t$ on ${\ensuremath{\mathbb{R}}}^d$ is defined as $$d_t (x_1, x_2) = \|\Psi_t (x_1) - \Psi_2 (x_2) \|_2 \,,$$ the metric that ${\ensuremath{\mathbb{R}}}^d$ inherits from ${\ensuremath{\mathbb{L}_2}}({\ensuremath{\mathbb{R}}}^d)$ via the embedding $\psi_t$. A calculation shows that $$\label{E:difdistance}
\begin{split}
d^2_t (x_1, x_2) &= \|\Psi_t (x_1)\|^2_2 + \|\Psi_t (x_2)\|^2_2
- 2G_{2t}(x_1,x_2) \\
&= 2 \left( \frac{1}{C_d (2t)} - G_{2t} (x_1, x_2) \right) \,,
\end{split}$$ where we used the fact that $\|\Psi_t (x)\|^2_2 =
1/ C_d (2t)$, for every $x \in {\ensuremath{\mathbb{R}}}^d$. Note that this implies that the metric space $({\ensuremath{\mathbb{R}}}^d, d_t)$ has finite diameter.
Diffusion distances on weighted networks may be defined in a similar way by invoking the graph Laplacian and the associated diffusion kernel. Let $v_1, \ldots, v_n$ be the nodes of a weighted network $K$. The weight of the edge between $v_i$ and $v_j$ is denoted $w_{ij}$, with the convention that $w_{ij} = 0$ if there is no edge between the nodes. We let $W$ be the $n\times n$ matrix whose $(i,j)$-entry is $w_{ij}$. The graph Laplacian is the $n \times n$ matrix $\Delta = D-W$, where $D$ is the diagonal matrix with $d_{ii}=\sum_{k=1}^{n}w_{ik}$, the sum of the weights of all edges incident with the node $v_i$ (cf.[@lux]). This definition is based on a finite difference discretization of the Laplacian, except for a sign that makes $\Delta$ a positive semi-definite, symmetric matrix. The diffusion (or heat) kernel at $t>0$ is the matrix $e^{-t\Delta}$.
Let $\xi$ be a probability distribution on the vertex set $V$ of $K$. If $\xi_i \geq 0$, $1 \leq i \leq n$, is the probability of the vertex $v_i$, we write $\xi$ as the vector $\xi = [\xi_1 \, \ldots \, \xi_n]^T \in {\ensuremath{\mathbb{R}}}^n$, where $T$ denotes transposition. Clearly, $\xi_1 + \ldots + \xi_n =1$. Note that $u = e^{-t\Delta} \xi$ solves the heat equation $\partial_t u = - \Delta u$ with initial condition $\xi$. (The negative sign is due to the convention made in the definition of $\Delta$.) Mass initially distributed according to $\xi$, whose diffusion is governed by the heat equation, has time $t$ distribution $u_t = e^{-t\Delta} \xi$. A point mass at the $i$th vertex is described by the vector $e_i \in {\ensuremath{\mathbb{R}}}^n$, whose $j$th entry is $\delta_{ij}$. Thus, $e^{-t\Delta} e_i$ may be viewed as a network analogue of the Gaussian $G_t (x, \cdot)$. For $t>0$, the diffusion mapping $\psi_t \colon V \to {\ensuremath{\mathbb{R}}}^n$ is defined by $\psi_t (v_i) = e^{-t\Delta} e_i$ and the time $t$ [*diffusion distance*]{} between the vertices $v_i$ and $v_j$ by $$d_t (i,j) = \|e^{-t\Delta} e_i - e^{-t\Delta} e_j\| \,,$$ where $\|\cdot\|$ denotes Euclidean norm. If $0 = \lambda_1 \leq \ldots \leq \lambda_n$ are the eigenvalues of $\Delta$ with orthonormal eigenvectors $\phi_1, \ldots, \phi_n$, then $$d_{t}^2(i,j)= \sum_{k=1}^{n}e^{-2\lambda_kt}\left(\phi_k(i)-\phi_k(j)\right)^2 \,,$$ where $\phi_k (i)$ denotes the $i$th component of $\phi_k$ [@coifman].
Diffusion Fréchet Functions {#S:efrechet}
===========================
The classical Fréchet function $V_\alpha$ of a probability measure $\alpha$ on ${\ensuremath{\mathbb{R}}}^d$ with finite second moment is a useful statistic. However, for complex distributions, such as those with a multimodal profile or more intricate geometry, their Fréchet functions usually fail to provide a good description of their shape. Aiming at more effective descriptors, we introduce diffusion Fréchet functions and show that they encode a wealth of information across a full range of spatial scales. At scale $t>0$, define the [*diffusion Fréchet function*]{} $V_{\alpha,t} \colon {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$ by $$V_{\alpha,t} (x) = \int_{{\ensuremath{\mathbb{R}}}^d}
d_t^2 (y,x) \, d\alpha (y) \,.$$ Note that $V_{\alpha,t}$ is defined for any Borel probability measure $\alpha$, not just those with finite second moment, because $({\ensuremath{\mathbb{R}}}^d, d_t)$ has finite diameter. Moreover, $V_{\alpha,t}$ is uniformly bounded. We also point out that this construction is distinct from the standard kernel trick that pushes $\alpha$ forward to a probability measure $(\psi_t)_\ast (\alpha)$ on ${\ensuremath{\mathbb{L}_2}}({\ensuremath{\mathbb{R}}}^d)$, where the usual Fréchet function of $(\psi_t)_\ast (\alpha)$ can be used to define such notions as an extrinsic mean of $\alpha$. $V_t$ is an intrinsic statistic and typically a function on ${\ensuremath{\mathbb{R}}}^d$ with a complex profile, as we shall see in the examples below. From and , it follows that $$\label{E:frechet3}
\begin{split}
V_{\alpha,t/2} (x) = \frac{2}{C_d (t)} - 2 \int_{{\ensuremath{\mathbb{R}}}^d}
G_{t} (x,y) \, d \alpha (y)
= \frac{2}{C_d (t)} - 2 \alpha_{t} (x) \,,
\end{split}$$ where $\alpha_{t} (x) = u(x, t)$, the solution of the heat equation $\partial_t u = \Delta u$ with initial condition $\alpha$. If $p_1, \ldots, p_n \in {\ensuremath{\mathbb{R}}}^d$ are data points sampled from $\alpha$, then the diffusion Fréchet function of the empirical measure $\alpha_n = \frac{1}{n} \sum_{i=1}^n \delta_{p_i}$ is closely related to the Gaussian density estimator $\widehat{\alpha}_{n, t} = \frac{1}{n} \sum_{i=1}^n G_{t} (x, p_i)$ derived from the sample points. Indeed, it follows from that $$\begin{split}
V_{\alpha_n,t/2} (x) = \frac{2}{C_d (t)} - 2 \int_{{\ensuremath{\mathbb{R}}}^d}
G_{t} (x,y) \, d \alpha_n (y)
= \frac{2}{C_d (t)} - 2 \widehat{\alpha}_{n,t} (x) \,.
\end{split}$$ Thus, diffusion Fréchet functions provide a new interpretation of Gaussian density estimators, essentially as second moments with respect to diffusion distances. This opens up interesting new perspectives. For example, the classical Fréchet function $V_\alpha \colon {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$ (see ) may be viewed as the trace of the covariance tensor field $\Sigma^\alpha \colon {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}^d
\otimes {\ensuremath{\mathbb{R}}}^d$ given by $$\Sigma_\alpha (x) = {\ensuremath{\mathbb{E}_{\alpha}\!\left[(y-x) \otimes (y-x)\right]}}
= \int_{{\ensuremath{\mathbb{R}}}^d} (y-x)\otimes (y-x) \, d \alpha (y) \,.$$ Thus, it is natural to ask: [*How to define diffusion covariance tensor fields $\Sigma_{\alpha,t}$ that capture the modes of variation of $\alpha$, about each $x \in {\ensuremath{\mathbb{R}}}^d$ at all scales, bearing a close relationship to $V_{\alpha,t}$?*]{} A multiscale approach to data along related lines has been developed in [@mmm13; @diaz1].
Here we consider the dataset highlighted in blue in Figure \[F:evolution\], comprising $n=400$ points $p_1, \ldots, p_n$ on the real line, grouped into two clusters. The figure shows the evolution of the diffusion Fréchet function for the empirical measure $\alpha_n = \sum_{i=1}^n \delta_{p_i} /n$ for increasing values of the scale parameter.
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![Evolution of the diffusion Fréchet function across scales.[]{data-label="F:evolution"}](diff-1d-0_005a "fig:"){width="0.4\linewidth"} ![Evolution of the diffusion Fréchet function across scales.[]{data-label="F:evolution"}](diff-1d-0_1a "fig:"){width="0.4\linewidth"}
$t = 0.005$ $t = 0.1$
![Evolution of the diffusion Fréchet function across scales.[]{data-label="F:evolution"}](diff-1d-4a "fig:"){width="0.4\linewidth"} ![Evolution of the diffusion Fréchet function across scales.[]{data-label="F:evolution"}](diff-1d-20a "fig:"){width="0.4\linewidth"}
$t = 4$ $t= 20$
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At scale $t=0.1$, the Fréchet function has two well defined “valleys”, each corresponding to a data cluster. At smaller scales, $V_{\alpha_n,t}$ captures finer properties of the organization of the data, whereas at larger scales $V_{\alpha_n,t}$ essentially views the data as comprising a single cluster. For probability distributions on the real line, the Fréchet function levels off to $1/\sqrt{2 \pi t}$, as $|x| \to \infty$. The local minima of $V_{\alpha,t}$ provide a generalization of the mean of the distribution and a summary of the organization of the data into sub-communities across multiple scales. Elucidating such organization in data in an easily interpretable manner is one of our principal aims.
In this example, we consider the dataset formed by $n = 1000$ points in ${\ensuremath{\mathbb{R}}}^2$, shown on the first panel of Figure\[F:frechet2d\]. The other panels show the diffusion Fréchet functions for the corresponding empirical measure $\alpha_n$, calculated at increasing scales, and the gradient field $-\nabla V_{\alpha_n}$ at $t=2$, whose behavior reflects the organization of the data at that scale.
[cc]{}
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![Data points in ${\ensuremath{\mathbb{R}}}^2$, heat maps of their diffusion Fréchet functions at increasing scales, and the gradient field at $t=2$.[]{data-label="F:frechet2d"}](data2-data "fig:"){width="0.14\linewidth"}
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![Data points in ${\ensuremath{\mathbb{R}}}^2$, heat maps of their diffusion Fréchet functions at increasing scales, and the gradient field at $t=2$.[]{data-label="F:frechet2d"}](data2-dff-0_5 "fig:"){width="0.14\linewidth"}
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![Data points in ${\ensuremath{\mathbb{R}}}^2$, heat maps of their diffusion Fréchet functions at increasing scales, and the gradient field at $t=2$.[]{data-label="F:frechet2d"}](data2-dff-2 "fig:"){width="0.14\linewidth"}
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data & $t = 0.5$ & $t = 2$\
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![Data points in ${\ensuremath{\mathbb{R}}}^2$, heat maps of their diffusion Fréchet functions at increasing scales, and the gradient field at $t=2$.[]{data-label="F:frechet2d"}](data2-dff-4_5 "fig:"){width="0.14\linewidth"}
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![Data points in ${\ensuremath{\mathbb{R}}}^2$, heat maps of their diffusion Fréchet functions at increasing scales, and the gradient field at $t=2$.[]{data-label="F:frechet2d"}](data2-dff-9 "fig:"){width="0.14\linewidth"}
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![Data points in ${\ensuremath{\mathbb{R}}}^2$, heat maps of their diffusion Fréchet functions at increasing scales, and the gradient field at $t=2$.[]{data-label="F:frechet2d"}](data2-dff-30 "fig:"){width="0.14\linewidth"}
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$t = 4.5$ & $t = 9$ & $t = 30$
&
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![Data points in ${\ensuremath{\mathbb{R}}}^2$, heat maps of their diffusion Fréchet functions at increasing scales, and the gradient field at $t=2$.[]{data-label="F:frechet2d"}](data2-gradient-2 "fig:"){width="0.33\linewidth"}
$t=2$
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This example illustrates the evolution of a dataset consisting of $n=3158$ points under the (negative) gradient flow of the Fréchet function of the associated empirical measure at scale $t=0.2$. Panel (a) in Figure \[F:flow\] shows the original data and the other panels show various stages of the evolution towards the attractors of the system.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Panel (a) shows the original data and panels (b)–(f) display various stages of the evolution of the data towards the attractors of the gradient flow of the Fréchet function at scale $t=0.2$.[]{data-label="F:flow"}](trjs_0_2_0 "fig:"){width="0.25\linewidth"} ![Panel (a) shows the original data and panels (b)–(f) display various stages of the evolution of the data towards the attractors of the gradient flow of the Fréchet function at scale $t=0.2$.[]{data-label="F:flow"}](trjs_0_2_5 "fig:"){width="0.25\linewidth"} ![Panel (a) shows the original data and panels (b)–(f) display various stages of the evolution of the data towards the attractors of the gradient flow of the Fréchet function at scale $t=0.2$.[]{data-label="F:flow"}](trjs_0_2_10 "fig:"){width="0.25\linewidth"}
(a) (b) (c)
![Panel (a) shows the original data and panels (b)–(f) display various stages of the evolution of the data towards the attractors of the gradient flow of the Fréchet function at scale $t=0.2$.[]{data-label="F:flow"}](trjs_0_2_15 "fig:"){width="0.25\linewidth"} ![Panel (a) shows the original data and panels (b)–(f) display various stages of the evolution of the data towards the attractors of the gradient flow of the Fréchet function at scale $t=0.2$.[]{data-label="F:flow"}](trjs_0_2_20 "fig:"){width="0.25\linewidth"} ![Panel (a) shows the original data and panels (b)–(f) display various stages of the evolution of the data towards the attractors of the gradient flow of the Fréchet function at scale $t=0.2$.[]{data-label="F:flow"}](trjs_0_2_35 "fig:"){width="0.25\linewidth"}
(d) (e) (f)
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We now prove stability results for $V_{\alpha,t}$ and its gradient field $\nabla V_{\alpha,t}$, which provide a basis for their use as robust functional statistics in data analysis. We begin by reviewing the definition of the Wasserstein distance between two probability measures.
A [*coupling*]{} between two probability measures $\alpha$ and $\beta$ is a probability measure $\mu$ on ${\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d$ such that $(p_1)_\ast (\mu) = \alpha$ and $(p_2)_\ast (\mu) = \beta$. Here, $p_1$ and $p_2$ denote the projections onto the first and second coordinates, respectively. The set of all such couplings is denoted $\Gamma (\alpha, \beta)$.
For each $p \in [1, \infty)$, let ${\ensuremath{\mathcal{P}}}_p ({\ensuremath{\mathbb{R}}}^d)$ denote the collection of all Borel probability measures $\alpha$ on ${\ensuremath{\mathbb{R}}}^d$ whose $p$th moment $M_p (\alpha) = \int \|y\|^p \, d\alpha (y)$ is finite.
\[D:wass\] Let $\alpha, \beta \in {\ensuremath{\mathcal{P}}}_p ({\ensuremath{\mathbb{R}}}^d)$. The $p$-Wasserstein distance $W_p (\alpha, \beta)$ is defined as $$\label{E:wass}
W_p (\alpha, \beta) = \left(\inf_\mu \iint\limits_{{\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d}
\|x-y\|^p d\mu(x,y)\right)^{1/p} \,,$$ where the infimum is taken over all $\mu \in \Gamma (\alpha, \beta)$.
It is well-known that the infimum in is realized by some $\mu \in \Gamma (\alpha, \beta)$ [@villani09]. Moreover, if $p > q \geq 1$, then ${\ensuremath{\mathcal{P}}}_p ({\ensuremath{\mathbb{R}}}^d) \subset {\ensuremath{\mathcal{P}}}_q ({\ensuremath{\mathbb{R}}}^d)$ and $W_q (\alpha, \beta) \leq W_p (\alpha, \beta)$.
The following lemma will be useful in the proof of the stability results. Part (a) of the lemma is a special case of Lemma 1 of [@diaz1]. Let $K_t \colon {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$ be the heat kernel centered at zero. In the notation used in , $K_t (y) = G_t (0,y)$.
\[L:gauss\] Let $\displaystyle C_d(t) = (4\pi t)^{d/2}$. For any $y_1, y_2 \in {\ensuremath{\mathbb{R}}}^d$ and $t>0$, we have:
- $\displaystyle \left| K_t (y_1) - K_t (y_2) \right| \leq
\frac{\|y_1- y_2\|}{C_d(t)\sqrt{2t\cdot e}}$ ;
- $\displaystyle \left \Vert y_1 K_t (y_1) - y_2 K_t(y_2) \right \Vert \leq
\frac{e+2}{e C_d (t)} \|y_1- y_2\|$.
\(a) For each $\xi \in[0,1]$, let $y(\xi) = \xi y_1 +(1-\xi) y_2$. Then, $$\begin{split}
\| K_t (y_1) - K_t (y_2) \| &=
\left| \int_0^1 \frac{d}{d\xi} K_t(y(\xi))\,d\xi \right|
\leq \int_0^1 \left| \frac{d}{d\xi} K_t(y(\xi)) \right| \, d\xi \\
&= \int_0^1 \left| \nabla K_t(y(\xi))\cdot (y_1- y_2) \right| \, d\xi \\
&\leq \|y_1- y_2\| \int_0^1 \left\| \nabla K_t(y(\xi)) \right\| \, d\xi \,.
\end{split}$$ Note that $$\| \nabla K_t(y) \| = \|\frac{y}{2t} K_t(y) \| \leq
\frac{\|y\|}{2tC_d (t)} \exp\left(-\frac{\Vert y\Vert^2}{4t}\right)
\leq \frac{1}{C_d (t) \sqrt{2t\cdot e}} \,.$$ In the last inequality we used the fact that $\|y\| \exp\left(-\frac{\Vert y\Vert^2}{4t}\right) \leq \sqrt{\frac{2t}{e}}$. Hence, $$\begin{split}
\left| K_t (y_1) - K_t (y_2) \right|
&\leq \frac{\|y_1- y_2\|}{C_d(t) \sqrt{2t\cdot e}} \,.
\end{split}$$
\(b) Using the same notation as in (a), $$\frac{d}{d\xi} \left[y K_t (y) \right] =
K_t (y) (y_1-y_2) - y \frac{K_t (y)}{2t} \, y \cdot (y_2-y_2) \,.$$ Hence, $$\label{E:diff}
\begin{split}
\left\| \frac{d}{d\xi} \left[y K_t (y) \right] \right\| &\leq
\left(K_t (y) + \frac{\|y\|^2}{2t} K_t (y) \right) \|y_2-y_2\| \\
&\leq \frac{1}{C_d (t)} \left(1 + \frac{2}{e}\right) \|y_1-y_2\| \,.
\end{split}$$ In the last inequality we used the facts that $K_t (y)
\leq 1/ C_d (t)$ and $\|y\|^2 K_t (y) \leq 4t/e C_d (t)$. Writing $$\label{E:diff1}
y_1 K_t (y_1) - y_2 K_t (y_2) = \int_0^1
\frac{d}{d\xi} \left[y K_t (y) \right] \, d\xi \,,$$ it follows from and that $$\|y_1 K_t (y_1) - y_2 K_t (y_2)\| \leq
\frac{1}{C_d (t)} \left(1 + \frac{2}{e}\right) \|y_1-y_2\| \,,$$ as claimed.
\[T:stability1\] Let $\alpha$ and $\beta$ be Borel probability measures on ${\ensuremath{\mathbb{R}}}^d$ with diffusion Fréchet functions $V_{\alpha,t}$ and $V_{\beta,t}$, $t>0$, respectively. If $\alpha, \beta \in {\ensuremath{\mathcal{P}}}_1 ({\ensuremath{\mathbb{R}}}^d)$, then $$\left\| V_{\alpha,t} - V_{\beta,t} \right\|_\infty \leq
\frac{1}{C_d (2t)\sqrt{t\cdot e}}\, W_1 (\alpha,\beta) \,.$$
Fix $t>0$ and $x \in {\ensuremath{\mathbb{R}}}^d$. Let $\mu \in \Gamma(\alpha,\beta)$ be a coupling such that $$\displaystyle\iint\limits_{{\ensuremath{\mathbb{R}}}^d \times
{\ensuremath{\mathbb{R}}}^d}\|z_1 - z_2\| d \mu(z_1, z_2) = W_1 (\alpha,\beta) \,.$$ Then, we may write $$\label{E:simplified1}
V_{\alpha,t} (x)=\iint\limits_{{\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d}
d^2_t(z_1,x)\, d\mu(z_1, z_2)$$ and $$\label{E:simplified2}
V_{\beta,t} (x)=\iint\limits_{{\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d}
d^2_t(z_2,x)\, d\mu(z_1, z_2).$$ By , $d^2_t(z,x)=2/C_d (2t) - 2 G_{2t}(z,x)$. Hence, $$\label{E:simplified3}
V_{\alpha,t} (x) - V_{\beta,t} (x) = -2 \iint\limits_{{\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d}
\left(G_{2t} (z_1,x) - G_{2t} (z_2,x)\right) d\mu(z_1, z_2) \,,$$ which implies that $$\label{E:simplified4}
\left \vert V_{\alpha,t} (x)- V_{\beta,t} (x) \right \vert \leq
2 \iint\limits_{{\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d}
\left\vert G_{2t}(z_1,x) - G_{2t}(z_2,x) \right\vert d\mu(z_1, z_2) \,.$$ After translating $\alpha$ and $\beta$, we may assume that $x=0$. Thus, by Lemma \[L:gauss\](a), $$\label{E:simplified5}
\begin{split}
\left| V_{\alpha,t} (x) - V_{\beta,t} (x) \right|
&\leq \frac{1}{C_d (2t)\sqrt{t\cdot e}}
\iint\limits_{{\ensuremath{\mathbb{R}}}^d \times {\ensuremath{\mathbb{R}}}^d} \left \Vert z_1-z_2 \right \Vert
d\mu(z_1, z_2) \\
&= \frac{1}{C_d (2t)\sqrt{t\cdot e}} \, W_1 (\alpha,\beta) \,,
\end{split}$$ as claimed.
\[T:stability1a\] Let $\alpha$ and $\beta$ be Borel probability measures on ${\ensuremath{\mathbb{R}}}^d$ with diffusion Fréchet functions $V_{\alpha,t}$ and $V_{\beta,t}$, $t>0$, respectively. If $\alpha, \beta \in {\ensuremath{\mathcal{P}}}_1 ({\ensuremath{\mathbb{R}}}^d)$, then $$\sup_{x \in {\ensuremath{\mathbb{R}}}^d} \left\| \nabla V_{\alpha,t} (x) -
\nabla V_{\beta,t} (x) \right\| \leq
\frac{e+2}{e C_d(2t)} \, W_1 (\alpha,\beta) \,.$$
Let $\mu \in \Gamma(\alpha, \beta)$ be a coupling that realizes $W_1 (\alpha, \beta)$. From , it follows that $$\begin{split}
\| \nabla V_{\alpha,t} (x) &- \nabla V_{\beta,t} (x) \|
\leq 2 \iint \left\| \nabla_x G_{2t}(z_1,x) - \nabla_x G_{2t}(z_2,x) \right\|
d\mu(z_1, z_2) \\
&\leq \frac{1}{2t} \iint\limits
\left\| (x-z_1) G_{2t}(z_1,x) - (x-z_2) G_{2t}(z_2,x) \right\|
d\mu(z_1, z_2) \,.
\end{split}$$ We may assume that $x=0$, so we rewrite the previous inequality as $$\| \nabla V_{\alpha,t} (x) - \nabla V_{\beta,t} (x) \|
\leq \frac{1}{2t} \iint
\left\| z_1 K_{2t}(z_1) - z_2 K_{2t}(z_2) \right\|
d\mu(z_1, z_2) \,.$$ Therefore, by Lemma \[L:gauss\](b), $$\begin{split}
\| \nabla V_{\alpha,t} (x) - \nabla V_{\beta,t} (x) \|
&\leq \frac{e+2}{e C_d(2t)} \iint \|z_1-z_2\| d\mu(z_1, z_2) \\
&= \frac{e+2}{e C_d(2t)} W_1 (\alpha, \beta) \,,
\end{split}$$ which proves the theorem.
Diffusion Fréchet Vectors on Networks {#S:nfrechet}
=====================================
In this section, we define a network analogue of diffusion Fréchet functions and prove a stability theorem. Let $\xi = [\xi_1 \, \ldots \, \xi_n]^T \in {\ensuremath{\mathbb{R}}}^n$ represent a probability distribution on the vertex set $V = \{v_1, \ldots, v_n\}$ of a weighted network $K$. For $t>0$, define the [*diffusion Fréchet vector*]{} (DFV) as the vector $F_{\xi,t} \in {\ensuremath{\mathbb{R}}}^n$ whose $i$th component is $$F_{\xi,t} (i) = \sum_{j=1}^n d_t^2 (i,j) \xi_j \,.$$
We illustrate the behavior of DFVs on a social network of frequent interactions among 62 dolphins in a community living off Doubtful Sound, New Zealand [@lusseau]. The network data was obtained from the UC Irvine Network Data Repository. All edges are given the same weight and we consider the uniform distribution on the vertex set. Figure \[F:dolphins\] shows maps of the diffusion Fréchet vector at multiple scales. As in the Euclidean case, the profiles of the DFVs reveal sub-communities in the network and their interactions.
------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Evolution across scales of the diffusion Fréchet vector for the uniform distribution.[]{data-label="F:dolphins"}](g-1sig-0_01 "fig:"){width="0.35\linewidth"} ![Evolution across scales of the diffusion Fréchet vector for the uniform distribution.[]{data-label="F:dolphins"}](g-5sig-0_0305 "fig:"){width="0.35\linewidth"}
$t = 0.01$ $t = 0.03$
![Evolution across scales of the diffusion Fréchet vector for the uniform distribution.[]{data-label="F:dolphins"}](g-9sig-0_0931 "fig:"){width="0.35\linewidth"} ![Evolution across scales of the diffusion Fréchet vector for the uniform distribution.[]{data-label="F:dolphins"}](g-13sig-0_284 "fig:"){width="0.35\linewidth"}
$t = 0.09$ $t= 0.3$
------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------
To state a stability theorem for DFVs, we first define the Wasserstein distance between two probability distributions on the vertex set $V$ of a weighted network $K$. This requires a base metric on $V$ to play a role similar to that of the Euclidean metric in . We use the [*commute-time distance*]{} between the nodes of a weighted network (cf. [@lovasz96]).
Let $K$ be a connected, weighted network with nodes $v_1, \ldots, v_n$, and let $\phi_1, \phi_2, \ldots, \phi_n$ be an orthonormal basis of ${\ensuremath{\mathbb{R}}}^n$ formed by eigenvectors of the graph Laplacian with eigenvalues $0 = \lambda_1 < \lambda_2 \leq \ldots
\leq \lambda_n$. The commute-time distance between $v_i$ and $v_j$ is defined as $$d_{CT} (i,j)= \left(
\sum_{k=2}^n \frac{1}{\lambda_k}\left(\phi_k(i)-\phi_k(j)\right)^2
\right)^{1/2}.$$
It is simple to verify that $d_{CT}^2 (i,j) = 2 \int_0^\infty d_t^2 (i,j) \, dt$.
Let $\xi, \zeta \in {\ensuremath{\mathbb{R}}}^n$ represent probability distributions on $V$. The $p$-Wasserstein distance, $p \geq 1$, between $\xi$ and $\zeta$ (with respect to the commute-time distance on $V$) is defined as $$W_p (\xi, \zeta) = \min_{\mu \in \Gamma (\alpha, \beta)}
\left( \sum_{j=1}^n \sum_{i=1}^n d_{CT}^p(i,j) \mu_{ij} \right)^{1/p} \,,
\label{E:wasserstein-netw}$$ where $\Gamma (\xi, \zeta)$ is the set of all probability measures $\mu$ on $V \times V$ satisfying $(p_1)_\ast (\mu) = \xi$ and $(p_2)_\ast (\mu) = \zeta$.
Let $\xi, \zeta \in {\ensuremath{\mathbb{R}}}^n$ be probability distributions on the vertex set of a connected, weighted network. For each $t>0$, their diffusion Fréchet vectors satisfy $$\|F_{\xi,t} - F_{\zeta,t} \|_\infty \leq
4 \sqrt{\frac{\operatorname{Tr}e^{-2t \Delta }-1}{2 et}} \, W_1 (\xi, \zeta) \,.$$
Fix $t>0$ and a node $v_\ell$. Let $\mu \in \Gamma(\xi,\zeta)$ be such that $$\sum_{j=1}^n \sum_{i=1}^n d_{CT}(i,j)\mu(i,j) = W_1 (\xi, \zeta)\,.$$ Since $\mu$ has marginals $\alpha$ and $\beta$, we may write $$F_{\xi,t}(\ell)=\sum_{j=1}^n \sum_{i=1}^n d_t^2(i,\ell) \mu_{i,j}
\quad \text{and} \quad
F_{\zeta,t}(\ell)=\sum_{j=1}^n \sum_{i=1}^n d_t^2(j,\ell) \mu_{i,j} \,,$$ which implies that $$\left\vert F_{\xi,t}(\ell) - F_{\zeta,t}(\ell) \right\vert \leq
\sum_{j=1}^n \sum_{i=1}^n \left| d_t^2(i, \ell) -
d_t^2(j, \ell)\right|\mu_{i,j} \,.$$ By Lemma \[L:commute\] below, $$\label{E:ineq1}
\begin{split}
\left\vert F_{\xi,t}(\ell) - F_{\zeta,t}(\ell) \right\vert \leq &
4\sqrt{\operatorname{Tr}e^{-2\Delta t}-1} \sum_{j=1}^n \sum_{i=1}^n d_t(i,j)\mu_{i,j} \,.
\end{split}$$ Observe that $$\label{E:ineq2}
\begin{split}
d_t^2(i,j) &= \sum_{k=1}^{n}e^{-2\lambda_kt}\left(\phi_k(i)-\phi_k(j)\right)^2 \\
&= \sum_{k=1}^{n}\lambda_k e^{-2\lambda_kt}\frac{1}{\lambda_k}
\left(\phi_k(i)-\phi_k(j)\right)^2 \leq \frac{1}{2e t} d_{CT}^2(i,j)
\end{split}$$ since $\lambda_k e^{-2 \lambda_k t} \leq \frac{1}{2et}$. The theorem follows from and .
\[L:commute\] Let $v_i,v_j,v_{\ell}$ be nodes of a connected, weighted network. For any $t>0$, $$\left\vert d_t^2(i,\ell)-d_t^2(j,\ell) \right\vert \leq
4\sqrt{\operatorname{Tr}e^{-2t \Delta} -1} \, d_t(i,j)\,,$$ where $\operatorname{Tr}$ denotes the trace operator.
Since the eigenfunction $\phi_1$ is constant, we may write the diffusion distance as $$\label{E:diffdist}
\begin{split}
d_t^2 (i,\ell) = \sum_{k=2}^n e^{-2\lambda_kt}
\left(\phi_k^2 (i) - 2 \phi_k (i) \phi_k(\ell) + \phi_k^2 (\ell) \right) \,.
\end{split}$$ Thus, $$\begin{split}
d_t^2(i,\ell)-d_t^2(j,\ell) &=
\sum_{k=2}^{n}e^{-2\lambda_kt}\left(\phi_k^2(i)-\phi_k^2(j)\right) \\
&-2 \sum_{k=2}^{n}e^{-2\lambda_kt}\phi_k(\ell)\left(\phi_k(i)-\phi_k(j)\right) \\
&= \sum_{k=2}^{n}e^{-2\lambda_kt} (\phi_k(i)+\phi_k(j)) (\phi_k(i)-\phi_k(j)) \\
&- 2\sum_{k=2}^{n}e^{-2\lambda_kt}\phi_k(\ell)\left(\phi_k(i)-\phi_k(j)\right) \,.
\end{split}$$ Since each $\phi_k$ has unit norm, $|\phi_k (\ell)| \leq 1$ and $|\phi_k (i) + \phi_k (j)| \leq 2$ . Therefore, $$\label{E:difflip}
\left\vert d_t^2(i,\ell)-d_t^2(j,\ell) \right\vert \leq
4 \sum_{k=2}^{n} e^{-2\lambda_kt} \left| \phi_k(i)-\phi_k(j) \right| \,.$$ The Cauchy-Schwarz inequality, applied to the vectors $a = \left( e^{-\lambda_2 t}, \ldots, e^{-\lambda_n t} \right)$ and $b = \left( e^{-\lambda_2 t} \left| \phi_2(i)-\phi_2 (j) \right|, \ldots,
e^{-\lambda_n t} \left| \phi_n(i)-\phi_n (j) \right| \right)$, yields $$\label{E:cs}
\begin{split}
\sum_{k=2}^{n} e^{-2\lambda_kt} \left| \phi_k(i)-\phi_k (j) \right|
&\leq \sqrt{\sum_{k=2}^{n}e^{-2\lambda_kt}} \,
\sqrt{\sum_{k=2}^{n}e^{-2\lambda_kt}\left(\phi_k(i)-\phi_k(j)\right)^2} \\
&= \sqrt{\operatorname{Tr}e^{-2t \Delta} -1} \, d_t(i,j) \,.
\end{split}$$ The lemma follows from and .
[*C. difficile*]{} Infection and Fecal Microbiota Transplantation {#S:cdi}
=================================================================
This section presents an application to analyses of microbiome data associated with [*Clostridium difficile*]{} infection (CDI). CDI kills thousands of patients every year in healthcare facilities [@kelly]. Traditionally, CDI is treated with antibiotics, but the drugs also attack other bacteria in the gut flora and this has been linked to recurrence of CDI in recovering patients [@vincent]. Fecal microbiota transplantation (FMT) is a promising alternative that shows recovery rates close to $90\%$ [@bakken; @gough]. Shahinas [*et al.*]{}[@shahinas] and Seekatz [*et al.*]{}[@seekatz] have used 16S rRNA sequencing to estimate the abundance of the various bacterial taxa living in the gut of healthy donors and CDI patients. These studies have reported a reduced diversity in the bacterial communities of CDI patients and abundance scores after FMT treatment that are closer to those for healthy donors. The studies, however, have focused on counts of bacterial taxa, disregarding interactions in the bacterial communities. To account for these, we employ diffusion Fréchet vectors to examine differences in pre-treatment and post-treatment fecal samples and the effect of FMT in the composition of the gut flora. The analysis is based on metagenomic data for 17 patients (paired pre-FMT and post-FMT) and 7 donor samples, selected from data collected by Lee [*et al.*]{}[@lee].
Metagenomic Data {#S:data}
----------------
The data comes from a subset of 94 patients treated with fecal microbiota transplantation by one of the authors [@lee], covering the period 2008–2012. From this, 17 patients were selected, not randomly, for sequencing. The protocol followed for obtaining consent consisted of first sending a written letter to each patient asking permission to further study their already collected stool samples. In the letter it was stated that there will be a followup telephone call to the patient verifying that they received the letter and whether they will provide consent 5-10 business days following the date of the mailing. All patients in this study were contacted, and all patients provided written informed consent. Furthermore, this study and permission protocol was approved by the Hamilton Integrated Research Ethics Board \#12-3683, the University of Guelph Research Ethics Board 12AU013 and the Florida State University Research Ethics IRB00000446.
All [*C. difficile*]{} infections were confirmed by in-hospital, real-time, polymerase chain reaction (PCR) testing for the toxin B gene. This study sequenced the forward V3-V5 region of the 16S rRNA gene from 17 CDI patients who were treated with FMT(s). A pre-FMT, a corresponding post-FMT, and 7 samples from four donors, corresponding altogether to 41 fecal samples were sequenced.
The bioinformatics software [mothur]{} was used as the primary means of processing and quality-filtering reads and calculating statistical indices of community structure; see [@rush] for a breakdown of the [mothur]{} processing pipeline.
Co-occurrence Networks {#S:network}
----------------------
This study is based on bacterial interactions at the phylum level. We model the (expected) interactions among the various phyla found in the healthy human gut by means of a co-occurrence network [@junker] in which each node represents a phylum. An edge between two phyla is weighted according to the correlation between their counts, estimated from samples taken from a group of healthy individuals. More precisely, let $v_1, \ldots, v_n$ be the nodes of the network and $\rho_{ij}$, $i \ne j$, be the correlation coefficient between the counts for the phyla represented by $v_i$ and $v_j$. As we are interested in sub-communities of interactive phyla, the edge between $v_i$ and $v_j$ is weighted by the absolute correlation $w_{ij} = |\rho_{ij}|$, disregarding whether the correlation is positive or negative. Since this construction typically yields a fully connected network, we use the locally adaptive network sparsification (LANS) technique [@foti] to simplify the network, retaining the most significant interactions (edges) while keeping the network connected. The following description of LANS is equivalent to that in [@foti]. Define $$F_{ij}=\frac{1}{n}\sum_{k=1}^{n}\mathbf{1}\{w_{ik}\leqslant w_{ij}\} \,,$$ where $\mathbf{1}\{w_{ik}\leqslant w_{ij}\}$ returns $1$ if $w_{ik}\leqslant w_{ij}$ and $0$ otherwise. For a given pair $(i,j)$, $F_{ij}$ is the probability that the absolute correlation between the counts for a random phylum and $v_i$ is no larger than $w_{ij}$. Observe that $F_{ij}$ may differ from $F_{ji}$. For $i \ne j$, the decision as to whether the edge between $v_i$ and $v_j$ is deleted or preserved is based on a “significance” level $0 \leqslant \alpha \leqslant 1$. The edge is preserved if $1-F_{ij} < \alpha$ or $1-F_{ji} < \alpha$. Larger values of $\alpha$ retain more edges of the network.
The model we develop is based on seven bacterial phyla: *Actinobacteria*, *Bacteroidetes*, *Firmicutes*, *Fusobacteria*, *Proteobacteria*, *Verrucomicrobia*, and a group of unidentified bacteria treated as a single phylum labeled Unclassified. Figure \[F:graph\] shows the co-ocurrence network obtained after LANS sparsification with $\alpha= 0.1$. Dark edge colors indicate a higher level of interaction; that is, larger edge weights. Note that there is a highly interactive sub-community comprising *Actinobacteria*, *Verrucomicrobia* and unclassified bacteria. This network is used in our analyses of variation in the structure of bacterial communities in samples from CDI patients, recovering patients and healthy individuals, as well as the effects of FMT. (The package *NetworkX* for the Python programming language was used to depict the network [@netw].)
![Bacterial phyla co-occurrence network for the human gut sparsified to significance level $\alpha = 0.1$ with the LANS method.[]{data-label="F:graph"}](graph_0_1){width="40.00000%"}
Microbiota Analysis {#S:analysis}
-------------------
The co-occurrence network constructed in Section \[S:network\] (Figure \[F:graph\]) from culture data for healthy donors provides a model for the expected interactions among the seven bacterial phyla considered in this study. We use the network and bacterial count data to produce a biomarker $\gamma_t$ that is effective in characterizing CDI and potentially in monitoring the effects of FMT treatment. To establish a baseline, we also construct a biomarker $\beta$ solely based on bacterial counts and compare it with $\gamma_t$.
Let $v_1, \ldots, v_7$ be the nodes of the co-occurrence network. For a gut culture sample $S$, let $\xi_i (S)$ be the frequency of $v_i$ in $S$. Clearly, $\xi_1 (S) + \ldots + \xi_7 (S) = 1$. Our first method of analysis is based directly on the probability distribution on the vertex set given by $$\xi (S) = \left( \xi_1(S) , \ldots , \xi_7 (S) \right) \in {\ensuremath{\mathbb{R}}}^7 \,.$$ To derive a scalar biomarker $\beta (S)$, we use culture samples from healthy individuals and CDI patients. We calculate their 7-dimensional frequency vectors $\xi (S)$ and use linear discriminant analysis (LDA) to learn an axis in ${\ensuremath{\mathbb{R}}}^7$ along which their scores $\beta (S)$ optimally discriminate healthy samples from those of CDI patients. For a sample $S$, $\beta (S)$ is the score of $\xi (S)$ along the learned axis. Next, we describe the biomarker $\gamma_t$. For a sample $S$, let $$F^S_t (i) = \sum_{j=1}^7 d^2_t (i,j) \xi_j (S)$$ be the diffusion Fréchet vector for the distribution $\xi (S)$. As before, using training data and applying LDA, we obtain $\gamma_t$.
Results {#S:results}
-------
Our analyses were based on 41 gut culture samples comprising 7 healthy donors, 17 pre-treatment CDI patients (pre-FMT), 4 post-treatment patients not in resolution (post-NR), and 13 post-treatment patients in resolution (post-R). Figure \[F:freq\] shows boxplots for each of the four groups of the distribution of the phylum frequency data obtained from metagenomic sequencing of the forty-one samples.
![Boxplot of bacterial phylum frequency in the human gut for healthy donors, pre-FMT patients, post-FMT patients not in resolution, and post-FMT patients in resolution.[]{data-label="F:freq"}](freqs_boxplot){width="4.5in"}
Due to the relatively small sample size, we formed a Healthy group comprising all samples from donors and post-FMT patients in resolution and a CDI group consisting of all samples from pre-FMT and post-FMT patients not in resolution. Inclusion of post-R samples in the Healthy group has the virtue of challenging the biomarkers $\beta$ and $\gamma_t$ to be sensitive to partial restoration to normal of the gut flora of recovering patients.
From the frequency data, we constructed $\beta$, as described in Section \[S:analysis\]. Linear discriminant analysis yielded an axis in ${\ensuremath{\mathbb{R}}}^7$ along a direction determined by a unit vector whose loadings are specified in Table \[T:loadings\]. The loadings revealed that $\beta$ captures a complex combination of the frequencies of the various phyla, with only *Fusobacteria* and *Verrucomicrobia* playing lesser roles due to their low frequencies.
Phylum LDA loadings for $\beta$ LDA loadings for $\gamma_t$
------------------- -------------------------- -----------------------------
*Firmicutes* -0.412 -0.079
*Proteobacteria* -0.644 0.059
*Bacteroidetes* 0.517 -0.623
*Actinobacteria* 0.267 -0.340
Unclassified 0.275 -0.443
*Fusobacteria* -0.010 -0.119
*Verrucomicrobia* 0.006 -0.525
: Loadings of the directions that determine the axes in 7-D space for $\beta$ and $\gamma_t$.[]{data-label="T:loadings"}
![Boxplot of the values of the diffusion Fréchet functions.[]{data-label="F:dffs"}](frechet_boxplot){width="4.5in"}
A similar analysis was carried out for $\gamma_t$. We tested a range of values for the significance level $\alpha$ used in network sparsification and the scale parameter $t$. The values $\alpha = 0.1$ and $t = 7.75$ were selected because they optimized the performance of $\gamma_t$ as measured by the area under its receiver operating characteristic (ROC) curve [@fawcett], a plot of the true positive rate (sensitivity) against the false positive rate (1 - specificity) at different threshold levels. Figure \[F:dffs\] shows boxplots of the values of the diffusion Fréchet function for the four groups. Table \[T:loadings\] shows the loadings for a unit vector in the direction of the axis in ${\ensuremath{\mathbb{R}}}^7$ space associated with $\gamma_t$ that indicate that the composition of the sub-communities associated with *Bacteroidetes* and *Verrucomicrobia* have a dominant role in characterizing CDI through $\gamma_t$, followed by Unclassified and *Actinobacteria*. Note that the model identifies *Verrucomicrobia* as a key player, whereas its contribution to $\beta$ is minor simply because its count is significantly smaller than the counts for other phyla.
Figure \[F:roc\] provides a comparison of the ROC curves for $\beta$ and $\gamma_t$, $t = 7.75$, demonstrating that $\gamma_t$ has a superior performance relative to $\beta$ in characterizing CDI and recovery from CDI.
![ROC curves for $\beta$ (black) and $\gamma_t$ (blue), for $t=7.75$.[]{data-label="F:roc"}](roc_both){width="2.5in"}
At the threshold value $a = -1.128$, $\gamma_t$ attained a true positive rate of $83\%$ and a false positive rate of $20\%$, as estimated from data for 20 healthy and 21 infected samples, respectively. At $a = -1.108$, the sensitivity increased to $91\%$ with a false positive rate of $25\%$. As samples from recovering CDI patients whose gut flora are only partially restored to normal have been included in the healthy group, we conclude that $\gamma_t$ also shows good sensitivity to the effects of FMT treatment. In this regard, we note that a significant portion of the false positives correspond to samples from post-FMT treatment patients in resolution whose gut flora are only partially restored to normal.
Concluding Remarks {#S:remarks}
==================
We introduced diffusion Fréchet functions and diffusion Fréchet vectors for probability measures on Euclidean space and weighted networks that integrate geometric information at multiple spatial scales, yielding a pathway to the quantification of their shapes. We proved that these functional statistics are stable with respect to the Wasserstein distance, providing a theoretical basis for their use in data analysis. To demonstrate the usefulness of these concepts, we discussed examples using simulated data and applied DFVs to the analysis of data associated with fecal microbiota transplantation that has been used as an alternative to antibiotics in treatment of recurrent CDI. Among other things, the method provides a technique for detecting the presence and studying the organization of sub-communities at different scales. The approach enables us to address such problems as quantification of structural variation in bacterial communities associated with a cohort of individuals (for example, bacterial communities in the gut of multiple individuals), as well as associations between structural changes, health and disease.
Diffusion Fréchet functions may be defined in the broader framework of metric spaces equipped with a diffusion kernel. However, their stability in this general setting remains to be investigated. As remarked in Section \[S:efrechet\], the present work suggests the development of refinements of diffusion Fréchet functions such as diffusion covariance tensor fields for Borel measures on Riemannian manifolds to make their geometric properties more readily accessible.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported in part by: NSF grants DBI-1262351 and DMS-1418007; CIHR-NSERC Collaborative Health Research Projects (413548-2012); and NSF under Grant DMS-1127914 to SAMSI.
References {#references .unnumbered}
==========
|
---
abstract: 'We give some conditions under which (uniform) convergence of a family of Dirichlet series to another Dirichlet series implies the convergence of their individual coefficients and/or exponents. We give some applications to some spectral zeta functions that arise in Riemannian geometry and physics.'
address:
- 'Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland'
- 'Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, GR-157 84, Athens, Greece '
author:
- Gunther Cornelissen
- Aristides Kontogeorgis
bibliography:
- 'spectral.bib'
date: ' (version 1.0)'
title: Convergence of families of Dirichlet series
---
Introduction
============
Suppose that we have pointwise convergence of a sequence of general Dirichlet series $$\label{D} D_n(s)=\sum_{\nu \geq 1} a_{n,\nu} e^{ -s \mu_{n,\nu}} \rightarrow D(s)=\sum_{\nu \geq 1} a_\nu e^{ -s \mu_\nu},$$ all of which converge absolutely in a common half plane ${\mathsf{Re}}(s)>\gamma$, with $a_*$ complex coefficients, and $\mu_*$ is a *strictly* increasing sequence of real numbers. In this paper, we study what this implies about “convergence” of the sequences $(a_{n,\nu})$ and $(\mu_{n,\nu})$. First, we consider the case of classical Dirichlet series, where $a_*=1$ and $\mu_* = \log \lambda_*$, and in Theorem \[main\], we prove that (\[D\]) is equivalent to the $\ell^1$-convergence of $(\lambda^\gamma_{n,\nu})$ to $(\lambda^\gamma_{\nu})$. In Theorem \[qwe\], we consider the case where $\mu_{n,\nu}=\mu_\nu$ is independent of $n$; then, for every $n$, $(a_{n,\nu})$ converges to $(a_n)$. The most general case is studied in Section \[general\] from the point of view of the Perron formula for Dirichlet series (where we give a concrete result under some hypotheses), and form the point of view of the Laplace-Stieltjes transform in Section \[L\], where we prove the equivalence to Lipschitz convergence of some step functions.
One application is to Riemannian geometry. If $\{X_n\}_{n=1}^\infty \cup \{X\}$ is a sequence of connected closed smooth Riemannian manifolds such that $d:=\sup \dim X_n$ is finite, convergence of their zeta functions implies convergence of their spectra. The application to Riemannian manifolds seems relevant in the light of proposals in physics to use the Laplace spectrum as parameter space in cosmological averaging problems [@Seriu], and as dynamical variables in classical gravity, inspired by particle models coupled to gravity in noncommutative geometry [@LandiRovelli], [@Connes]. We discuss this briefly in Section \[phys\], where we show how to use spectral zeta functions to introduce a metric on spaces of Riemannian manifolds up to isospectrality. By the analogy between manifolds and number fields, we similarly introduce a metric on the space of number fields up to arithmetic equivalence, using topology deduced from convergence of their Dedekind zeta functions.
Convergence of Dirichlet series
===============================
Let us once and for all introduce the following convenient notation: if $s \in \mathbf{C}$ and $\Lambda=(\lambda_\nu)_{\nu=1}^\infty$ is a sequence of positive real numbers, we denote by $\Lambda^s$ the sequence $(\lambda_\nu^s)_{\nu=1}^\infty$.
\[main\] Suppose that $$D_n(s)=\sum_{\nu \geq 1} \lambda_{n,\nu}^{-s} \mbox{ and } D(s):=\sum_{\nu \geq 1 } \lambda_\nu^{-s}$$ is a family of (generalized) Dirichlet series for $n=\emptyset, 1,2,\dots$, where, for each $n$, $\Lambda_n:=(\lambda_{n,\nu})_{\nu=1}^\infty$ forms a sequence of increasing positive real numbers with finite multiplicities. Assume that all series $D_n(s)$ are convergent in a common right half plane ${\mathsf{Re}}(s)>\gamma>0$. Then the following are equivalent:
1. As $n \rightarrow +\infty$, the functions $D_n(s)$ converge to $D(s)$, pointwise in $s$ with ${\mathsf{Re}}(s)>\gamma$;
2. For every fixed $\nu$, any bounded subsequence of $\{\lambda_{n,\nu}\}_{n=1}^\infty$ converges to the same element $\lambda \in \Lambda$, and $$\#\{ (\lambda_{n,\nu})_{n=1}^\infty: \lim_{n \rightarrow \infty}\lambda_{n,\nu}= \lambda \}=
\#\{ \lambda_\nu: \lambda_\nu=\lambda\}.$$
3. $\Lambda^{-\gamma}_n$ converges to $\Lambda^{-\gamma}$ in $\ell^1$.
The assumption that all series $D_n(s)$ are convergent in a common right half plane ${\mathsf{Re}}(s)>\gamma$ is a minimal necessary assumption, since if this is not the case, the questions we ask are void.
The series $D_n$ (when divergent at $s=0$), converges for ${\mathsf{Re}}(s)>\gamma_n$, where (see Chapter 1, Section 6 of [@Hardy]): $$\label{hardy1}
\gamma_n=\limsup_{\nu\rightarrow \infty}\frac{\log \nu }{\log \lambda_{n,\nu}}.$$ The hypothesis says that $\gamma:=\sup\gamma_n$ is finite.
The spectral zeta function $\zeta_X$ of a closed smooth Riemannian manifold converges absolutely for ${\mathsf{Re}}(s)>d/2$, where $d$ is the dimension of $X$ [@Rosenberg]. In the situation of Theorem \[Riem\], the assumption of a common half-plane of convergence hence follows from the fact that we assume that $\sup \dim(X_n)$ is finite. Hence Theorem \[Riem\] follows from Theorem \[main\].
Since we later want to interchange some limits, we will first prove:
\[U\] The sequences $$S_{n,N}(s):= \sum_{\nu=1}^N \frac{1}{\lambda_{n,\nu}^s}$$ converge to $D_n(s)$ for $N \rightarrow + \infty$ *uniformly in $n$*.
The uniform convergence means that $$\forall \epsilon>0, \exists N_0 \geq 1, \forall N>N_0, \;\;\;
\left|
\sum_{\nu=1}^N \frac{1}{\lambda_{n,\nu}^s}-D_n(s)
\right|\leq \epsilon$$ where $N_0$ *does not depend on $n$*. Now notice that by Equation (\[hardy1\]) we have $$\gamma_n= \lim \sup_{\nu \rightarrow \infty }\frac{ \log \nu}{\log \lambda_{n,\nu}}$$ and the series $D_n(s)$ converge for $\mathrm{Im}(s) \geq \gamma_n$. The assumption that there is a common half plane of convergence for all $D_n$ means that the sequence $\gamma_n$ is bounded by $\gamma=\sup \gamma_n$. The uniform convergence of $S_{n,N}(s)$ follows by repeating the argument found in the proof on page 7 of [@Hardy]. The proof there uses $\gamma_n$ for each $n$, but one may as well use (the same) $\gamma$ for all $n$.
Next, we will show that unbounded subsequences do not contribute to the limit. For this, suppose that, for some fixed $\kappa$, $(\lambda_{n_k,\kappa})_{k\in \mathbf{N}}$ is an unbouded subsequence, with $$\lim_{k\rightarrow \infty} \lambda_{n_k,\kappa}=\infty$$ and, by enlarging the subsequence if necessary, such that the sequence $$(\lambda_{n,\kappa})_{n\in (\mathbf{N}- \{n_k: k \in \mathbf{N} \}) }$$ is either bounded or the empty set. Now since $\lambda_{n_k,\kappa} \leq \lambda_{n_k,\mu}$ for $\mu \geq \kappa$, all sequences $(\lambda_{n_k,\mu})_{k\in \mathbf{N}}$ for $\mu \geq \kappa$ tend to infinity as well. Observe now that the series $$D^{\geq \kappa}_{n_k}(s):=\sum_{\nu=\kappa}^\infty \frac{1}{\lambda_{n_k,\nu}^s}$$ tends to the zero function as $n_k$ tends to infinity: $$\lim_{n_k\rightarrow \infty} D^{\geq \kappa}_{n_k}(s)= \sum_{\nu=\kappa}^\infty
\lim_{n_k\rightarrow \infty} \frac{1}{\lambda_{n_k,\nu}^s}=0,$$ for $s$ real positive (hence for all $s$ by analytic continuation). In the above equation we were allowed to interchange the order of the limits (in $n_k$ and the summation variable of $D_{n_k}$) since the series converge uniformly in $n_k$. Since we assume that $D_n$ is a convergent sequence of functions, it has the same limit as its the subsequence $D_{n_k}$.
Since we have now proven that unbounded subsequences do not contribute to the limit, we can assume that $\lambda_{n,\nu}$ is bounded in $n$, for all $\nu$, i.e., $$\forall n\in \mathbf{N}\;\; \lambda_{n,\nu} \leq c_\nu.$$ Then we can select a subsequence so that for all $\nu$, the limit $$\lim\limits_{k\rightarrow \infty} \lambda_{n_k,\nu}=\ell_\nu$$ exists. Not to overload notation, we will momentarily relabel the convergent subsequence $\lambda_{n_k,\nu}$ as $\lambda_{n,\nu}$. In particular, $\lambda_{n,1}$ converges to $\ell_1$. We will prove that $\lambda_{n,1}$ converges to $\lambda_1$. Let us rewrite $$D_n(s)=\frac{1}{\lambda_{n,1}^s}\left(\sum_{\nu=1}^\infty
\left(\frac{\lambda_{n,1}}{\lambda_{n,\nu}}\right)^s \right)$$ and $$D(s)=\frac{1}{\lambda_{1}^s}\left(\sum_{\nu=1}^\infty
\left(\frac{\lambda_{1}}{\lambda_{\nu}}\right)^s \right).$$ We now assume that *$s$ is an integer $s>\gamma$*. Since $D_n(s)\rightarrow D(s)$ we have that $$\label{11}
\lim_{n\rightarrow \infty}
\left(
\frac{\lambda_1}{\lambda_{n,1}}
\right)^s=
\frac{
\sum\limits_{\nu=1}^\infty
\left(\frac{\lambda_{1}}{\lambda_{\nu}}\right)^s
}
{
\lim_{n\rightarrow \infty} \sum\limits_{\nu=1}^\infty
\left(\frac{\lambda_{n,1}}{\lambda_{n,\nu}}\right)^s
}
\leq \sum_{\nu=1}^\infty
\left(\frac{\lambda_{1}}{\lambda_{\nu}}\right)^s$$ For the last inequality, we have used the fact that $\lambda_{n,\nu}>0$ and that the denominator is $\geq 1$.
Set $$\ell:=\lim_{n\rightarrow \infty} \frac{\lambda_1}{\lambda_{n,1}}.$$ We now consider the limit as $s \rightarrow \infty$ (along the integers) in Equation (\[11\]), to find $$\lim_{s\rightarrow \infty} \ell^s \leq
\#\{\lambda_i =\lambda_1 \}.$$
Now $$\lim_{s \rightarrow \infty} \ell^s=
\left\{
\begin{array}{ll}
1 & \mbox{ if } \ell=1 \\
0 & \mbox{ if } \ell<1 \\
\infty & \mbox{ if } \ell >1
\end{array}
\right..$$ Hence we find $\ell \leq 1$.
We also have the inequality $$\label{12}
\ell^s= \lim_{n\rightarrow \infty}
\left(
\frac{\lambda_1}{\lambda_{n,1}}
\right)^s=
\frac{
\sum\limits_{\nu=1}^\infty
\left(\frac{\lambda_{1}}{\lambda_{\nu}}\right)^s
}
{
\lim_{n\rightarrow \infty} \sum\limits_{\nu=1}^\infty
\left(\frac{\lambda_{n,1}}{\lambda_{n,\nu}}\right)^s
}
\geq
\frac{1}
{
\sum\limits_{\nu=1}^\infty
\left(\frac{\ell_1}{\ell_\nu }\right)^s
}.$$ In the inequality, we have used that we can interchange limit and summation in the denominator, by uniform convergence.
By taking the limit $s\rightarrow \infty$ (along the integers) we arrive at $$\lim_{s\rightarrow \infty} \ell^s \geq \frac{1}{\#\{ \ell_n=\ell_1 \}} >0.$$ We conclude from all the above that $\ell=1$, and hence that $$1= \frac{\#\{\lambda_i =\lambda_1 \}}{\#\{ \ell_n=\ell_1 \}}.$$ Now recall that we have relabelled before, so that we have actually shown that every convergent subsequence $(\lambda_{n_k,1})_{k\in \mathbf{N}}$ of $(\lambda_{n,1})_{n\in \mathbf{N}}$ tend to some limit, and since $\ell=1$ all these subsequences converge to the *same* limit $\lambda_{1}$. Therefore $(\lambda_n)_{n\in \mathbf{N}}$ itself is convergent to $\lambda_1$. We conclude that in general (viz., before erasing all unbounded subsequences), that every bounded subsequence of $(\lambda_n)_{n\in \mathbf{N}}$ converges to $\lambda_1$.
We now use an inductive argument to treat the general term. Namely, consider the Dirichlet series $$D^{\geq 2}_n(s):=D_n(s) -\lambda_{n,1}^{-s}$$ which (by what we have proven) converges to $$D^{\geq 2}(s):=D(s)-\lambda_1^{-s}.$$ These are still sequences of Dirichlet series of the same form, but with first eigenvalues $\lambda_{n,2}$ and $\lambda_2$. We can repeat the argument with this series, to conclude $\lambda_{n,2} \rightarrow \lambda_2$, etc.
This finishes the proof that (i) implies (ii).
Since we assume that ${\mathsf{Re}}(s)>\gamma$ is a common half plane of convergence of all series $D_n$ ($n=\emptyset,1,2,\dots$), the sums $\sum_{\nu=1}^\infty {\lambda_{n,\nu}^{-\gamma}}$ converge, and hence the sequences $\Lambda_n^{-\gamma}$ ($n=\emptyset,1,2,\dots$) belong to the Banach space $\ell^1$. We will now prove that $\Lambda^{-\gamma}_n\rightarrow \Lambda^{-\gamma}$ as elements of $\ell^1$.
In order to do so we have to prove that for every $\epsilon>0$ there is an $n_0 \in \mathbf{N}$ such that $n>n_0$ implies $$\sum_{\nu=1}^\infty \left|
\frac{1}{\lambda_{n,\nu}^{\gamma}} - \frac{1}{\lambda_\nu^{\gamma}}
\right| \leq \epsilon$$ It is known that if $(a_\nu)$ is a sequence of positive real numbers so that $\sum_{\nu=1}^\infty a_\nu$ converges, then all of its “tails” tend to zero: $$\lim_{N\rightarrow \infty} \sum_{\nu=N}^\infty a_\nu=0.$$ So given an $\epsilon> 0$ there is an $n_0$, which does not depend on $n$ (using the same $\gamma$ for all $n$), such that for $N\geq n_0$ $$\sum_{\nu=N}^\infty \left|
\frac{1}{\lambda_{n,\nu}^{\gamma}}\right| + \sum_{\nu=N}^\infty
\left|\frac{1}{\lambda_\nu^{\gamma}}
\right| <\epsilon/2.$$ Therefore, $$\begin{aligned}
\sum_{\nu=1}^\infty \left|
\frac{1}{\lambda_{n,\nu}^{\gamma}} - \frac{1}{\lambda_\nu^{\gamma}}
\right| &\leq&
\sum_{\nu=1}^N \left|
\frac{1}{\lambda_{n,\nu}^{\gamma}} - \frac{1}{\lambda_\nu^{\gamma}}
\right| +
\sum_{\nu=N}^\infty \left|
\frac{1}{\lambda_{n,\nu}^{\gamma}}\right| + \sum_{\nu=N}^\infty
\left|\frac{1}{\lambda_\nu^{\gamma}}
\right| \nonumber \\
&\leq &
\sum_{\nu=1}^N
\frac{|\lambda_\nu^{\gamma} -\lambda_{n,\nu}^{\gamma}|}
{\lambda_{n,\nu}^{\gamma}\lambda_\nu^{\gamma}} + \frac{\epsilon}{2} \nonumber \\
&\leq&
\sum_{\nu=1}^N
\frac{|\lambda_\nu^{\gamma} -\lambda_{n,\nu}^{\gamma}|}
{C} + \frac{\epsilon}{2}, \label{con-bound}\end{aligned}$$ where $$0\neq C=\inf_{1\leq \nu \leq N}
(\lambda_{1,\nu}^{\gamma}\lambda_\nu^{\gamma}) \leq
(\lambda_{n,\nu}^{\gamma}\lambda_\nu^{\gamma}).$$ Now the finite number ($\nu=1,\ldots,N$) of sequences $(\lambda_{n,\nu}^{\gamma})_{n\in \mathbf{N}}$ can be made to uniformly converge to $\lambda_\nu^{\gamma}$, that is, for every $\epsilon>0$ there is an $n_1$ such that $n>n_1$ implies
$$|\lambda_\nu^{\gamma} -\lambda_{n,\nu}^{\gamma}| \leq \frac{\epsilon C}{2N}$$ and inequality (\[con-bound\]) gives us the desired result for all $n\geq \max\{n_0,n_1\}$.
This proves that (ii) implies (iii). Finally, if $\Lambda_n^{-\gamma}$ converges to $\Lambda^{-\gamma}$, we have for every $s \in \mathbf{C}$ with ${\mathsf{Re}}(s)>\gamma$ that $\Lambda_n^{-s}$ converges to $\Lambda^{-s}$, and it follows easily that $D_n(s)$ converges to $D(s)$, pointwise in $s$. This proves that (iii) implies (i) and finishes the proof of the theorem.
Suppose $X=(X,g_X)$ and $Y=(Y,g_Y)$ are two isospectral connected smooth closed Riemannian manifolds, i.e., suppose their Laplace-Beltrami operators $\Delta_X$ and $\Delta_Y$ have the same spectrum with multiplicities $\Lambda_X=\Lambda_Y$ [@Rosenberg]. The spectrum $\Lambda_X$ is considered as a sequence $(\lambda_\nu)_{\nu=1}^\infty$ with $0 \leq \lambda_1 \leq \lambda_2 \leq \dots$, with finite repetitions.
The identity theorem for Dirichlet series [@Hardy] shows that such isospectrality can also be described as the manifolds having the same *zeta function* $\zeta_X = \zeta_Y$, where $$\zeta_{X}(s) := \mathrm{tr}(\Delta_X^{-s}) = \sum_{0 \neq \lambda \in \Lambda_X} \frac{1}{\lambda^s},$$ since connectedness implies that the zero eigenvalue has multiplicity one. In this context, Theorem \[main\] says the following, which is a “convergent” version of the identity theorem:
\[Riem\] Suppose $\{X_n\}_{n=1}^\infty$ is a sequence of connected closed smooth Riemannian manifolds such that $d:=\sup \dim X_n$ is finite, and suppose that $X$ is another closed smooth Riemannian manifold. Then the following statements are equivalent
1. For ${\mathsf{Re}}(s)>d/2$, the functions $\zeta_{X_n}(s)$ converge pointwise to $\zeta_X$;
2. For some $\gamma \in \mathbf{C}$ with $Re(\gamma)>d/2$, the sequence of eigenvalues $\Lambda^{-\gamma}_{X_n}$ converges to $\Lambda_X^{-\gamma}$ in $\ell^1$.
If the manifolds are closed and smooth and *of odd dimension*, but possibly disconnected, the equality $\zeta_X = \zeta_Y$ implies that also the multiplicity of the zero eigenvalue is equal for $X$ and $Y$, namely, it is minus the value at $0$ of the analytic continuation of $\zeta_X$ ([@Rosenberg], 5.2).
The circle of radius $r$ has $\Delta=-r^2 \partial^2_\theta$ (with $\theta\in [0,2\pi)$ the angle coordinate), spectrum $\lambda_{r,\nu} = r^{-2} \lceil \nu/2 \rceil^2$ and zeta function $\zeta_r(s)=r^{2s} \zeta(2s)$, where $\zeta$ is the Riemann zeta funtion. For varying $r \rightarrow r_0$, the convergence in the theorem happens for $\gamma>1/2$.
Already in the case of families of Riemannian manifolds, it can happen that $(\lambda_{n,\kappa})$ has unbounded subsequences for some fixed $\kappa$; for example, a family of circles whose radius tends to zero. However, for fixed $\kappa$, we have bounds on the eigenvalues of the form ([@BBG]) $$C_1 \sqrt[d]{\kappa^2} \leq \lambda_{n,\kappa} \leq
\frac{C_2}{\mathrm{vol}(X_n)} \sqrt[d]{\kappa^2},$$ where the constants $C_i$ depend on the dimension $d$, the diameter $D$, and a lower bound $R$ on the Ricci curvature of the manifolds under consideration. This implies that (at least if we fix the data $d,D$ and $R$, so we are in the Gromov precompact moduli space [@Gromov]) in unbounded subsequences, the volume should shrink to zero.
Applications: metric theories derived from Dirichlet series {#phys}
===========================================================
Distances in cosmology {#distances-in-cosmology .unnumbered}
----------------------
In connection with the averaging problem in cosmology and the question of topology change under evolution of the universe, Seriu [@Seriu] proposes to use eigenvalues of the Laplace-Beltrami operator on spatial sections of a cosmological model to construct a metric on the space of such Riemannian manifolds up to some notion of “large scale isospectrality”. More precisely, he raises two objections against the use of the plain difference of spectra as a measure: large energy contributions (corresponding to small scale geometry) should carry a lower weight, and the dominant weight should be put on the small spectrum (corresponding to large scale geometry); therefore, he introduces a cut-off $N$ and only compares the first $N$ eigenvalues. Secondly, the eigenvalue difference is not a dimensionless quantity, and because of this, he suggests comparing quotients of spectra. However, as $N \rightarrow +\infty$, his distance diverges.
From the above theorem, it also appears natural not to use a cut-off function, but rather use a distance *between the zeta functions* (which, like partition functions, give more weight to low energy in their region of convergence), considered as complex functions; here, one may use classical notions of distance between complex functions [@Conway] used in the study of limits of holomorphic or meromorphic functions. Also, the *quotient* of two zeta functions is a dimensionless function. Actually, a distance between Riemannian manifolds *up to isometry* was constructed by the first author and de Jong, who have furthermore given a spectral characterization of when a diffeomorphism of closed smooth Riemannian manifolds is an isometry, in terms of equality of more general zeta functions under pullback by the map [@CdJ]. Also this distance is based on the dimensionless object of quotients of zeta functions.
In conclusion, we propose the following function as a distance on suitable spaces of Riemannian geometries up to isospectrality:
Let $\mathcal{M}$ denote a space of Riemannian manifolds up to isospectrality, with $$\sup \{ \dim(X) \colon X \in \mathcal{M} \} <2 \gamma$$ finite. Then for any $X_1, X_2 \in \mathcal{M}$, the function $$d(X_1,X_2):= \sup_{\gamma < s < \gamma+1} \left| \log \left| \frac{\zeta_{X_1}(s)}{\zeta_{X_2}(s)} \right| \right|$$ where ${\mathsf{Re}}(s)>\gamma$ is common plane of convergence for the spectral zeta functions of $X_1$ and $X_2$, defines a metric on $\mathcal{M}$.
The function $d$ is positive, and if $d(X_1,X_2)=0$, then $|\zeta_{X_1}(s)|=|\zeta_{X_2}(s)|$ for all $s$ in the interval $]\gamma,\gamma+1[$. Since this set has accumulation points, and since the zeta function is positive real for such values of $s$, we find that $\zeta_{X_1} = \zeta_{X_2}$ as complex functions. Hence the main theorem (or the identity theorem for Dirichlet series) implies that $X_1$ and $X_2$ are isospectral. The function $d$ is symmetric, since $\left| \log(x^{-1}) \right| = \left| \log(x) \right|$. Finally, the triangle inequality follows from $$\frac{\zeta_{X_1}(s)}{\zeta_{X_3}(s)} = \frac{\zeta_{X_1}(s)}{\zeta_{X_2}(s)} \cdot \frac{\zeta_{X_2}(s)}{\zeta_{X_3}(s)}$$ and the usual properties of the absolute value.
This is a distance that weighs correctly the energy contributions, but does not depend on a cut-off, nor diverges if a cut-off tends to infinity. Also, convergence in the topology defined by this distance can be easily understood from Theorem \[main\].
Taking the supremum over $\gamma < s < \gamma+1$ is quite random, any set with an accumulation point and avoiding the poles of the zeta functions will do. Also, the distance $d$ can be replaced by $d/(1+d)$ to have it take values in the unit interval. The exact numerical values of the metric are not so relevant, but rather, their interrelation and the topology and uniformity that they induce.
If $S_r$ denotes a circle of radius $r$, then $$d(S_{r_1}, S_{r_2}) = 4 \left| \log(r_1/r_2) \right|.$$ This example shows that the distance can be non-differentiable in the parameter space of a family.
Let us compute the spectral distance between a sphere $S$ and a real projective space $\R\PP^2$ with the same volume $4 \pi$. The zeta functions are $$\zeta_{S}= \sum_{\nu=1}^\infty \frac{2 \nu +1}{\nu^s (\nu+1)^s} \mbox{ and } \zeta_{\R\PP^2}= \sum_{\nu=1}^\infty \frac{4 \nu +1}{\nu^s (2 \nu+1)^s}.$$ A numerical experiment suggests that the maximum in the distance formula is attained at $s=2$, and there we get $$d(S,\R\PP^2) = \log (4-\pi^2/3) \approx 0.342.$$
Eigenvalues as dynamical variables {#eigenvalues-as-dynamical-variables .unnumbered}
----------------------------------
Gravity coupled to matter can be given a spectral description using the framework of noncommutative geometry [@Connes]. Even by ignoring the matter part, one arives at an interesting description of classical gravity (general relativity) in terms of spectral data. These spectra form a diffeomorphism invariant set of coordinates on the space of manifolds, up to isospectrality. Diffeomorphism invariant coordinates are an important prerequisite for certain programmes to quantize gravity. In this way, spectra were used as dynamical variables for classical gravity by Landi and Rovelli [@LandiRovelli]. Our theorem shows that convergence in these spacetime variables is the same as convergence of classical Dirichlet series in complex analysis.
A distance on Number Fields {#a-distance-on-number-fields .unnumbered}
---------------------------
Consider the Dedekind-zeta function for a number field $K$ as $\zeta_K(s):=\sum N(I)^{-s}$, where the sum runs over all non-zero ideals $I$ of $\mathcal{O}_K$, the ring of integers of $K$, and $N(I)$ is the norm of the ideal $I$. For any fixed $a>0$, we can define a distance $$d(K_1,K_2) := \sup_{1 < s < 1+a} \left| \log \left| \frac{\zeta_{K_1}(s)}{\zeta_{K_2}(s)} \right| \right|.$$ For a prime number $p$ and a positive integer $f$, let $I_K(p,f)$ denote the number of ideals of $\mathcal{O}_K$ with norm $p^f$. Since $\zeta_K(s)$ admits an Euler product $$\zeta_K(s)=\prod_{P \in \mathrm{Spec}\mathcal{O}_K} \frac{1}{1-N(P)^{-s}},$$ we find $$d(K_1,K_2) = \sup_{1 < s < 1+a} \left| \sum_{p,f} \left(I_{K_1}(p,f)-I_{K_2}(p,f)\right) \log \left(1-p^{-fs}\right) \right|.$$
Let us estimate the distance between the field of rational numbers $\Q$ and a real quadratic field $\Q(\sqrt{D})$ for $D>0$. Let $I$ denote the primes inert in $\Q(\sqrt{D})$, $S$ the set of split primes and $R$ the set of ramified primes, i.e., the divisors of the squarefree part of $D$. Using the Euler products, we find $$\begin{aligned}
d(\Q,\Q(\sqrt{D}) &= \sup_{1 < s < 1+a} \left| \log \left| L(\chi_D,s)
\right| \right|
\\ & = \sup_{1 < s < 1+a} \left| \log \left|
\frac{
\prod\limits_{p \in S} \left(1-p^{-s}\right)^{-2}
\cdot \prod\limits_{p \in I} (1-p^{-2s})^{-1}
\cdot \prod\limits_{p \in R}
(1-p^{-s})^{-1}
}
{ \prod\limits_{p} (1-p^{-s})^{-1} }
\right| \right|.
\end{aligned}$$
As $D \rightarrow 1$, the distance goes to zero. If $D_i$ denotes the product of the first $i$ prime numbers, then $R$ increases to the set of all primes, so then $\lim\limits_{i \rightarrow + \infty} d(\Q,\Q(\sqrt{D_i})) =0$, too. We verified numerically in SAGE that $L(\chi_{D_i},1) \rightarrow 1$ as $i \rightarrow +\infty$; The figure is a plot of $\left| L(\chi_{D_i},1) \right|$ as a function of $i$.
{width="8cm"}
Series with general coefficients
================================
In this section, we study what happens if we have pointwise convergence of general Dirichlet series $$D_n(s)=\sum_{\nu \geq 1} a_{n,\nu} e^{ -s \mu_{n,\nu}} \rightarrow D(s)=\sum_{\nu \geq 1} a_\nu e^{ -s \mu_\nu},$$ all of which converge absolutely in a common half plane ${\mathsf{Re}}(s)>\gamma$, with $a_*$ complex coefficients, and $\mu_*$ is a *strictly* increasing sequence of real numbers. The previous case occurs when $\{\mu_*\} = \{\log \lambda_*\}$ and $a_*$ counts the multiplicities in $(\lambda_*)$. In this paper, we will not discuss subtleties that arise from such series that have a different region of convergence and absolute convergence. We start by discussing two special cases.
The first is when $\mu_{n,\nu}=\log \nu$ for all $n=\emptyset,1,2,\dots$. In this case, we set $z=e^{-s}$ and we get a (pointwise) convergence of Taylor series $$D_n(z) = \sum_{\nu \geq 0} a_{n,\nu} z^\nu \rightarrow D(z) = \sum_{\nu \geq 0} a_{\nu} z^\nu.$$ In this case, the individual series $D_*$ converge in $\Omega:=\{z>e^{-\gamma}\}$ to a holomorphic function (by assumption). Evaluation at zero gives $\lim_{n \rightarrow + \infty} a_{n,0} = a_0$, and we can proceed by induction to conclude that $$\lim_{n \rightarrow + \infty} a_{n,\nu} = a_\nu$$ for all $\nu$.
Alternatively, one can use the representation of the coefficients by a complex contour integral to deduce the result “in a more complicated way”. Namely, fix $\epsilon>0$, and let $n_0$ satisfy that $|D_n(z)-D(z)|<\epsilon$ for $n>n_0$, uniformly in $z \in K \subset \Omega$, where $K$ is a compact set. For a contour $C \subset K$ around $z=0$ (independent of $n$), we have $$|a_{n,\nu} - a_\nu| \leq \frac{1}{2 \pi} \int_{C} |D_n(z)-D(z)| |z|^{-n-1} dz \leq \varepsilon.$$
The reason for providing this second proof is that it leads us to the next special case, in which we use the analogue of the integral representation for the coefficients for general Dirichlet series, also called Perron’s formula. This formula gives a representation of the terms of a general Dirichlet series by integration over a vertical line in the complex plane, and since this integration domain, unlike the contour in the Taylor series proof, is not compact, we will need to work more to establish the result (or assume uniform convergence on an entire half-line, which seems too strong an assumption). This second special case occurs if $\mu_{n,\nu}$ is constant in $n$. Then we have the following result:
\[qwe\] Assume that $D_n$ ($n=\emptyset,1,2,\dots$) is a set of Dirichlet series that converge absolutely in a common half plane ${\mathsf{Re}}(s)>\gamma$, and such that $D_n(s) \rightarrow D(s)$ converges pointwise there. Assume that $\mu_{n,\nu}=\mu_\nu$ is independent of $n$. Then for every $n$, we have $$\lim_{n \rightarrow + \infty} a_{n,\nu} = a_\nu;$$ actually, for $\sigma_1>\gamma $, we have a convergence of sequences $$(a_{n,\nu} e^{-\sigma_1 \mu_\nu})_{\nu=1}^\infty \rightarrow (a_\nu e^{-\sigma_1 \mu_\nu})_{\nu=1}^\infty \mbox{ in } \ell^{\infty}.$$
Consider the difference $$B_n(s):= D_n(s)-D(s)=\sum_{\nu \geq 1} {b_{n,\nu}}{e^{-s \mu_\nu}},$$ where $$b_{n,\nu}:=a_{n,\nu}-a_{\nu}.$$
Now according to Theorem I.3.1 in [@mandel] we have the following integral representation for every $n$ and every fixed $\nu$: $$\begin{aligned}
\left|
b_{n,\nu}e^{- \sigma_1 \mu_\nu}
\right|
&= &
\left|
\lim_{T\rightarrow \infty} \frac{1}{T} \int_{0}^T B_n(\sigma_1 + i t) e^{\mu_\nu i t} dt
\right| \\
&\leq &
\lim_{T\rightarrow \infty} \frac{1}{T} \int_{0}^T \left| B_n(\sigma_1 + i t) \right| dt
$$ We could finish the proof here by assuming that $D_n$ converges uniformly to $D$ on the entire line ${\mathsf{Re}}(s)=\sigma_1$. However, we can avoid this (strong) hypothesis by proving the following lemma:
\[bound-lemma\] For every $\epsilon>0$ there is a $t_0 \in \mathbf{R}$ such that for $t \in \R$ and all $n$, $$|B_n(\sigma_1 + it)| \leq \epsilon + |B_n(\sigma_1+it_0)|.$$
We are then finished with the proof of Theorem \[qwe\], since now, given any $\epsilon>0$, the pointwise convergence at $t_0$ implies that there exists $n_0$ such that for all $n>n_0$, $$|B_n(\sigma_1 + it)| \leq \epsilon + |B_n(t_0)| \leq 2 \epsilon$$ and then the above inequality becomes $$\left|
b_{n,\nu}e^{- \sigma_1 \mu_\nu}
\right| \leq 2 \epsilon,$$ Since $\sigma_1$ and $\nu$ are fixed, $e^{- \sigma_1 \mu_\nu}$ is a non-zero constant, and this proves that $b_{n,\nu} \rightarrow 0$ as $n \rightarrow +\infty$. Since the $\epsilon$-bound holds uniformly in $\nu$, we do find the $\ell^\infty$ convergence as stated.
Since the series $D_n$ are absolutely convergent on a common half plane, their sequences of tails tend to zero uniformly in $n$, that is, for every $\epsilon>0$ there is an $N$ that is independent of $n$ such that $$\sum_{\nu=N+1}^\infty | a_{n,\nu} e^{-s \mu_\nu}|+ \sum_{\nu=N+1}^\infty
|a_\nu e^{-s \mu_\nu}| < \epsilon.$$ Hence $$|B_n(s)|
\leq \left| B_n^{\leq N}(s)
+D_n^{> N}(s) -D^{> N}(s) \right|
\leq \left| B_n^{\leq N}(s) \label{bb1}
\right| + \epsilon$$ We will now estimate the sum of the first $N$ terms on a vertical line ${\mathsf{Re}}(s)=\sigma_1$. Consider the function $f:\mathbf{R} \rightarrow \left(S^1\right)^{N}$ given by $$t\mapsto (e^{i t \mu_{1}}, \ldots, e^{i t \mu_{N}})$$ and the function $$F:\left(S^1\right)^{N} \rightarrow \mathbf{C}$$ sending $$(P_1,\ldots, P_N) \mapsto
\sum_{\nu=1}^N
e^{-\sigma_1 \mu_\nu} P_\nu.$$ The function $F$ is continuous on a compact set therefore it attains a maximal value $M$ at a point $A^0:=(P_1^0,\ldots, P_N^0)$.
There exists $t_0 \in \mathbf{R}$ such that all the numbers $\{t_0 \mu_\nu\}_{\nu=1}^\infty$ are irrational.
The set of multiples $\{ b \rho\}$ of a given real number $\rho \in \mathbf{R}$ such that $b\rho \in \mathbf{Q}$ is just $\frac{1}{\rho} \mathbf{Q}$ and this set is denumerable. A denumerable union of denumerable sets cannot exhaust the set of reals and the result follows.
This proves that the set $f(\mathbf{R})$ is dense in $\left(S^{1} \right)^{N}$. Therefore, for every $\delta>0$ there exists $t_0 \in \mathbf{R}$ such that $ |f(t_0)-A^0| \leq \delta$, and hence, since $F$ is continuous, $$\label{bb2}
|B_n^{\leq N}(t_0) -M|=|F(f(t_0))-F(A^0)|< \epsilon.$$ Since $M$ is the maximum, for all $t\in \mathbf{R}$, we have $$\label{bb3}
|B_n^{\leq N}(\sigma_1+it)| \leq M\leq |B_n^{\leq N}(\sigma_1+it_0)|+|B_n^{\leq N}(\sigma_1+it_0) -M|.$$ By equations (\[bb1\]), (\[bb2\]) and (\[bb3\]) we now have $$|B_n(\sigma_1 + i t)| \leq \epsilon + |B_n^{\leq N}(\sigma_1 + it)| \leq \epsilon + |B_n^{\leq N}(\sigma_1+it_0)|+|B_n^{\leq N}(\sigma_1+it_0) -M| \leq 2 \epsilon + |B_n^{\leq N}(\sigma_1+it_0)|,$$ and this finishes the proof of lemma \[bound-lemma\].
The proof of the lemma also completes the proof of the theorem.
Let $X$ denote a closed smooth Riemannian manifold and let $a \in C^{\infty}(X)$ denote a smooth function. Define a generalized Dirichlet series by $$\zeta_{X,a}:=\mathrm{tr}(a\Delta_X^{-s}),$$ cf. [@CdJ]. Then $$\zeta_{X,a} = \sum_{0 \neq \lambda \in \Lambda_X} \frac{1}{\lambda^s} \cdot \int_X a \sigma_{X,\lambda},$$ where $$\sigma_{X,\lambda}:=\sum_{\lambda \dashv \Psi} |\Psi|^2$$ is the sum of the elements $\Psi$ of an orthonormal basis of eigenfunctions that belong to the eigenvalue $\lambda$. In [@CdJ], it was proven that a diffeomorphism $\varphi \colon Y \rightarrow X$ between closed Riemannian manifolds *with simple spectrum* is an isometry precisely if $\zeta_{X,a} = \zeta_{Y,\varphi^*(a)}$ for all $a \in C^\infty(X)$ (and there is also a version if the spectrum is not simple).
Now assume that we have a compact manifold $X$ and a family $\{g_r\}$ ($r \in \R$) of isospectral metrics with simple eigenvalues on $X$ (cf. Gordon and Wilson [@GW] for the existence of such families). Denote by $\Psi_{r,\lambda}$ the normalized real eigenfunction for the metric $g_r$ corresponding to the eigenvalue $\lambda$. If all zeta functions converge in the sense that $$\label{ZZ} \zeta_{X,g_r,a} \rightarrow \zeta_{X,g_s,a} \mbox{ for all } a$$ then we find from the above result that $$\int a \Psi^2_{r,\lambda} d\mu_r \rightarrow \int a \Psi^2_{s,\lambda} d\mu_s$$ for all functions $a \in C^\infty(X)$, where $\mu_r$ is the measure belonging to the metric $g_r$.
Taking residues at $\dim(X)/2$ in (\[ZZ\]) for $a=1$, we find that the volume of $X$ in $g_r$ is constant, and then taking residues for general $a$, we find that for all $a \in C^\infty(X)$, $$\int a d\mu_r \rightarrow \int a d\mu_s.$$ Changing variables, we get that $$\int a \left(1-\frac{d\mu_r}{d\mu_s}\right) d\mu_s \rightarrow 0,$$ and hence that $\frac{d\mu_r}{d\mu_s} \rightarrow 1$. From the above theorem, we conclude that $\Psi^2_{r,\lambda}\frac{d\mu_r}{d\mu_s} \rightarrow \Psi^2_{s,\lambda},$ and hence that $$\Psi^2_{r,\lambda} \rightarrow \Psi^2_{s,\lambda},$$ a convergence of squared eigenfunctions.
General case {#general}
============
Finally, in the most general case of varying coefficients and varying exponents, we prove a theorem about accumulation points. First, we do some preparation.
\[strongI\] For a fixed strictly positive real function $g$, define for a real function $f$, the $g$-sup norm as $$||f||_{\infty,g}:=\sup_{x \in \R} \left|\frac{f(x)}{g(x)}\right|,$$ when it is defined. We say that a sequence of functions $\{f_n\}$ *converges multiplicatively* to a real function $f$ if there exists a strictly positive real function $g$ that is integrable with respect to the multiplicative Haar measure on $\R^*$ (i.e., such that $\int_{\R} g(x)\frac{dx}{|x|}<+\infty$), such that $$|| f_n(x) - f(x) ||_{\infty,g} \rightarrow 0$$ for $n \rightarrow + \infty$.
For $f$ a complex function defined for ${\mathsf{Re}}(s)=c$, and $x \in \R$, denote by $$I^c_x(f):= \int_{{\mathsf{Re}}(s)=c} f(s) e^{xs} \frac{ds}{s}.$$
The relevance of this integral for the theory of Dirichlet series lies in the following formula of Perron: if $D(s)=\sum_{\nu \geq 1} a_\nu e^{ -s \mu_\nu}$ is convergent for $s=\beta+i \gamma$ and $c>0$, $c > \beta$, $x\in \mathbf{R}$, $x \geq \beta$, then $$\sum_{\lambda_n \leq x} a_n =\frac{1}{2 \pi i} I^c_x(D),
$$ with the convention that the last summand on the left hand side is multiplied by $1/2$ if $x$ equals some $\lambda_\nu$.
Since $I^c_x(D)$ does not depend on $c$ once it satisfies the conditions for Perron’s formula, we will now write $I_x(D)$ for $I^c_x(D)$ with any suitable $c$.
\[inter\] If $\{f_n(c+it)\}$ converges multiplicatively to $f(c+it)$ in $t$, then for all $x \in \R$, $$\lim_n I_x(f_n)= I_x( \lim_n f_n)=I_x(f).$$
We have $$\begin{aligned}
\left| I_x(f_n)-I_x(f)\right| &\leq \int_{\R} \left| \frac{f_n(c+it)-f(c+it)}{
c+it} e^{x(c+it)} \right| dt \\
&\leq \int_{\R} \left| \frac{f_n(c+it)-f(c+it)}{c+it} e^{x(c+it)} \right| dt \\
&\leq e^{cx} \left( \int_{\R} \frac{g(t)}{\sqrt{c^2+t^2}}dt \right) \cdot ||f_n(c+it)-f(c+it)||_{\infty,g} \\
&\leq e^{cx} \left( \int_{\R} \frac{g(t)}{|t|}dt \right) \cdot ||f_n(c+it)-f(c+it)||_{\infty,g} \\
& \leq C \varepsilon,
\end{aligned}$$ with $C=e^{cx} \left( \int_{\R} \frac{g(t)}{|t|}dt \right)$ finite constant, for $n$ sufficiently large. This proves the desired result.
Before stating the main result of this section, we need to introduce some notation:
\[subseq\] Assume that all sequences $(\lambda_{n,j})_{n=1}^\infty$ are bounded. Let $ \ell_i^{(j)}$, $i\in I_j$ be the accumulation points of sequence $(\lambda_{n,j})_{n=1}^\infty$.
We consider a subsequence $n_k$ such that for all $j$ $ \lim\limits_{n_k\rightarrow \infty }\lambda_{n_k,j}=\ell_{i_j}^{(j)}$ for a selection $i_j \in I_j$. Notice that the sequences $(\lambda_{n_k,j})_{k=1}^\infty$ and $(\lambda_{n_k,j+1})_{k=1}^\infty$ satisfy $\lambda_{n_k,j} < \lambda_{n_k,j+1}$ but they can tend to the same accumulation point.
For the infinite vector of convergent sequences $\big( (\lambda_{n_k,j})_{k=1}^\infty \big)_{j\geq 1}$ converging to the infinite vector $\big( \ell_{i_j}^{(j)} \big)_{j\geq 1}$ we consider the sequence $m_1,m_2,\ldots,$ such that $$\ell_{i_1}^{(1)}=\ell_{i_2}^{(2)}=\cdots=\ell_{i_{m_1}}^{(m_1)},
\ell_{i_{m_1+1}}^{(m_1+1)}=\ell_{i_{m_1+2}}^{(m_1+2)}=\cdots=\ell_{i_{m_2}}^{(m_2)}, \mbox{ etc.}$$
$$\xymatrix{
(\lambda_{n_k,1}) \ar@{.}[r] \ar[d] & (\lambda_{n_k,m_1}) \ar[dl] &
(\lambda_{n_k,m_1+1}) \ar@{.}[r] \ar[d] & (\lambda_{n_k,m_2}) \ar[dl]\\
\ell_{i_1}^{(1)} & & \ell_{i_{m_1+1}}^{(m_1+1)} &
} \cdots$$
\[dif\] We use the notation of \[subseq\]. Assume that $D_n$ converges multiplicatively to $D(s)=\sum_{j \geq 1} a_j e^{-s \log \lambda_j}$. Then, $\lambda_j$ are accumulation points for some sequence $(\lambda_{n,j'})_{n=1}^\infty$.
Consider the set of subsequences $\big( (\lambda_{n,j})_{n=1}^\infty \big)_{j\geq 1}$ converging to the infinite vector $\big( \ell_{i_j}^{(j)} \big)_{j\geq 1}$. Suppose that the sequences $(\lambda_{n_k,j})_{n=1}^\infty$ for $j=m_\mu+1,\dots,m_{\mu+1}$ converge to $\ell$. Set $$A_{n_k}^{(\mu)}:=\sum_{j=m_\mu+1}^{m_{\mu+1}} a_{n_k,j}, \mbox{ for } \mu \geq 0.$$ Then, $$\label{sum-fo}
\lim A_{n_k}^{(\mu)}=\left\{
\begin{array}{l}
a_i \ \ \mbox{if } \ell=\lambda_i, \\
0 \ \ \mbox{ otherwise. }\\
\end{array}
\right.$$
Assume that the set of subsequences $\big( (\lambda_{n_k,j})_{k=1}^\infty \big)_{j\geq 1}$ converges to the set of accumulation points $(\ell_i^{(j)})$.
Consider the first eigenvalue $\lambda_1$ of $D$. If $\ell$ is the first element in the set $\ell_i^{(j)}$ that is smaller than $\lambda_1$ then by choosing $x$ such that $\ell < x < \lambda_1$, by Perron’s formula, we have that $I_x(D_{n_k})=\sum_{j=1}^{m_1} a_{n_k,j}=A_{n_k}^{(0)}$ should tend to $I_x(D)=0$ since $x<\lambda_1$. This proves that $A_{n_k}^{(0)}$ tends to zero as desired. We proceed now to the next accumulation point that is smaller than $\lambda_1$ and by the same argument we prove that $\lim\limits_{n_k\rightarrow \infty} \sum_{j=1}^{m_2} a_{n_k,j}=0$. Then, since the limit of the sum of the first $m_1$ terms tends to zero we have that $$\lim\limits_{n_k \rightarrow \infty} \sum_{j=m_1+1}^{m_2}a_{n_k,j}=0,$$ and so the desired result is proved for all $\ell< \lambda_1$.
We will prove now that $\lambda_1$ is an accumulation point. Indeed, for sufficiently small $\epsilon>0$ the quantity $$I_{\lambda_1-\epsilon}(D)-I_{\lambda_1+\epsilon}(D)=a_1 \neq 0.$$ Using the above equation and lemma \[inter\] we obtain $$\lim_{n_k \rightarrow \infty} \big(I_{\lambda_1-\epsilon}(D_{n_k})-I_{\lambda_1+\epsilon}(D_{n_k}) \big)=a_1=
\lim \sum_{\lambda_1-\epsilon < \lambda_{n_k,j} < \lambda_1+\epsilon} {a_{n_k,j}}.$$ So by taking small $\epsilon$ we can find a subsequence tending to $\lambda_1$ so $\lambda_1$ is one of the accumulation points of the sequence $\big( (\lambda_{n_k,j})_{k=1}^\infty \big)_{j\geq 1}$. Notice also that desired result of eq. (\[sum-fo\]) is also proved.
We continue the proof by induction by taking $\ell$ to be inside $\lambda_1$ and $\lambda_2$ so the corresponding sum tends to zero, then we take $\ell$ to be $\lambda_2$, then inside $\lambda_2$ and $\lambda_3$ etc.
Relation with Laplace-Stieltjes Transform {#L}
=========================================
The notion of Dirichlet series and Laplace transforms can be unified in terms of the Riemann-Stieltjes integrals (Widder [@Widder:29], compare [@Arendt:2001]).
Suppose $\omega \geq 0$ is a real number.
1. The space $\mathrm{Lip}_\omega$ is defined as the set of functions $F \colon \R_{ \geq 0} \rightarrow \R$ with bounded norm $$||F||_{\mathrm{Lip},\omega}:= \sup_{0 \leq s < t}\frac{ |F(t)-F(s) |}{(t-s)e^{\omega t}} < \infty.$$
2. The space $\mathrm{Wid}_\omega$ is defined as the space of smooth function $(\omega,\infty) \rightarrow \R$ with bounded norm $$||D||_{\mathrm{Wid},\omega}:= \sup_{{s>\omega}\atop {k \in \mathbf{N}}} \frac{(s-\omega)^{k+1}}{k!} \left| \frac{d^k D}{ds^k}(s) \right| < \infty.$$
The main result is now that the so-called *Laplace-Stieltjes transform* $$F \mapsto \int_0^\infty e^{-st} dF(t)$$ induces an isometric isomorphism $
\mathrm{Lip}_\omega \rightarrow \mathrm{Wid}_\omega
$ ([@Arendt:2001], Thm. 2.4.1).
Widder ([@Widder:29], Theorems 11.2 12.4) proved that a Dirichlet series of the form $D(s)=\sum_{\nu} a_\nu e^{-s \mu_\nu}$ convergent for ${\mathsf{Re}}(s)>\omega$ is in the space $\mathrm{Wid}_{\omega}$. Also, such $D$ is the Laplace-Stieltjes transform of $$F(t)=\sum_{\nu=0}^\infty a_\nu H(t-\mu_\nu),$$ where $H$ is the Heaviside step function. Thus, we immediately conclude the following:
\[final\] Suppose $D_n(s)=\sum_{\nu \geq 1} a_{n,\nu} e^{ -s \mu_{n,\nu}}$ is a sequence of Dirichlet series each converging absolutely in a common half plane ${\mathsf{Re}}(s)>\gamma$; then for any $\omega>\gamma$, $D_n$ converges to a Dirichlet series $D(s)=\sum_{\nu \geq 1} a_\nu e^{ -s \mu_\nu}$ in $\mathrm{Wid}_{\omega}$-norm if and only if $$\sum_{\nu=0}^\infty \big( a_{n,\nu} H(t-\mu_{n,\nu}) - a_\nu H(t-\mu_\nu) \big) \rightarrow 0$$ in $\mathrm{Lip}_\omega$-norm.
It would be interesting to deduce Theorem \[qwe\] and \[dif\] from Theorem \[final\].
|
---
abstract: 'Many low-dimensional materials are well described by integrable one-dimensional models such as the Hubbard model of electrons or the Heisenberg model of spins. However, the small perturbations to these models required to describe real materials are expected to have singular effects on transport quantities: integrable models often support dissipationless transport, while weak non-integrable terms lead to finite conductivities. We use matrix-product-state methods to obtain quantitative values of spin/electrical and thermal conductivities in an almost integrable gapless ($XXZ$-like) spin chain. At low temperatures, we observe power laws whose exponents are solely determined by the Luttinger liquid parameter. This indicates that our results are independent of the actual model under consideration.'
author:
- 'Y. Huang'
- 'C. Karrasch'
- 'J. E. Moore'
bibliography:
- 'scaling.bib'
title: Scaling of electrical and thermal conductivities in an almost integrable chain
---
The physics of many one-dimensional systems with idealized interactions is rather special: the quantum Hamiltonian has infinitely many independent conserved quantities that are sums of local operators. Such Hamiltonians are called “integrable” in analogy with classical Hamiltonian systems that decompose into independent action and angle variables. Examples relevant to experiments on crystalline materials include the Hubbard model of electrons and the XXZ model of spins; ultracold atomic systems can realize integrable continuum models of bosons. However, in all these cases it is expected that integrability is only an approximation to reality and that experimental systems have integrability-breaking perturbations which, while small, drastically change some of the physical properties.
Transport properties provide an experimentally important example of the effects of non-integrable perturbations. In integrable systems, parts of charge, spin, or energy currents are conserved, and thus transport is dissipationless even at non-zero temperature. This corresponds to a finite “Drude weight” $D$ in the frequency-dependent conductivity: [@betheT0; @bethezotos; @bethekluemper; @edfabian; @edmillis; @qmcsorella; @qmcgros; @Sirkerprl; @Sirker; @prosen; @karrasch; @qmctherm; @bethetherm; @fabianrev; @cher1; @sirkerw] () = 2 D () + \_[reg]{.nodecor}() . In reality, many quasi-one-dimensional systems are expected to be well described by integrable Hamiltonians *plus weak non-integrable perturbations*. The zero-frequency conductivity is regularized ($D=0$) by these perturbations [@integrab; @edmillis; @qmcsorella; @qmcgros; @roschandrei; @Jungprb; @Jung; @Jungth; @fabianrev; @edfabian; @qmctherm]. An experimental example is the large (but not dissipationless) anisotropic thermal transport observed in Sr$_{14}$Cu$_{24}$O$_{41}$ attributed to a long mean free path of quasi-1D magnons [@hess; @ott; @hlubek]. However, computing $\sigma_\textnormal{reg}(\omega)$ quantitatively for a microscopic non-integrable Hamiltonian is a challenging problem.
We study a generic gapless non-integrable system (a $XXZ$-like spin chain) using density matrix renormalization group (DMRG) methods, which were developed in the past few years to access finite-temperature dynamics of correlated systems. Using linear prediction, we extrapolate current correlations functions to large times. This allows to quantitatively observe the destruction of the thermal and electrical Drude weights; we compute the corresponding conductivities and analyze how they depend on temperature and the strength of the non-integrable perturbation. Our key observation is power law scaling behavior at low temperatures. The corresponding exponents are functions of the equilibrium Luttinger liquid parameter, which indicates that our results should hold for any gapless non-integrable model in which no conserved quantity protects the current. We compare our numerics to a low-energy field theory calculation using bosonization techniques which we obtain by adapting previous results [@Sirker; @Sirkerprl] to the non-integrable perturbation in our model.
![The phase diagram [@Alcaraz; @Okamoto] of (\[XXZ\]). The points share the same Luttinger liquid parameter $K\approx2.4$ computed by DMRG [@llpaper].[]{data-label="phase"}](phase.eps){width="0.7\linewidth"}
![The distributions of $\{E_{i+1}-E_i\}$, where $E_1\le E_2\le\ldots\le E_{4862}$ are the eigenenergies of (\[XXZ\]) for $L=18, \Delta=-0.8$ in the $S_z=1,k=2\pi/9$ sector (the total magnetization $S^z=\sum_{i=1}^LS_i^z$ and the momentum $k$ are conserved). The green curve and red curve are best-fit exponential and Wigner-Dyson (orthogonal ensemble) distributions respectively. Neither the Poisson nor the Wigner-Dyson distribution appears clearly if we do not restrict to a symmetry sector of the model. The crossover to Wigner-Dyson with increasing $h$ is observed independent of the choice of $\Delta$ and symmetry sector.[]{data-label="stat"}](stat.eps){width="0.7\linewidth"}
[*The model*]{}. We use the XXZ model in presence of a staggered magnetic field which breaks integrability (which we will demonstrate explicitly by considering level statistics). Its Hamiltonian is given by $H=\sum_{i=1}^L h_i$ with $$h_i = S_i^xS_{i+1}^x+S_i^yS_{i+1}^y+\Delta S_i^zS_{i+1}^z+\left(-1\right)^ihS_i^z .
\label{XXZ}$$ Without the staggered field, (\[XXZ\]) is gapless for $|\Delta|\le1$ and gapped for $|\Delta|>1$. Bosonization leads to the low-energy effective Hamiltonian for $|\Delta|\le1$ and infinitesimal $h$: $$\label{effective}\begin{split}
H=& \frac{v}{2}\int dx\left(\Pi^2+\left(\partial_x\phi\right)^2\right)+ch\int dx\cos\left(2\sqrt{\pi K}\phi\right)\\[1ex]
& + H_\textnormal{umklapp} + H_\textnormal{band curv.} + H_{\textnormal{higher terms in } h}~,
\end{split}$$ where $\Pi$ is the conjugate momentum of the bosonic field $\phi$ with the canonical commutation relation $[\phi(x),\Pi(y)]=i\delta(x-y)$. The first term in (\[effective\]) describes a Luttinger liquid. The Luttinger liquid parameter $K$ is given through Bethe ansatz: $2K\arccos(-\Delta)=\pi$, and the coefficients $v$ and $c$ are also known exactly [@Lukyanovprb; @Lukyanov]. As the scaling dimension of $h$ is $2-K$, the second term in (\[effective\]) is relevant and opens a gap for $K<2$ or $-\sqrt2/2<\Delta\le1$; the term is irrelevant and (\[XXZ\]) remains in the gapless Luttinger liquid phase for $K>2$ or $-1<\Delta<-\sqrt2/2$ (Fig. \[phase\]). In the regime where $h$ is relevant, its effects have been studied perturbatively [@affleck; @affleck2] and compared to experiments on spin diffusion in copper benzoate [@hsexp; @hsexp2].
Integrability is well-defined in classical mechanics, but the definition of its quantum counterpart remains a subject of debate [@Caux]. It is generally believed that the level spacing distribution (the distribution of the differences of the adjacent eigenenergies) is the exponential distribution for an integrable model, as levels appear as a Poisson process, and the Wigner-Dyson distribution for a nonintegrable model. Intuitively, two nearby levels in an integrable model likely have different values of at least one integrable quantity, and thus live in different sectors of Hilbert space that are independent of each other; hence their energies are uncorrelated. Non-integrable models do not have an extensive number of such sectors and show energy level repulsion. The belief has been verified numerically on a variety of models [@Rabson; @Santos]. We perform an exact diagonalization of (\[XXZ\]) with periodic boundary conditions. Fig. \[stat\] shows the level spacing distributions, and the crossover from Poissonian behavior at $h=0$ to the Wigner-Dyson distribution at nonzero $h$ is clear. Hence (\[XXZ\]) is nonintegrable for nonzero $h$.
[*Numerical approach*]{}. The DC charge (c) and heat (h) conductivities can be computed via the Kubo formula $$\sigma= \lim_{t_M\to\infty}\lim_{L\to\infty} \frac{1}{LT}\,\textnormal{Re} \int_0^{t_M} \langle J(t)J(0)\rangle\, dt~,
\label{Kubo}$$ where the corresponding current operators $J=\sum_{i=1}^L j_i$ are defined through a continuity equation [@edfabian]: $$\begin{split}
\partial_t h_i = j_{\textnormal{h},i}-j_{\textnormal{h},i+1}& ~\Rightarrow~ J_\textnormal{h} = i \sum_{i=2}^L [h_{i-1},h_i]~,\\
\partial_t S_i^z = j_{\textnormal{c},i}-j_{\textnormal{c},i+1}& ~\Rightarrow~ J_\textnormal{c} = i \sum_{i=2}^L [h_{i-1},S_i^z]~.
\end{split}\label{current}$$ The current correlation functions $\langle J(t)J(0)\rangle$ can be computed efficiently using the real-time finite-temperature density matrix renormalization group (DMRG) algorithm [@dmrgrev; @white; @frank; @white2; @vidal; @daley; @frank2; @mettts; @barthel1; @barthel2] introduced in [@karrasch]. DMRG is essentially controlled by the so-called discarded weight $\epsilon$ (we ensure that $\epsilon$ is chosen small enough and that $L$ is chosen large enough to obtain numerically-exact results in the thermodynamic limit). The simulation is stopped when the DMRG block Hilbert space dimension $\chi$ reaches $\chi\sim1000-1500$. This allows to access time scales $t\sim 10-20$ which are larger than the inherent microscopic scale $t=1$.
Results for $\langle J(t)J(0)\rangle$ are shown in Fig. \[corr\]. In the integrable case $h=0$, the heat and charge Drude weights $$\label{eq:ecorr}
D = \lim_{t\to\infty}\lim_{L\to\infty} \frac {\textnormal{Re } \langle J(t)J(0)\rangle}{2LT}$$ are finite. A nonzero $h>0$ renders the model nonintegrable; one expect that the current correlation functions decay to zero at large times and that the conductivities become finite. Our data is consistent with this. In order to compute the integral in Eq. (\[Kubo\]) quantitatively, $\langle J(t)J(0)\rangle$ needs to be extrapolated. The heat current correlation function at intermediate to large temperatures $0.5\leq T\leq\infty$ (see Fig. \[corr\](b)) can be fitted by a single exponential function $\exp(-\lambda t)$ [^1]. Oscillations develop at small $T$, but it is reasonable to assume that $\langle J(t)J(0)\rangle$ can be described by sums of exponentially decaying terms $\exp(-\lambda_n t +i\omega_nt)$ (the same holds for the charge current correlation function). This motivates us to use so-called linear prediction [@dmrgrev; @barthel1; @linpred] as an extrapolation procedure. Its stability can be tested by varying fit parameters (e.g. the number of terms or the fit interval) and by checking sum rules (see below); we can obtain accurate results for the heat conductivity at any $h$ and temperatures $0.2\lesssim T\leq \infty$ as well as for the charge conductivity at intermediate to large $h$ and small $T\lesssim 0.3$.
![Heat and charge current correlation functions $\langle J(t)J(0)\rangle$ at $\Delta=-0.85$. Their Fourier transform determines the corresponding conductivities through Eq. (\[Kubo\]). Our data is consistent with the following picture: The Drude weight (Eq. (\[eq:ecorr\])) is nonzero only in the integrable case $h=0$; a nonintegrable perturbation $h>0$ renders the conductivity finite. DMRG data (solid lines) is extrapolated using linear prediction (dashed lines). []{data-label="corr"}](currentcorr.eps){width="0.95\linewidth"}
{width="0.475\linewidth"} {width="0.475\linewidth"}
[*Scaling of the conductivities: numerical results*]{}. DMRG data for the heat and charge conductivities is shown in Fig. \[sigma\]. For fixed $T$ and small $h$, one expects $\sigma$ to diverge as [@Jungth] $$\sigma \sim h^{-2}~~\textnormal{for}~~h\to0~.$$ This is consistent with our results (for the thermal case see the inset to Fig. \[sigma\](a); data for $\Delta=-0.95$ is similar). At small $T$, $\sigma_c$ features power laws with nontrivial exponents: $$\sigma_c \sim T^{\alpha_c} ~~\textnormal{for}~~T\to0~.$$ Our model is a Luttinger liquid at low energies and one thus expects $\alpha_c$ to be a universal function of the Luttinger liquid parameter $K$ only. This is verified in Fig. \[sigma\](b) which shows $\sigma_c(T)$ for different parameter sets $(\Delta,h)$ which share the same $K$ ($K$ is a continuous function of $h$ and $\Delta$; it can be obtained from an independent ground state DMRG calculation,[@llpaper] and the parameters in Fig. \[fig:sigma\](b) are determined numerically such that they all correspond to the same $K$). The exponent $\alpha_c(K)$ varies strongly with $K$; it is consistent with the analytic predition $\alpha_c=3-2K$ established below (see the insets to Fig. \[sigma\](b)). The heat conductivity is only accessible for intermediate temperatures $T\gtrsim 0.2$; our data in this regime is almost independent of $K$ (see Fig. \[sigma\](a)), suggesting that we have not reached the limit of low $T$.
Since for $h=0$ the heat current is conserved by the Hamiltonian, the AC conductivity $\sigma_h(\omega,h=0)=2\pi D\delta(\omega)$ features a Drude peak only. By generalizing Eq. (\[Kubo\]) to finite frequencies [@Sirkerprl; @Sirker], we can straightforwardly compute $\sigma_h(\omega,h)$ and demonstrate that it indeed becomes a $\delta$-function series for $h\to0$; the frequency-integrated heat conductivity just yields the Drude weight (‘sum rule’). This is illustrated in Fig. \[drude\] and provides an independent test for the reliability of our extrapolation procedure.
[*Bosonization*]{}. We now present an analytic calculation of the low-temperature behavior of the charge conductivity using bosonization. We closely follow Refs. which derive a parameter-free result for $\sigma_\textnormal{c}$ that is supposed to be correct if no conserved quantity protects the Drude weight. The current operator (\[current\]) reads $J_c=-v\sqrt{K/\pi}\int dx\Pi$, and the Kubo formula for the AC conductivity reduces to $\sigma_c(\omega)=iK\omega\langle\phi\phi\rangle_r(\omega)/\pi$. The retarded correlation function $\langle\phi\phi\rangle_r$ is obtained by a perturbative field theory calculation to leading order in $h$, in $H_\textnormal{umklapp}$, and in $H_\textnormal{band curv.}$. The overall leading term governing the DC conductivity reads $\sigma_c=h^{-2}T^{3-2K}/C(K)$, which is consistent with Eq. (\[T\]) [^2]. This term can be attributed to the staggered field, i.e. the term which breaks integrability in the lattice model.
From the point of view of the field theory alone, the umklapp and staggered field terms have the same cosine structure; one of those terms is insufficient to break integrability of the *field theory*. One might thus question the above reasoning and expect different temperature exponents in the conductivity depending on which of the two cosines was the ‘integrability-breaking small perturbation’ on the integrable theory obtained by the other (i.e., different exponents for large and small $h$). As mentioned above, this picture is not supported by our DMRG data for the lattice model which agrees with the bosonization result over a broad range of values of $h$; however, we cannot rule out different behavior at even lower temperatures.
[*Scaling analysis*]{}. We finally carry out a simple scaling analysis. Combining several assumptions, which are likely to hold for other 1D models, we establish a scaling form for the conductivity in which all the exponents are determined up to one number, the scaling dimension of the integrability-breaking perturbation. For our model, this is consistent with the bosonization result (which also yields the scaling dimension).
It is reasonable to assume that Re$\langle J(t)J(0)\rangle/LT\approx A(\Delta,h,T)\exp(-\gamma(\Delta,h,T)t)$ at long time as correlations typically decay exponentially at finite $T$. The oscillation of this correlation function (see Fig. \[corr\]) of Re$\langle J(t)J(0)\rangle/LT$ is not taken into account, as it cancels out in computing the integral (\[Kubo\]). We also assume the amplitude $A(\Delta,h,T)$ does not vanish as $T\to0$ (note that the Drude weight $D(\Delta,h=0,T)$ is nonzero and continuous as $T\to0$). Then, (\[Kubo\]) implies $\sigma_c\sim\gamma^{-1}$, and $\sigma_c$ takes the scaling form $$\sigma_c(\Delta,h,T)=f(\Delta/T^{[\Delta]},h/T^{[h]})/T,
\label{scaling}$$ where $[\Delta]$ and $[h]$ are the scaling dimensions of $\Delta$ and $h$, respectively. Note that $\sigma_c\sim T^{-1}$ for $[\Delta]=[h]=0$ or at the phase transition $\Delta=-\sqrt2/2,h=o(1)$.
In the perturbative regime (i.e., infinitesimal $h$), $[\Delta]=0$ as there is no renormalization of $\Delta$ (it is exactly marginal). Then (\[scaling\]) simplifies to $\sigma_c(\Delta,h,T)=f(\Delta,h/T^{[h]})/T$. As one expects $\sigma$ to diverge as $h^{-2}$ by a golden-rule argument [@Jungth] unless this perturbation is inefficient in inducing scattering, we take $f(x)\sim x^{-2}$. Then $$\sigma_c\sim h^{-2}T^{2[h]-1}=h^{-2}T^{3-2K}
\label{T}$$ where in the second equality we have substituted the bosonization result $[h]=2-K$. Note that $\sigma_c\sim T^{-1}$ for $K=2$ or at the phase transition $\Delta=-\sqrt2/2,h=o(1)$.
The scaling analysis result is consistent with the bosonization calculation, which more convincingly justifies the assumptions in this section. However, it is worth pointing out that from the scaling analysis we still expect scaling of conductivity at low temperature in a gapless 1D system even when bosonization is inapplicable. In general a gapless 1D system with a single velocity of low-energy excitations will be described by a conformal field theory (CFT) at long distances, and such theories are effectively ballistic as left-moving and right-moving excitations decouple. We expect that the basic picture that conductivity is controlled by the leading irrelevant operator that induces scattering will still apply even when the CFT is more complicated than a free boson.
![AC heat conductivity. A Drude peak emerges as $h\to0$: The integrated conductivity is independent of $h$ (right inset) and equal to the Drude weight (left inset).[]{data-label="drude"}](drudepeak.eps){width="\linewidth"}
[*Outlook*]{}. Our work demonstrates an approach valid for many actual 1D materials, in which integrability-breaking terms are likely to be present but small. We studied one specific model but expect that our key result – a power-law scaling of the conductivity $\sigma\sim T^{\alpha}$ with a universal exponent determined by the Luttinger liquid parameter – should be a general qualitative feature of any gapless nonintegrable model in which no conserved operator protects the current. Quantitative results for other nonintegrable perturbations can be obtained by the numerical framework used in this paper. It should be possible to compute optical charge conductivities for comparison to experiments on conducting polymers and other systems.
On a more basic level, quantum critical transport in one dimension is controlled by the leading irrelevant operators if and only if those destroy integrability. In higher dimensions, quantum critical transport is different because the critical theory is believed to be non-integrable, and transport properties are actively being studied by methods from high-energy physics. Our results provide a constraint on these methods in a case where direct computation of transport coefficients is possible.
[*Acknowledgements*]{}: The authors thank F. Essler, F. Heidrich-Meisner, A. Rosch, J. Sirker, and M. Zaletel for useful comments and suggestions and acknowledge financial support from the Deutsche Forschungsgemeinschaft via KA3360-1/1 (C.K.), the Nanostructured Thermoelectrics program of LBNL (C.K.), the AFOSR MURI on “Control of Thermal and Electrical Transport" (J.E.M.), and ARO via the DARPA OLE program (Y.H.).
[^1]: An estimate of the Drude weight has been given for this model [@lima] under the assumption that thermal currents do not decay when $h > 0$; as shown in Fig. 3, we find that thermal currents decay and the Drude weight is zero.
[^2]: The prefactor can also be obtained easily: $ C(K)=\pi^{K-3}(1-K^{-1}/2)^{2K-1}\cos^2(\pi K/2)\sin^{1-2K}(\pi K^{-1}/2) \Gamma^2(K/2) \Gamma^2(1/2-K/2)\Gamma^{2K}\left(\frac{1}{4K-2}\right)\Gamma^{-2K}\left(\frac{1}{2-K^{-1}}\right)\exp\Big(2\int_0^{+\infty}\frac{dx}{x}\Big(1-(K-1)e^{-2x}-\frac{\sinh x}{\sinh((K^{-1}-1)x)+\sinh x}\Big)\Big)$; bosonization predicts $\sigma_c\approx33$ at $\Delta=-0.85,h=0.2,T=0.05$, compared to the numerical result $\sigma_c\approx57$ (Fig. \[sigma\]). Noting that $h=0.2$ is still not very small, the agreement is reasonable at all points in the regime $\Delta=-0.85,h=0.2,T\lesssim0.1$. and numerically we observe that $\partial\log\sigma_c/\partial\log h$ is slightly larger than $-2$ at $h=0.2$, implying that the agreement is better at smaller $h$.
|
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abstract: 'We perform [*ab initio*]{} plane wave supercell density functional calculations on three candidate models of the (3$\times$2) reconstruction of the $\beta$–SiC(001) surface. We find that the two-adlayer asymmetric-dimer model (TAADM) is unambiguously favored for all reasonable values of Si chemical potential. We then use structures derived from the TAADM parent to model the silicon lines that are observed when the (3$\times$2) reconstruction is annealed (the ($n\times$2) series of reconstructions), using a density-functional-tight-binding method. We find that as we increase $n$, and so separate the lines, a structural transition occurs in which the top addimer of the line flattens. We also find that associated with the separation of the lines is a large decrease in the HOMO-LUMO gap, and that the HOMO state becomes quasi-one-dimensional. These properties are qualititatively and quantitatively different from the electronic properties of the original (3$\times$2) reconstruction.'
address:
- |
Department of Physics and Astronomy, University College London,\
Gower Street, London WC1E 6BT, United Kingdom
- '$\dagger$Institut de Ciència de Materials de Barcelona - CSIC, Campus de la Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain'
author:
- 'S.A. Shevlin, A.J. Fisher[^1] and E. Hernandez$\dagger$'
title: 'Modeling the series of ($n\times$2) Si-rich reconstructions of $\beta$–SiC(001): a prospective atomic wire?'
---
Introduction
============
Silicon carbide has long been studied because of its technological potential for electronic devices [@park; @choyke]. While most semiconductor compounds exhibit only one stable phase at room temperature, SiC possesses many polytypes. Of these polytypes, cubic or $\beta$–SiC has demonstrated its potential for use in high-temperature, high-frequency and high-power electronic devices. The lattice parameter of $\beta$–SiC possesses a lattice mismatch of $\sim$20% when compared with the lattice parameters of silicon or diamond. This has several consequences: first that when $\beta$–SiC is grown as a film on a Si substrate, the interface between the two materials possesses an associated strain field. Second, an unreconstructed Si or C-terminated SiC surface is under compressive or tensile stress. This stress is a driving force for significant reconstructions on both surfaces.
Depending on the stoichiometry of the $\beta$–SiC(001) surface, a large variety of such reconstructions has been found. For silicon-rich surfaces (containing more silicon than the Si-terminated bulk structure), the series of reconstructions (3$\times$2), (5$\times$2), ...$(n\times$2) has been observed [@soukiassian97b; @hara90], while for less silicon-rich surfaces the $c(4\times$2) [@douillard98; @soukiassian97; @pizzagalli98; @lu98; @shek96] or (2$\times$1) reconstructions are observed [@powers92; @husken98]. For C-terminated surfaces, the $c(2\times$2) reconstruction is favoured [@powers91; @long96].
It has been found that annealing the highly silicon-rich (3$\times$2) surface at 1100 K for several minutes leads to the formation of a grating of very long ($\sim$1000 Å), very straight lines at the transition from the (3$\times$2) to c(4$\times$2) surface [@soukiassian97b; @soukiassian97c; @semond98; @douillard98b], with the separation between the lines of magnitude $\sim$6 Å. Additional heat treatment leads to the removal of this grating and the formation of widely separated atomic lines [@soukiassian97b; @semond98; @aristov99], which we refer to as the $(n\times$2) series of reconstructions. These complex ($n\times$2) reconstructions consist of two subunits: the structural elements corresponding to the area between the lines (believed to be the $c(4\times$2) reconstruction [@soukiassian97c]), and the structural elements corresponding to the lines themselves.
In STM images of these lines [@soukiassian97c; @semond98] the bright regions appear to be $\sim$9 Å wide and the separation of these bright features along the line appears to be $\sim$ 6 Å. The surface lattice vector of the $\beta$–SiC (001) surface is 3.08 Å. Thus it is believed that these lines are formed from the same structural components as the silicon-rich (3$\times$2) reconstruction, of which there are several models (see figures 1-3): the additional dimer row model (ADRM) [@yan95; @semond96], the double dimer row model (DDRM) [@dayan86; @hara94] and the two-adlayer asymmetric-dimer model (TAADM) [@lu99]. The ADRM has an excess Si surface coverage of 1/3 ML (or one addimer per surface unit cell) and has an orientation of (2$\times$3) compared to our other candidate models which have an orientation we term (3$\times$2), the DDRM possesses a coverage of 2/3 ML (or two addimers per surface unit cell), and the TAADM has a coverage of 1 ML (or three addimers per surface unit cell). In the interests of completeness we also mention the single dimer row model (SDRM) which is a unit cell which is a 90$^{\circ}$ rotation of the ADRM and which we find has a total energy 2.67 eV higher than the ADRM, and so is thermodynamically unfavorable compared to the ADRM; and the two-adlayer-asymmetric rotated dimer model (TAARDM) which is a unit cell which resembles the TAADM, but with the top addimer rotated so that it is parallel to the second level ad-dimers, and which we find has a total energy 0.68 eV higher than the TAADM and so is thermodynamically unfavorable when compared with the TAADM. We will not consider the SDRM or the TAARDM further.
In Section II we discuss the various calculational methods used in our simulations, and how we can compare the grand canonical potentials of the various models. In Section III, using [*ab initio* ]{} techniques, we present the structural results that we obtain for the (3$\times$2) models, the thermodynamic stability of each model, the electronic properties of and how they compare to experiment. In Section IV, we perform calculations on the series of ($n\times$2) reconstructions modeled by tight-binding, using the thermodynamically favoured (3$\times$2) model as the parent, and present the details of a change in the structure of the line when $n\geq$7. We then present our conclusions in Section V.
Method
======
First-principles total energy calculations
------------------------------------------
We performed the calculations using [*ab initio*]{} density functional theory using the projector augmented wave method (PAW) to handle the atomic cores [@blochl94]. We use the Perdew-Zunger parameterisation [@perdew81] of the Ceperley-Alder [@ceperley80] treatment of the uniform electron gas. We solve the Kohn-Sham equations using the Car-Parrinello algorithm [@car85]. The main idea of the PAW method is to split the all-electron (AE) wavefunction into three parts; $$|\Psi\rangle = |\tilde\Psi\rangle + \sum_i(|\phi_i\rangle - |\tilde\phi_i\rangle)\langle \tilde p_i|\tilde\Psi\rangle.$$ $|\Psi\rangle$ is the AE wave function and $|\tilde\Psi\rangle$ is a pseudo (PS) wavefunction analogous to the wave functions of the pseudopotential method, or the envelope functions of the linear method. The $|\phi_i\rangle$ are a set of AE partial waves within the core region, while the $|\tilde\phi_i\rangle$ are a set of smooth partial waves which coincide with the corresponding AE partial waves outside the core region. $|\tilde p_i\rangle$ are projector functions localised within the core region which obey the relation $\langle \tilde p_i|\tilde\phi_j\rangle$ = $\delta_{ij}$.
As there are several models for the (3$\times$2) surface unit cell we modelled all of them, using the surface slab technique with periodic boundary conditions. The simulation cells were six and ten atomic layers deep (six layers were used for preliminary checks, ten layers for results quoted here); the bottom two layers were held in their bulk configuration while the other layers were allowed to relax, with the bottom carbon layer passivated with hydrogen atoms. In our calculations we sampled four points of the Brillouin zone, corresponding to the $\Gamma$–point, the $J'$–point (in the \[$\overline{1}$10\] direction in real space), the $J$–point (in the \[110\] direction in real space) and the $K$–point (see figure 4). For our structural calculations we used a plane-wave cutoff of 20 Rydbergs, but for our electronic spectra and total energy calculations we increased the cutoff to 30 Rydbergs. The vacuum spacing between slabs was set to 6.08 Å .
The convergence of the simulation with respect to vacuum spacing and plane wave cutoff and $k$-point sampling was checked. It was found that increasing the vacuum spacing did not affect the resulting relaxed structure. The difference between a relaxed cell at a cutoff of 20 Rydbergs and 30 Rydbergs was negligible, with minor differences in surface dimer length ($\sim$0.10 Å at most) and almost identical ad-dimer length ($\sim$ 0.01 Å difference). Dimer buckling was found to be slightly increased for the higher cutoff compared to the case with the lower cutoff ($\sim$ 0.06 Å increase). We can thus say that our calculations are converged with respect to vacuum spacing and plane-wave cutoff.
Tight Binding Calculations
--------------------------
Tight binding methods (TB) [@goringe97] have the advantage of being computationally much less demanding than first-principles methods, while still affording relatively high accuracy. The disadvantage originates in the fact that the approximations involved are usually of a less controllable nature, and hence are more difficult to improve. Nevertheless, the extreme simplicity and sometimes surprising accuracy of TB methods make them a frequently used tool in computational condensed matter and materials science studies.
We have made use of the TB model known as [*Density Functional Tight Binding*]{} (DFTB), due to Porezag and coworkers [@porezag95]. This method goes beyond conventional TB schemes in several ways. First, an atomic-like basis set is explicitly employed. Because of this, the model that results is non-orthogonal. In empirical TB methods it is customary to work in terms of an underlying basis set (normally assumed to be orthogonal), but only the matrix elements of the Hamiltonian are used, and the basis set is never explicitly constructed. Secondly, the model is constructed by directly evaluating the Hamiltonian matrix elements within the framework of Density Functional Theory, albeit using a two-center approximation (i.e. neglecting environment effects), while in empirical TB models the matrix elements are adjusted either to empirical or theoretical data. For a detailed description of the model construction procedure, the reader should consult Ref. [@porezag95].
In DFTB, the total energy of the system under study is calculated as follows. The relative positions of the atoms in the system determine the values of the Hamiltonian and overlap matrix elements. From these, the Schrödinger equation for the electronic problem can be obtained in matrix form, and solved by the usual techniques (for example, by matrix diagonalization): $$\sum_{j\beta} C_{j\beta}^{(n)} (H_{i\alpha,j\beta} - E_n S_{i\alpha,j\beta} )
= 0.
\label{eq:schroedinger}$$ Here indices [*i*]{} and [*j*]{} label atoms, Greek indices label basis set functions, and [*n*]{} labels eigen-states and eigen-values. Once the eigen-value problem (\[eq:schroedinger\]) is solved, the [*band-structure energy*]{} is calculated: $$E_{bs} = 2 \sum_n^{occ} E_n,$$ where the sum extends over the occupied eigen-states, and the factor of two accounts for the degeneracy of spin. To complete the total energy of the system a repulsive potential contribution is added, $$E_{rep} = \sum_{j\neq i} V(\mid {\bf R}_j - {\bf R}_i\mid),$$ where ${\bf R}_i$ is the position of atom [*i*]{}. This potential accounts for the core-core repulsion and the double counting of the electron-electron energy implicit in the band-structure term (see Ref. [@goringe97]).
We have used a DFTB parametrisation consisting of four basis functions ($S, P_x, P_y, P_z$) per atom, corresponding to the four valence orbitals of carbon and silicon. Similar models have been used with great success by Porezag and coworkers [@porezag95] to study carbon clusters and hydrocarbons, as well as bulk crystalline phases of carbon, amorphous carbon [@kohler95], Si and SiH clusters [@frauenheim95], crystalline BN [@widany96], and SiC structures [@gutierrez96].
Thermodynamical considerations
------------------------------
We can use the total energies calculated from [*ab initio*]{} calculations to discover which of the four candidate models possesses the lowest grand canonical potential. We assume that the formation of the different structures is determined by thermodynamic factors but we cannot compare the total energies of different models with different stoichiometries. A common method of comparing structures is to find the chemical potential $\mu_{\rm Si}$ for the silicon adatoms and then compare the grand potentials [@qian88; @northup93] $$\label{one}
\Omega=E-N_{\rm Si}\mu_{\rm Si}.$$ However, because of the experimental conditions that produce the (3$\times$2) surface and because SiC is a two component system, it is nontrivial to calculate $\mu_{\rm Si}$. Nevertheless we can estimate a value for the range of chemical potential that the silicon ad-atoms experience. We define $\Delta E_{\rm Coh}^{\rm Si}$ in terms of $\mu_{\rm Si}$ by the relation $$\mu_{\rm Si} = E^{\rm atom}_{\rm Si} + \Delta E_{\rm Coh}^{\rm Si},$$ where $E^{\rm atom}_{\rm Si}$ is the energy of an isolated silicon atom (-3.80769 Hartrees in the PAW formalism); $\Delta E_{\rm Coh}^{\rm Si}$ is the corresponding value for the cohesive energy per silicon atom in our system.
We must decide what physical/chemical process determines the value of $\mu_{\rm Si}$ (and hence $\Delta E_{\rm Coh}^{\rm Si}$) in our system. We can make several choices. We choose to consider the incorporation of Si atoms into bulk SiC. $\Delta E_{\rm Coh}^{\rm Si}$ would then become the cohesive energy for a Si atom in SiC. However this cannot be calculated directly, because it is impossible to insert a single Si into SiC without creating a structural defect of some kind. We can, though, estimate the maximum range of $\mu_{\rm Si}$ that the Si ad-atoms experience [@zyweitz99]. We can write the chemical potential per unit cell of an ideal bulk system of SiC as $$\mu^{\rm bulk}_{\rm SiC}=\mu_{\rm Si}+\mu_{\rm C},$$ where $\mu_{\rm Si}$ and $\mu_{\rm C}$ are the (unknown) contributions from the Si and C atoms. In extremely Si-rich conditions, $\mu_{\rm Si}$ approaches the value of bulk Si. We can consider deviations from this bulk value $\Delta\mu_{\rm Si}$=$\mu_{\rm Si}-\mu^{\rm bulk}_{\rm Si}$ (with a corresponding definition for $\Delta\mu_{\rm C}$). The allowed range is determined by the heat of formation of the SiC compound $$\Delta H_f =\mu^{bulk}_{\rm SiC}-\mu^{bulk}_{\rm C}-\mu^{bulk}_{\rm Si} \\ =\left(\mu_{\rm Si}-\mu^{bulk}_{\rm Si}\right)+\left(\mu_{\rm C}-\mu^{bulk}_{\rm C}\right)=\Delta\mu_{\rm Si}+\Delta\mu_{\rm C}$$ The experimental value of $\Delta H_f$ is 0.72 eV [@kubaschewski79]. In a Si-rich environment $\Delta\mu_{\rm Si}=0$, and therefore $\Delta\mu_{\rm C}=\Delta H_f$. Similarly, in a C-rich environment $\Delta\mu_{\rm C}=0$ and $\Delta\mu_{\rm Si}=\Delta H_f$. This means that furthest deviation the chemical potential of Si can make from the value for bulk Si is 0.72 eV. We show our results as a function of $\Delta E^{\rm Si}_{\rm Coh}$ over the range from its value for bulk Si $\Delta E^{\rm Si}_{\rm Coh} ({\rm Si})$ to the value obtained by assuming that the cohesive energy of SiC can be equally divided between the carbon and silicon atoms, we call this value $\Delta E^{\rm Si}_{\rm Coh} ({\rm C})$.
3$\times$2 unit cell
====================
Structural
----------
It was found from structural calculations that all models we considered were locally stable, with silicon adatoms forming addimers. All addimers were found to be asymmetric (apart from the case of the TAADM, where the second layer addimers were found to be flat). The structure obtained for the ADRM is in agreement with previous [*ab initio*]{} [@yan95; @lu99; @pizzagalli99], and tight-binding work [@gutierrez99], with a strongly bound and asymmetric addimer. The structure obtained for the TAADM is in agreement with previous work [@lu99]. There is an alternating arrangement of first layer and second layer addimers as we look along \[$\overline{1}$10\], with the top layer addimer strongly bound and asymmetric, and the second level addimers strongly bound and flat. However, the structure obtained for the DDRM is in disagreement with both previous [*ab initio*]{} [@pizzagalli99], and tight-binding [@gutierrez99] results, as we find that both addimers are buckled and quite strongly bound. Our result corresponds to one of the DDRM models studied previously [@lu99] (the LAFM-DDRM structure). The other DDRM models were all found to be of higher total energy than this structure, in contradiction with previous work [@lu99]. The Si atoms of the first Si layer of the SiC proper were also found to be dimerised in all unit cells, in agreement with previous work [@lu99; @pizzagalli99; @gutierrez99; @kitabatake96; @kitabatake98]. Table I provides further information about the equilibrium structure of each model.
Thermodynamics
--------------
We use calculated [*ab initio*]{} total energies, to find $\Omega$ from equation $(\ref{one})$ between the various models. The results presented (see figure 5) are for our most converged simulations (30 Rydberg cutoff, 4 $k$-points sampled). We first of all start off with the assumption that the Si adatoms are in equilibrium with the bulk. As can be seen, all crossover points (that is the values of $\Delta E_{\rm Coh}^{\rm Si}$ for which $\Omega$ is equal for two separate models) are outside the maximum allowable range of $\Delta E_{\rm Coh}^{\rm Si}$ as determined by $\Delta H_f$. We can thus unambiguously apply our calculations of $\Omega$ to determine which model is thermodynamically preferred. Over the entire range of allowable chemical potentials, the TAADM model is favored.
Our results are in agreement with the previous work of Lu [*et al.*]{} [@lu99], where it is found that the TAADM is the thermodynamically preferred model throughout the entire allowable range of $\Delta E_{\rm Coh}^{\rm Si}({\rm SiC})$. Regarding the relative stability of the ADRM and DDRM , we agree with other work [@pizzagalli99; @gutierrez99], where it is suggested that the ADRM is the favoured model, although the TAADM is not considered. We find that the ADRM is only favored for $\Delta E_{\rm Coh}^{\rm Si}$ which are close to $\Delta E_{\rm Coh}^{\rm Si}({\rm SiC})$. But there is no evidence for a (2$\times$3) reconstruction which is less silicon-rich than the $c(4\times$2) reconstruction. As we find that the c(4$\times$2) reconstruction is only preferred over a narrow range of $\mu$ when the ADRM is included, (contrary to observations), we conclude that the ADRM does not occur in practice, as it would prevent the formation of the c(4$\times$2) reconstruction. We can therefore rule out the ADRM.
We have therefore found from our [*ab initio*]{} calculations of the grand potential that the TAADM is preferred over a large range of chemical potential and the other two models can be discounted, with the DDRM found to be never preferred and the ADRM only valid for a range of chemical potential which is non-physical.
Electronic
----------
We can calculate the differences (the ‘dispersion’) between the Kohn-Sham eigenvalues calculated at different points of the Brillouin zone, and compare these with recent photoemission experiments performed on the (3$\times$2) surface [@bermudez95; @lubbe98; @yeom97; @yeom98]. Our results, and equivalent results in the literature are presented in Table 2. The magnitude of dispersion in the majority of cases is approximately equal to the magnitude of dispersion measured [@yeom98] ($\sim$0.2eV). The exception is for the DDRM, where we find a large dispersion along $\Gamma-J$. We attribute this to our modeling of a perfectly ordered periodic (3$\times$2) surface, whereas the (3$\times$2) surface has been observed to be almost perfectly disordered [@hara96; @hara99], in the sense that there is no observable correlation between the addimer tilt in one (3$\times$2) unit cell, and the next (in either the $\times$3 or $\times$2 direction). The measured dispersion of the surface state bands is $\sim$0.2 eV along $\Gamma-J'$, with no measured dispersion along $J$ [@yeom98]. We find that the only model which reflects this anisotropy of dispersion is the ADRM, with the DDRM and the TAADM both possessing larger dispersions along $\Gamma-J$ and smaller dispersions along $\Gamma-J'$, contrary to experiment. As seen in Table II, there are some disagreements among the different calculations. We especially highlight the discrepancy between our results and those of Lu [*et al.*]{} [@lu99] for the energetically preferred TAADM model. However we find that the HOMO state for the TAADM exists mostly in the vacuum of the simulation slab, and suggest that as we use plane waves to describe our electron wavefunction, as opposed to the Gaussian orbitals used in other work [@lu99], our dispersion results more accurately describes the dispersion of the surface state.
Larger unit cells ($n\times$2)
==============================
We can model the ($n\times$2) series of reconstructions by the mixing of two sets of reconstructions, the (3$\times$2) and the $c(4\times$2). We use the thermodynamically favoured TAADM model as the parent reconstruction of the lines themselves. We use the favoured MRAD model [@lu98; @shevlin00] as the surface between the lines [@lu00]. Due to the large size of the reconstructions, [*ab initio*]{} methods are inefficient for modelling these surfaces. We therefore used the non-self-consistent density-functional tight-binding method to simulate these surfaces [@porezag95; @frauenheim95]. We performed simulations for $n$=3,5,7,9 and 11 sampling both the $\Gamma$ point and a (221) $k$–point mesh generated by the Monkhorst-Pack scheme [@monkhorst76]. It was found that structural calculations are converged for this choice of $k$–point sampling scheme. All simulation cells are ten atomic layers thick, with the bottom carbon layer terminated with hydrogen to avoid artificial charge transfer. The results can be neatly summarised in Figure 6.
As can be seen, a comparison of [*ab initio*]{} and tight-binding simulations for the (3$\times$2) surface unit cell shows that although the addimer bond length $a_1$ is different in the two cases, with the [*ab initio*]{} addimer length found to be 2.31Å and the tight-binding addimer length found to be 2.62Å , we find a similar buckling $\Delta z$, (0.58Å and 0.53Å respectively). We find that as we increase the width of the surface unit cell from $n$=3, to $n$=5, the asymmetry and length of the weak addimer $a_1$ remain approximately the same (this is qualitatively in agreement with previous theoretical work [@lu00], wherein it is found that the asymmetry and length of the top ad-dimer remains the same). However, we find that for $n\geq$7, the top addimer becomes symmetric and the addimer bond length becomes shorter with a bond length of 2.35Å . This matches STM topographs of the lines, which show that the lines are composed of symmetric units [@douillard98b; @derycke]. The flat addimer of the (7$\times$2) unit cell has a shorter bond length than the buckled addimer of the (3$\times$2) or (5$\times$2) because there is a stronger $\sigma$ bond and a much stronger $\pi$ bond. The bond angles between the top addimer and the top-adlayer-to-second-adlayer bonds all become $\sim 109.5^\circ$, compared to $\sim 85^\circ$ and $125^\circ$ for the buckled addimer case. It was also found that if the addimer is buckled and then left to relax in a unit cell where all other atoms are already in their relaxed positions, then the addimer remains buckled, albeit with a higher energy than the flat addimer. That is, the structural transition may be kinematically limited. The buckled addimer structure is 0.25 eV higher in total energy than the flat addimer structure; however we point out that this is less than the average thermal energy of all the surface silicon atoms at room temperature. The flat and buckled addimer structures could both be accessible to the surface.
Analysis of the bond order (off-diagonal density matrix elements $\rho^i_j$) between the top adatom and adatoms of the second adlayer show that as the addimer becomes flat this bond becomes much stronger, with larger $\sigma$ and $\pi$ components. Associated with this stronger bond between top adlayer and second adlayer is the loss of an electron from the top addimer. This electron is transferred to the surface silicon atoms directly below the addimer as shown in Figure 6.
We also find that coincident with the isolation of the separate lines, there is a change in the HOMO-LUMO (highest-occupied-molecular-orbital lowest-unoccupied-molecular- orbital) gap. Analysis of the electronic structure of the unit cell was performed using a (551) $k$–point Monkhorst-Pack net. Within our tight-binding formalism, we find that the HOMO-LUMO gap of the (3$\times$2) unit cell varies from 1.48 eV at the zone centre, to 1.14 eV at the zone boundary. For the (7$\times$2) unit cell, we find that the HOMO-LUMO gap is decreased by a considerable amount, from 0.87 eV at the zone centre to 0.30 eV at the zone boundary. We also find that the anisotropy of dispersion of the HOMO state changes; for the (3$\times$2) unit cell the HOMO state disperses by 0.05 eV along $\Gamma-J'$ and by 0.43 eV along $\Gamma-J$, while for the (7$\times$2) unit cell we find that the dispersion of the HOMO state is 0.43 eV along $\Gamma-J'$ and 0.05 eV along $\Gamma-J$. As the HOMO state is associated with the top addimer, and as this is now more strongly bound to the second level addimers, this means that there there are now strong connections along the line. Therefore the HOMO state of the isolated lines shows strong quasi-one-dimensional behaviour when compared with the interacting silicon lines that constitute the (3$\times$2) and (5$\times$2) reconstructions.
We observe three changes that happen together: that the HOMO state becomes quasi-one-dimensional in character, that the top adatom of the buckled addimer moves down, and that there is a transfer of an electron from the top adlayer to the top layer of the silicon carbide crystal proper. We offer one possible rationalisation for this structural transition: that as the lines become separated, the HOMO state becomes quasi-one-dimensional. This makes the HOMO state interact more strongly along the line, which means that there is an extra contribution to the bonds to the addimers of the second layer. This forces the top adatom closer to the surface, and flattens the buckle of the top addimer. As the dangling bond state rises in energy as the addimer becomes flat, a state of the top adatom becomes depopulated, and an electron leaves the top addimer, to reside in the silicon atoms of the top layer of the silicon carbide crystal directly below the addimer. However we are aware that the charge transfer which seems to be involved in this structural transition could be inaccurately treated, as the tight-binding method we use is a non-charge-self-consistent scheme.
Conclusions
===========
We have performed detailed electronic structure calculations on several different models of the silicon-rich (3$\times$2) reconstruction of cubic silicon carbide. It was found that the 1 ML TAADM model was the preferred model for the (3$\times$2) reconstruction over the entire range of allowable chemical potential. Our calculated dispersion values for the TAADM contradict experiment however, we find that the only model that matches the photoemission data [@yeom98] is the ADRM. The mapping between observed surface states and one electron Kohn-Sham eigenvalues is not well defined however. We conclude that the TAADM is the theoretically preferred model.
We have also used our DFT electronic structure calculations as the basis for a tight-binding analysis of the ($n\times$2) series of reconstructions, that is those reconstructions that correspond to the silicon lines observed on the $c(4\times$2) surface [@soukiassian97b; @soukiassian97c; @semond98; @douillard98b]. We find that there is a structural transition associated with a critical value of $n$, where if $n\geq$7, then the top addimer of the TAADM becomes flat. We find that the HOMO-LUMO gap associated with the increasing separation of the lines is reduced, so that within the tight-binding formalism the ($n\geq7\times$2) reconstructions are narrow bandgap semiconductors. We also find that the HOMO state associated with the flat addimer is now confined to the line and is strongly dispersed along the line, i.e. it displays quasi-one-dimensional behaviour. Our results thus show that the electronic and structural properties of the ($n\times$2) series of reconstructions, corresponding to widely separated silicon lines, are very different from the electronic and structural properties of the (3$\times$2) surface. Detailed experimental data on the physical properties of the ($n\times$2) series of reconstructions is needed to verify these theoretical results.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the UK Engineering and Physical Sciences Research Council. We would like to thank Hervé Ness, John Harding, David Bowler, Marshall Stoneham and Tony Harker for a number of helpful discussions.
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Private communication V. Derycke
Model Addimer bond length Buckling of addimer Surface dimer length
------- ------------------------------------------ --------------------- ------------------------------------------
ADRM 2.30 ($a_1$) 0.50 2.59 ($d_1$), 2.47 ($d_2$), 2.49 ($d_3$)
DDRM 2.35 ($a_1$), 2.34 ($a_2$) 0.51, 0.53 2.40 ($d_1$), 2.41 ($d_2$)
TAADM 2.31 ($a_1$), 2.44 ($a_2$), 2.42 ($a_3$) 0.58, 0.01, 0.00 2.41 ($d_1$), 2.41 ($d_2$)
: Equilibrium bond lengths for the addimers and buckling of addimers (magnitude) in our total energy calculations. All distances are in Å . See Figures 1-3 for more.[]{data-label="Table 1"}
Author ADRM DDRM TAADM
---------------------------------------- ----------------------------------------------------- ----------------------------------------------------- ----------------------------------------------
This work 0.18 ($\Gamma J'$), 0.00 ($\Gamma J$) 0.01 ($\Gamma J'$), 0.34 ($\Gamma J$) 0.13 ($\Gamma J'$), 0.21 ($\Gamma J$)
Pizzagalli [*et al*]{} [@pizzagalli99] $\leq$ 0.10 ($\Gamma J'$), $\leq$ 0.10 ($\Gamma J$) 1.00 ($\Gamma J'$), $\leq$ 0.10 ($\Gamma J$)
Lu [*et al*]{} [@lu99] 0.20 ($\Gamma J'$), $\sim$ 0.00 ($\Gamma J$) $\sim$ 0.00 ($\Gamma J'$), $\sim$ 0.50 ($\Gamma J$) 0.37 ($\Gamma J'$), $\sim$ 0.00 ($\Gamma J$)
: ‘Dispersion’ of HOMO state along \[$\overline{1}$10\] ($\Gamma-J'$) and \[110\] ($\Gamma-J$ directions, for various models. Also shown are values found in the literature. All values are in eV.
[^1]: Email [email protected]
|
---
abstract: 'Epistemic concepts, and in some cases epistemic logic, have been used in security research to formalize security properties of systems. This survey illustrates some of these uses by focusing on confidentiality in the context of cryptographic protocols, and in the context of multi-level security systems.'
author:
- 'Riccardo Pucella[^1]'
bibliography:
- 'knowledgeandsecurity.bib'
title: Knowledge and Security
---
*Security: the state of being free from danger or threat.*\
*(New Oxford American Dictionary)*
Introduction {#chap13:introduction}
============
A persistent intuition in some quarters of the security research community says that epistemic logic and, more generally, epistemic concepts are useful for reasoning about the security of systems. What grounds this intuition is that much work in the field is based on epistemic concepts—sometimes explicitly, but more often implicitly, by and large reinventing possible-worlds semantics for knowledge and belief.[^2]
Reasoning about the security of systems in practice amounts to establishing that those systems satisfy various security properties. A security property, roughly speaking, is a property of a system stating that the system is not vulnerable to a particular threat. Threats, in this context, are generally taken to be attacks by agents intent on subverting the system.
While what might be considered a threat—and therefore what security properties are meant to protect against—is in the eye of the beholder, several properties have historically been treated as security properties:
- **Data Confidentiality:** only authorized agents should have access to a piece of data; more generally, only authorized agents should be able to infer any information about a piece of data.
- **Data Integrity:** only authorized agents should have access to alter a piece of data.
- **Agent Authentication:** an agent should be able to prove her identity to another agent.
- **Data Authentication:** an agent should be able to determine the source of a piece of data.
- **Anonymity:** the identity of an agent or the source of a piece of data should be kept hidden except from authorized agents.
- **Message Non-repudiation:** the sender of a message should not be able to deny having sent the message.
These properties may seem intuitive on a first reading but they are vague and depend on terms that require clarification: *secret*, *authorized*, *access to a piece of data*, *source*, *identity*.
Epistemic concepts come into play when defining many of the terms that appear in the statements of security properties. Indeed, those terms can often be usefully understood in terms of knowledge: confidentiality can be read as *no agent except for authorized agents can know a piece of information*; authentication as *an agent knows the identity of the agent with whom she is interacting*, or *an agent knows the identity of the agent who sent the information*; anonymity as *no one knows the identity of the agent who performed a particular action*; and so on. While it is not the case that every security property can be read as an epistemic property, enough of them can to justify studying them *as* epistemic properties.
Epistemic logic and epistemic concepts play two roles in security research:
- **Definitional**: they are used to formalize security properties and concepts, and provide a clear semantic grounding for them. Epistemic logic may be explicitly used as an explanatory and definitional language for properties of interest.
- **Practical**: they are used to derive verification and enforcement techniques for security properties, that is, to either establish that a security property is true in a system, or to force a security property to be true in a system.
It is fair to say that after nearly three decades of research, epistemic logic has had several successes on the definitional front and somewhat fewer on the practical front. This is perhaps not surprising. While epistemic logic and epistemic concepts are well suited for definitions and for describing semantic models, verification of epistemic properties tends to be expensive, and tools for the verification of security properties in practice often approximate epistemic properties using properties that are easier to check, such as safety properties.[^3]
This chapter illustrates the use of epistemic logic and epistemic concepts for reasoning about security through the study of a specific security property, confidentiality. Not only is confidentiality a prime example of the use of knowledge to make a security property precise, but it has also been studied extensively from several perspectives. Moreover, many of the issues arising while studying confidentiality also arise for other security properties with an epistemic flavor.
Confidentiality is explored in two contexts: cryptographic protocols in [\[chap13:cryptoprotocols\]]{}, and multi-level security systems in [\[chap13:iflow\]]{}. Cryptographic protocols are communication protocols that use cryptography to protect information exchanged between agents in a system. While it may seem simple enough for Alice to send a confidential message to Betty by encrypting it, Alice and Betty need to share a common key for this to work. How is such a key distributed before communication may take place? Most cryptographic protocols involve key creation and distribution, and these are notoriously difficult to get right. Key distribution also forces the consideration of authentication as an additional security property. Cryptographic protocol analysis is the one field of security research that has explicitly and extensively used epistemic logic, and the bulk of this chapter is dedicated to that topic.
In multi-level security systems, one is generally interested in confidentiality guarantees even when information is used or released within the system during a computation. The standard example is that of a centralized system where agents have different security clearances and interact with data with different security classifications; the desired confidentiality guarantees ensure that classified data, no matter how it is manipulated by agents with an appropriate security clearance, never flows to an agent that does not have an appropriate security clearance. Most of the work in this field of security research uses epistemic concepts implicitly— the models use possible-worlds definitions of knowledge, but no epistemic logic is introduced. All reasoning is semantic reasoning in the models.
Security properties other than confidentiality are briefly discussed in [\[chap13:beyond\]]{}. The chapter concludes in [\[chap13:perspectives\]]{} with some personal views on the use of epistemic logic and epistemic concepts in security research. My observations should not be particularly controversial, but my main conclusion remains that progress beyond the current state of the art in security research—at least in security research that benefits from epistemic logic—will require a deeper understanding of resource-bounded knowledge, which is itself an active research area in epistemic logic.
All bibliographic references are postponed to [\[chap13:bibnotes\]]{}, where full references and additional details are given for topics covered in the main body of the chapter. It is worth noting that the literature on reasoning about security draws from several fields besides logic. For instance, much of the research on cryptographic protocols derives from earlier work in distributed computing. Similarly, recent research both on cryptographic protocol analysis and on confidentiality in multi-level systems is based on work in programming language semantics and static analysis. The interested reader is invited to follow the references given in [\[chap13:bibnotes\]]{} for details.
Cryptographic Protocols {#chap13:cryptoprotocols}
=======================
Cryptographic protocols are communication protocols—rules for exchanging messages between agents—that use cryptography to achieve a security goal such as authenticating one agent to another, or exchanging confidential messages.
Cryptographic protocols are a popular object of study for several reasons. First, they are concrete—they correspond to actual artifacts implemented and used in practice. Second, their theory extends that of distributed protocols and network protocols in general, which are themselves thoroughly studied.
Cryptographic protocols have characteristics that distinguish them from more general communication protocols. In particular, they
1. enforce security properties;
2. rely on cryptography;
3. execute in the presence of attackers that might attempt to subvert them.
Protocols can be analyzed concretely or symbolically. The concrete perspective views protocols as exchanging messages consisting of sequences of bits and subject to formatting requirements, which is the perspective used in most network protocols research. The symbolic perspective views protocols as exchanging messages consisting of symbols in some formal language, which is the perspective used in most distributed protocols research. The focus of this section is on symbolic cryptographic protocols analysis.
Protocols {#chap13:protocols}
---------
A common notation for protocols is to list the sequence of messages exchanged between the parties involved in the protocol, since the kinds of protocols studied rarely involve complex control flow.
A simple protocol between Alice and Betty (represented by $A$ and $B$) in which Alice sends message $m_1$ to Betty and Betty responds by sending message $m_2$ to Alice would be described by:
[sample]{} [1. & A & & B & : & m\_1]{}\
[2. & B & & A & : & m\_2]{}
The message sequence notation takes a global view of the protocol, describing the protocol from the outside, so to speak. An alternate way to describe a protocol is to specify the roles of the parties involved in the protocol. For protocol , for instance, there are two roles: the initiator role, who sends message $m_1$ to the receiver and waits for a response message, and the receiver role, who waits for a message to arrive from the initiator and responds with $m_2$.
A protocol executes in an environment, which details anything relevant to the execution of said protocol, such as the agents participating in the protocol, whether other instances of the protocol are also executed concurrently, the possible attackers and their capabilities. The result of executing a protocol in a given environment can be modeled by a set of traces, where a trace corresponds to a possible execution of the protocol. A trace is a sequence of global states. A global state records the local state of every agent involved in the protocol, as well as the state of the environment. This general description is compatible with most representations in the literature, and can be viewed as a Kripke structure by defining a suitable accessibility relation over the states of the system.
To illustrate protocols in general, and initiate the study of confidentiality, here are two simple protocols that achieve a specific form of confidentiality without requiring cryptography. One lesson to be drawn from these examples is that confidentiality in some cases can be achieved without complex operations.
The first protocol solves an instance of the following problem: how two agents may exchange secret information in the open, without an eavesdropping third agent learning about the information. The instance of the problem, called the Russian Cards problem, is pleasantly concrete and can be explained to children: Alice and Betty each draw three cards from a pack of seven cards, and Eve (the eavesdropper) gets the remaining card. Can players Alice and Betty learn each other’s cards without revealing that information to Eve? The restriction is that Alice and Betty can only make public announcements that Eve can hear.
Several protocols for solving the Russian Cards problem have been proposed; a fairly simple solution is the *Seven Hands protocol*. Recall that there are seven cards: three are dealt to Alice, three are dealt to Betty, and the last card is dealt to Eve. Call the cards dealt to Alice $a_1,a_2,a_3$, and the cards dealt to Betty $b_1,b_2,b_3$. The card dealt to Eve is $e$.
The Seven Hands protocol is a two-step protocol that Alice can use to tell her cards to Betty and learn Betty’s cards in response:
[seven-hands]{} [1. & A & & B & : & \_A]{}\
[2. & B & & A & : & \_B]{}
where ${\mathit{SH}}_A$ and ${\mathit{SH}}_B$ are the following specific messages:
1. Message ${\mathit{SH}}_A$ is constructed by Alice as follows. Alice first chooses a random renaming $W,X,Y,Z$ of the elements in $\{b_1,b_2,b_3,e\}$, that is, a random permutation of the four cards not in her hand. Message ${\mathit{SH}}_A$ then consists of the following seven subsets of cards, in some arbitrary order: $$\begin{gathered}
\{a_1,a_2,a_3\}\\
\{a_1,W,X\}\quad\{a_1,Y,Z\}\\
\{a_2,W,Y\}\quad\{a_2,X,Z\}\\
\{a_3,W,Z\}\quad\{a_3,X,Y\}\end{gathered}$$ These subsets are carefully chosen: for every possible hand of Betty, that is, for every possible subset $S$ of size three of $\{W,X,Y,Z\}$, there is exactly one set in ${\mathit{SH}}_A$ with which $S$ has an empty intersection, and that set is Alice’s hand $\{a_1,a_2,a_3\}$. Thus, upon receiving ${\mathit{SH}}_A$, Betty can identify Alice’s hand by examining the sets Alice sent and picking the one with which her own hand has an empty intersection, and in the process Betty learns Alice’s hand, and by elimination, Eve’s card.
2. Message ${\mathit{SH}}_B$, Betty’s response, is simply Eve’s card. Alice, upon receiving ${\mathit{SH}}_B$, knows her own hand and Eve’s card, and therefore can infer by elimination Betty’s hand.
At the end of the exchange, Alice knows Betty’s hand and Betty knows Alice’s hand, as required.
What about Eve? She does not learn anything about the cards in Alice or Betty’s hand. Indeed, after seeing Alice’s message, Eve has no information about Alice’s hand, since every card appears in exactly three of the sets Alice sent. There is no way for Eve to isolate which of those cards might be one of Alice’s. Furthermore, after seeing Betty’s message, all she has learned is her own card, which she already knew.
If we define $c{\,\mathord{\in}\,}i$ to be the primitive proposition *card $c$ is in player $i$’s hand* (where $A,B,E$ represent Alice, Betty, and Eve, respectively) then we expect that the following epistemic formula holds after the first message is received by Betty: $$K_B(a_1{\,\mathord{\in}\,}A) \land K_B(a_2{\,\mathord{\in}\,}A) \land K_B(a_3{\,\mathord{\in}\,}A),$$ that the following epistemic formula holds after the second message is received by Alice: $$K_A(b_1{\,\mathord{\in}\,}B) \land K_A(b_2{\,\mathord{\in}\,}B) \land K_B(b_3{\,\mathord{\in}\,}B),$$ and that the following formula holds after either of the messages is received: $$\begin{aligned}
& \neg K_E(a_1{\,\mathord{\in}\,}A) \land \neg K_E(a_2{\,\mathord{\in}\,}A) \land \neg K_E(a_3{\,\mathord{\in}\,}A)\\
& \qquad \land \neg K_E(b_1{\,\mathord{\in}\,}B)\land \neg K_E(b_2{\,\mathord{\in}\,}B)\land \neg K_E(b_3{\,\mathord{\in}\,}B).\end{aligned}$$ It is an easy exercise to construct the Kripke structures describing this scenario.
The Seven Hands protocol is ideally suited for epistemic reasoning via a possible-worlds semantics for knowledge, as it relies on combinatorial analysis. Its applicability, however, is limited.
The second protocol is a protocol to ensure anonymity, which is a form of confidentiality (see [\[chap13:beyond\]]{}). It does not rely on combinatorial analysis but rather on properties of the XOR operation.[^4] The Dining Cryptographers protocol was originally developed to solve the following problem. Suppose that Alice, Betty, and Charlene are three cryptographers having dinner at their favorite restaurant. Their waiter informs them that arrangements have been made for the bill to be paid anonymously by one party. That payer might be one of the cryptographers, but it might also be U.S. National Security Agency. The three cryptographers respect each other’s right to make an anonymous payment, but they would like to know whether the NSA is paying.
The following protocol can be used to satisfy the cryptographers’ curiosity and allow each of them to determine whether the NSA or one of her colleagues is paying, without revealing the identity of the payer in the latter case.
1. Every cryptographer $i$ flips a fair coin privately with her neighbor $j$ on her right: the Boolean result $T_{\{i,j\}}$ is ${\mathit{true}}$ if the coin lands tails, and ${\mathit{false}}$ if the coin lands heads. Thus, the cryptographers produce the Boolean results $T_{\{A,B\}}$, $T_{\{A,C\}}$, $T_{\{B,C\}}$; Alice sees $T_{\{A,B\}}$ and $T_{\{A,C\}}$; Betty sees $T_{\{A,B\}}$ and $T_{\{B,C\}}$; Charlene sees $T_{\{A,C\}}$ and $T_{\{B,C\}}$.
2. Every cryptographer $i$ computes a private Boolean value ${\mathit{Df}}_i$ as ${\mathit{true}}$ if the two coin tosses she has witnessed are different, and ${\mathit{false}}$ if they are the same. Thus, ${\mathit{Df}}_A =
T_{\{A,B\}}\oplus T_{\{A,C\}}$, ${\mathit{Df}}_B = T_{\{A,B\}}\oplus
T_{\{B,C\}}$, and ${\mathit{Df}}_C = T_{\{A,C\}}\oplus T_{\{B,C\}}$.
3. Every cryptographer $i$ publicly announces ${\mathit{Df}}_i$, except for the paying cryptographer (if there is one) who announces $\neg
{\mathit{Df}}_i$, the negation of ${\mathit{Df}}_i$.
Once the protocol is executed, any curious cryptographer interested in determining who paid for dinner simply has to take the XOR of all the announcements: if the result is ${\mathit{false}}$, then the NSA paid, and if the result is ${\mathit{true}}$, then one of the cryptographers paid.
To see why this is the case, consider the two possible scenarios. Suppose the NSA paid. Then the XOR of all the announcements is: $$\begin{aligned}
& {\mathit{Df}}_A \oplus {\mathit{Df}}_B \oplus {\mathit{Df}}_C \\
& \qquad = \left(T_{\{A,B\}}\oplus T_{\{A,C\}}\right) \oplus
\left(T_{\{B,C\}}\oplus T_{\{A,B\}}\right) \oplus
\left(T_{\{A,C\}}\oplus T_{\{B,C\}}\right) \\
& \qquad =
\left(T_{\{A,B\}}\oplus T_{\{A,B\}}\right) \oplus
\left(T_{\{B,C\}}\oplus T_{\{B,C\}}\right) \oplus
\left(T_{\{A,C\}}\oplus T_{\{A,C\}}\right) \\
& \qquad = {\mathit{false}} \oplus {\mathit{false}} \oplus {\mathit{false}} \\
& \qquad = {\mathit{false}}\end{aligned}$$ whereas if one of the cryptographers paid (without loss of generality, suppose it is Alice), then the XOR of all the announcements is: $$\begin{aligned}
& \neg {\mathit{Df}}_A \oplus {\mathit{Df}}_B \oplus {\mathit{Df}}_C \\
& \qquad =
\neg \left(T_{\{A,B\}}\oplus T_{\{A,C\}}\right) \oplus
\left(T_{\{B,C\}}\oplus T_{\{A,B\}}\right) \oplus
\left(T_{\{A,C\}}\oplus T_{\{B,C\}}\right) \\
& \qquad =
\left(\neg T_{\{A,B\}}\oplus T_{\{A,C\}}\right) \oplus
\left(T_{\{B,C\}}\oplus T_{\{A,B\}}\right) \oplus
\left(T_{\{A,C\}}\oplus T_{\{B,C\}}\right) \\
& \qquad =
\left(\neg T_{\{A,B\}}\oplus T_{\{A,B\}}\right) \oplus
\left(T_{\{B,C\}}\oplus T_{\{B,C\}}\right) \oplus
\left(T_{\{A,C\}}\oplus T_{\{A,C\}}\right) \\
& \qquad = {\mathit{true}} \oplus {\mathit{false}} \oplus {\mathit{false}} \\
& \qquad = {\mathit{true}}\end{aligned}$$ If one of the cryptographers paid, neither of the two other cryptographers will know which of her colleagues paid, since either possibility is compatible with what they can observe. Again, it is an easy exercise to construct the Kripke structures capturing these scenarios.
Cryptography {#chap13:crypto}
------------
While protocols such as the Seven Hands protocol and the Dining Cryptographers protocol enforce confidentiality by carefully constructing specific messages meant to convey specific information in a specific context, most cryptographic protocols rely on cryptography for confidentiality.
Cryptography seems a natural approach for confidentiality. After all, the whole point of cryptography is to hide information in such a way that only agents with a suitable key can access the information. And indeed, if the goal is for Alice to send message $m$ to Betty when Alice and Betty alone share a key to encrypt and decrypt messages, then the simplest protocol for confidential message exchange is simply for Alice to encrypt $m$ and send it to Betty. But how do Alice and Betty come to share a key in the first place? Distributing keys is tricky, because keys have to be sent to the right agents, in such a way that no other agent can get them.
Before addressing those problems, let us review the basics of cryptography. The reader is assumed to have been exposed to at least informal descriptions of cryptography. An encryption scheme is defined by a set of sourcetexts, a set of ciphertexts, a set of keys, and for every key $k$ an injective encryption function $e_k$ producing a ciphertext from a sourcetext and a decryption function $d_k$ producing a sourcetext from a ciphertext, with the property that $d_k(e_k(x))=x$ for all sourcetexts $x$. We often assume that ciphertexts and keys are included in sourcetexts to allow for nested encryption and encrypted keys.[^5]
There are two broad classes of encryption schemes, which differ in how keys are used for decryption. *Shared-key encryption schemes* require an agent to have a full key to both encrypt and decrypt a message. They tend to be efficient, and can often be implemented directly in hardware. *Public-key encryption schemes*, on the other hand, are set up so that an agent only needs to know part of a key (called the public key) to encrypt a message, while needing the full key to decrypt a message. The full key cannot be easily recovered from knowing only the public key. Public keys are generally made public (hence the name), so that any agent can encrypt a message intended for, say, Alice, by looking up and using Alice’s public key. Since only Alice has the full key, only she can decrypt that message. DES and AES are concrete examples of shared-key encryption schemes, while RSA and elliptic-curve encryption schemes are concrete examples of public-key encryption schemes.
Cryptographic protocols are needed with shared-key encryption schemes because agents need to share a key in order to exchange encrypted messages. How is such a shared key distributed? And how can agents make sure they are not tricked into sharing those keys with attackers? An additional difficulty is that when the same shared key is reused for every interaction between two agents, the content of all those interactions becomes available to an attacker that manages to learn that key. To minimize the impact of a key compromise, many systems create a fresh session key for any two agents that want to communicate, which exacerbates the key distribution problem.
Public-key encryption simplifies key distribution, since public keys can simply be published. Any agent wanting to send a confidential message to Alice has only to look up Alice’s public key and use it to encrypt her message. The problem, from Alice’s perspective, is that anyone can encrypt a message and send it to her, which means that if Alice wants to make sure that the encrypted message she received came from Betty, some sort of authentication mechanism is needed. Furthermore, all known public-key encryption schemes are computationally expensive, so a common approach is to have agents that want to exchange messages in a session first use public-key encryption to generate a session key for a shared-key encryption scheme that they use for their exchanged messages. Such a scenario requires authentication to ensure that agents are not tricked into communicating with an attacker.
#### Sample Cryptographic Protocols.
Most classical cryptographic protocols are designed to solve the problem of key distribution for shared-key encryption schemes, and of authentication for public-key encryption schemes. In these contexts, confidentiality and authentication are the key properties: confidentiality to enforce that distributed keys remain secret from attackers, and authentication to ensure that agents can establish the identity of the other agents involved in a message exchange.
This section presents two protocols, each illustrating different problems that can arise and highlighting vulnerabilities that attackers can exploit. (Attackers will be introduced more carefully in the next section.) The first protocol distributes session keys for a shared-key encryption scheme, while the second protocol aims at achieving mutual authentication for public-key encryption schemes. Not all of the problems illustrated will occur in every protocol, of course, nor are vulnerabilities in one context necessarily vulnerabilities in another context.
For the first protocol, consider the following situation. Suppose Alice wants to communicate with Betty, and there is a trusted server Serena who will generate a shared session key (for some shared-key encryption scheme) for them to use. Assume that every registered user of the system shares a distinct key with the trusted server in some shared-key encryption scheme; these keys for Alice and Betty are denoted $k_{{\mathit{AS}}}$ and $k_{{\mathit{BS}}}$.
The idea is for Serena to generate a fresh key and send it to both Alice and Betty. Sending it in the clear, however, would allow an eavesdropping attacker to read it and then use it to decrypt messages between Alice and Betty. Since Alice and Betty both share a key with Serena, one solution might be to use those keys to encrypt the session key sent to Alice and Betty, but this turns out to be difficult to implement in practice. Here is the problem. Alice, wanting to communicate with Betty, sends a message to Serena asking her to generate a session key, and Serena sends it to both Alice and Betty. As far as Betty is concerned, she receives a key with an indication that Alice will use it to send her messages. Betty now has to store the key and wait for Alice to send her messages encrypted with that key. If Alice wants to set up several concurrent communications with Betty, then Betty will have to match each incoming communication with the appropriate key, which is annoying at best and inefficient at worst. It turns out to be more efficient for Serena to send the fresh session key $k_{{\mathit{sess}}}$ to Alice, and for Alice to forward the key to Betty in her first message. This observation leads to the following protocol:
[prot1]{} [1. & A & & S & : & B]{}\
[2. & S & & A & : & B,[{k\_}\_[k\_]{}]{},[{k\_}\_[k\_]{}]{}]{}\
[3. & A & & B & : & A,[{k\_}\_[k\_]{}]{}]{}
Both Alice and Betty learn key $k_{{\mathit{sess}}}$, which is kept secret from eavesdroppers.
While this protocol might seem sufficient to distribute a key to both Alice and Betty, several things can go wrong in the presence of an insider attacker, that is, an attacker that is also a registered user of the system and has control over the network (i.e., can intercept and forge messages; see [\[chap13:attackers\]]{}).
The insider attacker, Isabel, can initiate a communication with trusted server Serena via her shared key $k_{{\mathit{IS}}}$ (Isabel is assumed to have such a key because she is a registered user of the system), and use the key to pose as Alice to Betty. Here is a sequence of messages exemplifying the attack, where the notation $I[A]$ denotes $I$ posing as $A$:[^6]
[attack2]{} [I & & S & : & B]{}\
[S & & I & : & B,[{k\_}\_[k\_]{}]{},[{k\_}\_[k\_]{}]{}]{}\
[I\[A\] & & B & : & A,[{k\_}\_[k\_]{}]{}]{}
Betty believes that she is sharing key $k_{{\mathit{sess}}}$ with Alice, while she is in fact sharing it with Isabel. This is a failure of authentication—the protocol does not authenticate the initiator to the responder.
Isabel can also trick Alice into believing she is talking to Betty, by posing as the server and intercepting messages between Alice and the server, as the following sequence of messages exemplifies:
[attack3]{} [A & & I\[S\] & : & B]{}\
[I\[A\] & & S & : & I]{}\
[S & & I\[A\] & : & I,[{k\_}\_[k\_]{}]{},[{k\_}\_[k\_]{}]{}]{}\
[I\[S\] & & A & : & B,[{k\_}\_[k\_]{}]{},[{k\_}\_[k\_]{}]{}]{}\
[A & & I\[B\] & : & A,[{k\_}\_[k\_]{}]{}]{}
This form of attack is commonly known as a man-in-the-middle attack. Isabel intercepts Alice’s message to the server, and turns around and sends a different message to the server posing as Alice. The response from the server is intercepted by Isabel, who crafts a suitable response back to Alice. Alice takes that response (which she believes is coming from the server) and sends it to Betty, but that message is intercepted by Isabel as well. Now, as far as Alice is concerned, she has successfully completed the protocol, and holds a key $k_{{\mathit{IS}}}$ that she believes she can use to communicate confidentially with Betty, while she is really communicating with Isabel.
How can we correct these vulnerabilities? One feature on which these attacks rely is that the identity of the intended parties for the keys in the protocol are easily forged by the attacker. So one fix is to bind the intended parties to the appropriate copies of the key. Here is an amended version of the protocol:
[prot4]{} [1. & A & & S & : & B]{}\
[2. & S & & A & : & B,[{B,k\_}\_[k\_]{}]{},[{A,k\_}\_[k\_]{}]{}]{}\
[3. & A & & B & : & [{A,k\_}\_[k\_]{}]{}]{}
When Alice receives her response from the server and decrypts her message ${\{B,k_{{\mathit{sess}}}\}_{k_{{\mathit{AS}}}}}$, she can verify that the key she meant Serena to create to communicate with Betty is in fact a key meant to communicate with Betty. This suffices to foil Isabel in attack . Similarly, when Betty receives her message from Alice containing ${\{A,k_{{\mathit{sess}}}\}_{k_{{\mathit{BS}}}}}$, she can verify that the key is meant to communicate with Alice. This suffices to foil attack .
Protocol now seems to work as intended. It does suffer from another potential vulnerability, though, one that is less directly threatening, but can still cause problems: it is susceptible to a replay attack. Here is the scenario. Suppose that Isabel eavesdrops on messages as Alice gets a session key $k_0$ from the trusted server to communicate with Betty, and holds on to messages ${\{B,k_0\}_{k_{{\mathit{AS}}}}}$ and ${\{A,k_0\}_{k_{{\mathit{BS}}}}}$. Suppose further that after a long delay Isabel manages to somehow obtain key $k_0$, perhaps by breaking into Alice’s or Betty’s computer, or by expending several months’ worth of effort to crack the encryption. Once Isabel has $k_0$, she can subvert an attempt by Alice to get a session key for communicating with Betty by simply intercepting the messages from Alice to Serena, and replaying the messages ${\{B,k_0\}_{k_{{\mathit{AS}}}}}$ and ${\{A,k_0\}_{k_{{\mathit{BS}}}}}$ she intercepted in the past. The following sequence of messages exemplifies this attack:
[attack5]{} [A & & I\[S\] & : & B]{}\
[I\[S\] & & A & : & B,[{B,k\_0}\_[k\_]{}]{},[{A,k\_0}\_[k\_]{}]{}]{}\
[A & & B & : & [{A,k\_0}\_[k\_]{}]{}]{}
The main point here is that Alice and Betty after this protocol interaction end up using key $k_0$ as their session key, but that key is one that Isabel knows, meaning that Isabel can decrypt every single message that Alice and Betty exchange in that session. So even though she does not have the shared keys $k_{{\mathit{AS}}}$ and $k_{{\mathit{BS}}}$, she has managed to trick Alice and Betty into using a key she knows.
Preventing this kind of replay attack requires ensuring that messages from earlier executions of the protocol cannot be used in later executions. One way to do that is to have every agent record every message they have ever sent and received, but that is too expensive to be practical. The common alternative is to use timestamps, or nonces. A nonce is a large random number, meant to be unpredictable and essentially unique—the likelihood that the same nonce occurs twice within two different sessions should be negligible. To fix protocol and prevent replay attacks, it suffices for Alice and Betty to generate nonces $n_A$ and $n_B$, respectively, and send them to trusted server Serena so that she can include them in her responses:
[prot6]{} [1. & A & & B & : & n\_A]{}\
[2. & B & & S & : & A,n\_A,n\_B]{}\
[3. & S & & A & : & [{B,k\_,n\_A}\_[k\_]{}]{},[{A,k\_,n\_B}\_[k\_]{}]{}]{}\
[4. & A & & B & : & [{A,k\_,n\_B}\_[k\_]{}]{}]{}
As long as Alice and Betty, when each receives her encrypted message containing the session key, both check that the nonce in the encrypted message is the one that they generated, then they can be confident that the encrypted messages have not been reused from earlier sessions.
Protocol now seems to work as intended and is not vulnerable to replay attacks. But it does not actually guarantee mutual authentication; that is, it does not guarantee to Alice that she is in fact talking to Betty when she believes she is, and to Betty that she is in fact talking to Alice when she believes she is. Consider the following attack, in which attacker Trudy is not an insider—she is not a registered user of the system—but has control of the network and thus can intercept and forge messages. Trudy poses as Betty by intercepting messages from Alice and forging responses:
[attack7]{} [A & & T\[B\] & : & n\_A]{}\
[T\[B\] & & S & : & A,n\_A,n\_T]{}\
[S & & A & : & [{B,k\_,n\_A}\_[k\_]{}]{},[{A,k\_,n\_T}\_[k\_]{}]{}]{}\
[A & & T\[B\] & : & [{A,k\_,n\_T}\_[k\_]{}]{}]{}
From Alice’s perspective, she has completed the protocol by exchanging messages with Betty, and holds a session key for sending confidential messages to Betty. But of course Alice has been communicating with Trudy, and Betty is not even aware of the exchange. Trudy cannot actually read the messages sent by Alice, so there is no breach of confidentiality, but Trudy has managed to trick Alice into believing she shares a session key with Betty. In terms of knowledge, Alice knows the session key, but she does not know that Betty does.
There is also a way for Trudy to trick Betty into believing she shares a session key with Alice, by posing as Alice:
[attack8]{} [T\[A\] & & B & : & n\_T]{}\
[B & & S & : & A,n\_T,n\_B]{}\
[S & & T\[A\] & : & [{B,k\_,n\_T}\_[k\_]{}]{},[{A,k\_,n\_B}\_[k\_]{}]{}]{}\
[T\[A\] & & B & : & [{A,k\_,n\_B}\_[k\_]{}]{}]{}
From Betty’s perspective, she has completed the protocol by exchanging messages with Alice, and holds a session key for sending confidential messages to Alice. But of course Betty has been communicating with Trudy, and Alice is not even aware of the exchange. In terms of knowledge, Betty knows the session key, but she does not know that Alice does.
Mutual authentication is achieved through an additional nonce exchange at the end of the protocol which uses the newly created session key:
[prot9]{} [1. & A & & B & : & n\_A]{}\
[2. & B & & S & : & A,n\_A,n\_B,n’\_B]{}\
[3. & S & & A & : & n’\_B,[{B,k\_,n\_A}\_[k\_]{}]{},[{A,k\_,n\_B}\_[k\_]{}]{}]{}\
[4. & A & & B & : & n’\_A,[{A,k\_,n\_B}\_[k\_]{}]{},[{A,n’\_B}\_[k\_]{}]{}]{}\
[5. & B & & A & : & [{B,n’\_A}\_[k\_]{}]{}]{}
This protocol now seems to work as intended without being vulnerable to replay attacks or authentication failures. How can this be guaranteed?
Intuitively, protocol is not susceptible to replay attacks because of the use of nonces: Alice can deduce that the first encrypted component of the third message was not reused from an earlier protocol execution, and Betty can deduce that the first encrypted component of the fourth message was not reused from an earlier protocol execution.
Similarly, mutual authentication in protocol follows from the use of shared keys: Alice can deduce that Serena created the first encrypted component of the third message; and Betty can deduce that Serena created the first encrypted component of the fourth message. Moreover, if Betty believes that $k_{{\mathit{sess}}}$ is a key known only to Alice and herself, then she can deduce that Alice created the second encrypted component in the fourth message, and similarly, if Alice believes that $k_{{\mathit{sess}}}$ is a key known only to Betty and herself, then she can deduce that Betty created the encrypted component in the fifth message.
The confidentiality of the session key requires the assumption that trusted server Serena is indeed trustworthy and creates keys that have not previously been used and distributed to other parties. If so, then Alice can deduce that the session key she receives in the third message is a confidential key for communicating with Betty, since Alice can also deduce that she has been executing the protocol with Betty. Similarly, Betty can deduce that the session key she receives in the fourth message is a confidential key for communicating with Alice, since Betty can deduce that she has been executing the protocol with Alice
In a precise sense, the goal of cryptographic protocol analysis is to prove these kind of properties formally, and many techniques have been developed which are surveyed below in [\[chap13:reasoning\]]{}.
The second cryptographic protocol uses a public-key encryption scheme to achieve mutual authentication:
[needham-schroeder]{} [1. & A & & B & : & [{A,n\_A}\_[\_B]{}]{}]{}\
[2. & B & & A & : & [{n\_A,n\_B}\_[\_A]{}]{}]{}\
[3. & A & & B & : & [{n\_B}\_[\_B]{}]{}]{}
where ${\mathit{pk}}_A$ and ${\mathit{pk}}_B$ are the public keys of Alice and Betty, respectively, and $n_A$ and $n_B$ are nonces.
Intuitively, when Alice receives her nonce $n_A$ back in the second message, she knows that Betty must have decrypted her first message at some point during the execution of the protocol (because only Betty could have decrypted the message that contained it), and similarly, when Betty receives her nonce $n_B$ back, she knows that Alice must have decrypted her second message. Note that $n_A$ and $n_B$ are kept confidential throughout the protocol, and that mutual authentication relies on that confidentiality.
Protocol , known as the Needham-Schroeder protocol, achieves mutual authentication even in the presence of an attacker that has control of the network and can intercept and forge messages. It is however vulnerable to insider attackers that are registered users of the system and have control of the network. For example, insider attacker Isabel can use an attempt by Alice to initiate an authentication session with her to trick unsuspecting Betty into believing that Alice is initiating an authentication session with her:
[attack-ns]{} [A & & I & : & [{A,n\_A}\_[\_I]{}]{}]{}\
[I\[A\] & & B & : & [{A,n\_A}\_[\_B]{}]{}]{}\
[B & & I\[A\] & : & [{n\_A,n\_B}\_[\_A]{}]{}]{}\
[I & & A & : & [{n\_A,n\_B}\_[\_A]{}]{}]{}\
[A & & I & : & [{n\_B}\_[\_I]{}]{}]{}\
[I\[A\] & & B & : & [{n\_B}\_[\_B]{}]{}]{}
From Alice’s perspective, she has managed to complete a mutual authentication session with Isabel, which was her goal all along. But Isabel also managed to complete an authentication session with Betty, tricking Betty into believing she is interacting with Alice.
There is a simple fix that eliminates that vulnerability:
[needham-schroeder-lowe]{} [1. & A & & B & : & [{A,n\_A}\_[\_B]{}]{}]{}\
[2. & B & & A & : & [{B,n\_A,n\_B}\_[\_A]{}]{}]{}\
[3. & A & & B & : & [{n\_B}\_[\_B]{}]{}]{}
It is interesting to see how the fix works: if Alice, during her mutual authentication attempt with Isabel, notices that the response message she receives from Isabel names a different agent than Isabel, then she can deduce that her authentication attempt is being subverted to try to confound another agent, and she can abort the authentication attempt at that point.
Attackers {#chap13:attackers}
---------
A distinguishing feature of cryptographic protocols, besides the use of cryptography, is that they are deployed in potentially hostile environments in which attackers may attempt to subvert the operations of the protocol.
Reasoning about cryptographic protocols, therefore, requires a threat model, describing the kind of attackers against which the cryptographic protocol should protect. Attackers commonly considered in the literature include:
- **Eavesdropping attackers:** assumed to be able to read all messages exchanged between agents. Eavesdropping attackers do not affect communication in any way, however, and remain hidden from other agents.
- **Active attackers:** assumed to have complete control over communications between agents, that is, able to read all messages as well as intercept them and forge new messages. They remain hidden from other agents, and thus no agent will intentionally attempt to communicate with an active attacker.
- **Insider attackers:** assumed to have complete control over communications between agents just like active attackers, but also considered legitimate registered users in their own right. They can therefore initiate interactions with other agents as themselves, and other agents can intentionally initiate interactions with them.[^7]
The class of insider attackers includes the class of active attackers, which itself includes the class of eavesdropping attackers. Thus, in that sense, an insider attacker is stronger than an active attacker which is stronger than an eavesdropping attacker. In practice, this means that a cryptographic protocol that is deemed secure in the presence of an insider attacker will remain so in the presence of active and eavesdropping attackers, and so on.
We saw several examples of attacks in [\[chap13:crypto\]]{}, performed by different kind of attackers. Most of the protocols in [\[chap13:crypto\]]{} achieve their goals in the presence of eavesdropping attackers, while some also achieve their goals in the presence of active attackers but fail in the presence of insider attackers. The Needham-Schroeder protocol , for instance, can be shown to satisfy mutual authentication in the presence of active attackers, but not in the presence of insider attackers—as exemplified by attack —while the variant protocol achieves mutual authentication even in the presence of insider attackers.
The attacks described in [\[chap13:crypto\]]{} took place at the level of the protocols themselves, and not at the level of the encryption schemes used by the protocols. But vulnerabilities in encryption schemes are also relevant: an attacker cracking an encrypted message from the trusted server to the agents in protocol will learn the session key, which will invalidate any confidentiality guarantees claimed for the protocol. Despite this, cryptographic protocol are typically analyzed independently from the details of the encryption scheme. The main reason is that it abstracts away from vulnerabilities specific to the encryption scheme used, leaving only those relating to the cryptographic protocol. Vulnerabilities in encryption schemes are usually independent of the cryptographic protocols that use them, and can be investigated separately. A vulnerability in the protocol will be a vulnerability no matter what encryption scheme is used, and requires a change in the protocol to correct the flaw.
The standard way to analyze cryptographic protocols independently of any encryption scheme is to use a *formal model of cryptography* that assumes perfect encryption leaking no information about encrypted content. It can be defined as the following symbolic encryption scheme. If $P$ is a set of plaintexts and $K$ is a set of keys, then the set of sourcetexts is taken to be the smallest set $S$ of symbolic terms containing $P$ and $K$ such that $(x,y)\in S$ and ${\{x\}_{k}}\in S$ when $x,y\in S$ and $k\in K$. Intuitively, $(x,y)$ represents the concatenation of $x$ and $y$, and ${\{x\}_{k}}$ represents the encryption of $x$ with key $k$. The ciphertexts are all sourcetexts of the form ${\{x\}_{k}}$. The symbolic encryption function $e_k(x)$ simply returns ${\{x\}_{k}}$, and the symbolic decryption function $d_k(x)$ returns $y$ if $x$ is ${\{y\}_{k}}$, and some special token **fail** otherwise.[^8]
In the context of analyzing protocols with a formal model of cryptography, attackers are usually modeled using *Dolev-Yao capabilities*. These capabilities go hand in hand with the symbolic aspect of formal models of cryptography. Intuitively, eavesdropping Dolev-Yao attackers can split up concatenated messages and decrypt them if they know the decryption key; active Dolev-Yao attackers can additionally create new messages by concatenating existing messages and encrypting them with known keys. Dolev-Yao attackers do not have the capability of cracking encryptions, nor can they access messages at the level of their component bits.
Modeling Knowledge {#chap13:knowledge}
------------------
The analyses in [\[chap13:crypto\]]{} show that various notions of knowledge arise rather naturally when reasoning informally about properties of cryptographic protocols. There are essentially two main kinds of knowledge described in the literature. In some frameworks, both kinds of knowledge are used.
#### Message Knowledge.
The first kind of knowledge, the most common and in some sense the most straightforward, tries to capture the notion of *knowing a message*.
There are several equivalent approaches to modeling this kind of knowledge, at least in a formal model of cryptography with Dolev-Yao capabilities. Intuitively, the idea is a constructive one: an attacker knows a message if she can construct that message from other messages she has received or intercepted. (Message knowledge in the context of confidentiality properties is often presented from the perspective of an attacker, since confidentiality is breached when the attacker comes to know a particular message.) In such a context, knowing a message is sometimes called *having a message*, *possessing a message*, or *seeing a message*.
Message knowledge may be described via the following sets. Let $H$ be a set of messages that the attacker has received or intercepted. The set ${\mathit{Parts}}(H)$, the set of all components of messages from $H$, is defined inductively by the following inference rules: $$\begin{gathered}
{ \begin{array}{c}
m \in H \\\hline
m\in{\mathit{Parts}}(H)
\end{array}}\qquad
{ \begin{array}{c}
{\{m\}_{k}}\in{\mathit{Parts}}(H) \\\hline
m\in{\mathit{Parts}}(H)
\end{array}}\\\displaybreak[0]
{ \begin{array}{c}
(m_1,m_2)\in{\mathit{Parts}}(H) \\\hline
m_1\in{\mathit{Parts}}(H)
\end{array}}\qquad
{ \begin{array}{c}
(m_1,m_2)\in{\mathit{Parts}}(H) \\\hline
m_2\in{\mathit{Parts}}(H)
\end{array}}\end{gathered}$$ We see that the content of all encrypted messages in $H$ is included in ${\mathit{Parts}}(H)$, even those that the attacker cannot decrypt. In a sense, ${\mathit{Parts}}(H)$ is an upper bound on messages the attacker can know. In contrast, the set ${\mathit{Analyzed}}(H)$ of messages that the attacker can actually see is more restricted: $$\begin{gathered}
{ \begin{array}{c}
m \in H \\\hline
m\in{\mathit{Analyzed}}(H)
\end{array}}\\\displaybreak[0]
{ \begin{array}{c}
(m_1,m_2)\in{\mathit{Analyzed}}(H) \\\hline
m_1\in{\mathit{Analyzed}}(H)
\end{array}}\qquad
{ \begin{array}{c}
(m_1,m_2)\in{\mathit{Analyzed}}(H) \\\hline
m_2\in{\mathit{Analyzed}}(H)
\end{array}}\\\displaybreak[0]
{ \begin{array}{c}
{\{m\}_{k}}\in{\mathit{Analyzed}}(H)\quad k\in{\mathit{Analyzed}}(H) \\\hline
m\in{\mathit{Analyzed}}(H)
\end{array}}\end{gathered}$$ Clearly, ${\mathit{Analyzed}}(H)\subseteq{\mathit{Parts}}(H)$. One definition of message knowledge is to say that an attacker knows message $m$ in a state where she has received or intercepted a set $H$ of messages if $m\in{\mathit{Analyzed}}(H)$. This is the *attacker knows what she can see* interpretation of message knowledge.
The best way to understand this concept of knowledge is to use a physical analogy: we can think of plaintext messages as stones, and encrypted messages as locked boxes. Encrypting a message means putting it in a box and locking it. A message is known if it can be held in one’s hands. An encrypted message is known because the box can be held. The content of an encrypted message is known only if the box can be opened (decrypted) and the content (a stone or another box) taken and held.
This form of message knowledge can be captured fairly easily in any logic without using heavy technical machinery, since the data required to define message knowledge is purely local. If we let ${\mathit{Messages}}_i(s)$ be the set of messages received or intercepted by agent $i$ in state $s$ of the system, then we can capture message knowledge via a proposition ${\mathit{knows}}_i(m)$, where $i$ is an agent and $m$ is a message, defined to be true at state $s$ if and only if $m\in{\mathit{Analyzed}}({\mathit{Messages}}_i(s))$.
Rather than using a dedicated proposition, another approach relies on a dedicated modal operator to capture message knowledge. Message knowledge as defined above can be seen as a form of *explicit knowledge*, often represented by a modal operator $X_i{\varphi}$, read *agent $i$ explicitly knows ${\varphi}$*. (Explicit knowledge is to be contrasted with the implicit knowledge captured by the possible-worlds definition of knowledge.) One form of explicit knowledge, *algorithmic knowledge*, uses a local algorithm stored in the local state of an agent to determine if ${\varphi}$ is explicitly known to that agent. Thus, $X_i{\varphi}$ is true at a state $s$ if the local algorithm of agent $i$ says that the agent knows ${\varphi}$ in state $s$. If we let proposition ${\mathit{part}}_i(m)$ be true at a state $s$ when $m\in{\mathit{Parts}}({\mathit{Messages}}_i(s))$, then it is a simple matter to define a local algorithm to check if $m\in{\mathit{Analyzed}}({\mathit{Messages}}_i(s))$ and capture knowledge of message $m$ via $X_i({\mathit{part}}_i(m))$: *agent $i$ explicitly knows that message $m$ is part of the messages she has received*. Thus, ${\mathit{part}}_i(m)$ may be true at a state while $X_i({\mathit{part}}_i(m))$ is false at that state if the message is encrypted with a key that agent $i$ does not know.
A variant of the *can see* interpretation of message knowledge is to consider instead the messages that an attacker can create. The set ${\mathit{Synthesized}}(H)$ of messages that the attacker can create from a set $H$ of messages is inductively defined by the following inference rules: $$\begin{gathered}
{ \begin{array}{c}
m \in H \\\hline
m\in{\mathit{Synthesized}}(H)
\end{array}}\\\displaybreak[0]
{ \begin{array}{c}
m_1\in{\mathit{Synthesized}}(H)\quad m_2\in{\mathit{Synthesized}}(H) \\\hline
(m_1,m_2)\in{\mathit{Synthesized}}(H)
\end{array}}\\\displaybreak[0]
{ \begin{array}{c}
m\in{\mathit{Synthesized}}(H)\quad k\in{\mathit{Synthesized}}(H) \\\hline
{\{m\}_{k}}\in{\mathit{Synthesized}}(H)
\end{array}}\end{gathered}$$ An alternative interpretation of message knowledge, the *attacker knows what she can send* interpretation, can be defined as: an attacker knows message $m$ in a state where she has received or intercepted a set $H$ of messages if $m\in{\mathit{Synthesized}}({\mathit{Analyzed}}(H))$. Since ${\mathit{Analyzed}}(H)\subseteq{\mathit{Synthesized}}({\mathit{Analyzed}}(H))$, everything an attacker can see she can also send.
The *can send* interpretation of message knowledge is tricky, because clearly any agent can send any plaintext and any key—this is akin to being able to send any password—and it is easy to inadvertently define nondeterministic attackers that can synthesize any message. The intent is for attackers to be able to send only messages based on those she has received or intercepted, but that is a restriction that can be difficult to justify. This suggests some subtleties in choosing the right definition of message knowledge.
Message knowledge, whether under the *can see* or *can send* interpretation, is severely constrained. It is knowledge of terms, as opposed to knowledge of facts—although terms can be facts, facts are more general than terms. Message knowledge is conducive to formal verification using a variety of techniques, mostly because it does not require anything but looking at the local state of an agent. Indeed, message knowledge is inherently local.
#### Possible-Worlds Knowledge.
The other kind of knowledge that arises in the study of cryptographic protocols is the standard possible-worlds definition of knowledge via an accessibility relation over the states of a structure. The Kripke structures interpreting knowledge are usually sets of traces of the protocol and the accessibility relation for agent $i$ is an equivalence relation over the states of the system that relates two states in which agent $i$ has the same local state (including having received or intercepted the same messages).
In the presence of cryptography, the standard accessibility relation, meant to capture when two states are indistinguishable to an agent, seems inappropriate. After all, the whole point of cryptography is to hide information—and in particular, most cryptographic definitions say that if an agent receives message $m_1$ encrypted with a key $k_1$ that she does not know and message $m_2$ encrypted with key $k_2$ that she also does not know, then that agent should be unable to distinguish the two messages, in the sense of being able to identify which is which. Thus, goes the argument, a state where an agent has received ${\{m_1\}_{k_1}}$ and a state where that agent has received ${\{m_2\}_{k_2}}$ instead should be indistinguishable if $k_1$ and $k_2$ are not known.
To capture a more appropriate definition of state indistinguishability, one approach is to filter the local states through a function that replaces all messages encrypted with an unknown key by a special token $\Box$. More precisely, if $H$ is a set of messages, we write $[H] = \{[m]^H : m\in H\}$, where $[m]^H$ is inductively defined as follows: $$\begin{aligned}
[m]^H & = m \qquad \text{if $m$ is a plaintext}\\\displaybreak[0]
[(m_1,m_2)]^H & = ([m_1]^H,[m_2]^H) \\\displaybreak[0]
[{\{m\}_{k}}]^H & = \begin{cases}
{\{[m]^H\}_{k}} & \text{if $k\in{\mathit{Analyzed}}(H)$}\\
\Box & \text{otherwise}
\end{cases}\end{aligned}$$ The revised equivalence relations $\sim^\Box_i$ through which knowledge is interpreted can now be defined to be $s\sim^\Box_i t$ if and only if $[{\mathit{Messages}}_i(s)]=[{\mathit{Messages}}_i(t)]$.[^9]
The definition $[-]^H$ above, which is typical, uses ${\mathit{Analyzed}}(-)$ to extract the keys that the agent knows. Alternate definitions can be given, from a simpler definition that looks for keys appearing directly in the local state, to a more complex recursive definition defined using possible-worlds knowledge.
Possible-worlds knowledge interpreted via an $\sim_i^\Box$ accessibility relation is general enough to express message knowledge. If we assume a class of propositions ${\mathit{part}}_i(m)$ as before, true at a state $s$ when $m\in{\mathit{Parts}}({\mathit{Messages}}_i(s))$, then formula $K_i({\mathit{part}}_i(m))$ says that agent $i$ knows message $m$—intuitively, she knows that $m$ is part of some message in her local state, and has access to it.
To see that $K_i({\mathit{part}}_i(m))$ corresponds to message knowledge as defined above, we can relate it to the definition of message knowledge in terms of a local ${\mathit{knows}}_i(m)$ proposition, using the *can see* interpretation of message knowledge. It is not difficult to show that if ${\mathit{knows}}_i(m)$ is true at state $s$, then $K_i({\mathit{part}}_i(m))$ must also be true at state $s$. The converse direction requires a suitable richness condition that guarantees that there are enough encrypted messages to compare: for every message ${\{m\}_{k}}$ received or intercepted where $k$ is not known to the agent, there should exist another state in which the agent has received ${\{m'\}_{k'}}$ for a different $m'$ and a different $k'$.[^10] Under such a richness condition, if $K_i({\mathit{part}}_i(m))$ is true at a state $s$, then ${\mathit{knows}}_i(m)$ is true at that same state $s$.
Thus, possible-worlds knowledge can be used to express message knowledge, and can also capture higher-order knowledge, that is, knowledge about general facts, including other agents’ knowledge. The informal analyses of [\[chap13:crypto\]]{} show that it makes sense to state that Alice may know that Betty knows the key. While Betty’s knowledge here is message knowledge (knowledge of the key) and therefore can be modeled with any of the approaches above, Alice’s knowledge is higher-order knowledge, knowledge about knowledge of another agent. Logics that allow reasoning about Alice’s knowledge of Betty’s knowledge of the key tend to rely on possible-worlds definitions of knowledge.
Reasoning about Cryptographic Protocols {#chap13:reasoning}
---------------------------------------
Several approaches have been developed for reasoning about cryptographic protocols. Most are not based on epistemic logic, but extend a classical propositional or first-order logic—possibly with temporal operators—with a simple form of message knowledge in the spirit of ${\mathit{knows}}_i(m)$. This allows them to leverage well-understood techniques for system analysis from the formal verification community and from the programming language community. Other approaches are explicitly epistemic in nature.
Techniques for reasoning about cryptographic protocols roughly split along two axes, each corresponding to a way of using logic to reason about protocols in general.
1. Reasoning can be performed either deductively using the proof theory of the logic (e.g., through deductions in a theorem prover), or semantically, using the models of the logic (e.g., through model checking).
2. Reasoning can be performed either directly on the description of the protocol—either taken as a sequence of messages or a program for each role in the protocol—or indirectly on the set of traces generated by protocol executions.
Comparing reasoning methods across these axes is difficult, as each have their advantages and their disadvantages.
#### The Inductive Method.
A good example of a deductive approach for reasoning about security protocols is the Inductive Method, based on inductive definitions in higher-order logic (a generalization of first-order predicate logic allowing quantification over arbitrary relations). These inductive definitions admit powerful induction principles which become the main proof technique used to establish confidentiality and authentication properties.
The Inductive Method is fairly characteristic of many deductive approaches to cryptographic protocol analysis: the deductive system is embedded in a powerful logic such as higher-order logic, and does not use epistemic concepts beyond a local definition of message knowledge equivalent to the use of a ${\mathit{knows}}_i(m)$ proposition.
The Inductive Method proper is based on defining a theory—a set of logical rules—for analyzing a given protocol. The theory for a protocol describes how to generate the protocol execution traces, where a trace is a sequence of events such as *$A$ sends $m$ to $B$*, represented by the predicate $\mathsf{Say}(A,B,m)$. Rules state which events can possibly follow a given sequence of events, thereby describing traces inductively. In general, there is a rule in the theory for every message in the protocol description. Rules inductively define a set $\mathsf{Prot}$ of traces representing all the possible traces of the protocol.
If we consider a theory for Protocol , a rule for message (1) would say: $$\begin{aligned}
& {\mathit{tr}} \in \mathsf{Prot} \\
& \qquad {\Rightarrow}{\langle {\mathit{tr}}, \textsf{Say}(A,B,{\{A,n_A\}_{{\mathit{pk}}_B}})\rangle}\in \mathsf{Prot}\end{aligned}$$ where ${\langle {\mathit{tr}}, e\rangle}$ adds event $e$ to trace ${\mathit{tr}}$. That is, if ${\mathit{tr}}$ represents a valid trace of the protocol, then that trace can be extended with the first message of a new protocol execution. Similarly, a rule for message (2) would say: $$\begin{aligned}
& {\mathit{tr}} \in \mathsf{Prot} \\
& \land \mathsf{Say}(A',B,{\{A,n_A\}_{{\mathit{pk}}_B}})\in{\mathit{tr}} \\
& \qquad {\Rightarrow}{\langle {\mathit{tr}}, \mathsf{Say}(B,A,{\{B,n_A,n_B\}_{{\mathit{pk}}_A}})\rangle}\in\mathsf{Prot}\end{aligned}$$ That is, if ${\mathit{tr}}$ is a valid trace of the protocol in which an agent has received the first message of a protocol execution, that agent can respond appropriately with the second message of the protocol execution.[^11] Rules are simply implications and conjunctions over a vocabulary of events.
The attacker $S$ is also defined by rules; these rules describe how attacker actions can extend traces with new events. For a Dolev-Yao attacker, these rules define a nondeterministic process that can intercept any message, decompose it into parts and decrypt it if the correct key is known, and that can create new messages from other messages it has observed. The theory includes inductive definitions for the ${\mathit{Analyzed}}$ and ${\mathit{Synthesized}}$ sets given in [\[chap13:knowledge\]]{}, as well as rules of the form $$\begin{aligned}
& {\mathit{tr}}\in\mathsf{Prot}\\
& \land m\in{\mathit{Synthesized}}({\mathit{Analyzed}}(\mathsf{Spied}({\mathit{tr}})))\\
& \land B\in\mathsf{Agents}\\
& \qquad {\Rightarrow}{\langle {\mathit{tr}}, \mathsf{Say}(S,B,m)\rangle} \in \mathsf{Prot}\end{aligned}$$ that states that if $m$ can be synthesized from the messages the attacker observed on trace ${\mathit{tr}}$ (captured by an inductively-defined set $\mathsf{Spied}({\mathit{tr}})$), then the attacker can add an event $\mathsf{Say}(S,B,m)$ for any agent $B$ to the trace.
The Inductive Method is geared for proving safety properties: for every state in every trace, that state is not a bad state. A protocol is proved correct by induction on the length of the traces: choosing the shortest sequence to a bad state, assuming all states earlier on the trace are good, then deriving a contradiction by showing that any state following these good states must be good itself.
A confidentiality property such as *the attacker never learns message $m$* is established by making sure that the attacker is unable to ever send message $m$, by proving the following formula: $$(\forall{\mathit{tr}}\in\mathsf{Prot})\,(\forall B\in\mathsf{Agents})\,\mathsf{Say}(S,B,m)\not\in{\mathit{tr}}$$ This is a *can send* interpretation of message knowledge. Indeed, according to the rules for the attacker, if the attacker knows message $m$ at any point during a trace, then there exists a extension of that trace where the attacker sends message $m$. Thus, showing that the attacker never learns message $m$ amounts to showing that there is no trace in which an event $\mathsf{Say}(S,B,m)$ appears, for any agent $B$.
Abstracting away from the details of the approach, the Inductive Method essentially relies on rules to describe the evolution of a protocol execution, and verifying a confidentiality property is reduced to verifying that a certain bad state is not reachable. Other approaches to cryptographic protocol analysis share this methodology, many of them using a logic programming language rather than higher-order logic to express protocol evolution rules; see [\[chap13:bibnotes\]]{}.
#### BAN Logic.
The Inductive Method relies on encoding rules for generating protocol execution traces in an expressive general logic suitable for automating inductive proofs. In contrast, the next approach, BAN Logic, is a logic tailored for reasoning about cryptographic protocols described as a sequence of message exchanges. It has the additional feature of including a higher-order *belief* operator as a primitive.
BAN Logic is a logic in the tradition of Hoare Logic, in that it advocates an axiomatic approach for reasoning about cryptographic protocols. BAN Logic tracks the evolution of beliefs during the execution of cryptographic protocol, and is described by a set of inference rules for deriving new beliefs from old. BAN Logic includes primitive formulas stating that $k$ is a shared key known only to $A$ and $B$ ($A{\mathrel{\stackrel{\scriptscriptstyle {k}}{\leftrightarrow}}}B$), that $m$ is a secret between $A$ and $B$ ($A{\mathrel{\stackrel{\scriptscriptstyle
{m}}{\rightleftharpoons}}}B$), that agent $A$ believes formula $F$ ($A{~{\mathsf{believes}}~}F$), that agent $A$ controls the truth of formula $F$ ($A{~{\mathsf{controls}}~}F$), that agent $A$ sent a message meaning $F$ ($A{~{\mathsf{said}}~}F$), that agent $A$ received and understood a message meaning $F$ ($A{~{\mathsf{sees}}~}F$), and that a message meaning $F$ was created during the current protocol execution (${{\mathsf{fresh}}}( F)$). The precise semantics of these formulas is given indirectly through inference rules, some of which are presented below.
BAN Logic assumes that agents can recognize when an encrypted message is one they have created themselves; encryption is in consequence written ${\{F\}_{k}}^i$, where $i$ denotes the agent who encrypted a message meaning $F$ with key $k$. (This also highlights another characteristic of BAN Logic: messages are formulas.)
Here are some of the inference rules of BAN Logic: $$\begin{aligned}
& \text{(R1)} &
& { \begin{array}{c}
A{~{\mathsf{believes}}~}B{\mathrel{\stackrel{\scriptscriptstyle {k}}{\leftrightarrow}}}A\quad A{~{\mathsf{sees}}~}\{ F\}^i_k \quad i\ne A \\\hline
A{~{\mathsf{believes}}~}B{~{\mathsf{said}}~}F
\end{array}} \\\displaybreak[0]
& \text{(R2)} &
& { \begin{array}{c}
A{~{\mathsf{believes}}~}B {~{\mathsf{said}}~}(F,F') \\\hline
A {~{\mathsf{believes}}~}B{~{\mathsf{said}}~}F
\end{array}}\\\displaybreak[0]
& \text{(R3)} &
& { \begin{array}{c}
A{~{\mathsf{believes}}~}{{\mathsf{fresh}}}( F)\quad A{~{\mathsf{believes}}~}(B{~{\mathsf{said}}~}F) \\\hline
A{~{\mathsf{believes}}~}B{~{\mathsf{believes}}~}F
\end{array}}\\\displaybreak[0]
& \text{(R4)} &
& { \begin{array}{c}
A{~{\mathsf{believes}}~}B{~{\mathsf{controls}}~}F \quad A{~{\mathsf{believes}}~}B{~{\mathsf{believes}}~}F \\\hline
A{~{\mathsf{believes}}~}F
\end{array}} \\\displaybreak[0]
& \text{(R5)} &
& { \begin{array}{c}
A{~{\mathsf{sees}}~}( F, F') \\\hline
A{~{\mathsf{sees}}~}F
\end{array}}\\\displaybreak[0]
& \text{(R6)} &
& { \begin{array}{c}
A{~{\mathsf{believes}}~}B{\mathrel{\stackrel{\scriptscriptstyle {k}}{\leftrightarrow}}}A\quad A{~{\mathsf{sees}}~}\{ F\}^i_k \quad i \ne A \\\hline
A{~{\mathsf{sees}}~}F
\end{array}}\\\displaybreak[0]
& \text{(R7)} &
& { \begin{array}{c}
A{~{\mathsf{believes}}~}{{\mathsf{fresh}}}( F) \\\hline
A{~{\mathsf{believes}}~}{{\mathsf{fresh}}}(( F', F))
\end{array}}\\\displaybreak[0]
& \text{(R8)} &
& { \begin{array}{c}
A {~{\mathsf{believes}}~}B {~{\mathsf{believes}}~}( F, F') \\\hline
A {~{\mathsf{believes}}~}B {~{\mathsf{believes}}~}F
\end{array}}\end{aligned}$$ Rule (R1), for instance, says that if agent $A$ believes that $k$ is shared only between $B$ and herself, and she receives a message encrypted with key $k$ that she did not encrypt herself, then she believes that $B$ sent the original message. Rule (R3) is an honesty rule: it says that agents send messages meaning $F$ only when they believe $F$. There are commutative variants of rules (R2), (R5), (R7), and (R8), as well as variants for more general tuples; there are also variants of (R8) for any level of nested belief.
BAN Logic does not attempt to model protocol execution traces. Reasoning is done directly on the sequence of messages in the description of the protocol. Because sequences of messages do not carry enough information to permit this kind of reasoning, a transformation known as *idealization* must be applied to the protocol. Roughly speaking, idealization consists of replacing the messages in the protocol by formulas of BAN Logic that capture the intent of each message. For instance, if agent $A$ sends key $k$ to agent $B$ with the intention of sharing a key that is known only to $A$, then a suitable idealization would have $A$ send the formula $A{\mathrel{\stackrel{\scriptscriptstyle {k}}{\leftrightarrow}}}B$ to $B$. Idealization is an annotation mechanism, and as such is somewhat subjective.
To illustrate reasoning in BAN Logic, consider the following simple protocol in which Alice sends a secret value $m_0$ to Betty encrypted with their shared key $k_{{\mathit{AB}}}$, along with a nonce exchange to convince $B$ that the message is not a replay of a message in a previous execution of the protocol (see [\[chap13:crypto\]]{}):
[prot-shared]{} [1. & A & & B & : & A]{}\
[2. & B & & A & : & n\_B]{}\
[3. & A & & B & : & A,[{m\_0,n\_B}\_[k\_]{}]{}]{}
A possible idealization of protocol would be:
[prot-shared-ideal]{} [3’. & A & & B & : & [{AB,n\_B}\_[k\_]{}]{}]{}
The first two messages in protocol carry information that BAN Logic does not use, so they are not present in the idealized protocol. The third message is idealized to $A$ sending formula $A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B$ to $B$ along with the nonce $n_B$, indicating that $A$ considers $m_0$ to be a secret at that point.
Reasoning about an idealized protocol consists in laying out the initial beliefs of the agents, and deriving new beliefs from those and from the messages exchanged between the agents, using the inference rules of the logic. For protocol , initial beliefs include that both parties believe that key $k_{\mathit{AB}}$ has not been compromised, that nonce $n_B$ has not already been used, and that message $m_0$ that $A$ wants to send to $B$ is initially secret. These initial beliefs are captured by the following formulas: $$\begin{aligned}
& A{~{\mathsf{believes}}~}A{\mathrel{\stackrel{\scriptscriptstyle {k_{\mathit{AB}}}}{\leftrightarrow}}}B \\
& B{~{\mathsf{believes}}~}A{\mathrel{\stackrel{\scriptscriptstyle {k_{\mathit{AB}}}}{\leftrightarrow}}}B \\
& B{~{\mathsf{believes}}~}{{\mathsf{fresh}}}(n_B)\\
& A{~{\mathsf{believes}}~}A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B\end{aligned}$$ We can derive new formulas from these initial beliefs in combination with the messages exchanged by the agents. The idea is to update this set of formulas after each protocol step: after an idealized step $A\rightarrow B: F$, which says that $B$ receives a message meaning $F$, we can add formula $B{~{\mathsf{sees}}~}F$ to the set of formulas, and we use the inference rules to derive additional formulas to add to the set.
For example, in idealized protocol , after message (3’), we add formula $$\label{e:Bseesencryption}
B {~{\mathsf{sees}}~}{\{A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B,n_B\}_{k_{{\mathit{AB}}}}}$$ to the set of initial beliefs. Along with the initial belief $B{~{\mathsf{believes}}~}A{\mathrel{\stackrel{\scriptscriptstyle {k_{\mathit{AB}}}}{\leftrightarrow}}}B$, formula allows us to apply inference rule (R1) to derive: $$\label{e:BbelievesAsaid}
B{~{\mathsf{believes}}~}A{~{\mathsf{said}}~}(A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B,n_B).$$ From the initial belief $B{~{\mathsf{believes}}~}{{\mathsf{fresh}}}(n_B)$, inference rule (R7) lets us derive that any message combined with $n_B$ must be fresh, and thus we can derive: $$\label{e:Bbelievesfresh}
B{~{\mathsf{believes}}~}{{\mathsf{fresh}}}(A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B,n_B).$$ Formula together with give us, via inference rule (R3): $$B{~{\mathsf{believes}}~}A{~{\mathsf{believes}}~}(A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B,n_B).$$ Via inference rule (R8), this yields: $$\label{e:BbelievesAbelieves}
B{~{\mathsf{believes}}~}A{~{\mathsf{believes}}~}A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B.$$ Thus, after the messages of the idealized protocol have been exchanged, $B$ believes that $A$ believes that $m_0$ is a secret between $A$ and $B$. This is about as much as we can expect.
We can say more if we are willing to assume that $B$ believes that the secrecy of $m_0$ is in fact controlled by $A$. If so, we can add the following formula to the set of initial beliefs: $$\label{e:BbelievesAcontrols}
B{~{\mathsf{believes}}~}A{~{\mathsf{controls}}~}A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B$$ and formulas and combine via inference rule (R4) to yield the stronger conclusion: $$B{~{\mathsf{believes}}~}A{\mathrel{\stackrel{\scriptscriptstyle
{m_0}}{\rightleftharpoons}}}B.$$ In other words, if $B$ believes that $A$ controls the secrecy of $m_0$ and also that $A$ believes $m_0$ to be secret, then after the protocol executes $B$ also believes that $m_0$ is a secret shared only with $A$.
Attackers are not explicit in BAN Logic. In a sense, an active Dolev-Yao attacker is implicitly encoded within the inference rules of the logic, but the focus of BAN Logic is reasoning about the belief of agents in the presence of an active attacker, as opposed to reasoning about the knowledge of an attacker. A successful attack in BAN Logic shows up as a failure to establish a desired belief for one of the agents following a protocol execution.
#### Temporal and Epistemic Temporal Logics.
Another class of approaches for reasoning about cryptographic protocols rely on a form of temporal logic to express desired properties of the protocol and show that they are true of a model representing the protocol—generally through a suitable representation of its traces. This is done through model-checking techniques to determine algorithmically whether a formula is true in the models representing the protocol. These model-checking techniques vary in terms of how the models are described: these can be either directly expressed by finite state machines, or through domain-specific languages.
The simplest approaches to cryptographic protocol analysis via temporal logics merely extend existing temporal-logic verification techniques. At least two challenges arise in these cases: modeling attackers, and expressing message knowledge. For attackers, while eavesdropping attackers do not affect the execution of protocols and therefore are comparatively easy to handle in standard temporal-logic verification frameworks, active attackers require work. In some cases, it is possible to simply encode an active attacker within the model using the tools of the framework. Message knowledge is usually dealt with by introducing a variant of a $\mathit{knows}_i(m)$ proposition.
In general, the logics themselves are completely straightforward: they are standard propositional or first-order temporal logics extended with a message knowledge predicate. All the action is in the interpretation of the message knowledge predicate, and in the construction of the models to account for the actions of active attackers. There is not much to say about those approaches as far as pertains to epistemic concepts, but they are popular in practice.
More interesting from an epistemic perspective are those frameworks relying on a temporal epistemic logic, that is, a logic with both temporal and epistemic operators. The MCK model-checker is an example of a verification framework that uses a linear-time temporal logic with epistemic operators to verify protocols that do not use cryptography, such as the Seven Hands or the Dining Cryptographers protocols of [\[chap13:protocols\]]{}. Protocols are described via finite state machines, and formulas express properties of paths through that finite state machine, each such path corresponding to a possible execution of the protocol.
As an example, consider the Dining Cryptographers protocol, which translates well to a finite state machine. States can be described using three agent-indexed Boolean variables ${\mathit{paid}}[i]$, ${\mathit{chan}}[i]$, and ${\mathit{df}}[i]$, where variable ${\mathit{paid}}[i]$ records whether agent $i$ paid;, variable ${\mathit{chan}}[i]$ is a communication channel used by agent $i$ to send the result of its coin toss to her right neighbor, and variable ${\mathit{df}}[i]$ records the announcement of ${\mathit{Df}}_i$ by agent $i$ at the end of the protocol. The initial states are all the states satisfying: $$\begin{aligned}
& (\neg paid[1] \land \neg paid[2] \land \neg paid[3]) \lor (paid[1]
\land \neg paid[2] \land \neg paid[3])\\
& \quad \lor (\neg paid[1] \land paid[2] \land \neg paid[3]) \lor (\neg
paid[1] \land \neg paid[2] \land paid[3]) \end{aligned}$$ Every agent executes the following program, where a single step of the program for each agent is executed in a transition of the state machine:
protocol diningcrypto (paid : observable Bool,
chan_left, chan_right : Bool,
df : observable Bool[])
coin_left, coin_right : observable Bool
begin
if True -> coin_right := True
[] True -> coin_right := False
fi;
chan_right.send(coin_right);
coin_left := chan_left.recv();
df[self] := coin_left xor coin_right xor paid;
end
Program `diningcrypto`[^12] is instantiated for every agent with suitable variables for the parameters: $$\begin{aligned}
& \text{agent 1 executes \texttt{diningcrypto (${\mathit{paid}}[1]$,${\mathit{chan}}[1]$,${\mathit{chan}}[2]$,${\mathit{df}}$)}}\\
& \text{agent 2 executes \texttt{diningcrypto (${\mathit{paid}}[2]$,${\mathit{chan}}[2]$,${\mathit{chan}}[3]$,${\mathit{df}}$)}}\\
& \text{agent 3 executes \texttt{diningcrypto (${\mathit{paid}}[3]$,${\mathit{chan}}[3]$,${\mathit{chan}}[1]$,${\mathit{df}}$)}}\end{aligned}$$ At the first state transition, every agent nondeterministically chooses a value for their coin toss into local variable `coin_right`; at the second state transition, the result of the coin toss is sent on the channel given as the `chan_right` parameter; at the third state transition, the local variable `coin_left` for each agent is updated to reflect the result of the coin toss received from the agent’s left neighbor; at the fourth state transition, variable ${\mathit{df}}$ is updated for every agent.
Given such a state machine, a formula expressing the anonymity of the payer from agent 1’s perspective can be written as: $$\mathbf{X}^4\begin{aligned}[t]& ( \neg {\mathit{paid}}[1]\\
& \qquad {\Rightarrow}(K_1 (\neg {\mathit{paid}}[1] \land \neg {\mathit{paid}}[2] \land \neg {\mathit{paid}}[3])) \\
& \qquad \qquad \lor (K_1 ({\mathit{paid}}[2] \lor {\mathit{paid}}[3])
\land \neg K_1 {\mathit{paid}}[2] \land \neg K_1 {\mathit{paid}}[3]))
\end{aligned}$$ where $\mathbf{X}^4$ is a temporal operator meaning *after four rounds*. This formula, which is true or false of an initial state, says that after the protocol terminates, if cryptographer 1 did not pay, then she either knows that no cryptographer paid, or she knows that one of the other two cryptographers paid but does not know which. (Formulas expressing anonymity from agent 2 and agent 3’s perspectives are similar.)
MCK has no built-in support for active attackers, so it cannot easily deal with cryptographic protocols even if we were to add a message knowledge primitive to the language that can deal with encrypted messages. Of course, it is possible to encode some attackers within the language that MCK provides for describing models, but the effect on the efficiency of model checking is unclear.
The theoretical underpinnings of model checking for temporal epistemic logic are fairly well understood, even though the problem has not been studied nearly as much as model checking for temporal logics. Message knowledge does not particularly complicate matters, once the choice of how to interpret message knowledge is made. Accounting for active attackers is more of an issue, since active attackers introduce additional actions into the model, increasing its size.
The main difficulty with model checking epistemic temporal logic is its inherent complexity. While model checking a standard epistemic logic such as S5 takes time polynomial in the size of the model, adding temporal operators and interpreting the logic over the possibly infinite paths in a finite state machine increases that complexity. For example, in the presence of perfect recall (when agents remember their full history) and synchrony (when agents have access to a global clock), the model-checking problem has non-elementary complexity if the logic includes an *until* temporal operator, and is PSPACE-complete otherwise. The problem tends to become PSPACE-complete when perfect recall is dropped. Progress has been made to control the complexity of model-checking epistemic temporal logics by a careful analysis of the complexity of specific classes of formulas that, while restricted, are still sufficiently expressive to capture interesting security properties, but much work remains to be done to make the resulting techniques efficient.
Information Flow in Multi-Level Systems {#chap13:iflow}
=======================================
Confidentiality in cryptographic protocols is mainly viewed through the lens of access control: some privilege (a key) is required in order to access the confidential data (the content of an encrypted message). An agent who has the key can access the content, an agent without the key cannot. Those access restrictions can control the release of information, but once that information is released there is nothing stopping it from being propagated by agents or by the system through error or malice, or because the released information is needed for the purpose of computations. For systems in which confidentiality is paramount, it is not sufficient to simply ensure that access to confidential data is controlled, there also needs to be a guarantee that even when the confidential data is released it does not land in unauthorized hands. These sorts of confidentiality guarantees require understanding the flow of information in a system.
Confidentiality in the presence of released information is usually studied in the context of systems in which all data are classified with a security level, and where agents have security clearances allowing them to access data at their security level or lower. For simplicity, only scenarios with security levels *high* and *low* (think *classified* and *unclassified* in military settings) will be considered. Intuitively, a high-security agent should be allowed to read both high- and low-security data, and a low-security agent should be allowed to read only low-security data. This is an example of security policy, which describes the forms of information flows that are allowed and those that are disallowed. Information flows that are disallowed capture the desired confidentiality guarantee.
As an example, imagine a commercial system such as a bank mainframe, where agents perform transactions via credit cards or online accounts. In such a system, credit card numbers and bank account numbers might be considered high-security data, and low-security agents should be prevented from accessing them. However, what about the last four digits of a credit card number? Even that information is often considered sensitive. What about a single digit? What about the digits frequency in any given credit card number? Because it is in general difficult to characterize exactly what kind of information about high-security data should not be leaked to low-security agents, it is often easier to prevent any kind of partial information disclosure.
The problem of preventing information disclosure is made more interesting, and more complicated, by the fact that information may not only flow directly from one point to another (e.g, by an agent sending a message to another, or by information being posted, or by updating an observable memory location) but may also flow indirectly from one point to another. Suppose that the commercial system described above sends an email to a central location whenever a transfer of more than one million dollars into a given account $A$ occurs. Anyone observing email traffic can see those emails being sent and learn that account $A$ now contains at least a million dollars. This is an extreme example, but it illustrates indirect information flow: information is gained not by directly observing an event, but by correlating an observation with the event.
Epistemic concepts arise naturally in this setting—a security policy saying that there is no flow of information from high-security data to low-security agents can be expressed as *low-security agents do not learn anything about high-security data*. Moreover, the definitions used in the literature essentially rely on a possible-worlds definition of knowledge within a specific class of models.
Two distinct models of information flow will be described. Both of these models are observational models: they define the kind of observations that agents can make about the system and about the activity of other agents. These observations form the basis of agents’ knowledge.
The first model considered takes a fairly abstract view of a system, as sequences of events such as inputs from agents, outputs to agents, internal computation, and so on. These events are the observations that agents can make. In such a setting, security policies regulate information about the occurrence of events. The second model considered is more concrete, and stems from practical work on defining verification techniques for information flow at the level of the source code implementing a system. In that model, observations take the form of content of memory locations that programs can manipulate.
Information Flow in Event Systems {#chap13:events}
---------------------------------
The first model of information flow uses sets of traces corresponding to the possible executions of the system. Every trace is a sequence of events; some of those events are high-security events (and only observable by high-security agents), and some of those events are low-security events (and observable by both high-security and low-security agents). The intuition is that a low-security agent, observing only the low-security events in a trace, should not be able to infer any information about the high-security events in a trace.
How can a low-security agent infer information? If we assume that the full set of traces of the system is known to all agents, then a low-security agent, upon observing a particular sequence of low-security events, can narrow down a set of possible traces that could be the actual trace by considering all the traces that are compatible with her view of the low-security events. By looking at those possible traces, she may infer information about high-security events. For instance, maybe a particular high-security event $e$ appears in every such possible trace, and thus she learns that high-security event $e$ has occurred. In the most extreme case, there may be a single trace compatible with her view of the low-security events, and therefore that low-security agent learns exactly which high-security events have occurred.
The model can be formalized using event systems. An event system is a tuple $S=(E,I,O,{\mathit{Tr}})$ where $E$ is a set of events, $I\subseteq E$ a set of input events, $O\subseteq E$ a set of output events, and ${\mathit{Tr}}\subseteq E^*$ a set of finite traces representing the possible executions of the system. Given a trace $\tau\in E^*$ and a subset $E'\subseteq E$ of events, we write $\tau{\rvert_{E'}}$ for the subtrace of $\tau$ consisting of events from $E'$ only.
We assign a security level to events in $E$ by partitioning them into low-security events $L$ and high-security events $H$: events in $I\cap L$ are low-security input events, events in $O\cap L$ are low-security output events, and so on.
A naive attempt at defining information flow in this setting might be to say that there is information flowing from high-security events to low-security agents if a low-security agent’s view of $\tau{\rvert_{L}}$ implies that at least one high-security event subsequence is not possible. In other words, seeing a particular sequence of low-security events rules out one possible high-security event subsequence. Formally, if we write ${\mathit{Tr}}{\rvert_{H}}$ for $\{\tau{\rvert_{H}}:\tau\in{\mathit{Tr}}\}$, information flows from high-security events to low-security agents if there is a trace $\tau\in{\mathit{Tr}}$ such that $\{\tau'{\rvert_{H}} :
\tau'{\rvert_{L}}=\tau{\rvert_{L}}\}\ne{\mathit{Tr}}{\rvert_{H}}$.
Such a definition turns out to be too strong—it is equivalent to separability described below—because it pinpoints information flows where there are none: since low-security events may influence high-security events, a particular subsequence of high-security events may be ruled out due to the influence of low-security events, and in that case there should be no information flow since the low-security agent could have already predicted that the high-security subsequence would have been ruled out. Intuitively, there is information flow when one high-security event subsequence that should be possible as far as the low-security agent expects is not in fact possible. This argument gives an inkling as to why the definition of information flow is not entirely trivial.
Security policies in event systems are often defined as closure properties of the set of traces. Security policies that historically were deemed interesting for the purpose of formalizing existing multi-level systems include the following:
- **Separability:** no nontrivial interaction between high-security events and low-security agents is possible because for any such interaction there is a trace with the same high-security events but different low-security events, and a trace with the same low-security events but different high-security events. Formally, for every pair of traces $\tau_1,\tau_2\in{\mathit{Tr}}$, there is a trace $\tau\in{\mathit{Tr}}$ such that $\tau{\rvert_{L}}=\tau_1{\rvert_{L}}$ and $\tau{\rvert_{H}}=\tau_2{\rvert_{H}}$.
- **Noninference:** a low-security agent cannot learn about the occurrence of high-security events because any trace, as far as the low-security agent can tell, could be a trace where there are no high-security events at all. Formally, for all traces $\tau\in{\mathit{Tr}}$, there is a trace $\tau'\in{\mathit{Tr}}$ such that $\tau{\rvert_{L}}=\tau'{\rvert_{L}}$ and $\tau'{\rvert_{H}}=\langle\,\rangle$.
- **Generalized Noninference:** A more lenient form of noninference, where a low-security agent cannot learn about the occurrence of high-security input events because any trace could be a trace where there are no high-security input events at all. Formally, for all traces $\tau\in{\mathit{Tr}}$, there is a trace $\tau'\in{\mathit{Tr}}$ such that $\tau{\rvert_{L}}=\tau'{\rvert_{L}}$ and $\tau'{\rvert_{(H\cap I)}}=\langle\,\rangle$.
- **Generalized Noninterference:** a low-security agent cannot learn about high-security input events, and high-security input events cannot influence low-security events. Formally, for all traces $\tau\in{\mathit{Tr}}$ and all traces $\tau'\in{\mathit{interleave}}((H\cap I)^*,\{\tau{\rvert_{L}}\})$, there is a $\tau''\in{\mathit{Tr}}$ such that $\tau''{\rvert_{L}}=\tau{\rvert_{L}}$ and $\tau''{\rvert_{(L\cup(H\cap I))}}=\tau'$. (Function ${\mathit{interleave}}(T,U)$ returns every possible interleaving of every trace from $T$ with every trace from $U$.)
A closure property says that if some traces are in the model, then other variations on these traces must also be in the model. This is clearly an epistemic property. Under a possible-worlds definition of knowledge, an agent knows a formula if that formula is true at all traces that the agent considers possible given her view of the system. In general, the fewer possible traces there are, the more facts can be known, since it it easier for a fact to be true at all possible traces if there are few of them. The closure properties ensure that there are enough possible traces from the perspective of a low-security agent to prevent a specific of class facts from being known.[^13]
Closure conditions on sets of traces are therefore just a way to enforce lack of knowledge, given a possible-worlds definition of knowledge. We can make this precise by viewing event systems as Kripke frames.
The accessibility relation of each agent depends on the agent’s security level. In the case of interest, a low-security agent is assigned an accessibility relation $\sim_L$ defined as $\tau_1\sim_L \tau_2$ if and only if $\tau_1{\rvert_{L}}=\tau_2{\rvert_{L}}$.
We identify a proposition with a set of traces, intuitively, those traces in which the proposition is true. As usual, conjunction is intersection of propositions, disjunction is union of propositions, negation is complementation of propositions with respect to the full set of traces in the event system, and implication is subset inclusion. To define the proposition *the low-security agent knows $P$*, we first define the low-security agent’s knowledge set of a trace $\tau$ as the set of all traces $\sim_L$-equivalent to $\tau$, ${\mathcal{K}}_L(\tau) = \{\tau' :
\tau\sim_L\tau'\}$. The proposition *the low-security agent knows $P$* can be defined in the usual way, as: $${\mathsf{K}}_L(P) = \{\tau : {\mathcal{K}}_L(\tau)\subseteq P\}$$ It is easy to see that ${\mathsf{K}}_L$ satisfies the usual S5 axioms, suitably modified to account for propositions being sets: $$\begin{aligned}
\textrm{(D)} \qquad & {\mathsf{K}}_L(P) \cap {\mathsf{K}}_L(Q) = {\mathsf{K}}_L(P\cap Q)\\
\textrm{(K)} \qquad & {\mathsf{K}}_L(P) \subseteq P\\
\textrm{(PI)} \qquad & {\mathsf{K}}_L(P) \subseteq {\mathsf{K}}_L({\mathsf{K}}_L(P))\\
\textrm{(NI)} \qquad & \neg{\mathsf{K}}_L(P) \subseteq {\mathsf{K}}_L(\neg{\mathsf{K}}_L(P))\end{aligned}$$ These properties are the set-theoretic analogues of *Distribution*, *Knowledge*, *Positive Introspection*, and *Negative Introspection*, respectively.
As an example, consider an event system $(E,I,O,{\mathit{Tr}})$ that satisfies generalized noninterference, and the proposition *high-security input event $e$ has occurred*. This proposition is represented by the set $P_e$ of all traces in ${\mathit{Tr}}$ in which $e$ occurs. The proposition *the low-security agent knows that $e$ has occurred* corresponds to the set of traces ${\mathsf{K}}_L(P_e)$. It is easy to check that because the system satisfies generalized noninterference, the set ${\mathsf{K}}_L(P_e)$ is empty, meaning that there is no trace on which the low-security agent ever knows $P_e$, that is, that $e$ has occurred. By way of contradiction, suppose that $\tau\in{\mathsf{K}}_L(P_e)$. By definition, $\tau\in{\mathsf{K}}_L(P_e)$ if and only if ${\mathcal{K}}_L(\tau)\subseteq P_e$. But the closure condition for generalized noninterference implies that there must exist a trace $\tau'\sim_L\tau$, that is, a trace in ${\mathcal{K}}_L(\tau)$, such that $e$ does not occur in $\tau'$. Thus, there is a trace in ${\mathcal{K}}_L(\tau)$ which is not in $P_e$, and ${\mathcal{K}}_L(\tau)\not\subseteq P_e$. Thus, $\tau\not\in{\mathsf{K}}_L(P_e)$, a contradiction.
This is a somewhat roundabout way to see that there is an implicit epistemic logic lurking which explains the notions of information flow security policies in event systems. It is certainly possible to make such a logic explicit by introducing a syntax and adding an interpretation to event systems, and study information flow in event systems from such a perspective.
The key point here is that event-system models of information flow and the expression of security policies in those models intrinsically use epistemic concepts, and all reasoning is essentially classical epistemic reasoning performed directly on the models.
Language-Based Noninterference {#chap13:languages}
------------------------------
A more concrete model for information flow is obtained by moving away from trace-based models of systems and relying instead on the program code implementing those systems.
Defining information flow at the level of programs has several advantages: the system is described in detail, information can be defined in terms of the data explicitly manipulated by the program, and enforcement can be automated; the latter turns out to be especially important given the complexity of modern computing systems which makes manual analysis often infeasible.
The observational model used by most language-based information-flow security research is not event-based, although it is still broadly concerned with input and output. The focus here is on information flow in imperative programs, which operates by changing the state of the environment as a program executes. The state of the environment is represented by a store holding values associated with variables. Variables can be read and written by programs. Every variable is tagged with a security level, describing the security level of the data it contains. A low-security agent can observe all low-security variables, but not the high-security ones. Inputs to programs are modeled as initial values of variables, while outputs are modeled as final value of variables: low-security inputs are initial value of low-security variables, and so on. The basic security policy generally considered is a form of noninterference: that low-security outputs do not reveal anything about high-security inputs, and that high-security inputs do not influence the value of low-security outputs.
Consider the following short programs: $$\begin{aligned}
\text{(P1)} \qquad & h := l + 1 \\
\text{(P2)} \qquad & l := h +1 \\
\text{(P3)} \qquad & {\mathsf{if}}~l=0~{\mathsf{then}}~h:=h+1~{\mathsf{else}}~l:=l+1 \\
\text{(P4)} \qquad & {\mathsf{if}}~h=0~{\mathsf{then}}~h:=h+1~{\mathsf{else}}~l:=l+1\end{aligned}$$ In all of these programs, variable $h$ is a high-security variable, and variable $l$ is a low-security variable.
A program executes in a store assigning initial values to variables, and execution steps modify the store until the program terminates in a final store. Several simplifying assumption are made: programs are deterministic, and programs always terminate. This is purely to keep the discussion and the technical machinery light. These restrictions can be lifted easily. Moreover, the programming language under consideration will not be described in detail; the sample programs should be intuitive enough.
How do we formalize noninterference in this setting? A store $\sigma$ is a mapping from variables $x$ to values $\sigma(x)$. We assume every variable $x$ is tagged with a security level ${\mathit{sec}}(x)\in\{L,H\}$. Let $\Sigma$ be the set of all possible stores. We model execution of a program $C$ using a function ${[\hspace{-.3ex}[C]\hspace{-.3ex}]}:\Sigma\longrightarrow\Sigma$ from initial stores to final stores. Thus, executing program $C$ in store $\sigma$ yields a final store ${[\hspace{-.3ex}[C]\hspace{-.3ex}]}(\sigma)$. For example, executing program (P1) in store $\langle l\mapsto 5,h\mapsto 10\rangle$ yields store $\langle l\mapsto 5,
h\mapsto 6\rangle$, and executing program (P3) in store $\langle l\mapsto 5,h\mapsto 10\rangle$ yields store $\langle l\mapsto 6,
h\mapsto 10\rangle$.
Two stores $\sigma_1$ and $\sigma_2$ are $L$-equivalent, written $\sigma_1\approx_L \sigma_2$, if they assign the same values to the same low-security variables: $\sigma_1\approx_L\sigma_2$ if and only if for all variables $x$ with ${\mathit{sec}}(x)=L$, $\sigma_1(x)=\sigma_2(x)$. A program $C$ satisfies noninterference if executing $C$ in two $L$-equivalent states (that is, in two states that a low-security agent cannot distinguish) yields two $L$-equivalent states: for all $\sigma_1$ and $\sigma_2$, if $\sigma_1\approx_L\sigma_2$, then ${[\hspace{-.3ex}[C]\hspace{-.3ex}]}(\sigma_1)\approx_L{[\hspace{-.3ex}[C]\hspace{-.3ex}]}(\sigma_2)$.[^14]
How do programs (P1–4) fare under this definition of noninterference? Program (P1) clearly satisfies noninterference, since the final value of low-security variable $l$ does not depend on the value of any high-level variable, while program (P2) clearly does not. The other two programs are more interesting. The final value of low-security variable $l$ in program (P3) only depends on the initial value of $l$, and thus we expect (P3) to satisfy noninterference, and it does. Program (P4), however, does not, as we can see by executing the program in stores $\langle l\mapsto 0, h\mapsto 0\rangle$ and $\langle l\mapsto 0, h\mapsto 1\rangle$, both $\approx_L$-equivalent, but which yield stores $\langle l\mapsto 0, h\mapsto 1\rangle$ and $\langle l\mapsto 1,h\mapsto 1\rangle$, respectively, two stores that cannot be $L$-equivalent since they differ in the value they assign to variable $l$. And indeed, observing the final value of $l$ reveals information about the initial value of $h$.
Noninterference is usually established by a static analysis of the program code, which approximates the flow of information through a program before execution. While the details of the static analyses are interesting in their own right, they have little to do with epistemic logic beyond providing an approach to verifying a specific kind of epistemic property in a specific context.
Recent work on language-based information-flow security has highlighted the practical importance of declassification, that is, the controlled release of high-security data to low-security agents. The problem of password-based authentication illustrates the need for such release: when a low-security agent tries to authenticate herself as a high-security agent, she may be presented with a login screen asking for the password of the high-security agent. That password should of course be considered high-security information. However, the login screen leaks information, since entering an incorrect password will reveal that the attempted password is not the right password, thereby leaking a small amount of information about the correct password. The leak is small, but it exists, and because of it the login screen does not satisfy the above definition of noninterference. Defining a suitable notion of security policy that allows such small release of information while still preventing more important information flow is a complex problem.
While the concepts underlying information-flow security are clearly epistemic in nature—taking stores as possible worlds and $L$-equivalence as an accessibility relation for low-security agents—there is no real demand for an explicit epistemic logic in which to describe policies. One reason is that it is in general difficult to precisely nail down, in a given system, what high-security information should be kept from low-security agents. It is simply easier to ask that no information be leaked to low-security agents. This *no information* condition is easier to state semantically than through an explicit logical language—not learning any information in the sense of noninterference can be stated straightforwardly as a relationship between equivalence relations, while if we were to use an epistemic logic, we would have to say something along the lines of *for all formulas ${\varphi}$ that do not depend only on the state of the low-security agent, $\neg K_L{\varphi}$* where $K_L$ expresses the knowledge of that low-security agent. The latter is patently clunkier to work with. It may be the case that an explicit epistemic logic would be more useful in the context of declassification, where not all information needs to be kept from low-security agents.
Beyond Confidentiality {#chap13:beyond}
======================
The focus of this chapter has been on confidentiality, because it is by far the most studied security property. It is not only important, it also underpins several other security properties. Other related properties are also relevant.
#### Anonymity.
A specific form of confidentiality is anonymity, where the information to be kept secret is the association between actions and agents who perform them. Anonymity has been studied using epistemic logic, and several related definitions have been proposed and debated.
To discuss anonymity, we need to be able to talk about actions and agents who perform them. Let $\delta(i, a)$ be a proposition interpreted as *agent $i$ performed action $a$*.
The simplest definition of anonymity is lack of knowledge: action $a$ performed by agent $i$ is minimally anonymous with respect to agent $j$ if agent $j$ does not know that agent $i$ performed $a$. This can be captured by the formula $$\neg K_j\delta(i,a).$$
Minimal anonymity is, well, minimal. It does not rule out that agent $j$ may narrow down the list of possible agents that performed $a$ to agent $i$ and one other agent. Stronger forms of anonymity can be defined: action $a$ performed by agent $i$ is totally anonymous with respect to agent $j$ if, as far as agent $j$ is concerned, action $a$ could have been performed by any agent in the system (except for agent $j$). This can be captured by the formula $$\delta(i,a){\Rightarrow}\bigwedge_{i'\ne j}P_j\delta(i',a)$$ where $P_i{\varphi}$ is the usual dual to knowledge, $\neg K_i(\neg{\varphi})$, read as *agent $i$ considers ${\varphi}$ possible*.
Total anonymity is at the other extreme on the spectrum from minimal anonymity; it is a very strong requirement. Intermediate definitions can be obtained by requiring that actions be anonymous only up to a given set of agents—sometimes called an anonymity set: action $a$ performed by agent $i$ is anonymous up to $I$ with respect to agent $j$ if, as far as agent $j$ is concerned, action $a$ could have been performed by any agent in $I$. This can be captured by the formula: $$\delta(i,a){\Rightarrow}\bigwedge_{i'\in I}P_j\delta(i',a).$$
As an example of this last definition of anonymity, note that it can be used to describe the anonymity provided by the Dining Cryptographers protocol from [\[chap13:protocols\]]{}. Recall that if one of the cryptographers paid, the Dining Cryptographers protocol guarantees that each of the non-paying cryptographers think it possible that any of the cryptographers but herself paid. In other words, if $C=\{{\mathit{Alice}},{\mathit{Betty}},{\mathit{Charlene}}\}$ are the cryptographers and if cryptographer $i$ paid, then the protocol guarantees that the paying action is anonymous up to $C\setminus\{j\}$ with respect to cryptographer $j$, as long as $j\ne i$.
#### Coercion Resistance.
Voting protocols are protocols in which anonymity plays an important role. Voting protocols furthermore satisfy other interesting security properties. Aside from secrecy of votes (that every voter’s choice should be private, and observers should not be able to figure out who voted how), other properties include fairness (voters do not have any knowledge of the distribution of votes until the final tallies are announced), verifiability (every voter should be able to check whether her vote was counted properly), and receipt freeness (no voter has the means to prove to another that she has voted in a particular manner).
This last property, receipt freeness, is particularly interesting in terms of epistemic content. Roughly speaking, receipt freeness says that a voter Alice cannot prove to a potential coercer Corinna that she voted in a particular way. This is the case even if Alice wishes to cooperate with Corinna; receipt freeness guarantees that such cooperation cannot lead to anything because it will be impossible for Corinna to be certain how Alice voted. In that sense, receipt freeness goes further than secrecy of votes. Even if Alice tells Corinna that she voted a certain way, Corinna has no way to verify Alice’s assertion, and Alice has no way to convince her.
Coercion resistance is closely related to receipt freeness but is slightly stronger. Intuitively, a voting protocol is coercion resistant if it prevents voter coercion and vote buying even by active coercers: a coercer should not be able to influence the behavior of a voter. Coercion resistance can be modeled epistemically, although the details of the modeling is subtle, and many important details will be skipped in the description below. Part of the difficulty and subtlety is that the idea of coercion means changing how a voter behaves based on a coercer’s desired outcome or goal, which needs to be modeled somehow.
One formalization of coercion resistance uses a model of voting protocols based on traces, where some specific agents are highlighted: a voter that the coercer tries to influence (called the coerced voter), the coercer, and the remaining agents and authorities, assumed to be honest. Every voter in the system votes according to a voting strategy, which in the case of honest voters is the strategy corresponding to the voting protocol.
The formalization assumes that every voter has a specific voting goal, formally captured by the set of traces in which that voter successfully votes according to her desired voting goal. The coercer, however, is intent on affecting the coerced voter—for instance, to coerce a vote for a given candidate, or perhaps to coerce a vote away from a given candidate. To coerce a voter, the coercer hands the coerced voter a particular strategy that will fulfill the coercer’s goals instead of the coerced voter’s. For instance, the coercer’s strategy may simply be one that forwards all messages to and from the coercer, effectively making the coerced voter a proxy for the coercer.
Let $V$ be the space of possible strategies that voters and coercers can follow. Coercion resistance can be defined by saying that for every possible strategy $v\in V$, there is another strategy $v'\in V$ that the coerced voter can use instead of $v$ with the property that: (1) the voter always achieves her goal by using $v'$, and (2) the coercer does not know whether the coerced voter used strategy $v$ or $v'$. In other words, in every trace in which the coerced voter uses strategy $v$, the coercer considers it possible, given her view of the trace, that the coerced voter is using strategy $v'$ instead. Conversely, in every trace in which the coerced voter uses strategy $v'$, the coercer considers it possible that the coerced voter is using strategy $v$. So, the coercer cannot know whether the coerced voter followed the coercer’s instructions (i.e., used $v$) or tried to achieve her own goal (i.e., used $v'$). As in the case of information flow in event systems in [\[chap13:events\]]{}, the definition of coercion resistance is a form of closure property on traces, which corresponds to lack of knowledge in the expected way, where knowledge is captured by an indistinguishability relation on states based on the coercer’s observations.
#### Zero Knowledge.
The property *an agent does not learn anything about something*, as embodied in information-flow security policies and other forms of confidentiality, is generally modeled using an indistinguishability relation over states and enforced by making sure that there are enough states to prevent the confidential information from being known by unauthorized agents.
Another approach to modeling and enforcing this lack of learning is demonstrated by *zero knowledge interactive proof systems*. An interactive proof system for a string language $L$ is a two-party system $(P,V)$ in which a prover $P$ tries to convince a verifier $V$ that some string $x$ is in $L$ through a sequence of message exchanges amounting to an interactive proof of $x\in L$. Classically, the prover is assumed to be infinitely powerful, while the verifier is assumed to be a probabilistic polynomial-time Turing machine. An interactive proof system has the property that if $x\in
L$, the conversation between $P$ and $V$ will show $x\in L$ with high probability, and if $x\not\in L$, the conversation between *any* prover and $V$ will show $x\in L$ with low probability. (The details for why the second condition refers to any prover rather than just $P$ is beyond the scope of this discussion.)
An interactive proof system for $L$ is zero knowledge if whenever $x\in L$ holds the verifier is able to generate *on its own* the conversations it would have had with the prover during an interactive proof of $x\in L$. The intuition here is that the verifier does not learn anything from a conversation with the prover (other than $x\in L$) if it can learn exactly the same thing by generating that whole conversation itself. Thus, the only knowledge gained by the verifier is that which the prover initially set out to prove.
Zero knowledge interactive proof systems rely on indistinguishability, but not indistinguishability among a large set of states. Rather, it is indistinguishability between two scenarios: a scenario where the verifier interacts with the prover, and a scenario where the verifier does not interact with the prover but instead simulates a complete interaction with the prover. This simulation paradigm, a core notion in modern theoretical computer science, says roughly that an agent does not gain any knowledge from interacting with the outside world if she can achieve the same results without interacting with the outside world.
To give a sense of the kinds of definitions that arise in this context, here is one formal definition of perfect zero knowledge:
> Let $(P,V)$ be an interactive proof system for $L$, where $P$ (the prover) is an interactive Turing machine and $V$ (the verifier) is a probabilistic polynomial-time interactive Turing machine. System $(P,V)$ is *perfect zero-knowledge* if for every probabilistic polynomial-time interactive Turing machine $V^*$ there is a probabilistic polynomial-time Turing machine $M^*$ (the simulator) such that for every $x\in L$ the following two random variables are identically distributed:
>
> 1. the output of $V^*$ interacting with $P$ on common input $x$;
>
> 2. the output of machine $M^*$ on input $x$.
>
While the details are beyond the scope of this chapter, the intuition behind this definition is to have, for every possible verifier $V^*$ (and not only $V$) interacting with $P$, a machine $M^*$ that can simulate the interaction of $V^*$ and $P$ even though it does not have access to the prover $P$. The existence of such simulators implies that $V^*$ does not gain any knowledge from $P$.
This gives a different epistemic foundation for confidentiality, one that is intimately tied to computation and its complexity. The relationship with classical epistemic logic is essentially unexplored.
Perspectives {#chap13:perspectives}
============
The preceding sections illustrate how extensively epistemic concepts, explicitly framed as an epistemic logic or not, have been applied to security research. Whether the application of these concepts has been successful is a more subjective question.
In a certain sense, the problems described in this chapter are solved problems by now. Confidentiality and authentication in cryptographic protocol analysis under a formal model of cryptography and Dolev-Yao attackers, for example, can be checked quite efficiently with a vast array of methods, at least for common security properties, and the definitions used approximate the epistemic definitions quite closely.
So what are the remaining challenges in cryptographic protocol analysis, and has epistemic logic a role to play? The most challenging aspect of cryptographic protocol analysis is to move beyond Dolev-Yao attackers and beyond formal models of cryptography, towards more concrete models of cryptography.
Moving beyond a Dolev-Yao attacker requires shifting the notion of message knowledge to use richer algebras of message with more operations. Directions that have been explored include providing attackers with the ability to perform offline dictionary attacks, working with an XOR operation, or even number-based operations such as exponentiation. One problem is that when the algebra of messages is subject to too many algebraic properties, determining whether an attacker knows a message may quickly become undecidable. Even when message knowledge for an attacker is decidable, it may still be too complex for efficient reasoning. It is not entirely clear how epistemic concepts can help solve problems in that arena.
Moving from a formal model of cryptography to a concrete model, one that reflects real encryption schemes more accurately using sequences of bits and computational indistinguishability, requires completely shifting the approach to cryptographic protocol analysis.
Formal models of cryptography work by abstracting away the *one-way security* property of encryption schemes—that it is computationally hard to recover the sourcetext from a ciphertext without knowing the encryption key. More concrete models of cryptography rely on stronger properties than one-way security, properties such as *semantic security*, which intuitively says that if any information about a message $m$ can be computed by an efficient algorithm given the ciphertext $e_k(m)$ for a random $k$ and $m$ chosen according to an arbitrary probability distribution, that same information can be computed without knowing the ciphertext. In other words, the ciphertext $e_k(m)$ offers no advantage in computing information about some message $m$ chosen from an arbitrary probability distribution.
The definition of semantic security is reminiscent of the definition of zero knowledge interactive proof systems in [\[chap13:beyond\]]{}, and it is no accident, as they both rely on a simulation paradigm to express the fact that no knowledge is gained. As in the case of zero knowledge interactive proof systems, there is a clear epistemic component to the definition of semantic security, one to which classical epistemic logic has not been applied.
The main difficulty with applying classical epistemic logic to concrete models of cryptography is that these models take attackers to be probabilistic polynomial-time Turing machines, and take security properties to be probabilistic properties relative to those probabilistic polynomial-time Turing machines. This means that an epistemic approach to concrete models of cryptography needs to be probabilistic as well as computationally bounded. The former is not a problem, since probabilistic reasoning shares much of the same foundations as epistemic reasoning. But the latter is more complicated. Concrete models of cryptography are not based on impossibility, but on computational hardness. And while possible-worlds definitions of knowledge are well suited to talking about impossible versus possible outcomes, they fare less well at talking about difficult versus easy outcomes.
The trouble that possible-worlds definitions of knowledge run into when trying to incorporate a notion of computational difficulty is really the problem of logical omniscience in epistemic logic under a different guise. Agents, under standard possible-worlds definitions of knowledge, know all tautologies, and know all logical consequences of their knowledge: if $K{\varphi}$ is true and ${\varphi}{\Rightarrow}\psi$ is valid, then $K\psi$ is also true. Any normal epistemic operator will satisfy these properties, and in particular, any epistemic logic based on Kripke structures will satisfy these properties. Normality does not deal well with computational difficulty, because while it may be computationally difficult to establish that ${\varphi}{\Rightarrow}\psi$ is valid, a normal modal logic will happily derive all knowledge-based consequences of that valid formula. It would seem that giving a satisfactory epistemic account of concrete models of cryptography requires a non-normal epistemic logic, one that supports a form of resource-bounded knowledge. Resource-bounded knowledge is not well understood, and logics for resource-bounded knowledge still feel too immature to form a solid basis for reasoning about concrete models of cryptography.
Leaving aside concrete models of cryptography, it is almost impossible to discuss epistemic logic in the context of cryptographic protocols without addressing the issue of BAN Logic. BAN Logic is an interesting and original use of logic, developed to prove cryptographic protocol properties manually by paralleling informal arguments for protocol correctness.
BAN Logic has spilled a lot of virtual ink. Aside from its technical limitations—it requires a protocol idealization step that remains outside the purview of the logic but affects the results of analysis—the logic is considered somewhat *passé*. Other approaches we saw in [\[chap13:reasoning\]]{} operate in the same space, namely analyzing cryptographic protocols under a formal model of cryptography in the presence of Dolev-Yao attackers, and most are less limited and more easily automated. Other approaches, such as Protocol Composition Logic, even advocate Hoare-style reasoning about the protocol text from within the logic, just like BAN Logic.
My perspective on BAN Logic is that it tried to do something which has not really been tried since, something that remains a sort of litmus test for our understanding of security in cryptographic protocols: identifying high-level primitives that capture relevant concepts for security, high-level primitives that match our intuitive understanding of security properties, those same intuitions that guide our design of cryptographic protocols in the first place. We do not have such high-level primitives in any other framework, all of which tend to work at much lower levels of abstraction. The primitives in BAN Logic are intuitively attractive, but poorly understood. The continuing conversation on BAN Logic is a reminder that we still do not completely understand the basic concepts and basic terms needed to discuss cryptographic protocols, and I think BAN Logic remains relevant, if only as a nagging voice telling us that we have not quite gotten it right yet.
Many of the issues that arise when trying to push cryptographic protocol analysis from a formal model of cryptography to a more concrete model also come up in the context of information-flow security. As mentioned in [\[chap13:languages\]]{}, recent work has turned to the question of declassification, or controlled release of information. The reason for this is purely pragmatic: most applications need to release some kind of information in order to do any useful work, even under a lax interpretation of noninterference.
But it does not take long to see that even a controlled release of information can lead to unwanted release of information in the aggregate. Returning to the password-login problem from [\[chap13:languages\]]{}, it is clear that every wrong attempt at entering a password leaks some information, something that needs to happen if the login screen is to operate properly. But of course, repeated attempts at checking the password will eventually lead to the correct password as the only remaining possibility, which is a severe undesirable release of information. Security policies controlling declassification therefore seem to require a way to account for more quantitative notions of leakage which aggregate over time, something that symbolic approaches to information-flow security have difficulty handling well.
Modeling information flow quantitatively can be seen as a move from reasoning about information as a monolithic unit to reasoning about information as a resource. Once we make that leap, other resources affecting information flow start suggesting themselves. For example, execution time can leak information. Consider the simple program: $$\begin{aligned}
& {\mathsf{if}}~\text{\textit{(high-security Boolean variable)}}\\
& \qquad{\mathsf{then}}~\text{\textit{fast code}}\\
& \qquad{\mathsf{else}}~\text{\textit{slow code}}\end{aligned}$$ By observing the execution time of the program, we can determine the value of the high-security Boolean variable. This example is rather silly, of course, but it illustrates the point that information leakage can occur based on observations of other resources than simply the state of memory.
What about the combination of information flow and cryptography? After all, in practice, systems do use cryptography internally to help keep data confidential. Encrypted data can presumably be written on shared storage (which might be easier to manage than storage segregated into high-security and low-security storage) or moved online, or in general given to low-security agents without information being released, as long as they do not have the key or the resources to decrypt. Accounting for cryptography in information-flow security raises questions similar to those in cryptographic protocol analysis concerning what models of cryptography to use and how to account for the cryptographic capabilities of attackers. It also raises difficulties similar to those in cryptographic protocol analysis when trying to move from a formal model of cryptography to a concrete model, including how to provide an epistemic foundation for information flow using a resource-bounded definition of knowledge.
#### Conclusion.
Epistemic concepts are central to many aspects of reasoning about security. In some cases, these epistemic concepts may even naturally take expression in a *bona fide* epistemic logic that can be used to formalize the reasoning. But whether an epistemic logic is used or not, the underlying concepts are clearly epistemic. In particular, the notion of truth at all possible worlds reappears in many different guises throughout the literature.
Research in security analysis has reached a sort of convergence point around the use of symbolic methods. The challenge seems to be to move beyond this convergence point, and such a move requires taking resources seriously: realistic definitions of security rely on the notion that exploiting a vulnerability should require more resources (time, power, information) than are realistically available to an attacker. In epistemic terms, what is needed is a reasonably well-behaved definition of resource-bounded knowledge, itself an active area of research in epistemic logic. It would appear, then, that advances in epistemic logic may well help increase our ability to reason about security in direct ways.
#### Acknowledgments.
Thanks to Aslan Askarov, Philippe Balbiani, Stephen Chong, and Vicky Weissman for comments on an early draft of this chapter.
Bibliographic Notes and Further Reading {#chap13:bibnotes}
=======================================
For the basics of epistemic logic, both the syntax and the semantics, the reader is referred to the introductory chapter of the current volume. For the sake of making this chapter as self-contained as possible, most of the background material can be usefully obtained from the textbooks of Fagin, Halpern, Moses, and Vardi [@r:fagin95] and Meyer and Van der Hoek [@r:meyer95]. The possible-worlds definition of knowledge used throughout this chapter is simply the view that knowledge is truth at all worlds that an agent considers as possible alternatives to the current world, a view which goes back to Hintikka [@r:hintikka62].
#### Cryptographic Protocols.
While the focus of the section is on symbolic cryptographic protocol analysis, cryptographic protocols can also be studied from the perspective of more computationally-driven cryptography, of the kind described in [\[chap13:perspectives\]]{}; see Goldreich [@r:goldreich04]. The Russian Cards problem, which was first presented at the Moscow Mathematic Olympiads in 2000, is described formally and studied from an epistemic perspective by Van Ditmarsch [@r:ditmarsch03]. The problem has been used as a benchmark for several epistemic logic model checkers [@r:ditmarsch06]. The Dining Cryptographers problem and the corresponding protocol is described by Chaum [@r:chaum88]. It was proved correct in an epistemic temporal logic model checker by Van der Meyden and Su [@r:meyden04].
For a good overview of classical cryptography along with some perspectives on protocols, see Stinson [@r:stinson95] and Schneier [@r:schneier96]; both volumes contain descriptions of DES, AES, RSA, and elliptic-curve cryptography. Goldreich [@r:goldreich01; @r:goldreich04] is also introductory, but from the perspective of modern computational cryptography.
For a good high-level survey of the kind of problems surrounding the design and deployment of cryptographic protocols, see Anderson and Needham [@r:anderson95], then follow up with Abadi and Needham’s [@r:abadi96a] prudent engineering practices. The key distribution protocol used as the first example in [\[chap13:crypto\]]{} is related to the Yahalom protocol described by Burrows, Abadi, and Needham [@r:burrows90]. The Needham-Schroeder protocol was first described in Needham and Schroeder [@r:needham78]. The man-in-the-middle attack on the Needham-Schroeder protocol in the presence of an insider attacker was pointed out by Lowe [@r:lowe95], and the fix was analyzed by Lowe [@r:lowe96].
The Dolev-Yao model of the attacker given in [\[chap13:attackers\]]{} is due to Dolev and Yao [@r:dolev83].
The formal definition of message knowledge via ${\mathit{Analyzed}}$ and ${\mathit{Synthesized}}$ sets is taken from Paulson [@r:paulson98]. Equivalent definitions are given in nearly every formal system for reasoning about cryptographic protocols in a formal model of cryptography. Message knowledge can be defined using a local deductive system, which makes it fit nicely within the deductive knowledge framework of Konolige [@r:konolige86]—see also Pucella [@r:pucella06c]. More generally, message knowledge is a form of algorithmic knowledge [@r:halpern94], that is, a local form of knowledge that relies on an algorithm to compute what an agent knows based on the local state of the agent. In the case of a Dolev-Yao attacker, this local algorithm simply computes the sets of analyzed and synthesized messages [@r:halpern12].
Another way of defining message knowledge is the hidden automorphism model, due to Merritt [@r:merritt83], which is a form of possible-worlds knowledge. While it never gained much traction, it has been used in later work by Toussaint and Wolper [@r:toussaint89] and also in the logic of Bieber [@r:bieber90]. It uses algebraic presentations of encryption schemes called cryptoalgebras. There is a unique surjective cryptoalgebra homomorphism from the free cryptoalgebra over a set of plaintexts and keys to any cryptoalgebra over the same plaintexts and keys which acts as the identity on plaintexts and keys. Message knowledge in a given cryptoalgebra $C$ is knowledge of the structure of messages as given by that surjective homomorphism from the free cryptoalgebra to $C$. A revealed reduct is a subset of $C$ that the agent has seen. A state of knowledge with respect to revealed reduct $R$ is a set of of mappings $f$ from the free cryptoalgebra to $C$ that are homomorphisms on $f^{-1}(R)$. In this context, an agent knows message $m$ if the agent knows the representation of message $m$, meaning that $m$ is the image of the same free cryptoalgebra term under every mapping in the state of knowledge of the agent. Thus, if an agent receives ${\{m_1\}_{k}}$ and ${\{m_2\}_{k}}$ but does not receive $k$, then only ${\{m_1\}_{k}}$ and ${\{m_2\}_{k}}$ are in the revealed reduct; the agent may consider any distinct messages $m'_1$ and $m'_2$ to map to ${\{m_1\}_{k}}$ and ${\{m_2\}_{k}}$ after encryption with $k$, since any such mapping will act as a homomorphism on the pre-image of the revealed reduct.
Possible-worlds definitions of knowledge in the presence of cryptography are problematic because cryptography affects what agents can observe, and this impacts the definition of the accessibility relation between worlds. The idea of replacing encrypted messages in the local state of agents by a token goes back to Abadi and Tuttle’s semantics for BAN Logic [@r:abadi91]. Treating encrypted messages as tokens while still allowing agents to distinguish different encrypted messages is less common, but has been used at least by Askarov and Sabelfeld [@r:askarov07] and Askarov, Hedin, and Sabelfeld [@r:askarov08] in the context of information flow.
There are several frameworks for formally reasoning about cryptographic protocols, and I shall not list them all here. But I hope to provide enough pointers to the literature to ensure that the important ones are covered. For an early survey on the state of the art in formal reasoning about cryptographic protocols until 1995, see Meadows [@r:meadows95].
The Inductive Method described in [\[chap13:reasoning\]]{} is due to Paulson [@r:paulson98], and is built atop the Isabelle logical framework [@r:paulson94], a framework for higher-order logic. BAN Logic is introduced by Burrows, Abadi, and Needham [@r:burrows90], who use it to perform an analysis of several existing protocols in the literature. The logic courted controversy pretty much right from the start [@r:nessett90; @r:burrows90a]. Probably the most talked-about problem with BAN Logic is the lack of an independently-motivated semantics which would ensure that statements of the logic match operational intuition. Without such a semantics, it is difficult to argue for the reasonableness of the result of a BAN Logic analysis, except for the pragmatic observation that failure to prove a statement in BAN Logic often indicates a problem with the cryptographic protocol. Abadi and Tuttle [@r:abadi91] attempt to remedy the situation by defining a semantics for BAN Logic. Successor logics extending or modifying BAN Logic generally start with a variant of the Abadi-Tuttle semantics [@r:syverson90; @r:gong90; @r:oorschot93a; @r:syverson94; @r:wedel96; @r:stubblebine96]. Contemporary epistemic logic alternatives to BAN Logic were also developed, using a semantics in terms of protocol execution, but they never really took hold [@r:bieber90; @r:moser90].
The model checker MCK is described by Gammie and Van der Meyden [@r:gammie04], and was used to analyze the Dining Cryptographers protocol [@r:meyden04] as well as the Seven Hands protocol for the Russian Cards problem [@r:ditmarsch06]. TDL is an alternative epistemic temporal logic for reasoning about cryptographic protocols with a model checker developed by Penczek and Lomuscio [@r:lomuscio06], based on a earlier model checker [@r:raimondi04]. TDL is a branching-time temporal epistemic logic extended with a message knowledge primitive in addition to standard possible-worlds knowledge for expressing higher-order knowledge, and does not provide explicit support for attackers in its modeling language. The model-checking complexity results mentioned are due to Van der Meyden and Shilov [@r:meyden99]; see also Engelhardt, Gammie, and Van der Meyden [@r:meyden07] and Huang and Van der Meyden [@r:meyden10].
Another epistemic logic which forms the basis for reasoning about cryptographic protocol is Dynamic Epistemic Logic (DEL) [@r:gerbrandy99]. DEL is an epistemic logic of broadcast announcements which includes formulas of the form $[\rho]_i{\varphi}$, read *${\varphi}$ holds after agent $i$ broadcasts formula $\rho$*, where $\rho$ is a formula in a propositional epistemic sublanguage. (The actual syntax of DEL is slightly different.) Agents may broadcast that they know a fact, and this broadcast affects the knowledge of other agents. Kripke structures are used to capture the state of knowledge of agents at a point in time, and agent $i$ announcing $\rho$ will change Kripke structure $M$ representing the current state of knowledge of all agents into a Kripke structure $M^{\rho,i}$ representing the new state of knowledge that obtains. Dynamic Epistemic Logic has been used to analyze the Seven Hands protocol in great detail [@r:ditmarsch03]. Extensions to handle cryptography are described by Hommersom, Meyer, and De Vink [@r:hommersom04], as well as Van Eijck and Orzan [@r:vaneijck07].
Process calculi, starting with the spi calculus [@r:gordon99] and later the applied pi calculus [@r:abadi01], have been particularly successful tools for reasoning about cryptographic protocols. These use either observational equivalence to show that a process implementation of the protocol is equivalent to another process that clearly satisfies the required properties, or static analysis such as type checking to check the properties [@r:gordon03]. Epistemic logics defined against models obtained from processes are given by Chadha, Delaune, and Kremer [@r:chadha09] and Toninho and Caires [@r:toninho09]. Another process calculus, CSP, has also proved popular as a foundation for reasoning about cryptographic protocols [@r:lowe98; @r:ryan00].
Finally, other approaches rely on logic programming ideas: the NRL protocol analyzer [@r:meadows96], Multiset Rewriting [@r:cervesato99], and ProVerif [@r:blanchet01]. Thayer, Herzog, and Guttman [@r:thayer99] introduce a distinct semantic model for protocols, strand spaces, which has some advantages over traces. Syverson [@r:syverson99] develops an authentication logic on top of strand spaces, while Halpern and Pucella [@r:halpern03d] investigate the suitability of strand spaces as a basis for epistemic reasoning.
#### Information Flow.
Bell and LaPadula [@r:bell73a; @r:bell73b] were among the first to develop mandatory access control, and introducing the idea of attaching security levels to data to enforce confidentiality.
Early work on information flow security mostly focused on event traces, and tried to describe both closure conditions on traces, as well as unwinding conditions that would allow one to check that a set of event traces satisfies the security condition. Separability was defined by McLean [@r:mclean94], noninference by O’Halloran [@r:oHalloran90], generalized noninference by McLean [@r:mclean94], and generalized noninterference by McCullough [@r:mccullough87; @r:mccullough88] following the work of Goguen and Meseguer [@r:goguen82; @r:goguen84]. Other definitions of information-flow security for event systems are given by Sutherland [@r:sutherland86] and Wittbold and Johnson [@r:wittbold90]. A modern approach to information-flow security in event systems is described by Mantel [@r:mantel00]. The set-theoretic description of the knowledge operator is taken from Halpern [@r:halpern99], but appears in various guises in the economics literature [@r:aumann89]. Halpern and O’Neill [@r:halpern08] layer an explicit epistemic language on top of the event models re-expressed as Kripke structures, and show that the resulting logic can capture common definitions of confidentiality in event systems.
Denning and Denning [@r:denning77] first pointed out that programming languages are a useful setting for reasoning about information flow by observing that static analysis can be used to identify and control information flow. Most recent work on information-flow security from a programming language perspective goes back to Heintze and Riecke’s Secure Lambda Calculus [@r:heintze98] in a functional language setting, and Smith and Volpano [@r:smith98] in an imperative language setting. Honda, Vasconcelos, and Yoshida [@r:honda00] give a similar development in the context of a process calculus. Sabelfeld and Myers [@r:sabelfeld03] give a survey and overview of the state of the field up to 2003. Balliu, Dam, and Le Guernic [@r:balliu11; @r:balliu12] offer a rare use of an explicit epistemic temporal logic to reason about information-flow security. Sabelfeld and Sands [@r:sabelfeld09] give a good overview of the issues involved in declassification for language-based information flow. Askarov and Sabelfeld [@r:askarov07] use an epistemic logic in the context of declassification. Chong [@r:chong10] uses a form of algorithmic knowledge to model information release requirements.
#### Beyond Confidentiality.
Protocols for anonymous communication generally rely on a cloud of intermediaries that prevent information about the identity of the original sender to be isolated; Crowds is an example of such a protocol [@r:reiter98]. Anonymity has been well studied as an instance of confidentiality [@r:hughes04; @r:garcia05]. The explicit connection with epistemic logic was made by Halpern and O’Neill [@r:halpern05d], which is the source of the definitions in [\[chap13:beyond\]]{}. An early analysis of anonymity via epistemic logic is given by Syverson and Stubblebine [@r:syverson99a].
Anonymity is an important component of voting protocols. Van Eijck and Orzan [@r:vaneijck07] prove anonymity for a specific voting protocol using epistemic logic. More general analyses of voting protocols with epistemic logic have also been attempted [@r:baskar07; @r:kusters09]. The model of coercion resistance in [\[chap13:beyond\]]{} is from Küsters and Truderung [@r:kusters09].
Zero knowledge interactive proof systems were introduced by Goldwasser, Micali, and Rackoff [@r:goldwasser89] and have become an important tool in theoretical computer science. A good overview is given by Goldreich [@r:goldreich01]. Halpern, Moses, and Tuttle [@r:halpern88] give an epistemically-motivated analysis of zero knowledge interactive proof systems using a computationally-bounded definition of knowledge devised by Moses [@r:moses88].
Another context in which epistemic concepts—or perhaps more accurately, epistemic vocabulary—appear is that of authorization and trust management. Credential-based authorization policies can be used to control access to resources by requiring agents to present appropriate credentials (such as certificates) proving that they are allowed access. Because systems that rely on credential-based authorization policies are often decentralized systems, meaning that there is no central clearinghouse for determining for every authorization request whether an agent has the appropriate credentials, the entire approach relies on a web of trust between agents and credentials. Since in many such systems credentials can be delegated—an agent may allow another agent to act on her behalf—not only can credential checking become complicated, but authorization policies themselves become nontrivial to analyze to determine contradictions (an action being both allowed and forbidden by the policy under certain conditions) or coverage (a class of actions remaining unregulated by the policy under certain conditions). Where do epistemic concepts come up in such a scenario? Authorization logics from the one introduced by Abadi, Burrows, Lampson, and Plotkin [@r:abadi93] to the recent NAL [@r:schneider11] have been described as logics of belief, and are somewhat reminiscent of BAN Logic. One of their basic primitives is a formula $A~
\mathsf{says}~F$, which as a credential means that $A$ believes and is accountable for the truth of $F$. Delegation, for example, is captured by a formula $(A~\mathsf{says}~F) {\Rightarrow}(B~\mathsf{says}~G)$. This form of belief is entirely axiomatic, just like belief in BAN Logic.
#### Perspectives.
Ryan and Schneider [@r:ryan98] have extended the Dolev-Yao model of attackers with an XOR operation; Millen and Shmatikov [@r:millen03] with products and enough exponentiation to model the Diffie-Hellman key-establishment protocol [@r:diffie76]; and Lowe [@r:lowe02] and later Corin, Doumen, and Etalle [@r:corin05] and Baudet [@r:baudet05] with the ability to mount offline dictionary attacks. As described by Halpern and Pucella [@r:halpern12], many of these can be expressed using algorithmic knowledge, at least in the context of eavesdropping attackers. More generally, extending Dolev-Yao with additional operations can best be studied using equational theories, that is, equations induced by looking at the algebra of the additional operations; see for example Abadi and Cortier [@r:abadi04] and Chevalier and Rusinowitch [@r:chevalier08].
While it would be distracting to discuss the back and forth over BAN Logic in the decades since its inception, I will point out that recent work by Cohen and Dam has taken a serious look at the logic with modern eyes, and highlighted both interesting interpretations as well as subtleties [@r:cohen05; @r:cohen05a]. The protocol composition logic PCL of Datta, Derek, Mitchell, and Roy [@r:datta07], which builds on earlier work by Durgin, Mitchell, and Pavlovic [@r:durgin03], is a modern attempt at devising a logic for Hoare-style reasoning about cryptographic protocols.
A good overview of concrete models of cryptography is given by Goldreich [@r:goldreich01]. Semantic security, among others, is studied by Bellare, Chor, Goldreich, and Schnorr [@r:bellare98]. The relationship between formal models of cryptography and concrete models—how well does the former approximate the latter?—has been explored by Abadi and Rogaway [@r:abadi02a], and later extended by Micciancio and Warinschi [@r:micciancio04], among others. Backes, Hofheinz, and Unruh [@r:backes09] provide a good overview.
Approaches to analyze cryptographic protocols in a concrete model of cryptography have been developed [@r:lincoln98; @r:mitchell01]. In recent years some of the approaches for analyzing cryptographic protocols in a formal model of cryptography have been modified to work with a concrete model of cryptography, such as PCL [@r:datta05] and ProVerif [@r:blanchet08]. In some cases, cryptographic protocol analysis in a concrete model relies on extending indistinguishability over states to indistinguishability over the whole protocol [@r:datta04].
Defining a notion of resource-bounded knowledge that does not suffer from the logical omniscience problem is an ongoing research project in the epistemic logic community, and various approaches have been advocated, each with its advantages and its deficiencies: algorithmic knowledge [@r:halpern94], impossible possible worlds [@r:hintikka75], awareness [@r:fagin88]. A comparison between the approaches in terms of expressiveness and pragmatics appears in Halpern and Pucella [@r:halpern11].
Information flow in probabilistic programs was first investigated by Gray and Syverson [@r:gray98] using probabilistic multiagent systems [@r:halpern93]. Backes and Pfitzmann [@r:backes02] study it in a more computational setting. Smith [@r:smith09] presents some of the tools that need to be considered to analyze the kind of partial information leakage occurring in the password-checking example. Preliminary work on information flow in the presence of cryptography includes Laud [@r:laud03], Hutter and Schairer [@r:hutter04], and Askarov, Hedin, and Sabelfeld [@r:askarov08].
[^1]: Author’s email: `[email protected]`. To appear in *Handbook of Logics for Knowledge and Belief*.
[^2]: This chapter assumes from the reader a basic knowledge of epistemic logic and its Hintikka-style possible-worlds semantics; see [\[chap13:bibnotes\]]{} for references. Furthermore, to simplify the exposition, the term *epistemic* is used to refer both to knowledge and to belief throughout.
[^3]: A safety property is a property of the form *a bad state is never reached in any execution of the system*, for some definition of *bad state*. A safety property can be checked by examining every possible execution independently of any other; in contrast, checking an epistemic property requires examining every possible execution in the context of all other possible executions.
[^4]: XOR (exclusive or) is a binary Boolean operation $\oplus$ defined by taking $b_1 \oplus b_2$ to be true if and only if exactly one of $b_1$ or $b_2$ is true. It is associative and commutative.
[^5]: This section considers deterministic encryption schemes only, ignoring probabilistic encryption schemes.
[^6]: We can assume that every message has a *from* and *to* field—think email—and that these can be forged. Isabel posing as Alice means that Isabel sends a message and forges the *from* field of the message to hold Alice’s name.
[^7]: A fourth class of attackers, less commonly considered, shares characteristics with both eavesdropping attackers and insider attackers: dishonest agents are assumed not to have control over the network but may attempt to subvert the protocol while acting within the limits imposed on legitimate users.
[^8]: The symbolic decryption function embodies an assumption that encrypted messages have enough redundancy for an agent to determine when decryption is successful.
[^9]: This definition does not account for the possibility that an agent, even if she does not know the content of an encrypted message, may still recognize that she has already seen that encrypted message. (This is an issue when encryption is deterministic, so that encrypting $m$ with key $k$ always yields the same string of bits.) One approach is to refine the definition so that every encryption ${\{m\}_{k}}$ is replaced by a unique token $\Box_{m,k}$.
[^10]: To see the need for a richness condition, if there is a single state in which agent $i$ has received an encrypted message, then $K_i({\mathit{part}}_i(m))$ holds vacuously when $m$ is the content of the encrypted message.
[^11]: These rules are simplifications. Actual rules would contain appropriate quantification and additional side conditions to ensure that $A$ and $B$ are different agents, that nonces do not clash, and so on.
[^12]: The `observable` annotation is used to derive the indistinguishability relation: two states are indistinguishable to agent $i$ if the observable variables of the program executed by agent $i$ have the same value in both states. The construct nondeterministically executes one of its branches with an associated condition that evaluates to true. Variable `self` is assigned the name of the agent executing the program.
[^13]: As in [\[chap13:cryptoprotocols\]]{}, this does not take probabilistic information into account.
[^14]: Another way of understanding this definition is that it requires the relation on stores induced by program execution to be a refinement of $L$-equivalence $\approx_L$. If we define ${[\hspace{-.3ex}[C]\hspace{-.3ex}]}_{\approx_L}$ as the relation $\{({[\hspace{-.3ex}[C]\hspace{-.3ex}]}(\sigma_1),{[\hspace{-.3ex}[C]\hspace{-.3ex}]}(\sigma_2)) : \sigma_1\approx_L\sigma_2\}$, then the noninterference condition can be rephrased as ${[\hspace{-.3ex}[C]\hspace{-.3ex}]}_{\approx_L}\subseteq{}\approx_L$.
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abstract: 'We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in [@CK]. It possesses a triangular structure $\g = \n_+ \oplus \mathbb{C}.d \oplus \n_-$, like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” $\lambda \in \mathbb{C}$. We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.'
address: 'Department of Mathematics Boston University, Boston, MA 02215'
author:
- Matthew Szczesny
date: October 2007
title: 'On the structure and representations of the insertion-elimination Lie algebra'
---
[^1]
Introduction
============
The insertion-elimination Lie algebra $\g$ was introduced in [@CK] as a means of encoding the combinatorics of inserting and collapsing subgraphs of Feynman graphs, and the ways the two operations interact. A more abstract and universal description of these two operations is given in terms of rooted trees, which encode the hierarchy of subdivergences within a given Feynman graph, and it is this description that we adopt in this paper. More precisely, $\g$ is generated by two sets of operators $\{ D^+_t \}$, and $\{ D^-_t \}$, where $t$ runs over the set of all rooted trees, together with a grading operator $d$. In [@CK] $\g$ was defined in terms of its action on a natural representation $\CT$, where the latter denotes the vector space spanned by rooted trees. For $s \in \CT$, $D^+_t.s$ is a linear combination of the trees obtained by attaching $t$ to $s$ in all possible ways, whereas $D^-_t.s$ is a linear combination of all the trees obtained by pruning the tree $t$ from branches of $s$. $\n_+= \{ D^+_t \}$ and $\n_-=\{ D^-_t\}$ form two isomorphic nilpotent Lie subalgebras, and $\g$ has a triangular structure $$\g = \n_+ \oplus \mathbb{C}.d \oplus \n_-$$ as well as a natural $\mathbb{Z}$–grading by the number of vertices of the tree $t$. The Hopf algebra $U(\n_{\pm})$ is dual to Kreimer’s Hopf algebra of rooted trees [@K].
This note aims to estblish a few basic facts regarding the structure and representation theory of $\g$. We begin by showing that $\g$ is simple, which together with its infinite-dimensionality implies that it has no non-trivial finite-dimensional representations, and that any non-trivial representation is necessarily faithful. We then proceed to develop a highest-weight theory for $\g$ along the lines of [@K1; @K2]. In particular, we show that every irreducible highest-weight representation of $\g$ is a quotient of a Verma-like module, and that these are generically irreducible.
One can define a larger, “two-parameter” version of the insertion-elimination Lie algebra $\widetilde{\g}$, where operators are labelled by pairs of trees $D_{t_1, t_2}$ (roughly speaking, in acting on $\CT$, this operator replaces occurrences of $t_1$ by $t_2$). In the special case of ladder trees, $\widetilde{\g}$ was studied in [@M; @KM1; @KM2]. The finite-dimensional representations of the nilpotent subalgebras $\n_{\pm}$ as well as many other aspects of the Hopf algebra $U(\n_{\pm})$ were studied in [@F].
[**Acknowledgements:**]{} The author would like to thank Dirk Kreimer for many illuminating conversations and explanations of renormalization as well as related topics. This work was supported by NSF grant DMS-0401619.
The insertion-elimination Lie algebra on rooted trees {#basicfacts}
=====================================================
In this section, we review the construction of the insertion-elimination Lie algebra introduced in [@CK], with some of the notational conventions introduced in [@M].
Let $\T$ denote the set of rooted trees. An element $t \in \T$ is a tree (finite, one-dimensional contractible simplicial complex), with a distinguished vertex $r(t)$, called the root of $t$. Let $V(t)$ and $E(t)$ denote the set of vertices and edges of $t$, and let $$| t | = \# V(t)$$ Let $\CT$ denote the vector space spanned by rooted trees. It is naturally graded, $$\label{CT}
\CT = \bigoplus_{n \in \mathbb{Z}_{\geq 0}} \CT_n$$ where $\CT_n = \operatorname{span}\{ t \in \mathbb{T} \vert |t| = n \}$. $\CT_0$ is spanned by the empty tree, which we denote by $\bf{1}$. We have
$\CT_0 = <1> $ $\CT_1=<\bullet> $ $\CT_2=$ $<$ $>$
$\CT_3 = $ $<$ , $>$
where $<,>$ denotes span, and the root is the vertex at the top. If $e \in E(t)$, by a *cut along* $e$ we mean the operation of cutting $e$ from $t$. This divides $t$ into two components - $R_c(t)$ containing the root, and $P_e(t)$, the remaining one. $R_e(t)$ and $P_e (t)$ are naturally rooted trees, with $r(R_c (t)) = r(t)$ and $r(P_e (t)) = $ (endpoint of e). Note that $V(t) = V(R_e(t)) \cup V(P_e(t))$.
Let $\g$ denote the Lie algebra with generators $D^+_t, D^-_t, d$, $t \in \T$, and relations $$\label{rel1}[D^+_{t_1}, D^{+}_{t_2}] = \sum_{v \in V(t_2)} D^{+}_{t_2 \cup_v t_1} - \sum_{v \in V(t_1)} D^{+}_{t_1 \cup_v t_2}$$ $$\label{rel2} [D^-_{t_1}, D^{-}_{t_2}] = \sum_{v \in V(t_1)} D^{-}_{t_1 \cup_v t_2}
- \sum_{v \in V(t_2)} D^{-}_{t_2 \cup_v t_1}$$ $$\label{rel3} [D^-_{t_1}, D^{+}_{t_2}] = \sum_{t \in \T} \alpha(t_1, t_2;t) D^{+}_t + \sum_{t \in T} \beta(t_1,t_2;t) D^{-}_t$$ $$\label{rel4} [D^{-}_t, D^{+}_t] = d$$ $$\label{rel5} [d, D^{-}_t] = - |t| D^{-}_t$$ $$\label{rel6} [d, D^{+}_t] = |t| D^{+}_t$$
where for $s, t \in \T$, and $v \in V(s)$ $s \cup_v t$ denotes the rooted tree obtained by joining the root of $t$ to $s$ at the vertex $v$ via a single edge, and
- $\alpha(t_t, t_2; t) = \# \{ e \in E(t_2) | R_e(t_2) = t, \, \, P_e(t_2) = t_1 \}$
- $\beta(t_1, t_2; t) = \# \{ e \in E(t_1) | R_e(t_1) = t, \, \, P_e(t_1) = t_2 \}$
Thus, for example $$\begin{aligned}
\psset{levelsep=0.3cm, treesep=0.3cm}
[D^+_{\bullet}, D^+_{ \pstree{\Tr{\bullet}}{\Tr{\bullet}\Tr{\bullet}} }] &= D^+_{\pstree{\Tr{\bullet}}{\Tr{\bullet}
\Tr{\bullet} \Tr{\bullet }}} + 2 D^{+}_{ \pstree{\Tr{\bullet}}{\pstree{\Tr{\bullet}}{\Tr{\bullet}}\Tr{\bullet}} } - D^+_{\pstree{\Tr{\bullet}}{\pstree{\Tr{\bullet}}{ \Tr{\bullet} \Tr{\bullet}} }} \\
\psset{levelsep=0.3cm, treesep=0.3cm}
[D^-_{\bullet}, D^-_{ \pstree{\Tr{\bullet}}{\Tr{\bullet}\Tr{\bullet}} }] &= -D^-_{\pstree{\Tr{\bullet}}{\Tr{\bullet}
\Tr{\bullet} \Tr{\bullet }}} - 2 D^{-}_{ \pstree{\Tr{\bullet}}{\pstree{\Tr{\bullet}}{\Tr{\bullet}}\Tr{\bullet}} } + D^-_{\pstree{\Tr{\bullet}}{\pstree{\Tr{\bullet}}{ \Tr{\bullet} \Tr{\bullet}} }} \\
\psset{levelsep=0.3cm, treesep=0.3cm}
[D^-_{\bullet}, D^+_{ \pstree{\Tr{\bullet}}{\Tr{\bullet}\Tr{\bullet}} }] &= 2 D^{+}_{ \pstree{\Tr{\bullet}}{\Tr{\bullet}}}\end{aligned}$$
$\g$ acts naturally on $\CT$ as follows. If $s \in \T$, viewed as an element of $\CT$, and $t \in \T$, then $$D^{+}_t (s) = \sum_{v \in V(s)} s \cup_v t$$ $$D^{-}_t (s) = \sum_{e \in E(s), P_e(s) = t} R_e (s)$$ $$d (s) = |s| s$$
Structure of $\g$
=================
Let $\n_+$ and $ \n_{-}$ be the Lie subalgebras s of $\g$ generated by $D^+_{t}$ and $D^-_t$ , $t \in \T$. We have a triangular decomposition $$\label{triangular}
\g = \n_+ \oplus \mathbb{C}. d \oplus \n_{-}$$ The relations \[rel4\], \[rel5\], and \[rel6\] imply that for every $t \in \T$ $$\g^t = <D^{+}_t, D^{-}_t, d>$$ forms a Lie subalgebra isomorphic to $\mathfrak{sl}_2$. We have that $\g_t \cap \g_s = \mathbb{C}. d$ if $s \ne t$. Assigning degree $|t|$ to $D^{+}_t$, $-|t|$ to $D^{-}_t$, and $0$ to $d$ equips $\g$ with a $\mathbb{Z}$–grading. $$\g = \bigoplus_{n \in \mathbb{Z}} \g_n$$ $\g$ possesses an involution $\iota$, with $$\iota(D^+_t)=D^-_t \hspace{1in} \iota(D^-_t)=D^{+}_t \hspace{1in} \iota(d) = - d$$ Thus $\iota$ is a gradation-reversing Lie algebra automorphism exchanging $\n_{+}$ and $\n_{-}$.
\[gissimple\] $\g$ is a simple Lie algebra
Suppose that $\I \subset \g$ is a proper Lie ideal. If $x \in \I$, let $x = \sum_i x_i, \; \; x_i \in \g_i$ be its decomposition into homogenous components. We have $$[d,x] = \sum_{n} n x_n$$ which implies that $x_n \in I$ for every $n$ (because the Vandermonde determinant is invertible) i.e. $\I = \oplus_{n \in \mathbb{Z}} (\I \cap \g_n)$. Suppose now that $x_n \in \g_n, \; n > 0 $. We can write $x_n$ as a linear combination of $n$–vertex rooted trees $$\label{xdecomp}
x_n = \sum_{t \in \T_n} \alpha_t \cdot t$$ We proceed to show that $D^+_{\bullet} \in \I$, where $\bullet$ is the rooted tree with one vertex. Let $S(x_n) \subset \T_n$ be the subset of n-vertex trees occurring with a non-zero $\alpha_t$ in \[xdecomp\]. Given a rooted tree $t$, let $St(t)$ denote the set of rooted trees obtained by removing all the edges emanating from the root. Let $$St(x_n) = \bigcup_{s \in S(x_n)} St(s)$$ and let $\xi \in St(x_n)$ be of maximal degree. It is easy to see that $[D^-_\xi, x_n]$ is a non-zero element of $\g_{n-|\xi|}$. Starting with $x_n \in \n_+, \; x_n \neq 0$, and repeating this process if necessary, we eventually obtain a non-zero element of $\g_1 = < D^+_{\bullet} >$. Now, $[D^-_{\bullet}, D^+_{\bullet}] = d$, and since $[d, \g]=\g]$, this implies $\I=\g$. We have thus shown that if $\I$ is proper, then $$\I \cap \n_+ = 0$$ Applying $\iota$ shows that $\I \cap \n_ - 0$ as well, and it is clear that $\I \cap \mathbb{C}.d = 0$.
We can now use this result to deduce a couple of facts about the representation theory of $\g$.
If $V$ is a non-trivial representation of $\g$, then $V$ is faithful.
$\g$ has no non-trivial finite-dimensional representations.
The latter can also be easily deduced by analyzing the action of the $\mathfrak{sl}_2$ subalgebras $\g^t$ as follows. Suppose that $V$ is a finite-dimensional representation of $\g$. To show that $V$ is trivial, it suffices to show that it restricts to a trivial representation of $\g^t$ for every $t \in \T$. This in turn, will follow if we can show that for a *single* tree $t \in \T$, $\g^t$ acts trivially, because this implies that $d$ acts trivially, and $\mathbb{C}.d \subset \g^t$ plays the role of the Cartan subalgebra. Let $$V = \bigoplus_{i=1\cdots k} V_{\delta_i}$$ be a decomposition of $V$ into $d$–eigenspaces - i.e. if $v \in V_{\delta_i}$, then $d.v_i = \delta_i v$. Since $V$ is finite-dimensional, the set $\{\delta_i \}$ is bounded, and so lies in a disc of radius $R$ in $\mathbb{C}$. If $v \in V_{\delta_i}$ then $[d,D^{+}_t] = |t| D^{+}_t$ implies that $D^{+}_t . v \in V_{\delta_i+|t|}$. Choosing a $t \in \T$ such that $|t| > 2R$ shows that $D^{+}_t . v = 0$ for every $v \in V$.
Lowest-weight representations of $\g$
-------------------------------------
We begin by examining the “defining” representation $\CT$ of $\g$ introduced in section \[basicfacts\]. Its decomposition into $d$–eigenspaces is given by \[CT\]. Given a representation $V$ of $\g$ on which $d$ is diagonalizable, with finite-dimensional eigenspaces, and writing $$V = \bigoplus_{\delta} V_{\delta}$$ for this decomposition, we define the emph[character]{} of $V$, $char(V,q)$ to be the formal series $$char(V,q) = \sum_\delta dim(V_\delta) q^\delta$$ The case $V=\CT$, where $dim(V_n)$ is the number of rooted trees on $n$ vertices, suggests that representations of $\g$ may contain interesting combinatorial information. The triangular structure \[triangular\] of $\g$ suggests that a theory of highest– or lowest–weight representations may be appropriate.
\[lw\] We say that a representation $V$ of $\g$ is *lowest–weight* if the following properties hold
1. $V = \oplus V_{\delta}$ is a direct sum of finite-dimensional eigenspaces for $d$.
2. The eigenvalues $\delta$ are bounded in the sense that there exists $L \in \mathbb{R}$ such that $Re(\delta) \geq L$.
We call the $\delta$ the *weights* of the representation, and category of such representations $\OO$. If $V \in \OO$, we say $v \in V_{\delta}$ is a *lowest-weight vector* if $\n_- v = 0$. Since $D^{-}_t$ decreases the weight of a vector by $|t|$, and the weights all lie in a half-plane, it is clear that every $V \in \OO$ contains a lowest-weight vector.
Recall that a representation $V$ of $\g$ is *indecomposable* if it cannot be written as $V = V_1 \oplus V_2$ for two non-zero representations. Let $U(\mathfrak{h})$ denote the universal enveloping algebra of a Lie algebra $\mathfrak{h}$.
If $v \in V_\lambda$ is a lowest-weight vector, then $U(\n_+).v$ is an indecomposable representation of $\g$
$U(\g).v$ is clearly the smallest sub-representation of $V$ containing $v$. The decomposition \[triangular\] together with the PBW theorem implies that $$U(\g) = U(\n_+)\otimes \mathbb{C}[d] \otimes U(\n_-)$$ Because $v$ is a lowest-weight vector, $\mathbb{C}[d] \otimes U(\n_-).v = \mathbb{C}.v$. It follows that $U(\g).v = U(\n_+).v$. That the latter is indecomposable follows from the fact that in $U(\n_+).v$, the weight space corresponding to $\lambda$ is one-dimensional, and so if $U(\n_+).v=V_1\oplus V_2$, then $v \in V_1$ or $v \in V_2$.
Observe that $$U(\n_+).v = \oplus (U(\n_+).v)_{\lambda+k}, \; \; k \in \mathbb{Z}_{\geq 0}$$ where $(U(\n_+).v)_{\lambda+k}$ is spanned by monomials of the form $$\label{PBW}
D^{+}_{t_1} D^{+}_{t_2} \cdots D^{+}_{t_i} . v$$ with $|t_1|+\cdots |t_i| = k$.
The category $\mc{O}$ contains Verma-like modules. For $\lambda \in \mathbb{C}$, let $\mathbb{C}_{\lambda}$ denote the one-dimensional representation of $\mathbb{C}.d\oplus \n_{-}$ on which $\n_-$ acts trivially, and $d$ acts by multiplication by $\lambda$.
The $\g$–module $$W(\lambda) = U(\g) \underset{\mathbb{C}[d] \otimes U(\n_-)}{\otimes} \mathbb{C_\lambda}$$ will be called *the Verma module* of lowest weight $\lambda$.
Choosing an ordering on trees yields a PBW basis for $\n_+$, and thus also a basis of the form \[PBW\] for $W(\lambda)$.
Given a representation $V \in \mc{O}$, and a lowest weight vector $v \in V_\lambda$, we obtain a map of representations $$\label{univprop}
W(\lambda) \mapsto V$$ $$\mathbf{1} \mapsto v$$
If $V \in \mc{O}$ is an irreducible representation, then $V$ is the quotient of a Verma module.
Since $V \in \mc{O}$, $V$ possesses a lowest-weight vector $v \in V_\lambda$ for some $\lambda \in \mathbb{C}$. Since $V$ is irreducible, $V = U(\g).v=U(\n_+).v$. The latter is a quotient of $W(\lambda)$.
We have $$\begin{aligned}
Char(W(\lambda)) &= q^{\lambda} \sum_{n \in \mathbb{Z}_{\geq 0}} dim(\CT_n) q^{n} \\
&= q^{\lambda} \prod_{n \in \mathbb{Z}_{\geq 0}} \frac{1}{{(1 - q^n)}^{P(n)}}\\\end{aligned}$$ where $P(n)$ is the number of primitive elements of degree $n$ in $\mathcal{H}_{K}$.
Irreducibility of $W(\lambda)$
-------------------------------
It is a natural question whether $W(\lambda)$ is irreducible. In this section we prove the following result:
For $\lambda$ outside a countable subset of $\mathbb{C}$ containing $0$, $W(\lambda)$ is irreducible.
Let $v \neq 0$ be a basis for $W(\lambda)_\lambda$. $W(\lambda)$ contains a proper sub-representation if and only if contains a lowest-weight vector $w$ such that $w \notin \mathbb{C}.v$. In $W(0)$, $D^+_{\bullet} .v \in W(0)_1$ is a lowest-weight vector, since $$D^-_{\bullet} D^{+}_{\bullet}.v = D^{+}_{\bullet} D^-_{\bullet}.v + d.v = 0$$ and $D^{-}_t.v =0$ for all $t \in \T$ with $|t| \geq 2$ by degree considerations. It follows that $W(0)$ is not irreducible.
If $I=(t_1, \cdots, t_k)$ is a $k$–tuple of trees such that $$t_1 \preceq t_2 \preceq \cdots \preceq t_k$$ in the chosen order, let $D^+_I.v$ denote the vector $$\label{vv}
D^+_{t_k} \cdots D^+_{t_1}.v \; \in W(\lambda)$$ $w \in W(\lambda)_{\lambda+n}$ is a lowest-weight vector if and only if $$\label{lw}
D^-_t.w=0$$ for all $t$ such that $|t| \leq n$. Writing $w$ in the basis \[vv\] $$w = \sum_{|I|=n} \alpha_I D^+_{I}.v$$ the conditions \[lw\] translate into a system of equations for the coefficients $\alpha_I$. For example, if $w \in W(\lambda)_{\lambda+2}$, then $$\psset{levelsep=0.3cm, treesep=0.3cm}
w=\alpha_1 D^+_{ \pstree{\Tr{\bullet}}{\Tr{\bullet}}}.v + \alpha_2 D^{+}_{\bullet} D^{+}_{\bullet}.v$$ and conditions $D^-_{\pstree{\Tr{\bullet}}{\Tr{\bullet}}}.v =0 $, $D^-_{\bullet}.w=0$ translate into $$\begin{aligned}
\lambda \alpha_1 + \lambda \alpha_2 &= 0 \\
\alpha_1 + (2\lambda + 1) \alpha_2 &=0\end{aligned}$$ The determinant of the corresponding matrix is $2 \lambda^2$, and so for $\lambda \neq 0$, there is no lowest-weight vector $w \in W(\lambda)_{\lambda+2}$. For a general $n$, the system can be written in the form $$(A + \lambda B) [\alpha_I] = 0$$ where $A$ and $B$ are matrices whose entries are non-negative integers. Let $$f_n (\lambda) = dim(Ker(A+\lambda B))$$ Then for every $r \in \mathbb{N}$ $$S_{n,r} = \{ \lambda \in \mathbb{C} | f_n(\lambda) \geq r \}$$ if proper, is a finite subset of $\mathbb{C}$, since the condition is equivalent to the vanishing a finite collection of sub-determinants, each of which is a polynomial in $\lambda$. The set of $\lambda \in \mathbb{C}$ for which $W(\lambda)$ is irreducible is therefore $$\bigcup_{n \in \mathbb{N}} \{ \mathbb{C} \backslash S_{n,1} \}$$ The theorem will follow if $S_{n,1}$ is proper for each $n \in \mathbb{N}$. This follows from the following Lemma.
\[Z1\] $Z(1)$ is irreducible.
We begin by examining the representation $\CT$. The degree $0$ subspace $\mathbb{C}.1$ is a trivial representation of $\g$. Let $\M$ denote the quotient $\CT/\mathbb{C}.1$. It is easily seen that the exact sequence $$0 \mapsto \mathbb{C} \mapsto \CT \mapsto \M \mapsto 0$$ is non-split. $\M$ has highest weight $1$, and the subspace $\M_1$ can be identified with the span of the tree on one vertex $\bullet$. By the universal property of Verma modules, \[univprop\] we have a map $$\label{surject}
W(1) \mapsto M$$ sending the lowest-weight vector of $W(1)$ to $\bullet$. Now, $W(1)_n$ is spanned by all vectors \[vv\] such that $|t_1|+ \cdots |t_k| = n-1$, and so can be identified with the set of forests on $n-1$ vertices, while $M_n$ can be identified with $\CT_n$. The operation of adding a root to a forest on $n-1$ vertices to produce a rooted tree with $n$ vertices yields an isomorphism $W(1)_{n} \cong M_n$. Thus, if the map \[surject\] is a surjection, it is an isomorphism. This in turn, follows from the fact that $M$ is irreducible.
It suffices to show that $M_n$ contains no lowest-weight vectors for $n > 1$. This follows from an argument similar to the one used to prove \[gissimple\]. Let $w \in M_n$, and write $$w = \alpha_1 t_1 + \cdots \alpha_k t_k$$ where $|t_i|=n$ and we may assume that $\alpha_i \neq 0$. In the notation of \[gissimple\], let $\xi \in St(w)$ be of maximal degree. Then $$D^-_{\xi}. w \neq 0$$ Thus, $\M$ is irreducible, and hence isomorphic to $W(1)$ by the map \[surject\].
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Kac, V. G. *Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras*, World Scientific, 1988.
Kac, V. G. *Infinite Dimensional Lie Algebras*, Cambridge University Press, 1994.
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abstract: 'Based on special geometry, we consider corrections to $N=2$ extremal black-hole solutions and their entropies originating from higher-order derivative terms in $N=2$ supergravity. These corrections are described by a holomorphic function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp(2$n$+2) vectors. We apply the formalism to $N=2$ type-IIA Calabi-Yau string compactifications and compare our results to recent related results in the literature.'
---
-5mm ====
**Higher-Order Black-Hole Solutions**
**in $N=2$ Supergravity and Calabi-Yau String Backgrounds**
**Klaus Behrndt$^1$[^1] , Gabriel Lopes Cardoso$^2$[^2] , Bernard de Wit$^2$[^3] ,**
**Dieter Lüst$^1$[^4] , Thomas Mohaupt$^3$[^5] and Wafic A. Sabra$^4$[^6]**
*Dedicated to the memory of Constance Caris*
One of the celebrated successes within the recent non-perturbative understanding of string theory and M-theory is the matching of the thermodynamic Bekenstein-Hawking black-hole entropy with the microscopic entropy based on the counting of the relevant D-brane configurations which carry the same charges as the black hole [@StromVafa; @CallanMal]. This comparison works nicely for type-II string, respectively, M-theory backgrounds which break half of the supersymmetries, i.e. exhibit $N=4$ supersymmetry in four dimensions. In this paper we will discuss charged black-hole solutions and their entropies in the context of $N=2$ supergravity in four dimensions. Four-dimensional $N=2$ supersymmetric vacua are obtained by compactifying the type-II string on a Calabi-Yau threefold $CY_3$ or by M-theory compactification on $CY_3\times S^1$. In addition, the heterotic string on $K3\times T^2$ also leads to $N=2$ supersymmetry in four dimensions and the heterotic and the type-II vacua are expected to be related by string-string duality [@KachruVafa].
Extremal, charged, $N=2$ black holes, their entropies and also the corresponding brane configurations were discussed in several recent papers [@FerrKallStro; @BCDWKLM; @rey; @BehrndtLuestSabra; @BehLu; @msw; @vafa; @sen]. The macroscopic Bekenstein-Hawking entropy ${\cal S}_{\rm BH}$, i.e. the area $A$ of the black-hole horizon, can be obtained as the minimum of the graviphoton charge $Z$ [@FerrKallStro], \_[BH]{}= [A4]{}=|Z\_[min]{}|\^2 , \[area\] where, as a solution of the minimization procedure, the entropy as well as the scalar fields (moduli) at the horizon depend only on the electric and magnetic black hole charges, but not on the asymptotic boundary values of the moduli fields. This procedure has been used extensively to determine the entropy of $N=2$ black holes for type-II compactifications on Calabi-Yau threefolds [@BCDWKLM]. One of the key features of extremal $N=2$ black-hole solutions is that the moduli depend in general on $r$, but show a fixed-point behaviour at the horizon. This fixed-point behaviour is implied by the fact that, at the horizon, full $N=2$ supersymmetry is restored; at the horizon the metric is equal to the Bertotti-Robinson metric, corresponding to the $AdS_2\times S^2$ geometry. This metric can be described by a $(0,4)$ superconformal field theory [@Strominger]. The extremal black hole can be regarded as a soliton solution which interpolates between two fully $N=2$ supersymmetric vacua, namely corresponding to $AdS_2\times
S^2$ at the horizon and flat Minkowski spacetime at spatial infinity.
The $N=2$ black holes together with their entropies which were considered so far, appeared as solutions of the equations of motion of $N=2$ Maxwell-Einstein supergravity action, where the bosonic part of the action contains terms with at most two space-time derivates (i.e., the Einstein action, gauge kinetic terms and the scalar non-linear $\sigma$-model). This part of the $N=2$ supergravity action can be encoded in a holomorphic prepotential $F(X)$, which is a function of the scalar fields $X$ belonging to the vector multiplets. However, the $N=2$ effective action of strings and M-theory contains in addition an infinite number of higher-derivative terms involving higher-order products of the Riemann tensor and the vector field strengths. A particularly interesting subset of these couplings in $N=2$ supergravity can be again described by a holomophic function $F(X,W^2)$ [@BergshoeffdeRoodeWit; @AntonGavaNarTayl; @Bershadskyetal; @deWit; @DWCLMR; @Moore] , where the additional chiral superfield $W$ is the Weyl superfield, comprising the covariant quantities of conformal supergravity. Its lowest component is the graviphoton field strength (in the form of an auxiliary tensor field $T_{\mu \nu}^-$), while the Weyl tensor appears at the $\theta^2$ level. The aim of this paper is, using the superconformal calculus, to study the $N=2$ black-hole solutions and the corresponding entropies for higher order $N=2$ supergravity based on the holomorphic function $F(X,W^2)$. We treat $W^2$ as a new chiral background and expand the black-hole solutions as a power series in $W^2$. As an interesting example we compute the entropy of the charged $N=2$ black holes in type-II compactifications on a Calabi-Yau threefold. At the end we compare our results to a recent computation of the microscopic Calabi-Yau black hole in M-theory [@msw; @vafa].
Let us start by recalling the $N=2$ black-hole solutions and their entropies in the case where the holomorphic function $F(X)$ does not depend on the Weyl multiplet. The bosonic $N=2$ supergravity action coupled to $n$ vector multiplets is given by S\_[N=2]{}= \^4x , \[action\] where the $z^A$ (with $A=1\dots, n$) denote complex scalar fields, and ${F}^{\pm I \, \mu\nu}$ (with $I=0,\dots ,n$) are the (anti-)selfdual abelian field strengths (including the graviphoton field strength). An intrinsic definition of a special Kähler manifold [@c] can be given in terms of a flat $(2n+2)$-dimensional symplectic bundle over the $(2n)$-dimensional Kähler-Hodge manifold, with the covariantly holomorphic sections V=,\[section\] obeying the symplectic constraint i|V, V=i (|X\^I F\_I-|F\_I X\^I)=1 .\[constr\] Usually the $F_I$ can be expressed in terms of a holomorphic prepotential $F(X)$, homogenous of degree two, via $F_I=\partial F(X)/\partial X^I$. The field-dependent gauge couplings in (\[action\]) can then also be expressed in terms of derivatives of $F$. The constraint (\[constr\]) can be solved by introducing the projective holomorphic sections $X^I(z)$, which are related to the $X^I$ according to X\^I =e\^[[12]{}K(z,|z)]{}X\^I(z), K(z,|z)=-i|X\^I(z)F\_I(X\^I(z)) -iX\^I(z)|F\_I(|X\^I(|z)).\[kp\] Here $K(z,\bar z)$ is the Kähler potential which gives rise to the metric $g_{A\bar B}$. The holomorphic sections transform under projective transformations $X^I(z)\to \exp[f(z)]\,X^I(z)$, which induce a Kähler transformation on the Kähler potential $K$ and a U(1) transformation on the section $V$, K(z,|z)K(z,|z)-f(z)-|f(|z), V(z,|z) \^[[12]{}(f(z)-|f(| z))]{}V(z,|z). \[kweight\] Besides $V$, the magnetic/electric field strengths $(F_{\mu\nu}^I, G_{\mu\nu I})$ also consitute a symplectic vector. Here $G_{{\mu}{\nu}I}$ is generally defined by $G^+_{{\mu}{\nu}I}(x) = -4i g^{-1/2}\, {\delta}S/{\delta}F^{+{\mu}{\nu}I}$. Consequently, also the corresponding magnetic/electric charges $Q=(p^I, q_I)$ transform as a symplectic vector.
In terms of these symplectic vectors the stationary solutions have been discussed in [@BehrndtLuestSabra] in a fixed Kähler gauge. The generalized Maxwell equations can be solved in terms of $2n+2$ harmonic functions, which therefore also transform as a symplectic vector ($m,n=1,2,3$), F\_[mn]{}\^I=12\_[mnp]{}\_pH\^I(r), G\_[mn I]{}=12\_[mnp]{}\_pH\_I(r) .\[gaugeh\] Throughout this paper we assume that the metric can be brought in the form [@tod] ds\^2 = - e\^[2U]{} dt\^2 + e\^[-2 U]{} dx\^m dx\^m ,\[metrican\] where $U$ is a function of the radial coordinate $r=\sqrt{x^m x^m}$. The harmonic functions can be parametrized as H\^I(r) = h\^I+[p\^Ir]{}, H\_I(r)=h\_I+[q\_Ir]{}, \[harmonic\] and we write the corresponding symplectic vector as $H(r)=(\tilde
H^I(r), H_I(r)) = h + Q/r$.
Once one has identified the various symplectic vectors that play a role in the solutions, it follows from symplectic covariance that these vectors should satisfy a certain proportionality relation. The simplest possibility is to assume that $V$ and $H$ are directly proportional to each other. Because $H$ is real and invariant under U(1) transformations, there is a complex proportionality factor, which we denote by $Z$. Hence we define a U(1)-invariant symplectic vector (here we use the homogeneity property of the function $F$), =|Z V=(Y\^I, F\_I(Y)) ,\[newsec\] so that $Y^I=\bar ZX^I$, and assume (r) -|(r) = i H(r). \[stable\] This equation determines $Z$, Z(r)= - H\_I(r) X\^I + H\^I(r) F\_I(X), Z(r)\^2 = i |(r) ,(r). \[Z\] The first of these equations indicates that $Z(r)$ is related to the auxiliary field $T_{\mu \nu}^-$. This relation may not hold when $F$ depends on $W^2$. The equations (\[stable\]), which we call the stabilization equations, also govern the $r$-dependence of the scalar moduli fields: $z^A(r)=Y^A(r)/ Y^0(r)$. So the constants $(\tilde h^I,h_I)$ just determine the asymptotic values of the scalars at $r=\infty$. In order to obtain an asymptotically flat metric with standard normalization, these constants must fullfill some constraints. Near the horizon ($r\approx 0$), (\[stable\]) takes the form used in [@BCDWKLM] and $Z$ becomes proportional to the holomorphic BPS mass ${\cal M}(z) = q_I X^I(z) - p^I F_I (X(z)$.
When in addition we make the symplectically invariant ansatz $e^{-2U} = Z \bar Z$, it can be shown that the solution preserves half the supersymmetries, except at the horizon and at spatial infinity, where supersymmetry is unbroken. [From]{} the form of the static solution at the horizon ($r\rightarrow 0$) we can easily derive its macroscopic entropy. Specifically the Bekenstein-Hawking entropy is given by \_[BH]{}&=&(r\^2e\^[-2U]{})\_[r=0]{}=(r\^2Z|Z)\_[r=0]{}\
&=&i(|Y\^I\_[hor]{}F\_I(Y\_[hor]{})-|F\_I (|Y\_[hor]{})Y\^I\_[hor]{}),\[entropy\] where the symplectic vector $\Pi$ at the horizon, (r), Y\^I(r) ,\[qhor\] is determined by the following set of stabilization equations: \_[hor]{}-|\_[hor]{} = i Q.\[stabhor\] So we see that the entropy as well as the scalar fields $z^A_{\rm hor}=
Y^A_{\rm hor}/ Y^0_{\rm hor}$ depend only on the magnetic/electric charges $(p^I,q_I)$. It is useful to note that the set of stabilization equations (\[stabhor\]) is equivalent to the minimization of $Z$ with respect to the moduli fields [@FerrKallStro]. As already said, at the horizon $r=0$ full $N=2$ supersymmetry is recovered. Also note that in the same way one can construct more general stationary solutions, such as rotating $N=2$ black holes, multi-centered black holes, TAUB-NUT spaces, etc. [@BehrndtLuestSabra].
As an example, consider a type-IIA compactification on a Calabi-Yau 3-fold. The number of vector superfields is given as $n =h^{(1,1)}$. The prepotential, which is purely classical, contains the Calabi-Yau intersection numbers of the 4-cycles, $C_{ABC}$, and, as $\alpha'$-corrections, the Euler number $\chi$ and the rational instanton numbers $n^r$. Hence the black-hole solutions will depend in general on all these topological quantities [@BCDWKLM]. However, for a large Calabi-Yau volume, only the part from the intersection numbers survives and the $N=2$ prepotential is given by F(Y)=D\_[ABC]{}[Y\^AY\^BY\^CY\^0]{},D\_[ABC]{}=- 16 C\_[ABC]{} . \[cprep\] Based on this prepotential we consider in the following a class of non-axionic black-hole solutions (that is, solutions with purely imaginary moduli fields) with only non-vanishing charges $q_0$ and $p^A$ ($A=1,\dots ,h^{(1,1)}$). So only the harmonic functions $H_0(r)$ and $\tilde H^A(r)$ are nonvanishing. This charged configuration corresponds, in the type-IIA compactification, to the intersection of three D4-branes, wrapped over the internal Calabi-Yau 4-cycles and hence carrying magnetic charges $p^A$, plus one D0-brane with electric charge $q_0$. In the corresponding M-theory picture these black holes originate from the wrapping of three M5-branes, intersecting over a common string, plus an M-theory wave solution (momentum along the common string). For the solution indicated above, the four-dimensional metric of the extremal black-hole solutions is given by [@BehrndtLuestSabra] e\^[-2U(r)]{}=2 . \[IIAmetric\] The scalars at the horizon are determined as z\^A\_[hor]{}=[Y\^A\_[hor]{}Y\^0\_[hor]{}]{},Y\^A\_[hor]{}=12 i p\^A, Y\^0\_[hor]{}=12 ,D=D\_[ABC]{}p\^Ap\^Bp\^C .\[IIAscal\] Finally, the corresponding macroscopic entropy takes the form [@BCDWKLM; @BehCaKaMo] \_[BH]{}=2.\[IIAentropy\]
Let us now turn to the discussion of $N=2$ black-hole solutions in the presence of higher-derivative terms in the $N=2$ supergravity action. In that situation the function $F$ depends on both $X$ and $W^2$ and is still holomorphic and homogeneous of second degree. Just like the chiral superfield, whose lowest component is $X^I$, $W_{{\mu}{\nu}}$ is also a reduced chiral superfield. Therefore it has the same chiral and Weyl weights as $X^I$. Thus, $W^2= W_{{\mu}{\nu}}W^{{\mu}{\nu}}$ is a scalar chiral multiplet with Weyl and chiral weight equal to twice that of the $X^I$. Hence the homogeneity of $F$ implies X\^I F\_I(X,W\^2) + 2 W\^2 [F(X,W\^2) W\^2]{} = 2 F(X,W\^2) . The lowest-$\theta$ component of the superfield $W$ contains the auxiliary tensor field that previously took the form of the graviphoton field strength (up to fermionic terms). However, in the case at hand the Lagrangian is more complicated, so that this tensor can only be evaluated as a power series in terms of external momenta divided by the Planck mass. Similar comments apply to all the auxiliary fields. Nevertheless we can still use the superconformal $N=2$ multiplet calculus [@BergshoeffdeRoodeWit]. Obviously, this case is much more complicated and we do not attempt to give a full treatment of the solutions here. A detailed discussion of these solutions will appear elsewhere. In the following we will instead rely on symplectic covariance to analyze the immediate consequences of the $W$-dependence of $F$ and discuss its implication for the black-hole entropy.
Our strategy will be to expand $F(X, W^2)$ as a power series in $W^2$ as follows, F(X\^I,W\^2)=\_[g=0]{}\^F\^[(g)]{}(X\^I)W\^[2g]{},\[fg\] where $F^{(0)}$ is nothing else than the prepotential discussed before. This expansion will allow us to make contact with the microscopic results of [@msw; @vafa], where a suitable expansion of the microscopic entropy was proposed. [From]{} the fact that the superfield $W$ contains the Weyl tensor at order $\theta^2$, the $W^2$ dependence leads, for instance, to terms proportional to the square of the Weyl tensor and quadratic in the abelian field strengths, times powers of the tensor field $T$. Nevertheless, these terms are all precisely encoded in (\[fg\]).
For type-II compactifications on a Calabi-Yau space, the terms involving $F^{(g)}(X)$ arise at $g$-loop order, whereas in the dual heterotic vacua the $F^{(g)}$ appear at one loop and also contain non-perturbative corrections. In passing we note that the physical couplings are in general non-holomorphic, where the holomorphic anomalies are governed by a set of recursive holomorphic anomaly equations [@AntonGavaNarTayl; @Bershadskyetal; @DWCLMR].
The presence of the chiral background $W^2$ will not modify the special geometry features that we discussed before, provided one now uses the [*full*]{} function $F$ with the $W^2$ dependence included. So, the section $V$, which transforms as a vector under Sp$(2n+2)$, now takes the form V==.\[sectionw\] In order for the Einstein term to be canonical, $V$ has to obey again the symplectic constraint (\[constr\]), so that $\bar X^I F_I(X,W^2)-X^I\bar F_I(\bar X,
\bar W^2)= -i$. Using the U(1)-invariant combinations $Y^I=\bar ZX^I$ and $\bar Z^2W^2$ we can define a U(1)-invariant symplectic vector as =|Z V= =,\[newsecw\] where we used the expansion $F(Y,\bar Z^2W^2)=\sum_{g=0}F^{(g)}(Y)(\bar Z^2
W^2)^g$. The factor $Z$ will again be determined by the stabilization equations and will thus depend on $W^2$. Second, the field strength $G_{\mu\nu I}$, which together with $F_{\mu\nu}^I$ forms a symplectic pair, is still defined by the derivative with respect to the abelian field strength of the full action, and therefore modified by the $W$-dependence of $F$. Prior to eliminating the auxiliary fields, this action is at most quadratic in the field strengths, and $G_{{\mu}{\nu}I}^\pm$ is generally parametrized as G\^+\_[I]{}=|F\_[IJ]{}(|X,|W\^2)F\^[+J]{}\_ + [O]{}\_[ I]{}\^+, G\^-\_[I]{}= F\_[IJ]{}(X, W\^2)F\^[-J]{}\_ + [O]{}\_[I]{}\^- , \[defG\] where ${\cal
O}^\pm_{\mu\nu I}$ represents bosonic and fermionic moment couplings to the vector fields, such that the Bianchi identities and the field equations read $\partial^\nu(F^+- F^-)_{\mu\nu}^I = \partial^\nu(G^+- G^-)_{\mu\nu I} =0$. Again, it is crucial to include the full dependence on the Weyl multiplet, also in the tensors $G^\pm_{\mu\nu I}$ and ${\cal
O}^\pm_{\mu\nu I}$. The reason is that the symplectic reparametrizations are linked to the full equations of motion for the vector fields (which involve the Weyl multiplet) and not to (parts of) the Lagrangian. The modification of the field strength $G_{\mu\nu
I}$ can be interpreted as having the effect that the electric charges $q_I$ (which, together with the magnetic charges $p^I$ form the symplectic charge vector $Q$) get modified in the chiral $W^2$ “medium", compared to their original “microscopic” values $q^{(0)}_I$. Below we will comment on the relevance of this observation. Extremal $N=2$ black-hole solutions in the presence of higher-derivative interactions must again preserve half the supersymmetries, except for $r=0,\infty$, but obviously this condition is now much harder to solve. Nevertheless it is possible to show that the (tangent-space) derivative of the moduli with respect to the radial variable is still vanishing at the horizon, indicating the expected fixed-point behaviour. Rather than solving the equations for the full black-hole solution, we impose the stabilization equations. For the metric we make again the ansatz (\[metrican\]) where $e^{-2U}$ takes the same form in terms of $\Pi$, which now incorporates the modifications due to the background, e\^[-2U]{}&=&Z|Z=i(|Y\^IF\_I(Y,|Z\^2W\^2)-Y\^I|F\_I(|Y, Z\^2|W\^2)) .\[metricw\] The stabilization equations now read = i = i + [ir]{} ,\[solutw\] where the harmonic functions characterize the values of the field strengths according to (\[gaugeh\]), just as before, except that the field strength $G_{{\mu}{\nu}I}$ incorporates the modifications due to the background.
We are now particularly interested in the behavior of our solution at the horizon $r\rightarrow 0$, that is we would like to compute the corrected expression for the black-hole entropy. Without any $W$-dependence in $F$, previous calculations show that the quantity ${\bar Z}^2 W^2=\bar Z^2(T_{\mu\nu}^-T^{-\mu\nu})$ has the following behaviour near the horizon: \^2 W\^2 = + [O]{}(r\^0) .\[zerow\] Equation (\[zerow\]) could get modified when $F$ depends on $W$. These corrections should then be viewed as the back reaction of the non-trivial $W^2$-background on the black-hole solution. We will work to leading order and therefore assume that possible corrections to (\[zerow\]) can be neglected to that order. Thus, we will simply use equation (\[zerow\]) at the horizon, so that the modified $N=2$ black-hole entropy is then given by: \_[BH]{}&=&(r\^2e\^[-2U]{})\_[r=0]{}= (r\^2Z|Z)\_[r=0]{}\
& =& i ( [|Y]{}\_[hor]{}\^I F\_I(Y\_[hor]{}, 1) - Y\^I\_[hor]{} [|F]{}\_I([| Y]{}\_[hor]{}, 1) ) . \[entro\] As before, the symplectic vector $\Pi_{\rm hor}$ is given as in equation (\[stabhor\]), where now, however, $\Pi_{\rm hor}$ has a non-trivial dependence on the $W^2$-background. So we have Y\^I\_[hor]{}-[|Y]{}\^I\_[hor]{}=i p\^I, F\_I(Y\_[hor]{},1)-|F\_I([|Y]{}\_[hor]{},1)= iq\_I . \[stabilw\] Consequently, the modified black hole entropy only depends again on the magnetic/electric charges.
For concreteness, let us again discuss the type-IIA compactification on a Calabi-Yau 3-fold in the limit of large radii, which amounts to suppressing all $\alpha'$-corrections. We are, in particular, interested in the contribution from $F^{(1)}$, which arises at one loop in the type-IIA string. This term is of topological origin; it is related to a one-loop $R^4$ term in the ten-dimensional effective IIA action [@Mina]. For the function $F(Y,\bar Z^2W^2)$ we thus take F(Y, [|Z]{}\^2 W\^2) = F\^[(0)]{}(Y) + F\^[(1)]{}(Y) |Z\^2 W\^2 = D\_[ABC]{} -1[24]{} c\_[2A]{} [|Z]{}\^2 W\^2 .\[ourf0f1\] Here the $c_{2A}$ are the second Chern class numbers of the Calabi-Yau 3-fold. For simplicity we will consider again axion-free black holes with $p^0 =0, q_A =0$. Then the stabilization equations (\[stabilw\]) have the solution Y\^A\_[hor]{} = 12 [i]{} p\^A , Y\^0\_[hor]{} = . \[yhorf\] Inserting (\[yhorf\]) into (\[entro\]) yields \_[BH]{} = 2 .\[wrong\] Equation (\[wrong\]) is only to be trusted to linear order in $c_{2A}$, because we have not included the back reaction. Expanding (\[wrong\]) to lowest order in $c_{2A}$ yields \_[BH]{} = 2 + 2 ( - ) c\_[2A]{} p\^A + . \[mentro\] The terms linear in $c_{2A}$ thus cancel out! Thus, when expressed in terms of the charges $Q=(p^I,q_I)$, there is no correction to ${\cal S}_{\rm BH}$ to lowest order in $c_{2A}$!. Although this is a rather striking result, whose significance is not quite clear to us at the moment, it seems to disagree with the microscopic findings of [@msw; @vafa], as we will now discuss.
We would like to compare the macroscopic entropy formula (\[mentro\]) with the microscopic entropy formula recently computed in [@msw; @vafa] for certain compactifications of M-theory and type-IIA theory on Calabi-Yau 3-folds. In [@msw; @vafa] the microscopic entropy was, to all orders in $c_{2A}$, found to be \_[micro]{}=2 .\[malmicro\] Expanding (\[malmicro\]) to lowest order in $c_{2A}$ yields \_[micro]{}=2+ 2 1[12]{}c\_[2A]{}p\^A + .\[entr1msw\] The correction in (\[entr1msw\]) was then matched [@msw; @vafa] with a correction to the effective action involving $R^2$-type terms with a coefficient function $F^{(1)}$.
The approach used above to obtain the macroscopic entropy (\[mentro\]) is different from the approach used in [@msw; @vafa] for obtaining the macroscopic formula. A comparison of the results of both approaches, that is of (\[entr1msw\]) and (\[mentro\]), indicates that in order to obtain matching of both results, the macroscopic electric charge $q_0$ appearing in (\[mentro\]) cannot be identical to the electric $q_0$ appearing in (\[entr1msw\]). This is one way of explaining the discrepancy. If we denote the charge $q_0$ appearing in (\[entr1msw\]) by $q_0^{(0)}$, then matching of (\[entr1msw\]) and (\[mentro\]) can be achieved provided that the $q_0$ appearing in (\[mentro\]) is related to $q_0^{(0)}$ as follows: q\_0 = q\_0\^[(0)]{} (1+[c\_[2A]{}p\^A6D]{}) . \[mimaq\] As alluded to earlier, this can be interpreted as a modification of the “microscopic” charge $q_0^{(0)}$ due to the $W^2$ medium. This interpretation can be further motivated by noting that if one defines $q_0^{(0)}$ to be given by q\_I\^[(0)]{} -i ( F\_I\^[(0)]{}(Y\_[hor]{}) - [|F]{}\_I\^[(0)]{}([|Y]{}\_[hor]{}) )= q\^I + i \_[g 1]{} ( F\_I\^[(g)]{}(Y\_[hor]{}) - [|F]{}\_I\^[(g)]{}([|Y]{}\_[hor]{}) ) , \[qmicro\] then insertion of (\[yhorf\]) into (\[qmicro\]) precisely yields (\[mimaq\]). Equation (\[qmicro\]) makes it clear that, in the presence of a $W^2$ medium, the charges $q_0$ and $q_0^{(0)}$ cannot be identical. Since the “microscopic” charge $q_0^{(0)}$ has the interpretation of quantized momentum around a circle [@msw], it is integer valued. We thus note that the macroscopic charge $q_0$ given in (\[mimaq\]), which is measured at spatial infinity, is not integer valued in the presence of a $W^2$ medium.
Given that the equations (\[entr1msw\]) and (\[mentro\]) agree (provided the relation (\[mimaq\]) holds), it is then conceivable that the full macroscopic entropy, derived from the metric given in (\[metricw\]), also agrees with the full microscopic entropy (\[malmicro\]). In order to calculate the full macroscopic entropy from (\[metricw\]), one will have to take into account a possible back reaction of the non-trivial $W^2$ background on the black hole solution, in particular the corrections to the quantity $(\bar Z^2W^2)$ at the horizon (\[zerow\]). In the presence of such corrections, the quantity $(\bar Z^2W^2)_{\rm hor}$ will presumably also depend on the magnetic/electric charges, that is $(\bar Z^2W^2)_{\rm hor}={f(p,q)\over r^2}$. In addition the higher functions $F^{(g)}$ ($g>1$) might also contribute to the entropy. Finally, even non-holomorphic corrections to the higher-derivative effective action might play a role. At the end, let us note that if one inserts (\[mimaq\]) into (\[malmicro\]), the microscopic entropy takes again the very simple form: \_[micro]{}=2. So with this charge “renormalization" one rediscovers the zero-th order entropy (\[IIAentropy\]). It is tempting to speculate that this feature is related to the enhancement of the supersymmetry at the horizon, namely to the fact that at the horizon we still have an $AdS_2\times S^2$ geometry, even after including all higher-derivative terms.
[**Acknowledgements**]{}
This work is supported by the Deutsche Forschungsgemeinschaft (DFG) and by the European Commission TMR programme ERBFMRX-CT96-0045 in which Humboldt-University Berlin and Utrecht University participate. W.A.S is partially supported by DESY-Zeuthen.
[**Note Added**]{}
In [@wald] it has been pointed out that there are modifications to the Bekenstein-Hawking entropy formula in the presence of higher curvature terms. In this paper we have not considered such modifications. It is not clear at present what their contribution to our result is. We would like to thank Serge Massar for raising this issue and for bringing references [@wald] to our attention. B. d. W. thanks Robert Myers for useful discussions regarding this topic.
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address: '\[0pt\]\[0pt\][Fakultät für Mathematik, Universität Wien, Vienna, Austria]{}'
author:
- 'Alexander R. Miller'
title: Note on parity and the irreducible characters of the symmetric group
---
1em
Introduction {#Introduction:Section .unnumbered}
============
The object of this short note is to prove a theorem and present a conjecture for the number of even entries in the character table of the symmetric group $S_n$.
\[Main:Theorem\] The number of even entries in the character table of $S_n$ is even.
\[Main:Conjecture\] The proportion of the character table of $S_n$ covered by even entries tends to $1$ as $n\to \infty$.
Theorem \[Main:Theorem\] is proved in Section \[Theorem:Section\]. Conjecture \[Main:Conjecture\] is discussed in Section \[Conjectures:Section\]. To support Conjecture \[Main:Conjecture\] we write down in Table \[EO:Table\] the number of even entries and odd entries in the character table of $S_n$ for $1\leq n\leq 76$. See Figure \[Figure:Plot\]. Another table (Table \[Other:Table\]) in Section \[Conjectures:Section\] suggest a more general phenomenon.
\[General:Conjecture\] The proportion of the character table of $S_n$ covered by entries divisible by a given prime number $p$ tends to $1$ as $n\to\infty$.
(0,0-.004)–(0,1)–(2+.01,1)–(2+.01,0-.004)–cycle; iin [2,4,6,8]{} (0,i/10) – (.03,i/10); iin [2,4,6,8]{} at (0-.05,i/10) [**0.i**]{}; at (0-.05,1) [**1.0**]{}; at (0-.05,0-.002) [**0.0**]{}; iin [10,20,...,76]{} (2\*i/76,0-.004)–(2\*i/76,.03-.004); iin [0,10,...,76]{} at (2\*i/76,0-.004-.05) [**i**]{}; /in [ 1/ 0.000000, 2/ 0.000000, 3/ 0.222222, 4/ 0.240000, 5/ 0.326530, 6/ 0.363636, 7/ 0.400000, 8/ 0.549587, 9/ 0.564445, 10/ 0.547619, 11/ 0.581633, 12/ 0.598414, 13/ 0.597392, 14/ 0.635720, 15/ 0.621578, 16/ 0.702611, 17/ 0.701470, 18/ 0.695632, 19/ 0.711920, 20/ 0.729827, 21/ 0.727850, 22/ 0.756931, 23/ 0.755566, 24/ 0.761004, 25/ 0.766302, 26/ 0.776543, 27/ 0.773131, 28/ 0.791316, 29/ 0.785669, 30/ 0.791326, 31/ 0.790687, 32/ 0.808667, 33/ 0.803730, 34/ 0.810735, 35/ 0.811763, 36/ 0.815064, 37/ 0.815565, 38/ 0.824422, 39/ 0.822188, 40/ 0.827024, 41/ 0.827150, 42/ 0.832165, 43/ 0.830679, 44/ 0.837467, 45/ 0.835640, 46/ 0.839228, 47/ 0.839611, 48/ 0.844193, 49/ 0.843245, 50/ 0.847102, 51/ 0.847389, 52/ 0.849612, 53/ 0.850410, 54/ 0.853485, 55/ 0.853151, 56/ 0.855968, 57/ 0.856350, 58/ 0.858603, 59/ 0.858999, 60/ 0.860868, 61/ 0.860969, 62/ 0.862982, 63/ 0.863306, 64/ 0.865487, 65/ 0.865570, 66/ 0.866821, 67/ 0.867150, 68/ 0.869040, 69/ 0.869127, 70/ 0.870139, 71/ 0.870653, 72/ 0.871719, 73/ 0.871960, 74/ 0.873174, 75/ 0.873273, 76/ 0.874223 ]{} ([(2\*)/76]{}, ) circle (0.005);
Proof of Theorem \[Main:Theorem\] {#Theorem:Section}
=================================
Let $p_n$ be the number of partitions of $n$. Here a partition of $n$ is a sequence of positive integers $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ such that $\lambda_1\geq \lambda_2\geq \ldots\geq \lambda_\ell$ and $\lambda_1+\lambda_2+\ldots+\lambda_\ell=n$. The conjugate of $\lambda$ is the partition $\lambda'$ whose parts are $\lambda_i'=\#\{j : i\leq \lambda_j\}$ for $1\leq i\leq \lambda_1$. Conjugation is the involution ${\lambda\mapsto \lambda'}$. The fixed points of this involution are self-conjugate partitions. Self-conjugate partitions $\lambda$ of $n$ are in one-to-one correspondence with partitions $\mu$ of $n$ into odd distinct parts via $\lambda\mapsto \mu$ where $\mu_i=2(\lambda_i-i)+1$ for $i$ such that $1\leq i\leq \lambda_i$.
Let $O_n$ be the number of odd entries in the character table of $S_n$. Then $$\label{Odd:Eq}
O_n\equiv \sum_g\sum_{\chiup}{\chiup}(g)\equiv \sum_g\sum_{\chiup}{\chiup}(g)^2\pmod{2}$$ where the two outer sums run over a set of representatives $g$ for the conjugacy classes and the two inner sums run over the irreducible characters ${\chiup}$. Moreover one of the orthogonality relations [@F] tells us that $$\label{Orth:Eq}
\sum_{{\chiup}}{\chiup}(g)^2=1^{m_1}m_1!2^{m_2}m_2!\ldots n^{m_n}m_n!$$ for $m_p$ the number of cycles of period $p$ in the cycle decomposition of $g$. Together and imply $$O_n\equiv OD_n \pmod{2}$$ where $OD_n$ is the number of partitions of $n$ into odd distinct parts. Let $SC_n$ be the number of self-conjugate partitions of $n$ so that $SC_n=OD_n$ and hence $$\label{O:SC:Eq}
O_n\equiv SC_n\pmod{2}.$$
Let $E_n$ be the number of even entries in the caracter table of $S_n$. Then $$\label{O:E:Eq}
O_n+E_n=p_n^2\equiv p_n\pmod{2}.$$ Together and imply $$E_n\equiv p_n-SC_n\pmod{2}.$$ But $p_n-SC_n\equiv 0\pmod{2}$ because conjugation restricts to a fixed-point-free involution on the set of non-self-conjugate partitions of $n$.
Remarks and some tables {#Conjectures:Section}
=======================
This section contains some tables and remarks. The main object is Table \[EO:Table\] for the number of even entries in the character table of $S_n$.
Remarks {#Remarks:Section}
-------
Let ${\chiup}(\mu)$ be short for the constant value ${\chiup}(g)$ of the irreducible character ${\chiup}$ of $S_n$ on the class consisting of all permutations $g\in S_n$ for which the periods of the disjoint cycles form the partition $\mu$.
###
In terms of the probability that an entry ${\chiup}(\mu)$ is even when chosen uniformly at random from the character table of the symmetric group $S_n$ Conjecture \[Main:Conjecture\] says $${\rm Prob}(\,{\chiup}(\mu)\text{ is even}\,)\to 1\text{ as }n\to\infty.$$ This parity bias becomes even more striking when compared with the distribution of signs in the character table of $S_n$ (cf. [@M Question 3]). See Figure \[Figure:PM:Plot\] and Table \[PMZ:Table\].
${\rm Prob}(\,{\chiup}(\mu)>0 \mid {\chiup}(\mu)\neq 0\,)\to 1/2$ as $n\to\infty$.
(0,0-.004)–(0,1)–(2+.01,1)–(2+.01,0-.004)–cycle; iin [0.25,0.5,0.75]{} (0,i) – (.03,i); iin [0.25,0.5,0.75]{} at (0-.05,i) [**i**]{}; at (0-.05,1) [**1.0**]{}; at (0-.05,0-.002) [**0.0**]{}; iin [5,10,...,38]{} (4\*i/76,0-.004)–(4\*i/76,.03-.004); iin [0,5,...,38]{} at (4\*i/76,0-.004-.05) [**i**]{}; /in [ 1 / 1.00000, 2 / 0.750000, 3 / 0.750000, 4 / 0.666667, 5 / 0.666667, 6 / 0.630435, 7 / 0.576470, 8 / 0.586102, 9 / 0.580101, 10 / 0.554421, 11 / 0.550047, 12 / 0.546094, 13 / 0.533494, 14 / 0.535560, 15 / 0.524393, 16 / 0.521590, 17 / 0.520837, 18 / 0.518988, 19 / 0.513955, 20 / 0.512761, 21 / 0.511580, 22 / 0.510413, 23 / 0.508808, 24 / 0.507993, 25 / 0.506464, 26 / 0.506354, 27 / 0.505301, 28 / 0.504786, 29 / 0.504102, 30 / 0.503858, 31 / 0.503163, 32 / 0.503088, 33 / 0.502617, 34 / 0.502368, 35 / 0.502168, 36 / 0.501929, 37 / 0.501712, 38 / 0.501601 ]{} ([(4\*)/76]{}, ) circle (0.0075); /in [ 1 / 0.000000, 2 / 0.250000, 3 / 0.250000, 4 / 0.333333, 5 / 0.333333, 6 / 0.369565, 7 / 0.423530, 8 / 0.413898, 9 / 0.419899, 10 / 0.445578, 11 / 0.449953, 12 / 0.453907, 13 / 0.466506, 14 / 0.464440, 15 / 0.475607, 16 / 0.478410, 17 / 0.479163, 18 / 0.481013, 19 / 0.486044, 20 / 0.487239, 21 / 0.488420, 22 / 0.489587, 23 / 0.491192, 24 / 0.492007, 25 / 0.493536, 26 / 0.493646, 27 / 0.494699, 28 / 0.495214, 29 / 0.495897, 30 / 0.496142, 31 / 0.496837, 32 / 0.496912, 33 / 0.497383, 34 / 0.497632, 35 / 0.497831, 36 / 0.498070, 37 / 0.498290, 38 / 0.498398 ]{} ([(4\*)/76]{}, ) circle (0.0075); (0,0.5)–(2+.01,0.5);
###
Conjecture \[General:Conjecture\] implies that for any integer number $d$ one has $${\rm Prob}(\,{\chiup}(\mu)\equiv 0\,\, ({\rm mod}\ d)\,)\to 1\text{ as }n\to\infty.$$ Figure \[Figure:Plot\] suggests that there is a sharper statement. See for example Figure \[Figure:Comp\].
(0,0-.004)–(0,1)–(2+.01,1)–(2+.01,0-.004)–cycle; iin [2,4,6,8]{} (0,i/10) – (.03,i/10); iin [2,4,6,8]{} at (0-.05,i/10) [**0.i**]{}; at (0-.05,1) [**1.0**]{}; at (0-.05,0-.002) [**0.0**]{}; iin [10,20,...,76]{} (2\*i/76,0-.004)–(2\*i/76,.03-.004); iin [0,10,...,76]{} at (2\*i/76,0-.004-.05) [**i**]{}; /in [ 2/ 0.000000, 3/ 0.222222, 4/ 0.240000, 5/ 0.326530, 6/ 0.363636, 7/ 0.400000, 8/ 0.549587, 9/ 0.564445, 10/ 0.547619, 11/ 0.581633, 12/ 0.598414, 13/ 0.597392, 14/ 0.635720, 15/ 0.621578, 16/ 0.702611, 17/ 0.701470, 18/ 0.695632, 19/ 0.711920, 20/ 0.729827, 21/ 0.727850, 22/ 0.756931, 23/ 0.755566, 24/ 0.761004, 25/ 0.766302, 26/ 0.776543, 27/ 0.773131, 28/ 0.791316, 29/ 0.785669, 30/ 0.791326, 31/ 0.790687, 32/ 0.808667, 33/ 0.803730, 34/ 0.810735, 35/ 0.811763, 36/ 0.815064, 37/ 0.815565, 38/ 0.824422, 39/ 0.822188, 40/ 0.827024, 41/ 0.827150, 42/ 0.832165, 43/ 0.830679, 44/ 0.837467, 45/ 0.835640, 46/ 0.839228, 47/ 0.839611, 48/ 0.844193, 49/ 0.843245, 50/ 0.847102, 51/ 0.847389, 52/ 0.849612, 53/ 0.850410, 54/ 0.853485, 55/ 0.853151, 56/ 0.855968, 57/ 0.856350, 58/ 0.858603, 59/ 0.858999, 60/ 0.860868, 61/ 0.860969, 62/ 0.862982, 63/ 0.863306, 64/ 0.865487, 65/ 0.865570, 66/ 0.866821, 67/ 0.867150, 68/ 0.869040, 69/ 0.869127, 70/ 0.870139, 71/ 0.870653, 72/ 0.871719, 73/ 0.871960, 74/ 0.873174, 75/ 0.873273, 76/ 0.874223 ]{} ([(2\*)/76]{}, ) circle (0.005); in [2.0,2.2,...,75.8]{} ([(2\*)/76]{},[(2/3.1459)\*rad(atan((/2)\^(1/2)-1))]{}) – ([(2\*(+0.2))/76]{},[(2/3.1459)\*rad(atan(((+0.2)/2)\^(1/2)-1))]{});
Tables {#Table:Section}
------
$n$ [no. of evens]{} [no. of odds]{}
------ ------------------ -----------------
$1$ $0$ $1$
$2$ $0$ $4$
$3$ $2$ $7$
$4$ $6$ $19$
$5$ $16$ $33$
$6$ $44$ $77$
$7$ $90$ $135$
$8$ $266$ $218$
$9$ $508$ $392$
$10$ $966$ $798$
$11$ $1824$ $1312$
$12$ $3548$ $2381$
$13$ $6094$ $4107$
$14$ $11586$ $6639$
$15$ $19254$ $11722$
$16$ $37492$ $15869$
$17$ $61876$ $26333$
$18$ $103110$ $45115$
$19$ $170932$ $69168$
$20$ $286916$ $106213$
$21$ $456554$ $170710$
$22$ $759962$ $244042$
$23$ $1190034$ $384991$
$24$ $1887766$ $592859$
$25$ $2937820$ $895944$
$26$ $4608084$ $1326012$
$27$ $7004646$ $2055454$
$28$ $10938762$ $2884762$
$29$ $16372732$ $4466493$
$30$ $24851432$ $6553384$
$31$ $37014368$ $9798596$
$32$ $56368810$ $13336991$
$33$ $82688102$ $20192347$
$34$ $122855526$ $28680574$
$35$ $179808396$ $41695293$
$36$ $263406424$ $59766105$
$37$ $381814902$ $86344867$
$38$ $557951490$ $118828735$
: Number of even entries and number of odd entries in the character table of $S_n$ for $1\leq n\leq 76$.[]{data-label="EO:Table"}
$n$ [no. of evens]{} [no. of odds]{}
------ ------------------ ------------------
$39$ $799580980$ $172923245$
$40$ $1152977342$ $241148902$
$41$ $1644080076$ $343563813$
$42$ $2352923494$ $474550782$
$43$ $3324344208$ $677609913$
$44$ $4732761850$ $918518775$
$45$ $6639049122$ $1305820834$
$46$ $9351080036$ $1791411328$
$47$ $13067332410$ $2496228106$
$48$ $18309958344$ $3379378185$
$49$ $25390864566$ $4720061059$
$50$ $35331180090$ $6377078986$
$51$ $48786461562$ $8786181687$
$52$ $67367826002$ $11924538919$
$53$ $92571070272$ $16283394489$
$54$ $127268025536$ $21847658489$
$55$ $173744388742$ $29905639434$
$56$ $237567368138$ $39975105191$
$57$ $323002974632$ $54182161084$
$58$ $439208932802$ $72330715598$
$59$ $594363393060$ $97561119340$
$60$ $804101537262$ $129956924827$
$61$ $1082902860136$ $174870604889$
$62$ $1458789177232$ $231616447104$
$63$ $1956705210484$ $309822028517$
$64$ $2625259647972$ $408015408928$
$65$ $3505898738012$ $544490965352$
$66$ $4679753246976$ $718991943424$
$67$ $6226771093726$ $953962042995$
$68$ $8285512851154$ $1248594579071$
$69$ $10979998587386$ $1653369791639$
$70$ $14541318538948$ $2170163830076$
$71$ $19209876952108$ $2853857859917$
$72$ $25351409083192$ $3730699401897$
$73$ $33363529811282$ $4899218593439$
$74$ $43886589872232$ $6374420377768$
$75$ $57554118617836$ $8352091755860$
$76$ $75434276878574$ $10852934727707$
: Number of even entries and number of odd entries in the character table of $S_n$ for $1\leq n\leq 76$.[]{data-label="EO:Table"}
$n$ pos. neg.
----- ------- -------
1 1 0
2 3 1
3 6 2
4 14 7
5 26 13
6 58 34
7 98 72
8 194 137
9 344 249
10 652 524
11 1165 953
12 2020 1679
13 3552 3106
14 6077 5270
15 10362 9398
16 17080 15666
17 28570 26284
18 46836 43409
19 77045 72861
: Number of positive entries and number of negative entries in the character table of $S_n$ for ${1\leq n\leq 38}$.[]{data-label="PMZ:Table"}
$n$ pos. neg.
----- ----------- -----------
20 122013 115940
21 198461 189476
22 310602 297929
23 494008 476904
24 767237 743094
25 1205391 1174624
26 1828252 1782368
27 2846995 2787256
28 4277605 4196505
29 6520106 6413986
30 9795470 9645485
31 14738493 14553197
32 21750402 21483398
33 32582580 32243250
34 47614253 47165359
35 70213289 69606943
36 102477724 101689585
37 149340038 148321445
38 215267489 213892988
: Number of positive entries and number of negative entries in the character table of $S_n$ for ${1\leq n\leq 38}$.[]{data-label="PMZ:Table"}
$n$ $d=3$ $d=4$ $d=5$ $d=6$ $d=7$
----- -------- -------- -------- -------- --------
1,2 0 0 0 0 0
3 1 1 1 1 1
4 6 4 4 4 4
5 11 12 12 11 10
6 39 30 35 29 29
7 73 61 64 59 63
8 181 187 178 163 168
9 426 368 336 352 339
10 803 681 726 643 660
11 1456 1272 1219 1188 1147
12 3138 2722 2668 2542 2503
13 5289 4532 4359 4135 3989
14 9980 8443 8332 8088 8031
15 16935 14067 14173 13363 13108
16 29669 27733 25351 25171 24066
17 49768 45156 42136 42202 39316
18 88645 77206 72601 73047 68206
19 139983 126447 115972 116635 108050
: Number of entries $\equiv 0\pmod d$ in the character table of $S_n$ for $3\leq d\leq 7$ and $1\leq n\leq 19$.[]{data-label="Other:Table"}
{#section-2 .unnumbered}
[2]{} G. Frobenius, Über die Charaktere der symmetrischen Gruppe. Sitzungsberichte Akad. Berlin (1900) 516–534.
A. R. Miller, The probability that a character value is zero for the symmetric group. *Math. Z.* [**277**]{} (2014) 1011–1015.
|
[ Ángel Paredes$^1$, David Novoa$^2$, Daniele Tommasini$^1$ and Héctor Mas$^1$]{}\
*$^1$ Departamento de Física Aplicada, Universidade de Vigo, As Lagoas s/n, Ourense, ES-32004 Spain;\
$^2$ Max Planck Institute for the Science of Light, Günther-Scharowsky Str. 1, 91058 Erlangen, Germany.*\
.2cm [Corresponding author: [[email protected]]{}]{}
**Abstract**
Introduction
============
Since the invention of chirped pulse amplification [@CPA], the achievable peak intensity of laser light has increased by more than eight orders of magnitude. The current record intensity, achieved at HERCULES few years ago [@HERCULES2008], is $2\times10^{22}$ W/cm$^2$, and it may be improved by an order of magnitude by focusing Petawatt (PW) laser pulses close to the diffraction limit. Such enormous intensities are obtained by squeezing the laser pulses both in space and in time, packing a huge number of photons ($\sim10^{20}$ for a PW laser of duration 30 fs and wavelength $\lambda_0=800$nm) in a volume of the order of a few $\mu$m$^3$.
These new sources of radiation have very relevant implications in many fields, such as charged particle acceleration, fast ignition of fusion targets, laboratory simulation of astrophysical conditions and experimental probing of extreme physical regimes [@mourou06; @qvac-reviews]. In addition, they have been proposed as new tools to test the quantum polarization properties of the vacuum [@PPSVsearch] and search for new physics, such as axion-like or mini charged particles [@new_physics].
Recently, it has been suggested [@pressure] that they can also be used to gauge Extreme High-Vacuum (XHV) [@redhead98; @redhead; @calcatelli], corresponding to pressures $p<10^{-10}$ Pa. Having a set-up without the electric field of usual ionization gauges may be useful to circumvent their limitations—see also [@Chen87; @haffner] for alternative approaches to XHV gauging. This application of ultra intense lasers to vacuum science is of topical interest since the number and availability of such facilities is expected to increase at a very significant rate in the near future. Moreover, many of the experiments that have been proposed to search vacuum polarization effects and new physics at such facilities, as cited above [@PPSVsearch; @new_physics], require the generation and calibration of XHV to control the background noise stemming from the interaction of the laser pulse with the classical vacuum and compute the final sensitivities for the signal. The fact that this can be done using the ultra intense laser itself is a most welcome result.
The idea behind this proposal is fairly simple: photons from the laser pulse are scattered by electrons in the vacuum chamber. The number of scattered photons is directly proportional to the electron density and therefore to the pressure. Background noise can in principle be kept below the signal by appropriately synchronizing the measurements to the passage of the pulse through the detection region. However, even though the physical principle behind the technique is rather straightforward, the key question to be answered regarding its viability is whether the photon signal is strong enough to be measured. This sets a lower limit for the measurable electron density in a given facility and with a given photon detection system. Here, we provide a complete analysis of the process, computing the expected rates and spectra both for linear and circular polarizations of the laser pulses, taking into account the effect of the time envelope in a slowly-varying envelope approximation. We also design a realistic experimental configuration allowing for the implementation of the idea and compute the corresponding geometric efficiencies. Finally, we develop an optimization procedure for this Photonic Gauge of Extreme High Vacuum at high repetition rate Petawatt and multi-Petawatt laser facilities, such as VEGA, JuSPARC and ELI.
The outline of this work is the following: In section \[sec II\], we use a slowly-varying envelope approximation (in space and time) to compute the average number of nonlinear-Thomson-scattered photons by an electron from an intense Gaussian-shaped pulse of light. In particular, we study the impact of the use of circular polarization and the corrections involved after taking into account the time envelope of the pulses. In section \[sec: collect\], we take into account that in an eventual XHV measurement, only a fraction of the scattered photons may be actually measured and therefore discuss the geometric efficiency in terms of a few simple parameters. This allows to develop an optimization procedure for a Photonic Gauge of XHV. Section \[sec: quant\] is devoted to give some quantitative estimates of the possibilities of detection of scattered photons at present and future PW and multi-PW facilities. Section \[sec: discussion\] addresses several further questions such as the maximum pressure this photonic gauge might potentially handle, the possibility of using table-top high-intensity lasers for the vacuum measurements and the actual spectrum of scattered radiation we might expect in a realistic situation. In section \[sec: conclusions\] we present our conclusions. Some technical details are relegated to two appendices.
Number of scattered photons per pulse {#sec II}
=====================================
The dominant interaction of an ultra-intense beam with an extremely rarefied gas is nonlinear, relativistic, Thomson scattering [@NTS]. In this section, we will use the results of [@sarachik] to estimate the number of scattered photons when a pulse traverses a vacuum chamber in which we assume there is a uniform number of non-relativistic free electrons per unit volume $n_e$. The pulse will be modelled as a standard Gaussian beam in the transverse direction (wavelength $\lambda_0$, beam waist $w_0$, peak intensity $I_0$) $$I=I_0\left(\frac{w_0}{w(z)}\right)^2e^{-\frac{2r^2}{w(z)^2}},
\label{Igauss}$$ with $w(z)=w_0\sqrt{1+z^2/z_R^2}$, where the Rayleigh range is $z_R=\pi\,w_0^2/\lambda_0$. For simplicity, a sharp time envelope of duration $\tau$, such that the pulse energy is given by $E_{pulse}=\tau\,I_0\pi\,w_0^2/2$, will be considered. At the end of the section, the consequences of non-trivial time envelopes will be discussed.
In the following, most of the equations will be given in terms of a dimensionless parameter $q$, related to the intensity $I$ as $$q^2 = \frac{2I\,r_0 \lambda_0^2}{\pi \,m_e c^3},
\label{q2}$$ where $r_0\approx 2.82\times 10^{-15}$m is the classical electron radius and $m_e$ is the electron mass. $q \approx 1$ signals the onset of relativistic effects, while for $q\ll 1$ linear Thomson scattering is a good approximation. In order to catch a glimpse of realistic values at present day facilities, let us consider a one PW peak power infrared pulse with $\lambda_0=800$nm. Taking $w_0=1\mu$m (near to the diffraction limit), we find $I_0 \approx 0.6 \times 10^{23}$ W/cm$^2$ corresponding to $q_0 \approx 170$ whereas for $w_0=20\mu$m, the peak intensity is $I_0 \approx 1.6 \times 10^{20}$ W/cm$^2$ and $q_0 \approx 8.6$.
Introducing dimensionless quantities $\rho=r/w_0$ and $\xi=z/z_R$, we can write the position-dependent value of $q$ for a Gaussian beam as $$q^2 (\rho,\xi)= q_0^2 \frac{1}{1+\xi^2}\exp\left(-\frac{2\rho^2}{1+\xi^2}\right).
\label{qpos}$$
Relativistic Thomson scattering {#sec: RelThom}
-------------------------------
The differential cross section for the relativistic Thomson scattering of plane wave radiation by the electrons of a gas has been computed analytically long ago by Sarachik and Schappert [@sarachik]. The computation neglects quantum effects, $n\,h\,c/\lambda_0 \ll m_e c^2$, where $n$ is the harmonic order, and radiation reaction, $q_0^2 \ll \lambda_0 / r_0$, conditions which are always met at optical frequencies. The results for plane waves can be used in a realistic set-up depending on whether a kind of slowly-varying envelope approximation is sound. This requires the number of optical periods in the pulse to be large, $\tau \gg \lambda_0/c$, and the transverse excursion of the electron to be small compared to the beam radius, i.e. $w_0 \gg q_0 \lambda_0/2\pi$. This latter condition results in $$w_0 \gg \frac{\lambda_0}{\pi}\left(\frac{E_{pulse}r_0}{m_ec^3\tau}\right)^\frac14.
\label{w0cond}$$ In the rest of this subsection, we review part of the results of [@sarachik] are fix the notation.
In the laboratory frame, the power scattered per unit solid angle in harmonic $n$ when a plane wave hits a free electron at rest can be written as $$\frac{dP^{(n)}}{d\Omega}=\frac{e^2 c}{8\epsilon_0\lambda_0^2}f^{(n)}$$ in SI units. The spherical coordinates $\theta\in[0,\pi]$, $\varphi\in[0,2\pi]$ are chosen in such a way that $\theta=0$ corresponds to forward scattering. The form of $f^{(n)}$ depends on the polarization of the beam. Hereafter we will analyze linear and circular polarization. To do so, we define $${\cal M}=1+\frac12 q^2\sin^2\left(\frac{\theta}{2}\right)\,.$$ For linear polarization, the function $f^{(n)}$ reads then $$\begin{aligned}
f_l^{(n)}=\frac{q^2n^2}{{\cal M}^4}
\Bigg[\left( 1-\frac{(1+\frac12 q^2) \cos^2\alpha}{{\cal M}^2} \right)(F_1^n)^2+\nonumber\\
-\frac{q\,\cos\alpha (\cos\theta - \frac12 q^2 \sin^2(\theta/2))}{2{\cal M}^2}
F_1^n F_2^n+\frac{q^2 \sin^2 \theta}{16 {\cal M}^2 }(F_2^n)^2\Bigg],
\label{fn}\end{aligned}$$ where $\cos\alpha=\sin\theta\,\cos\varphi$, the polarization axis corresponds to $\varphi=0,\pi$ and the following functions have been introduced $$\begin{aligned}
F_s^n=\sum_{l=-\infty}^{+\infty}J_l\left(\frac{n\,q^2\sin^2(\theta/2)}{4{\cal M}}\right)\times\nonumber\\
\left[J_{2l+n+s}\left(\frac{q\,n\,\cos\alpha}{{\cal M}}\right)+J_{2l+n-s}\left(\frac{q\,n\,\cos\alpha}{{\cal M}}\right)\right],\nonumber\end{aligned}$$ where the $J_l$ are Bessel functions of the first kind.
The result for circular polarization is $$f_c^{(n)}=\frac{2q^2n^2}{{\cal M}^4}
\Bigg[\frac{2(\cos\theta - \frac{q^2}{2}\sin^2(\theta/2))^2}{q^2\sin^2\theta}
J_n^2(n\,\Theta)+J_n'^2(n\,\Theta)\Bigg],
\label{fncirc}$$ where $$\Theta = \frac{q\,\sin\theta}{\sqrt2\,{\cal M}}.$$
In the laboratory frame, the $n$’th-harmonic frequency is not just a multiple of the incident one. Instead, the following relation holds, $$\lambda^{(n)}={\cal M}\lambda_0/n.
\label{lambdan}$$
All these expressions are valid for electrons initially at rest — equivalent expressions for rapid electrons for linear and circular polarization were computed in [@salamin], [@salamin2]. We stress that even if the electrons reach relativistic velocities while the pulse is passing, they remain slow afterwards, since typically no net energy can be transferred to them [@lawson].
Photons scattered per electron from a plane wave {#sec: Gamma}
------------------------------------------------
Regarding Eq. (\[lambdan\]), the laboratory frame energy of $n$’th harmonic photons is given by $\frac{h\,c\,n}{\lambda_0{\cal M}}$ and thus depends on the intensity of the incident wave and the scattering angle. The number of photons scattered per unit time and per unit solid angle is $\frac{dP^{(n)}}{d\Omega}\frac{{\cal M}\,\lambda_0}{h\,c\,n}$. If a plane wave of duration $\tau$ impinges on a single electron, the number of scattered photons for the $n$’th harmonic is $$N_{\gamma,pw}^{(n)}=\tau \frac{e^2}{8\epsilon_0\lambda_0 h}\Gamma^{(n)}(q),
\label{ngp}$$ with $$\Gamma^{(n)}(q)=\frac{1}{n}\int_0^{2\pi}\int_0^\pi f^{(n)} {\cal M} \sin\theta d\theta d\varphi.
\label{GammaDef}$$
For large $q$, all $\Gamma^{(n)}(q)$ tend asymptotically to some constant. Figure \[figGamma\] shows the results of the numerical integration of the expression in Eq. (\[GammaDef\]) for linear and circular polarization. Linear polarization produces somewhat more $n=1$ photons when $q$ is larger than one, whereas higher harmonics are slightly enhanced by circular polarization.
![ The function $\Gamma^{(n)}(q)$ found by numerical integration for linear (solid lines) and circular (dashed lines) polarization, for n=1,…,4, from top to bottom. []{data-label="figGamma"}](fig1.eps){width="60.00000%"}
For $q\rightarrow 0$, both linear and circular polarizations yield similar values of $\Gamma^{(n)}(q)$. In that limit, the agreement between the curves corresponding to different polarizations is better as $n$ gets reduced (see Fig. \[figGamma\]). This assertion is further confirmed by the lower limit of the analytical representations of $\Gamma^{(n)}(q)$, which are obtained by fitting the curves displayed in Fig. \[figGamma\] to quotients of polynomials (see appendix A).
Photons scattered from a Gaussian pulse {#sec: gaussi}
---------------------------------------
As noted in [@pressure], in order to find the total number of photons scattered from a realistic pulse, it is crucial to take into account its finite transverse profile. The previous results for plane waves are useful since, in the spirit of the slowly-varying envelope approximation introduced in section \[sec: RelThom\], a suitable Gaussian profile can be considered, [*locally*]{}, as plane.
The number of photons scattered from a Gaussian pulse is given by an integral of the plane wave result over the non-trivial profile, namely $N_\gamma^{(n)} = n_e \int N_{\gamma,pw}^{(n)}(q) d^3 \vec x$, where $n_e$ is the number of electrons per unit volume and $N_{\gamma,pw}^{(n)}(q)$ is given in Eq. (\[ngp\]). The parameter $q$ depends on the point of space according to Eq. (\[qpos\]). Using the coordinates $\rho,\xi$ defined at the beginning of this section, we can write: $$N_\gamma^{(n)}= {\cal K}\int_{-\infty}^\infty\int_0^\infty \rho\,\Gamma^{(n)}(q) d\rho d\xi,
\label{Ngamma}$$ where ${\cal K}$ is a dimensionless quantity, $${\cal K}=\frac12 n_e\,c\,\tau\,\pi^2w_0^4\lambda_0^{-2}\alpha,
\label{calK}$$ where $\alpha=e^2/4\pi\,\epsilon_0 \hbar c \approx 1/137$ is the fine structure constant. Notice that $N_\gamma^{(n)}/{\cal K}$ depends only on $n$ and $q_0$, and can be straightforwardly computed numerically.
As shown in [@pressure], $N_\gamma^{(n)}/{\cal K}$ grows as $q_0^3 \sim I_0^\frac32 \sim w_0^{-3}$ for large values of the laser peak intensity. Thus, if the remaining parameters are fixed, the number of scattered photons grows linearly with the beam waist radius $w_0$ (for small enough $w_0$). This behaviour changes for large $w_0$ when the intensities are low so that harmonic production is suppressed. In fact, just considering the asymptotic behaviour $\Gamma^{(n)} \propto q^{2n}$ for small values of $q$, we readily find that the integrand in Eq. (\[Ngamma\]) is proportional to $q_0^{2n} \propto w_0^{-2n}$. Taking into account the factor ${\cal K}$, we conclude that $N_\gamma^{(n)} \propto w_0^{4-2n}$ for large $w_0$. This asymptotic dependence holds independently of the chosen polarization. These rough arguments qualitatively explain the behaviour depicted in Fig. \[figNg\], where a sample numerical computation of the number of scattered photons $N_\gamma^{(n)}$ as a function of the beam waist is shown. In particular, the number of photons produced in the second harmonic $n=2$ tends to a constant value, i.e., they do not display any further dependence on $w_0$ for large waists. For higher harmonics $n\geq3$, the signal drops with increasing $w_0$, as predicted by the analysis developed in this section. The situation for $n=1$ is subtler since the integral in Eq. (\[Ngamma\]) diverges in this case. It is plain that such result is not physical since the scattering region is always limited. For the plots included in Fig. \[figNg\], the integral has been cut at $q=0.01$, the value associated to the barrier suppression regime, below which the electron of a hydrogen atom cannot be approximated as free any more (for $\lambda_0=800$nm) [@barriersupH]. As we can appreciate from the figure, $N_\gamma^{(1)}$ grows monotonically with $w_0$.
![ The number of scattered photons from a Gaussian pulse as a function of the beam waist. The following parameters have been fixed for the laser beam $E_{pulse}=30$J, $\tau=30$fs, $\lambda=800$nm. The pressure has been fixed to $p=0.5\times 10^{-11}$Pa and the temperature to $T=300K$ such that $n_e=2p/k_B T \approx 2.4$mm$^{-3}$, where the factor of 2 comes from considering molecular hydrogen with two electrons per molecule. The y-axis (x-axis) is given in logarithmic (linear) scale. []{data-label="figNg"}](fig2.eps){width="60.00000%"}
From the figure, we might be tempted to conclude that the optimal value of $w_0$ for the measurement of tiny pressures by detecting harmonic-$n$ photons would be the one in which $N_\gamma^{(n)}$ reaches its maximum. Nevertheless, this statement is naive for at least three reasons. First, it could prove costly or unfeasible to manipulate ultra-high power pulses in order to achieve too large waists. Second, the larger the $w_0$, the larger the region in which the vacuum gauging is taking place. In a chamber with differential vacuum, it would be impossible to measure the vacuum confined in a small region if $w_0$ is too large. Third, producing more photons does not mean that a larger signal can be measured. If the region where the scattering is taking place is extensive, it might be impossible to set up an efficient system to collect the emitted photons. We will turn to these questions in section \[sec: collect\].
Non-trivial time envelope {#sec: envelope}
-------------------------
Let us comment on the effect of considering a more realistic non-trivial time envelope instead of a sharp rectangular pulse. We will show that the number of scattered photons does only depend mildly on the envelope and the (simpler) computations of the previous subsections capture the quantitative results up to a factor of order $1$. To take into account the time envelope, we describe the spatio-temporal dependent intensity as $I_{te}=g(\tilde t) I$, where $I$ is given in Eq. (\[Igauss\]) and $t$ should be understood as $\tilde t=t-z/c$. If the time envelope varies mildly within a light cycle ($\frac{dg}{d\tilde t}\ll \omega g$), a slowly-varying envelope approximation is valid [@gibbon] and we may simply use the expression above by including time dependence in $q$. It is useful to define a time envelope correction parameter as the quotient $$\kappa\equiv
\frac{N_\gamma^{(n)}|_{te}}{N_\gamma^{(n)}|_{ss}}=\frac
{\int \rho\,\Gamma^{(n)}\left(\sqrt{g(\tilde t)} q(\rho,\xi)\right) d\tilde t\,d\rho\,d\xi}
{\tau\,\int \rho\,\Gamma^{(n)} \left(q(\rho,\xi)\right) d\rho\,d\xi},$$ where $N_\gamma^{(n)}|_{ss}$ refers to the computation with a sharp step time envelope, as in section \[sec: gaussi\]. Given the form of the time envelope, $\kappa$ only depends on $q_0$ and $n$. As examples, let us consider a Gaussian $g(\tilde t)=e^{-\pi \tilde t^2/\tau^2}$ and a hyperbolic secant $g(\tilde t)=$ sech$(\pi\,\tilde t/\tau)$, chosen such that $g(0)=1$ and $\int_{-\infty}^\infty g(\tilde t)d\tilde t=\tau$. In figure \[fig:te\], the value of $\kappa$ as a function of $q_0$ is plotted for these two time envelopes and for different harmonics, considering linear polarization. The plots for circular polarization are not shown since they are practically coincident with the linear polarization ones.
![ Multiplicative correction due to non-trivial time envelopes to the number of scattered photons as a function of $q_0$. On top, the result for the Gaussian envelope and below for hyperbolic secant envelope. []{data-label="fig:te"}](fig3a.eps){width="60.00000%"}
![ Multiplicative correction due to non-trivial time envelopes to the number of scattered photons as a function of $q_0$. On top, the result for the Gaussian envelope and below for hyperbolic secant envelope. []{data-label="fig:te"}](fig3b.eps){width="60.00000%"}
The conclusion is that the sharp step envelope overestimates the number of scattered photons by a factor of order $1$. The correction factors depend on the shape of the actual envelope, the harmonic number and the peak intensity, with typical values around $0.7$ or $0.8$ (see Fig. \[fig:te\]).
Photon collection and geometric efficiency {#sec: collect}
==========================================
Up to now, we have computed how many photons are scattered from a given pulse traversing a vacuum chamber. In this section, we will discuss how many might be actually measured in a realistic experiment. Since, in any case, the signal from XHV will be very low, it is essential to use single-photon detectors, which can achieve remarkable quantum efficiencies with state-of-art technology [@detectors1; @detectors2]. Typically, the size of the active region of this kind of detectors is of the order of a few microns. Then, since the Thomson scattered photons have a wide angular distribution [@pressure], it is essential to introduce a suitable optical system in order to have efficient photon collection.
The situation is analogous to the detection of Hyper-Rayleigh scattering, a well-established technique for the characterization of nonlinear optical properties of different materials, in particular molecular hyperpolarizabilities [@HRS]: a laser beam traverses the substance to be studied producing anisotropic faint radiation in a multiple of the incident frequency (usually, incoherent second harmonic) which can be collected and measured by an optical system which concentrates part of the emitted light into a photomultiplier. The typical photon collection system — see for instance [@HRS2] — is schematically depicted in Fig. \[fig:hrs\]. As it can be appreciated in the figure, a parabolic mirror captures the photons that are counter-scattered with respect to the position of the single-photon detector, thus enhancing the signal accordingly. An optical system made by filters and lenses allows then to couple most of the scattered light into the detector. A similar arrangement may be used for the measurement of nonlinear Thomson photons in a vacuum chamber.
![ Sketch of a typical scheme for an efficient photon collection system. M: concave mirror, FoV: field of view, FF: frequency filter, LS1, LS2: lens systems, SPD: single-photon detector. []{data-label="fig:hrs"}](fig4.eps){width="0.5\columnwidth"}
The scheme of Fig.\[fig:hrs\] is the simplest one can envision, but it may be possible to upgrade it in order to increase the efficiency and/or anticipate possible problems. One possibility is to include a second device like the one in the figure at a different angle in the transverse plane. Another option is to include a multi-mode optical fiber in order to couple the outcome of the optical system to the photon detector. This would allow to place the photon counter away from the experimental zone in order to shield it from eventual secondary radiation, e.g., X-rays, and to reduce undesired background.
Hereafter, we will assume that the optical system can be parametrized by its [*field of view*]{} (FoV), namely, the size of the $z$-region from which scattered photons can be collected, and its [*numerical aperture*]{} (NA), which provides an angular cut for the photons entering the optical system. We also assume that the [*depth of field*]{} (DOF), namely the size of the region that is transverse to the laser beam in the direction of the photon-collecting lens system, does not restrict the detected signal. This approximation is justified if DOF $\gg w_0$. We stress that it is only necessary to count the number of photons and not to resolve the location where they were originated.
The field of view and geometric efficiency {#sec: FOV}
------------------------------------------
Only a fraction of the photons given in Eq. (\[Ngamma\]) are scattered inside the FoV of the detection system. Assuming that the center of the active detection region coincides with the beam focus, we obtain $$N_{\gamma,FoV}^{(n)}= {\cal K} \int_{-\xi_m}^{\xi_m}
\int_0^\infty \rho\,{\Gamma}^{(n)}(q) d\rho d\xi.
\label{Ngammadr}$$ The integration in $\rho$ is still formally taken up to infinity since typically the beam is concentrated in a submillimeter region in the transverse plane which is assumed to be within the DOF of the collection system in that direction. Our goal in this section is to estimate the geometric efficiency factor associated to the finiteness of the FoV $N_{\gamma,FoV}^{(n)}/N_{\gamma}^{(n)}$ and to discuss the role of $w_0$. Notice that $w_0$ enters the expression in Eq. (\[Ngammadr\]) in three different ways: in the expression of ${\cal K}$ (recall Eq. (\[calK\])), in the value of $q_0$ which affects $q$ through Eq. (\[q2\]) and in the value of the limit value of the integral $\xi_m = z_m / z_R = z_m \lambda_0 /
\pi w_0^2$ where $z_m$ is half the FoV. Two examples of the numerically computed value of $N_{\gamma,FoV}^{(n)}$ as a function of $w_0$ are presented in Fig. \[figNgdr\].
![ The number of scattered photons within the region $z\in (-z_m,z_m)$ from a Gaussian pulse as a function of the beam waist. The parameters of the laser beam and $n_e$ are fixed as in Fig. \[figNg\]. The plot on the left corresponds to $z_m=5$mm and the one on the right to $z_m=20$mm. Again, the solid lines correspond to linear polarization and the dashed lines to circular polarization. The y-axis (x-axis) is given in logarithmic (linear) scale. []{data-label="figNgdr"}](fig5a.eps "fig:"){width="49.00000%"} ![ The number of scattered photons within the region $z\in (-z_m,z_m)$ from a Gaussian pulse as a function of the beam waist. The parameters of the laser beam and $n_e$ are fixed as in Fig. \[figNg\]. The plot on the left corresponds to $z_m=5$mm and the one on the right to $z_m=20$mm. Again, the solid lines correspond to linear polarization and the dashed lines to circular polarization. The y-axis (x-axis) is given in logarithmic (linear) scale. []{data-label="figNgdr"}](fig5b.eps "fig:"){width="49.00000%"}
As compared to Fig. \[figNg\], we can observe that the dependence of $N_{\gamma,FoV}^{(n)}$ on $w_0$ is much milder than that of $N_{\gamma}^{(n)}$. Qualitatively, this can be understood as follows for $n>1$: for moderate values of $w_0$, the integral in Eq. (\[Ngamma\]) is, roughly, proportional to $q_0^3 \propto w_0^{-3}$ (multiplying by the prefactor ${\cal K}\propto w_0^4$, we find $N_{\gamma}^{(n)} \propto w_0$). The reason is that the integral is proportional to the volume (in $\rho-\xi$ coordinates) of the region where significant dispersion takes place. The limiting values of $\xi$ and $\rho$ are proportional to $q_0$ and thus the volume is proportional to $q_0^3$ [@pressure]. When the integral is cut as in Eq. (\[Ngammadr\]) and $\xi_m$ lies within the significant dispersion region, the integral in $\rho$ still picks up a $q_0^2\propto w_0^{-2}$ factor and the integral in $\xi$ is roughly proportional to $\xi_m \propto z_R^{-1} \propto w_0^{-2}$. Thus, the $w_0^4$ in ${\cal K}$ cancels out the $w_0^{-4}$ from the integral and the dependence of $N_{\gamma,FoV}^{(n)}$ on $\omega_0$ is approximately flat. When $w_0$ is very small, however, $\xi_m$ becomes very large meaning that the region where $q$ is large enough to have significant harmonic production is very small and is comprised within $\xi < \xi_m$. Then $N_{\gamma,FoV}^{(n)} \propto w_0$ for small $w_0$. This can be appreciated in the plot of the right in Fig. \[figNgdr\].
Finally, it is interesting to plot the fraction of photons which are indeed scattered within the FoV of the detection system, namely the geometric efficiency associated to the FoV, $$\epsilon_{FoV}^{(n)}=\frac{N_{\gamma,FoV}^{(n)}}{N_{\gamma}^{(n)}}.$$ Notice that this is just the quotient of the quantities plotted in Fig. \[figNgdr\] and Fig. \[figNg\]. A sample computation is displayed in Fig. \[figquotient\]. The fraction is larger for higher harmonics because the scattering region gets more and more concentrated around the beam focus.
![ Fraction of photons scattered within the FoV. The physical parameter are those of Fig. \[figNg\] together with $z_m=5 $mm. The region $w_0< 25\mu$m is enlarged for $n=1$ in the inset. []{data-label="figquotient"}](fig6.eps){width="60.00000%"}
This plot permits to give an order of magnitude for one of the factors entering the geometric efficiency. For instance, for $n=2$, $w_0=15\mu$m with $E_{pulse}=30$J, $\tau=30$fs, $\lambda_0=800$nm, FoV=10mm, a 10% photons are scattered from the region from which they do enter the light-collection system. Obviously, this depends strongly on the FoV itself, see Fig. \[figNgdr\].
For simplicity, up to now we have assumed the center of the photon collection system to be at $\xi=0$. However, as a direct consequence of considering the evolution of the Gaussian beam profile, this is not always the optimal choice. In order to get some qualitative insight, let us model (for $n>1$) $\Gamma^{(n)}(q)\approx b_n \Theta(q-q_{step,n})$ where $b_n, q_{step,n}$ are constants and $\Theta(x)$ is Heaviside function. Then, it is straightforward to check that the quantity $\int_0^\infty \rho\,{\Gamma}^{(n)}(q) d\rho$ has two maxima at $\xi_c \approx \pm \sqrt{\frac{q_0^2}{e\,q_{step,n}^2-1}}$. This simple argument qualitatively captures the behaviour depicted in Fig. \[fig:bump\], which can be found by direct numerical integration.
![ Distribution along the longitudinal direction $z$ of the scattered radiation. The plot has been made taking $E_{pulse}=30$J, $\lambda_0=800$nm, $\tau=30$fs, $w_0=15\mu$m. In order to present all plots in the same graph, the $n=1$ profile was divided by 5. []{data-label="fig:bump"}](fig7.eps){width="60.00000%"}
By placing the detection system around the corresponding $\xi_c$, we obtain $$N_{\gamma,FoV}^{(n)}= {\cal K} \int_{\xi_c-\xi_m}^{\xi_c+\xi_m}
\int_0^\infty \rho\,{\Gamma}^{(n)}(q) d\rho d\xi,$$ which is in general larger than the expression of Eq. (\[Ngammadr\]). Thus, by appropriately displacing the detection system with respect to the beam focus, the geometric efficiency factor can be increased to some extent. We have performed an analysis of an example, using the actual values of $\Gamma^{(n)}(q)$. The results showing the optimal value of $z_c = \xi_c \pi w_0^2 / \lambda_0$ and the increase in the geometric efficiency as compared to Fig. \[figNgdr\] are displayed in Fig. \[fig:displaced\].
![ Left: Optimal value of $z_c$ for the different harmonics as a function of the waist radius. Right: Plot of the factor by which efficiency is increased, as compared to placing the detector centered around $z=0$. Parameters are as in Figs. \[figNg\], \[figquotient\]. []{data-label="fig:displaced"}](fig8a.eps "fig:"){width="49.00000%"} ![ Left: Optimal value of $z_c$ for the different harmonics as a function of the waist radius. Right: Plot of the factor by which efficiency is increased, as compared to placing the detector centered around $z=0$. Parameters are as in Figs. \[figNg\], \[figquotient\]. []{data-label="fig:displaced"}](fig8b.eps "fig:"){width="49.00000%"}
Formally, for $n=1$, the optimal value of $z$ would be $z_c\rightarrow\infty$, since the associated profile asymptotes to a constant in Fig. \[fig:bump\]. However, for convenience, $z_c$ corresponding to $n=1$ will be taken to coincide with that for $n=2$. On the other hand, the discontinuities exhibited by the curves $n=3,4$ in Fig. \[fig:displaced\] can be explained by the following argument. The displacement of the detector is useful to capture one of the bumps of the radiation distribution, as shown in Fig. \[fig:bump\]. However, when the bumps for both positive and negative values of $z$ come close enough and, as a consequence, can be included within the FoV of the detector, the optimal choice is simply to take $z_c=0$. In particular, for a FoV$=10$mm, that would be the case of the $n=4$ lines of Fig. \[fig:bump\].
The optimized geometric efficiency associated to the FoV is then given by the result of Fig. \[figquotient\] multiplied by the enhancement factor that can be achieved by displacing the detector with respect to the focus (see Fig. \[fig:displaced\]). For instance, let us consider a laser featuring $E_{pulse}=30$J, $\tau=30$fs, $\lambda=800$nm $w_0=15\mu$m and a detector with FoV=10mm. For such a system, the efficiency for the collection of $n=2$ photons would then be optimum (e.g., $\epsilon_{FoV}^{(2)}\approx 0.16$ in this particular case) for $z_c=6.85$mm.
Numerical aperture. Angular acceptance {#sec: apert}
--------------------------------------
Among all photons coming from the FoV, only those within certain angular cuts are effectively captured by the collection system. The quantity defining the angular acceptance of the optical system is NA. Assuming that the refractive index of the medium is 1, it is defined as NA $= \sin \tilde \theta_i$, such that the radiation with $\tilde \theta < \tilde
\theta_i$ is measured. $\tilde \theta$ is defined as the angle between the photon direction and the axis joining the scattering region to the center of the optical system and thus is not the $\theta$ used in the previous sections. The goal of this section is to estimate the fraction of scattered photons lying within the NA, thus finding the corresponding factor for the geometric efficiency of the photon collection system.
The number of photons scattered per unit solid angle by a Gaussian beam traversing a vacuum chamber is easily derived from the expressions given in section \[sec II\], $$\frac{dN_\gamma^{(n)}}{d\Omega}={\cal K}\int_{-\infty}^\infty \int_0^\infty
\frac{1}{n}\rho {\cal M} f^{(n)} d\rho d\xi.
\label{angular}$$
The efficiency associated to the numerical aperture would then be $$\epsilon_{NA}^{(n)}=\frac{2\int_{\tilde \theta<\tilde \theta_i}\left(dN_\gamma^{(n)}/d\Omega\right)d\Omega}
{\int_\Omega\left(dN_\gamma^{(n)}/d\Omega\right)d\Omega},
\label{ENA1}$$ where the integral in the denominator is taken over the full solid angle. The factor of 2 in the numerator comes from considering both the integrals in $\tilde \theta \in [0,\tilde\theta_i]$ and in $\tilde \theta \in [\pi-\tilde\theta_i,\pi]$ because of the mirror placed opposite to the detector (see Fig. \[fig:hrs\]).
The expression Eq. (\[ENA1\]) can be directly evaluated numerically in any particular case. In fact, we have verified that the angular distribution of Eq. (\[angular\]) is rather accurately approximated by the low $q$ angular dependence of the integrand. We can then obtain a simple estimate of $\epsilon_{NA}^{(n)}$, depending only on the polarization state, the harmonic number, and the numerical aperture. Defining $$\Xi^{(n)}(\theta,\varphi) \equiv \frac{1}{n}\lim_{q\to 0} \frac{f^{(n)}}{q^{2n}},
\label{Xidef}$$ Eq. (\[ENA1\]) reads $$\epsilon_{NA}^{(n)}\approx\frac{2\int_{\tilde \theta<\tilde \theta_i}\Xi^{(n)}d\Omega}
{\int_\Omega\Xi^{(n)}d\Omega}.
\label{ENA2}$$ The cases of circular and linear polarization are discussed separately below.
### Circular polarization
By expanding Eq. (\[fncirc\]), we find $$\Xi_c^{(n)}=\frac{2^{2-3n}n^{2n-1}}{(n-1)!}(1+\cos^2\theta)(\sin \theta)^{2n-2},
\label{Xicirc}$$ which gives $$\int_\Omega\Xi_c^{(n)}d\Omega = \frac{2^{4-n}\pi(1+n)n^{2n}}{(2n+1)!}.$$ In order to compute the numerator of Eq. (\[ENA2\]), let us define a new set of spherical coordinates obtained by a rotation of $\pi/2$ with respect to the $x$-axis, $$\begin{aligned}
\cos \theta &=& \sin\tilde \theta \sin \tilde \varphi\,\,,\nonumber\\
\tan \varphi &=& - \cot \tilde \theta \sec \tilde \varphi \,.
\label{coord1} \end{aligned}$$ This amounts to placing the detection system ($\tilde \theta=0$) along the $y$-axis. By inserting the expression given in Eq. (\[coord1\]) into Eq. (\[Xicirc\]) and Eq. (\[ENA2\]), estimates for $\epsilon_{NA}^{(n)}$ can be found. For instance, if $NA=0.5$, meaning $\tilde\theta_i = \pi/6$, we obtain $\epsilon_{NA}^{(1)}\approx 0.11$, $\epsilon_{NA}^{(2)}\approx 0.17$, $\epsilon_{NA}^{(3)}\approx 0.20$, $\epsilon_{NA}^{(4)}\approx 0.23$.
### Linear polarization
This case is more complicated than the previous one because of the $\varphi$-dependence of the differential cross section and the cumbersome form of Eq. (\[fn\]). The angle $\beta$ between the polarization direction and the location of the detector has to be properly chosen. All computational details are relegated to appendix B, whereas only the estimates of the geometric efficiency factor related to angular acceptance are quoted here. Assuming NA=0.5, namely $\tilde\theta_i = \pi/6$, they are $\epsilon_{NA}^{(1)}\approx 0.19$, $\epsilon_{NA}^{(2)}\approx 0.19$, $\epsilon_{NA}^{(3)}\approx 0.31$ and $\epsilon_{NA}^{(4)}\approx 0.49$. Notice that they are larger than the efficiencies that can be achieved with circular polarization. This is due to the breaking of the azimuthal symmetry, implying that the distribution of scattered power is more inhomogeneous over the solid angle. We can profit from this fact by suitably choosing the location of the photon collection system.
Summary and an example
----------------------
Let us summarize the main results of section \[sec: collect\]. Once given the characteristics of a laser pulse ($E_{pulse}$, $\tau$, $\lambda_0$) and of a photon collection system (its FoV and NA), the waist radius of the beam and the position of the detector, both in the longitudinal direction $z_c$ and its angular position in the transverse plane $\beta$ can be optimized. In section \[sec: apert\] and appendix B, we give an estimate of the optimal $\beta$ and the efficiency associated to the angular acceptance. Section \[sec: FOV\] discusses how to choose values of $w_0$ and $z_c$ and gives the quantitative results for the geometric efficiency associated to the FoV in a sample case. It does not seem possible to provide a simple estimate for these quantities as a function of all the input parameters.
It is worth mentioning that the full geometric efficiency is not exactly the product of $\epsilon_{FoV}$ and $\epsilon_{NA}$ since, in an actual computation, both cuts should be taken into account simultaneously. However, we have discussed their computations separately for clarity of exposition. In any case, the error we make by splitting the computation in this fashion is not large, although it depends on the particular case. For instance, for the quoted case with linear polarization with $E_{pulse}=30$J, $\tau=30$fs, $\lambda=800$nm, FoV = 10mm, $w_0=15\mu$m, $z_c=6.85$mm, $NA=0.5$, $\beta=0.84$ the efficiencies given above are $\epsilon_{FoV}^{(2)}=0.163$ and $\epsilon_{NA}^{(2)}=0.19$, such that $\epsilon_{FoV}^{(2)}
\epsilon_{NA}^{(2)}=0.031$. This should be compared with the computation including directly in the integrals both cuts which gives $\epsilon_{geom}=0.030$.
Some quantitative estimates {#sec: quant}
===========================
One of the conclusions of the previous sections is that the beam polarization affects only mildly the number of scattered photons, see Figs. \[figNg\]-\[fig:displaced\]. In contrast, it modifies more severely the angular distribution of radiation and, therefore, the number of photons that propagate within the numerical aperture of the detector. This distribution is more inhomogeneous for linear polarization and this fact permits to enhance the geometric efficiency by suitably placing the detection system, see section \[sec: apert\]. In the following we will concentrate on linear polarization. It was shown in [@pressure] that in this case the number of scattered photons per pulse is $N_\gamma^{(n)} \approx c_n {\cal K}\,q_0^3$ where the $c_n$ are coefficients that can be computed numerically, $c_1\approx 275$, $c_2\approx 1.3$, $c_3\approx 0.22$, $c_4\approx 0.088$. This expression is valid for large $q_0$, corresponding to the region of small $w_0$ in Fig. \[figNg\], in which $N_\gamma^{(n)} \propto w_0$. A laser with repetition rate $r_r$ operating for a time interval $\Delta t$ produces $\Delta t\,r_r
$ pulses. Under conditions of XHV, the number of detected photons is proportional to the electron density: $$N_{\gamma,det}^{(n)} = {\cal A}\,n_e.
\label{Ngresult2}$$ The value of the proportionality constant can be found by substituting the values for ${\cal K}$ and $q_0$ in the expression for $N_\gamma^{(n)}$ given above [@pressure], so that $${\cal A} \approx \frac{4c_n}{\pi} (\Delta t\,r_r)
\alpha\,\frac{w_0 \lambda_0 r_0^{3/2}}{(c\,\tau)^{\frac12}}
\left(\frac{E_{pulse}}{m_e c^2}\right)^{\frac32}f\,\kappa\,.
\label{Ngresult3}$$ The parameter $\kappa$ is the correction due to a non-trivial time envelope and will be fixed to a typical value 0.8, see section \[sec: envelope\]. The efficiency factor $f \approx \epsilon_{geom} \epsilon_q \epsilon_\lambda$ is the efficiency factor including the geometric efficiency (see section \[sec: collect\]), the quantum efficiency of the detector and the cuts imposed by the frequency filter.
Furthermore, $n_e$ is proportional to the pressure, $$n_e = \eta\frac{p} {k_B T},
\label{eta}$$ where $\eta$ is the average number of weakly bound electrons per molecule [@pressure] — namely, those in the barrier suppression regime. It depends on the atomic and molecular composition of the remnant gas in the vacuum chamber. In a canonical XHV, its value would be $\eta \approx 2$ since it is mostly composed of hydrogen molecules [@bryant1965; @fernandez2012].
The goal of this section is to provide estimates of these quantities for three PW facilities that will be available in the near future, namely VEGA [@VEGA], JuSPARC [@JuSPARC; @JuSPARC2], and a 10 PW branch of the ELI project [@ELI], which have been chosen because of their sizable repetition rates, see table \[tab:1\].
**Facility** **$P_p$** (PW) **$E_{pulse}$** (J) **$\tau$** (fs) **$\lambda_0$** (nm) **$r_r$** (Hz)
-------------- ---------------- --------------------- ----------------- ---------------------- ----------------
VEGA 1 30 30 800 1
JuSPARC 1.5 45 30 800 1
ELI 10 PW 10 300 30 800 0.1
: A few facilities that will operate in the near future.[]{data-label="tab:1"}
In all cases, we will assume band-pass filters for each harmonic as $\lambda_1$(nm) $\in [800,1200]$, $\lambda_2$(nm) $\in [400,800]$, $\lambda_3$(nm) $\in [267,400]$, $\lambda_4$(nm) $\in [200,267]$ (recall that the photon wavelength is shifted by a $q$-dependent factor, Eq. (\[lambdan\])). This choice should be adjusted for a particular detector and frequency filter, see an enlarged discussion in section \[sec: discussion\]. We consider an experiment running for $\Delta t=$ 1 day. A detection system with FoV = 10mm, NA = 0.5 and an average quantum efficiency of $\epsilon_q=0.25$ within the allowed wavelength bands will be considered. These are sample values intended to be representative and to provide a reasonable estimate for realistic situations.
The results are summarized in table \[tab:2\]. Harmonics $n=1,\dots,4$ are considered in each case, the position of the detector along $z$ is optimized as explained in section \[sec: FOV\] and the angle $\beta$ is chosen in each case as in appendix B. The waist radii, chosen to comply with (\[w0cond\]), are taken to be $w_0=15\mu$m, $w_0=15\mu$m, $w_0=27.5\mu$m for VEGA, JuSPARC and ELI 10, respectively, ($q_0\approx 11.5$, $q_0\approx 14.1$, $q_0\approx 19.8$). The efficiency factors $\epsilon_{geom}$, $\epsilon_\lambda$ and the proportionality factor ${\cal A}$ of equations (\[Ngresult2\]), (\[Ngresult3\]) are found by computing the appropriate numerical integrals.
**Facility** n **$z_c$** (mm) **$\epsilon_{geom}$** **$\epsilon_\lambda$** **${\cal A}$** (mm$^3$)
-------------- --- ---------------- ----------------------- ------------------------ -------------------------
VEGA 1 6.85 $6.5\times 10^{-4}$ 0.94 113
2 6.85 $3.0\times 10^{-2}$ 0.96 25
3 5.2 $7.2\times 10^{-2} $ 0.68 7.6
4 0 $9.6\times 10^{-2}$ 0.16 0.9
JuSPARC 1 8.4 $ 5.4\times 10^{-4}$ 0.94 172
2 8.4 $2.6\times 10^{-2} $ 0.97 41
3 6.4 $6.8\times 10^{-2} $ 0.65 12
4 2.35 $7.2\times 10^{-2}$ 0.19 1.5
ELI 10 1 4.35 $1.2\times 10^{-4}$ 0.95 122
2 4.35 $6.2\times 10^{-3}$ 1.0 32
3 3.3 $1.8\times 10^{-2}$ 0.53 8.4
4 1.25 $1.1\times 10^{-2} $ 0.12 0.46
: Estimates for three future facilities.[]{data-label="tab:2"}
The first observation is that, even if the geometric efficiency is much lower for $n=1$, the majority of photons reaching the detector are of this fundamental harmonic. Nevertheless, the difference is less than one order of magnitude with respect to $n=2$. Gauging the pressure by looking at this second harmonic would have several assets: it would help to avoid possible undesired background of photons from the main beam reaching the detector without having been ’Thomson scattered’ and also to reduce other sources of background such as thermal noise. Moreover, photon detectors typically reach higher quantum efficiencies with smaller dark counts in the visible than in the IR, although that can depend on the detector itself, see [@eisaman] for a review of single photon detectors. Recall that in table \[tab:2\], the same quantum efficiency was assumed in all cases. On the other hand, the separate measurement of [*both*]{} the $n=1$ and $n=2$ harmonics can be used to self-calibrate the procedure.
Let us first estimate the minimum pressure that could be gauged, in principle, in a one day experiment at the three mentioned facilities by detecting the $n=1$ photons. Since extreme vacuum is mostly formed by molecular hydrogen, we take $\eta=2$ in Eq. (\[eta\]). We require that the average number of photons measured in the detection period is at least 10. Then $p_{min}\approx 10 k_B T / (2 {\cal A})$, where the values of ${\cal A}$ are given in table \[tab:2\]. For VEGA, we obtain $p_{min}\approx 1.8\times 10^{-13}$Pa at room temperature $T=300$K or $p_{min}\approx 2.3\times 10^{-15}$Pa at liquid He temperature $T=4$K. For JuSPARC, $p_{min}\approx 1.2\times 10^{-13}$Pa at $T=300$K or $p_{min}\approx 1.5\times 10^{-15}$Pa at $T=4$K. For ELI 10, $p_{min}\approx 1.6\times 10^{-13}$Pa at $T=300$K or $p_{min}\approx 2.2\times 10^{-15}$Pa at $T=4$K. It should be noted that these results may be improved, leading to the possible measurement of even lower pressures, by using a different setup allowing for a greater geometric efficiency. The theoretical limit can be found by multiplying the results of Ref. [@pressure] including the time envelop correction that we have computed above. In any case, the optimization procedure that we have developed above can be straightforwardly generalized to any given geometry.
Let us consider the case in which the second harmonic, $n=2$, is used to gauge the vacuum, and find the estimates of the minimum pressure that could be gauged in one day in the three mentioned facilities. For VEGA, we obtain $p_{min}\approx 8\times 10^{-13}$Pa at room temperature $T=300$K or $p_{min}\approx 1.1\times 10^{-14}$Pa at liquid He temperature $T=4$K. For JuSPARC, $p_{min}\approx 5\times 10^{-13}$Pa at $T=300$K or $p_{min}\approx 0.7\times 10^{-14}$Pa at $T=4$K. For ELI 10, $p_{min}\approx 6.5\times 10^{-13}$Pa at $T=300$K or $p_{min}\approx 0.9\times 10^{-14}$Pa at $T=4$K.
Discussion {#sec: discussion}
==========
In this section, a few interesting questions that have been left out of the general discussion are addressed.
For linear polarization, the trajectory of an electron extracted from an atom by the electromagnetic field passes near the ion during its oscillation, opening the possibility of electron-nucleus recombination with the associated photon emission. This harmonic-generating phenomenon has not been taken into account in the discussion. The safest possibility is to introduce a slight ellipticity in the beam polarization in order to reduce the probability of this circumstance to happen, while keeping nearly unchanged the angular distribution of the Thomson radiation.
We have discussed the minimum pressure that can be gauged in a given situation, associated to having a detectable signal of photons. Another interesting question is which would be the [*maximum*]{} measurable pressure. The method presented in this note would be useful as long as the pressure and the number of scattered photons remain proportional to each other. This can only break down when the density of active electrons is high enough to introduce collective effects. A extremely conservative estimate would be to compare the volume per electron ($n_e^{-1}=k_B T/\eta p$) to the volume of the laser pulse (roughly $\frac{\pi}{2}w_0^2 c\,\tau$). Taking values $w_0=15\mu$m, $\tau=30$fs, $T=300$K, $\eta=2$ gives $p\approx 10^{-6}$Pa. This value is in the so-called high vacuum regime in which pressure can be measured with great precision with standard techniques. In fact, comparing in this regime laser measurements with standard ones would be a valuable benchmark calibration of the method.
It is conceivable to design ultra-high or extreme vacuum gauges using table-top terawatt lasers rather than PW facilities. These could find more applications since the cost of the required device would be orders of magnitude lower. The reduced power would be compensated, at least partially, by larger repetition rates. However, even if the general idea presented here would hold, the actual computations would not. For beams far from the diffraction limit, terawatt lasers would yield $q_0<1$, i.e., intensities out of the relativistic regime. For instance, with $P_p=3$TW, $w_0=15\mu$m, $\lambda_0=800$nm, we obtain $q_0^2\approx 0.4$. Harmonic production would be suppressed and expressions like Eqs. (\[Ngresult2\])-(\[Ngresult3\]) would fail. Moreover, for pulse durations down to the few-cycle limit, the approximation of slowly-varying envelope considered throughout this paper would no longer hold — see for instance [@mackenroth]. The exploration of such limiting case, although interesting, lies beyond the scope of the present work.
Finally, it is worth discussing the wavelength spectrum of the scattered photons. In laboratory frame, the spectral distribution that can be computed with the expression used above is rather broad [@pressure]. It would be further broadened by at least two additional effects which are enhanced for short pulses: the width of the incoming laser pulse itself and the departure from the results of [@sarachik] when the envelope is not slowly-varying [@krafft; @Gao]. The spectra for the different harmonics can be overlapping, producing a sort of supercontinuum. In fact, splitting the results in harmonics is just a convenient computational artifact, while the physical measurable result is the sum of all them. In that sense, the results presented in table \[tab:2\] are lower limits since they only include the first (larger) contribution in the harmonic sum for the different wavelength bands. Notice that this overlap is harmless for the proposed pressure gauge since the total signal remains proportional to the number of scattering electrons. It is obvious that photon detectors with broad efficiency curves would be necessary. Once given a curve in a particular case, the computations shown above can be generalized by properly including it in the integrals, instead of assuming a constant quantum efficiency and a sharp band-pass filter.
Conclusions {#sec: conclusions}
===========
The availability of ultra-short and ultra-intense laser pulses opens the possibility of gauging extreme vacuum pressure by photon counting. The huge photon concentration in these pulses allows to overcome, in the long run, the scantiness of scattering centers in extremely rarefied gases. The shortness of the pulses allows to synchronize the measurements with the pulse passage and to eliminate (or, at least, dramatically reduce) the undesired background by gating in time the signal produced by the photon detectors. Moreover, for the high intensities that can be obtained with focused PW laser beams corresponding to $q \gtrsim 1$, a significant quantity of radiation is non-linearly Thomson-scattered in harmonics $n>1$. The selective detection of only these higher harmonics can be used to significantly reduce any possible background coming from the possible deviations of the original beam from the axially-centered Gaussian distribution. We have considered a typical photon collection system and shown how to optimize the vacuum gauge accuracy by properly placing the detectors. Within this realistic setup, we have obtained optimized geometric efficiencies of the order of a few percent for $n>1$. We have also shown that these results hold for any choice of polarization of the incoming pulse, with numerical variations of the order of the unity. With these assumptions, pressures of the order of $p=10^{-13}-10^{-12}$ Pa at room temperature can be measured in a one-day experiment at VEGA, JuSPARC or ELI 10, assuming that such conditions can be created and maintained during this time. This same procedure can be also applied to more encompassing dispositions of the detectors, that can lead to greater geometrical efficiencies and may eventually allow to lower the limiting pressure that can be achieved.
Upgrading and understanding the classical vacuum may be crucial for experiments trying to explore properties of the quantum vacuum [@qvac-reviews; @PPSVsearch; @new_physics]. Apart from gauging the pressure, nonlinear Thomson scattering might also be useful for beam characterization [@krafft] since its detection can be a probe of the focusing region where it is impossible to introduce any direct characterization system. Har-Shemesh and Di Piazza have proposed to employ it to provide indirect measurements of the peak intensity [@peak] and, in the same spirit, the possibility of studying beam profiles or time envelopes is worth investigating. Hopefully, the computations presented here could be instrumental in this direction.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank M. Büscher, D. González-Díaz, J. Hernández-Toro, J. A. Pérez-Hernández, A. Peralta, L. Roso and C. Ruiz for useful discussions. A.P. is supported by the Ramón y Cajal program. D.T. thanks the InterTech group of Valencia Politechnical University for hospitality during a research visit that was supported by the Salvador de Madariaga program of the Spanish Government. The work of A.P. and D.T. is supported by Xunta de Galicia through grant EM2013/002.
Approximate expressions for the $\Gamma^{(n)}(q)$
=================================================
The functions $\Gamma^{(n)}(q)$ defined in section \[sec: Gamma\] are important tools in the computation of the number of photons scattered by the nonlinear Thomson effect. The formal expressions are rather involved and can only be evaluated numerically. Nevertheless, we have checked that they can be well approximated by simple quotients of polynomials for the values of $n$ considered, see Fig. \[figGamma\]. The error of the approximation is under 1% in most of the range and, in fact, plotting the expressions below in Fig. \[figGamma\] would display lines not distinguishable from the numerical results. Only even powers of $q$ are considered since, formally, $\Gamma^{(n)}(q)=\Gamma^{(n)}(-q)$.
For linear polarization $$\begin{aligned}
\Gamma^{(1)}_l(q)&\approx& \frac{8\pi}{3}\frac{q^2(1+0.414 q^2)}{1+1.33 q^2 + 0.497q^4}\,\,,
\nonumber\\
\Gamma^{(2)}_l(q)&\approx& \frac{7\pi}{5}\frac{q^4(1+0.454 q^2)}{1+2.18 q^2 +
1.63 q^4 + 0.539 q^6}\,\,,
\nonumber\\
\Gamma^{(3)}_l(q)&\approx& \frac{207\pi}{224}\frac{q^6}{1+2.97q^2+1.66 q^4 +1.13 q^6}\,\,,
\nonumber\\
\Gamma^{(4)}_l(q)&\approx & \frac{1081\pi}{1620}\frac{q^8}{1+2.79q^2+ 4.79q^4 + 2.07q^6 + 1.05q^8}\,\,.
\nonumber\end{aligned}$$ For circular polarization $$\begin{aligned}
\Gamma^{(1)}_c(q)&\approx& \frac{8\pi}{3}\frac{q^2(1+0.249 q^2)}{1+1.20 q^2 + 0.370q^4}\,\,,
\nonumber\\
\Gamma^{(2)}_c(q)&\approx& \frac{8\pi}{5}\frac{q^4(1+0.246 q^2)}{1+1.98 q^2 +
1.34 q^4 + 0.330 q^6}\,\,,
\nonumber\\
\Gamma^{(3)}_c(q)&\approx& \frac{81\pi}{70}\frac{q^6(1+0.245 q^2)}{1+2.74q^2+2.94 q^4 +1.44 q^6
+0.305 q^8}\,\,,
\nonumber\\
\Gamma^{(4)}_c(q)&\approx & \frac{512\pi}{567}\frac{q^8}{1+1.17 q^2+ 7.05 q^4 + 1.55 q^6 + 1.16 q^8}\,\,.
\nonumber\end{aligned}$$ We have taken into account that the leading term of all $\Gamma^{(n)}(q)$ for small $q$ is of order $q^{2n}$. Its coefficient can be straightforwardly computed by Taylor expansion and has been inserted in the expressions above. The rest of coefficients have been fitted to the data found from numerical integrals.
Geometric efficiency related to numerical aperture for linear polarization
==========================================================================
It is possible to write down explicit expressions for the $\Xi^{(n)}$ defined in Eq. (\[Xidef\]) by expanding (\[fn\]): $$\begin{aligned}
\Xi_l^{(1)}&=&\sin^2\alpha\,\,,\nonumber\\
\Xi_l^{(2)}&=&\cos^2\alpha(2\sin^2\alpha- \cos\theta) + \frac18 \sin^2\theta
\,\,,\nonumber\\
\Xi_l^{(3)}&=&\frac{27}{64}\Big(\frac14\sin^2\alpha(1-\cos\theta-6\cos^2\alpha)^2+\nonumber\\
&+& \sin^2\theta \cos^2\alpha + \cos \theta \cos^2\alpha (1-\cos\theta-6\cos^2\alpha)
\Big)\,\,,\nonumber\\
\Xi_l^{(4)}&=&\sin^2\alpha \cos^2\alpha (1-\cos\theta-\frac83\cos^2\alpha)^2+\nonumber\\
&-&\frac14 \cos\theta \cos^2\alpha (1-\cos\theta-\frac83\cos^2\alpha)\nonumber \\
&&(1-\cos\theta-8\cos^2\alpha)+ \nonumber\\
&+& \frac{1}{64}\sin^2 \theta (1-\cos\theta-8\cos^2\alpha)^2.\end{aligned}$$ Equivalent expressions for $n=1,2,3$ were given in [@sarachik] with two typos that were corrected in [@castillo].
Since in the case of linear polarization the azimuthal symmetry is broken, it is convenient to choose the angular position of the detection system with respect to the polarization direction in order to optimize the detection of photons. Consider a new set of spherical coordinates by first performing a $\pi/2$-rotation around the $x$-axis followed by a $\beta$-rotation around the new $y$-axis, $$\begin{aligned}
\theta&=&\arccos \left(\sin\tilde \theta \,\sin \tilde \varphi\right)\,\,,\nonumber\\
\varphi &=&- \arctan \left(\frac{\sin\beta \sin\tilde \theta\cos\tilde\varphi+\cos\beta
\cos\tilde \theta}{\cos\beta \sin\tilde \theta\cos\tilde\varphi-\sin\beta
\cos\tilde \theta}\right).\end{aligned}$$ For $\beta=0$, Eq. (\[coord1\]) is recovered. For a given value of $\beta$ and the numerical aperture $\sin \tilde \theta_i$, the efficiency (\[ENA2\]) can be computed. Then, it is straightforward to find the value of $\beta$ which optimizes $\epsilon_{NA}^{(n)}$ and which indicates where the optical system should be placed. The optimal $\beta$ depends on the harmonic number and also on the numerical aperture. For instance, fixing $NA=0.5$, namely $\tilde\theta_i=\pi/6$, we find $\beta=0$ for $n=1$, $\beta\approx 0.84$ for $n=2$ and $\beta=\pi/2$ for $n=3$ and $n=4$. The corresponding values of the efficiency are quoted in section \[sec: apert\] of the main text. The value $\beta=0$ for $n=1$ could be expected since, for linear Thomson scattering, the maximum of the scattered radiation is emitted perpendicular to the polarization direction. Nevertheless, that is not the case for harmonic generation from nonlinear Thomson scattering.
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|
---
author:
- Chris Bowman
- Rowena Paget
title: The uniqueness of plethystic factorisation
---
Introduction {#introduction .unnumbered}
============
Let $s_\lambda$ and $s_\mu$ denote the Schur functions labelled by the partitions ${\lambda}$ and $\mu$. There are three ways of “multiplying" this pair of functions together in order to obtain a new symmetric function; these are the Littlewood–Richardson, Kronecker, and plethysm products. The primary purpose of this paper is to address the most fundamental question one can ask of such a product: [*“does it factorise uniquely?"*]{}. For the Littlewood–Richardson product, this question was answered by Rajan [@Rajan]. We solve this question for the most difficult and mysterious of these products, the plethysm product (which we denote $\circ$) as follows.
Let $\mu,\nu, \pi,\rho$ be arbitrary partitions. If $
s_\nu\circ s_\mu = s_\rho \circ s_\pi
$ then either $\nu=\rho$ and $\mu=\pi$; or we are in one of five exceptional cases, $$\begin{array}{llll}
& s_{(2,1^2)}\circ s_{(1)}= s_{(1^2)}\circ s_{(1^2)},
\quad\quad\quad
& s_{(3,1)}\circ s_{(1)}= s_{(1^2)}\circ s_{(2)},
\\
&s_{(2,1^2)}\circ s_{(2)} =
s_{(1^2)}\circ s_{(3,1)}, &
s_{(2,1^2)}\circ s_{(1^2)} =
s_{(1^2)}\circ s_{(2,1^2)}, \\
&s_\nu \circ s_{(1)}=s_{(1)}\circ s_\nu.
\end{array}$$
In general, the decomposition of a plethysm product will have very, very many constituents. We ask: [*“when is the plethysm product of two Schur functions indecomposable?"*]{}. We prove that in fact such a product is always decomposable, and even inhomogeneous, except for some obvious exceptions. The analogous result for the Kronecker product was obtained by Bessenrodt and Kleshchev [@BK].
Let $\mu,\nu $ be partitions. The product $
s_\nu\circ s_\mu
$ is decomposable and inhomogeneous except in the following exceptional cases: $$s_{(1^2)}\circ s_{(1^2)}= s_{(2,1^2)},
\quad
s_{(1^2)}\circ s_{(2)}= s_{(3,1)},
\quad
s_\nu \circ s_{(1)}=s_\nu=
s_{(1)}\circ s_\nu .$$
Understanding and decomposing the Kronecker and plethystic products of pairs of Schur functions was identified by Richard Stanley as two of the most important open problems in algebraic combinatorics [@Sta00 Problems 9 & 10]. Almost nothing is known about general constituents of plethysm products; however the maximal terms in the dominance ordering are now well-understood [@PW]. Our proof of Theorems A and B proceeds by careful analysis of these maximal terms.
Outside of combinatorics, plethysm products arise naturally in the representation theory of symmetric and general linear groups. In quantum information theory, the positivity of constituents in a plethysm product of two Schur functions is equivalent to the existence of quantum states with certain spectra, margins, and occupation numbers [@MR2421478; @MR2745569]. Decomposing Kronecker and plethystic products of Schur functions is the central plank of Geometric Complexity Theory, an approach that seeks to settle the P versus NP problem [@MR1861288]; this approach was recently shown to require not only knowledge of the positivity but also precise information on the actual multiplicities of the constituents of the products $s_\nu \circ s_\mu$ [@MR3868002].
Partitions, symmetric functions\
and maximal terms in plethysm
================================
We define a [composition]{} $\lambda\vDash n$ to be a finite sequence of non-negative integers $ (\lambda_1,\lambda_2, \ldots)$ whose sum, $|\lambda| = \lambda_1+\lambda_2 + \dots$, equals $n$. If the sequence $(\lambda_1,\lambda_2, \ldots)$ is weakly decreasing, we say that ${\lambda}$ is a [partition]{} and write ${\lambda}\vdash n$. Given $\lambda $ a partition of $n$, the [Young diagram]{} is defined to be the configuration of nodes $$[{\lambda}]=\{(r,c) \mid 1{\leqslant}c{\leqslant}\lambda_r\}.$$ We say that a partition is [linear]{} if it consists only of one row, or one column. The conjugate partition, ${\lambda}^T$, is the partition obtained by interchanging the rows and columns of ${\lambda}$. The number of non-zero parts of a partition, ${\lambda}$, is called its [length]{}, $\ell(\lambda)$; the size of the largest part is called the [width]{}, $w({\lambda})$; the sum of all the parts of $\lambda$ is called its size.
Given two partitions ${\lambda}$ and $\mu$, we let ${\lambda}+ \mu$ and ${\lambda}\sqcup \mu$ denote the partitions obtained by adding the partition horizontally and vertically respectively. In more detail $${\lambda}+\mu=({\lambda}_1+\mu_1,{\lambda}_2+\mu_2,{\lambda}_3+\mu_3,\dots )$$ and ${\lambda}\sqcup\mu$ is the partition whose multiset of parts is the disjoint union of the multisets of parts of ${\lambda}$ and $\mu$. We have that $${\lambda}\sqcup\mu= ({\lambda}^T+\mu^T)^T.$$ Finally we remark that, in this paper, the partition ${\lambda}\sqcup\mu$ is usually equal to $$({\lambda}_1,{\lambda}_2,\dots,{\lambda}_{\ell({\lambda})},\mu_1,\mu_2,\dots,\mu_{\ell(\mu)}).$$ In other words, we often do not need to reorder the multisets of parts — this is simply because ${\lambda}_{\ell({\lambda})}\ge \mu_1$ in most cases.
We now recall the [dominance ordering]{} on partitions. Let ${\lambda},\mu$ be partitions. We write ${\lambda}{\trianglerighteqslant}\mu$ if $$\sum_{1{\leqslant}i {\leqslant}k}{\lambda}_i {\geqslant}\sum_{1{\leqslant}i {\leqslant}k}\mu_i \text{ for all } k{\geqslant}1.$$ If ${\lambda}{\trianglerighteqslant}\mu$ and ${\lambda}\neq \mu$ we write ${\lambda}\rhd \mu$. The dominance ordering is a partial ordering on the set of partitions of a given size. This partial order can be refined into a total ordering as follows: we write ${\lambda}\succ \mu$ if $$\text{ ${\lambda}_k >\mu_k$ for some $k{\geqslant}1$ and }
{\lambda}_i = \mu_i \text{ for all } 1{\leqslant}i {\leqslant}k-1.$$ We refer to $\succ $ as the [lexicographic ordering]{}. We now define the [transpose-lexicographic]{} ordering as follows: $${\lambda}\succ_T\mu \text{ if and only if }
{\lambda}^T\succ \mu^T .$$ We emphasise that this total ordering is not simply the opposite ordering to the lexicographic ordering; minimality with respect to $\succ $ is not equivalent to maximality with respect to $\succ_T$.
Let $\lambda $ be a partition of $n$. A [Young tableau of shape $ \lambda$]{} may be defined as a map ${\mathsf{t}}: [{\lambda}] \to {{\mathbb N}}.$ Recall that the tableau ${\mathsf{t}}$ is [semistandard]{} if ${\mathsf{t}}(r,c-1){\leqslant}{\mathsf{t}}(r,c)$ and ${\mathsf{t}}(r-1,c)< {\mathsf{t}}(r,c)$ for all $(r,c)\in [{\lambda}]$. We let ${\mathsf{t}}_k = |\{ (r,c)\in [{\lambda}] \mid {\mathsf{t}}(r,c)=k\}|$ for $k\in {{\mathbb N}}$. We refer to the composition $\alpha=({\mathsf{t}}_1,{\mathsf{t}}_2,{\mathsf{t}}_3,\dots)$ as the [weight]{} of the tableau ${\mathsf{t}}$. We denote the set of all tableaux of shape ${\lambda}$ by ${{\rm SStd}}_{{\mathbb N}}({\lambda})$, and the subset of those having weight $\alpha$ by ${{\rm SStd}}_{{\mathbb N}}({\lambda},\alpha)$. The [Schur function]{} $s_{\lambda}$, for ${\lambda}$ a partition of $n$, may be defined as follows: $$s_{\lambda}= \sum_{
\begin{subarray}c
\alpha \vDash n
\end{subarray}
}
| {{\rm SStd}}_{{\mathbb N}}({\lambda},\alpha)
| x^\alpha
\qquad
\text{ where} \qquad x^\alpha= x_1^{\alpha_1} x_2^{\alpha_2} x_3^{\alpha_3}\dots .$$
The [plethysm product]{} of two symmetric functions is defined in [@Sta99 Chapter 7, A2.6] or [@MR3443860 Chapter I.8]. The plethysm product of two Schur functions is again a symmetric function and so can be rewritten as a linear combination of Schur functions: $$s_{\nu}\circ s_{\mu}
=\sum_\alpha p(\nu,\mu,\alpha) s_{\alpha}$$ such that $p(\nu,\mu,\alpha) {\geqslant}0$. We say that the product is [homogeneous]{} if there is precisely one partition, $\alpha$, such that $p(\nu,\mu,\alpha) > 0$; we say that the product is [indecomposable]{} if, in addition, $p(\nu,\mu,\alpha) =1$. We now recall the role conjugation – often called the $\omega$ involution – plays in plethysm (see, for example, [@MR3443860 Ex. 1, Chapter I.8]). For $\mu\vdash m$, $\nu \vdash n$, and $\alpha\vdash mn$ we have that $$\label{conjugate}
p( \nu,\mu,\alpha)=
\begin{cases}
p( \nu,\mu^T,\alpha^T) &\text{if $m$ is even}\\
p( \nu^T,\mu^T,\alpha^T) &\text{if $m$ is odd.}
\end{cases}$$ Throughout this paper we shall let $\mu,\nu, \pi,\rho$ be partitions of $m, n, p$ and $q$ respectively. In order to keep track of the effect of this conjugation when comparing products $s_\nu \circ s_\mu $ and $s_\rho \circ s_\pi $, we set $$\nu^M=\begin{cases}
\nu &\text{if $m$ is even}\\
\nu^T &\text{if $m$ is odd}
\end{cases}
\qquad
\rho^P=\begin{cases}
\rho &\text{if $p$ is even}\\
\rho^T &\text{if $p$ is odd}
\end{cases}$$ Given a total ordering, $>$, on partitions we let $${ \rm max}_{> } (s_{\nu}\circ s_{\mu})$$ denote the unique partition, $\lambda$, such that $p(\nu,\mu,\lambda)\neq 0$ and $p(\nu,\mu,\alpha)=0$ for all $\alpha > \lambda$. We shall use this with both the lexicographic $\succ$ and transpose-lexicographic $\succ_T$ orderings . By \[conjugate\] we have that $${ \rm max}_{\succ_T } (s_{\nu } \circ s_{\mu })
= ({ \rm max}_{\succ } (s_{\nu^M} \circ s_{\mu^T}))^T .$$
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The following theorems will be incredibly important in our arguments.
\[pppppppppp\] Let $\mu $, $\nu $ be partitions of $m$ and $n$ respectively. The unique maximal terms of $s_\nu \circ s_\mu$ in the lexicographic and transpose lexicographic ordering are as follows : $${ \rm max}_{\succ } (s_\nu \circ s_\mu) =
(n\mu_1,n\mu_2,\dots, n\mu_{\ell(\mu) -1},n\mu_{\ell(\mu)}-n+\nu_1, \nu_2, \dots ,\nu_{\ell(\nu)}) ,$$ $${ \rm max}_{\succ_T } (s_\nu \circ s_\mu) =
(n\mu_1^T,n\mu_2^T,\dots, n\mu^T_{\mu_1 -1},n\mu^T_{\mu_1}-n+\nu_1^M, \nu_2^M, \dots ,\nu_{\ell(\nu^M)}^M) )^T.$$ Moreover, we have that $$p(\nu,\mu,{ \rm max}_{\succ } (s_\nu \circ s_\mu) )=1=
p(\nu,\mu, { \rm max}_{\succ_T } (s_{\nu } \circ s_{\mu }) ).$$
\[PWex\] When $\mu=(m)$, Theorem \[PW\] shows that $${ \rm max}_{\succ } (s_\nu \circ s_{(m)}) = (nm-n) + \nu, \quad
{ \rm max}_{\succ_T } (s_\nu \circ s_{(m)}) =
((n^{m-1}) \sqcup \nu^M)^T.$$
Sometimes we shall use the dominance ordering $\rhd $ to compare the summands of $s_{\nu}\circ s_{\mu}$, and then there will, in general, be many (incomparable) maximal partitions. To understand these summands, we require some further definitions. We place a lexicographic ordering, $\prec$, on the set of semistandard Young tableaux as follows. Let ${\mathsf{S}}\neq{\mathsf{T}}$ be semistandard $\mu$-tableaux, and consider the leftmost column in which ${\mathsf{S}}$ and ${\mathsf{T}}$ differ. We write ${\mathsf{S}}\prec {\mathsf{T}}$ if the greatest entry not appearing in both columns lies in ${\mathsf{T}}$. Following [@deBPW Definition 1.4], we define a [plethystic tableau of shape $ \mu^\nu$]{} and weight $\alpha$ to be a map $${\mathsf{T}}: [\nu] \to {{\rm SStd}}_{{\mathbb N}}(\mu)$$ such that the total number of occurrences of $k$ in the tableau entries of ${\mathsf{T}}$ is $\alpha_k$ for each $k$. We say that such a tableau is [semistandard]{} if ${\mathsf{T}}(r,c-1) \preceq {\mathsf{T}}(r,c)$ and ${\mathsf{T}}(r-1,c) \prec {\mathsf{T}}(r,c)$ for all $(r,c)\in [\nu]$. We denote the set of all plethystic tableaux of shape $\mu^\nu$ and weight $\alpha$ by by ${{\rm PStd}}(\mu^\nu,
\alpha)$.
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\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}} \end{scope}
\draw[very thick] (-0.5,0.5)--(8.5,0.5)--(8.5,-2.5)--(5.5,-2.5)--(5.5,-5.5)--(-0.5,-5.5)--(-0.5,0.5);
\draw[very thick] (5.5,-2.5)--(-0.5,-2.5);
\draw[very thick] (5.5,-2.5)--(5.5,0.5);
\draw[very thick] (2.5,-5.5)--(2.5,0.5);
\end{tikzpicture}$$
\[PW\]The maximal partitions $\alpha$ in the dominance order such that $s_\alpha$ is a constituent of $s_\nu \circ s_\mu$ are precisely the maximal weights of the plethystic semistandard tableaux of shape $\mu^\nu$. Moreover if $\alpha$ is such a maximal partition then $p( \nu,\mu,\alpha)$ is equal to $|{{\rm PStd}}(\mu^\nu,
\alpha)|$.
Finally, we recall the one known case in which every term in a plethystic product is both maximal and minimal in the dominance ordering. Given $\alpha$ a partition of $n$ with distinct parts, we let $2[\alpha]$ denote the unique partition of $2n$ whose leading diagonal hook-lengths are $2\alpha_1, \dots, 2\alpha_{\ell(\alpha)}$ and whose $i\textsuperscript{th}$ row has length $\alpha_i+i$ for $1{\leqslant}i {\leqslant}\ell(\alpha)$. (An example follows.) We have the decomposition $$\label{maxminall}
s_{(1^n)}\circ s_{(2)}=\sum_{\alpha} s_{2[\alpha]},$$ where the sum is over all partitions $\alpha $ of $n$ into distinct parts. This decomposition is given in [@MR3483115 Corollary 8.6] and [@MR3443860 I. 8, Exercise 6(d)]. We observe that for $n>2$ this product is never homogeneous (for example $\alpha=(n)$ and $\alpha=(n-1,1)$ both label summands).
For $n=5$ the decomposition obtained is $$s_{(1^5)}\circ s_{(2)}=
s_{2[(3,2)]}+ s_{2[(4,1)]}+s_{2[(5)]}
= s_{(4^2,2)}+s_{(5,3,1^2)}+s_{(6,1^4)}.$$ We picture these partitions (and the manner in which they are formed) in \[2alpha\] below. We remark that $$s_{(1^5)}\circ s_{(1^2)}=
s_{(4^2,2)^T}+s_{(5,3,1^2)^T}+s_{(6,1^4)^T}
=s_{(3^2,2^2) }+s_{(4,2^2,1^2) }+s_{(5,1^5) }$$ by \[conjugate\] simply because $m=2$ is even.
$$\begin{minipage}{3cm} \begin{tikzpicture} [scale=0.5]
\fill[cyan!30](1,0)--++(0:3) --++(-90:2)--++(180:2)--++(90:1)--++(180:1);
\fill[white!30](0,0)--++(-90:3)--++(0:2) --++(90:2) --++(180:1) --++(90:1) ;
\draw[very thick](0,0)--++(0:4)--++(-90:1) --++(180:3) --++(90:1) ;
\draw[very thick](0,0)--++(-90:3)--++(0:1) --++(90:2) ;
\draw[very thick](0,0)--++(-90:3)--++(0:2) --++(90:2) --++(-90:1) --++(0:2) --++(90:1) ;
\clip(0,0)--++(0:4)--++(-90:2)--++(180:2)--++(-90:1)
--++(180:2)--++(90:3)
;
\path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture} \end{minipage}\qquad
\begin{minipage}{3cm} \begin{tikzpicture} [scale=0.5]
\draw[very thick,fill=white!30](0,0)--++(-90:4)--++(0:1)--++(90:2)--++(0:1)
--++(90:1)--++(180:1) --++(90:1);
\draw[very thick,fill=cyan!30](0,0)--++(0:5)--++(-90:1)--++(180:2)--++(-90:1)--++(180:1)--++(90:1)--++(180:1)--++(90:1);
\draw[very thick](1,-1) rectangle (3,-2);
\clip(0,0)--++(0:5)--++(-90:1)--++(180:2)--++(-90:1)
--++(180:2) --++(-90:2) --++(180:1)--++(90:5)
;
\path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture} \end{minipage}
\qquad
\begin{minipage}{3cm} \begin{tikzpicture} [scale=0.5]
\draw[very thick]
(0,0)--++(0:6)--++(-90:1)--++(180:5)--++(-90:4)
--++(180:1) --++(90:5);
\draw[very thick,fill=cyan!30]
(1,0)--++(0:5)--++(-90:1)--++(180:5)--++(90:1)
;
\draw[very thick,fill=white!30]
(0,0)--++(-90:5)--++(0:1)--++(90:5)--++(180:1)
;
\clip(0,0)--++(0:6)--++(-90:1)--++(180:5)--++(-90:4)
--++(180:1) --++(90:5) ;
\path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture} \end{minipage}$$
Decomposability and homogeneity of plethysm
============================================
In this section, we prove Theorem B of the introduction: namely we classify all decomposable/homogeneous plethystic products of Schur functions. This also serves to remove the homogeneous products from consideration in the proof of Theorem A.
\[homog\] Let $\mu,\nu $ be partitions of $m$ and $n$, respectively. The product $
s_\nu\circ s_\mu
$ is decomposable and inhomogeneous except in the following cases: $$s_{(1^2)}\circ s_{(1^2)}= s_{(2,1^2)},
\quad
s_{(1^2)}\circ s_{(2)}= s_{(3,1)},
\quad
s_\nu \circ s_{(1)}=s_\nu,
\quad
s_{(1)}\circ s_\mu =s_\mu.$$
That the listed products are homogeneous is obvious. We assume that $m, n \neq~1$ and $$\label{hellotheremr}
{\rm max}_{\succ}(s_\nu\circ s_\mu ) ={\rm max}_{\succ_T}(s_\nu\circ s_\mu ).$$ We shall show that this implies that $\nu =(1^2)$ and $\mu \vdash 2$. We first assume that $\mu $ is non-linear, that is $\mu$ is neither $(m)$ nor $(1^m)$. We set $k=\ell(\mu )$. We draw a horizontal line across the Young diagrams of $ {\rm max}_{\succ}(s_\nu\circ s_\mu )$ and $ ({\rm max}_{\succ}(s_{\nu^M}\circ s_{\mu^T} ))^T$ so that the partitions below each of these lines each have strictly fewer than $n$ nodes in total and are maximal with respect to this property. For $ {\rm max}_{\succ}(s_\nu\circ s_\mu )$, this line is drawn between the $k\textsuperscript{th}$ and $(k+1)\textsuperscript{th}$ rows (even though the $(k+1)\textsuperscript{th}$ row might be zero). For $ ({\rm max}_{\succ}(s_{\nu^M}\circ s_{\mu^T} ))^T$, this line is drawn at some point after the $(n(k-1)+1)^\textsuperscript{th}$ row. Since $ k < n(k-1)+1$ for $n>1$, we see that ${\rm max}_{\succ}(s_\nu\circ s_\mu )
\neq ({\rm max}_{\succ}(s_{\nu^M}\circ s_{\mu^T} ))^T$ as required.
It remains to consider the case that $\mu$ is linear and we assume (by conjugating if necessary) that $\mu=(m)$. Then, as we saw in Example \[PWex\], $${\rm max}_{\succ}(s_\nu\circ s_{(m)})=(mn-n)+ \nu, \quad
({\rm max}_{\succ}(s_{\nu^M}\circ s_{(1^m)} ))^T = ( {(m-1)}^n)+(\nu^M)^T.$$ Therefore row $n$ of $ {\rm max}_{\succ}(s_\nu\circ s_{(m)})$ has length $\nu_n$ which is at most 1, and row $n$ of $({\rm max}_{\succ}(s_{\nu^M}\circ s_{(1^m)} ))^T$ has length at least $m-1$. Since we are considering only $m \ge 2$, we conclude that $m=2$ and $\nu_n=1$, that is $\nu=(1^n)$. From the closed formula for the decomposition of $s_{(1^n)}\circ s_{(2)}$ in \[maxminall\], and the resulting decomposition of its plethystic conjugate $s_{(1^n)}\circ s_{(1^2)}$, we observe that the product is homogeneous if and only if $n=1,2$.
If $s_\nu \circ s_{(1)}=s_\rho\circ s_\pi$ or $s_{(1)} \circ s_{\mu}=s_\rho\circ s_\pi$ then either: $\pi=(1^2)$ and $\rho$ is a partition of 2; or at least one of $\rho$ or $\pi$ has size 1.
Therefore in the remainder of the paper, we can and will assume that none of the indexing partitions in our plethystic products are equal to $(1)\vdash 1$.
Unique factorisation of plethysm
=================================
A quick scan of the diagrams in \[maxlex\] tells us that the maximal terms in the product under the lexicographic and transpose-lexicographic orderings encode a great deal of information concerning the multiplicands of the product. We might even think that these maximal terms are enough to uniquely determine the multiplicands. In fact, this is not the case (as the following example shows).
Consider the plethysm products $$s_{(3^3,2,1)} \circ s_{(1^2) }
\qquad\text{and}\quad
s_{(2,1)}\circ s_{(4,1^4)}.$$ Both have the same maximal terms in the lexicographic and transpose-lexicographic orderings, namely those labelled by $(12,3^3,2,1)$ and $(15,3^2,2,1)^T$. \[fig1,fig2\] depict how these two partitions can be seen to be maximal in the lexicographic and transpose-lexicographic orderings using \[pppppppppp\].
$$\begin{minipage}{3.7cm} \begin{tikzpicture} [scale=0.3]
\draw[thick](0,0)--++(0:12)--++(-90:1)--++(180:9)--++(-90:3)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:6) ;
\clip(0,0)--++(0:12)--++(-90:1)--++(180:9)--++(-90:3)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:6) ;
\path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin ] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture}
\end{minipage}
=\; \begin{minipage}{3.7cm}
\begin{tikzpicture} [scale=0.3]
\fill[cyan!50](0,0)--++(0:12)--++(-90:1)--++(180:12);
\draw[thick](0,0)--++(0:12)--++(-90:1)--++(180:9)--++(-90:3)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:6) ;
\clip(0,0)--++(0:12)--++(-90:1)--++(180:9)--++(-90:3)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:6) ;
\path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin ] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture}
\end{minipage}= \; \begin{minipage}{3.7cm}
\begin{tikzpicture} [scale=0.3]
\fill[cyan!50](0,0)--++(0:12)--++(-90:1)--++(180:9)--++(-90:3)
--++(180:3)--++(90:4) ;
\draw[thick](0,0)--++(0:12)--++(-90:1)--++(180:9)--++(-90:3)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:6) ;
\clip(0,0)--++(0:12)--++(-90:1)--++(180:9)--++(-90:3)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:6) ;
\path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin ] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture}
\end{minipage}$$
$$\begin{minipage}{4.6cm} \begin{tikzpicture} [scale=0.3]
\draw[thick](0,0)--++(0:15)--++(-90:1)--++(180:12)--++(-90:2)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:5) ;
\clip(0,0)--++(0:15)--++(-90:1)--++(180:12)--++(-90:2)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:5) ; \path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin ] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture}
\end{minipage}
= \;\begin{minipage}{4.6cm} \begin{tikzpicture} [scale=0.3]
\fill[cyan!50](0,0)--++(0:12)--++(-90:1)--++(180:12);
\draw[thick](0,0)--++(0:15)--++(-90:1)--++(180:12)--++(-90:2)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:5) ;
\clip(0,0)--++(0:15)--++(-90:1)--++(180:12)--++(-90:2)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:5) ; \path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin ] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture}
\end{minipage} = \;\begin{minipage}{4.6cm} \begin{tikzpicture} [scale=0.3]
\fill[cyan!50](0,0)--++(0:15)--++(-90:1)--++(180:12)--++(-90:2)--++(180:3);
\draw[thick](0,0)--++(0:15)--++(-90:1)--++(180:12)--++(-90:2)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:5) ;
\clip(0,0)--++(0:15)--++(-90:1)--++(180:12)--++(-90:2)
--++(180:1)--++(-90:1)
--++(180:1)--++(-90:1)
--++(180:1)--++(90:5) ; \path(0,0) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:10cm) coordinate (ca\i);
\path (b\i)++(0:10cm) coordinate (cb\i);
\draw[thin ] (a\i) -- (ca\i) (b\i) -- (cb\i); }
\path(0.5,-0.5) coordinate (origin);
\foreach \i in {1,...,19}
{
\path (origin)++(0:1*\i cm) coordinate (a\i);
\path (origin)++(-90:1*\i cm) coordinate (b\i);
\path (a\i)++(-90:1cm) coordinate (ca\i);
\path (ca\i)++(-90:1cm) coordinate (cca\i);
\path (b\i)++(0:1cm) coordinate (cb\i);
\path (cb\i)++(0:1cm) coordinate (ccb\i);
}
\end{tikzpicture}
\end{minipage}$$
This puts a scupper on our plans to determine uniqueness solely using maximal terms in the [*lexicographic*]{} and [*transpose-lexicographic*]{} orderings. Now, we notice that the plethysm products $s_{(3^3,2,1)}\circ s_{(1^2) }$ and $ s_{(2,1)}\circ s_{(4,1^4)}
$ can still be distinguished by looking at the maximal terms for both products in the [*dominance ordering*]{}. For example, $ (11, 4, 4, 3, 2)$ labels a maximal term that appears in $s_{(3^3,2,1)} \circ s_{(1^2) }$ but it is not a maximal term in and $s_{(2,1)}\circ s_{(4,1^4)}$. Similarly, $ (11, 4, 3, 3, 3)$ labels a maximal term in $s_{(2,1)}\circ s_{(4,1^4)}$ but not in $s_{(3^3,2,1)} \circ s_{(1^2) }$.
Our method of proof will proceed to distinguish plethysm products by first using maximal terms in the lexicographic ordering and only when necessary considering the broader family of terms which are maximal in the dominance ordering. We first consider the case where $\mu$ consists of a single row.
\[Rowena\] Let $\mu,\nu, \pi,\rho$ be partitions of $m,n,p,q>1$ respectively. We suppose that $\mu=(m)$. If $$s_\nu\circ s_\mu = s_\rho \circ s_\pi$$ then either $\nu=\rho$ and $\mu=\pi$ or we are in the exceptional case $$s_{(2,1^2)}\circ s_{(2)} =
s_{(1^2)}\circ s_{(3,1)}.$$
From the set-up, we know $mn=pq$. We set $\ell(\pi)=c+1$ for some $c{\geqslant}0$. By assumption, we have that $$\begin{aligned}
\label{1}
{\rm max}_{\succ}(s_\nu \circ s_{(m)})
&=
{\rm max}_{\succ}(s_\rho \circ s_\pi)\\ \label{2}
{\rm max}_{\succ}(s_{\nu^M} \circ s_{(1^m)})
&=
{\rm max}_{\succ}(s_{\rho^P} \circ s_{\pi^T}).
\end{aligned}$$ As a warm-up, we first consider the case where $\pi$ is linear. If $\mu=(m)$ and $\pi=(p)$ then (see Example \[PWex\]) \[2\] says that $(n^{m-1}) \sqcup (\nu^M)^T=(q^{p-1}) \sqcup (\rho^P)^T$. By comparing widths we deduce that $q=n$. This implies $m=p$ and then $\nu=\rho$. Now, suppose that $\mu=(m)$ and $\pi=(1^p)$. Then $ {\rm max}_{\succ}(s_\nu \circ s_{(m)})=(nm-n)+\nu$ which, as $m {\geqslant}2$ and $\nu$ has size $n$, has final column of length 1. For \[1\] to hold, the same to be true of $ {\rm max}_{\succ}(s_\rho \circ s_(1^p))=(q^{p-1}) \sqcup \rho$; this implies $p=2$. Similarly, comparing the final columns of ${\rm max}_{\succ}(s_{\nu^M} \circ s_{(1^m)}) = (n^{m-1}) \sqcup \nu^M$ and $ {\rm max}_{\succ}(s_{\rho^P} \circ s_{(p)}) = (qp-q)+\rho^P$ also shows that $m=2$. Hence $n=q$ and we obtain a contradiction from comparing the widths of $(n) \sqcup \nu^M$ and $(q)+\rho^M$.
We now assume that $\pi$ is non-linear so $\pi_1>1$ and $c>0$. By \[2\], $$\label{bigbox}
\color{black}(n^{m-1}) \sqcup \nu^M
\color{black}= (q\pi_1^T,q\pi_2^T,\dots, q\pi_{\pi_1-1}^T,q\pi_{\pi_1}^T-q+\rho_1^M,\rho_2^M,\dots ).$$ Since $m{\geqslant}2$ and $\pi_1>1$, it follows that $n=q\pi_1^T=q(c+1)$ and, as $mn=pq$, $p=(1+c)m$. If $\nu^M=(n)$ then the left hand side of \[bigbox\] is $(n^m)$. Since $q<n$, comparing the width in \[bigbox\] shows that $\rho^P=(q)$ and that $\pi = (m^{c+1})$. This implies that ${\rm max}_\succ(s_\nu\circ s_{(m)})$ is a hook partition whereas ${\rm max}_\succ ( s_{\rho} \circ s_{ (m^{c+1} )} )$ has second row of width at least $ q(m-1)>1$, a contradiction. Therefore we can assume that $\nu^M\neq (n)$. Then \[bigbox\] implies that the first $m-1$ rows of $\pi^T$ are all equal to $n/q=c+1$ and therefore $\pi=((m-1)^{c+1}) + \pi'$ for some $\pi'\vdash c+1$. In particular, $\pi_1-\pi_2 {\leqslant}c+1$. We now consider \[1\]: the difference between the first and second rows of $ {\rm max}_{\succ}(s_\nu \circ s_\mu)$ is $$\left((m-1)n+\nu_1\right)-\nu_2$$ whereas the difference between the first and second rows of $ {\rm max}_{\succ}(s_\rho \circ s_\pi)$ is less than or equal to $q\times (\pi_1-\pi_2 +1) = n+q$. Therefore the necessary inequality $$(m-1)n+\nu_1 -\nu_2 {\leqslant}n+q$$ implies that $m=2$ (since $q<n$). For the remainder of the proof $\mu=(2)$ and $\pi= (1^{c+1}) + \pi'
\vdash 2(c+1)$ and therefore $\rho^P=\rho$ and $\nu^M=\nu$.
We first consider the case $c>1$. Here we have that $\ell(\pi)=c+1>2$ and so the difference between the first and second rows of $ {\rm max}_{\succ}(s_\rho \circ s_\pi)$ is $q\times (\pi_1-\pi_2)=q(\pi'_1-\pi'_2)\le q(1+c)=n $. On the other hand, for $ {\rm max}_{\succ}(s_\nu \circ s_{(m)})=(n)+\nu$ the difference is at least $n$. For equality, we require $\pi'=(c+1)$, that is $\pi=(c+1, 1^c)$. Then \[1\] becomes $(n)+\nu=(q(c+1)+q, q^{c-1}) \sqcup \rho$ and we find $\nu=(q^c) \sqcup \rho$. We now employ the dominance ordering to examine the case $$\pi =(c+2,1^c) \qquad \nu=(q^c) \sqcup \rho.$$
A necessary condition for $ {{\rm PStd}}( (c+2,1^{c})^{\rho}, \alpha))\neq \emptyset $ is that $\alpha_1+\alpha_2{\leqslant}q(c+3)$. To see this, simply note that if ${\mathsf{S}}\in {{\rm PStd}}( (c+2,1^{c})^{\rho}, \alpha))$, then $${\mathsf{S}}: [\rho] \to {{\rm SStd}}_{{\mathbb N}}((c+2,1^{c}))$$ and the maximum number of entries equal to 1 or 2 in a semistandard Young tableaux of shape $(c+2,1^{c})$ is equal to $(c+2)+1=c+3$ (the sum of the lengths of the first and second rows of $(c+2,1^{c})$). Thus $p(\rho, (c+2,1^c),\alpha) =0$ for any $\alpha$ such that $\alpha_1+\alpha_2 >q(c+3)$ by \[PW\]. We shall now construct a plethystic tableau ${\mathsf{S}}\in {{\rm PStd}}((2)^{(q^{c}) \sqcup \rho},\beta)$ with $\beta_1+\beta_2>q(c+3)$. This tableau will either be of maximal possible weight or there exists another plethystic tableau of the same shape but of weight $\beta' \rhd \beta$; in either case, for a partition for $\gamma \in \{\beta,\beta'\}$, $0 \neq p((q^c) \sqcup \rho,(2),\gamma )$ whereas $p(\rho,(c+2,1^c),\gamma ) =0$ (by \[PW\]), providing us with the necessary contradiction. Let ${\mathsf{T}}\in {{\rm PStd}}((2)^{ (q^c) \sqcup \rho}, \beta)$ be the plethystic tableau such that $${\mathsf{T}}(a,b)=
\begin{cases}
\gyoung(2;2) &\text{if $(a,b)$ is the least dominant (lowest) removable node of $(q^{c}) \sqcup \rho$} \\
\gyoung(1;a) &\text{otherwise}.
\end{cases}$$ This tableau has weight $\beta$ with $\beta_1=q(c+2)-1$ and $\beta_2=q+2$ and so $\beta_1+\beta_2=
q(c+3)+1$ as required.
Finally, we consider the case $c=1$. Here $\mu=(2)$ and $\pi\vdash 2(c+1)=4$ is either $(3,1)$ or $(2,2)$. In the $(2^2)$ case, comparing the widths of the partition on the left and right of \[1\] we see that $\nu_1=0$, a contradiction. In the $(3,1)$ case, comparison of maximal terms again reveals that $\nu=(q) \sqcup \rho$. Now $$s_{\rho}\circ s_{(3,1)}=
s_{\rho}\circ (s_{(1^2)}\circ s_{(2)})=
(s_{\rho}\circ s_{(1^2)})\circ s_{(2)}.$$ We observe that ${\rm max}_\succ(s_\rho\circ s_{(1^2)})=(q) \sqcup \rho$, but $s_\rho\circ s_{(1^2)}$ is decomposable unless $\rho=(1^2)$ by \[homog\]. For $\rho \neq (1^2)$, we deduce that $s_{(q) \sqcup \rho}\circ s_{(2)}$ is properly contained in $s_{\rho}\circ s_{(3,1)}$. Thus we have $q=2$, $\rho=(1^2)$ and $\nu=(2,1^2)$, as required.
We may conjugate (applying \[conjugate\]) to complete the case where $\mu$ is linear.
\[one\_col\] Let $\mu,\nu, \pi,\rho$ be partitions of $m,n,p,q>1$ respectively. We suppose that $\mu=(1^m)$. If $$s_\nu\circ s_\mu = s_\rho \circ s_\pi$$ then either $\nu=\rho$ and $\mu=\pi$ or we are in the exceptional case $$s_{(2,1^2)}\circ s_{(1^2)} =
s_{(1^2)}\circ s_{(2,1^2)}.$$
Let $\mu,\nu, \pi,\rho$ be arbitrary partitions of $m,n,p,q>1$ respectively. We now consider what the condition $$\label{rowenaslabel}
{\rm max}_{\succ}(s_\nu \circ s_\mu)
=
{\rm max}_{\succ}(s_\rho \circ s_\pi)$$ tells us about this quadruple of partitions. We first suppose that $\ell(\mu)=\ell(\pi)=k$. This implies that $\ell(\nu)=\ell(\rho)=\ell$, say. Furthermore, $$\begin{aligned}
\label{max}
\begin{split} &( n\mu_1,n\mu_2,\dots, n\mu_{k-1}, n\mu_k-n+\nu_1, \nu_2, \dots, \nu_\ell)
\\=
&( q\pi_1,q\pi_2,\dots, q\pi_{k-1}, q\pi_k-q+\rho_1, \rho_2, \dots, \rho_\ell).
\end{split}\end{aligned}$$ We set $d=\gcd(n,q)$, $e=\gcd(m,p)$ and set $n=n'd$, $q=q'd$, $m=m'e$, $p=p'e$. Since $mn=pq$, we note that $m'n'ed=p'q'ed$ and so $m'n'=p'q'$. Since $m'$ and $p'$ are coprime, as are $n'$ and $q'$, it follows that $m'=q'$ and $p'=n'$. Thus $$m=q'e \quad
n=n'd \quad
q=q'd
\quad
p=n'e.$$ From \[max\], we observe that $n\mu_i=\pi_i q$ implies $n'\mu_i = q' \pi_i$, and so we can set $\alpha_i:=\tfrac{\mu_i}{q'}
=\tfrac{\pi_i} {n'}
\in{{\mathbb N}}$ for all $1{\leqslant}i {\leqslant}k-1$. Now, $\mu \vdash m=q'e$ and so the final row length satisfies $$\mu_k=q'e - \sum_{i=1}^{k-1} q' \alpha_i = q'
\underbrace{\left(e- \sum_{i=1}^{k-1} \alpha_i \right) }_{\alpha_k}.$$ We have a partition $(\alpha_1,\dots,\alpha_k)\vdash e$ with $q'\alpha=\mu$, and, in a similar fashion, we deduce that $n'\alpha=\pi$. Without loss of generality, we now assume that $n{\geqslant}q$. We plug in our equalities $\pi=n'\alpha$ and $\mu=q'\alpha$ back into \[max\] and to show that $$\rho_i=\nu_i\textrm{ for $i \ge 2$ and } \nu_1 = (n-q)+\rho_1.$$ We immediately obtain the following corollary.
Let $\mu,\nu, \pi,\rho$ be partitions of $m,n,p,q>1$, respectively. We suppose that $\ell(\pi)=\ell(\mu)$. If $$s_\nu\circ s_\mu = s_\rho \circ s_\pi$$ then $\nu=\rho$ and $\mu=\pi$.
By the discussion above, we know that we are dealing with a quadruple $$\mu=q'\alpha,
\qquad
\nu=\rho+(n-q),
\qquad \pi=n'\alpha,
\qquad
\rho .$$ Comparing the width of the partitions on the left and right of $${\rm max}_{\succ}(s_{\nu^M} \circ s_{\mu^T})=
{\rm max}_{\succ}(s_{\rho^P} \circ s_{\pi^T})$$ we deduce that $\ell(\mu)n=\ell(\pi)q$. Thus $n=q$, $\nu=\rho$, $q'=n'$ and thus $\mu=\pi$, as required.
We now consider the case where the lengths of the partitions $\mu$ and $\pi$ (and hence $\nu$ and $\rho$) differ. We suppose (without loss of generality) that $\ell(\mu)<\ell(\pi)$. We set $\ell(\mu)=k$ and $\ell(\pi)=k+c$ for some $c{\geqslant}1$. Thus $\ell(\rho)+c=\ell(\nu)=\ell$, say. Observe that $
{\rm max}_{\succ}(s_\nu \circ s_\mu)
=
{\rm max}_{\succ}(s_\rho \circ s_\pi)
$ if and only if the partitions $$\scalefont{0.9}\begin{array}{ccccccccccccccccccc}\label{max2}
\!\!\!\!\!(n\mu_1 &\dots& n\mu_{k-1}& n\mu_k-n+\nu_1& \nu_2& \dots &\nu_c &\nu_{c+1} &\nu_{c+2} &\dots & \nu_\ell)
\\
\!\!\!\!\!(q\pi_1&\dots& q\pi_{k-1}& q\pi_k &q\pi_{k+1} & \dots &q\pi_{k+c-1} &q \pi_{k+c}-q+\rho_1 &\rho_2 &\dots & \rho_{\ell-c}).
\end{array}$$ coincide. We deduce that $$\label{use1}
\mu=q'(\alpha_1,\dots,\alpha_k), \qquad
\pi = n' (\alpha_1,\dots, \alpha_{k-1}) \sqcup {(\pi_k,\dots, \pi_{k+c})}$$ for $ \alpha \vdash e $, $(\pi_k,\dots, \pi_{k+c})\vdash n'\alpha_k$ and $$\label{use2} \nu=(q\pi_k - n(q'\alpha_{k}-1)) \sqcup q(\pi_{k+1},\dots, \pi_{k+c-1})\sqcup ( q(\pi_{k+c}-1)+\rho_1)\sqcup (\rho_2, \rho_3, \ldots \rho_{\ell-c})$$ and, in order for $\nu$ to be a partition, we need $$q\pi_k - n(q'\alpha_{k}-1) {\geqslant}q \pi_{k+1}$$ which, rearranging, gives $$q(\pi_k - \pi_{k+1}) {\geqslant}n(\mu_k-1).$$ We are now ready to complete our proof of Theorem A.
\[Rowena\] Let $\mu,\nu, \pi,\rho$ be partitions of $m,n,p,q>1$, respectively. We suppose that both $\mu$ and $\pi$ are non-linear and $\ell(\pi)>\ell(\mu)$. If $$s_\nu\circ s_\mu = s_\rho \circ s_\pi$$ then $\nu=\rho$ and $\mu=\pi$.
We set $\ell(\mu)=k{\geqslant}2$ and $\ell(\pi)=k+c$ for $c{\geqslant}1$. We first see what can be deduced from $ {\rm max}_{\succ}(s_\nu \circ s_\mu)
=
{\rm max}_{\succ}(s_\rho \circ s_\pi)
$. From \[use1\] we have that $$\label{max1}\mu=q'(\alpha_1,\dots,\alpha_k) \qquad
\pi = n' (\alpha_1,\dots, \alpha_{k-1}) \sqcup {(\pi_k,\dots, \pi_{k+c})}$$ for $ \alpha \vdash e $ and $(\pi_k,\dots, \pi_{k+c})\vdash n'\alpha_k$, and, from \[use2\], we deduce that $|\rho|<|\nu|$ and so $q=q'd<n'd=n$ which implies $q'<n'$. From \[use1\] this implies that $\mu_1=q'\alpha_1<n'\alpha_1=\pi_1$ in other words $\ell(\mu^T) < \ell(\pi^T)$.
We now see what can be deduced from $ {\rm max}_{\succ}(s_{\nu^M} \circ s_{\mu^T})
=
{\rm max}_{\succ}(s_{\rho^P} \circ s_{\pi^T})
$. We have already concluded that $\ell(\mu^T)< \ell(\pi^T)$. Therefore applying \[use1\] (but with the partitions $\mu^T$, $\nu^M$, $\pi^T$ and $\rho^P$) we deduce that $$\label{max2}\mu^T=q'(\beta_1,\dots, \beta_{\mu_1})
\qquad
\pi^T= n' (\beta_1,\dots, \beta_{\mu_1-1}) \sqcup (\pi^T_{\mu_1},\dots, \pi^T_{\pi_1})$$ for some $\beta\vdash e$ and $ (\pi^T_{\mu_1},\dots, \pi^T_{\pi_1})\vdash n'\beta_{\mu_1}$.
From \[max1,max2\] we deduce that $\mu$ can be built from boxes of size $q'\times q'$. In other words, $$\mu=q'(\underbrace{\gamma_1,\gamma_1,\dots, \gamma_1}_{q'},\underbrace{ \gamma_2,\gamma_2,\dots, \gamma_2}_{q'}, \dots).$$ for some $\gamma\vdash m /q'^{2}$. Since $\gamma$ might have repeated parts, we write $\gamma$ in the form $$\gamma=(a_1^{b_1},a_2^{b_2},\dots, a_x^{b_x})$$ where $a_1>a_2>\dots >a_x$, so $$\gamma^T=({(b_1+\dots+b_x )}^{a_x},{(b_1+\dots+b_{x-1})}^{a_{x-1}-a_x},\dots, {b_1}^{a_1-a_2 }).$$ Now, \[max1\] reveals that $$\label{pi}
\pi = (\underbrace{n'a_1,\dots ,n'a_1}_{b_1q'},\underbrace{n'a_2,\dots ,n'a_2}_{b_2q'},\dots,
\underbrace{n'a_x,\dots ,n'a_x}_{b_xq'-1},
\color{black}\pi_k,
\color{black} \dots , \pi_{k+c})$$ where $(\pi_k,
\dots , \pi_{k+c})\vdash n'a_x$ and, from \[max2\], $$\begin{aligned}
\label{pit}\begin{split}
\pi^T = &( ({n'{(b_1+\dots+b_x )}})^{ q' a_x},({n'{(b_1+\dots+b_{x-1} )}})^ { q'( a_x-a_{x-1})}, \dots
(n'b_1)^{q'(a_1-a_2)-1}) \sqcup
\\ &
( \pi^T_{\mu_1}, \dots , \pi^T_{\pi_1})
\end{split}\end{aligned}$$ where $(\pi^T_{\mu_1}, \dots , \pi^T_{\pi_1})\vdash n'b_1$. By looking at the first row of $\pi^T$ we deduce that provided $x\ne 1$ the last part of $\pi$ is $q'a_x$ and that it appears with multiplicity $n'b_x$. This implies that $$(\color{black}
\pi_k, \color{black}
\dots, \pi_{k+c})
=
(\dots, \underbrace{q'a_x,\dots q'a_x}_{n'b_x})\vdash n'a_x.$$ But the sum over these final $n'b_x$ rows is $
q'a_a \times n' b_x
$ which implies $q'=1$ and $b_x=1$ and that $$(\color{black}
\pi_k, \color{black}
\dots, \pi_{k+c})
=
( \underbrace{ a_x, \dots , a_x}_{n' })\vdash n'a_x.$$ Now we input this into \[pi\] to deduce that $$\ell(\pi)
=b_1+\dots +b_x -1+n'.$$ On the other hand by \[pit\] we know that $$\ell(\pi)=\pi^T_1=
n' (b_1+\dots+b_x )$$ Therefore $$n'(b_1+\dots+b_x -1)
=
b_1+b_2+\dots +b_x-1$$ and thus $n'=1$ or $b_1+b_2+\dots +b_x =1$. If $n'=1$ then $n=q$, contrary to our earlier observation that $q<n$. If $b_1+b_2+\dots +b_x =1$, then $\ell(\gamma)=\ell(\alpha)=\ell(\mu)=1$, contrary to our assumption that $\mu$ is non-linear.
Finally, it remains to consider the $x=1$ case. This is the case in which $\gamma=(a^b)$ is a rectangle. Here we have that $
\mu=q'(a^{bq'}), \mu^T=q'(b^{aq'})
$ and $$\begin{aligned}
\label{1111}
\pi = ((n'a)^{q'b-1})\sqcup( \pi_k,\dots ,\pi_{k+c}) \quad &\text{for $( \pi_k,\dots ,\pi_{k+c})\vdash n'a$}
\\ \label{2222}
\pi = ({(q'a-1)}^{n'b})+( \pi_{\mu_1}^T,\dots ,\pi_{\pi_1}^T)^T \quad
&\text{for $( \pi_{\mu_1}^T,\dots ,\pi_{\pi_1}^T)\vdash n'b$. }\end{aligned}$$ Now, recall that $q'<n'$; and so $$q'b-1<q'b <n'b$$ and so the rectangle in \[1111\] is at least 2 rows shorter than that in \[2222\]. This implies that $q'=1$ and $a$ or $b=1$ and so $\mu$ is linear, a contradiction.
We have now classified all possible equalities between products $
s_\nu\circ s_\mu = s_\rho \circ s_\pi
$ where neither, one, or both of $\pi$ and $\mu$ are linear partitions. This completes the proof of Theorem A.
We would like to thank Cedric Lecouvey for bringing to our attention the question of unique factorisability of products of Schur functions and in particular for introducing us to Rajan’s result for Littlewood–Richardson products.
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---
abstract: 'We prove that $G_{n,p=c/n}$ has a $k$-regular subgraph if $c$ is at least [$e^{-\Theta(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an non-empty $k$-core.]{} In particular, this pins down the [threshold for the ]{} appearance of a $k$-regular subgraph to a window [of size $e^{-\Theta(k)}$.]{}'
author:
- 'Dieter Mitsche[^1]'
- 'Michael Molloy[^2]'
- 'Paweł Prałat[^3]'
title: '$k$-regular subgraphs near the $k$-core threshold of a random graph'
---
c ¶ §
\[lemma\][Theorem]{} \[lemma\][Corollary]{} \[lemma\][Claim]{} \[lemma\][Remark]{} \[lemma\][Observation]{} \[lemma\][Proposition]{} \[lemma\][Definition]{}
In this paper, we study the threshold for [the Erdős-Rényi random graph model]{} $G_{n,p=c/n}$ to have a $k$-regular subgraph where $k$ is fixed. This problem was first studied by Bollobás, Kim and Verstraëte [@bkv] who proved, amongst other things, that $G_{n,p=c/n}$ w.h.p.[^4] has a $k$-regular subgraph when $c$ is at least roughly $4k$. Letzter [@let] proved that this threshold is sharp[^5].
This problem is reminiscent of the $k$-core, a maximal subgraph with minimum degree at least $k$. Pittel, Spencer and Wormald [@Pittel99] established the threshold for $G_{n,p}$ to have a [non-empty]{} $k$-core to be a specific constant $c_k=k+o(k)$ (we specify $c_k$ more precisely in (\[eq:ck\]) below). This provides a lower bound on the threshold for a $k$-regular subgraph, and it is natural to ask:
[**Question:**]{} Is the threshold for a $k$-regular subgraph equal to the $k$-core threshold?
Bollobás, Kim and Verstraëte [@bkv] proved that the answer is “No” for $k=3$ and conjectured that it is “No” for all $k\geq 4$. On the other hand, Pretti and Weigt [@pw] provided a non-rigorous analysis and claimed that it indicates the answer is “Yes” for $k\geq 4$.
Prałat, Verstraëte, and Wormald [@Pawel] proved that the $(k+2)$-core of $G_{n,p}$ (if it is non-empty) contains a $k$-regular spanning subgraph. Chan and Molloy [@Mike] proved the same for the $(k+1)$-core. So the $k$-regular subgraph threshold is at most $c_{k+1}\approx c_k+1$. We will reduce this bound to within an exponentially small distance (as a function of $k$) from $c_k$:
\[mt\] For $k$ a sufficiently large constant, and for any $c\geq c_k+e^{-k/300}$, $G_{n,p=c/n}$ contains a $k$-regular subgraph.
It is not hard to see that the $k$-core cannot have a $k$-regular [*spanning*]{} subgraph; for example it has many vertices of degree $k+1$ whose neighbours all have degree $k$. Our approach is to start with the $k$-core and repeatedly remove such vertices, along with other problematic vertices. We will then apply a classic theorem of Tutte to show that what remains has a spanning $k$-regular subgraph. The aforementioned papers [@Pawel; @Mike] applied Tutte’s theorem to the $(k+2)$- and $(k+1)$-core.
[[ The $k$-core is well known to have size [[ $(1-o_k(1))n$]{}]{}, and we will show that we only remove $o_k(1)n$ vertices (see Remark \[rokn\]). So the $k$-regular subgraph that we obtain will have size [[ $(1-o_k(1))n$]{}]{}. ]{}]{}
The new arguments required in this work are (i) stripping the $k$-core down to something to which Tutte’s theorem can be applied and (ii) applying Tutte’s theorem to it. [The first part requires a delicate variant of the configuration model (see the discussion at the beginning of Section \[stsp\]),]{} whereas the presence of degree $k$ vertices brings new challenges to the second part.
The number of problematic vertices, as described above, is linear in $n$. Furthermore, removing them from the $k$-core will cause a linear number of vertices to have their degrees drop below $k$. It is not surprising that if $c$ is too close to $c_k$, then what remains will have [[ an empty]{}]{} $k$-core, and so this argument will not work unless $c$ is bounded away from $c_k$. Fortunately, when $k$ is large, the number of problematic vertices is very small (but linear in $n$): $e^{-\Theta(k)} n$. So we only need $c$ to be bounded away from $c_k$ by an exponentially small distance [(in terms of $k$)]{}. Furthermore, the subgraph that we show to have a $k$-factor consists of all but $e^{-\Theta(k)}n$ vertices of the $k$-core [[ (see Remark \[rokn\]).]{}]{} [This is consistent with a result of]{} Gao [@Gao] [who]{} proved that any $k$-regular subgraph must contain all but at most $\e_k n$ vertices of the $k$-core where $\e_k\rightarrow 0$ as $k$ grows.
[ [*Organization of the paper:*]{} We begin, in section 1, by presenting Tutte’s condition. Section 2 contains some brief probabilistic tools. The stripping procedure to find our subgraph is given and analyzed in section 3; this is most of the work. Finally, in section 4, we show how to prove that the subgraph satisfies Tutte’s condition and thus has a spanning $k$-regular subgraph. ]{}
Tutte’s condition
=================
We begin by presenting Tutte’s theorem for establishing that a graph has a $k$-regular spanning subgraph. Recall that a $k$-regular spanning subgraph is called a [*$k$-factor*]{}. Let $\G$ be a graph with minimum degree at least $k$.
\[dlh\] $L=L(\G)$ is the set of [*low vertices*]{} of $\G$, i.e. the vertices $v$ with $d_{\G}(v)=k$, and $H=H(\G)$ is the set of [*high vertices*]{} of $G$, i.e. the vertices $v$ with $d_{\G}(v)\geq k+1$. For any set of vertices $Z$, we use $Z_L,Z_H$ to denote $Z\cap L$, respectively $Z\cap H$.
[[ ]{}]{} For any $S \subseteq V(\G)$ we use $e(S)$ to denote the number of edges of $\G$ with both endpoints in $S$. For any disjoint $S,T \subseteq V({{\color{black} \G}})$, we use $e(S,T)$ to denote the number of edges of $G$ from $S$ to $T$ and $q(S,T)$ to denote the number of components $Q$ of ${{\color{black} \G}} \setminus (S\cup T)$ such that $k|Q|$ and $e(Q,T)$ have different parity. [[ Throughout the paper, we use $d_A(v)$ to denote the number of neighbours that $v$, a vertex, has in $A$, a subset of the vertices.]{}]{} [[ Furthermore, we refer to the [*total degree*]{} of $S$ as the sum of degrees of the vertices in $S$.]{}]{}
\[thm:Tutte\] A graph $\G$ with minimum degree at least $k\geq1$ has a $k$-factor if and only if for every pair of disjoint sets $S,T \subseteq V(\G)$, $$k|S| \ge q(S,T) + k|T|- \sum_{v \in T} d_{\G \setminus S}(v).$$
\[Tutte-condition\] A graph $\G$ with minimum degree at least $k\geq 1$ has a $k$-factor if [[ and only if]{}]{} for every pair of disjoint sets $S,T \subseteq V(\G)$, $$\label{etutte2}
k|S|+\sum_{v\in T_H}(d_{\G}(v)-k) \ge q(S,T)+e(S,T).$$
[[[**Proof.**]{}]{}]{} Rearranging the terms in Theorem \[thm:Tutte\], we obtain that the condition given there for the existence of a $k$-factor is equivalent to every pair of disjoint sets $S,T \subseteq V(\G)$ satisfying: $$\begin{aligned}
\sum_{v \in T} d_{\G}(v) + k|S| &\ge &q(S,T)+k|T|+e(S,T)\\
\mbox{i.e.} \qquad \sum_{v \in T} (d_{\G}(v)-k) + k|S| &\ge &q(S,T)+e(S,T)\end{aligned}$$ which is equivalent to (\[etutte2\]) since $d_{\G}(v)=k$ for all $v\in T_L$.
In all but one case, we will in fact show that $S,T$ satisfy the stronger condition, which implies (\[etutte2\]) since $d_{\G}(v)\geq k+1$ for all $v\in T_H$: $$\label{etutte}
k|S|+|T_H| \ge q(S,T)+e(S,T).$$
[**Remark:**]{} In previous papers [@Mike; @Pawel] Theorem \[thm:Tutte\] was applied to subgraphs with minimum degree at least $k+1$, specifically the $(k+1)$-core [@Mike] and the $(k+2)$-core [@Pawel]. So it sufficed to prove the weaker bound $k|S|+|T| \ge q(S,T)+e(S,T)$.
We begin by showing that in Corollary \[Tutte-condition\] we may assume that $ S \subseteq H$ and every component counted by $q(S,T)$ has a high vertex:
\[lem:minimality\] A graph $\G$ with minimum degree $k\geq 1$ has a $k$-factor if [[ and only if]{}]{} (\[etutte2\]) holds for every pair of disjoint sets $S,T \subseteq V(\G)$ satisfying:
- $S \subseteq H$; and
- every component $Q$ counted by $q(S,T)$ satisfies $Q_H\neq\emptyset$.
[[**Proof.**]{}]{}[We will prove that if (\[etutte2\]) holds for every $S,T$ satisfying (M1) and (M2) then (\[etutte2\]) holds for every $S,T$. So Corollary \[Tutte-condition\] implies that $\G$ has a $k$-factor. To do this, we show that if $S,T$ violate (\[etutte2\]) and violate either (M1) or (M2) then we can modify $S,T$ so that (M1) and (M2) both hold but (\[etutte2\]) is still violated. [[ This proves]{}]{} our lemma.]{}
Suppose there exist two disjoint [sets]{} $S,T \subseteq V(\G)$ [that violate both (\[etutte2\]) and (M1). So there exists some $u \in S_L$.]{} We will show that after moving $u$ to $V(\G) \setminus (S \cup T)$, [(\[etutte2\])]{} will still fail for the new pair of sets. By applying this procedure iteratively for every $u \in {S_L}$, [we obtain two sets violating (\[etutte2\]) and satisfying (M1)]{}.
Let $S':=S \setminus \{u\}$. Note first that $k|S'|=k|S|-k$ and $e(S',T)=e(S,T)-d_T(u)$. Now, since $d_{\G}(u)=k$, there are at most $(k-d_T(u))$ neighbours of $u$ in $V(\G) \setminus (S \cup T)$ (observe that some neighbours of $u$ might be in $S$). In the worst case, all neighbours of $u$ in $V(\G) \setminus (S \cup T)$ belong to different components all contributing to $q(S,T)$; moreover, after moving $u$ to $V(\G) \setminus (S \cup T)$, they form one connected component of $V(\G) \setminus (S \cup T)$, and so do not contribute to $q(S',T)$ anymore. In any case, $q(S',T) \ge q(S,T) - (k - d_T(u))$. Since $|T_H|$ is left unchanged by switching from $S$ to $S'$, we have $$\begin{aligned}
k|S'|+ {\sum_{v\in T_H}(d_{\G}(v)-k)}&=k|S|+ {\sum_{v\in T_H}(d_{\G}(v)-k)}-k \\
&< q(S,T)+e(S,T)-k \\
& \le q(S',T)+(k-d_T(u)) +e(S',T)+d_T(u) -k \\
&= q(S',T)+e(S',T),\end{aligned}$$ and hence (\[etutte2\]) still fails.
Now suppose there exist two disjoint $S,T \subseteq V(\G)$ [violating (\[etutte2\])]{} and which satisfy (M1) but [not]{} (M2). Let $Q$ be a component of $\G\bk (S\cup T)$ such that $k|Q|$ and $e(Q,T)$ have different parity and with $Q_H=\emptyset$. Note that $e(Q, V(\G)\bk Q)=e(Q, S \cup T)$ has the same parity as $\sum_{u\in Q} d(u)=k|Q|$ since [$Q_H=\emptyset$]{}. So $e(Q, V(\G)\bk Q)$ has a different parity than $e(Q,T)$ and thus $e(Q,S)\neq 0$. Now we move $Q$ into $T$; i.e. we set $T'=T\cup Q$. Since $Q\subseteq L$, we have $T'_H=T_H$. Since $Q$ is a component of $\G\bk (S\cup T)$, this move does not affect whether any other component counts towards $q$; i.e. $q(S,T')= q(S,T)-1$. Moreover, as we argued above, $e(Q,S)>0$ and so $e(S,T')\geq e(S,T)+1$. Combining this with $k|S|+ {\sum_{v\in T_H}(d_{\G}(v)-k)} < q(S,T)+e(S,T)$ yields $k|S|+ {\sum_{v\in T'_H}(d_{\G}(v)-k)} < q(S,T')+e(S,T')$ and hence (\[etutte2\]) still fails. Clearly, $S$ has not changed and hence (M1) still holds. Repeated applications result in a pair $S,T$ that violates (\[etutte2\]) and satisfies (M1) and (M2).
Probabilistic preliminaries {#spp}
===========================
[[ We use ${\mbox{\bf Bin}}(\ell,p)$ to denote the binomial random variable with $\ell$ trials and success probability $p$. We use ${\mbox{\bf Po}}(x)$ to denote the Poisson variable with mean $x$.]{}]{}
Pittel, Spencer and Wormald [@Pittel99] established the $k$-core threshold to be: $$\label{eck}
c_k=\min_{x>0}\frac{x}{1-e^{-x}\sum_{i=0}^{k-2}\frac{x^i}{i!}}.$$ In [@Pawel] the asymptotic value of $c_k$ is determined up to an additive ${O(1/\log k) =} o_k(\cdot)$ term. Setting $q_k=\log k-\log (2\pi)$, we have $$\label{eq:ck}
c_k =k+(kq_k)^{1/2}+\left( \frac {k}{q_k} \right)^{1/2} + \frac{q_k-1}{3}+ O \left( \frac {1}{\log k} \right).$$
We will use the following well-known bounds on tail probabilities known as Chernoff’s bound (see, for example, [@JLR], Theorem 2.1). [[ Let $X$ be distributed as ${\mbox{\bf Bin}}(\ell,p)$, so]{}]{} $\E[X]=\mu=p\ell$. Then, $$\begin{aligned}
\label{chernoff:low}
\Pr{ (X \le \mu - t)} \le \exp \left( -\frac{t^2}{2\mu} \right)
\end{aligned}$$ and $$\begin{aligned}
\label{chernoff:up}
\Pr{ (X \ge \mu + t) } \le \exp\left(-\frac{t^2}{2(\mu+t/3)}\right).
\end{aligned}$$ In addition, all of the above bounds hold for the general case in which $X=\sum_{i=1}^{\ell} X_i$ and $X_i$ is the Bernoulli random variable with parameter $p_i$ with (possibly) different $p_i$’s.
The [[ Azuma-Hoeffding]{}]{} inequality can be generalized to include random variables close to martingales. One of our proofs, proof of Lemma \[lstop\], will use the supermartingale method of Pittel et al. [@Pittel99], as described in [@Wormald-DE Corollary 4.1]. Let $G_0, G_1,\ldots, G_\ell$ be a random process and let $X_i$ be a random variable determined by $G_0, G_1, \ldots, G_i$, $0 \le i \le \ell$. Suppose that for some [[ real constant $b$ and real positive constants $c_1,\ldots ,c_{\ell}$]{}]{}, $$\ex (X_i - X_{i-1} | G_0, G_1, \ldots, G_{i-1}) < b \quad \text{ and } \quad |X_i - X_{i-1}| \le c_i$$ for each $1 \le i \le \ell$. Then, for every $\alpha > 0$, $$\label{eq:HA-inequality2}
{\mbox{\bf Pr}}\left[ \text{For some } i \text{ with } 0 \le i \le \ell : X_i - X_0 \ge ib+\alpha \right] \le \exp \left(-\frac{\alpha^2}{2 \sum_j c_j^2} \right).$$
[[ Throughout the paper, we often omit floor and ceiling signs.]{}]{}
Finding the subgraph {#sfts}
====================
[Let $k \in {{\mathbb N}}$]{} and set $$\label{ebeta}
\b= e^{-k/200}.$$ We begin with a random graph $G=G_{n,p}$ with $p=c/n$ for some constant $c$ satisfying $$\label{eq:conditions_for_c}
c_k+ k^{10} \beta =: \cmin \le c \le \cmax := c_k + k^{-1/2},$$ where $c_k$ is the threshold of the emergence of the $k$-core. Our goal is to find [(for $k$ sufficiently large)]{} a subgraph $K$ of the $k$-core with certain properties, which will ensure that it has a $k$-factor.
Our first property is simply a degree requirement. Of course, $K$ must have minimum degree at least $k$; for technical reasons, it will help if the maximum degree is bounded by a constant; we arbitrarily chose [this to be]{} $2k$. In the introduction, we noted that $K$ cannot have any vertex of degree greater than $k$ whose neighbours all have degree $k$. It is not hard to build similar problematic local structures; it turns out that we can eliminate all of them by not allowing any vertex of degree greater than $k$ that has many neighbours of degree $k$; our second property enforces this. Our third property simply says that $K$ has linear size. Our final property is a trivial necessary condition for having a $k$-factor.
- for every vertex $v\in K$, $k \le d_K(v) \le 2k$;
- for every vertex $v\in K$ with $d_K(v)\geq k+1$, we have $|\{w \in N_{{K}}(v): d_K(w)=k\}| \le \frac{9}{10}k$;
- $|K|\geq \frac{n}{3}$;
- $k|K|$ is even.
(In fact, we will be able to find an induced subgraph $K$ of $G$ satisfying these properties.)
The stripping procedure {#stsp}
-----------------------
[In order to achieve our goal, we are going to use a carefully designed stripping procedure during which one vertex is removed in each step until a subgraph $K$ satisfying properties (K1), (K2), and (K3) remains. It will be easy to modify $K$ to enforce the final property (K4), if necessary, at the end.]{}
We found property (K2) to be particularly challenging to enforce. The typical approach to this sort of problem is to repeatedly remove a vertex if it violates one of (K1-3). Often one can argue that at every step, the remaining graph is uniformly random conditional on its degree sequence (for example, this happens when analyzing the $k$-core stripping process). In some situations, the vertex set is initially partitioned into a fixed number of parts, and one must condition on the number of remaining neighbours each vertex has in each part; this is more complicated but in principle not much more difficult than conditioning on the degree sequence. In the present situation, enforcing [[ (K2)]{}]{} requires conditioning on the number of remaining neighbours each vertex has in $W$, the set of vertices of degree $k$. However, this is not an initial partition; $W$ changes during the process. This made our analysis more difficult.
In dealing with this problem, it helps to partition $W$ into $W_0$, the vertices that initially have degree $k$, and $W_1$, the vertices whose degrees change to $k$ during the process. $W_1$ is much smaller than $W_0$ and so we can afford to delete vertices if they have at least [*two*]{} neighbours in $W_1$ rather than at least $\frac{9}{10}k$. This simpler deletion rule helps us deal with the fact that $W_1$ is changing throughout our stripping process.
We begin with the $k$-core of $G$, as any [subgraph]{} $K$ satisfying (K1) must be a subgraph of the $k$-core. The $k$-core of a graph can be found by repeatedly deleting vertices of degree less than $k$ from the graph so this [can be viewed as an initial phase of our stripping procedure.]{} We continue stripping the graph (as explained below) and, throughout our procedure, we partition the vertex set as follows: $$\begin{aligned}
W_0&=& \mbox{ the vertices in the remaining graph that had degree $k$ in the $k$-core of $G$;}\\
W_1&=& \mbox{ the vertices of degree at most $k$ in the remaining graph that are not in $W_0$;}\\
R&=& \mbox{ the vertices of degree greater than $k$ in the remaining graph.}\end{aligned}$$ Note that vertices may move from $R$ to $W_1$ during our procedure, but no vertex leaves $W_0$ [[ or $W_1$]{}]{} unless it is deleted.
The following definition governs the stripping procedure.
\[def:deletable\] We say a vertex $v$ is [*deletable*]{} if in the initial $k$-core:
- $\deg(v)>2k$; [[ or]{}]{}
- $v\notin W_0$ (that is, $\deg(v) \ge k+1$) and $v$ has at least $\hf k$ neighbours in $W_0$;
or if in the remaining graph:
- $\deg(v)<k$; [[ or]{}]{}
- $v\in R$ and $v$ has at least two neighbours that are in $W_1$; or
- $v\in W_1$ and $v$ has a neighbour that is either (i) in $R$ and deletable, or (ii) in $W_1$.
Furthermore,
- once a vertex becomes deletable it remains deletable.
[**Remarks:**]{} [Let us make three remarks.]{}
- Deleting vertices in $W_1$ with non-deletable neighbours in $W_1$ is not required for properties (K1), (K2), and (K3); we only delete them because it helps with our analysis.
- In many [[ similar]{}]{} stripping processes (for example, the $k$-core process), we have the property that the subgraph we eventually obtain does not depend on the order in which vertices are deleted. That is not true for our procedure—whether a vertex ever becomes deletable can depend on the deletion order. However, our goal is only to obtain a subgraph with the desired properties (K1), (K2), and (K3) and so this works for our purpose.
- Deleting a vertex that is deletable may cause some non-deletable vertices to become deletable which, in turn, might force more non-deletable vertices to change their status. However, this “domino effect” will eventually stop (possibly with all remaining vertices marked as deletable) because of (D6).
At any point of the algorithm, we use $Q$ to denote the set of deletable vertices [[ that have not yet been removed]{}]{}. Recall from (\[ebeta\]) that $\b= e^{-k/200}$. We will show that we reach $Q=\emptyset$ within $\b n$ steps. It will be convenient to force our stripping procedure to halt after $\b n$ steps regardless of whether it has reached [the desired property (that is, $Q=\emptyset$)]{}.
Now, we are ready to introduce our stripping procedure. [[ This procedure can be applied to any graph $G$. Much of the work in this paper will be to analyze what happens when it is applied to $G=G_{n,p}$.]{}]{}
[**STRIP**]{}
1. Begin with the $k$-core [of $G$]{}, and initialize $Q$ to be [[ the set of]{}]{} vertices $v$ with $\deg(v)>2k$ or $v\notin W_0$ and $v$ has at least $\hf k$ neighbours in $W_0$; [[ i.e. for which (D1) or (D2) hold.]{}]{}
2. Until $Q=\emptyset$ or until we have run $\b n$ iterations, let $v$ be the next vertex in $Q$, according to a specific fixed vertex ordering. Let [[ $N(v)$ be the set of neighbours of $v$ remaining at this point.]{}]{}
1. Remove $v$ from the graph (and from $Q$).
2. If any $u \in {{\color{black} N(v)\cap R}}$ now has degree at most $k$, then move $u$ from $R$ to $W_1$.
3. If any vertex $w\notin Q$ is now deletable, place $w$ into $Q$.
[**Clarification:**]{} In step 2c, $w$ does not leave whichever of $W_0,W_1,R$ it was in. So [[ for example, $w$ could be in]{}]{} $Q\cap W_0$.
[**Remark:**]{} Note that initially no vertex has degree less than $k$ and $W_1 = \emptyset$ so, indeed, all initially deletable vertices are added to $Q$ in step 1.
[The following observation is an immediate consequence of Definition \[def:deletable\]. Indeed, parts (a) and (b) follow from (D5); part (c) follows from (D4).]{}
\[ob1\] The following properties hold [[ at the beginning of every iteration]{}]{}.
1. If $u\in W_1$ then $u$ has no neighbours in $W_1\bk Q$.
2. If $u\in R\cap Q$ then $u$ has no neighbours in $W_1\bk Q$.
3. If $u\in R\bk Q$ then $u$ has at most one neighbour in $W_1$.
[Here are two more straightforward but useful observations.]{}
\[oq4k2\] During any iteration of Step 2, at most $4k^2$ vertices enter $Q$.
[**Proof:**]{} This follows by noting: (i) the deleted vertex $v$ has at most $2k$ neighbours $u$; (ii) if $u$ enters $W_1$ then $u$ has at most $k-1$ other neighbours $z$; (iii) if a vertex $w$ becomes deletable then it is either one of the $u,z$ mentioned in (ii) or $w\in W_1$ and $w$ is a neighbour of a $z\in N(u)\cap R$ which becomes deleteable; (iv) each such $z$ has at most one neighbour $w\neq u$ that is in $W_1$, else $z$ would have already been in $Q$.
\[ob2\] If [the stripping procedure]{} terminates with $Q=\emptyset$, then in the remaining subgraph: the degree [of every vertex]{} is in ${{\color{black} \{k, k+1, \ldots ,2k\}}}$ and [each]{} vertex in $R$ has [at most]{} ${{\color{black} \hf k}}$ neighbours of degree $k$ [(provided that $k \ge 3$)]{}. Thus, the remaining subgraph satisfies [properties]{} (K1) and (K2).
Configuration models {#scm}
--------------------
We model the $k$-core of $G=G_{n,p=c/n}$ with the configuration model introduced by Bollobás [@Bollobas], and inspired by Bender and Canfield [@Bender]. We are given the degree sequence of a graph (that is, the degree of each vertex). We take $\deg(v)$ copies of each vertex $v$ and then choose a uniform pairing of those vertex-copies. Treating each pair as an edge gives a multigraph. It is well-known (see, for example, the result of McKay [@McKay]) that the multigraph [(for a degree sequence meeting some mild conditions; these conditions are met in our application)]{} is simple with probability tending to a positive constant, and it follows that if a property holds for a uniformly random configuration then [[ it]{}]{} holds for a uniformly random simple graph on the same degree sequence; see the survey of Wormald [@Wormald_models] for more on this, and for a history of the configuration model and other related models.
Throughout our analysis, we often refer to a pair in the configuration as an edge in the corresponding multigraph. A sub-configuration is simply a subset of the pairs in a configuration, and thus yields a subgraph of the multigraph.
We define $C$ to be the $k$-core of $G_{n,p=c/n}$. We expose the degree sequence $\cald$ of $C$ and then define $\La$ to be a uniform configuration with degree sequence $\cald$. We will prove:
\[mtc\] W.h.p. STRIP terminates with $Q=\emptyset$ when run on $\La$.
As described above, we can transfer our results on random configurations to random simple graphs thus obtaining:
\[mtsg\] W.h.p. STRIP terminates with $Q=\emptyset$ when run on $C$.
$k$-core properties
-------------------
We will require the following properties of the configuration $\La$.
[**Setup for Lemma \[lw0\]:**]{} $k$ is a sufficiently large constant, and $c_k < c \le \cmax = c_k + k^{-1/2}$. $C$ is the $k$-core of $G_{n,p=c/n}$. $\La$ is a uniform configuration with the same degree sequence as $C$. [Finally,]{} $W_0,R$ are as defined in Section \[stsp\].
\[lw0\] [[ W.h.p. ]{}]{} before the first iteration of STRIP:
1. $|\La|> {0.99} n$;
2. ${ 0.99} \frac{n}{k} < |W_0| < {1.01} \frac{n}{k}$;
3. the total degree of [[ the set of]{}]{} vertices with degree greater than $2k$ is at most [$e^{-k/6}n$]{};
4. there are at least $\frac{n}{{5}k}$ edges with both endpoints in $W_0$;
5. there are at least ${\frac {1}{2}} n$ edges from $W_0$ to $R$;
6. there are at least ${\frac{1}{ 3}}k n$ edges with both endpoints in $R$;
7. $C$ has at most [$e^{-k/3}n$]{} vertices of degree at most $2k$ and with at least $\hf k$ neighbours in $W_0$;
8. at least $\frac{n}{{200}}$ vertices in $R$ have no neighbours in $W_0$.
[[ The proof of Lemma \[lw0\] is straightforward, but lengthy. So we defer the proof to Section \[sec:lw0\].]{}]{}
\[ck3\] When STRIP terminates, the remaining subgraph has size at least $\frac{n}{3}$; that is, it satisfies (K3).
[[**Proof.**]{}]{}This follows from Lemma \[lw0\](a), since ${0.99}n-\b n \ge \frac{n}{3}$, [for $k$ sufficiently large]{}.
Stripping a configuration {#ssac}
-------------------------
[[ We will apply STRIP to the random configuration $\La$. In order to analyze this process, We must carefully track the information that is exposed. If the partition $W_0,W_1,R$ were fixed throughout our procedure, then we would simply expose the number of remaining neighbours that each vertex has in each part. But because $W_1$ and $R$ change during the process, our exposure is more delicate. We found that the best way to deal with this complication includes exposing many of the remaining edges involving $W_1$ and $R$.]{}]{}
\[def:RW\] Suppose the vertices of a configuration are partitioned into $\calr,\calw_0,\calw_1$. We define the [*RW-information*]{} to be:
- for each vertex $v\in \calw_0$, $\deg_{\calw_1\cup\calr}(v), \deg_{\calw_0}(v)$;
- for each vertex $v\in \calw_1\cup \calr$, $\deg_{\calr}(v)$, $\deg_{\calw_0}(v)$, $\deg_{\calw_1}(v)$;
- all vertex-copy pairs that have one vertex-copy in $\calw_1$ and the other in $\calw_1\cup \calr$.
A priori, it is not obvious that this is the information we should expose. For example, for each $v\in \calw_0$, it may seem more natural to expose $\deg_{\calr}(v)$ and $\deg_{\calw_1}(v)$ rather than $\deg_{\calw_1\cup\calr}(v)$. But this precise set of information is what we needed to make our analysis work.
We restate STRIP here, describing how it runs on a configuration and adding details about how we expose the pairs of the configuration. Recall that pairs of vertex-copies correspond to edges in a graph, so when we remove a vertex-copy we do not [[ necessarily]{}]{} remove the actual vertex or any other copies of the vertex.
Recall the procedure STRIP and the definitions of $W_0,W_1,R$ and [[ being]{}]{} deletable from Section \[stsp\].
[**STRIP2**]{}
1. [[ Begin with $\La$ and set $\calw_0=W_0,\calw_1=\emptyset,\calr=V(\La)\setminus W_0$. (Note that these sets are $W_0,W_1,R$ respectively at the beginning of our procedure.) Now expose the RW-information.]{}]{}
2. Initialize $Q$ to be all vertices $v$ with $\deg(v)>2k$ or $v\notin W_0$ and $v$ has at least $\hf k$ neighbours in $W_0$.
3. Until $Q=\emptyset$ or until we have run $\b n$ iterations, let $v$ be the next vertex in $Q$, according to a specific fixed vertex ordering.
1. Expose the partners of every vertex-copy of $v$ (of course, if they are not already exposed). Let [[ $N(v)$]{}]{} be the set of neighbours of $v$ [[ remaining at this point.]{}]{}
2. Remove $v$ from $\La$ (and from $Q$), along with all vertex-copies of $v$ and their partners.
3. If any $u \in {{\color{black} N(v)\cap R}}$ has its degree decreased to at most $k$, then
1. move $u$ from $R$ to $W_1$;
2. [[ expose the vertex-copies of $u$ that have partners in $W_1\cup R$; for each such vertex-copy, expose its partner.]{}]{}
4. If any vertex $w\notin Q$ is now deletable, place $w$ into $Q$.
5. [[ Set $\calw_0=W_0,\calw_1=W_1,\calr=R$ and expose the RW-information of the remaining configuration.]{}]{}
The exposure of the RW-information in step [[ 3]{}]{}(e) is redundant; that information was in fact exposed during previous steps. We state step [[ 3]{}]{}(e) in this way to be explicit about the fact that it is exposed.
We define $W_0(i),W_1(i), R(i)$, and $Q(i)$ to be those vertex sets at the end of iteration $i$ of STRIP2. We define $\Psi(i)$ to be the RW-information of the remaining configuration with partition $\calr=R(i), \calw_0=W_0(i)$, and $\calw_1=W_1(i)$; i.e. the RW-information exposed during step 3[[ (e)]{}]{}. $W_0(0), W_1(0), R(0), Q(0), \Psi(0)$ are the sets and RW-information from steps 1 and 2.
\[orw\] We expose enough information to carry out each step of STRIP2.
[[**Proof.**]{}]{}$\Psi(0)$ specifies the vertices of $Q$ in step 2. Now consider iteration $i$ of step 3. The exposed partners of the vertex-copies of $v$ allow us to loop through the vertices $u$ in Step 3(c), and the RW-information $\Psi(i-1)$ tells us the degree of each such $u$.
To determine which vertices $w$ become deleteable: We know whether $w$ satisfies (D3) from $\Psi(i-1)$ and the number of vertex-copies of $w$ that were removed in previous steps. If $w\in R$ now satisfies (D4) then $w$ had 0 neighbours [[ (or 1, respectively)]{}]{} in $W_1$ at the end of iteration $i-1$ (we know this from $\Psi(i-1)$, and at least 2 neighbours [[ (or 1, respectively)]{}]{} of $w$ moved into $W_1$ during step 3(c) (we know this because the neighbours of those vertices in $R$ were exposed in step 3(c)). If $w\in W_1$ now satisfies (D5) then first note that we have exposed all neighbours of $w$ that are in $W_1\cup R$, either in Step 3(c) if $w$ entered $W_1$ during this iteration, or in $\Psi(i-1)$ otherwise. Any such neighbour in $R$ that is now deleteable satisfies (D4); we have already described how we know whether it is deleteable. This is enough to determine whether $w$ satisfies (D5). [**Clarification:**]{} It is important to note that in the definition of RW-information, $\calr,\calw_0,\calw_1$ are not necessarily set to be the sets $W_0,W_1,R$ at some point during STRIP2; they can be any partition of the vertices. This arises in the proof of Lemma \[luni\] below; in particular, it is the reason that we need to prove Claim 2.
Given a particular partition into $\calr,\calw_0,\calw_1$, and RW-information, $\Psi$, let $\Omega_{\Psi}$ be the set of all configurations on vertex set $\calr\cup \calw_0\cup \calw_1$ with RW-information $\Psi$; that is, all configurations in which each $v$ has $\deg_{\calr}(v),\deg_{\calw_0}(v), \deg_{\calw_1}(v), \deg_{\calw_1\cup\calr}(v)$ equal to the values prescribed in $\Psi$, and where the set of pairs with one copy in $\calw_1$ and the other in $\calw_1\cup \calr$ is as prescribed in $\Psi$.
The next lemma allows us to analyze STRIP2 by treating the configuration remaining after iteration $t$ as being uniformly selected from $\Omega_{\Psi(t)}$.
\[luni\] For any $t\geq 0$, and any possible set $\Psi(t)$: every configuration in $\Omega_{\Psi(t)}$ is equally likely to be the subconfiguration remaining after $t$ iterations.
[[**Proof.**]{}]{}Let $H$ be [[ the subconfiguration induced by the vertex-copies that remain after $t$ iterations of STRIP2]{}]{}. Consider any $H'\in\Omega_{\Psi(t)}$, and form $\La'$ from $\La$ by replacing the pairs of $H$ with the pairs of $H'$.
[*Claim 1: Each vertex $v$ has the same degree in both $\La$ and $\La'$.* ]{}
[*Proof:*]{} Indeed, [[ $\La,\La'$]{}]{} differ only on the pairs of $H,H'$ and, by construction, $v$ has the same degree in $H'$ as in $H$. The claim holds.
Claim 1 says that $\La,\La'$ have the same degree sequence [$\cald$]{} and so both are equally likely to be chosen as our initial configuration. The remainder of this proof will establish that if we apply $t$ iterations of STRIP to $\La'$ we will obtain $H'$. This will imply that $H,H'$ are both equally likely to be what remains after applying $t$ iterations of STRIP2 to the our initial random configuration. This [will finish the proof of]{} the lemma.
We define $H(i), H'(i)$ to be the sub-configurations remaining after applying $i$ iterations of STRIP2 to $\La,\La'$, respectively. So $H=H(t)$ and we wish to show that $H'=H'(t)$. Note that this does not follow [[ immediately]{}]{} from the fact that $H'\in\Omega_{\Psi(t)}$. [[ $\Psi(t)$ specifies, for example, that each vertex $v\in H$ has the same number of neighbours from $\calr=R(t)$ in both $\La$ and $\La'$ at the beginning of the procedure. But it is not obvious that, after running $t$ iterations of STRIP2, $\calr$ is the set of remaining vertices with degree greater than $k$. Claim 2, below, argues that this is indeed the case.]{}]{}
We let $W_0'(i), W_1'(i), R'(i), Q'(i)$ denote the vertex sets $W_0,W_1,R,Q$ after applying $i$ iterations of STRIP2 to $\La'$. In what follows, we use $\deg(v), \deg_R(v)$, etc. to denote degrees in $\La$ and $\deg'(v), \deg'_R(v)$, etc. to denote degrees in $\La'$.
[*Claim 2: for every $0\leq i\leq t$, $W_0(i)=W_0'(i), W_1(i)=W_1'(i), R(i)=R'(i)$, and $Q(i)=Q'(i)$.*]{}
[*Proof:*]{} By definition, $W_1(0)=W_1'(0)=\emptyset$. Claim 1 implies that $W_0(0)=W_0'(0)$ and $R(0)=R'(0)$. To prove that $Q(0)=Q'(0)$ we apply Claim 1 and we also need to argue that each vertex $v$ has the same number of neighbours in $W_0(0){=W_0'(0)}$ in both $\La$ and $\La'$. Note that $W_0(t)\subseteq W_0{(0)}$. The number of neighbours $v$ has in $W_0(t)$ is specified by $\Psi(t)$ to be the same in both $\La,\La'$, and the set of edges from $v$ to $W_0{(0)}\bk W_0(t)$ is identical in both $\La,\La'$ as $W_0{(0)}\bk W_0(t)$ is not in $H$.
Having established the base case, we proceed by induction [on $i$]{}. Suppose that Claim 2 holds for some $i<t$. Since $Q(i)=Q'(i)$, the [[ $(i+1)$-st]{}]{} vertex deleted is the same for both $\La,\La'$, since we choose the next deletable vertex according to the same ordering in both procedures; let $v$ be that vertex. So the vertex set of the subgraph after $i+1$ steps is the same in both procedures. Furthermore, $v$ [[ is not in]{}]{} $H$ as it is removed during the first $t$ iterations on $\La$. [[ Thus]{}]{} the set of pairs including a copy of $v$ is the same in both $\La,\La'$ and hence in both remaining configurations.
The key observation is that all decisions made in STRIP2 are based on sets of pairs that are equal in $H(i), H'(i)$. This implies that we will make the same changes to the various vertex sets in both configurations. Indeed, the decisions made are determined entirely by:
- Pairs containing copies of $v$: these are the same in $H(i),H'(i)$ since $v\notin H$.
- [[ The decision as to whether $u$ is moved to $W_1$ in step 3(c) is determined by $\deg(u)$. Every vertex has the same degree in what remains of both $\La$ and $\La'$]{}]{} by Claim 1 and the fact that all pairs removed during the first $t$ steps of STRIP2 are the same in both $\La,\La'$, as they each include at least one vertex not in $H$.
- The decision as to whether $w$ enters $Q$ in Step 3(d) is determined by [[ whether $w$ is deleteable. Whether $w$ satisfies (D3) is determined by $\deg(v)$ which is the same in both processes as described above. Whether $w$ is now in $W_0,W_1$ or $R$ was decided in step 3(c) and so is the same in both processes. Whether $w\in R$ satisfies (D4) is determined by the number of pairs containing a copy of $w$ and a vertex-copy in $W_1$; all such pairs are the same in both $\La$ and $\La'$, either because the pair is removed during the first $t$ iterations, or because it is a pair between $W_1(t)$ and $R(t)\cup W_1(t)$ and hence is specified by $\Psi(t)$. The argument for whether $w\in W_1$ satisfies (D5) is the same.]{}]{}
So STRIP2 makes the same decisions, and carries out the same steps during iteration $i+1$ on both $\La$ and $\La'$. This yields the claim for iteration $i+1$ [and so the proof of the claim is finished]{}.
Therefore STRIP[2]{} removes the same sequence of vertices for the first $t$ iterations on $\La$ and on $\La'$, and so $H'$ is what remains after $t$ iterations on $C$. [This finishes the proof of the lemma as explained above.]{}
Lemma \[luni\] allows us to analyze the configuration remaining after any iteration $t$ of STRIP[2]{} by taking a uniform member of $\Omega_{\Psi(t)}$. This is fairly simple as the members of $\Omega_{\Psi(t)}$ all decompose into the union of a few configurations which can be analyzed independently using the configuration model. Given any three disjoint vertex sets $\calw_0,\calw_1,\calr$, a set of RW-information $\Psi$, and a configuration $\La\in\Omega_{\Psi}$ we define:
- $\La_{\calw_0}\subset \La_{\Psi}$ - the sub-configuration induced by $\calw_0$;
- $\La_{\calw_1}\subset \La_{\Psi}$ - the sub-configuration induced by $\calw_1$;
- $\La_{\calr}\subset \La_{\Psi}$ - the sub-configuration induced by $\calr$;
- $\La_{\calw_0,\calw_1\cup\calr}\subset \La_{\Psi}$ - the bipartite sub-configuration induced by $\calw_0,\calw_1\cup\calr$;
- $\La_{\calw_1,\calr}\subset \La_{\Psi}$ - the bipartite sub-configuration induced by $\calw_1,\calr$.
Note that $\Psi$ specifies the vertex-copies and pairs of $\La_{\calw_1}, \La_{\calw_1,\calr}$; thus it also specifies which vertex-copies in $\calw_1$ are in $\La_{\calw_0,\calw_1\cup\calr}$. To select a uniform member of $\La_{\Psi}$, we can select the other three configurations independently; that is,
1. For each vertex $v\in \calr$, choose a uniform partition of the vertex-copies of $v$ not paired with copies in $\calw_1$ into those that will be paired with copies in $\calw_0,\calr$, according to $\deg_{\calw_0}(v), \deg_{\calr}(v)$.
2. For each vertex $v\in \calw_0$, choose a uniform partition of the vertex-copies of $v$ into those that will be paired with copies in $\calw_0,\calw_1\cup\calr$, according to $\deg_{\calw_0}(v), \deg_{\calw_1\cup\calr}(v)$.
3. Choose $\La_{\calw_0}$ by taking a uniform matching on the appropriately selected vertex-copies in $\calw_0$.
4. Choose $\La_{\calr}$ by taking a uniform matching on the appropriately selected vertex-copies in $\calr$.
5. Choose $\La_{\calw_0,\calw_1\cup\calr}$ by taking a uniform bipartite matching on the appropriately selected vertex-copies in $\calw_0,\calr$ and the vertex-copies of $\calw_1$ that are (implicitly) specified by $\Psi$ to be paired with $\calw_0$.
To see that this yields a uniform member of $\La_{\Psi}$, note that there is a bijection between the union of these choices and the members of $\La_{\Psi}$; note also that the number of choices for steps 3,4,5 is independent of the partitions chosen in steps 1,2.
In every use of this model, we will use it to analyze the configuration remaining at a particular point during STRIP2. We will set $\calw_0=W_0, \calw_1=W_1, \calr=R$ and so we will use the notation, for example, $\La_{W_1,R}$.
The crucial parameter in our analysis is the branching parameter that comes from exploring $W_0{(0)}$ with a branching process. It is well-known that for any $c>c_k$, $W_0{(0)}$ is subcritical and so this parameter is less than [[ 1]{}]{} (see eg. [@mrbranch; @sato]). We need to show how far away it is from [[ 1]{}]{} when $c={\cmin}$. This bound is the only reason that we require $c\geq {\cmin}$ rather than $c>c_k$ in (\[eq:conditions\_for\_c\]).
\[lbr\] Let $k \in {{\mathbb N}}$ be a sufficiently large constant and set $\a=k^{9}\b$. Let $G=G_{n,p}$ be a random graph with $p=c/n$ for some constant $c_k + k\a = \cmin \le c \le \cmax = c_k + k^{-1/2}$. Then, w.h.p. in the $k$-core of $G$, i.e. at step $i=0$ of STRIP: $$\label{elbr}
\frac{\sum_{u\in W_0}\deg_{W_0}(u)(\deg_{W_0}(u)-1)}{\sum_{u\in W_0}\deg_{W_0}(u)} < 1-\a.$$
[[**Proof.**]{}]{}Recall from (\[ebeta\]) that $\b=e^{-k/200}$ and so $\a$ is very small.
We are going to use the notation and observations used in the proof of Lemma \[lw0\]. See (\[eq:c\]) for the definition of $f(x)$ and $x$, (\[eq:xk\]) for the definition of $x_k$, and (\[eq:deg\_distribution\]) for the degree distribution of the $k$-core. In particular, recall that $c=f(x)$ and $c_k=f(x_k)$. Our first step is to use the fact that $c$ is bounded away from $c_k$, to bound $x$ away from $x_k$. We set $$x=x_k+\d.$$
[*Claim:*]{} $\delta >k \alpha$.
[*Proof:*]{} Recall from (\[edbound\]) that $c\leq\cmax$ implies $\d\leq\log k$. Also recall that $\e=o_k(1)$ was defined in (\[eq:error2\]) and used in (\[eq:bound\_for\_f’\]). Since $\a<(\log k)/k$, and since $\e+k\a=o_k(1)$, it follows from (\[eq:bound\_for\_f’\]) that over the range $[x_k,x_k+k\a]$, $f'$ does not exceed $(\e+k\a)\sqrt{(\log k)/k}+O(1/\sqrt{k})<\sqrt{(\log k)/k}$ for $k$ sufficiently large. Therefore, $f(x_k+k\a)< f(x_k) + (k\a)\cdot \sqrt{(\log k)/k}<c_k+k\a=\cmin \leq c$. Since $c=f(x)=f(x_k+\d)$ and $f$ is increasing, this implies the claim.
We consider the configuration $\La$ with the same degree sequence as the $k$-core of $G$. Recall that $W_0$ is the set of vertices with degree exactly $k$ in $\La$. We could determine the degree sequence of the subconfiguration induced by $W_0$ and then bound the LHS of (\[elbr\]), but we obtain a simpler calculation by considering the following experiment: choose a uniformly random vertex-copy [[ $\sigma$]{}]{} from $W_0$ conditional on [[ $\sigma$]{}]{} being paired in $\La$ with another copy from $W_0$; let $u\in W_0$ be the vertex of which [[ $\sigma$]{}]{} is a copy. Set $Z$ to be the number of other copies of $u$ that are paired with vertex-copies in $W_0$. Note that ${\mbox{\bf E}}(Z)$ is the LHS of (\[elbr\]).
One way to choose [[ $\sigma$]{}]{} is to repeatedly take a uniform vertex-copy from $W_0$ and expose its partner; halt the first time that the partner is also in $W_0$. By Lemma \[lw0\](b), [[ ]{}]{} a linear proportion of the copies are in $W_0$ and so we only expose $o(n)$ [[ pairs of vertex-copies]{}]{} before halting. Now, having found $u$, we expose the partners of the remaining $k-1$ copies of $u$. So ${\mbox{\bf E}}(Z)$ is simply the expected number of these partners that are in $W_0$. [[ Since a vertex in $W_0$ has degree $k$, $Z \le k$ always holds, and on the other hand ${\mbox{\bf E}}(Z)=\Omega(1)$, as by Lemma \[lw0\](b), $|W_0| \ge 0.99n/k$, and in such case ${\mbox{\bf E}}(Z)=\Omega(1)$. Hence, the first order term of ${\mbox{\bf E}}(Z)$ stems from the event when Lemma \[lw0\](b) holds and only $o(n)$]{}]{} [[ pairs of vertex-copies]{}]{} were exposed before halting. [[ In that case, the probability that a particular copy of $u$ selects a partner in $W_0$ is simply the total degree of $W_0$ divided by the total degree of $\Lambda$ plus a $o(1)$ term for the exposed pairs. Applying (\[eq:deg\_distribution\]) to obtain this ratio, we find that $E(Z)=(1+o(1))g(x)$ where]{}]{}
$$g(x) := (k-1)\cdot \frac{k \ e^{-x}\ \frac{x^k}{k!}}{\sum_{i \ge k} i\ e^{-x}\ \frac{x^i}{i!}} =\frac{k(k-1)\ e^{-x} \ \frac{x^{k}}{k!}}{x \cdot e^{-x} \sum_{i \ge k-1}\frac{x^i}{i!}}
= \frac{k(k-1)\ e^{-x} \ \frac{x^{k}}{k!}}{x \cdot \Pr{ ({\mbox{\bf Po}}(x) \ge k-1). }}$$
As mentioned in the proof of Lemma \[lw0\][[ (b)]{}]{}, $x_k$ minimizes the function $f(x)$. [[ Using the fact that]{}]{} $f'(x_k)=0$, it is a simple exercise to show that $g(x_k) = 1$; see [@Molloy_cores] for the details. Hence, using the fact that $\Pr{ ({\mbox{\bf Po}}(x) \ge k-1) } \ge \Pr{ ({\mbox{\bf Po}}(x_k) \ge k-1) }$ for $x>x_k$, it follows from (\[eq:error1\]) that $$\begin{aligned}
g(x) & \le \frac{k (k-1)\ e^{-x_k}\ \frac{x_k^k}{k!} \left( 1 - \delta (\log k/k)^{1/2} (1+o_k(1)) \right) }{(x_k + \delta) \ \Pr{ ({\mbox{\bf Po}}(x_k) \ge k-1) }}\\
& = g(x_k) \left( 1 - \delta (\log k/k)^{1/2} (1+o_k(1)) \right) \\
& \le 1 - \frac {1}{2} \delta (\log k/k)^{1/2}< 1-\alpha, \qquad\qquad\qquad\qquad\mbox{ by {{\color{black} our Claim}}}\end{aligned}$$ provided $k$ is large enough.
The procedure terminates quickly
--------------------------------
In this section, we will prove that STRIP2 terminates with $Q=\emptyset$ when run on $\La$. In order to show that $Q$ reaches $\emptyset$ within $\b n$ iterations, we will keep track of a weighted sum of the total degree of the vertices in $Q$, and we will show that this parameter drifts towards zero.
Within this parameter, the change in the number of edges from $Q$ to $W_0$ is the most delicate. In particular, the most sensitive part of our process is avoiding cascades that could be formed when the deletion of vertices in $W_0\cap Q$ causes too many other vertices in $W_0$ to be added to $Q$. (Roughly speaking, Lemma \[lbr\] ensures that such cascades do not occur.) So we place a high weight on the number of edges from $Q\cap W_0$ to $W_0$. We place an even higher weight on the edges from $Q\bk W_0$ to $W_0$; these edges play a different role in the analysis because they initiate the potential cascades. [[ We express this weighted sum with the following variables; each refers to the sets $W_0,W_1,R$ at the end of iteration $i$ of STRIP2.]{}]{} $$\begin{aligned}
A_i&=& \sum_{v\in Q\cap W_0} \deg_{W_0}(v)\\
B_i&=& \sum_{v\in Q\bk W_0} \deg_{W_0}(v)\\
D_i&=& \sum_{v\in Q} \deg_{W_1\cup R}(v)\\
X_i&=&A_i +kB_i + {{\color{black} k^7}}\b D_i \end{aligned}$$ Note that, [indeed]{}, since $\b= e^{-k/200} < k^{-10}$, the edges counted by $A_i$ and $B_i$ have much higher weights in $X_i$ than those counted by $D_i$.
We will prove that $X_i$ has a negative drift, and that it [[ reaches]{}]{} zero before $\b n$ iterations of STRIP2. At that point, if any vertices remain in $Q$ then they all have degree zero and so will be removed without any new vertices being added to $Q$.
\[obxx\] [[ W.h.p. ]{}]{} throughout STRIP2, we always have:
1. $|W_1|\leq 3k\b n$;
2. ${{\color{black} e(R, W_0)}}\geq \frac{n}{3}$;
3. ${{\color{black} e(R,R)}}\geq \frac{kn}{4}$;
4. $|Q|\leq {{\color{black} 5k^2}}\b n$.
[[**Proof.**]{}]{} [[he observation follows by noting that, since STRIP2 carries out a very small number of iterations and hence deletes a very small number of vertices, these parameters will not change much from their initial values as provided in Lemma \[lw0\]. [[ In more detail]{}]{}, STRIP2 runs for]{}]{} at most $\b n$ iterations, where $\b=e^{-k/200}$ from (\[ebeta\]), and in each iteration we delete one vertex.
[[ Initially, $W_1=\emptyset$ and every vertex that moves to $W_1$ is the neighbour of a deleted vertex. So $|W_1|$ is bounded by the total degree of the deleted vertices.]{}]{} The total degree of the vertices from the initial set $Q$ is at most [$e^{-k/6}n + (2k)e^{-k/3}n < k\b n$]{} by Lemma \[lw0\](c,g). Every other deleted vertex has degree at most $2k$ and so the total degree of all deleted vertices is at most $(2k) \b n+k\b n=3k\b n$; [[ this proves part (a).]{}]{}
[[ For part (b), observe that by Lemma \[lw0\](e), initially, $e(R,W_0) \ge \frac12 n$. Since by the proof of part (a) the total degree of all deleted vertices is at most $3k \b n$, and the number of edges deleted is bounded by the total degree of all deleted vertices, at any time $e(R, W_0) \ge \frac12n - 3k\b n \ge \frac13 n$. By an analogous argument, this time using Lemma \[lw0\](f), $e(R,R) \ge \frac{nk}{3} - 3k\b n \ge \frac{nk}{4}$, and part (c) follows. ]{}]{} Finally, for part (d): [[ Lemma \[lw0\](c,g) implies that the size of the initial set $Q$ is at most $(e^{-k/6}+ e^{-k/3})n<\b n$. Observation \[oq4k2\] says that at most $4k^2$ vertices are added to $Q$ during each of the at most $\b n$ steps. So $|Q|$ can never exceed $\b n+4k^2\b n<5k^2\b n$.]{}]{}
The key parameter in bounding the drift of $X$ is [controlled by the following lemmas.]{}
[[ If $\s$ is a vertex-copy, then we use $u(\s)$ to denote the vertex that $\s$ is a copy of.]{}]{}
\[lq0\] [[ W.h.p. at every]{}]{} iteration $i\leq \b n$ of STRIP2, [[ the RW-information $\Psi_i$ is such that:]{}]{} if a vertex-copy $\s$ is chosen uniformly at random from the remaining $W_0$-copies in ${{\color{black} \La}}$, then $$\ex\left(\deg_{W_0}(u(\s))\right)<\left(2-\frac{\a}{2}\right),$$ where $\a=k^{9}\b$ is from Lemma \[lbr\].
[[**Proof.**]{}]{} The expectation we are bounding is [equal to]{}: $$\label{ed1}
\sum_{u\in W_0}\deg_{W_0}(u)^2/\sum_{u\in W_0}\deg_{W_0}(u).$$ By Lemma \[lbr\], at iteration $i=0$ this is $$\frac{\sum_{u\in W_0}\deg_{W_0}(u)(\deg_{W_0}(u)-1)+\sum_{u\in W_0}\deg_{W_0}(u)}{\sum_{u\in W_0}\deg_{W_0}(u)} < 1-\a +1 = 2-\a.$$ At iteration $1\leq i\leq \b n$, each of the $i$ vertices that have been deleted decreases the denominator of (\[ed1\]) by at most $2k$ (the largest effect is when we remove a vertex of $W_0$ with $k$ neighbours in $W_0$) and so the denominator has decreased by at most $2k\b n$. The numerator has not increased. Since the denominator was initially at least $n/(5k)$ by Lemma \[lw0\](d), the value of (\[ed1\]) is at most $$(2-\a) \cdot \frac {n/(5k)}{n/(5k) - 2k\b n} = \frac{2-\a}{1-\frac{2k\b n}{n/5k}} < (2-\a)(1+10 k^2\b)<2-\frac{\a}{2},$$ as $\a=k^{9}\b$.
This brings up to our key lemma:
\[ldriftx\] [[ W.h.p. at every iteration $i\leq \b n$ of STRIP2, the RW-information $\Psi_i$ is such that:]{}]{} if the vertex $v\in Q$ that is deleted has degree at least one, then $$\ex(X_{i}-X_{i-1})\leq -\hf {{\color{black} k^7}}\b.$$
[**Remark:**]{} If $v$ has degree zero then $X_{{i}}=X_{{i-1}}$ [(deterministically)]{} as during that iteration of STRIP2, no vertex-copies will be removed from the configuration, and no vertices will join $Q$.
First note that $X_{i}$ can only increase through vertices joining $Q$. So most of this analysis focuses on the expected number of vertices that are added to $Q$. [[ This analysis relies on the fact that all RW-information is specified deterministically by $\Psi$ (see Definition \[def:RW\]), and that the configuration is uniform amongst all configurations with that RW-information (see Lemma \[luni\]).]{}]{}
$\Psi$ specifies that vertex $v$ [is incident to]{} [[ $\deg_{W_0}(v)$]{}]{} edges to $W_0$, [[ $\deg_{W_1}(v)$]{}]{} edges to $W_{1}$, [[ $\deg_{R}(v)$]{}]{} edges to $R$ [[ and/or $\deg_{W_1\cup R}(v)$ edges to $W_1\cup R$ depending on whether $v\in W_1 \cup R$ or $v\in W_0$.]{}]{} We consider the effect on $X_{{i}}-X_{{i-1}}$ of deleting each of these pairs; [[ specifically, the effect of deleting a copy of $v$ and a copy of some $u\in N(v)$ which is specified to be in]{}]{} [[ $W_0,W_1, R$ or $W_1\cup R$.]{}]{}
[*Case 1: $v\in W_1$.*]{} [[ Subcase $u \in W_1$:]{}]{} The removed pair is not random; it is specified [[ by $\Psi$, as $\Psi$ specifies all pairs with one member in $W_1$ and the other in $W_1 \cup R$. ]{}]{} By Observation \[ob1\](a), [[ $u$]{}]{} is already in $Q$. So no new vertices are added to $Q$ and $D_{i}$ decreases by [[ exactly]{}]{} two. [[ Thus, the deletion of $uv$ causes $X$ to change by]{}]{} $$\partial X=-2{{\color{black} k^7}}\b.$$
[[ Subcase $u \in R$]{}]{}: The removed pair is not random; it is specified by $\Psi$. By Observation \[ob1\](c), [[ $u$ is]{}]{} already in $Q$ or has no other neighbours in $W_1$; either way, any neighbours that $u$ has in $W_1$ are already in $Q$. Also, those neighbours are specified by $\Psi$ (since they are edges from $R$ to $W_1$) and so they do not need to be exposed.
Thus, any new vertices that are added to $Q$ are the result of the neighbours of $u$ that are in $R$, and so we turn our attention to those neighbours. [Note that the degree of $u$ is at least $k+1$ before the removal of the pair of copies $uv$ (since $u \in R$); if $u$ has degree at least $k+2$ before the removal, no more vertices will be moved to $Q$.]{} So for the remainder of the analysis, we will assume that $u$ has degree $k+1$ [[ before the removal of $uv$]{}]{}. Thus $u$ moves to $W_1$ and we expose its at most $k$ neighbours in $W_1\cup R$. There are two ways that the choice of one such partner, $w$, can cause vertices to be added to $Q$:
1. [[ $w\in R$ and]{}]{} $w \in Q$. Then $u$ is added to $Q$ if it was not already in $Q$. [[ (Note that if $w\in W_1$ then $u$ is already in $Q$ as it had two neighbours in $W_1$.)]{}]{}
2. [[ $w\in R$]{}]{}, $w\notin Q$ and has exactly one neighbour $w'\neq u$ in $W_1$. So $w$ is added to $Q$, and also $w'$ is added to $Q$ if it was not already in $Q$. If $w$ has more than one such neighbour $w' \in W_1$, $w'\neq u$, then $w$ and all such neighbours would already be in $Q$.
[[ Both of those situations require $w\in R$, so to analyze the effect of those possibilities we consider the at most $k$ neighbours $u$ has in $R$.]{}]{} To expose a neighbour [[ $w$]{}]{} of $u$ in $R$, we choose the partner of a [[ copy of $u$]{}]{} in $\La_R$; that is, we choose a uniform vertex-copy in $\La_R$; $w$ is the vertex containing that copy.
Note that, since $w\in R$, the edges from $w$ to $W_1$ are specified by $\Psi$, and so we do not need to expose any new partners of copies of $w$ to determine whether $w$ has a neighbour $w'\in W_1$. [[ Furthermore, the vertices $w$ that would result in additions to $Q$ as in (a,b) above are specified by $\Psi$. So to bound the expected change in $X$, we bound the number of copies of such $w$.]{}]{} By Observation \[obxx\](a,c,d):\
1. There are at least $kn/2$ vertex-copies in $\La_R$ to choose from.
2. At most $2k\cdot {{\color{black} 5k^2}} \b n + \b n < {{\color{black} 11k^3}} \b n$ of them are copies of vertices in $R\cap Q$ (as the total degree of vertices with degree greater than $2k$ is at most ${e^{-k/6} n <} \b n$ by Lemma \[lw0\](c)). [[ If we choose one of these copies then we have situation (a) and so up to one vertex could be added to $Q$.]{}]{}
3. At most $6 k^3 \b n$ of them are copies of the at most $3 k^2 \b n$ neighbours of $W_1$ not already in $Q$ (as each such neighbour has degree at most $2k$). [[ If we choose one of these copies then we have situation (b) and so up to two vertices could be added to $Q$.]{}]{}
[[ Recall that in this portion of the analysis, $\deg(u)=k+1$ and one of $u$’s neighbours, $v$, has been deleted. ]{}]{} So the expected number of vertices added to $Q$ is at most $k \cdot {{\color{black} 23k^3}} \b n/(kn/{2}) \leq {{\color{black} 46k^3}}\b$. Each vertex added to $Q$ will increase $X$ by at most $2k^2$ (the extreme case is if it has $2k$ neighbours in $W_0$). In addition, the removal of the pair $uv$ causes $D_{i}$ to decrease by at least one. So the expected change to $X$ by the removal of the pair $uv$ is at most: $$\label{eqx1}
\partial X \leq -{{\color{black} k^7}}\b + 2k^2\cdot {{\color{black} 46 k^3}}\b <-\hf {{\color{black} k^7}}\b,$$ [provided $k$ is large enough.]{}
[[ Subcase $u \in W_0$]{}]{}: We remove a copy of $v$ and its partner [[ in $\La_{W_0,W_1\cup R}$; that copy]{}]{} is selected by choosing a uniform vertex-copy from the $W_0$-copies in $\La_{W_0,W_1\cup R}$. [[ This specifies $u$, the vertex to which the copy belongs, and $u$ is added]{}]{} to $Q$. No other vertices are added to $Q$ in this step. $A_{i}+D_{i}$ can increase by at most $k-1$, as the vertex of $W_0$ added to $Q$ has remaining degree at most $k-1$. On the other hand, the removal of the pair causes $B_{i}$ to decrease by one. So [[ the deletion of $uv$ causes $X$ to change by $$\partial X \leq -k + (k-1) = -1.$$ ]{}]{}
[*Case 2: $v\in R$.*]{} [[ Subcase $u \in W_1$]{}]{}: The removed pair is not random; it is specified by $\Psi$. By Observation \[ob1\](b), the other endpoint of the edge is already in $Q$. So no new vertices are added to $Q$ and $D_{i}$ decreases by exactly two. [[ Thus, as in Case 1, the deletion of $uv$ causes $X$ to change by $$\partial X=-2{{\color{black} k^7}}\b.$$ ]{}]{}
[[ Subcase $u \in R$:]{}]{}
There are two differences between this subcase, and the corresponding subcase in Case 1.
1. The number of vertices $u\in N(v)$ that are in $R$ is specified by $\Psi$, but the actual vertices are not specified by $\Psi$ - they are selected randomly.
2. It is possible that $u\notin Q$ and $u$ has a neighbour in $W_1$.
As in [[ Case 1, subcase $u\in R$,]{}]{} the degree of $u$ prior to the removal of $uv$ is at least $k+1$. If that degree at least $k+2$, then no vertex is moved to $Q$ as a result of the deletion of $uv$. So we assume that degree is $k+1$. Thus $u$ moves to $W_1$ and we expose its at most $k$ neighbours in $W_1\cup R$. There are three ways that the choice of one such partner, $w$, can cause vertices to be added to $Q$. The first two, arising when $w\in R$, are the same as (a,b) from Case 1, and the analysis of the effect of those possibilites on the expected change in $X$ is the same as in Case 1. The third is:
\(c) $w\in W_1$. In that case, $u$ moves to $Q$ and so does $w$ if it is not already in $Q$. Note that if $u$ has more than one neighbour in $W_1$ then $u$ and those neighbours were already in $Q$.
We expose each of the $\deg_R(v)$ neighbours of $v$ in $R$ by choosing a vertex-copy uniformly from those in $\La_R$; $u$ is the vertex containing that copy.
The set of vertices $u$ whose choice would result in situation (c) is specified by $\Psi$, and there are at most $|N(W_1)|\leq |W_1|\times k$ such vertices. Each such vertex has at most $k$ copies in $\La_R$, as it has degree $k+1$ and one neighbour in $W_1$, so by Observation \[obxx\](a). there are at most $3k^3\b n$ copies of such vertices in $\La_R$. By Observation \[obxx\](c), there are at least $kn/2$ vertex-copies in $\La_R$ to choose from. So the probability that our choice of $u$ results in situation (c) is at most [[ $3k^3\b n/(\hf kn)=6k^2\b$]{}]{}. If we choose such a copy then up to two vertices will move to $Q$, and each can increase $X$ by at most $2k^2$. So the expected impact on $X$ [[ by exposing the vertex $u$]{}]{} is at most [[ $2k^2 \cdot 6k^2\b = 12k^4\b$.]{}]{}
[[ After exposing $u$, we next expose $N_R(u)$ in order to determine whether any vertices were added to $Q$ because of situations $a,b$. The same analysis as for (\[eqx1\]) shows that the expected impact on $X$ of this step is at most $-{{\color{black} k^7}}\b + 2k^2\times {46 k^3}\b$. As in the calculation for (\[eqx1\]), the removal of $uv$ causes $D_i$ to decrease by 1. ]{}]{} Putting this together, the expected change in $X$ resulting from the exposure and removal of the edge $uv$ is at most $$\label{eqx2}
\partial X \leq {{\color{black} -{{\color{black} k^7}}\b + 2k^2\cdot {46 k^3}\b + 12k^4\b<-{\frac{1}{ 2}}k^7\b}},$$ provided $k$ is large enough.
[[ Subcase $u \in W_0$:]{}]{} The same analysis as in Case 1, [[ subcase $u\in W_0$,]{}]{} shows that regardless of which vertex is selected, the change on $X$ will be $\partial X \leq -1$.
[*Case 3: $v\in W_0$.*]{} [[ In this case, $\Psi$ specifies the number of edges from $v$ to $W_1\cup R$, but does not specify how many go to $W_1$ and how many go to $R$.]{}]{}
[[ Subcase $u \in W_1\cup R$]{}]{}: The edge is a pair from $\La_{W_0,W_1\cup R}$ and so we choose [[ a vertex-copy uniformly from the $W_1\cup R$-copies in $\La_{W_0,W_1\cup R}$; $u$ is the vertex containing that copy.]{}]{}
[[ If we choose $u\in R$ then situations (a,b,c) above are the ways in which this can cause $Q$ to increase. We can apply the same analysis as for (\[eqx2\]).]{}]{} The only difference is that the number of $R$-copies in $\La_{W_0,W_1\cup R}$ is at least $n/{3}$ rather than at least $kn/{2}$, as we apply Observation \[obxx\](b) rather than Observation \[obxx\](c). The result is that the expected change to $X$ conditioning on the selected $u$ being in $R$ is at most $$\partial X \leq {{\color{black} -{k^7}\b + 2k^2\cdot {69 k^4}\b + 18k^5 \b}}.$$
If we choose $u\in W_1$, then $u$ will enter $Q$, if it is not already in $Q$; no other vertices will enter $Q$. By Observation \[obxx\](a,b) there are at least $n/{3}$ $R$-copies to choose from and at most $3k^2\b n$ $W_1$-copies. So the probability that we select [$u \in W_1$]{} is at most $3k^2\b n/ (n/{3})={9} k^2\b$. If $u$ is added to $Q$ then at most this will increase $B_i$ by $k-1$ and thus increase $X$ by $k(k-1)<k^2$. Putting this together, the expected change in $X$ resulting from the removal of the pair $uv$ is at most: [[ $$\partial X \leq 9k^2\b\cdot k^2+(1-9k^2\b)\cdot\left( -{k^7}\b + 2k^2\cdot {69 k^4}\b + 18k^5 \b\right)
\leq -\hf k^7\b,$$ ]{}]{} provided $k$ is large enough.
[[ Subcase $u \in W_0$]{}]{}: The edge is a pair from $\La_{W_0}$ and so [[ we choose a uniform vertex-copy from $\La_{W_0}$; $u$ is the vertex containing that copy.]{}]{} By Lemma \[lq0\], the expected increase in $A_{i}$ [[ from adding $u$ to $Q\cap W_0$]{}]{} is at most $2- \frac {\a}{2}$ where $\a=k^{9}\b$. The increase in $D_i$ is at most $k$, and deleting the pair $uv$ decreases $A_{i}$ by two. Putting this together, $$\ex(X_{i}-X_{i-1}) \leq -2 + \left( 2- \frac {\a}{2} \right) + k\cdot {{\color{black} k^7}}\b < - \frac {\a}{4} < -\hf {{\color{black} k^7}}\b.$$
So in every case, the deletion of a copy of $v$ and its partner [[ $u$]{}]{} results in an expected change in $X$ of less than $-\hf {{\color{black} k^7}}\b$. [[ Therefore $$\ex(X_{i}-X_{i-1})\leq -\hf {{\color{black} k^7}}\b\cdot \deg(v).$$]{}]{} Since $v$ has degree at least one, this yields the lemma.
Our bounds on the drift of $X_{i}$ and the initial size of $Q$ imply that our procedure stops quickly.
\[lstop\] W.h.p. STRIP2 halts with $Q=\emptyset$ within $\b n$ iterations.
[[**Proof.**]{}]{}We begin by showing that we reach $X_i=0$ long before step $i=\b n$.
As we argued in the proof of [[ Observation]{}]{} \[obxx\], Lemma \[lw0\](c,g) [[ implies]{}]{} that at iteration $i=0$, the total degree of the vertices in $Q$ is at most $(e^{-k/6}+2k\cdot e^{-k/3})n$ and so we have $X_0\leq k (e^{-k/6}+2k\cdot e^{-k/3})n<e^{-k/10}n$. Recall from (\[ebeta\]) that $\b=e^{-k/200}$.
By Lemma \[ldriftx\], for every $1 \le i \le \b n$, $$\E{ (X_{i+1}-X_{i})} \le -b, \qquad \mbox{ where } b:=\frac12 {{\color{black} k^7}} \beta.$$ Note also that (deterministically), for every $1 \le i \le \b n$, $|X_{i+1}-X_i| \le (2k)(4k^2) = 8k^3$, since the degree of a vertex not belonging to $Q$ is at most $2k$, and [[ by Observation \[oq4k2\] ]{}]{}we add at most $4k^2$ vertices to $Q$ in each iteration.
We will use the Martingale inequality from the end of Section \[spp\]. We cannot apply this directly to $X_i$ since $X_i$ stops changing when it reaches zero. So instead, we couple $X_i$ to a process which is allowed to drop below zero. We define $X_{i+1}'=X_{i+1}$ whenever $X_{i}>0$, and if $X_i=0$ then $X_{i+1}'=X'_i-1$ with probability $b$ and $X'_{i+1}=X_i'$ otherwise. So for all $i$ we have $\E{ (X'_{i+1}-X'_{i})} \le -b$ and $|X'_{i+1}-X_i'| \le 8k^3$. Setting $i^* := {{\color{black} \frac{\b}{k^3 }}}n$, we have $$\ex(X'_{i^*})\leq X'_0-i^*b< e^{-k/10}n- \frac12 {{\color{black} k^4}} \beta^2 n<- e^{-k/50}n.$$ Applying (\[eq:HA-inequality2\]) with ${{\color{black} \alpha=e^{-k/50}n}}$, [[ $\ell=i^*$ and $c_1,\ldots, c_\ell=8k^3$]{}]{} yields $${\mbox{\bf Pr}}(X_{{i^*}}>0) \le {\mbox{\bf Pr}}(X'_{{i^*}}>0) \leq \exp \left( - \frac {(e^{-k/50} n)^2}{\frac{2\b}{{{\color{black} k^3}}} n(8k^3)^2} \right) = \exp( - \Omega(n) ) = o(1).$$ At this point, $|Q| \le e^{-k/10}n+{{\color{black} 4k^2}}i^* <{{\color{black} \frac{5\b}{k}}}n $, since [[ by Observation \[oq4k2\]]{}]{} we add at most $4k^2$ vertices to $Q$ in each iteration. If $X_{i^*}=0$ then there are no edges from $Q$ to the remaining vertices outside of $Q$. From that point on, no vertices will be added to $Q$ and so STRIP2 halts after $|Q|$ further steps. So the total number of steps is at most $i^*+{{\color{black} \frac{5\b}{k}}}n<\b n$.
This proves Lemma \[mtc\] since carrying out STRIP2 performs the same steps as carrying out STRIP on $\La$ (the only difference is that STRIP2 also exposes some information). Since properties that hold on $\La$ also hold on $C$, the $k$-core of $G_{n,p=c/n}$, (see the discussion in Section \[scm\]), this yields Lemma \[mtsg\].
Enforcing (K4) {#sk4}
--------------
Observation \[ob2\], Lemma \[mtsg\], Corollary \[ck3\] and Lemma \[lstop\] imply that STRIP produces a subgraph $K$ satisfying properties (K1), (K2) and (K3). It only remains to enforce:
(K4) $k|K|$ is even.
To do so, we prove that the output of STRIP has a vertex $v$ which can be deleted without violating (K1), (K2) or (K3). Thus, if (K4) does not hold, then we remove $v$ to obtain our desired subgraph $K$.
\[lk4\] W.h.p. the output of STRIP contains a vertex of degree greater than $k$ whose neighbours all have degree greater than $k$.
[[**Proof.**]{}]{}We argue that the lemma holds when running STRIP2 on $\La$. It immediately follows for running STRIP on $C$.
By Lemma \[lw0\](h), initially at least $\frac{n}{{200}}$ vertices of $R$ have no neighbours in $W_0$. Since we remove at most one vertex per iteration, at most $\b n$ of these vertices are not in the output. By Observation \[obxx\](a), the output satisfies $|W_1| \le 3k\b n$. Each vertex in $W_1$ has degree at most $k$ [(in fact, at this point exactly equal to $k$)]{}, and so at most $3k^2\b n$ members of $R$ have a neighbour in $W_1$. This implies the lemma as $\frac{n}{200}-\b n-3k^2\b n>0$ for $k$ sufficiently large.
$K$ has a $k$-factor
====================
We focus on the subgraph $K$ which we obtained in the previous section by running STRIP on the $k$-core of our random graph, and then possibly deleting one vertex in Section \[sk4\]. We will apply Lemma \[lem:minimality\] with $\G:=K$ to prove that $K$ has a $k$-factor. First we establish some more random graph properties.
Further random graph properties
-------------------------------
Recall our setting: we begin with the random graph $G=G_{n,p=c/n}$ where $ c_k+k^{10}\b \le c \le c_k+k^{-1/2}$ (see (\[ebeta\]), (\[eq:conditions\_for\_c\])). $K$ is the subgraph of the $k$-core of $G$ obtained after applying **STRIP**, and possibly deleting one more vertex. [[ We have shown that $K$ satisfies properties (K1–4) (see Section \[sk4\]). We will require the following additional properties:]{}]{}
\[lem:properties\] There exist constants $\g,\e_0 >0,k_0 {\in \mathbb{N}}$ such that for any $k\geq k_0$, $K$ satisfies:
- For every $Y \subseteq V(K)$ with $|Y| \le 10 \epsilon_0 n$, $e(Y) < \frac{ k |Y|}{6000}$.
- For every $Y \subseteq V(K)$ with $|Y|\le \frac12 {{\color{black} |K|}}$, $e(Y, V(K) \setminus Y) \ge \gamma k |Y|.$
- For every disjoint pair of sets $X,Y \subseteq V(K)$ with $|X| \ge \frac{1}{200}|Y|$ and $|Y| \le \epsilon_0 n$, $e(X,Y) < \frac12 \gamma k |X|$.
- For every disjoint pair of sets $X, Y \subseteq V(K)$ with $|X|+|Y| \le \epsilon_0 n$, $e(X,Y) < \left( 1 +\frac{1}{2000}\right) |N(X) \cap Y|+ \frac{ k}{100}|X|$.
- For every disjoint pair of sets $S,T \subseteq V(K)$ with $|T|< \frac{1}{10}\epsilon_0 n$ and $|S| > \frac{9}{10}\epsilon_0 n$, $e(S,T) < \frac34 k |S|.$
- For every disjoint pair of sets $S,T \subseteq V(K)$ with $|T| \ge \frac{1}{10}\epsilon_0 n$, we have $e(S,T) \le k|S|+\frac{3}{4}\sqrt{k \log k}|T|$ and $\sum_{v \in T}d(v) > (k+\frac{7}{8}\sqrt{k \log k})|T|$.
Note that $\g,\e_0$ do not depend on $k$.
These properties all correspond to very similar properties in [@Mike; @Pawel], with the exception of [(P1)]{} which is very standard in random graph theory. There are no new ideas here, and the proof is a bit lengthy. So we defer the proof of Lemma \[lem:properties\] to Section \[sec:properties\].
Verifying Tutte’s condition
---------------------------
We now assemble our pieces to show that $K$ has a $k$-factor. Recall from Definition \[dlh\] that $L,H$ are the vertices of degree $k$ and at least $k+1$, respectively, and that, eg., $T_L, T_H$ denote $T\cap L, T\cap H$, respectively.
As we proved in Section \[sfts\], the subgraph $K$ has the following properties. Note that (K2) is rephrased using the notation of Definition \[dlh\].
- for every vertex $v\in K$, $k \le d_K(v) \le 2 k$;
- every vertex $v\in H$ has at most $\frac{9}{10}k$ neighbours in $L$;
- $|K|\geq \frac{n}{3}$;
- $k|K|$ is even.
[[ In the previous section, we showed that $K$ satisfies properties (P1–6), stated above.]{}]{}
Recall also Tutte’s condition : A graph $\G$ with minimum degree at least $k\geq 1$ has a $k$-factor if [[ and only if]{}]{} for every pair of disjoint sets $S,T \subseteq V(\G)$, $$k|S|+\sum_{v\in T_H}(d_{\G}(v)-k) \ge q(S,T)+e(S,T),$$ and recall also [[ from Lemma \[lem:minimality\]]{}]{} that a graph $\G$ with minimum degree $k\geq 1$ has a $k$-factor if [[ and only if]{}]{} this inequality holds for every pair of disjoint sets $S,T \subseteq V(\G)$ satisfying:
- $S \subseteq H$; and
- every component $Q$ counted by $q(S,T)$ satisfies $Q_H\neq\emptyset$.
By our previous lemmas, it will suffice to prove:
\[lkfactor\] If $K$ satisfies properties (K1–4,P1–[6]{}) then for every pair of disjoint sets $S,T \subseteq V(K)$ satisfying (M1–2) inequality (\[etutte2\]) holds.
In all but one case, we actually prove (\[etutte\]); it will be useful to restate it: $$k|S|+|T_H| \ge q(S,T)+e(S,T),$$ where $e(S,T)$ is the number of edges from $S$ to $T$ and $q(S,T)$ is the number of components $Q$ of $K \setminus (S\cup T)$ such that $k|Q|$ and $e(Q,T)$ have different parity. Recall that (\[etutte\]) implies (\[etutte2\]) since every vertex in $T_H$ has degree at least $k+1$.
[[[**Proof.**]{}]{}]{} Let $\epsilon_0,\d >0$ be the constants implied by Lemma \[lem:properties\]. [Recall that $\e_0,\d$ are independent of $k$ (see Remark 1 following the statement of Lemma \[lem:properties\]). So when we lower bound $k$ in what follows, our lower bounds can be in terms of $\e_0,\d$. We will assume that $k\geq k_0$ from Lemma \[lem:properties\] and so we can assume that properties (P1–6) all hold.]{}
We will consider two cases depending on the size of $S \cup T$.
*Case 1: $|S|+|T| \le \epsilon_0 n$:*\
Recall that $q(S,T)$ [[ is]{}]{} the number of connected components $Q$ of $K \setminus (S \cup T)$ such that $k|Q|$ and $e(Q,T)$ have different parities. Denote by $X$ the union of all vertices belonging to connected components $Q $ of $K \setminus (S \cup T)$ that contribute to $q(S,T)$, other than a largest component of $K \setminus (S \cup T)$ (this largest component might or might not contribute to $q(S,T)$, but is neglected in any case; [if more than one component is largest, we pick one of them arbitrarily]{}). As $K \setminus (S \cup T)$ has at most one component of size at least $\hf |K|$, we apply [(P2)]{} setting $Y$ to be any component in $X$. [[ N]{}]{}oting that all edges from $Y$ to $V(K)\bk Y$ are edges from $X$ to $S\cup T$ [[ and s]{}]{}umming over all such components yields: $$\label{exp3}
e(X, {{\color{black} S\cup T}}) \ge \gamma k |X|.$$ If $|X| > {\frac{1}{ 200}}(|S|+|T|)$ then we can apply [(P3)]{} with $(X,Y):=(X,S \cup T)$ to obtain $e(X,S\cup T)<\hf\g k |X|$ which contradicts (\[exp3\]). So we have: $$\label{ex200}
|X| \leq {\frac{1}{ 200}}(|S|+|T|).$$
Recalling that $S_H=S$ by (M1), we now turn our attention to the [[ vertices of $H$]{}]{}. (\[ex200\]) and the fact that we are in Case 1 imply that $|S|+|T_H|+|X_H|\leq |S|+|T|+|X|<2\e_0 n$ and so we can apply [(P1)]{} to obtain: $$\label{ep0}
e(S\cup T_H \cup X_H)< \frac{k}{6000} |S\cup T_H \cup X_H|.$$ [[ (K1-2)]{}]{} imply that each vertex in $X_H$ has at least $\frac{k}{10}$ neighbours in $H$. Every such neighbour must be in $X\cup S\cup T$ and so $e(X_H,S\cup T_H)\geq \frac{k}{10}|X_H|-2e(X_H)$. This yields $${{\color{black} e(S\cup T_H \cup X_H)\geq e(X_H)+e(S,T_H)+ e(X_H,S\cup T_H)\geq e(S,T_H)+ \frac{k}{10}|X_H|-e(X_H)}},$$ which combined with (\[ep0\]) and [(P1)]{} applied to $X:=X_H$ gives: $$\frac{k}{6000} (|S|+ |T_H| +| X_H|) \geq e(S,T_H)+ \left(\frac{k}{10}-\frac{k}{6000}\right)|X_H|\geq \frac{599k}{6000}|X_H|.$$ Rearranging allows us to replace (\[ex200\]) with a bound only involving the high vertices: $$\label{exh}
|X_H| \leq \frac{1}{400}(|S|+|T_H|).$$
(M2) implies that every component counted by $q(S,T)$, except possibly the one excluded from $X$, contains a vertex of $X_H$. So $q(S,T)\leq |X_H| +1$. This allows us to bound the RHS of (\[etutte\]) as: $$\begin{aligned}
\nonumber q(S,&T)+e(S,T)\\
\nonumber &\leq |X_H|+e(S,T_H) +e(S,T_L) +1\\
\nonumber &\leq \frac{1}{400}(|S|+|T_H|) + \left(1+\frac{1}{2000}\right)|N(S)\cap T_H| +\frac{k}{100}|S| +\frac{9k}{10}|S| +1\\
\nonumber & \qquad\qquad\qquad\qquad\qquad\mbox{ by (\ref{exh}), {(P4)}, (M1) and (K2)}\\
&\le \frac{1}{400}|T_H| + \left(1+\frac{1}{2000}\right)|N(S)\cap T_H| +\frac{92k}{100}|S| +1
\label{esth}\end{aligned}$$
We now split this case into 3 subcases.
*Case 1a: $S=\emptyset$.*\
In this case our goal is to show $|T_H|\geq q(S,T)$. [[ By , since $e(S,T)=0, |S|=0, |N(S) \cap T_H|=0$, we have $q(S,T)\leq 1 + \frac{1}{400}|T_H|$.]{}]{} This yields $|T_H|\geq q(S,T)$ if $|T_H|\geq 1$.
Thus we can assume $|T_H|=0$ [[ and so $q(S,T)\leq 1$, i.e. there is at most one component $Q$ in $K\bk (S\cup T)$ which is counted by $q(S,T)$.]{}]{} Since $S=\emptyset$, $e(T,Q)=e(T, K\bk T)$ which has the same parity as ${2e(T)+e(T,Q)=}\sum_{v\in T}d_K(v)=k|T|$ since $T_H=\emptyset$. (K4) implies that $k|T|$ has the same parity as $k|Q|$. Therefore $k|Q|$ has the same parity as [$e(Q,T)$ and so]{} $q(S,T)=0$. Therefore the LHS and RHS of (\[etutte\]) are both 0 and so [the desired inequality]{} holds.
*Case 1b: $|T_H|\leq 3|N(S)\cap T_H|$ and $S\neq\emptyset$.*\
(\[esth\]) and $|N(S)\cap T_H|\leq |T_H|$ imply $$\begin{aligned}
\nonumber q(S,T)+e(S,T)
&\leq& 1+ k|S| + |T_H| - \left( \frac{k}{20}|S| - \frac{1}{300}|T_H| \right) -\frac{k}{40}|S|\\
&\leq& k|S| + |T_H| - \left( \frac{k}{20}|S| - \frac{1}{300}|T_H| \right)
\label{ec1b}\end{aligned}$$ for $k\geq40$ since $S\neq\emptyset$. (K1) implies that $|N(S)\cap T_H|\leq |N(S)| \leq 2 k|S|$. So since we are in Case 1b: $$\frac{1}{300}|T_H| \leq{\frac{1}{ 100}}|N(S)\cap T_H| \leq \frac{k}{50}|S| \leq \frac{k}{20}|S|.$$ This and (\[ec1b\]) imply (\[etutte\]).
*Case 1c: $|T_H| > 3|N(S)\cap T_H|$ and $S\neq\emptyset$.*\
(\[esth\]) implies $$\begin{aligned}
q(S,T)+e(S,T)
&\leq& 1+ \frac{19k}{20}|S| + \frac{1}{400}|T_H| + 2|N_S\cap T_H|\\
&\leq&k|S|+|T_H|\end{aligned}$$ for $k\geq20$ since $S\neq\emptyset$. This is (\[etutte\]).
*Case 2: $|S|+|T| \ge \epsilon_0 n: $*\
This case follows the arguments of Cases 3 and 4 of [@Pawel]. We reproduce them here:
If $|T|<{\frac{1}{ 10}}\e_0 n$ then $|S|>\frac{9}{10}\e_0 n$ and so $e(S,T)< \frac{3}{4}k|S|$ by [(P5)]{}. We also have $q(S,T)<n<{\frac{1}{ 4}}k|S|$ for $k>\frac{40}{9\e_0}$, and this yields (\[etutte\]).
If $|T|\geq{\frac{1}{ 10}}\e_0 n$ then we use the bound $q(S,T)<n<\frac{1}{16}\sqrt{k\log k} |T|$ for $k> 25600 /\e_0^2$, and then the two parts of [(P6)]{} combine to give (\[etutte2\]). This is the only case where we prove (\[etutte2\]) directly rather than (\[etutte\]).
Our main theorem follows immediately:
[**Proof of Theorem \[mt\]**]{} We prove in Section \[sfts\] that, if $c_k+ {k^{10} \b} \leq c \leq c_k+k^{-1/2}$ then [[ ]{}]{} the subgraph $K$ obtained by STRIP (and possibly deleting one additional vertex) satisfies properties [[ (K1-4)]{}]{}. Lemma \[lem:properties\] establishes that [[ ]{}]{} $K$ satisfies properties [[ (P1-6)]{}]{}. So Lemma \[lkfactor\] implies that [[ ]{}]{} every pair of disjoint vertex sets $S,T\subseteq V(K)$ satisfying [[ (M1-2)]{}]{} also satisfy (\[etutte2\]). Lemma \[lem:minimality\] now establishes that $K$ has a $k$-factor and so Theorem \[mt\] holds for $c_k+ {k^{10} \b} \leq c \leq c_k+k^{-1/2}$. Containing a $k$-regular subgraph is a monotone [increasing]{} property and [$c_k+e^{-k/300} \ge c_k+k^{10}\b$ for $k$ sufficiently large]{}, so Theorem \[mt\] holds for all $c\geq c_k+e^{-k/300}$.
\[rokn\] The proof of Lemma \[lw0\](a), below, shows that the size of the $k$-core in fact is [[ $(1-o_k(1))n$]{}]{}. So after removing at most $\b n=e^{-k/200} n$ vertices (Lemma \[lstop\]), the $k$-regular subgraph we obtain is of size [[ $(1-o_k(1))n$.]{}]{}
Proof of Lemma \[lw0\] {#sec:lw0}
======================
To complete the paper, all that remains is two deferred proofs. First, we present the proof of Lemma \[lw0\], where we establish some straightforward properties of the $k$-core. Recall the statement:
[**Setup for Lemma \[lw0\]:**]{}
*$k$ is a sufficiently large constant, and $c_k < c \le \cmax = c_k + k^{-1/2}$. $C$ is the $k$-core of $G_{n,p=c/n}$. $\La$ is a uniform configuration with the same degree sequence as $C$. [Finally,]{} $W_0,R$ are as defined in Section \[stsp\]. [[ For the convenience of the reader, we state the lemma again.]{}]{}*
[**Lemma \[lw0\]:**]{} [W.h.p. ]{}before the first iteration of STRIP:
1. $|\La|> {0.99} n$;
2. ${ 0.99} \frac{n}{k} < |W_0| < {1.01} \frac{n}{k}$;
3. the total degree of [the set of]{} vertices with degree greater than $2k$ is at most [$e^{-k/6}n$]{};
4. there are at least $\frac{n}{{5}k}$ edges with both endpoints in $W_0$;
5. there are at least ${\frac {1}{2}} n$ edges from $W_0$ to $R$;
6. there are at least ${\frac{1}{ 3}}k n$ edges with both endpoints in $R$;
7. $C$ has at most [$e^{-k/3}n$]{} vertices of degree at most $2k$ and with at least $\hf k$ neighbours in $W_0$;
8. at least $\frac{n}{{200}}$ vertices in $R$ have no neighbours in $W_0$.
[[**Proof.**]{}]{}First recall from Section \[stsp\] that, before the first iteration of STRIP, $W_0$ is the set of vertices with degree $k$ in $\La$, and $R$ is the set of vertices with degree greater than $k$ in $\La$.
Part (a) is well-known; the size of the $k$-core approaches $n$ as $k$ grows (see, for example, the results of Molloy [@Molloy_cores] or Gao [@Gao]). Indeed, in [@Molloy_cores] it is proved that the $k$-core has size ${\z} n + o(n)$, [[ where]{}]{} $$\z = \z(c) = 1 - e^{-x} \sum_{i=0}^{k-1} \frac {x^i}{i!} = e^{-x} \sum_{i\ge k} \frac {x^i}{i!} = \Pr {( {\mbox{\bf Po}}(x) \ge k )},$$ where $x=x(c)$ is the greatest solution to $$\label{eq:c}
c = f(x) := \frac{x}{1-e^{-x}\sum_{i=0}^{k-2} x^i/i!}.$$ Recall from (\[eck\]) that $c_k$ is the minimum value of $f$ over all $x>0$. Simple analysis of $f$ shows that there is exactly one value of $x$ for which $f(x)=c_k$; we denote that value by $x_k$. [[ Moreover]{}]{}, for every $c>c_k$ there are exactly two solutions for $x$. It is straightforward to verify that $f'(x) \ge 0$ for $x \ge x_k$ and so for $c\geq c_k$ we have $x_k\le x(c) \le x(\cmax)$.
[Recall from (\[eq:ck\]) that $q_k=\log k-\log (2\pi)$. [@Pawel] shows]{} that $$\label{eq:xk}
x_k =k+(kq_k)^{1/2}+\frac{q_k}{3}-1+ o_k(1),$$ (note that $x_k$ is denoted as $\lambda_k$ in [@Pawel]). It follows that $$\z = \Pr{ ( {\mbox{\bf Po}}(x) \ge k )} \ge \Pr {( {\mbox{\bf Po}}(x_k) \ge k )} \to 1,$$ as $k \to \infty$. Part (a) holds for $k$ large enough.
Corollary 3 of [@cainwor] establishes that for any constant $i\geq k$, the number of vertices of degree $i$ in the $k$-core is $\la_i n +o(n)$ where $$\label{eq:deg_distribution}
\la_i= \Pr {( {\mbox{\bf Po}}(x) = i )} = \frac {e^{-x} x^i}{i!}.$$ In particular, $|W_0| = \frac{e^{-x}x^{k}}{k!}n +o(n)$, and so to prove part (b) we will estimate $\frac{e^{-x}x^{k}}{k!}$.
Below, setting $\d=\d(c)=x(c)-x_k$, we will prove that $c\leq \cmax$ implies $$\label{edbound}
\d\leq \log k.$$ So for now, we will restrict our attention to $x = x_k + \delta$ with $0 \le \delta \le \log k$. Using $1+y = \exp \left( y + O(y^2) \right)$, and $1/(1+y) = 1 - y + O(y^2)$ we get that $$\begin{aligned}
\frac{e^{-x}x^{k}}{k!} &= \frac{e^{-x_k-\delta}x_k^{k}}{k!} \ \left( 1 + \frac {\delta}{x_k} \right)^k = \frac{e^{-x_k}x_k^{k}}{k!} \ \exp \left( -\delta + \frac {\delta k}{x_k} + O \left( \frac {\delta^2 k}{x_k^2} \right) \right) \nonumber \\
&= \frac{e^{-x_k}x_k^{k}}{k!} \ \exp \left( -\delta + \delta \left( 1 - (q_k/k)^{1/2} + O(q_k/k) + O(\delta / k) \right) \right) \qquad\qquad \mbox{ by~(\ref{eq:xk}) }\nonumber \\
&= \frac{e^{-x_k}x_k^{k}}{k!} \ \exp \left( - \delta (q_k/k)^{1/2} + O(\delta \log k / k) \right)\qquad\qquad\qquad\qquad \mbox{ since } q_k=\log k +O(1)\nonumber \\
&= \frac{e^{-x_k}x_k^{k}}{k!} \left( 1 - \delta (q_k/k)^{1/2} + O(\log^3 k / k) \right) \label{eq:error1} \\
& = \frac{e^{-x_k}x_k^{k}}{k!} (1+o_k(1)), \qquad\qquad\qquad\qquad\qquad\qquad\mbox{ for }\d\leq\log k.\nonumber\end{aligned}$$ Using Stirling’s formula, (\[eq:xk\]) and the fact that $1+y=\exp \left( y-y^2/2+y^3/3+O(y^4) \right)$, we get for some $\epsilon = \epsilon(k) = o_k(1)$ $$\begin{aligned}
\frac{e^{-x_k}x_k^{k}}{k!} &= \frac{e^{-x_k}}{\sqrt{2\pi k}}\left(\frac{ex_k}{k}\right)^{k} \left( 1+O(k^{-1})\right) \nonumber\\
&= \frac{e^{-(kq_k)^{1/2}-\frac{q_k}{3}+1-\epsilon}}{\sqrt{2\pi k}}\left(1+\frac{(kq_k)^{1/2}+\frac{q_k}{3}-1+\epsilon}{k} \right)^{k} (1+O(k^{-1})) \nonumber \\
&= \frac{ \exp \left( - \frac{q_k}{2} + (1-\e)(q_k/k)^{1/2} + O( q_k^2 / k ) \right) }{\sqrt{2\pi k}} \nonumber\\
& =\frac{1+ (1-\e) (q_k/k)^{1/2} + O( \log^2 k / k ) }{k} \ \ \mbox{ since } q_k=\log(k/2\pi) \mbox{ and } e^y=1+y+O(y^2) \label{eq:error2} \\
&=\frac{1+ o_k(1)}{k}. \nonumber $$ Therefore $$\label{eok1}
\frac{e^{-x}x^{k}}{k!}=\frac{1+ o_k(1)}{k}\qquad\qquad\mbox{ for $x=x_k+\d$ with } 0\leq\d\leq\log k.$$ Next we prove that our upper bound $c \le \cmax = c_k + k^{-1/2}$ implies (\[edbound\]) and so (\[eok1\]) establishes part (b). To do this we estimate the derivative of $f(x)$ over the range $0\leq \d\leq \log k$ to show that at $\d = \log k$ and $x=x_k+\d$ we have $c(x)=f(x)>\cmax$. Rewriting (\[eq:c\]) as: $$f(x) = \frac{x}{e^{-x}\sum_{i\geq k-1}\frac{x^i}{i!}},$$ the derivative is $$\begin{aligned}
f'(x) &=& \frac { e^{-x} \sum_{i \ge k-1} \frac {x^i}{i!} - x \left(-e^{-x} \sum_{i \ge k-1} \frac {x^i}{i!} +e^{-x} \sum_{i \ge k-1} \frac {x^{i-1}}{(i-1)!}\right)} { \left( e^{-x} \sum_{i \ge k-1} \frac {x^i}{i!} \right)^2} \nonumber \\
&=& \frac { e^{-x} \sum_{i \ge k-1} \frac {x^i}{i!} - x e^{-x} \frac {x^{k-2}}{(k-2)!} } { \left( e^{-x} \sum_{i \ge k-1} \frac {x^i}{i!} \right)^2}. \label{ed0}\end{aligned}$$ Recalling $q_k=\log k-\log(2\pi)$, [[ and that $x=x_k+\d$ with]{}]{} $0\leq \d\leq \log k$, [[ we have]{}]{} $$\begin{aligned}
x e^{-x} \frac {x^{k-2}}{(k-2)!} &= e^{-x} \frac {x^{k}}{k!} \cdot \frac {k(k-1)}{x}
= \frac{e^{-x_k}x_k^{k}}{k!} \left( 1 - \delta (q_k/k)^{1/2} + O(\log^3 k / k) \right) \frac {k(k-1)}{x}
\qquad\mbox{ by~(\ref{eq:error1})} \nonumber \\
&= \left( 1 + (1- \e - \delta) (q_k/k)^{1/2} + O(\log^3 k/k) \right) \frac {k-1}{x}
\qquad \qquad\qquad\qquad\qquad\qquad \mbox{ by~(\ref{eq:error2})} \nonumber \\
&= \left( 1 + (1- \e - \delta) (q_k/k)^{1/2} + O(\log^3 k/k) \right)
\left( 1 -(q_k/k)^{1/2} + O({{\color{black} \log k}}/k) \right) \nonumber \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \mbox{ by~(\ref{eq:xk}) and since } {\frac{1}{ 1+y}} = 1 - y + O(y^2) \nonumber \\
&= 1 - (\e +\delta) (q_k/k)^{1/2} + O(\log^3 k/k).\label{ed7}\end{aligned}$$ Standard bounds for the tail probabilities of a Poisson random variable (see eg. [@AS Theorem A.1.15]), along with (\[eok1\]) yield that for $0\leq \d\leq \log k$ we have $$\Pr{ \left( {\mbox{\bf Po}}(x) \le k-2 \right) }
= O \left( k^{-1/2} \right),$$ and so $$\label{ed2}
e^{-x} \sum_{i \ge k-1} \frac {x^i}{i!} = 1 - \Pr{ \left( {\mbox{\bf Po}}(x) \le k-2 \right) } = 1 - O \left( k^{-1/2} \right).$$ Substituting (\[ed7\]) and (\[ed2\]) [[ into (\[ed0\])]{}]{} and recalling $q_k=\log k +O(1)$ yields $$\begin{aligned}
f'(x) &=& (\e +\delta) (q_k/k)^{1/2} + O \left( k^{-1/2} \right) \nonumber \\
&=& (\e +\delta) (\log k/k)^{1/2} + O \left( k^{-1/2} \right),
\mbox{ for $x=x_k+\d$ and } 0\leq\d\leq\log k. \label{eq:bound_for_f'}\end{aligned}$$ Therefore, recalling that $f(x_k)=c_k$ and since $\e=o_k(1)=o(\d)$ we have for $k$ sufficiently large $$f(x_k + \log k) = f(x_k) + \Theta( (\log k)^2 )\cdot (\log k / k)^{1/2} +(\log k)\cdot O \left( k^{-1/2} \right)
> c_k + k^{-1/2}=\cmax.$$ [[ (Note that integration of would have given the same result.)]{}]{} Since $f(x)$ is monotone increasing for $x>x_k$, it follows that $x(c)\leq x_k+{\log k}$ for all $c_k\leq c\leq \cmax$, thus proving (\[edbound\]). Therefore (\[eok1\]) and (\[eq:deg\_distribution\]) yield part (b).
For part (c) it is easier to show the desired property for $G$ instead of $C$; the conclusion for $C$ will trivially follow. The degree of each vertex in $G$ is a random variable $X$ with the binomial distribution ${\mbox{\bf Bin}}(n-1,c/n)$ with $\E[X] \le c < 1.1 k$, provided that $k$ is large enough. Hence, [[ applying with $t=0.9k+\ell$]{}]{}, we get that the expected total degree of all vertices with degree greater than $2k$ is at most $$\begin{aligned}
n \sum_{\ell \ge 1} (2k + \ell) \Pr{ ( X \ge 2k + \ell )} & \le n \sum_{\ell \ge 1} (2k+\ell) \exp \left( - \frac {(0.9k+\ell)^2}{{{\color{black} 2 (1.1k+ (0.9k+\ell)/3)}} } \right) \\
& \le n \sum_{\ell \ge 1} (2k+\ell) \exp \left( - \frac {k+\ell}{4} \right) = O( k e^{-k/4} n ) < e^{-k/5} n,\end{aligned}$$ provided $k$ is sufficiently large. Since the concentration can be proved with a straightforward concentration argument using, for example, Azuma’s Inequality or an easy second moment argument, we omit the details. This establishes part (c).
For the remaining parts, let us first observe that the total degree of all vertices in $G$ is $2 {\mbox{\bf Bin}}\left( {n \choose 2}, c/n \right)$ with expectation $c(n-1) < 1.01 kn$, provided $k$ is sufficiently large. Hence, by Chernoff’s bound, it is at most $1.02 kn$, and this upper bound clearly holds for the total degree of the vertices of $C$. On the other hand, part (a) implies that it is at least $0.99 kn$.
For parts (d,e,f), [[ we return to analyzing $\La$ directly. We will]{}]{} focus on the partners of the vertex-copies in $W_0$. Part (b) implies that the total degree of the vertices in $W_0$ is between $0.99n$ and $1.01n$. Expose the partners in $\La$ of the vertex-copies of $W_0$, one at a time. At step $i\leq 1.01 n$, the probability that the partner chosen is in $W_0$ is between $p_i$ and $q_i$, where $$\label{epiqi}
p_i = \max \left\{ \frac {0.99n - 2i}{1.02kn}, 0 \right\}
\text{ and }
q_i = \max \left\{ \frac {1.01n - i}{0.99kn-2i}, 0 \right\} \le \max \left\{ \frac {1.01n - i}{0.98kn}, 0 \right\},$$ provided $k$ is sufficiently large. Hence, the number of vertex-copies in $W_0$ whose partner is in $W_0$ can be stochastically lower/upper bounded by the two sums of independent Bernoulli random variables: the first one with parameters $p_i$ and the second one with $q_i$. The expected value of the first sum is more than $0.24 n / k$, and the expectation of the second one is at less than $0.53 n / k$. The concentration follows immediately from Chernoff’s bound and we get that the number of edges with both endpoints in $W_0$ is at least $\frac {n}{5k}$ and at most $\frac {n}{k}$, which finishes part (d).
Now, parts (e) and (f) follow deterministically. The number of edges from $W_0$ to $C\bk W_0$ is at least $0.99 n - 2 \frac {n}{k} \ge \frac {1}{2} n$ for $k$ large enough. Finally, there are at least $(0.99 kn - 2 \cdot 1.01 n)/2$ edges with both endpoints in $C\bk W_0$ which is more than $\frac {1}{3} kn$ for $k$ large enough.
Part (g) is slightly more complicated. For a contradiction, suppose that there [[ is a set $T$ with $|T|=e^{-k/3}n$ such that every vertex in $T$ is]{}]{} of degree at most $2k$ and [[ has]{}]{} at least $\hf k$ neighbours in $W_0$. (Note that some of them might be from $W_0$.) [[ We will first show that t]{}]{}his implies (deterministically) that there exists a [[ subset $S\subseteq T$ of size at least $\hf e^{-k/3}n$ and]{}]{} with at least $\frac {1}{8} k e^{-k/3}n$ edges between $S$ and $W_0 \setminus S$.
To prove this, we consider the average of $|E(S,W_0\setminus S)|$ over all subsets $S\subset T$ of size ${\mbox{$\lceil{ \hf|T|}\rceil$}}$. Consider any edge $uv$ with $u\in T$ and $v\in W_0$; there are at least ${\frac{1}{ 4}}k|T|$ such edges (if $u,v$ are both in $T\cap W_0$ then the edge $uv$ is only counted once). A very simple count shows that $uv\in E(S,W_0\setminus S)$ for at least half of the subsets $S$ of size ${\mbox{$\lceil{ \hf|T|}\rceil$}}$; there are four cases corresponding to the parity of $|T|$ and whether $v\in T$. Therefore, $$\sum_{S\subset T, |S|={\mbox{$\lceil{ \hf|T|}\rceil$}}}|E(S,W_0\setminus S)|\geq{\frac{1}{ 8}}k|T|{|T|\choose {\mbox{$\lceil{ \hf|T|}\rceil$}}},$$ and so at least one such set $S$ has $|E(S,W_0\setminus S)|\geq{\frac{1}{ 8}}k|T|$.
Next, we show that no such set $S$ exists in $\Lambda$.
Indeed, let us fix [[ a]{}]{} set $S$ and expose [[ the]{}]{} partners of all vertex-copies from $W_0 \setminus S$ in $\La$. Arguing similarly as for (\[epiqi\]), the number of edges from $W_0 \setminus S$ to $S$ can be stochastically upper bounded by the binomial random variable $$X \sim {\mbox{\bf Bin}}\left( 1.01 n, \frac {2k|S|}{0.98 kn} \right), \quad \text{ with } \quad \E[X] < 3 e^{-k/3} n.$$ It follows from Chernoff’s bound that the expected number of sets $S$ with a large number of edges to $W_0 \setminus S$ is at most $$\begin{aligned}
{n \choose \hf {{\color{black} \lceil}}e^{-k/3}n{{\color{black} \rceil}}} & \Pr{ \left( X \ge \frac {ke^{-k/3}}{8} n \right) } \le \left( {{\color{black} 3}}e^{1+k/3} \right)^{\hf {{\color{black} \lceil}}e^{-k/3}n{{\color{black} \rceil}}} \exp \left( - 1.4 \cdot \frac {k e^{-k/3}}{8} n \right) = o(1),\end{aligned}$$ provided that $k$ is sufficiently large. Part (g) follows from Markov’s inequality.
Finally, let us move to part (h). We showed earlier that the total degree of the vertices of $C$ is at least $0.99kn$ and the total degree of the vertices in $W_0$ is at most $1.01n$. It follows from parts (a), (b), and (c) that there are at least $0.98n$ vertices in $R$ that are of degree at most $2k$ (and, of course, at least $k+1$). We pick (arbitrarily) $0.13n$ of them and expose partners of all corresponding vertex-copies. Note that, regardless of the history of the process, the probability that a given vertex of degree $\ell \le 2k$ has no neighbour in $W_0$ is at least $$\left( 1 - \frac {1.01n}{0.99 kn - (2k)(0.13n)} \right)^{\ell} \ge \left( 1 - \frac {1.4}{k} \right)^{2k} \ge e^{-3},$$ provided that $k$ is sufficiently large. Hence, the number of vertices in $R$ that have no neighbours in $W_0$ is bounded from below by the random variable $X \sim {\mbox{\bf Bin}}(0.13 n, e^{-3})$ with $\E[X] = 0.13 e^{-3} n > 0.006 n$. Part (h) holds by Chernoff’s bound and the proof of the lemma is finished.
Proof of Lemma \[lem:properties\] {#sec:properties}
=================================
Our final piece is the deferred proof regarding properties of the subgraph $K$ obtained by our stripping procedure. Recall our setting: we begin with the random graph $G=G_{n,p=c/n}$ where $ c_k+k^{10}\b \le c \le c_k+k^{-1/2}$ (see (\[ebeta\]), (\[eq:conditions\_for\_c\])). $K$ is the subgraph of the $k$-core of $G$ obtained after applying **STRIP**, and possibly deleting one more vertex. We repeat the statement of the lemma:
[**Lemma \[lem:properties\]**]{} There exist constants $\g,\e_0 >0,k_0 {\in \mathbb{N}}$ such that for any $k\geq k_0$, $K$ satisfies:
- For every $Y \subseteq V(K)$ with $|Y| \le 10 \epsilon_0 n$, $e(Y) < \frac{ k |Y|}{6000}$.
- For every $Y \subseteq V(K)$ with $|Y|\le \frac12 |K|$, $e(Y, V(K) \setminus Y) \ge \gamma k |Y|.$
- For every disjoint pair of sets $X,Y \subseteq V(K)$ with $|X| \ge \frac{1}{200}|Y|$ and $|Y| \le \epsilon_0 n$, $e(X,Y) < \frac12 \gamma k |X|$.
- For every disjoint pair of sets $X, Y \subseteq V(K)$ with $|X|+|Y| \le \epsilon_0 n$, $e(X,Y) < \left( 1 +\frac{1}{2000}\right) |N(X) \cap Y|+ \frac{ k}{100}|X|$.
- For every disjoint pair of sets $S,T \subseteq V(K)$ with $|T|< \frac{1}{10}\epsilon_0 n$ and $|S| > \frac{9}{10}\epsilon_0 n$, $e(S,T) < \frac34 k |S|.$
- For every disjoint pair of sets $S,T \subseteq V(K)$ with $|T| \ge \frac{1}{10}\epsilon_0 n$, we have $e(S,T) \le k|S|+\frac{3}{4}\sqrt{k \log k}|T|$ and $\sum_{v \in T}d(v) > (k+\frac{7}{8}\sqrt{k \log k})|T|$.
[[[**Proof.**]{}]{}]{} For every property except [(P2)]{}, we actually show that it holds in $G$, and so we can work in the $G_{n,p}$ model. For each property, we will show that it holds if $\e_0$ is sufficiently small. Thus we can take a value of $\e_0$ that is sufficiently small for all properties.
The following property follows from Lemma 3 in [@Pawel], where they show that in fact there is no such subgraph in $G$:
(P0) For every $Y \subseteq V(K)$ with $|Y| \le 2\log n/(e c \log \log n)$, $e(Y) \le |Y|.$
Now, [(P1)]{} follows from a very standard first moment argument applied to $G=G_{n,p=c/n}$, so long as $\e_0$ is sufficiently small. For $Y \subseteq V(K)$ with $|Y|=s \le 2\log n/(e c \log \log n)$, the statement follows immediately by Property [(P0)]{}, for $k\geq 6000$. For larger values of $s$, and for $k\geq 1f$ and since $k<c<2k$, the expected number of sets $Y\subset V(G)$ with $|Y|=s$ for which $Y$ contains at least $ \frac{k s}{6000}$ edges in $G$ is at most $$\begin{aligned}
\binom{n}{s} \binom{\binom{s}{2}}{\frac{ ks}{6000}} p^{\frac{ ks}{6000}} &\le& \left(\frac{ne}{s}\right)^s \left(\frac{300cse}{ kn}\right)^{\frac{ks}{6000}} \le \left(\frac{ne}{s}\right)^s \left(\frac{6000se}{ n}\right)^{2s} \\
&=& \left(\frac{6000^2 e^3 s}{ n}\right)^{s}
< 2^{-s},\end{aligned}$$ for $s\leq \e_0n$ so long as $\e_0<1/(2\times6000^2 e^3)$.
Summing over all $2\log n/(e c \log \log n) < s \le 10 \epsilon_0 n$, we obtain that the expected number of sets that fail the desired property is $O(2^{-2\log n/(e c \log \log n)}) = o(1)$. Property [(P1)]{} now follows by Markov’s inequality.
For property [(P2) we consider two cases]{}:
[*Case 1: $|Y|\leq10\e_0n$.*]{} Since each vertex in $Y$ has degree at least $k$, and applying [(P1)]{} we know $e(Y,V(K)\bk Y)\geq k|Y|-2e(Y)\geq k|Y| - k|Y|/3000>\hf k |Y|$.
[*Case 2: $|Y|>10\e_0n$.*]{} Lemma 2 of [@Pawel] proves that in the $k$-core $C$, $e(Y,V(C)\bk Y)\geq \g'k|Y|$ for some constant $\g'>0$ independent of $k$. (In fact, they prove this for the $(k+2)$-core but the same proof applies to the $k$-core; the main tool is Lemma 5.3 of [Benjamini, Kozma and Wormald]{} [@Benjamini].) By Lemma \[mtsg\], $|C\bk K|\leq \b n$. Since $K$ is an induced subgraph of $C$ and every vertex of $Y$ has degree at most $2k$ (by property (K1)), this implies that $e(Y,V(K)\bk Y)\geq \g'k|Y|-2k\b n$. This is at least $\hf \g' k|Y|$ if $k$ is sufficiently large so that $2k\b < 5\g'\e_0$ (recall $\b=e^{-k/200}$). So [(P2)]{} holds for $\g=\hf\g'$.
Property [(P3)]{} follows from Property (P4) of [@Mike]. There it is shown that in $G(n,p)$ with $p=c/n$ and $0 < c < 2k$ there is no subgraph satisfying the desired property. Their inequality is not strict, but it can be clearly made strict. Their proof holds for any $\g>0$ so long as $\e_0$ is sufficiently small in terms of $\g$. Thus, we can use the same value of $\g$ as in property [(P2)]{}.
For property [(P4)]{}, it clearly suffices to only consider disjoint sets $X,Y\neq\emptyset$ for which $N(X) \cap Y=Y.$ [[ So it suffices to prove that every disjoint pair of sets $X,Y \subseteq V(G)$ with $G \in \mathcal{G}(n,p)$ with $p=c/n$ and $0 < c < 2k$ and $2\leq |X|+|Y| \le \epsilon_0 n$ satisfies $$\label{eq:P5}
e(X,Y) \le {{\color{black} \left(1+\frac{1}{2000}\right)}}|Y|+ \frac{k}{100}|X|.$$ ]{}]{}
The argument is essentially the same as for Lemma 2.5 of [@Mike], but with different constants; we give it here for the sake of completeness.
Let $\s n=|X|$ and $\t n=|Y|$. For any choice of $\sigma, \tau$, the expected number of sets $X,Y$ in $G$ violating is at most $$\begin{aligned}
\nonumber \binom{n}{\sigma n} & \binom{n}{\tau n}\binom{(\sigma n)(\tau n)}{(1+\frac{1}{2000}) \tau n+ \frac{k}{100}\sigma n}\left(\frac{c}{n}\right)^{(1+\frac{1}{2000} )\tau n+ \frac{ k}{100}\sigma n} \\
& \le \left( \frac{e}{\sigma}\right)^{\sigma n}\left( \frac{e}{\tau}\right)^{\tau n} \left( \frac{e \sigma \tau c}{(1+\frac{1}{2000} )\tau +\frac{ k}{100}\sigma}\right)^{(1+\frac{1}{2000} )\tau n+\frac{ k}{100}\sigma n}.
\label{ep5}\end{aligned}$$ If $\s n, \t n$ are both less than $\sqrt{n}$; i.e. $\s,\t<n^{-1/2}$ then (\[ep5\]) is at most $$\begin{aligned}
\nonumber
\left( \frac{e}{\sigma}\right)^{\sigma n} & \left( \frac{e}{\tau}\right)^{\tau n}
\left( \frac{e \sigma \tau c}{\left(1+\frac{1}{2000} \right)\tau} \right)^{\frac{ k}{100}\sigma n}
\left( \frac{e \sigma \tau c}{\frac{ k}{100}\sigma}\right)^{(1+\frac{1}{2000} )\tau n}\\
\nonumber
&= \left( \frac{e}{\sigma}\right)^{\sigma n}
\left( \frac{e \sigma c}{1+\frac{1}{2000} } \right)^{\frac{ k}{100}\sigma n}
\left( \frac{e}{\tau}\right)^{\tau n}
\left( \frac{100e \tau c}{k}\right)^{(1+\frac{1}{2000} )\tau n}\\
\nonumber & <(A n^{-1/2})^{(\frac{ k}{100}-1)\sigma n} (B n^{-1/2})^{\frac{1}{2000}\tau n}
\qquad\qquad \mbox{for {{\color{black} some}} constants $A,B$}\\
&<n^{-\frac{1}{4}\frac{1}{2000}(\s n + \t n)}, \qquad\qquad\qquad\qquad\mbox{ for $k>200$.}
\label{ep5b}
\end{aligned}$$ For general $\s n, \t n$, using $c < 2k$ and $\tau < \epsilon_0$ for $\e_0$ sufficiently small, we see that the [[ base of the exponent in the]{}]{} third factor of (\[ep5\]) is at most $$\frac{e \sigma \tau c}{(1+\frac{1}{2000}) \tau +\frac{k}{100}\sigma}< \frac{e\sigma \tau c}{\frac{ k}{100}\sigma} < 200e\tau < \left(\frac{\tau}{e^2}\right)^{\frac{1}{ 1+\frac{1}{2000}}}.$$ If $\sigma \ge e^{- k/200}$, then for $k\geq 400$ we have $$\frac{e \sigma \tau c}{\left(1+\frac{1}{2000} \right)\tau + \frac{k}{100}\sigma}<\left(\frac{\tau}{e^2}\right)^{\frac{1}{ 1+\frac{1}{2000}}} <e^{-1} \leq\left(\frac{\sigma}{e^2}\right)^{\frac{100}{ k}},$$ while if $\sigma < e^{-k/200}$, then for $k\geq 40000$ we have $$\frac{e \sigma \tau c}{\left(1+\frac{1}{2000} \right)\tau + \frac{k}{100}\sigma}< \frac{e\sigma \tau c}{\left(1+\frac{1}{2000} \right)\tau} < ec \sigma^{1/2}\sigma^{1/2} < 2ek e^{-k/400}\sigma^{1/2}< \sigma^{1/2} < \left(\frac{\sigma}{e^2} \right)^{\frac{100}{k}}.$$ Hence, the expected number of disjoint sets $X,Y$ with $|X|=\sigma n$, $|Y|=\tau n$ [[ and $e(X,Y) \ge \left(1+\frac{1}{2000} \right){{\color{black} |Y|}}+\frac{k}{100}|X|$]{}]{} is at most $$\label{ep5c}
\left( \frac{e}{\sigma}\right)^{\sigma n}\left( \frac{e}{\tau}\right)^{\tau n}
\left(\frac{\sigma}{e^2} \right)^{\frac{100}{ k} \frac{k}{100}\sigma n}\left( \frac{\tau}{e^2}\right)^{\frac{1}{ 1+\frac{1}{2000}}\left(1 +\frac{1}{2000} \right)\tau n}=\left(\frac{1}{e}\right)^{|X|+|Y|}.$$ For every choice of $y=|X|+|Y|$, there are $y-1$ choices for $|X|, |Y|\geq 1$. Applying (\[ep5b\]) and (\[ep5c\]), the expected number of sets violating [[ (\[eq:P5\])]{}]{} is at most $$\sum_{y=2}^{\sqrt{n}} (y-1)n^{-\frac{1}{4}\frac{1}{2000}y} ~~+~~
\sum_{y=\sqrt{n}}^{\epsilon_0 n} (y-1) \left(\frac{1}{e}\right)^{y}=o(1),$$ and, by Markov’s inequality, [[ (\[eq:P5\]) holds and thus Property [(P4)]{} holds.]{}]{}
Property [(P5)]{} follows in the same way as property (P8) of [@Mike], which used proofs taken from [@Pawel]. The proofs hold for any $\e_0$ sufficiently small so long as $k$ is sufficiently large in terms of $\e_0$.
Property [(P6)]{} comes from Case 4 of the proof of Theorem 1 in [@Pawel]. The first part is equation (14) of that paper with $\e:={\frac{1}{ 4}}$; it is easy to check that their proof goes through with that value of $\e$. The second part is from the line preceding (14) with $\e:={\frac{1}{ 8}}$; that line holds for every $\e$ so long as $k$ is sufficiently large in terms of $\e$. In both cases, the proof analyzes the degree sequence of $G_{n,p}$ and so holds for every subset $S,T$ of $G_{n,p}$ so long as the vertices of $T$ have degree at least $k$. There is a minor difference in that they have $c>c_{k+2}$ rather than $c>c_{k}$ but this has no significant effect on the proof. Again, the proofs hold for any $\e_0$ sufficiently small so long as $k$ is sufficiently large in terms of $\e_0$.
Acknowledgements {#acknowledgements .unnumbered}
================
Part of this research was conducted while M. Molloy was an Invited Professor at the École Normale Supérieure, Paris and while D. Mitsche was visiting Ryerson University. The authors are supported by NSERC Discovery grants and an NSERC Engage grant. [[ We are grateful to two referees who read this paper very carefully and provided many helpful comments and corrections.]{}]{}
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[^1]: Universite de Nice Sophia-Antipolis, Nice, France.
[^2]: Department of Computer Science, University of Toronto, Toronto, ON, Canada.
[^3]: Department of Mathematics, Ryerson University, Toronto, ON, Canada.
[^4]: A property is said to hold [*with high probability ()*]{} if it holds with probability tending to one as $n\rightarrow\infty$.
[^5]: Meaning that there is a function $\r_k(n)$ such that for any $\e>0$, $G_{n,p=c/n}$ w.h.p. has no $k$-regular subgraph for $c=\r_k(n)-\e$ and has a $k$-regular subgraph for $c=\r_k(n)+\e$.
|
---
abstract: |
In this paper we introduce a technique, called [*rim surgery*]{}, which can change a smooth embedding of an orientable surface $\Sig$ of positive genus and nonnegative self-intersection in a smooth $4$-manifold $X$ while leaving the topological embedding unchanged. This is accomplished by replacing the tubular neighborhood of a particular nullhomologous torus in $X$ with $S^1\x E(K)$, where $E(K)$ is the exterior of a knot $K\subset S^3$. The smooth change can be detected easily for certain pairs $(X,\Sig)$ called [*SW-pairs*]{}. For example, $(X,\Sig)$ is an SW-pair if $\Sig$ is a symplectically and primitively embedded surface with positive genus and nonnegative self-intersection in a simply connected symplectic 4-manifold $X$. We prove the following theorem:
[**Theorem.**]{} [*Consider any SW-pair $(X,\Sig)$. For each knot $K\subset S^3$ there is a surface $\Sig_K\subset X$ such that the pairs $(X,\Sig_K)$ and $(X,\Sig)$ are homeomorphic. However, if $K_1$ and $K_2$ are two knots for which there is a diffeomorphism of pairs $(X, \Sig_{K_1})\to (X,\Sig_{K_2})$, then their Alexander polynomials are equal: $\DD_{K_1}(t)=\DD_{K_2}(t)$.*]{}
address:
- 'Department of Mathematics, Michigan State University East Lansing, Michigan 48824'
- 'Department of Mathematics, University of California Irvine, California 92697'
author:
- Ronald Fintushel
- 'Ronald J. Stern'
title: ' Surfaces in 4-Manifolds'
---
.525cm
[^1]
Introduction\[Intro\]
=====================
We say that a surface $\Sig$ is [*primitively embedded*]{} in a simply connected smooth 4-manifold $X$ if $\Sig$ is smoothly embedded with $\pi_1(X\setminus \Sig)=0$. In particular, by Alexander duality, $\Sig$ must represent a primitive homology class $[\Sig] \in
H_2(X;\Z)$. In general, any smoothly embedded (connected) surface $S$ in a simply connected smooth 4-manifold $X$ with $[S]\ne 0$ has the property that the surface $\Sig$ which represents the homology class $[S]-[E]$ in $X\#\CPb$ and which is obtained by tubing together the surface $S$ with the exceptional sphere $E$ of $\CPb$ is primitively embedded (since the surface $\Sig$ transversally intersects the sphere $E$ in one point).
Given a primitively embedded positive genus surface $\Sig$ in $X$, in the first part of this paper we shall construct for each knot $K$ in $S^3$ a smoothly embedded surface $\Sig_K$ in $X$ which is [*$\Sig$-compatible*]{}; i.e. $[\Sig]=[\Sig_K]$ and there is a homeomorphism $(X, \Sig)\to (X,\Sig_K)$. This construction will have two properties. The first is that $(X,\Sig_{\text{unknot}})=(X,\Sig)$. The main result of this paper is the second property: under suitable hypotheses on the pair $(X,\Sigma)$, if $K_1$ and $K_2$ are two knots in $S^3$ and if there is a diffeomorphism $(X, \Sig_{K_1})\to (X,\Sig_{K_2})$, then $K_1$ and $K_2$ have the same symmetric Alexander polynomial, i.e. $\DD_{K_1}(t)=\DD_{K_2}(t)$. As a special case we show:
\[sympthm\] Let $X$ be a simply connected symplectic 4-manifold and $\Sig$ a symplectically and primitively embedded surface with positive genus and nonnegative self-intersection. If $K_1$ and $K_2$ are knots in $S^3$ and if there is a diffeomorphism of pairs $(X, \Sig_{K_1})\to (X,\Sig_{K_2})$, then $\DD_{K_1}(t)=\DD_{K_2}(t)$. Furthermore, if $\DD_K(t)\ne 1$, then $\Sig_K$ is not smoothly ambient isotopic to a symplectic submanifold of $X$.
For example, Theorem \[sympthm\] applies to the $K3$ surface where $\Sig$ is a generic elliptic fiber. It also applies to surfaces of the form $S - E$ in $\CP \# \CPb$, where $S$ is any positive genus symplectically embedded surface in $\CP$.
The outline of this paper is as follows. In §2 we shall construct the surfaces $\Sig_K$ with $[\Sig_K]=[\Sig]$ and show that if $\pi_1(X)=\pi_1(X\setminus \Sig)=0$, there is a homeomorphism of $(X, \Sig)$ with $(X,\Sig_K)$, i.e. $\Sig_K$ is $\Sig$-compatible. We give two descriptions of $\Sig_K$. One is explicit, while the other describes how to obtain $\Sig_K$ by removing a tubular neighborhood $T^2\x D^2$ of a homologically trivial torus in a tubular neighborhood of $\Sig$ and replacing it with $S^1 \x E(K)$, where $E(K)$ is the exterior of the knot $K$ in $S^3$. This is reminiscent of our construction in [@FS] where we performed the same operation on homologically essential tori. There, the Alexander polynomial $\DD_K(t)$ of $K$ detected a change in the diffeomorphism type of the ambient manifold $X$. Here, we shall show that $\DD_K(t)$ detects a change in the diffeomorphism type of the embedding of $\Sig$ in $X$.
If the self-intersection of $\Sig$ is $n\ge 0$, then in $X_n=X\#n\CPb$ consider the surface $\Sig_n=\Sig-\sum_{j=1}^n E_j$ (resp. $\Sig_{n,K}=\Sig_K-\sum_{j=1}^n E_j$) obtained from $\Sig$ (resp. $\Sig_K$) by tubing together with the exceptional spheres $E_j$, $j=1,\dots,n$, of the copies of $\CPb$ in $X_n$. If there is a diffeomorphism $H:(X, \Sig_{K_1})\to (X,\Sig_{K_2})$, then there is a diffeomorphism $H_n:(X_n, \Sig_{n,K_1})\to (X_n,\Sig_{n,K_2})$. For each genus $g\ge1$ we construct in §3 a standard pair $(Y_g, S_g)$, with the properties that $Y_g$ is a Kahler surface, $S_g$ is a primitively embedded genus $g$ Riemann surface in $Y_g$, and the torus used to construct $S_{g,K}=(S_g)_K$ is contained in a cusp neighborhood. Then in §4 we will study [*SW-pairs*]{}, i.e. pairs $(X,\Sig)$ where $X$ is a smooth simply connected 4-manifold, $\Sig$ is a primitively embedded genus $g$ surface with self-intersection $n\ge 0$, and the fiber sum of $X_n$ and $Y_g$ along the surfaces $\Sig_n$ and $S_g$ has a nontrivial Seiberg-Witten invariant $\sw_{X_n\#_{\Sig_n=S_g} Y_g}\ne 0$. The point here is that the nullhomologous torus used to construct the surface $\Sig_K$ in $X$ still resides in $X_n\#_{\Sig_n=S_g} Y_g$ and is now homologically essential and is contained in a cusp neighborhood. It will also follow that if $X$ is a symplectic 4-manifold and $\Sig$ is a symplectically and primitively embedded surface with nonnegative self-intersection, then $(X,\Sig)$ is a SW-pair.
In §5 we use in a straightforward fashion the results of [@FS] to show that the Alexander polynomial of $K$ distinguishes the $\Sig_K$ for SW-pairs, and we complete the proof our main theorem:
\[mainthm\] Consider any $\sw$-pair $(X,\Sig)$. If $K_1$ and $K_2$ are two knots in $S^3$ and if there is a diffeomorphism of pairs $(X, \Sig_{K_1})\to (X,\Sig_{K_2})$, then $\DD_{K_1}(t)=\DD_{K_2}(t)$.
Finally, in §6 we complete the proof of Theorem \[sympthm\] by showing that in the case that $\Sig$ is sympletically embedded in $X$ and $\DD_K(t)\ne
1$, then $\Sig_K$ is not smoothly ambient isotopic to a symplectic submanifold of $X$.
We conclude this introduction with two conjectures. The first conjecture is that, under the hypothesis of Theorem \[mainthm\], there is a diffeomorphism $(X, \Sig_{K_1})\to (X,\Sig_{K_2})$ if and only if the knots $K_1$ and $K_2$ are isotopic. In particular, this conjecture would imply that the study of the equivalence clases of $\Sig$-compatible surfaces under diffeomorphism is at least as complicated as classical knot theory. The second conjecture is a finiteness conjecture: given a symplectic 4-manifold $X$ and a symplectic submanifold $\Sig$, we conjecture that there are only finitely many distinct smooth isotopy classes of symplectic submanifolds $\Sig^\prime$ which are topologically isotopic to $\Sig$.
The construction of $\Sig_K$
============================
Let $X$ be a smooth $4$-manifold which contains a smoothly embedded surface $\Sig$ with genus $g>0$. Then there is a diffeomorphism $$h:\Sig \to T^2\#\cdots\# T^2 =
(T^2\setminus D^2)\cup (T^2\setminus(D^2 \amalg D^2))\cup \cdots\cup
(T^2\setminus
D^2).$$ Let $C\subset \Sig$ be a curve whose image under $h$ is the curve $S^1\x\{\text{pt} \} \subset T^2\setminus D^2 = (S^1\x S^1)\setminus D^2$ in the first $T^2\setminus D^2$ summand of $h(\Sig)$. Keep in mind that, since there are many such diffeomorphisms $h$, there are many such curves $C$. Given a knot $K$ in $S^3$ we shall give two different constructions of a surface $\Sig_{K,C}$. The first is an explicit construction, while the second shows how to obtain $\Sig_{K,C}$ by what we call a [*[rim surgery]{}*]{}, a surgical operation on a particular homologically trivial torus in a neighborhood of $\Sig$. It is this second construction that will allow us to compute appropriate invariants to distinguish the surfaces $\Sig_{K,C}$.
An explicit description of $\Sig_{K,C}$
---------------------------------------
Viewing $S^1$ as the union of two arcs $A_1$ and $A_2$, we have $$\begin{aligned}
T^2\setminus D^2 &=& (S^1\x S^1)\setminus D^2\\&=&((A_1\cup
A_2)\x(A_1\cup A_2))
\setminus (A_1 \times A_1)
\\&=&(A_2\x S^1)\cup (A_1 \x A_2)\end{aligned}$$ with $h(C)=A_2\x\{{\text{pt}}\}\cup A_1\x\{{\text{pt}}\}$. Now the normal bundle of $\Sig$ in $X$ when restricted to $T^2\setminus D^2\subset \Sigma$ is trivial, hence it is diffeomorphic to $$((A_2\x S^1)\cup (A_1 \x A_2))\x D^2=((A_2\x D^2)\x S^1 )\cup ((A_1\x
D^2) \x A_2)).$$ Furthermore, under this diffeomorphism, the inclusion $$(T^2\setminus D^2)\x \{0\} \subset (T^2\setminus D^2)\x D^2$$ becomes $$(A_2\x \{0\})\x S^1)\cup ((A_1 \x \{0\})\x A_2) \subset ((A_2\x D^2)\x
S^1)\cup
((A_1\x D^2)\x A_2).$$ Now tie a knot $K$ in the arc $(A_2\x \{0\}) \subset (A_2\x D^2)$ to obtain a knotted arc $A_K$ and to obtain a new punctured torus $$\begin{aligned}
T_K\setminus D^2 &= &(A_K\x S^1)\cup ((A_1 \x \{0\})\x A_2)
\\&\subset&((A_2\x D^2)\x S^1)\cup ((A_1 \x D^2)\x A_2)\end{aligned}$$ with $$\partial (T_K\setminus D^2)= \partial (T\setminus
D^2).$$ Then let $$\Sig_{K,C}
= (T_K\setminus D^2)\cup(T^2\setminus(D^2\amalg D^2))\cup\cdots\cup
(T^2\setminus D^2)
\subset N(\Sigma)\subset X.$$
A description of $\Sig_{K,C}$ via rim surgery
---------------------------------------------
Keeping the notation above, we first recall how, via a 3-manifold surgery, we can tie a knot $K$ in the arc $(A_2\x \{0\}) \subset (A_2\x D^2)$. In short, we just remove a small tubular neighborhood in $A_2\x D^2$ of a pushed-in copy $\g$ of the meridional circle $\{0\}\x S^1 \subset A_2\x D^2$ and sew in the exterior of the knot $K$ in $S^3$ so that the meridian of $K$ is identified with $\g$. This has the effect of tying a knot in the arc $A_2\x\{0\}\subset A_2\x D^2$. More specifically, consider the standard embedding of the solid torus $A=(A_1\cup A_2)\x D^2=S^1\x D^2$ in $S^3$ with complementary solid torus $B=D^2\x S^1$ with core $C^\prime = \{0\} \x S^1\subset D^2 \x S^1$. In $A\setminus C=(S^1 \x
D^2)\setminus
C=S^1\x S^1\x (0,1]=(A_1 \cup A_2)\x S^1\x (0,1]$, consider the circle $\g= \{t\} \x S^1 \x \{\frac{1}{2}\}$, with $t \in A_2$, and with tubular neighborhood $N(\g)\subset A\setminus C$. The curve $\g$ is isotopic in $S^3\setminus C$ to the core $C^\prime$ of $B$. We denote by $\g^\prime $ the curve $\g$ pushed off into $\partial N(\g)$ so that the linking number in $S^3$ of $\g$ and $\g^\prime$ is zero. For later reference, note that $D= (A\setminus N(\g))\cup B$ is again diffeomorphic to a solid torus. (It is the exterior of the unknot $\g \subset A \subset S^3$.) The core of $D$ is isotopic (in $D$) to $C$.
Let $M_K$ be the 3-manifold obtained by performing $0$-framed surgery on $K$. Then the meridian $m$ of $K$ is a circle in $M_K$ and has a canonical framing in $M_K$; we denote a tubular neighborhood of $m$ in $M_K$ by $m\x D^2$. Let $S_K$ denote the 3-manifold $$S_K=(A\setminus N(\g)) \cup (M_K\setminus (m\x D^2)).$$ The two pieces are glued together so as to identify $\g^\prime$ with $m$. In other words, we remove $N(\g)$ and sew in the exterior $E(K)$ of the knot $K$ in $S^3$. Note that the core $C$ of the solid torus $A$ is untouched by this operation, so $C \subset S_K$. Also, the boundary $\partial A$ of $A$ and the set $G=A_1\x D^2 \subset (A_1\cup A_2)\x D^2 \subset A$ remain untouched and thus can be viewed as subsets of $S_K$.
\[knot\] There is a diffeomorphism $h: S_K \to A$ which is the identity on $G$ and on the boundary. Furthermore, $h(C)$ is the knotted core $K \subset A$.
In $S^3=A \cup B$, the above operation replaces a tubular neighborhood of the unknot $\g \subset A \subset S^3$ with the exterior $E(K)$ of the knot $K$ in $S^3$. Thus there is a diffeomorphism $h: E(K)\cup D \to A\cup B=S^3$ sending the core circle of $D$ to the knot $K$. Now $C'\subset B\subset E(K)\cup D$ is unknotted, since in $D$, the curve $C'$ is isotopic to $\g'$, which bounds a disk. Thus $S_K$, which is the complement of a tubular neighborhood of $C'$, is an unknotted solid torus in $S^3=E(K)\cup D$. Furthermore, as we have noted above, $C$ is isotopic to the core of $D$; so $C\subset S_K$ is the knot $K$. Thus there is a diffeomorphism $h: S_K \to
S^1\x D^2$ which is the identity on the boundary. After an isotopy rel boundary we can arrange that $h(G)=G$.
To obtain $\Sig_{K,C}$ we cross everything with $S^1$; i.e. remove the neighborhood $N(\g)\x S^1 \subset (A_2\x D^2) \x S^1 \subset N(\Sig)$ of the (nullhomologous) torus $\g\x S^1 \subset (A_2\x D^2) \x S^1 \subset N(\Sig)$ and sew in $E(K)\x
S^1$ as above on the $E(K)$ factor and the identity on the $S^1$ factor. We refer to this as a [*rim surgery*]{} on $\Sigma$. Notice that this construction does not change the ambient manifold $X$. Except where it is absolutely necessary to keep track of the curve $C$, we shall suppress it from our notation and abbreviate $\Sig_{K,C}$ as $\Sig_K$.
The complement of $\Sig_K$
--------------------------
From the construction, it is clear that if the complement of $\Sig$ in $X$ is simply connected, then so is the complement of $\Sig_K$ in $X$, since the meridian of the knot (which is identified with the boundary of the normal fiber to $\Sig$) normally generates the fundamental group of the exterior of $K$. Now there is a map $f: E(K)\to B\cong D^2\x S^1$ which induces isomorphisms on homology and restricts to a homeomorphism $\bd
E(K)\to\bd B$ taking the class of a meridian to $[\{ {\text{pt}} \}\x S^1]$ and the class of a longitude to $[\bd D^2\x\{ {\text{pt}}\} ]$. The map $f\x {\text
{id}}_{S^1}$ on $E(K)\x S^1$ extends via the identity to a homotopy equivalence $X\setminus N(\Sig_K)\to X\setminus N(\Sig)$ which restricts to a homeomorphism $\bd N(\Sig_K)\to \bd N(\Sig)$. Then topological surgery [@Fr; @B] guarantees the existence of a homeomorphism $h:(X,\Sig)\to
(X,\Sig_K)$.
If $\pi_1(X\setminus\Sig)\ne 0$, it is not clear when $X\setminus \Sig_K$ is homeomorphic (or even homotopy equivalent) to $X\setminus \Sig_K$. We avoid such issues in this paper and only deal with the case where $\pi_1(X\setminus\Sig)=0$. However, as already noted; the surface $\Sig -
E$ in $X\#
\CPb$ obtained by tubing together the surface $\Sig$ with the exceptional sphere $E$ of $\CPb$ is primitively embedded; so there is a homeomorphism $h:(X\# \CPb, \Sig-E)\to (X\# \CPb,\Sig_K-E)$. In summary:
Let $X$ be a simply connected smooth 4-manifold with a primitively embedded surface $\Sig$. Then for each knot $K$ in $S^3$, the above construction produces a $\Sig$-compatible surface $\Sig_K$.
The standard pair $(Y_g,S_g)$
==============================
Let $g>0$. In this section we shall construct a simply connected smooth 4-manifold $Y_g$ and a primitive embedding of $S_g$, the surface of genus $g$, in $Y_g$ such that the torus used in the previous section to construct the $S_g$-compatible embedding $(S_g)_K=S_{g,K}$ is contained in a cusp neighborhood.
To this end, consider the $(2,2g+1)$-torus knot $T(2,2g+1)$. It is a fibered knot whose fiber is a punctured genus $g$ surface and whose monodromy $t'$ is periodic of order $4g+2$. If we attach a 2-handle to $\bd B^4$ along $T(2,2g+1)$ with framing $0$, we obtain a manifold $C(g)$ which fibers over the 2-disk with generic fiber a Riemann surface $S_g$ of genus $g$ and whose monodromy map $t$, induced from $t'$, is a periodic holomorphic map $t:S_g\to S_g$ of order $4g+2$. The singular fiber is the topologically (non-locally flatly) embedded sphere obtained from the cone in $B^4$ on the torus knot $T(2,2g+1)$ union the core of the 2-handle. Now consider the fibration over the punctured 2-sphere obtained from gluing together $4g+2$ such neighborhoods $C(g)$ along a neighborhood of a fiber in the boundary of $C(g)$. This is a complex surface, and the monodromy is trivial around a loop which contains in its interior the images of all the singular fibers. Thus we may compactify this manifold to obtain a complex surface $Y_g$ which is holomorphically fibered over $S^2$. For example, $Y_1$ is the rational elliptic surface $\CP\#9\CPb$. In fact, $Y_g$ is just the Milnor fiber of the Brieskorn singularity $\Sigma(2,2g+1,4g+1)$ union a generalized nucleus consisting of the 4-manifold obtained as the trace of the 0-framed surgery on $T(2,2g+1)$ and a $-1$ surgery on a meridian [@Fuller]. The fibration $\pi: Y_g\to S^2$ has a holomorphic section which is a sphere $\Lambda$ of self-intersection $-1$ (the sphere obtained by the $-1$-surgery above (cf. [@Fuller]). This proves that $\pi_1(Y_g\setminus S_g)=0$; so $S_g$ is a primitively embedded surface with self-intersection $0$.
Let $T$ denote the torus in $S_g\x D^2$ on which we perform a rim surgery in order to obtain the surface $S_{g,K}$. We wish to see that $T$ lies in a cusp neighborhood. A cusp neighborhood is nothing more than the regular neighborhood of a torus together with two vanishing cyles, one for each generating circle in the torus. The torus $T$ has the form $T=\g\x\t$ where $\t$ is a closed curve on $S_g$ and $\g = \{{\text {pt}}\}\x
(\{\frac12\}\x\bd D^2)$. The curve $\t$ is one of the generating circles for $H_1(S_g;\Z)$ with a dual circle $\s$. The curve $\g$ spans a $-1$-disk contained in $\Lambda$. The curve $\t$ degenerates to a point on the singular fiber in $C(g)$. Thus we see both required vanishing cycles.
SW-pairs
========
Recall that the Seiberg-Witten invariant $\sw_X$ of a smooth closed oriented $4$-manifold $X$ with $b ^+>1$ is an integer valued function which is defined on the set of $spin^{\, c}$ structures over $X$, (cf. [@W]). In case $H_1(X;\Z)$ has no 2-torsion, there is a natural identification of the $spin^{\, c}$ structures of $X$ with the characteristic elements of $H^2(X;\Z)$. In this case we view the Seiberg-Witten invariant as $$\sw_X: \lbrace k\in H^2(X,\Z)|k\equiv w_2(TX)\pmod2)\rbrace
\rightarrow \Z.$$ The Seiberg-Witten invariant $\sw_X$ is a smooth invariant whose sign depends on an orientation of $H^0(X;\R)\otimes\det H_+^2(X;\R)\otimes \det H^1(X;\R)$. If $\sw_X(\b)\neq
0$, then we call $\b$ a [*[basic class]{}*]{} of $X$. It is a fundamental fact that the set of basic classes is finite. If $\b$ is a basic class, then so is $-\b$ with $$\sw_X(-\b)=(-1)^{(\text{e}+\text{sign})(X)/4}\,\sw_X(\b)$$ where $\text{e}(X)$ is the Euler number and $\text{sign}(X)$ is the signature of $X$.
As in [@FS] we need to view the Seiberg-Witten invariant as a Laurent polynomial. To do this, let $\{\pm \b_1,\dots,\pm \b_n\}$ be the set of nonzero basic classes for $X$. We my then view the Seiberg-Witten invariant of $X$ as the ‘symmetric’ Laurent polynomial $$\sw_X = b_0+\sum_{j=1}^n
b_j(t_j+(-1)^{(\text{e}+\text{sign})(X)/4}\,t_j^{-1})$$ where $b_0=\sw_X(0)$, $b_j=\sw_X(\b_j)$ and $t_j=\exp(\b_j)$.
Now let $\Sig$ be genus $g>0$ primitively embedded surface in the simply connected 4-manifold $X$. If the self-intersection of $\Sig$ is $n\ge
0$, then in $X_n=X\#n\CPb$, consider the surface $\Sig_n=\Sig-\sum_{j=1}^n E_j$ (resp. $\Sig_{n,K}=\Sig_K-\sum_{j=1}^n E_j$) obtained from $\Sig$ (resp. $\Sig_K$) by tubing together with the exceptional spheres $E_j$, $j=1,\dots,n$, of the $\CPb$ in $X_n$. Note that the fiber sum $X_n\#_{\Sig_n=S_g} Y_g$ of $X_n$ and $Y_g$ along the surfaces $\Sig_n$ and $S_g$ has $b^+>1$. An [*SW-pair*]{} is such a pair $(X,\Sig)$ which satisfies the property that the Seiberg-Witten invariant $\sw_{X_n\#_{\Sig_n=S_g} Y_g}\ne 0$.
As we have pointed out earlier, there are several curves $C$ that can be used to construct the surfaces $\Sig_{K,C}$, and there are potentially several different fiber sums that can be performed in the construction of $X_n\#_{\Sig_n=S_g} Y_g$. We pin down our choice of $C$ by declaring it to be the image of the curve $\s$ from §3 under the diffeomorphism used in the construction of the fiber sum. A simple Mayer-Vietoris argument shows that in $X_n\#_{\Sig_n=S_g} Y_g$ the rim torus (equivalently $\g\x\t$) becomes homologically essential and is contained in a cusp neighborhood. Thus our results from [@FS] apply.
SW-pairs and the Alexander polynomial \[proof\]
===============================================
We are now in a position to prove our main theorem:
[**[Theorem 1.2.]{}**]{} [*Consider any $\sw$-pair $(X,\Sig)$. If $K_1$ and $K_2$ are two knots in $S^3$ and if there is a diffeomorphism of pairs $(X,\Sig_{K_1})\to (X,\Sig_{K_2})$, then $\DD_{K_1}(t)=\DD_{K_2}(t)$.*]{}
With notation as above, we have a diffeomorphism $(X_n,\Sig_{n,K_1})\to (X_n,\Sig_{n,K_2})$. Then there is a diffeomorphism $$Z_1=X_n\#_{\Sig_{n,K_1}=S_g}Y_g\to Z_2=X_n\#_{\Sig_{n,K_2}=S_g}Y_g.$$ It follows from [@FS] that $\sw_{Z_i}=\sw_{X_n\#_{\Sig_n=S_g} Y_g}\cdot\DD_{K_i}(t)$ for $t=\exp(2[T])$, where $T$ denotes the rim torus. Since $(X,\Sig)$ is a $\sw$-pair, and since $[T]\ne 0$ in $H_2(Z_i;\Z)$ we must have $\DD_{K_1}(t)=\DD_{K_2}(t)$.
Rim surgery on symplectically embedded surfaces
===============================================
We conclude with a proof of our claim of the introduction.
[**[Theorem 1.1.]{}**]{} [*Let $X$ be a simply connected symplectic 4-manifold and $\Sig$ a symplectically and primitively embedded surface with positive genus and nonnegative self-intersection. If $K_1$ and $K_2$ are knots in $S^3$ and if there is a diffeomorphism of pairs $(X,\Sig_{K_1})\to (X,\Sig_{K_2})$, then $\DD_{K_1}(t)=\DD_{K_2}(t)$. Furthermore, if $\DD_K(t)\ne 1$, then $\Sig_K$ is not smoothly ambient isotopic to a symplectic submanifold of $X$.*]{}
Since $\Sig$ and $S_g$ are symplectic submanifolds of $X$ and $Y_g$, the fiber sum $X_n\#_{\Sig_n=S_g}Y_g$ is also a symplectic manifold [@Gompf]. Thus $\sw_{X_n\#_{\Sig_n=S_g}Y_g}\ne 0$ [@T1]; so $(X,\Sig)$ forms an SW-pair. This proves the first statement of the theorem.
Next, suppose that $\Sig_K$ is smoothly ambient isotopic to a symplectic submanifold $\Sig'$ of $X$. This isotopy carries the rim torus $T$ to a rim torus $T'$ of $\Sig'$. We have $$\sw_{X_n\#_{\Sig'_n=S_g}Y_g}=\sw_{X_n\#_{\Sig_{n,K}=S_g}Y_g}=
\sw_{X_n\#_{\Sig_n=S_g}Y_g}\cdot\DD_K(t)
\label{invt}$$ with $t=\exp(2[T'])$ when this expression is viewed as $\sw_{X_n\#_{\Sig'_n=S_g}Y_g}$. As above, $[T']\ne 0$ in $H_2(X_n\#_{\Sig'_n=S_g}Y_g;\Z)$.
Symplectic forms $\o_X$ on $X_n$ (with respect to which $\Sig'_n$ is symplectic) and $\o_Y$ on $Y_g$ induce a symplectic form $\o$ on the symplectic fiber sum $X_n\#_{\Sig'_n=S_g}Y_g$ which agrees with $\o_X$ and $\o_Y$ away from the region where the manifolds are glued together. In particular, since $T'$ is nullhomologous in $X_n$, we have $\la\o,T'\ra = \la\o_X,T'\ra=0$. Now implies that the basic classes of $X_n\#_{\Sig'_n=S_g}Y_g$ are exactly the classes $b+2mT'$ where $b$ is a basic class of $X_n\#_{\Sig_n=S_g}Y_g$ and $t^m$ has a nonzero coefficient in $\DD_K(t)$. Thus the basic classes of $X_n\#_{\Sig'_n=S_g}Y_g$ can be grouped into collections ${\mathcal{C}}_{b}=\{ b+2mT'\}$, and if $\DD_K(t)\ne 1$ then each ${\mathcal{C}}_{b}$ contains more than one basic class. Note, however, that $\la\o,b+2mT'\ra=\la\o,b\ra$. Now Taubes has shown [@T2] that the canonical class $\k$ of a symplectic manifold with $b^+>1$ is the basic class which is characterized by the condition $\la\o,\k\ra >\la\o,b'\ra$ for any other basic class $b'$. But this is impossible for $X_n\#_{\Sig_n=S_g}Y_g$ since each ${\mathcal{C}}_{b}$ contains more than one class.
[999]{}
S. Boyer, [*Simply-connected 4-manifolds with a given boundary*]{}, Trans. Amer. Math. Soc., **298**, (1986), 331–357.
R. Fintushel and R. Stern, [*Knots, links, and 4-manifolds*]{}, 1996 preprint.
M. Freedman, [*The topology of four–dimensional manifolds*]{}, J. Diff. Geo. **17** (1982), 357–432.
T. Fuller, [*Generalized Nuclei*]{}, UCI Preprint 1997.
R. Gompf, [*A new construction of symplectic manifolds*]{}, Ann. Math. [**142**]{} (1995), 527–595.
C. Taubes, [*The Seiberg-Witten invariants and symplectic forms*]{}, Math. Res. Letters [**1**]{} (1994), 809–822.
C. Taubes, [*More constraints on symplectic manifolds from Seiberg-Witten invariants*]{}, Math. Res. Letters [**2**]{} (1995), 9–14.
E. Witten, [*Monopoles and four-manifolds*]{}, Math. Res. Letters [**1**]{} (1994), 769–796.
[^1]: The first author was partially supported NSF Grant DMS9401032 and the second author by NSF Grant DMS9626330
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abstract: 'For some non-linear field theories which allow for soliton solutions, submodels with infinitely many conservation laws can be defined. Here we investigate the symmetries of the submodels, where in some cases we find a symmetry enhancement for the submodels, whereas in others we do not.'
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ø 2[u\^2]{}
[**Knot soliton models, submodels, and their symmetries**]{}
1 true cm C. Adam$^{1a}$ and J. Sánchez-Guillén$^{1b}$
$^1$[Departamento de Física de Partículas,\
Facultad de Física, Universidad de Santiago, and\
Instituto Galego de Fisica de Altas Enerxias (IGFAE)\
E-15782 Santiago de Compostela, Spain]{}\
${}$\
0.2cm
$^a$[[email protected]]{}
$^b$[[email protected]]{}
Introduction
============
Non-linear field theories with a two-dimensional target space and base space $\RR \times \RR^d$ ($d+1$ dimensional space-time) can give rise to point like (vortex like) solitons for $d=2$, or to line like (knot like) solitons for $d=3$, provided that the fields are required to approach a fixed, constant value at spatial infinity (e.g., to render the energy finite), compactifying thereby the base space $\RR^d$. Especially, for $d=3$ some models with knot solitons have received considerable attention recently and, further, such models have applications both in condensed matter [@BFN1; @Bab1] and elementary particle physics [@FN2; @FNW1]. Here we concretely consider models where the target space is the two-sphere $S^2$. Their solitons can be classified by the homotopy groups $\pi_2 (S^2)=\ZZ$ (winding number, for vortex type solitons) and $\pi_3 (S^2) =\ZZ$ (Hopf index, for knot type solitons), respectively.
The fields of the theories may be parametrized by a three-component unit vector ${\bf n}: \, \RR \times \RR^d \to S^2$, ${\bf n}^2 =1$, or via the stereographic projection = ( u+|u , -i ( u-|u ) , u|u -1 ) , u = \[stereo\] by a complex scalar field $u$.
All models which we study can be constructed from the two Lagrangian densities \[cp1\] [L]{}\_2 = and \_4 = . In two space dimensions we consider the Baby Skyrme model ${\cal L}_{\rm BS}
={\cal L}_2$, whereas in three space dimensions we will consider the Faddeev–Niemi model [@Fad; @FN1] with Lagrangian \[FN-L\] [L]{}\_[FN]{} = [L]{}\_2 - \_4 (here $\lambda $ is a dimensionful coupling constant), the Nicole model \[Ni-La\] [L]{}\_[Ni]{}= ([L]{}\_2)\^ (for which the one known soliton solution was found by Nicole, [@Ni]), and the AFZ model \_[AFZ]{} = -([L]{}\_4)\^ , for which infinitely many soliton solutions have been found by Aratyn, Ferreira and Zimerman (=AFZ) [@AFZ1; @AFZ2]. All four models circumvent Derrick’s theorem and allow for static soliton solutions, either by being spatially scale invariant (the Baby Skyrme, the Nicole, and the AFZ model), or by involving two terms with opposite scaling behaviour (the Faddeev–Niemi model).
All four models (Baby Skyrme, Faddeev–Niemi, AFZ and Nicole) have the same target space $S^2$, therefore they have some common properties. For instance, all Lagrangians are invariant under modular transformations u , a|a + b|b =1. Furthermore, the same area-preserving diffeomorphisms on the target space $S^2$ can be defined for all models, but this does not imply that they are symmetries for all four field theories. In fact, only the AFZ model has the area-preserving diffeomorphisms as symmetries [@BF1; @FR1]. For the other three models the generators of the area-preserving diffeomorphisms do not generate symmetries and the corresponding Noether currents are not conserved. However, it was realized in the study of higher-dimensional integrability within the generalization of the zero curvature representation, [@AFSG], that these Noether currents [*are*]{} conserved for submodels of all three models defined by the additional condition \[eik-eq\] \^u\_u =0 , i.e., the complex eikonal equation. Therefore, these submodels have infinitely many conserved charges. On the other hand, their symmetries have to be determined independently, because the complex eikonal equation is not of the Euler–Lagrange type, i.e., it does not follow from an action, and the Noether theorem does not apply to the submodels. This symmetry determination is the main purpose of our talk.
In Section 2 we give a very brief survey of the issue of integrability in higher dimensions and of the resulting infinitely many conservation laws (i.e., conserved currents). In Section 3 we introduce a general class of Lagrangians (to which, of course, all models mentioned above belong) which provide a Lagrangian realization of the infinitely many conserved currents of Section 2. Further, we explain the geometric significance of these currents and their conservation. In Section 4 we briefly investigate the symmetries of the static equations of motion (which are the relevant ones for solitons) for the three submodels (of the Baby Skyrme, Faddeev–Niemi and Nicole models). Section 5 contains our conclusions.
Brief survey of integrability in higher dimensions
==================================================
In [@AFSG] a generalization of the zero curvature condition of Zakharov and Shabat in 1+1 dimensional integrable models was introduced in order to extend the concept of integrability to field theories in higher dimensions. In its original formulation, this condition was a zero curvature in a generalized loop space which leads to very non-local expressions when re-expressed in terms of fields over ordinary space-time. In the same paper, however, a [*local*]{} condition realizing this generalized zero curvature condition was given, which we want to describe briefly here. We choose a non-semisimple Lie algebra $\tilde {\cal G}$ which is the direct sum of a (possibly, but not necessarily semi-simple) Lie algebra ${\cal G}$ and an abelian ideal ${\cal P}$, i.e., = [G]{} + [P]{} where ${\cal P}$ may, e.g., be a (in general, reducible) representation of ${\cal G}$, in which case we have = f\^[abc]{} T\^c & ,& \[T\^a, P\^n \] = R\^[mn]{}(T\^a) P\^m\
[\[]{} P\^m , P\^n [\]]{} = 0 & , & T\^a ,P\^m and $R^{mn}(T^a)$ are matrices in the representation ${\cal P}$. Further, we choose a flat connection $A_\mu \in {\cal G}$, i.e., \_A\_- \_A\_+ \[ A\_, A\_\] =0 A\_= g\^[-1]{} \_g where $g \in {\rm \bf G}$ and ${\rm \bf G}$ is the Lie group of which ${\cal G}$ is the Lie algebra. Finally, we need a covariantly conserved, vector-valued element of the abelian ideal, $B_\mu \in {\cal P}$, i.e., \^B\_+ \[A\^, B\_\] =0, then there exist the conserved currents J\_g B\_g\^[-1]{} , \^J\_=0 as may be checked easily. If this construction holds for ${\rm dim} ({\cal P}) =\infty$ then we have infinitly many conserved currents.
For our purposes we now specialize to ${\rm\bf G}= SU(2)$ and choose as the group element $g$ a fixed, given function of the field $u: \RR^d \times \RR \to \CC$ and its complex conjugate, g=g(u , |u) SU(2). Essentially, $g$ takes values on the equatorial two-sphere contained within $SU(2)$ when $u$ takes values in $\CC$ (for an explicit expression see [@AFSG]). The representations $P^m$ are now just the standard representations $P^{(l,m)}$ of $SU(2)$ where $l$ and $m$ are the angular momentum and magnetic quantum numbers, respectively. Further we restrict to $m=\pm 1$, i.e., B\_= \_l c\_l B\_\^l , B\^l\_= K\_P\^[(l,1)]{} + |K\_P\^[(l,-1)]{} where the $c_l$ are arbitrary real constants (making the abelian ideal infinite-dimensional), and $K_\mu (u ,\bar u ,u_\mu ,\bar u_\mu )$ is a given function of the field variables $u,\bar u$ and its first derivatives (in principle, also of higher derivatives, but we do not consider this possibility here). Here and below we use the notation $\partial_\mu u \equiv u_\mu$. Different choices for $K_\mu$ correspond to different field theories, as we shall see. Further, $K_\mu$ has to obey the reality condition \[re-cond\] (|u\_K\^) =0. For the so chosen $B^l_\mu$, the corresponding currents $J_\mu^l = g B^l_\mu g^{-1}$ are equivalent to the currents \[J\^G\] J\_\^G = i(K\_G\_u - |K\_G\_[|u]{}) for an arbitrary [*real*]{} function $G(u , \bar u)$ ($G_u \equiv
\partial_u G$), see [@BF1]. If all $J_\mu^l$ are conserved, then $J_\mu^G$ is conserved for arbitrary functions $G$. In the next Section, we shall present a Lagrangian realization of these integrability concepts.
Lagrangian realization of conserved currents
============================================
We introduce the class of Lagrangian densities \[g-lan\] [L]{} (u ,|u ,u\_,|u\_) = F(a,b,c) where a=u|u ,b=u\_|u\^ ,c= (u\_|u\^)\^2 - u\_\^2 |u\_\^2 and $F$ is at this moment an arbitrary real function of its arguments. The phase symmetry $u\ra e^{i\alpha} u$ for a constant $\alpha \in \RR$ holds by construction. For the vector-valued function $K^\mu$ we choose \[k-mu\] K\^= f(a) |\^where $f$ is a real function of its argument, and $\Pi^\mu$ and $\bar \Pi^\mu$ are the conjugate four-momenta of $u$ and $\bar u$, i.e. ($u_\mu \equiv \partial_\mu u$, $F_b \equiv
\partial_b F$, etc.), \_\_[u\^]{} = |u\^F\_b + 2 (u\^|u\_|u\_- |u\_\^2 u\_)F\_c. $K^\mu$ in Eq. (\[k-mu\]) automatically obeys the reality condition (\[re-cond\]) for real Lagrangian densities. For the divergence $\partial^\mu J^G_\mu$ of the current (\[J\^G\]) we find \[div-jg\] \^J\^G\_& = & if ( \[( M’ |u G\_u + G\_[uu]{} ) u\_\^2 - ( M’ u G\_[|u]{} + G\_[|u|u]{}) |u\_\^2 \] F\_b .\
&& . + (uG\_u - |u G\_[|u]{}) \[ M’ (bF\_b + 2 cF\_c ) +F\_a \] ) where $M \equiv \ln f$, $M' \equiv \partial_a M$, and we used the equations of motion \[eom\] \^\_=[L]{}\_u = |u F\_a . Now we want to study under which circumstances the divergence (\[div-jg\]) vanishes (for a more detailed discussion we refer to Ref. [@ASG3]). If no constraints are imposed neither on the Lagrangian nor on the allowed class of fields $u$, then we find the two equations for $G$, \[geq1\] uG\_u - |u G\_[|u]{}=0, and \[geq2\] M\_a |u G\_u + G\_[uu]{} =0 \_u \[f(u|u) G\_u\]=0 , with the solution G\_u = k where $k$ is a real constant. The corresponding current $J^G_\mu$ is the Noether current for the phase transformation $u \ra e^{i \alpha}u $ which is a symmetry by construction.
Next we make the second term in (\[div-jg\]) vanish by imposing on the Lagrangian the condition M\_a (bF\_b + 2 cF\_c ) + F\_a =0 with the general solution \[sol-cha\] F(a,b,c)= F (,) which has an interpretation in terms of the target space geometry. In fact, introduce the two real target space coordinates $\xi^\alpha$ via $u=\xi^1 +i\xi^2$ and the target space metric w.r.t. to the coordinates $\xi^\alpha$ via g\_ f\^[-1]{} \_ & & [det]{}(g\_) f\^[-2]{}\
\_= f\^[-1]{} \_ && where $\epsilon_{\alpha \beta}$ is the usual antisymmetric symbol in two dimensions. Then the expressions on which $\tilde F$ may depend can be written as &=& g\_ () \^\^\_\^\
&=& \_ \_ \^\^\_\^\^\^\_\^i.e., they depend on the target space metric and on the determinant of the target space metric, respectively. Let us point out here that all models of Section 1 are of this type, i.e., ${\cal L}_2 = b/f$ and $ {\cal L}_4 = c/f^2$ for $ f=(1+a)^2$ (the target space metric of the two-sphere).
To make the first term in Eq. (\[div-jg\]) vanish, as well, we may either continue to impose Eq. (\[geq2\]), which is solved by those $G$ which generate the target space isometries for the given target space metric (i.e., the given function $f$). Or we may restrict the Lagrangian further by imposing $F_b \equiv 0 \, \Rightarrow \, F=\tilde F(c/f^2)$. Then we have no restriction on $G$ at all, and it follows that these unrestricted $G$ generate the area-preserving diffeomorphisms on target space. This is precisely the case for the AFZ model.
Alternatively, we may make the first term in Eq. (\[div-jg\]) vanish by imposing restrictions on the allowed field configurations $u$. In this case the currents $J^G_\mu$ are still the Noether currents of area-preserving diffeomorphisms, but these transformations are no longer symmetry transformations of the pertinent Lagrangians, in general. Concretely, we require that $u$ obeys the complex eikonal equation (\[eik-eq\]), which defines therefore submodels for all models of the type (\[sol-cha\]) with infinitely many conserved charges.
\[Remark: we might require, instead, that the field $u$ obeys the (in general nonlinear) first order PDE which follows from the condition $F_b =0$ in cases when this condition does not hold identically (i.e., for Lagrangians which do depend on the term $b=u^\mu \bar u_\mu$). This type of (“generalized”) integrability condition, which depends, however, on the chosen Lagrangian, has been discussed in [@Wer3], [@ASG3].\]
Symmetries of the static equations
==================================
Here we just want to present the results of the calculation of all geometric symmetries (point symmetries) of the static equations, i.e., the static equations of motion (e.o.m.) for the full models (Baby Skyrme, Nicole and Faddeev–Niemi), and the static equations of motion plus the static eikonal equation for the corresponding submodels (observe that the static complex eikonal equation does have nontrivial solutions, in contrast to its real counterpart, see, e.g., [@Ada1]). The method of prolongations has been used for all symmetry calculations. Concretely, the symmetries of the submodels are calculated by first calculating the on-shell symmetries of the static eikonal equation. In a next step the on-shell symmetries of the static second order equations are calculated, where the second order equations consist of the equations of motion plus the prolongations of the static eikonal equation (i.e., the second order equations that follow by acting with total derivatives on the complex eikonal equation). For the calculations we refer to [@ASG2]. Here we give a detailed discussion of the results, which are displayed in Table 1.
--------------- ---------------- --------------------- -----------
$\infty $ many geometric solutions
model conserv. laws symmetries known
Baby Skyrme yes$^a$ $C_2 \times SU(2) $ yes
submodel yes $C_2 \times C_2 $ yes
Nicole no $C_3 \times SU(2)$ yes
submodel yes $C_3 \times SU(2)$ yes
Faddeev–Niemi no $E_3 \times SU(2)$ yes$^b$
submodel yes $E_3 \times SU(2)$ no
--------------- ---------------- --------------------- -----------
: Some results for the three soliton models and their submodels. $C_d \; \ldots \; $ conformal group in $d$ dimensions. $E_d \; \ldots \; $ Euclidean group (translations and rotations) in $d$ dimensions. ${}^a $due to the infinite-dimensional base space symmetries $C_2$. ${}^b $known only numerically
For the Baby Skyrme model we find that the full static model has a point symmetry group which is a direct product of base space and target space symmetries, where the group of base space symmetries is the conformal group in two dimensions $C_2$, and the group of target space symmetries is the group $SU(2)$. On the other hand, the submodel has the point symmetry group $C_2 \times C_2$, i.e., the conformal group also in target space. Therefore, the submodel has more symmetry in the case of the Baby Skyrme model, although the additional symmetry is not related to the area-preserving diffeomorphisms. Further, there exist static solutions to the submodel. In fact, [*all*]{} soliton solutions of the Baby Skyrme model are, at the same time, also solutions of the submodel.
For the Nicole model we find that the group of point symmetries of the static e.o.m. is again a direct product of base space and target space symmetry groups, where the base space symmetry group is $C_3$, the conformal group in three dimensions, and the target space symmetry group is $SU(2)$. Further, the static submodel has exactly the same symmetry group $C_3 \times SU(2)$ as the full Nicole model. For the Nicole model only one analytical soliton solution is known, but this solution solves the static eikonal equation, as well, and is, therefore, also a solution of the submodel [@Ni; @ASG2].
For the Faddeev–Niemi model the situation is similar. Again, the static submodel has exactly the same symmetries as the full static model, and the symmetry group is a direct product of the Euclidean group in three dimensions $E_3$ in base space (i.e., rotations and translations) and of the group $SU(2)$ in target space. For the full Faddeev–Niemi model soliton solutions are known only numerically [@FN2], [@GH] - [@HiSa]. It is not known whether the submodel does or does not have solutions.
Conclusions
===========
In this talk we gave a brief survey of the generalized zero curvature representation of [@AFSG], which leads to a generalization of integrability to higher-dimensional non-linear field theories. Then we introduced a class of Lagrangian field theories parametrized by a complex field variable $u$, where this concept of integrability is realized by providing infinitely many conserved currents either for the full theory or for the submodel defined by the eikonal equation $(\partial u)^2 =0$. Further, these currents may be interpreted as Noether currents of area-preserving diffeomorphisms on the target space where $u$ takes its values. For some relevant models within this class of field theories, which allow for soliton solutions, we then presented the results of a thorough analysis of the symmetries of their submodels.
The general result for all cases is that the area-preserving diffeomorphisms are not symmetries of any eikonal submodel. Also, the three-dimensional submodels of Faddeev–Niemi and Nicole have no additional symmetries compared to the full theories.
The Baby Skyrme model is special, as the restriction does have an intriguing additional symmetry. This can be important as there is not much difference of the solutions of the full model and the restriction, at least for the static case. We remind that the method can be easily extended to include the time dependence.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partly supported by MCyT(Spain) and FEDER (FPA2002-01161), Incentivos from Xunta de Galicia and the EC network “EUCLID”. Further, CA acknowledges support from the Austrian START award project FWF-Y-137-TEC and from the FWF project P161 05 NO 5 of N.J. Mauser.
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abstract: 'We study the conditional distribution $K^N_k(z | p)$ of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point $p$ (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros but we show that it has quite a different small distance behavior. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension one. To prove this, we give universal scaling asymptotics for $K^N_k(z | p)$ around $p$. The key tool is the conditional [Szegő ]{}kernel and its scaling asymptotics.'
address:
- 'Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA'
- 'Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA'
- 'Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA'
author:
- Bernard Shiffman
- Steve Zelditch
- Qi Zhong
date: 'May 22, 2010'
title: 'Random zeros on complex manifolds: conditional expectations'
---
Introduction
============
In this paper we study the conditional expected distribution of zeros of a Gaussian random system $\{s_1, \dots,
s_k\}$ of $k \leq m$ polynomials of degree $N$ in $m$ variables, given that the polynomials $s_j$ vanish at a point $p\in M$, or at a finite set of points $\{p_1, \dots, p_r\}$. More generally, we consider systems of holomorphic sections of a degree $N$ positive line bundle $L^N \to M_m $ over a compact [Kähler ]{}manifold of dimension $m$. The conditional expected distribution is the current $ K_k^N(z |p)\in{\mathcal{D}}'^{k,k}(M)$ given by $$\label{CONDE}\Big( K_k^N(z | p), {\varphi}\Big) : = {{\mathbf E}}_N
\Big[(Z_{s_1, \dots, s_k}
,{\varphi})\Big|s_1(p)=\cdots=s_k(p)=0\Big],\quad
\mbox{for }\ {\varphi}\in{\mathcal{D}}^{m-k,m-k}(M)
\,.$$ Here, $Z_{s_1, \dots, s_k}$ is the $(k, k)$ current of integration over the simultaneous zeros of the sections; i.e., its pairing with a smooth test form ${\varphi}\in {\mathcal{D}}^{m - k, m -k}(M)$ is the integral $\int_{Z_{s_1, \dots, s_k}} {\varphi}$ of the test form over the joint zero set. The expectation ${{\mathbf E}}_N$ is the standard Gaussian conditional expectation on $\prod_1^kH^0(M, L^N)$, which we condition on the linear random variable $(s_1,\dots,s_k) \mapsto (s_1(p),\dots,s_k(p))$ that evaluates the sections at the point $p$ (see Definition \[defcondk\]).
We show that $K_k^N(z | p) $ is a smooth $(k, k)$ form away from $p$ (Lemma \[product\]), and we determine its asymptotics, both unscaled and scaled, as $N \to
\infty$. Our main result, Theorem \[scaled\] (for $k=m$) and Theorem \[all codim\] (for $k<m$), is that the scaling limit of $K_k^N(z |p)$ around the point $p$ is the conditional expected distribution $K_{km}^\infty (z | 0)$ of joint zeros given a zero at $z = 0$ in the Bargmann-Fock ensemble of entire holomorphic functions on ${{\mathbb C}}^m$, and we give an explicit formula for $K_{km}^\infty (z | 0)$. Thus, the scaling limit is universal.
Our study of $K_k^N(z | p) $ is parallel to our study of the two-point correlation function $K_{2k}^N(z, p)$ for joint zeros in our prior work with P. Bleher [@BSZ; @BSZ2]. There we showed that $K_{2k}^N(z, p)$ similarly has a scaling limit given by the pair correlation function $K^\infty _{2km}(z, 0)$ of zeros in the Bargmann-Fock ensemble. Both $K_{km}^N(z |p) $ and $K_{2 km}^N(z,
p)$ measure a probability density of finding simultaneous zeros at $z$ and at $p$: $K_{km}^N(z | p) $ is the result of conditioning in a Gaussian space (see e.g. [@J], Chapter 9.3), while $K_{2 km}^N(z, p)$ is a natural conditioning from the viewpoint of random point processes (see §\[PT\]). Of special interest is the case $k=m$ where the joint zeros are (almost surely) points. In this case, the scaling limit (Bargmann-Fock) conditional density $K_{mm}^\infty (z | 0)$ and pair correlation density $K_{2mm}^\infty (z, 0)$ turn out to have quite different short distance behavior, as discussed in §\[short\] below.
To state our results, we need to recall the definition of a Gaussian random system of holomorphic sections of a line bundle. We let $(L, h) \to (M, \omega_h)$ be a positive Hermitian holomorphic line bundle over a compact complex manifold with [Kähler ]{}form $\omega_h=\frac
i2\Theta_h$. We then let $H^0(M, L^N)$ denote the space of holomorphic sections of the $N$-th tensor power of $L$. A special case is when $M = {{{\mathbb C}}{{\mathbb P}}}^m$, and $L = {\mathcal{O}}(1)$ (the hyperplane section line bundle), in which case $H^0({{{\mathbb C}}{{\mathbb P}}}^m,
{\mathcal{O}}(N))$ is the space of homogenous polynomials of degree $N$. As recalled in $\S \ref{BACKGROUND}$, the Hermitian metric $h$ on $L$ induces inner products on $H^0(M,L^N)$ and these induce a Gaussian measure $\gamma_h^N$ on $H^0(M,L^N)$. A Gaussian random system is a choice of $k$ independent Gaussian random sections, i.e. we endow $\prod_{j = 1}^k H^0(M, L^N)$ with the product measure. We refer to $(\prod_{j = 1}^k H^0(M, L^N), \prod_{j=1}^k
\gamma_h^N)$ as the [*Hermitian Gaussian ensemble*]{} induced by $h$. We let ${{\mathbf E}}_N={{\mathbf E}}_{(\prod\gamma_h^N)}$ denote the expected value with respect to $\prod
\gamma_h^N$. Given $s_1, \dots, s_k \in H^0(M, L^N)$ we denote by $Z_{s_1,
\dots, s_k}$ the current of integration over the zero set $\{z \in
M: s_1(z) = \cdots = s_k(z) = 0\}$. Further background is given in §\[BACKGROUND\] and in [@SZ; @BSZ; @SZa].
Our first result gives the asymptotics as $N \to \infty$ of the conditional expectation of the zero current (\[CONDE\]) of one section. It shows that conditioning on $s(P) = 0$ only modifies the unconditional zero current by a term of order $N^{-m}$, where $m=\dim M$.
\[unscaled\] Let $(L, h) \to (M, \omega_h)$ be a positive Hermitian holomorphic line bundle over a compact complex manifold of dimension $m$ with [Kähler ]{}form $\omega_h=\frac i2\Theta_h$, and let $(H^0(M,L^N),
\gamma_{h}^N)$ be the Hermitian Gaussian ensemble. Let let $p_1,\dots, p_r$ be distinct points of $M$. Then for all test forms ${\varphi}\in
{\mathcal{D}}^{m-1,m-1}(M)$, we have $${{\mathbf E}}_N\Big[(Z_s,{\varphi})\Big|s(p_1)=\cdots=s(p_r)=0\Big]={{\mathbf E}}_N(Z_s,{\varphi}) -C_m
\,N^{-m}\sum_{j=1}^k\frac {i{\partial{\bar\partial}}{\varphi}(p_j)}{{\Omega}_M(p_j)}+O(N^{-m-1/2+{\varepsilon}}),$$ where ${\Omega}_M= \frac 1{m!}{\omega}_h^m$ is the volume form of $M$, and $C_m={{\textstyle \frac 12}}\pi^{m-1}\,\zeta(m+1)$.
As mentioned above, the interesting problem is to rescale the zeros around a fixed point $z_0$. When $k = m$ the joint zeros of the system are almost surely a discrete set of points which are $\frac{1}{\sqrt{N}}$-dense. Hence, we rescale a $\frac{C}{\sqrt{N}}$-ball around $z_0$ by $\sqrt{N}$ to make scaled zeros a unit apart on average from their nearest neighbors. If $z_0 \not = p_j$ for any $j$, the scaled limit density is just the unconditioned scaled density, so we only consider the case where $z_0 = p_{j_0}$ for some $j_0$. Then the other conditioning points $p_j, j \not= j_0$, become irrelevant to the leading order term, so we only consider the scaled conditional expectation with one conditioning point. Our main result is the following scaling asymptotics
\[scaled\] Let $(L,h)\to (M,{\omega}_h)$ and $(H^0(M,L^N),
\gamma_{h}^N)$ be as in Theorem \[unscaled\], and let $p\in M$. Choose normal coordinates $z=(z_1,\dots,z_m):M_0,p\to {{\mathbb C}}^m,0$ on a neighborhood $M_0$ of $p$, and let $\tau_N={\sqrt{N}}\,z:M_0\to{{\mathbb C}}^m$ denote the scaled coordinate map.
Let $K_m^N(z|p)$ be the conditional expected zero distribution given by and Definition \[defcondk\]. Then for a smooth test function ${\varphi}\in{\mathcal{D}}({{\mathbb C}}^m)$, we have $$\begin{gathered}
\left(K^N_m(z|p)\,,\,{\varphi}\circ \tau_N(z)\right)\\=\ {\varphi}(0)\ +\ \int_{{{\mathbb C}}^m{\smallsetminus}\{0\}} {\varphi}(u)\left(
\frac i{2\pi}{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2}) +|u|^2\right]\right)^m
\ +\ O(N^{-1/2+{\varepsilon}})\,,\end{gathered}$$ where $u=(u_1,\dots,u_m)$ denotes the coordinates in ${{\mathbb C}}^m$.
In §\[proof2\], we give a similar result (Theorem \[all codim\]) for the conditional expected joint zero current $K_k^N(z|p)$ of joint zeros of codimension $k<m$.
Theorem \[scaled\] may be reformulated (without the remainder estimate) as the following weak limit formula for currents:
\[MAINCOR\] Under the hypotheses and notation of Theorem \[scaled\], $$\begin{aligned}
\tau_{N*}\left(K^N_m(z|p)\right)\ \to\ K_{mm}^\infty(u|0) & {\buildrel {\operatorname{def}}\over =}& {\delta}_0(u)+ \left(
\frac i{2\pi}{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2}) +|u|^2\right]\right)^m
\\& =& {\delta}_0(u)+ \frac
{1-(1+|u|^2)e^{-|u|^2}}{(1-e^{-|u|^2})^{m+1}}\left(\frac
i{2\pi}{\partial{\bar\partial}}|u|^2\right)^m\end{aligned}$$ weakly in ${\mathcal{D}}'^{m,m}({{\mathbb C}}^m)$, as $N\to\infty$.
The term $\delta_0(u)$ comes of course from the certainty of finding a zero at $p$ given the condition. The form $\left(\frac
i{2\pi}{\partial{\bar\partial}}|u|^2\right)^m$ is the scaling limit of the unconditioned distribution of zeros.
It follows from the proof that $K_{mm}^\infty(u|0)$ is the conditional density of common zeros of $m$ independent random functions in the Bargmann-Fock ensemble of holomorphic functions on ${{\mathbb C}}^m$ of the form $$f(u) = \sum_{J \in {{\mathbb N}}^m}\frac{c_{J}}{\sqrt{J!}}\,
u^J \;,$$ where the coefficients $c_{J}$ are independent complex Gaussian random variables with mean 0 and variance 1. The monomials $\frac{\pi^{-m/2}}{\sqrt{J!}}\,
u^J$ form a complete orthonormal basis of the Bargmann-Fock space of holomorphic functions that are in $L^2({{\mathbb C}}^m, e^{- |z|^2} dz)$, where $dz$ denotes Lebesgue measure. (We note that $f(u)$ is a.s. not in $L^2({{\mathbb C}}^m, e^{- |z|^2} dz)$; instead, $f(u)$ is of finite order 2 in the sense of Nevanlinna theory. For further discussion of the Bargmann-Fock ensemble, see [@BSZ] and §6 of the first version (arXiv:math/0608743v1) of [@SZa].)
Short distance behavior of the conditional density {#short}
--------------------------------------------------
As in the case of the pair correlation function, Corollary \[MAINCOR\] determines the short distance behavior of the conditional density of zeros around the conditioning point.
Before describing the results for the conditional density, let us recall the results in [@BSZ; @BSZ2] for the pair correlation function of zeros. The correlation function $K_{nk}^N(z_1, \dots,
z_n)$ is the probability density of finding zeros of a system of $k$ sections at the $n$ points $z_1, \dots,
z_n$. For purposes of comparison to the conditional density, we are interested in the pair correlation density $K_{2m}^N(z_1,z_2)$ for a full system of $k = m$ sections. It gives the probability density of finding a pair of zeros of the system at $(z_1, z_2)$. The scaling limit $$\label{slcd}\kappa_{mm}(|u|):= \lim
_{N\to\infty}K^N_{1k}(p)^{-2} K_{2m}^N(p, p
+\frac{u}{{\sqrt{N}}})\,$$ measures the asymptotic probability of finding zeros at $p, p + \frac{u}{\sqrt{N}}$. As the notation indicates, it depends only on the distance $r = |u|$ between the scaled points in the scaled metric around $p$. For small values of $r$, it is proved in [@BSZ; @BSZ2] that $$\label{leading} \kappa_{mm}(r)= \frac{m+1}{4}
r^{4-2m} + O(r^{8-2m})\,,\qquad\mbox{as }\ r\to 0\,.$$ This shows that the pair correlation function exhibits a striking dimensional dependence: When $m = 1, \kappa_{mm}(r) \to 0$ as $r \to 0$ and one has “zero repulsion.” When $m = 2$, $\kappa_{mm}(r) \to
3/4$ as $r \to 0$ and zeros neither repel nor attract. With $m
\geq 3$, $\kappa_{mm}(r) \nearrow \infty$ as $r \to 0$ and there joint zeros tend to cluster, i.e. it is more likely to find a zero at a small distance $r$ from another zero than at a small distance $r$ from a given point.
The probability (density) of finding a pair of scaled zeros at $(p, p + \frac{u}{\sqrt{N}})$ sounds similar to finding a second zero at $p + \frac{u}{\sqrt{N}}$ if there is a zero at $p$, i.e. the conditional probability density. Hence one might expect the scaled conditional probability to resemble the scaled correlation function. But Corollary \[MAINCOR\] tells a different story. We ignore the term $\delta_0$ (again) since it arises trivially from the conditioning and only consider the behavior of the coefficient $$\label{leadingb}
\kappa_{m}^{\operatorname{cond}}(|u|) : = \frac {1-(1+|
u|^2)e^{-|u|^2}}{(1-e^{-|u|
^2})^{m+1}} \sim \frac{1}{2}\;|u|^{2 - 2 m}$$ of the scaling limit conditional distribution with respect to the Lebesgue density $\left(\frac i{2\pi}{\partial{\bar\partial}}|u|^2\right)^m$ near $u = 0$. The shift of the exponent down by $2$ in comparison to equation (\[leading\]) has the effect of shifting the dimensional description down by one: In dimension one, the coefficient is asymptotic to $\frac{1}{2}$ and therefore resembles the neutral situation in our description of the pair correlation function. Thus we do not see ‘repulsion’ in the one dimensional conditional density. In dimension two, the conditional density (\[leadingb\]) is asymptotic to $\frac{1}{2} |u|^{-2}$, and there is a singularly enhanced probability of finding a zero near $p$ similar to that for the pair correlation function in dimension three; and so on in higher dimensions.
The following graphs illustrate the different behavior of these two conditional zero distrbutions in low dimensions:
{height="2in"}{height="2in"}\
[$\kappa_{1}^{\operatorname{cond}}(r)
$ $\kappa_{2}^{\operatorname{cond}}(r)$]{}\
{height="2in"} {height="2in"}{height="2in"}\
It is well known that conditioning on an event of probability zero depends on the random variable used to define the event. So there is no paradox, but possibly some surprise, in the fact that the two conditional distributions are so different. See §\[comparison\] for further discussion of the comparison of the pair correlation and the conditional density.
\[BACKGROUND\] Background
=========================
We begin with some notation and basic properties of sections of holomorphic line bundles, Gaussian measures. The notation is the same as in [@BSZ; @SZ2; @SZa].
Complex Geometry
----------------
We denote by $(L,h)\to M$ a Hermitian holomorphic line bundle over a compact [Kähler ]{}manifold $M$ of dimension $m$, where $h$ is a smooth Hermitian metric with positive curvature form $$\Theta_h=-\partial\bar\partial\log \|e_L\|^2_h\,.$$ Here, $e_L$ is a local non-vanishing holomorphic section of $L$ over an open set $U\subset M$, and $\|e_L\|_h=h(e_L,e_L)^{1/2}$ is the $h$-norm of $e_L$. As in [@SZa], we give $M$ the Hermitian metric corresponding to the Kähler form ${\omega}_h=\frac{\sqrt{-1}}{2}\Theta_h$ and the induced Riemannian volume form $${\Omega}_M=\frac 1{m!}\,{\omega}_h^m.$$
We denote by $H^0(M,L^N)$ the space of holomorphic sections of $L^N=L^{\otimes N}$. The metric $h$ induces Hermitian metrics $h^N$ on $L^N$ given by $\|s^{\otimes N}\|_{h^N}=\|s\|^N_h.$ We give $H^0(M, L^N)$ the Hermitian inner product $$\label{inner product}
\langle s_1,s_2\rangle=\int_Mh^N(s_1,s_2)\,{\Omega}_M \ \ \ \ (s_1,s_2\in
H^0(M,L^N)),$$ and we write $\|s\|=\langle s,s\rangle^{1/2}.$
For a holomorphic section $s\in H^0(M, L^N)$, we let $Z_s\in{\mathcal{D}}'^{1,1}(M)$ denote the current of integration over the zero divisor of $s$: $$(Z_s, {\varphi})=\int_{Z_s}{\varphi}, \ \ \ {\varphi}\in {\mathcal{D}}^{m-1,m-1}(M),$$ where ${\mathcal{D}}^{m-1,m-1}(M)$ denotes the set of compactly supported $(m-1,m-1)$ forms on $M$. (If $M$ has dimension 1, then ${\varphi}$ is a compactly supported smooth function.) For $s=ge_L$ on an open set $U\subset M$, the Poincaré-Lelong formula states that $$\label{PL}
Z_s=\frac i\pi\partial\bar\partial\log
|g|=\frac i\pi \partial\bar\partial\log||s||_{h^N} +\frac N\pi{\omega}_h.$$
### \[SZEGO\] The Szegö kernel
Let $\Pi_N$: $L^2(M,L^N)\to H^0(M,L^N)$ denote the [Szegő ]{}projector with kernel $
\Pi_N$ given by $$\Pi_N(z,w)=\sum^{d_N}_{j=1}S^N_j(z)\otimes \overline{S^N_j(w)}\in L^N_z\otimes
\overline L^N_w\,,$$ where $\{S^N_j\}_{1\le j\le d_N}$ is an orthonomal basis of $H^0(M,L^N)$.
We shall use the [*normalized [Szegő ]{}kernel*]{} $$\label{PN} P_N(z,w):=
\frac{\|\Pi_N(z,w)\|_{h^N}}{\|\Pi_N(z,z)\|_{h^N}^{1/2}\,
\|\Pi_N(w,w)\|_{h^N}^{1/2}}\;.$$ (Note that $\|\Pi_N(z,w)\|_{h^N} = \sum\|S^N_j(z)\|_{h^N(z)}\,\|S^N_j(w)\|
_{h^N(w)}$, which equals the absolute value of the [Szegő ]{}kernel lifted to the associated circle bundle, as described in [@SZ2; @SZa].)
We have the ${\mathcal{C}}^\infty$ diagonal asymptotics for the [Szegő ]{}kernel ([@C; @Z]): $$\label{Zel} \|\Pi_N(z,z)\|_{h^N} = \frac {N^m}{\pi^m} +
O(N^{m-1})\;.$$ Off-diagonal estimates for the normalized [Szegő ]{}kernel $P_N$ were given in [@SZa], using the off-diagonal asymptotics for $\Pi_N$ from [@BSZ; @SZ2]. These estimates are of two types:\
1) ‘far-off-diagonal’ asymptotics (Proposition 2.6 in [@SZa]): For $b>\sqrt{j+2k}$, $j,k\ge 0$, we have $$\label{far} \nabla^j
P_N(z,w)=O(N^{-k})\qquad \mbox{uniformly for }\ d(z,w)\ge
b\,\sqrt{\frac {\log N}{N}} \;.$$ (Here, $\nabla^j$ stands for the $j$-th covariant derivative.)\
2) ‘near-diagonal’ asymptotics (Propositions 2.7–2.8 in [@SZa]): Let $ z_0\in M$. For ${\varepsilon},b>0$, there are constants $C_j=C_j({M,{\varepsilon},b})$, $j \ge 2$, independent of the point $z_0$, such that $$\label{near}\textstyle P_N\left(z_0+\frac u{{\sqrt{N}}},z_0 +\frac
v{{\sqrt{N}}}\right) =
e^{-\frac 12 |u-v|^2}[1 + R_N(u,v)]\;,$$ where $$\label{nearr} \begin{array}{c}|R_N(u,v)|\le \frac {C_2}2\,|u-v|
^2N^{-1/2+{\varepsilon}}\,, \quad
|\nabla R_N(u)| \le C_2\,|u-v|\,N^{-1/2+{\varepsilon}}\,,
\\[8pt] |\nabla^jR_N(u,v)|\le C_j\,N^{-1/2+{\varepsilon}}\quad j\ge 2\,,\end{array}$$ for $|u|+|v|<b\sqrt{\log N}$. (Here, $u, v$ are normal coordinates near $z_0$.)
The limit on the right side of (\[near\]) is the normalized [Szegő ]{}kernel for the Bargmann-Fock ensemble (see [@BSZ]). This is why the scaling limits of the correlation functions and conditional densities coincide with those of the Bargmann-Fock ensemble.
Probability
-----------
If $V$ is a finite dimensional complex vector space, we shall associate a complex Gaussian probability measure ${\gamma}$ to each Hermitian inner product on $V$ as follows: Choose an orthonormal basis $v_1,\dots,v_n$ for the inner product and define ${\gamma}$ by $$d\gamma(v)=\frac{1}{\pi^{n}}e^{-|a|^2}d_{2n}a, \ \
s=\sum^{n}_{j=1}a_jv_j\in V\,,$$ where $d_{2n}a$ denotes $2n$-dimensional Lebesgue measure. This Gaussian is characterized by the property that the $2n$ real variables ${{\operatorname{Re}\,}}a_j$, ${{\operatorname{Im}\,}}a_j$ ($j=0,....,d_N$) are independent random variables with mean 0 and variance $\frac12$; i.e., $$\mathbf{E}_{\gamma}a_j=0, \ \ \mathbf{E}_{\gamma}a_ja_k=0, \ \ \mathbf{E}_{\gamma}a_j\bar a_k=
\delta_{jk}\,.$$ Here and throughout this article, $\mathbf{E}_{\gamma}$ denotes expectation with respect to the probability measure ${\gamma}$: $\mathbf{E}_{\gamma}{\varphi}=\int{\varphi}\,d\gamma$. Clearly, ${\gamma}$ does not depend on the choice of orthonormal basis, and each (nondegenerate) complex Gaussian measure on $V$ is associated with a unique (positive definite) Hermitian inner product on $V$.
In particular, we give $H^0(M,L^N)$ the complex Gaussian probability measure ${\gamma}_h$ induced by the inner product ; i.e., $$d\gamma_h(s)=\frac{1}{\pi^{d_N+1}}e^{-|a|^2}da, \ \
s=\sum^{d_N}_{j=1}a_jS^N_j,$$ where $\{S^N_j:1\leq j\leq d_N\}$ is an orthonormal basis for $H^0(M,L^N)$ with respect to (\[inner product\]). The probability space $(H^0(M,L^N),{\gamma}_N)$ is called the [*Hermitian Gaussian ensemble*]{}. We regard the currents $Z_s$ (resp. measures $|Z_s|$), as current-valued (resp. measure-valued) random variables on $(H^0(M,L^N),{\gamma}_N)$; i.e., for each test form (resp. function) ${\varphi}$, ($Z_s,{\varphi}$) (resp. ($|Z_s|,{\varphi}$)) is a complex-valued random variable.
Since the zero current $Z_s$ is unchanged when $s$ is multiplied by an element of ${{\mathbb C}}^*$, our results remain the same if we instead regard $Z_s$ as a random variable on the unit sphere $SH^0(M,L^N)$ with Haar probability measure. We prefer to use Gaussian measures in order to facilitate computations.
### Holomorphic Gaussian random fields
Gaussian random fields are determined by their two-point functions or covariance functions. We are mainly interested in the case where the fields are holomorphic sections of $L^N$; i.e, our probability space is a subspace ${\mathcal{S}}$ of the space $ H^0(M, L^N)$ of holomorphic sections of $L^N$ and the probability measure on ${\mathcal{S}}$ is the Gaussian measure induced by the inner product . If we pick an orthonormal basis $\{S_j\}_{1\le j\le m}$ of ${\mathcal{S}}$ with respect to (\[inner product\]), then we may write $s
=\sum^{n}_{j=0}a_jS_j $, where the coordinates $a_j$ are i.i.d. complex Gaussian random variables. The two point function $$\Pi_{\mathcal{S}}(z,w):={{\mathbf E}}_{\mathcal{S}}\left(s(z)\otimes\overline{s(w)}\right)=
\sum^{n}_{j=1}S_j(z)\otimes \overline{S_j(w)}$$ is the kernel of the orthogonal projection onto ${\mathcal{S}}$, and equals the [Szegő ]{}kernel $\Pi_N(z,w)$ when ${\mathcal{S}}= H^0(M, L^N)$. The expected zero current ${{\mathbf E}}_
{\mathcal{S}}\big(Z_s\big)$ for random sections $s\in{\mathcal{S}}$ is given by the [*probabilistic Poincaré-Lelong formula:*]{}
\[important lemma\] [[@SZ]]{} Let $(L,h)\to M$ be a Hermitian holomorphic line bundle over a compact complex manifold $M$ and let $
{\mathcal{S}}\subset H^0(M,L^N)$ be a Gaussian random field with two-point function $
\Pi_{\mathcal{S}}(z,w)$. Then $${{\mathbf E}}_{\mathcal{S}}\big(Z_s\big) = \frac i{2\pi} {\partial{\bar\partial}}\log\|\Pi_{\mathcal{S}}(z,z)\|_{h^N} +
\frac N{2\pi}\,\sqrt{-1}\,\Theta_h\,.$$
This lemma was given in [@SZ Prop. 3.1] and [@SZa Prop. 2.1] with slightly different hypotheses. For convenience, we include a proof below.
Let $\{S_j\}_{1\le j\le n}$ be a basis of ${\mathcal{S}}$ such that $s\in{\mathcal{S}}$ is of the form $s=\sum_{j=1}^n a_j S_j$, where the $a_j$ are independent standard complex Gaussian random variables, as above. We then have $\|\Pi_{\mathcal{S}}(z,z)\|
_{h^N}= \sum_{j=1}^n\|S_j\|^2_{h^N}$. For any $s\in {\mathcal{S}}$, we write $$s=\sum_{j=1}^n a_jS_j=\langle a,F\rangle e_L^{\otimes
N},$$ where $e_L$ is a local non-vanishing holomorphic section of $L$, $S_j=f_j\,e_L^{\otimes N}$, and $F=(f_1,...,f_n).$ We then write $F(z)=|F(z)|U(z)$ so that $|U(z)|\equiv 1$ and $$\log|\langle a,F\rangle |=\log|F|+\log|\langle a,U\rangle|.$$ A key point is that ${{\mathbf E}}(\log|\langle a,U\rangle|)$ is independent of $z$, and hence ${{\mathbf E}}(d\log|\langle a,U\rangle|)=0$. We note that $U$ is well-defined a.e. on $ M\times {\mathcal{S}}$; namely, it is defined whenever $s(z)\neq 0$.
Write $d{\gamma}= \frac 1{\pi^n}e^{-|a|^2}\,da$. By , we have $$\begin{aligned}
({{\mathbf E}}Z_s,{\varphi})&= {{\mathbf E}}\left(\frac{\sqrt{-1}}{\pi} {\partial{\bar\partial}}\log|\langle a,F\rangle |,
{\varphi}\right)\
=\ \frac{\sqrt{-1}}{\pi}\int_{{{\mathbb C}}^{n}}(\log|\langle a,F\rangle
|,\partial\bar\partial
{\varphi})\,d\gamma\\
&=\frac{\sqrt{-1}}{\pi}\int_{{{\mathbb C}}^{d_N}}(\log|F|,\partial\bar
\partial
{\varphi})\,d\gamma+\frac{\sqrt{-1}}{N}\int_{{{\mathbb C}}^n}(\log|\langle a,U
\rangle|,\partial\bar\partial{\varphi})\,d\gamma,\end{aligned}$$ for all test forms ${\varphi}\in {\mathcal{D}}^{m-1,m-1}(M)$. The first term is independent of $a$, so we may remove the Gaussian integral. The vanishing of the second term follows by noting that $$\begin{aligned}
\int_{{{\mathbb C}}^{n}}(\log|\langle a,U\rangle|,\partial\bar\partial{\varphi})\,d\gamma&=
\int_{{{\mathbb C}}^{n}}\,d\gamma\int_M\log|\langle a,U\rangle|\,\partial\bar\partial{\varphi}\
\
&=\int_M\int_{{{\mathbb C}}^{n}}\log|\langle a,U\rangle|\,d\gamma\,\partial\bar\partial
{\varphi}=0,\end{aligned}$$ since $\int\log|\langle a,U\rangle|\,d\gamma=\frac{1}{\pi}\int_{{{\mathbb C}}}\log|a_0|e^{-|a_0|
^2}\,da_0$ is constant, by the ${{\rm U}}(n)$-invariance of $d\gamma$. Fubini’s Theorem can be applied above since $$\int_{M\times{{\mathbb C}}^{n}}\big|\log|\langle a,U\rangle|\,\partial\bar\partial{\varphi}\big|\,d\gamma=\frac{1}{\pi}\int_{{{\mathbb C}}}\big|\log|a_0|\,\big|e^{-|a_0|^2}\,da_0\,
\int_M
|\partial\bar\partial{\varphi}|<+\infty.$$ Thus $$\begin{aligned}
{{\mathbf E}}Z_s &=&\frac{\sqrt{-1}}{2\pi}\partial\bar\partial\log|F|^2\ =\
\frac{\sqrt{-1}}{2\pi}\partial\bar\partial\left(\log\sum^{n}_{j=1}\|S_j\|^2_h
-\log\|e_L\|^2_h\right)\\&=& \frac {\sqrt{-1}}{2\pi} {\partial{\bar\partial}}\log\|\Pi_
{\mathcal{S}}(z,z)\|_h +\frac {\sqrt{-1}}{2\pi}\Theta_h\,.\end{aligned}$$
Conditioning on the values of a random variable {#conditioning}
===============================================
In this section, we give a precise definition of the conditional expected zero current ${{\mathbf E}}\big(Z_{s_1,\dots,s_k}\big|s_1(p)=v_1,\,\dots,\,s_k(p)=v_k\big)$ (Definition \[defcondk\]) and give a number of its properties. In particular, we give a formula for ${{\mathbf E}}\big(Z_s\big|s(p_1)=\cdots=s(p_r)=0\big)$ in terms of the conditional [Szegő ]{}kernel (Lemma \[linear\]).
The Leray form {#LERAY}
--------------
We first give a general formula for the conditional expectation ${{\mathbf E}}(X|Y=y)$ of a continuous random variable $X$ with respect to a smooth random variable $Y$ when $y$ is a regular value of $Y$. Our discussion differs from the standard expositions, which do not tend to assume random variables to be smooth.
We begin by recalling the definition of the conditional expectations ${{\mathbf E}}(X|{\mathcal{F}})$ of a random variable $X$ on a probability space $({\Omega},{\mathcal{A}},P)$ given a sub-$\sigma$-algebra ${\mathcal{F}}\subset{\mathcal{A}}$:
Let $X$ be a random variable $X$ with finite first moment (i.e., $X\in L^1$) on a probability space $({\Omega},{\mathcal{A}},P)$, and let ${\mathcal{F}}\subset{\mathcal{A}}$ be a $\sigma$-algebra. The conditional expectation is a random variable $E(X|{\mathcal{F}})\in L^1({\Omega},P)$ satisfying:
- ${{\mathbf E}}(X|{\mathcal{F}})$ is measurable with respect to ${\mathcal{F}}$;
- For all sets $A\in{\mathcal{F}}$, $\ \int_A {{\mathbf E}}(X|{\mathcal{F}})\,dP =\int_A X\,dP$.
The existence and uniqueness (in $L^1$) of $E(X|{\mathcal{F}})$ is a standard fact (e.g., [@K Th. 6.1]).
In this paper, we are interested in the conditional expectation ${{\mathbf E}}(X|\sigma(Y ))$ of a continuous random variable $X$ on a manifold ${\Omega}$ with respect to a smooth random variable $Y:{\Omega}\to{{\mathbb R}}^k$. Here, $\sigma(Y )$ denotes the $\sigma$-algebra generated by $Y$, i.e. the pull-backs by $Y$ of the Borel sets in ${{\mathbb R}}^k$; $\sigma(Y )$ is generated by the sublevel sets $\{Y_j\leq t_j , j
= 1, \dots , k\}$. The condition that ${{\mathbf E}}(X|\sigma(Y ))$ is measurable with respect to $\sigma(Y)$ implies that it is constant on the level sets of $Y$. We then write $${{\mathbf E}}(X|Y=y)\ :=\ {{\mathbf E}}(X|\sigma(Y ))(x)\,,\quad x\in Y{^{-1}}(y)\,.$$ We call $ {{\mathbf E}}(X|Y=y)$ the [*conditional expectation of $X$ given that $Y=y$.*]{} We note that the function $y\mapsto {{\mathbf E}}(X|Y=y)$ is in $L^1({{\mathbb R}}^k,Y_*P)$, and is not necessarily well-defined at each point $y$. However, in the cases of interest to us, $ {{\mathbf E}}(X|Y=y)$ will be a continuous function.
To give a geometrical description of ${{\mathbf E}}(X|Y )$, we use the language of Gelfand-Leray forms:
Let $Y : {\Omega}\to {{\mathbb R}}^k$ be a ${\mathcal{C}}^\infty$ submersion where ${\Omega}$ is an oriented $n$-dimensional manifold. Let $\nu\in {\mathcal{E}}^n$, e.g. a volume form. The Gelfand-Leray form ${\mathcal{L}}(\nu,Y,y ) \in {\mathcal{E}}^{n-k}(Y^{-1}(y))
$ on the level set $\{Y = y\}$ is given by $${\mathcal{L}}(\nu,Y,y)\wedge dY_1 \wedge\cdots\wedge dY_k = \nu \ \ \mbox{on }\ {Y^{-1}
(y)}\,,\ \ \ i.e.,\ {\mathcal{L}}(\nu,Y,y) = \frac{\nu}{dY_1\wedge\cdots\wedge dY_k
}\bigg|_{Y{^{-1}}(y)} .$$
Conditional expectation of a random variable is a form of averaging. The following Proposition shows this explicitly: it amounts to averaging $X$ over the level sets of $Y$.
\[Leray\] Let $\nu\in{\mathcal{E}}^n({\Omega})$ be a smooth probability measure on a manifold ${\Omega}$. Let $Y : {\Omega}\to {{\mathbb R}}^k$ be a ${\mathcal{C}}^\infty$ submersion, and let $X\in L^1({\Omega},\nu)$. Then $${{\mathbf E}}(X|Y=y) = \frac{\int_{Y =y} X \,{\mathcal{L}}(\nu,Y,y ) } {\int_{Y =y} {\mathcal{L}}(\nu,
Y,y)}.$$
We first note that $$\int_{y\in {{\mathbb R}}^k}\left(\int_{Y^{-1}(y)}|X|\,{\mathcal{L}}(\nu,Y,y )\right)dy_1\cdots
dy_k=\int_X|X|\,\nu=1\,,$$ and hence $\int_{Y^{-1}(y)}|X|\,{\mathcal{L}}(\nu,Y,y )<+\infty$ for almost all $y\in
{{\mathbb R}}^k$. Furthermore $\int_{Y^{-1}(y)}{\mathcal{L}}(\nu,Y,y )>0$ for $Y_*\nu$-almost all $y\in{{\mathbb R}}^k$, and therefore ${{\mathbf E}}(X|Y=y)$ is well defined for $Y_*\nu$-almost all $y$. Now let $${\widetilde}E(x) = \frac{\int_{Y =Y(x)} X \,{\mathcal{L}}(\nu,Y,Y(x) ) } {\int_{Y
=Y(x)} {\mathcal{L}}(\nu, Y,Y(x))}\,, \quad \mbox{for\ } \nu\mbox{-almost all\ }\ x\in
X.$$ The function ${\widetilde}E$ is measurable with respect to $\sigma(Y )$ since it is the pull-back by $Y$ of a measurable function on ${{\mathbb R}}^k
$.
The only other thing to check is that $\int_A {\widetilde}E\,\nu =
\int_A X\,\nu$ for all $A\in{\mathcal{F}}$. It suffices to check this for sets $A$ of the form $Y^{-1}(R)$ where $R$ is a rectangle in ${{\mathbb R}}^k$. But then by the change of variables formula and Fubini’s theorem, $$\begin{gathered}
\int_{Y^{-1}(R)}{\widetilde}E\,\nu = \int_{y\in R}\left(\int_{Y^{-1}
(y)}{\widetilde}E\,{\mathcal{L}}(\nu,Y,y )\right)dy_1\cdots dy_k\\= \int_{y\in R}
\left(\int_{Y^{-1}(y)}X\,{\mathcal{L}}(\nu,Y,y )\right)dy_1\cdots dy_k = \int_{Y^{-1}
(R)}
X\,d\nu.\end{gathered}$$ By uniqueness of the conditional expectation, we then conclude that ${\widetilde}E = {{\mathbf E}}(X|\sigma(Y ))$.
Let ${\Omega}={{\mathbb C}}^n$ with Gaussian probability measure $d{\gamma}_n=\pi^{-
n}e^{-|a|^2}\,da$. Let $\pi_k:{{\mathbb C}}^n\to{{\mathbb C}}^k$ be the projection $
\pi_k(a_1,\dots,a_n)=(a_1,\dots,a_k)$. For $y\in{{\mathbb C}}^k$ we have $${\mathcal{L}}(d{\gamma}_n,\pi_k,y)=\frac 1{\pi^k}\,e^{-(|y_1|^2+\cdots+|y_k|^2)}\,d{\gamma}_{n-
k}(a_{k+1},\dots,a_n)\,,$$ where $$d{\gamma}_{n-k}(a_{k+1},\dots,a_n)=e^{-(|a_{k+1}|^2+\cdots + |a_n|^2)}\,
\left(\frac i{2\pi}\right)^{n-k} da_{k+1}\wedge d \bar a_{k+1}\wedge\cdots
\wedge da_n\wedge d \bar a_n$$ is the standard complex Gaussian measure on ${{\mathbb C}}^{n-k}$. For a bounded random variable $X$ on ${{\mathbb C}}^n$, let $X_y$ be the random variable on ${{\mathbb C}}^{n-k}$ given by $X_y(a')=X(y,a')$ for $a'\in{{\mathbb C}}^{n-
k}$. By Proposition \[Leray\], we then have $$\label{condgauss} {{\mathbf E}}_{{\gamma}_n}(X|\pi_k=y)\ =\ {{\mathbf E}}_{{\gamma}_{n-k}}(X_y)
\,.$$
This example leads us to the following definition:
\[defcg\] Let ${\gamma}$ be a complex Gaussian measure on a finite dimensional complex space $V$, and let $W$ be a subspace of $V$. We define the [*conditional Gaussian measure*]{} ${\gamma}_W$ on $W$ to be the Gaussian measure associated with the Hermitian inner product on $W$ induced by the inner product on $V$ associated with ${\gamma}$.
The terminology of Definition \[defcg\] is justified by the following proposition, which we shall use to define the expected zero current conditioned on the value of a random holomorphic section at a point or points:
\[special\]Let $T:{{\mathbb C}}^n\to V$ be a linear map onto a complex vector space $V$. Let $E$ be a closed subset of ${{\mathbb C}}^n$ such that $E\cap T
{^{-1}}(y)$ has Lebesgue measure $0$ in $T{^{-1}}(y)$ for all $y\in V$. Let $X$ be a bounded random variable on ${{\mathbb C}}^n$ such that $X|({{\mathbb C}}^n{\smallsetminus}E)$ is continuous. Then ${{\mathbf E}}_{{\gamma}_n}(X|T=y)$ is continuous on ${{\mathbb C}}^k$. Furthermore $${{\mathbf E}}_{{\gamma}_n}(X|T=0)\ =\ {{\mathbf E}}_{{\gamma}_{\ker T}}(X')\,,$$ where $X'$ is the restriction of $X$ to $\ker T$ and ${\gamma}_{\ker T}$ is the conditional Gaussian measure on $\ker T$ as defined above.
Let $k=\dim V$. We can assume without loss of generality that $
\ker T= \{0\}\times {{\mathbb C}}^{n-k}$. Then the map $T$ has the same fibers as the projection $\pi_k(a_1,\dots,a_n)=(a_1,\dots,a_k)$, and thus $\sigma(T)=
\sigma(\pi_k)$. Hence we can assume without loss of generality that $V={{\mathbb C}}^k$ and $T=\pi_k$.
Fix $y_0\in{{\mathbb C}}^k$ and let ${\varepsilon}>0$ be arbitrary. Choose a compact set $K\subset
{{\mathbb C}}^{n-k}$ such that $(\{y_0\}\times K)\cap E=\emptyset$ and ${\gamma}_{n-k}({{\mathbb C}}^{n-k}{\smallsetminus}K) <{\varepsilon}/\sup |X|
$. Since $E$ is closed, $(\{y\}\times K)\cap E=\emptyset$, for $y$ sufficiently close to $y_0$. As above, we let $X_y(a')=X(y,a')$ for $a'\in{{\mathbb C}}^{n-k}$. Since $X_y\to X_{y_0}$ uniformly on $K$, we have $$\label{l1}\lim_{y\to y_0}\int_K X_y\,d{\gamma}_{n-k} =
\int_KX_{y_0}\,d{\gamma}_{n-k}\;.$$
It follows from that $$\label{l2} \left|{{\mathbf E}}_{{\gamma}_n}(X|\pi_k=y) - \int_KX_y\,d{\gamma}_{n-k}
\right|
= \left|\int_{{{\mathbb C}}^{n-k}{\smallsetminus}K}X_y\,d{\gamma}_{n-k}\right|<{\varepsilon}\,,$$ for all $y\in{{\mathbb C}}^k$. The first conclusion is an immediate consequence of – and the formula for ${{\mathbf E}}_{{\gamma}_n}(X|T=0)$ follows from with $y=0$.
Conditioning on the values of sections
--------------------------------------
We now state precisely what is meant by the expected zeros conditioned on sections having specific values at one or several points on the manifold:
\[defconditioning\] Let $(L,h)$ be a positive Hermitian holomorphic line bundle over a compact [Kähler ]{}manifold $M$ with [Kähler ]{}form $
{\omega}_h$. Let $p_1,\dots,p_r$ be distinct points of $M$. Let $N\gg 0$ and give $H^0(M,L^N)$ the induced Hermitian Gaussian measure ${\gamma}_N$. Let $v_j\in
L^N_{p_j}$, for $1\le j\le r$. We let $$T: H^0(M,L^N)\to L^N_{p_1}\oplus\cdots
\oplus L^N_{p_r}\,,\qquad s\mapsto s(p_1)\oplus\cdots\oplus s(p_r)\,.$$ The expected zero current ${{\mathbf E}}\big(Z_s\big|s(p_1)=v_1,\,\dots,\,s(p_r)=v_r\big)
$ conditioned on the section taking the fixed values $v_j$ at the points $p_j$ is defined by: $$\bigg({{\mathbf E}}_N\big(Z_s\big|s(p_1)=v_1,\,\dots,\,s(p_r)=v_r\big)\,,\,{\varphi}\bigg)\ =\
{{\mathbf E}}_{{\gamma}_N}\big((Z_s,{\varphi})\big|T=
v_1\oplus\cdots\oplus v_r\big)\,,$$ for smooth test forms ${\varphi}\in
{\mathcal{D}}^{m-1,m-1}(M)$.
\[continuity1\] The mapping $$v_1\oplus\cdots\oplus v_r \mapsto {{\mathbf E}}_N\big(Z_s\big|s(p_1)=v_1,\,
\dots,\,s(p_r)=v_r\big)$$ is a continuous map from $L^N_{p_1}\oplus\cdots
\oplus L^N_{p_r}$ to $ {\mathcal{D}}'^{1,1}(M)$.
Let $N$ be sufficiently large so that $T$ is surjective. Let $
{\varphi}\in {\mathcal{D}}^{m-1,m-1}(M)$ be a smooth test form, and consider the random variable $X(s)=(Z_s,{\varphi})$ on $H^0(M,L^N){\smallsetminus}\{0\}$. By [@St Th. 3.8] applied to the projection $$\{(s,z)\in H^0(M,L^N)\times M: s(z)=0\}\to
H^0(M,L^N)\,,$$ the random variable $X$ is continuous on $H^0(M,L^N){\smallsetminus}\{0\}$. Furthermore, $X$ is bounded, since we have by , $$|X(s)| \le (\sup
\|{\varphi}\|) \,(Z_s,{\omega}^{m-1}) = \frac N\pi (\sup \|{\varphi}\|) \,\int_M{\omega}_h^m\,,$$ The conclusion follows from Proposition \[special\] with $E=\{0\}$.
We could just as well condition on the section having specific derivatives, or specific $k$-jets, at specific points. At the end of this section, we discuss the conditional zero currents of simultaneous sections.
We are particularly interested in the case where the $v_j$ all vanish. In this case, the conditional expected current ${{\mathbf E}}_N\big(Z_s\big|s(p_1)=
\cdots=s(p_r)=0\big)$ is well-defined and we have:
\[linear\] Let $(L,h)\to (M,{\omega}_h)$ and $(H^0(M,L^N),
\gamma_{h})$ be as in Theorem \[scaled\]. Let $p_1,\dots,p_r$ be distinct points of $M$ and let $H_N^{p_1\cdots p_r}\subset H^0(M,L^N)$ denote the space of holomorphic sections of $L^N$ vanishing at the points $p_1,\dots,p_r$. Then $${{\mathbf E}}_N\big(Z_s\big|s(p_1)=\cdots=s(p_r)=0\big)\ =\ {{\mathbf E}}_{{\gamma}_N^{p_1\cdots p_r}}(Z_s)\ =\ \frac i{2\pi} {\partial{\bar\partial}}\log\|
\Pi_N^{p_1\cdots p_r}(z,z)\|_{h^N} +\frac N{\pi}\,{\omega}_h\,,$$ where ${\gamma}_N^{p_1\cdots p_r}$ is the conditional Gaussian measure on $H_N^{p_1\cdots p_r}$, and $\Pi_N^{p_1\cdots p_r}$ is the [Szegő ]{}kernel for the orthogonal projection onto $H_N^{p_1\cdots p_r}$.
Let ${\varphi}\in {\mathcal{D}}^{m-1,m-1}(M)$ be a smooth test form. By Proposition \[special\], $$\bigg({{\mathbf E}}_N\big(Z_s\big|s(p_1)=\cdots=s(p_r)=0\big)\,,\,{\varphi}\bigg)\ =\ {{\mathbf E}}_N
\big((Z_s,{\varphi})\big|T=
0\big)\ =\ {{\mathbf E}}_{{\gamma}_N^{p_1\cdots p_r}}(Z_s,{\varphi})\,,$$ where $T$ is as in Definition \[defconditioning\]. By Lemma \[important lemma\] with ${\mathcal{S}}= H_N^{p_1\cdots p_r}$, we then have $${{\mathbf E}}_{{\gamma}_N^{p_1\cdots p_r}}(Z_s,{\varphi}) =\ \left(\frac i{2\pi}
{\partial{\bar\partial}}\log\|\Pi_N^{p_1\cdots p_r}(z,z)\|_{h^N} +\frac N{\pi}\,{\omega}_h\,,\,{\varphi}\right).$$
Recalling the definition of $P_N$ from (\[PN\]), we now prove:
\[cond\] We have $$\begin{aligned}
{{\mathbf E}}_N\big(Z_s\big|s(p)=0\big) &={{\mathbf E}}_N\big(Z_s\big)+\frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 -
P_N(z,p)^2\right)\,,\end{aligned}$$
As above, we let $H^p_N\subset
H^0(M, L^N)$ denote the space of holomorphic sections vanishing at $p$. Let $\{S^p_{Nj}:j=1,...,d_N-1\}$ be an orthonormal basis of $H^p_N$. The Szegö projection $\Pi_N^p$ is given by $$\Pi^p_N(z,w)=\sum S^p_{Nj}(z)\otimes \overline{S^p_{Nj}(w)}\,.$$ By Lemma \[linear\] with $r=1$, we have $$\label{conditionalPL} {{\mathbf E}}_N\big(Z_s\big|s(p)=0\big) = \frac
i{2\pi} {\partial{\bar\partial}}\log\|\Pi_N^p(z,z)\|_{h^N} +\frac N\pi {\omega}_h\,.$$
To give a formula for $\Pi_N^p(z,z)$, we consider the [*coherent state*]{} at $p$, $\Phi^p_N(z)$ defined as follows: Let $$\label{coherent}{\widehat}\Phi^p_N(z):=\frac{\Pi_N(z,p)}{\|\Pi_N(p,p)
\|_{h^N}^{1/2}} \in H^0(M,L^N)\otimes\overline L^N_p\,,$$ We choose a unit vector $e_p\in L_p$, and we let $\Phi^p_N\in
H^0(M,L^N)$ be given by $$\label{coherent1}{\widehat}\Phi^p_N(z) = \Phi^p_N(z) \otimes
\overline{e^{\otimes N}_p}\,.$$
The coherent state $\Phi^p_N$ is orthogonal to $H^p_N$, because $$\label{orthog}s\in H^p_N \ \implies\ \|\Pi_N(p,p)\|_{h^N}^{1/2}
{\left\langle}s,{\widehat}\Phi_N^p{\right\rangle}=\int_M\Pi_N(p,z)\,s(z)\,{\Omega}_M(z)=s(p)=0$$ Furthermore, $\|\Phi_N^p\|_{h^N}^2=1$, and hence $\{S^p_{Nj}:j=1,...,d_N-1\}\cup\{\Phi^p_N\}$ forms an orthonormal basis for $H^0(M,L^N)$. Therefore $$\label{cond szego}\Pi^p_N(z,w)=\Pi_N(z,w)-\Phi^p_{N}(z)\otimes
\overline{\Phi^p_{N}(w)}\,,$$ and in particular $$\label{condszegodiag}\|\Pi^p_N(z,z)\|_{h^N}= \|\Pi_N(z,z)\|
_{h^N}- \|\Phi^p_{N}(z)\|^2_{h^N}\,.$$ Thus, by , $$\begin{aligned}
\log \|\Pi_N^p(z,z)\|_{h^N} & = & \log \left( \|\Pi_N(z,z)
\|_{h^N} - \frac{\|\Pi_N(z,p)\|_{h^N}^2}{\|\Pi_N(p,p)\|_{h^N}}\right)\nonumber
\\
& = & \log \|\Pi_N(z,z)\|_{h^N} + \log \left( 1 -
P_N(z,p)^2\right) . \end{aligned}$$
By and , $$\begin{aligned}
\label{conda}
{{\mathbf E}}_N\big(Z_s\big|s(p)=0\big) &=\frac i{2\pi}{\partial{\bar\partial}}\log
\|\Pi_N(z,z)\|_{h^N} +\frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 -
P_N(z,p)^2\right)+\frac N\pi {\omega}_h\notag\\
&={{\mathbf E}}_N\big(Z_s\big)+\frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 -
P_N(z,p)^2\right)\,,\end{aligned}$$ concluding the proof of the Proposition.
Theorem \[scaled\] involves the conditional zero current of a system of random sections, which we now define precisely:
\[defcondk\] Let $(L,h)$ be a positive Hermitian holomorphic line bundle over a compact [Kähler ]{}manifold $M$ with [Kähler ]{}form ${\omega}_h$, let $1\le k\le
m=\dim M$, and let $p\in M$. Let $N\gg 0$ and give $H^0(M,L^N)$ the induced Hermitian Gaussian measure ${\gamma}_N$. We let $$T: \bigoplus^kH^0(M,L^N)\to \oplus^kL^N_p\,,$$ where $\bigoplus^kV$ denotes $k$-tuples in $V$. The conditional expected zero current ${{\mathbf E}}_N\big(Z_{s_1,\dots,s_k}\big|
s_1(p)=v_1,\,\dots,\,s_k(p)=v_k\big)$ is defined by: $$\bigg({{\mathbf E}}_N\big(Z_{s_1,\dots,s_k}\big|s_1(p)=v_1,\,\dots,\,s_k(p)=v_k\big)\,,\,
{\varphi}\bigg)\ =\ {{\mathbf E}}_{{\gamma}_N^k}\big((Z_{s_1,\dots,s_k},{\varphi})\big|T=(v_1,\dots,v_k)
\big)\,,$$ for smooth test forms ${\varphi}\in {\mathcal{D}}^{m-k,m-k}(M)$. The conditional expected zero distrbution is the current $$K_k^N(z|p):= {{\mathbf E}}_N\big(Z_{s_1,\dots,s_k}\big|
s_1(p)=0,\,\dots,\,s_k(p)=0\big)\,,$$ which is well defined according to the following lemma.
\[continuity\] For $N\gg 0$, the mapping $$(v_1,\dots,v_k)
\mapsto{{\mathbf E}}_N\big(Z_{s_1,\dots,s_k}\big|s_1(p)=v_1,\,\dots,\,s_k(p)=v_k\big)$$ is a continuous map from $\bigoplus^kL^N_p$ to ${\mathcal{D}}'^{m-k,m-k}(M)$.
Let $$E=\{(s_1,\dots,s_k)\in \bigoplus^kH^0(M,L^N):\dim Z_{s_1,\dots,s_k} = n-k\}
\,.$$ Since $L$ is ample, for $N$ sufficiently large, $E\cap T{^{-1}}(v_1,\dots,v_k)$ is a proper algebraic subvariety of $T{^{-1}}(v_1,\dots,v_k)$ and hence has Lebesgue measure 0 in $T{^{-1}}(v_1,\dots,v_k)$, for all $(v_1,\dots,v_k)\in \oplus^kL^N_p
$. Then Proposition \[special\] applies with ${{\mathbb C}}^n$ replaced by $
\oplus^kH^0(M,L^N)$, and continuity follows exactly as in the proof of Lemma \[continuity1\].
Proof of Theorem \[unscaled\] {#proof1}
=============================
Proof for $k=1$
---------------
We first prove Theorem \[unscaled\] when the condition is that $s(p) = 0$ for a single point $p$.
Let ${\varphi}\in {\mathcal{D}}'^{m-1,m-1}(M)$ be a smooth test form. By Proposition \[cond\], we have $$\label{cond1} \Big( {{\mathbf E}}_N(Z_s:s(p)=0),{\varphi}\Big)\ =\
({{\mathbf E}}_N Z_s,{\varphi}) +\int_M \log \left( 1 - P_N(z,p)^2\right) \frac
i{2\pi} {\partial{\bar\partial}}{\varphi}\,.$$
Away from the diagonal, we can write $\log \left( 1 -
P_N(z,p)^2\right)=P_N(z,p)^2+\frac12P_N(z,p)^4+\cdots$, and we have by , $$\label{ddbar1} \log \left( 1 - P_N(z,p)^2\right) = O(N^{-
m-2})\qquad \mbox{uniformly for }\ d(z,p)\ge
b\,\sqrt{\frac {\log N}{N}},$$ where $b=\sqrt{2m+6}$. Furthermore by , we have
$$\begin{aligned}
&\int_M \log \left( 1 - P_N(z,p)^2\right) \frac i{2\pi}
{\partial{\bar\partial}}{\varphi}=\int_{d(z,p)\leq b\sqrt{\frac{\log N}{N}}} \log \left( 1 -
P_N(z,p)^2\right) \frac i{2\pi} {\partial{\bar\partial}}{\varphi}+O(N^{-m-2})\,.\end{aligned}$$
Using local normal coordinates $(w_1,\dots,w_m)$ centered at $p$, we write $$\frac i{2\pi}{\partial{\bar\partial}}{\varphi}= \psi(w)\,{\Omega}_0(w)\,,\qquad {\Omega}_0(w)=
\left( \frac i2\right)^m dw_1\wedge d\bar w_1\wedge \cdots \wedge dw_m\wedge
d\bar w_m\,.$$ Recalling , we then have $$\begin{aligned}
\int_M \log &\left( 1 - P_N(z,p)^2\right) \frac i{2\pi}
{\partial{\bar\partial}}{\varphi}\notag\\& =\int_{|w|\leq b\sqrt{\frac{\log N}{N}}}\log
\left[1-P_N(p+w,p)^2\right]\psi(w)\,{\Omega}_0(w)+O(N^{-m-2})\notag\\&
=N^{-m}\int_{|u|\leq b\sqrt{\log N}}\log\left[1-P_N\left(p+\frac u{{\sqrt{N}}},p
\right)^2\right]\psi\left(\frac u{{\sqrt{N}}}\right){\Omega}(u)+O(N^{-m-2})\,.
\label{intM}\end{aligned}$$
Let $$\label{Lambda} \Lambda_N(z,p)= -\log
P_N(z,p)\;.$$ so that $$\label{USEFUL} \log \left( 1 -
P_N(z,p)^2\right)=Y\circ \Lambda_N(z,p)\;,$$ where $$\label{Y}Y({\lambda}):= \log (1-e^{-2{\lambda}})\quad \mbox{for }\
{\lambda}>0 .$$ By –, $$\label{RN0}\Lambda_N\left(p+\frac u{{\sqrt{N}}}\,,p\right)
={{{\textstyle \frac 12}}|u|^2} + {\widetilde}R_N(u) \;,$$ where $$\label{RN}{\widetilde}R_N(u)=-\log[1+R_N(u,0)]=O(|u|^2N^{-1/2+{\varepsilon}})\quad \mbox{for }\
|u|<b\sqrt{\log N}\;.$$ We note that $$\label{Yest}
0<-Y({\lambda}) = - \log(1-e^{-2{\lambda}}) \le \left(1+\log^+
\frac1{{\lambda}}\right)\;,$$ $$\label{Y'}Y'({\lambda})= \frac
2{e^{2{\lambda}}-1}\le \frac 1{{\lambda}},\quad \mbox{for }\ {\lambda}>1\;.$$ Hence by –, $$\label{L} \log\left[1-P_N
\left(p+\frac u{{\sqrt{N}}},p\right)^2\right] = \log \left(1- e^{-|u|^2}\right) +
O(N^{-1/2+{\varepsilon}})\qquad \mbox{for }\ |u|<b\sqrt{\log N}\,.$$ Since $
\psi\left(\frac u{{\sqrt{N}}}\right) =\psi(0)+O\left(\frac u{{\sqrt{N}}}\right)$, we then have $$\begin{gathered}
\log\left[1-P_N\left(p+\frac u{{\sqrt{N}}},p\right)^2\right]\psi
\left(\frac u{{\sqrt{N}}}\right) = \psi(0)\log \left(1- e^{-|u|^2}\right) +
O(N^{-1/2+{\varepsilon}})\\ +\frac 1{\sqrt{N}}\,O\left(|u|\,|\log (1- e^{-|u|^2})|\right)
\qquad \mbox{for }\ |u|<b\sqrt{\log N}\,.\end{gathered}$$ Since $O\left((\log N)^mN^{-1/2+{\varepsilon}}\right)=O(N^{-1/2+2{\varepsilon}})$ and $|u|\log
(1- e^{-|u|^2}) \in L^1({{\mathbb C}}^m)$, we conclude that $$\begin{gathered}
\int_{|u|\leq b\sqrt{\log N}}\log\left[1-P_N\left(p+\frac
u{{\sqrt{N}}},p\right)^2\right]\psi\left(\frac u{{\sqrt{N}}}\right){\Omega}_0(u) \\= \psi(0)
\int_{|u|\leq b\sqrt{\log N}}\log\left[1-e^{-|u|^2}\right]{\Omega}_0(u) + O(N^{-1/2+
{\varepsilon}}).\end{gathered}$$ We note that $$\int_{|u|\geq b\sqrt{\log N}}\log\left[1-e^{-|u|^2}\right]
{\Omega}_0(u) = \frac{2\pi^m}{(m-1)!}\int_{b\sqrt{\log N}}^{+\infty} \log(1-
e^{r^2})r^{2m-1}\,dr = O\left(N^{-b^2/2}\right)\,.$$ Since $b>1$, we then have $$\begin{gathered}
\label{intest}\int_{|u|\leq b\sqrt{\log N}}\log\left[1-P_N
\left(p+\frac u{{\sqrt{N}}},p\right)^2\right]\psi\left(\frac u{{\sqrt{N}}}\right)
{\Omega}_0(u) \\= \psi(0) \int_{{{\mathbb C}}^m}\log\left[1-e^{-|u|^2}\right]{\Omega}_0(u) +
O(N^{-1/2+{\varepsilon}}).\end{gathered}$$
Combining , and , we have $$\label{cond2} \Big( {{\mathbf E}}(Z_s:s(p)=0),{\varphi}\Big)\ =\ ({{\mathbf E}}Z_s,{\varphi})
+ N^{-m} \psi(0) \int_{{{\mathbb C}}^m}\log\left[1-e^{-|u|^2}\right]{\Omega}_0(u) + O(N^{-
m-1/2+{\varepsilon}})\,.$$ We note that$$\label{psi0}
\psi(0)=\frac 1{2\pi}\,\frac {i{\partial{\bar\partial}}{\varphi}(p)}{{\Omega}_M(p)}$$ and $$\begin{aligned}
\int_{{{\mathbb C}}^m}\log\left[1-e^{-|u|^2}\right]{\Omega}_0(u)&=&
\frac{2\pi^m}{(m-1)!}\int_0^{+\infty} \log(1-e^{-r^2})r^{2m-1}\,dr \notag\\&=&
\frac{\pi^m}{(m-1)!}\int_0^{+\infty} \log(1-e^{-t})t^{m-1}\,dt\notag \\&=&-
\frac{\pi^m}{(m-1)!}\sum_{n=1}^{+\infty}\int_0^{+\infty} \frac{e^{-nt}}
{n}t^{m-1}\,dt\notag\\
&=& -\frac{\pi^m}{(m-1)!}\sum _{n=1}^{+\infty}\frac {(m-1)!}{n^{m+1}} \ =\ -
\pi^m\, \zeta(m+1)\,.
\label{zeta}\end{aligned}$$ The the one-point case ($k=1$) of Theorem \[unscaled\] follows by substituting – into .
The multi-point case
--------------------
We now condition on vanishing at $k$ points $p_1,\dots,p_k$.
We let $H^V_N\subset H^0(M, L^N)$ denote the space of holomorphic sections vanishing at the points $p_1,\dots,p_k$. Let $\Phi_N^{p_j}$ be the coherent state at $p_j$ (given by –) for $j=1,\dots,k$. By , a section $s\in H^0(M,L^N)$ vanishes at $p_j
$ if and only if $s$ is orthogonal to $\Phi_N^{p_j}$. Thus $H^0(M,L^N)= H^V_N
\oplus Span\{\Phi_N^{p_j}\}$. Let $$T:H^0(M,L^N)\to L_{p_1}^N\oplus\cdots
\oplus L_{p_k}^N\,,\quad s\mapsto s(p_1)\oplus\cdots s(p_k)\,,$$ so that $\ker
T= H^V_N$. By Lemma \[linear\], the conditional expectation is given by $$\label{conditional2} {{\mathbf E}}_N\big(Z_s\big|s(p_1)=
\cdots=s(p_k)=0\big) = \frac i{2\pi} {\partial{\bar\partial}}\log\|\Pi_N^V(z,z)\|_{h^N} +\frac N
\pi {\omega}_h\,,$$ where $\Pi_N^V$ is the conditional [Szegő ]{}kernel for the projection onto $\Pi_N^V$. We let $\Pi_N^\perp(z,w)$ denote the kernel for the orthogonal projection onto $(H^V_N)^\perp=Span\{\Phi_N^{p_j}\}$, so that $$\label{Pi-orthog} \Pi_N^V(z,w)=\Pi_N(z,w)-\Pi_N^\perp(z,w)\;.$$
Recalling –, we have $$\begin{aligned}
{\left\langle}\Phi_N^{p_i}, \Phi_N^{p_j}{\right\rangle}\,
\overline{e_{p_i}^{\otimes N}} \otimes e_{p_j}^{\otimes N} &=&
\frac{ {\left\langle}\sum_{\alpha}S^N_{\alpha}(z)\otimes \overline
{S^N_{\alpha}(p_i)}\,,\,\sum_\beta S^N_{\beta}(z)\otimes \overline
{S^N_{\beta}(p_j)}{\right\rangle}}
{\|\Pi_N(p_i,p_i)\|_{h_N}^{1/2}\|\Pi_N(p_j,p_j)\|_{h_N}^{1/2}}\\[6pt]& = &
\frac{\overline{\Pi_N(p_i,p_j)}}
{\|\Pi_N(p_i,p_i)\|_{h_N}^{1/2}\|\Pi_N(p_j,p_j)\|_{h_N}^{1/2}}
\ ,\end{aligned}$$ and therefore by , $$\label{inner} \left|{\left\langle}\Phi_N^{p_i}, \Phi_N^{p_j}{\right\rangle}\right|
= P_N(p_i,p_j)= {\delta}_i^j+O(N^{-\infty})\,.$$ In particular the $
\Phi_N^j$ are linearly independent, for $N\gg 0$. Let $${\left\langle}\Phi_N^{p_i},
\Phi_N^{p_j}{\right\rangle}={\delta}_i^j +W_{ij}\,.$$ By , $W_{ij}=O(N^{-\infty})
$. Let us now replace the basis $\{\Phi_N^{p_j}\}$ of $(H_N^V)^\perp$ by an orthonormal basis $\{\Psi_N^{j}\}$, and write $$\Psi_N^{i}=\sum_{j=1}^k A_{ij}\,\Phi_N^{p_j}\,.$$ Then $${\delta}_i^j ={\left\langle}\Psi^i_N,\Psi^j_N {\right\rangle}= \sum_{{\alpha},{\beta}} {\left\langle}A_{i{\alpha}}
\Phi^{p_{\alpha}},
A_{j{\beta}} \Phi^{p_{\beta}}{\right\rangle}= \sum_{{\alpha},{\beta}} A_{i{\alpha}}\overline A_{j{\beta}}({\delta}_{\alpha}^
{\beta}+W_{{\alpha}{\beta}})\,,$$ or $I=A(I+W)A^*$.
We have $$\Pi_N^\perp(z,z)= \sum \Psi_N^j(z)\otimes \overline{ \Psi_N^j(z)}=
\sum_{j,{\alpha},{\beta}}A_{j{\alpha}} \overline A_{j{\beta}}\Phi_N^{p_{\alpha}}\otimes \overline
{\Phi_N^{p_{\beta}}}=
\sum_{j{\beta}}B_{{\alpha}{\beta}}\Phi_N^{p_{\alpha}}\otimes \overline {\Phi_N^{p_{\beta}}}\,,$$ where $$\label{B}B={}^t\!A\,\overline A= {}^t(A^*\,A)= {}^t(I+W)
{^{-1}}= I+O(N^{-\infty})\,.$$ The final equality in follows by noting that $$\|W\|_{HS}=\eta<1\implies \|(I+W){^{-1}}-I\|_{HS}=\|W-W^2+W^3+\cdots\|_{HS} \le
\eta+\eta^2+\eta^3+\cdots = \frac \eta{1-\eta},$$ where $\|W\|
_{HS}=[\mbox{Trace}(WW^*)]^{1/2}$ denotes the Hilbert-Schmidt norm. Therefore $$\|\Pi_N^\perp(z,z)\| = \sum_{j=1}^k\|\Phi_N^{p_j}(z)\|^2+O(N^{-
\infty})\,.$$ Repeating the argument of the 1-point case, we then obtain $$\begin{aligned}
\label{condk}
{{\mathbf E}}_N\big(Z_s\big|s(p_1)=\cdots=s(p_k)=0\big) =\ ({{\mathbf E}}_N Z_s,{\varphi}) + \log \left( 1 -
\sum P_N(z,p_j)^2\right) +O(N^{-\infty}).\end{aligned}$$ It suffices to verify the theorem in a neighborhood of an arbitrary point $z_0\in M$. If $z_0\not\in \{p_1,\dots,p_k\}$, then $\log \left( 1 - \sum
P_N(z,p_j)^2\right)=O(N^{-\infty})$ in a neighborhood of $z_0$, and the formula trivially holds. Now suppose $z_0=p_1$, for example. Then $$\log \left( 1 - \sum P_N(z,p_j)^2\right)=\log \left( 1 - P_N(z,p_1)^2\right)
+O(N^{-\infty})$$ near $p_1$ and the conclusion holds there by the computation in the 1-point case.
Proof of Theorem \[scaled\]: The scaled conditional expectation {#proof2}
===============================================================
In this section we shall prove Theorem \[scaled\] together with the following analogous result on the scaling asymptotics of conditional expected zero currents of dimension $\ge 1$:
\[all codim\] Let $1\le k\le m-1$. Let $(L,h)\to (M,{\omega}_h)$ and $(H^0(M,L^N),\gamma_{h}^N)$ be as in Theorem \[unscaled\]. Let $p\in M$, and choose normal coordinates $z=(z_1,\dots,z_m):M_0,p\to {{\mathbb C}}^m,0$ on a neighborhood $M_0$ of $p$. Let $\tau_N={\sqrt{N}}\,z:M_0\to{{\mathbb C}}^m$ be the scaled coordinate map. Then for a smooth test form ${\varphi}\in{\mathcal{D}}^{m-k,m-k}({{\mathbb C}}^m)$, we have $$\Big( K^N_k(z|p),\tau_N^*{\varphi}\Big)= \int_{{{\mathbb C}}^m{\smallsetminus}\{0\}} {\varphi}\wedge
\left(
\frac i{2\pi}{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2})
+|u|^2\right]\right)^k \ +\ O(N^{-1/2+{\varepsilon}})\,,$$ and thus $$\tau_{N*}\Big( K^N_k(z|p)\Big)
\to K_{km}^\infty(u|0):=\left(
\frac i{2\pi}{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2}) +|u|^2\right]\right)^k\,,$$ where $u=(u_1,\dots,u_m)$ denotes the coordinates in ${{\mathbb C}}^m$.
Just as in Theorem \[scaled\], $K_{km}^\infty(u|0)$ is the conditional expected zero current of $k$ independent random functions in the Bargmann-Fock ensemble on ${{\mathbb C}}^m$.
To prove Theorems \[scaled\] and \[all codim\], we first note that by and Proposition \[cond\], we have $$\begin{aligned}
\label{cond3}
K^N_1(z|p)
&=\frac i{2\pi}{\partial{\bar\partial}}\log \|\Pi_N(z,z)\|_{h^N} +\frac i{2\pi} {\partial{\bar\partial}}\log
\left( 1 -
P_N(z,p)^2\right)+\frac N\pi {\omega}_h\notag \\
&=\frac N\pi {\omega}_h+\frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 -
P_N(z,p)^2\right)+O(N {^{-1}})\,.\end{aligned}$$ In normal coordinates $(z_1,\dots,z_m)$ about $p $, we have $$\label{om}{\omega}_h=\frac i2 \sum g_{jl}dz_j\wedge d
\bar z_l\,,\quad g_{jl}(z)={\delta}_j^l +O(|z|).$$ Changing variables to $u_j={\sqrt{N}}z_j$ gives $$\label{omega}\frac N\pi {\omega}_h =\frac i{2\pi}\sum g_{jl}
\left(\frac u{\sqrt{N}}\right)\, du_j\wedge d\bar u_l = \frac i{2\pi}
{\partial{\bar\partial}}|u|^2 + \sum O(|u|N^{-1/2})\, du_j\wedge d\bar
u_l\,.$$
We can now easily verify the one dimensional case of Theorem \[scaled\]: Let $m=1$. By , and , we have $$\Big( K^N_1(z|p),\tau_N^* {\varphi}\Big)\ =\ \frac i{2\pi}\int_{{\mathbb C}}\left[\log(1-
e^{-|u|^2}) +|u|^2\right] {\partial{\bar\partial}}{\varphi}+O(N^{-1/2+{\varepsilon}})\,$$ for a smooth test function ${\varphi}\in{\mathcal{D}}({{\mathbb C}})$. By Green’s formula, $$\begin{aligned}
\int_{|u|>{\varepsilon}} \left[\log(1-e^{-|u|^2}) +|u|^2\right]{\partial{\bar\partial}}{\varphi}&=& \int_{|u|>{\varepsilon}}{\varphi}\,{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2}) +|u|^2\right]\\&& -
\frac{i{\varepsilon}^2}{1-e^{-{\varepsilon}^2}}\int_{|u|={\varepsilon}}{\varphi}\,d\theta+O({\varepsilon}\log{\varepsilon})\\ &\to &
\int_{{\mathbb C}}{\varphi}\,{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2}) +|u|^2\right]-2\pi i\,{\varphi}(0)\,,\end{aligned}$$ which yields Theorem \[scaled\] for $k=m=1$.
For the dimension $m>1$ cases, we first derive some pointwise formulas on $M
{\smallsetminus}\{p\}$: Let ${\Lambda}_N(z)={\Lambda}_N(z,p) = -\log P_N(z,p)$. Recalling , we have
$$\begin{aligned}
{\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right) &=& {\partial{\bar\partial}}(Y\circ
{\Lambda}_N)\ = \ Y''({\Lambda}_N)\,{\partial}{\Lambda}_N\wedge{\bar\partial}{\Lambda}_N +Y'({\Lambda}_N)\,{\partial{\bar\partial}}{\Lambda}_N\\
&=& -\frac{4\,e^{-2{\Lambda}_N}}{(1-e^{-2{\Lambda}_N})^2}\,{\partial}{\Lambda}_N\wedge{\bar\partial}{\Lambda}_N +\frac
2{e^{2{\Lambda}_N}-1}\,{\partial{\bar\partial}}{\Lambda}_N
\,.\end{aligned}$$
By and –, we have $${\Lambda}_N= {{\textstyle \frac 12}}|u|^2 + O(|u|^2N^{-1/2+{\varepsilon}})\,,\quad \frac {{\partial}{\Lambda}_N}{{\partial}\bar
u_j}= {{\textstyle \frac 12}}u_j + O(|u|N^{-1/2+{\varepsilon}})
\,,\quad \frac {{\partial}^2{\Lambda}_N}{{\partial}u_j{\partial}\bar u_l}= {{\textstyle \frac 12}}{\delta}_j^l+ O(N^{-1/2+{\varepsilon}})
\,.$$ Thus $${\bar\partial}{\Lambda}_N= {{\textstyle \frac 12}}\sum \big[u_j +O(|u|N^{-1/2+{\varepsilon}})\big]\,d\bar u_j\,,$$ and $${\partial{\bar\partial}}{\Lambda}_N = \left({{\textstyle \frac 12}}{\partial{\bar\partial}}|u|^2 + \sum c_{jl} du_j\wedge d\bar u_l
\right),\quad c_{jl}=O(N^{-1/2+{\varepsilon}}).$$ Since $Y^{(j)}({\lambda})=O({\lambda}^{-j})$ for $0<{\lambda}<1$, we then have $$\begin{aligned}
{\bar\partial}\log \left( 1 - P_N(z,p)^2\right)&=& \big[Y'({{\textstyle \frac 12}}|u|^2)
+ O(|u|^{-2}N^{-1/2+{\varepsilon}})\big]\big[{{\textstyle \frac 12}}{\bar\partial}|u|^2 +\sum O(|u|N^{-1/2+{\varepsilon}})d
\bar u_j\big]\notag\\
&=& \frac 1{e^{|u|^2}-1}\,{\bar\partial}|u|^2 +\sum O(|u|^{-1}N^{-1/2+{\varepsilon}})\, d\bar u_j
\,,\label{dbarlog}
\\ {\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right)&=&-\frac{e^{-|u|^2}}{(1-e^{-|u|
^2})^2}\,{\partial}|u|^2\wedge {\bar\partial}|u|^2+\frac 1{e^{|u|^2}-1}\,{\partial{\bar\partial}}|u|^2\notag\\&&
+\sum O(|u|^{-2}N^{-1/2+{\varepsilon}})\,du_j\wedge d\bar u_k\notag\\&=&{\partial{\bar\partial}}\log(1-
e^{-|u|^2})+\sum O(|u|^{-2}N^{-1/2+{\varepsilon}})\,du_j\wedge d\bar u_l\,,
\label{ddbarlog}\end{aligned}$$ for $0<|u|<b$. Therefore by , and , $$\begin{aligned}
K^N_1(z|p) &=&\frac i{2\pi}\, \frac 1{1-e^{-|
u|^2}}\left[
-\frac{e^{-|u|^2}}{1-e^{-|u|^2}}\,{\partial}|u|^2\wedge{\bar\partial}|u|^2 +{\partial{\bar\partial}}|u|^2\right]
\notag\\&& +\sum
O(|u|^{-2}N^{-1/2+{\varepsilon}})\,du_j\wedge d\bar u_k\notag\\ &=&\frac i{2\pi}{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2})
+|u|^2\right]+\sum O(|u|^{-2}N^{-1/2+{\varepsilon}})\,du_j\wedge d\bar u_l,\quad
\label{EZs}\end{aligned}$$ for $0<|u|<b$.
We shall use the following notation: If $R\in{\mathcal{D}}'^r(M)$ is a current of order 0 (i.e., its coefficients are given locally by measures), we write $R=R_{sing}+R_{ac }$, where $R_{sing}$ is supported on a set of (volume) measure 0, and the coefficients of $R_{ac }$ are in $L^1_{loc}$. We also let $\|R\|$ denote the total variation measure of $R$: $$(\|R\|,\psi):= \sup \{|(R,\eta)|: \eta\in {\mathcal{D}}^{2m-r}(M), |\eta|\le \psi\},
\quad \mbox{for }\ \psi\in{\mathcal{D}}(M)\,.$$
\[product\] The conditional expected zero distributions are given by $$\begin{aligned}
K^N_k(z|p)& =& \left[\frac i{2\pi} {\partial{\bar\partial}}\log\|\Pi_N(z,z)\|_{h^N}
+ \frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right)+\frac N\pi {\omega}_h
\right]^k_{ac}\\&& \mbox{for }\ 1\le k\le m-1\,,\\[8pt]K^N_m(z|p)& =& {\delta}_p+\left[\frac i{2\pi} {\partial{\bar\partial}}\log\|\Pi_N(z,z)\|_{h^N}
+ \frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right)+\frac N\pi {\omega}_h
\right]^m_{ac}\,.\end{aligned}$$ In particular, the currents $K^N_k(z|p)$ are smooth forms on $M{\smallsetminus}\{p\}$ for $1\le k\le m$, and only the top-degree current $K^N_m(z|p)$ has point mass at $p$.
Let $$T:H^0(M,L^N)^k\to (L^{\otimes N}_p)^k\,,\quad
(s_1,\dots,s_k)\mapsto(s_1(p),\dots,s_k(p))\,.$$ By Proposition \[special\] and Definition \[defcondk\], $$\left(K^N_k(z|p),{\varphi}\right)= {{\mathbf E}}_{({\gamma}^p_N)^k}(Z_{s_1,\dots,s_k},{\varphi})\,,$$ for ${\varphi}\in
{\mathcal{D}}^{m-k,m-k}(M{\smallsetminus}\{p\})$, where ${\gamma}^p_N$ is the conditional Gaussian on $H^p_N$.
Next, we shall apply Proposition 2.2 in [@SZa] to show that $$\label{E^k} K^N_k(z|p)=
{{\mathbf E}}_{({\gamma}^p_N)^k}
(Z_{s_1,\dots,s_k}) = \left[{{\mathbf E}}_{{\gamma}^p_N}Z_{s}\right]^{\wedge
k}=\big[K^N_1(z|p)\big]^{\wedge k} \quad
\mbox{on }\ M{\smallsetminus}\{p\}\,.$$ We cannot apply Proposition 2.2 in [@SZa] directly, since all sections of $H^p_N$ vanish at $p$ by definition, so $H^p_N$ is not base point free. Instead, we shall apply this result to the blowup ${\widetilde}M$ of $p$. Let $\pi:{\widetilde}M\to M$ be the blowup map, and let $E=\pi{^{-1}}(p)$ denote the exceptional divisor. Let ${\widetilde}L\to {\widetilde}M$ denote the pullback of $L$, and let ${\mathcal{O}}(-E)$ denote the line bundle over ${\widetilde}M$ whose local sections are holomorphic functions vanishing on $E$ (see [@GH pp. 136–137]). Thus we have isomorphisms $$\label{pistar}\tau_N:H_N^p\buildrel{\approx}\over\to H^0({\widetilde}M,{\widetilde}L^N\otimes {\mathcal{O}}(-E))\,,\qquad \tau_N(s)=s\circ \pi\,.$$ (Surjectivity follows from Hartogs’ extension theorem; see, e.g., [@GH p. 7].)
Let ${\mathcal{I}}_p\subset {\mathcal{O}}_M$ denote the maximal ideal sheaf of $\{p\}$. From the long exact cohomology sequence $$\cdots\to H^0(M,{\mathcal{O}}(L^N) )\to H^0(M,{\mathcal{O}}(L^N)\otimes ({\mathcal{O}}_M/{\mathcal{I}}^2_p))\to
H^1(M,{\mathcal{O}}(L^N)\otimes{\mathcal{I}}^2_p)\to\cdots$$ and the Kodaira vanishing theorem, it follows that $H^1(M,{\mathcal{O}}(L^N)\otimes{\mathcal{I}}^2_p)=0$ and thus there exist sections of $L^N$ with arbitrary 1-jet at $p$, for $N$ sufficiently large (see, e.g., [@SS Theorem (5.1)]). Therefore ${\widetilde}L^N\otimes{\mathcal{O}}(-E)$ is base point free.
We give $H^0({\widetilde}M,{\widetilde}L^N\otimes{\mathcal{O}}(-E))$ the Gaussian measure ${\widetilde}{\gamma}_N:=\tau_{N*}{\gamma}^p_N$. By [@SZa Prop. 2.1–2.2] applied to the line bundle ${\widetilde}L^N\otimes{\mathcal{O}}(-E)\to {\widetilde}M$ and the space ${\mathcal{S}}=H^0({\widetilde}M,{\widetilde}L^N\otimes{\mathcal{O}}(-E))$, we have ${{\mathbf E}}_{({\widetilde}{\gamma}_N)^k}\left(Z_{\tilde
s_1,\dots,\tilde s_k}\right)= \left({{\mathbf E}}_{{\widetilde}{\gamma}_N} Z_{\tilde
s_1}\right)^{\wedge k}$ (where the $\tilde s_j$ are independent random sections in ${\mathcal{S}}$). Equation then follows by identifying ${\widetilde}M{\smallsetminus}E$ with $M{\smallsetminus}\{p\}$ and $H^0({\widetilde}M,{\widetilde}L^N\otimes{\mathcal{O}}(-E))$ with $H_N^p$. By equations and , we then have $$\begin{aligned}
K^N_k(z|p)& =& \left[\frac i{2\pi} {\partial{\bar\partial}}\log\|\Pi_N(z,z)\|_{h^N}
+ \frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right)+\frac N\pi {\omega}_h
\right]^k\\&& \mbox {on }\ M{\smallsetminus}\{p\}\,, \quad \mbox{for }\ 1\le k\le m\,.\end{aligned}$$
Since $K^N_k(z|p)$ is a current of order 0, to complete the proof of the lemma it suffices to show that
1. $\|K^N_k(z|p)\|(\{p\}) = 0$ for $k<m$,
2. $K^N_m(z|p)(\{p\}) =1$.
We first verify (ii): Let $\{{\varphi}_n\}$ be a decreasing sequence of smooth functions on $M$ such that $0\le{\varphi}_n\le 1$ and ${\varphi}_n\to \chi_{\{p\}}$ as $n\to\infty$. We consider the random variables $X^m_n:(H_p^N)^m\to {{\mathbb R}}$ given by $$X^m_n({{\bf s}})=(Z_{{\bf s}},{\varphi}_n)\,,\quad {{\bf s}}=(s_1,\dots,s_m)\,.$$ Every $m$-tuple ${{\bf s}}\in (H^p_N)^m$ has a zero at $p$ by definition, and almost all ${{\bf s}}$ have only simple zeros; therefore $X^m_n({{\bf s}})\to Z_{{\bf s}}(\{p\}) = 1$ a.s. Furthermore $1\le X^m_n({{\bf s}})\le (Z_{{\bf s}},1) = N^mc_1(L)^m$. Therefore by dominated convergence, $$K^N_m(z|p)(\{p\}) =\lim_{n\to\infty} (K^N_m(z|p),{\varphi}_n) = \lim_{n\to\infty} \int X^m_n\,d({\gamma}^p_N)^m= \int \lim_{n\to\infty} X^m_n\,d({\gamma}^p_N)^m=1\,.$$ To verify (i), we note that $\|K^N_k(z|p)\|\,{\Omega}_M\le C K^N_k(z|p)\wedge {\omega}_h^{m-k}$ (where the constant $C$ depends only on $k$ and $m$), and thus it suffices to show that
1. $ \big(K^N_k(z|p)\wedge {\omega}_h^{m-k}\big)
(\{p\})=0$ for $k<m$.
For $k<m$, we let $$X^k_n({{\bf s}})=(Z_{{\bf s}}\wedge {\omega}_h^{m-k},{\varphi}_n)\le \pi^{m-k}N^mc_1(L)^m\,,\quad {{\bf s}}=(s_1,\dots,s_k)\,,$$ where ${\varphi}_n$ is as before. But this time, $X^k_n({{\bf s}})=\int_{Z_{{\bf s}}}{\varphi}_n {\omega}_h^{m-k}\to 0$ a.s. Equation (i$'$) now follows exactly as before. (Equation (i) is also an immediate consequence of Federer’s support theorem for locally flat currents [@Fe 4.1.20].)
We now complete the proof of Theorem \[all codim\]: By Lemma \[product\] and the asymptotic formula , we have $$\begin{aligned}
K^N_k(z|p)& =& K^N_k(z|p)_{ac }\\&=&\left[\frac i{2\pi} {\partial{\bar\partial}}\log\|\Pi_N(z,z)\|_{h^N}
+ \frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right)+\frac N\pi {\omega}_h
\right]^k_{ac }\\&=& \left(\frac i{2\pi}\right)^k
\left[-\frac{e^{-|u|^2}}{(1-e^{-|u|^2})^2}\,{\partial}|u|^2\wedge{\bar\partial}|u|^2 +
\frac{{\partial{\bar\partial}}|u|^2}{(1-e^{-|u|^2})} \right]^{k} \\&&\quad +\ \sum O(
|u|^{-2k}N^{-1/2+{\varepsilon}})du_{j_1}\wedge du_{l_1}\wedge\cdots \wedge du_{j_k}\wedge
du_{l_k}\,.\end{aligned}$$
Therefore, $$\begin{aligned}
\left(K^N_k(z|p)\,,\,\tau_N^*{\varphi}\right)&=&\int_{M_0{\smallsetminus}\{p\}}
\left[\frac i{2\pi} {\partial{\bar\partial}}\log\|\Pi_N(z,z)\|_{h^N}
+ \frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right)+\frac N\pi {\omega}_h
\right]^k\wedge \tau_N^*{\varphi}\notag\\&=& \left(\frac i{2\pi}\right)^k
\int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}\left[-\frac{e^{-|u|^2}}{(1-e^{-|u|^2})^2}\,{\partial}|u|^2\wedge
{\bar\partial}|u|^2 +\frac{{\partial{\bar\partial}}|u|^2}{(1-e^{-|u|^2})} \right]^{k}\wedge{\varphi}\notag
\\&&\quad +\ N^{-1/2+{\varepsilon}}\|{\varphi}\|_\infty \int_{{{\operatorname{Supp\,}}}({\varphi})}O(|u|^{-2k})\,(i
{\partial{\bar\partial}}|u|^2)^m\,,\label{best}\end{aligned}$$ which verifies Theorem \[all codim\].
To prove Theorem \[scaled\], we need to integrate by parts, since if $k=m$, the integral in the last line of does not a priori converge. To begin the proof, by Lemma \[product\] we have $$\begin{gathered}
\big(K^N_m(z|p)\,,\,{\varphi}\circ\tau_N\big)={\varphi}(0)\\+
\int_{M_0{\smallsetminus}\{p\}}{\varphi}\big(\sqrt N\,z\big)\left[\frac i{2\pi} {\partial{\bar\partial}}\log\|
\Pi_N(z,z)\|_{h^N}
+ \frac i{2\pi} {\partial{\bar\partial}}\log \left( 1 - P_N(z,p)^2\right)+\frac N\pi {\omega}_h
\right]^m.\label{withdelta}\end{gathered}$$ Writing $$\label{rho}{\omega}_h=\frac i2 {\partial{\bar\partial}}\rho\,,\qquad \rho(z)
= |z|^2+O(|z|^3)\,,$$ we then have $$\label{withdelta1} \big(K^N_m(z|p)\,,\,{\varphi}\circ\tau_N\big)=
{\varphi}(0)+ \int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\varphi}\,\cdot \left(\frac i{2\pi}{\partial{\bar\partial}}f_N\right)^m
\,,$$ where $$f_N(u)=\log \left\|\Pi_N\left(\frac u {\sqrt{N}},\frac u {\sqrt{N}}\right)\right\|
_{h^N} -m\log (N/\pi)
+\log \left( 1 - P_N\left(\frac u{\sqrt{N}},0\right)^2\right)+N\rho\left(\frac u
{\sqrt{N}}\right).$$ By , and , $$\label{fN}f_N(u)= \log \left(1- e^{-|u|
^2}\right) +|u|^2+ O(N^{-1/2+{\varepsilon}})\,.$$ Again recalling , we have $$\label{ddfN}{\partial{\bar\partial}}f_N= -\frac{e^{-|u|^2}}{(1-e^{-|u|^2})^2}\,
{\partial}|u|^2\wedge{\bar\partial}|u|^2 +\frac{{\partial{\bar\partial}}|u|^2}{(1-e^{-|u|^2})} + O(
|u|^{-2}N^{-1/2+{\varepsilon}})\,.$$
We now integrate by parts. Let $$\label{alpha}{\alpha}_N= f_N \,({\partial{\bar\partial}}f_N)^{m-1}\,.$$ Then for ${\delta}>0$, $$\int_{|u|>{\delta}}{\varphi}\,{\partial{\bar\partial}}{\alpha}_N =\int_{|u|>{\delta}} {\alpha}_N\wedge
{\partial{\bar\partial}}{\varphi}+\frac i2\int_{|u|={\delta}}( {\varphi}\,d^c{\alpha}_N -{\alpha}_N\wedge d^c{\varphi})\,,$$ where $d^c=i({\bar\partial}-{\partial})$. By –, $$\label{allalpha} {\alpha}_N={\alpha}_\infty + O\left(|u|^{-2m+2}\log(|u|
+|u|{^{-1}})N^{-1/2+{\varepsilon}}\right)\,,$$ where $$\begin{aligned}
{\alpha}_\infty &=&\left[\log(1-e^{-|u|^2}) +|u|^2\right] \left\{
{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2}) +|u|^2\right]\right\}^{m-1}
\\&=&
\left[\log(1-e^{-|u|^2}) +|u|^2\right]
\left[-\frac{e^{-|u|^2}}{(1-e^{-|u|^2})^2}\,{\partial}|u|^2\wedge{\bar\partial}|u|^2 +
\frac{{\partial{\bar\partial}}|u|^2}{(1-e^{-|u|^2})} \right]^{m-1}.\end{aligned}$$ In particular, $$\label{|alpha|}{\alpha}_N=O\left(|u|^{-2m+2}\log(|u|+|u|{^{-1}})\right)
\,,$$ and therefore $$\lim_{{\delta}\to 0} \int_{|u|={\delta}}{\alpha}_N\wedge d^c{\varphi}= 0\,.$$
Futhermore, by and , $$d^c{\alpha}_N = \frac {d^c|u|^2 \wedge ({\partial{\bar\partial}}|u|^2)^{m-1}}{(1-e^{-|u|^2})^m} +O
\left(|u|^{-2m+1}N^{-1/2+{\varepsilon}}\right).$$ Therefore, $$\begin{aligned}
\left(\frac i{2\pi}\right)^m \frac i2
\int_{|u|={\delta}}
{\varphi}\,d^c{\alpha}_N &=& -\frac {{\delta}\cdot {\delta}^{2m-1}}{(1-e^{-
{\delta}^2})^m}\mbox{Average}_{|u|={\delta}}({\varphi}) +O(N^{-1/2+{\varepsilon}})\sup_{|u|={\delta}}|{\varphi}|
\\&\to & -{\varphi}(0)\,\left[1+O(N^{-1/2+{\varepsilon}})\right]\,.\end{aligned}$$
Thus, $$\label{byparts} \left(\frac i{2\pi}\right)^m
\int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\varphi}\,{\partial{\bar\partial}}{\alpha}_N = \left(\frac i{2\pi}\right)^m\int_{{{\mathbb C}}^m
{\smallsetminus}\{0\}}{\alpha}_N\,{\partial{\bar\partial}}{\varphi}-{\varphi}(0)\,\left[1+O(N^{-1/2+{\varepsilon}})\right]\,.$$ ([*Remark:*]{} In fact, it follows from Demailly’s comparison theorem for generalized Lelong numbers [@Dem Theorem 7.1], applied to the plurisubharmonic functions $f_N(u)$ and $
\log|u|^2$ and closed positive current $T=1$, that the two measures $i^m {\partial{\bar\partial}}\alpha_N$ and $i^m {\partial{\bar\partial}}\log|u|^2$ impart the same mass to the point $0$, and therefore we have the precise identity $$\left(\frac i{2\pi}\right)^m
\int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\varphi}\,{\partial{\bar\partial}}{\alpha}_N = \left(\frac i{2\pi}\right)^m\int_{{{\mathbb C}}^m
{\smallsetminus}\{0\}}{\alpha}_N\,{\partial{\bar\partial}}{\varphi}-{\varphi}(0)\,.$$ However, suffices for our purposes.)
Combining , and , $$\begin{aligned}
\big(K^N_m(z|p)\,,\,{\varphi}\circ\tau_N\big) &=&
\left(\frac i{2\pi}\right)^m \int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\alpha}_N\,{\partial{\bar\partial}}{\varphi}+O(N^{-1/2+
{\varepsilon}})\\ &=&\left(\frac i{2\pi}\right)^m \int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\alpha}_\infty\,{\partial{\bar\partial}}{\varphi}+O(N^{-1/2+{\varepsilon}}).\end{aligned}$$
Repeating the integration by parts argument using ${\alpha}_\infty$ (or by the above comparison theorem of Demailly [@Dem]), we conclude that $$\left(\frac i{2\pi}\right)^m
\int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\varphi}\,{\partial{\bar\partial}}{\alpha}_\infty = \left(\frac i{2\pi}\right)^m
\int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\alpha}_\infty\,{\partial{\bar\partial}}{\varphi}-{\varphi}(0)\,.$$ Therefore $$\begin{aligned}
\big(K^N_m(z|p)\,,\,{\varphi}\circ\tau_N\big) &=&{\varphi}(0) +
\left(\frac i{2\pi}\right)^m \int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\varphi}{\partial{\bar\partial}}{\alpha}_\infty
+O(N^{-1/2+{\varepsilon}})\\&=&
{\varphi}(0) +\left(\frac i{2\pi}\right)^m \int_{{{\mathbb C}}^m{\smallsetminus}\{0\}}{\varphi}(u)\left\{
{\partial{\bar\partial}}\left[\log(1-e^{-|u|^2}) +|u|^2\right]\right\}^m\\&& \qquad
+O(N^{-1/2+{\varepsilon}})\,,\end{aligned}$$ which completes the proof of Theorem \[scaled\].
Comparison of pair correlation density and conditional density {#comparison}
==============================================================
We conclude with further discussion of the comparison between the pair correlation function and conditional Gaussian density of zeros.
Comparison in dimension one {#PT}
---------------------------
We now explain the sense in which the pair correlation $K^N_{1m}(p)^{-2} K_{2m}^N(z, p)$ of [@BSZ; @BSZ2] may be viewed as a conditional probability density.
We begin with the case of polynomials, i.e. $M={{{\mathbb C}}{{\mathbb P}}}^1$. The possible zero sets of a random polynomial form the configuration space $$({{{\mathbb C}}{{\mathbb P}}}^1)^{(N)} = Sym^{N} {{{\mathbb C}}{{\mathbb P}}}^1 :=
\underbrace{{{{\mathbb C}}{{\mathbb P}}}^1\times\cdots\times {{{\mathbb C}}{{\mathbb P}}}^1}_N /S_{N}$$ of $N$ points of ${{{\mathbb C}}{{\mathbb P}}}^1$, where $S_N$ is the symmetric group on $N$ letters. We define the joint probability current of zeros as the pushforward $$\label{JPCDEF}\vec K_N^N(\zeta_1, \dots, \zeta_N) : =
{\mathcal{D}}_* \gamma_h^N$$ of the Gaussian measure on the space ${\mathcal{P}}_N$ of polynomials of degree $N$ under the ‘zero set’ map $
{\mathcal{D}}: {\mathcal{P}}_N \to ({{{\mathbb C}}{{\mathbb P}}}^1)^{(N)}$ taking $s_N $ to its zero set. An explicit formula for it in local coordinates is
$$\begin{aligned}
\label{eq-030209b}
\vec K_N^N(\zeta_1, \dots, \zeta_N) & = & \frac{1}{Z_N(h)}
\frac{|\Delta(\zeta_1, \dots, \zeta_N)|^2 d_2 \zeta_1 \cdots d_2
\zeta_N}{\left(\int_{{{{\mathbb C}}{{\mathbb P}}}^1} \prod_{j = 1}^N |(z - \zeta_j)|^2 e^{-
N {\varphi}(z)} d\nu(z) \right)^{N+1}},\end{aligned}$$
where $Z_N(h)$ is a normalizing constant. We refer to [@ZZ] for further details.
As in [@D §5.4, (5.39)], the pair correlation function is obtained from the joint probability distribution by integrating out all but two variables. If we fix the second variable of $K_{2 1}^N(z, p)$ at $p$ and divide by the density $K_{11}^N(p)$ of zeros at $p$, we obtain the same density as if we fixed the first variable $\zeta_1 = p$ of the density of $\vec
K_N^N(\zeta_1, \dots, \zeta_N)$, integrated out the last $N - 2$ variables and divided by the density at $p$. But fixing $\zeta_1 =
p$ and dividing by $K_{11}^N(p) d_2 \zeta_1$ is the conditional probability distribution of zeros defined by the random variable $\zeta_1$. Thus in dimension one, $K^N_{11}(p)^{-2} K_{21}^N(z, p)$ is the conditional density of zeros at $z$ given a zero at $p$ if we condition using $\zeta_1 = p$ in the configuration space picture. This use of the term ‘conditional expectation of zeros given a zero at $p$’ can be found, e.g. in [@So].
Comparison in higher dimensions
-------------------------------
The above configuration space approach is difficult to generalize to higher dimensions and full systems of polynomials. In particular, it is difficult even to describe the configuration of joint zeros of a system as a subset of the symmetric product. Indeed, the number of simultaneous zeros of $m$ sections is almost surely $c_1(L)^mN^m$ so the variety $C_N$ of configurations of simultaneous zeros is a subvariety of the symmetric product $M^{(c_1(L)^mN^m)}$. Since $C_N$ is the image of the zero set map $${\mathcal{D}}:G(m,H^0(N,L^N))\to M^{(c_1(L)^mN^m)}$$from the Grassmannian of $m$-dimensional subspaces of $H^0(N,L^N)$, its dimension (given by the Riemann-Roch formula) is quite small compared with the dimension of the symmetric product:$$\dim
C_N=\frac{c_1(L)^m}{(m-1)!}N^m+O(N^{m-1})\sim \frac 1{m!}\dim
M^{(c_1(L)^mN^m)}\,.$$ Under the zero set map, the probability measure on systems pushes forward to $C_N$, but to our knowledge there is no explicit formula as (\[eq-030209b\]).
We now provide an intuitive and informal comparison of the two scaling limits without using our explicit formulas. Let $B_{\delta}(p) \subset {{\mathbb C}}^m$ be the ball of radius $\delta$ around $p$, let ${{\bf s}}= (s_1, \dots, s_m)$ be an $m$-tuple of independent random sections in $H^0(M,L^N)$, and let Prob denote the probability measure $({\gamma}_h^N)^m$ on the space of $m$-tuples ${{\bf s}}$. We define the events, $$U_{\delta}^p = \{{{\bf s}}: {{\bf s}}\; \mbox{has a zero in } \;
B_{\delta}(p)\}, \;\;\; U_{{\varepsilon}}^q = \{{{\bf s}}: {{\bf s}}\; \mbox{has
a zero in } \; B_{{\varepsilon}}(q)\}.$$
Now the probability interpretation of the pair correlation function is based on the fact that, as $\delta, {\varepsilon}\to 0$, $$\int_{ B_{\delta}(P)
\times B_{{\varepsilon}}(q)} {{\mathbf E}}\big[Z_{{{\bf s}}}(z) Z_{{{\bf s}}}(w)\big] = \mbox{Prob}(U_{\delta}^p \cap U_{{\varepsilon}}^q) \big[1+o(1)\big] \;,$$ since the probability of having two or more zeros in a small ball is small compared with the probability of having one zero.
It follows that $$\lim_{{\varepsilon}, \delta
\to 0} \frac{1}{{{\operatorname{Vol}}}( B_{\delta}(p)) \times {{\operatorname{Vol}}}(B_{{\varepsilon}}(q))}
\mbox{Prob}(U_{\delta}^p \cap U_{{\varepsilon}}^q) = K^\infty _{2mm}(p, q).$$ Similarly, $$\lim_{\delta \to 0} \mbox{Prob} ( U_{\delta}^p)
\simeq \frac{1}{{{\operatorname{Vol}}}B_{\delta}(p)} \int_{ B_{\delta}(p)} {{\mathbf E}}Z_{{{\bf s}}}(z) = K^\infty _{1mm}(p). \;$$ Hence, as ${\varepsilon}, \delta \to 0$, $$\mbox{Prob}(U_{{\varepsilon}}^q |
U_{\delta}^p) \simeq \frac{\left(\int_{ B_{\delta}(p) \times
B_{{\varepsilon}}(q)} {{\mathbf E}}Z_{{{\bf s}}}(z) Z_{{{\bf s}}}(w)\right)}{ \left(\int_{
B_{\delta}(p)} {{\mathbf E}}Z_{{{\bf s}}}(z) \right)} = \frac{\left(\int_{
B_{\delta}(p) \times B_{{\varepsilon}}(q)} K^\infty _{2mm}(z,w)\right)}{
\left(\int_{ B_{\delta}(p)}K^\infty _{1mm}(z) \right)},$$ so that $$\lim_{{\varepsilon}, \delta \to 0}\frac{1}{{{\operatorname{Vol}}}B_{{\varepsilon}}(q)} \mbox{Prob}
(U_{{\varepsilon}}^q |
U_{\delta}^p) = \frac{K^\infty _{2 mm}(p,q)}{K^\infty _{1mm}(p)}.$$
By comparison, $$K_1^\infty (q|p) = \lim_{{\varepsilon}\to 0}\frac{1}{{{\operatorname{Vol}}}B_{{\varepsilon}}(q)} \mbox{Prob}(U_{{\varepsilon}}^q | \; {{\bf s}}(p) = 0) =
\lim_{{\varepsilon}, \delta \to 0} \frac{1}{{{\operatorname{Vol}}}B_{{\varepsilon}}(q)}
\frac{\mbox{Prob}(U_{{\varepsilon}}^q \cap {\mathcal{F}}_{\delta}^p )}{\mbox{Prob}
({\mathcal{F}}_{\delta}^p)},$$ where $${\mathcal{F}}_{\delta}^p =\left \{(s_1,\dots,s_m: \left(
\sum |s_j(p)|^2_{h^N}\right)^{1/2}<{\delta}\right\}\,.$$
Thus, the difference between the Gaussian conditional density and the pair correlation density corresponds to the difference between the family of systems ${\mathcal{F}}_{\delta}^p $ and the family of systems $U_{{\varepsilon}}^p $. This comparison of the pair correlation density and the Gaussian conditional density shows that in a probabilistic sense, the conditions ‘${{\bf s}}(p)$ is small’ and ‘${{\mathbf s}}$ has a zero near $p$’ are mutually singular.
Comparison of the conditional expectation and pair correlation in codimension 1 {#compare1}
-------------------------------------------------------------------------------
We take a different approach to comparing $K_1^N(z | p) $ and $K_{2 1}^N(z, p)$: The scaling asymptotics of $K_1^N(z | p) $ and $K_{2 1}^N(z, p)$ are both given by universal expressions in the normalized [Szegő ]{}kernel or two-point function $P_N(z,w)$ (defined in (\[PN\])). This is to be expected since the two-point function is the only invariant of a Gaussian random field. Indeed, Proposition \[cond\] says that $$\label{RELCOND} K_1^N(z|p)
= {{\mathbf E}}_N\big(Z_s\big) + \frac 1{2\pi} (i{\partial{\bar\partial}})_z\; Y (- \log P_N(z,p))\,,$$ where $Y({\lambda})= \log (1-e^{-2{\lambda}})$ (recall ).
We now review the approach to the pair correlation current $K^N_{21}(z,p)$ given in [@SZa]. The [*pair correlation current*]{} of zeros $Z_s$ is given by ${{\mathbf E}}_N\big(Z_s \boxtimes Z_s
\big)$, and the [*variance current*]{} is given by $$\label{vc}
{{\bf Var}}_N\big(Z_s\big): =
{{\mathbf E}}_N\big(Z_s \boxtimes Z_s \big)
- {{\mathbf E}}_N\big(Z_s\big)\boxtimes {{\mathbf E}}_N
\big(Z_s\big)\in {\mathcal{D}}'^{2k,2k}(M\times M).$$ Here we write $$S\boxtimes T = \pi_1^*S \wedge \pi_2^*T \in {\mathcal{D}}'^{p+q}(M\times
M)\;, \qquad \mbox{for }\ S\in {\mathcal{D}}'^p(M),\ T\in {\mathcal{D}}'^q(M)\;,$$ where $\pi_1,\pi_2:M\times M\to M$ are the projections to the first and second factors, respectively.
In [@SZa], the first two authors gave a [*pluri-bipotential*]{} for the variance current in codimension one, i.e. a function $Q_N\in L^1(M\times M)$ such that $$\label{varcur}{\bf Var}_N\big(Z_{s}\big)=
(i{\partial{\bar\partial}})_z\,(i{\partial{\bar\partial}})_w \,Q_N(z,w) \;.$$ The bipotential $Q_N:M\times M\to
[0,+\infty)$ is given by $$\label{qN} Q_N(z,w)= {\widetilde}G(P_N(z,w))\,,\quad {\widetilde}G(t) = -\frac 1{4\pi^2}
\int_0^{t^2} \frac{\log(1-s)}{s}\,ds\;.$$
The analogue to for the pair correlation current can be written $$\label{BIPOT1}
K_{21}^N(z,p) ={{\mathbf E}}_N\big(Z_{s} \boxtimes Z_{s} \big)
= {{\mathbf E}}_N\big(Z_{s}\big)\boxtimes {{\mathbf E}}_N \big(Z_{s}\big) + {\partial{\bar\partial}}_z
{\partial{\bar\partial}}_p F (- \log P_N(z,p)),$$ where $F$ is the anti-derivative of the function $\frac 1{2\pi^2}Y$: $$\label{Gtilde1} F(\lambda) = {\widetilde}G(e^{-{\lambda}}) = -\frac
1{2\pi^2} \int_{\lambda}^\infty \log(1-e^{-2s})\,ds\;,\qquad {\lambda}\ge
0\;$$ That is, $\frac 1{2\pi^2}Y (- \log P_N(z,p))$ is the relative potential between the conditioned and unconditioned distribution of zeros, while $ F (- \log P_N(z,p))$ is the relative [*bi-potential*]{} for the pair correlation current $ {{\mathbf E}}_N\big(Z_{s} \boxtimes Z_{s} \big) $.
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|
---
author:
- Michael Benzaquen
- Thomas Salez
- Elie Raphaël
title: 'Intermediate Asymptotics of the Capillary-Driven Thin Film Equation'
---
Dimensional analysis is well understood through the Vaschy-Buckingham $\Pi$ theorem [@Buckingham1914]. Historically, this framework has already led to remarkably important results such as the expression of the hydrodynamical drag force on a sphere by Reynolds [@Reynolds1895], the theory of turbulence by Kolmogorov [@Kolmogorov1942; @Kolmogorov1941], and the estimation of the nuclear explosion power by Taylor [@Taylor1950a; @Taylor1950b]. Moreover, dimensional analysis is directly connected to the fundamental concepts of *scaling* and *self-similarity*, that appear in numerous situations such as fractals and diffusion [@Fourier1822].
The powerful theory of *intermediate asymptotics* developed in particular by Barenblatt goes one step further in understanding the deeper meaning of self-similarity [@Barenblatt1996]. In nonlinear problems, one may wonder what is the interest of finding exact particular solutions as there is no principle of superposition. Nonetheless, in certain cases, the self-similar solutions obtained for *idealised* problems, or idealised initial conditions, represent the intermediate asymptotic regimes of the solutions of more general *non-idealised* problems. Following Zeldovich in the foreword of [@Barenblatt1996], one might even say that intermediate asymptotics is the key that somehow replaces the superposition principle in nonlinear physics. In other words, by paying the price of a loss of information at short times, one obtains a certain generality at intermediate times. Even so, such an asymptotic behaviour must be proved for any given initial condition, which often turns out to be a difficult task.
Furthermore, in the afterword of his book Barenblatt states [@Barenblatt1996]: *“However, there exist many problems of recognised importance where this technique has not yet been fully explained, but for which results of substantial value can be expected from its application.”* One of these open problems is precisely the *capillary-driven thin film equation* of interest [@Blossey2012; @Craster2009; @Oron1997]: $$\partial_T H+\partial_X\left(H^3\,\partial_X^{\,3} H\right)=0\ .
\label{TFE_AD}$$ It governs the capillary evolution of the profile $H(X,T)$ of the free surface of a thin viscous liquid film: as soon as the curvature is nonconstant, this profile is unstable as the Laplace pressure drives a flow that is mediated by viscosity. Despite many efforts, this equation remains only partially solved. Nevertheless, in the past few years, several analytical [@Bowen2006; @Myers1998; @Salez2012] and numerical studies [@Bertozzi1998; @Salez2012a] have been performed in order to gain insights into this mathematical problem.
![Schematics of the intermediate asymptotics of the capillary-driven thin film equation (see Eq. ). No matter the initial condition, any summable profile converges in time towards a universal self-similar attractor.[]{data-label="Fig1"}](Fig1.pdf)
It should be stressed that Eq. is of tremendous importance in a variety of scientific fields such as polymer physics, physiology, biophysics, micro-electronics, surface chemistry, thermodynamics and hydrodynamics, since thin films are involved in modern mechanical and optical engineering processes, through lubrication, paints and coating. Gaining a complete understanding of these systems is a key step towards the development of molecular electronics, biomimetics, superadhesion and self-cleaning surfaces. Moreover, this equation may be crucial for understanding the nanorheology of ultra-thin polymer films, for which enhanced mobility effects have been predicted [@Brochard2000], before being related to entanglements networks [@Si2005], and observed in various experimental configurations such as: confinement [@Barbero2009; @Bodiguel2006; @Jones1999; @Shin2007], glassy state [@Fakhraai2008], and dewetting onto slippery substrates [@Baumchen2009; @Munch2005]. It may also govern the surface instabilities and pattern formation [@Amarandei2012; @Closa2011; @Mukherjee2011], through the film preparation by spin-coating [@Baumchen2012; @Raegen2010; @Reiter2001; @Stillwagon1990]. From all these examples, we understand the necessity of further exploring the solutions of the capillary-driven thin film equation. In particular, as for the diffusion equation [@Fourier1822], it would be interesting to characterise the convergence of the solutions to some asymptotic self-similar regimes [@Stone2012], as depicted in Fig. \[Fig1\], as well as the existence of possible exotic self-similarities [@Sekimoto2012].
The present article is divided into three parts. In the first one, we recall the main ingredients of the physical model that describes two-dimensional capillary-driven flows. In the second one, we linearise the governing equation and we derive the Green’s function that enables to calculate the general solution of the linear problem for any summable initial condition. In particular, we study the self-similar asymptotics of this solution (see Fig. \[Fig1\]). In the third part, we extend these ideas to the nonlinear equation through numerical solutions for compact-support initial profiles.
Physical model
==============
In this part, we describe the model and derive the capillary-driven thin film equation, within the lubrication approximation, for two-dimensional viscous flows. The evolution in time $t$ of a thin viscous film described by a profile of height $z=h(x,t)$ can be understood from the Laplace pressure $p(x,t)$, which arises due to curvature at the free interface. Considering small curvature gradients, one can write: $$\begin{aligned}
p(x,t)&\simeq&-\gamma\,\partial_x^{\,2} h\ ,
\label{Laplace}\end{aligned}$$ where $\gamma$ is the liquid-air surface tension. In the lubrication approximation, the Stokes equation along the $x$ horizontal direction is given by [@Landau1987]: $$\begin{aligned}
\partial_xp&=&\eta\,\partial_{z}^{\,2}v\ ,
\label{Stokes}\end{aligned}$$ where $v(x,z,t)$ is the horizontal velocity and $\eta$ is the shear viscosity. We assume no slip at the solid-liquid interface and no stress at the liquid-air interface, so that:
\[CondLim\] v |\_[z=0]{}&=&0\
\_zv |\_[z=h]{}&=&0 .
The pressure being independent of the vertical coordinate $z$, Eq. together with Eq. lead to a Poiseuille flow of the form: $$\begin{aligned}
v(x,z,t)&=&\frac{1}{2\eta}\,(z^2-2hz)\, \partial_x p\ .
\label{Poiseuille}\end{aligned}$$ Conservation of volume can be expressed as: $$\begin{aligned}
\partial_th&=&-\partial_x\int_0^h\textrm{d}z \,v\ .
\label{Mass}\end{aligned}$$ Combining Eqs. , and leads to the two-dimensional capillary-driven thin film equation: $$\begin{aligned}
\partial_th+\frac{\gamma}{3\eta}\,\partial_x\left( h^3\,\partial_x^{\,3}h \right)&=&0\ .
\label{TFE1}\end{aligned}$$ Finally, Eq. can be nondimensionalised by letting:
h&=&H h\_0\
x&=&Xh\_0\
t&=&T ,
where $h_0$ is the reference height at infinity. This leads to the dimensionless equation introduced above in Eq. . This equation can be linearised by letting: $$\label{defdelta}
H(X,T)=1+\Delta(X,T)\ ,$$ and by assuming $\Delta\ll 1$: $$\begin{aligned}
\partial_T\Delta+\partial_X^{\,\,4}\Delta&=&0\ .
\label{LTFE_AD}\end{aligned}$$ The linear case corresponds to a situation in which the surface of a flat film is only slightly perturbed. Note that Eq. also describes surface diffusion phenomena leading to flattening of solid surfaces [@Mullins1958; @Zhu2011], kinetic growth [@Krug1993; @Lai1991; @Wolf1990], or grooving [@Mullins1957; @Robertson1971]. Therefore, the following results apply to a broader class of physical situations.
Solving the linear equation
===========================
In this part, we solve Eq. for $T>0$ and we characterise its solutions. The linear study is divided into four paragraphs. In the first one, we derive the Green’s function and show that it is self-similar at all positive times. In the second one, we give the general formal solution and exploit it through two particular canonical examples. In the third one, we study the uniform convergence in time of the rescaled general solution towards the rescaled self-similar Green’s function. In the fourth one, we conclude the discussion on the linear case with some general remarks.
Green’s function and self-similarity
------------------------------------
Since Eq. (\[LTFE\_AD\]) is a linear partial differential equation, it can be solved by calculating the Green’s function $\mathcal G(X,T)$. This object is defined by: $$\begin{aligned}
\left[\partial_T+\partial_X^{\,\,4}\right]\,\mathcal G(X,T)&=&\delta(X,T)\ ,
\label{Green1}\end{aligned}$$ where $\delta$ denotes the Dirac distribution. Fourier transforms are defined as follows:
\[E:gp\] G(X,T)&=&K (K,)e\^[i(T +KX)]{} \[E:gp1\]\
(K,T)&=& (K,)e\^[iT]{}\[E:gp2\]\
(K,)&=&X T [G]{}(X,T)e\^[-i(T +KX)]{} .\[E:gp3\]
Writing the Fourier transform, as defined in Eq. (\[E:gp1\]c), of Eq. (\[Green1\]) leads to: $$\begin{aligned}
\label{fourgreen}
\hat {\mathcal G}(K,\Omega)&=&\frac{1}{i\Omega +K^4}\ .\end{aligned}$$ Using Eq. (\[E:gp1\]b) and Eq. (\[fourgreen\]), one obtains: $$\tilde{\mathcal G}(K,T)=\textrm{Res}\left(\frac{e^{i\Omega T}}{\Omega -iK^4}\, ;\,iK^4 \right)\, \Theta(T)\ ,
\label{Caus}$$ where Res$(f;z^*)$ denotes the complex residue of the function $f$ at $z=z^*$, and where $\Theta$ is the Heaviside function ensuring causality. Finally, expressing the residue and performing the inverse Fourier transform, defined in Eq. (\[E:gp1\]a), one obtains the Green’s function: $$\begin{aligned}
\label{greenx}
\mathcal G(X,T)&=&\frac{1}{2\pi}\int \textrm{d}K \, e^{-K^4T} \,e^{iKX}\ ,\end{aligned}$$ which is consistent with previous studies [@Bowen2006; @Mullins1957]. Then, at finite time, let us change variables through:
X&=&UT\^[1/4]{}\[CDV1\]\
K&=&QT\^[-1/4]{}\
G(X,T)&=&(U,T) . \[CDV2\]
This gives: $$\begin{aligned}
\breve{\mathcal{G}}(U,T)
&=& \frac 1{T^{1/4}}\ \phi(U)\, \Theta(T)\ ,\end{aligned}$$ where we introduced the auxiliary function: $$\label{auxi}
\phi(U)=\frac{1}{2\pi}\,\int \textrm{d}Q\,{e^{-Q^4}e^{iQU}}\ .$$
![Auxiliary function $\phi(U)={T^{1/4}}\ \breve{\mathcal G}(U,T)$, for positive times as given in Eq. , where $\breve{\mathcal G}(U,T)$ is the Green’s function of the dimensionless linear two-dimensional capillary-driven thin film equation given in Eq. .[]{data-label="Fig2"}](Fig2)
The Green’s function is thus *self-similar of the first kind* [@Barenblatt1996], at all positive times. Furthermore, the function $\phi$ is given for all $U\in \mathbb R$ by: $$\begin{aligned}
\phi(U)&=& \frac1{\pi} \,\Gamma\left(\frac{5}{4}\right)\ _0H_{2}\left(\left\{ \frac12,\frac34\right\},\left(\frac{U}{4}\right)^4\right)\nonumber\\&-&\frac1{8\pi}\,U^2 \,\Gamma\left(\frac{3}{4}\right)\ _0H_{2}\left(\left\{ \frac54,\frac32\right\},\left(\frac{U}{4}\right)^4\right)\ ,
\label{HPG}\end{aligned}$$ where the $(0,2)$-hypergeometric function is defined as [@Abramowitz1965; @Gradshteyn1965]: $$\begin{aligned}
_0H_{2}\left(\left\{ a,b\right\},w\right)&=&\sum_{k\geq0} \frac{1}{(a)_k(b)_k}\,\frac{w^k}{k!}\ ,\end{aligned}$$ with the Pochhammer notation $(.)_k$ for the rising factorial. The function $\phi$ is plotted in Fig. \[Fig2\]. Other than the oscillatory behaviour which is directly related to the fourth spatial derivative of Eq. (\[LTFE\_AD\]), this solution is qualitatively close to the point-source solution of the heat equation [@Fourier1822], for which the same analytical treatment would lead to the well-known Green’s function and to its specific self-similar variable $XT^{-1/2}$.
General solution
----------------
For a given summable[^1] initial condition, $\Delta(X,0)=\Delta_0(X)$, the solution $\Delta(X,T)$ of Eq. (\[LTFE\_AD\]) is given by the spatial convolution of $\Delta_0(X)$ with the Green’s function: $$\begin{aligned}
\Delta(X,T)&=&\int \textrm{d}Y\,\mathcal G(X-Y,T)\,\Delta_0(Y)\ .
\label{Sol_xt}\end{aligned}$$ Interestingly, Eq. (\[Sol\_xt\]) implies that the Green’s function of the problem is *exactly* the point-source solution obtained from an initial Dirac spatial distribution: $\Delta_0(Y)=\delta(Y)$. Let us once again change variables through Eq. (\[CDV1\]). Then, Eq. (\[greenx\]) and Eq. (\[Sol\_xt\]) lead to: $$\begin{aligned}
\label{Gen_sol}
\Delta(X,T)&=&\breve{\Delta}(U,T)\\ =\frac{1}{2\pi\ T^{1/4}}&&\int \textrm{d}Q\,e^{-Q^4}\,e^{iQU}\int \textrm{d}Y\,e^{-iQY/T^{1/4}} \,\Delta_0(Y)\nonumber\ .\end{aligned}$$ The particular case of an initial *stepped film*[^2] was studied in detail in a previous communication [@Salez2012]. In the present study, we have calculated the solutions for various summable initial conditions. We present two of them corresponding to canonical illustrations: a gate function (see Fig. \[Fig3\]) and a gaussian function (see Fig. \[Fig4\]).
![Normalised analytical solution (see Eq. ) of the dimensionless linear two-dimensional capillary-driven thin film equation (see Eq. ) as a function of the self-similar variable $U$ introduced in Eq. . The initial profile is defined as a gate function of width unity, as shown in the inset. The solution is plotted for different dimensionless times and compared to its normalised asymptotic attractor given in Eq. .[]{data-label="Fig3"}](Fig3)
In each case, we plot the normalised analytical solution of Eq. (\[LTFE\_AD\]) given in Eq. , as a function of the self-similar variable $U$ introduced in Eq. (\[CDV1\]), the initial profile at $T=0$ being shown in the inset. The solution is plotted for different dimensionless times. For comparison, we also plot the normalised function $\phi(U)/\phi(0)$, as given in Eq. . As one can see, both profiles seem to converge in time towards the later function that we shall henceforward call a *universal self-similar attractor* (see Fig. \[Fig1\]). This convergence statement will be addressed in the next paragraph.
Uniform convergence to the self-similar attractor
-------------------------------------------------
For clarity, we shall restrict here to summable initial profiles that have non-zero algebraic volume, that is: $$\begin{aligned}
\label{cond_mom}
\mathcal{M}_0\,\,=\int \textrm{d}X\,\Delta_0(X)&\,\,\neq\,\,&0\ .\end{aligned}$$ The extension of the following results to the specific case of zero initial algebraic volume is understood as well and will be addressed in the next paragraph. For the time being, let us introduce the function: $$\begin{aligned}
f(U,T)&=&\, \frac{T^{1/4}}{\mathcal{M}_0}\ \breve{\Delta}(U,T)\ .
\label{Pour_dev1}\end{aligned}$$
![Normalised analytical solution (see Eq. ) of the dimensionless linear two-dimensional capillary-driven thin film equation (see Eq. ) as a function of the self-similar variable $U$ introduced in Eq. . The initial profile is defined as a gaussian function of volume $0.1$, as shown in the inset. The solution is plotted for different dimensionless times and compared to its normalised asymptotic attractor given in Eq. .[]{data-label="Fig4"}](Fig4)
According to Eq. (\[auxi\]), Eq. (\[Gen\_sol\]) and Eq. (\[Pour\_dev1\]), for all $T>0$ and for all $U\in \mathbb R$, one has: $$\left| f(U,T)-\phi(U) \right| \leq \frac{a(T)}{2\pi|\mathcal{M}_0|}\ ,$$ where we introduced: $$a(T)= \int \textrm{d}Q\,e^{-Q^4}\int \textrm{d}Y\,\left|e^{-iQY/T^{1/4}}-1\right|\,\left|\Delta_0(Y) \right|\ .$$ Therefore, one gets for all $T>0$: $$\begin{aligned}
\left|\left| f(U,T)-\phi(U) \,\right|\right|_{\infty,U} &\leq & \frac{a(T)}{2\pi|\mathcal{M}_0|}\ ,\end{aligned}$$ where $|| ... ||_{\infty,U}$ is the uniform norm[^3] with respect to the variable $U$. In order to conclude that the function $f$ is uniformly convergent[^4] in time towards $\phi$, it remains to show that: $\displaystyle \lim_{T\rightarrow \infty} a(T) =0$. For this purpose, let us consider the auxiliary function defined by: $$\begin{aligned}
{m(Y,T)=\left(e^{-iQY/T^{1/4}}-1\right) \,\Delta_0(Y)}\ ,\end{aligned}$$ for all $Y\in \mathbb R$, and for all $T>0$. This function naturally converges to the zero function when $T\rightarrow \infty$. In addition, for all $T>0$, and for all $Y\in \mathbb R$, one has: $$\begin{aligned}
\label{condi1}
\left|m(Y,T)\right|&\leq& 2\,\left|\Delta_0(Y )\right|\ ,\end{aligned}$$ where the right-hand side is a summable function. Invoking Lebesgue’s continuity theorem leads to: $$\begin{aligned}
\label{lim1}
\lim_{T\rightarrow\infty} \int \textrm{d}Y\,m(Y,T)&=&0\ .\end{aligned}$$ Similarly, we then consider the second auxiliary function defined by: $$\begin{aligned}
g(Q,T)=e^{-Q^4}\,\int \textrm{d}Y\,m(Y,T)\ ,\end{aligned}$$ for all $Q\in \mathbb R$, and for all $T>0$. It converges to the zero function when $T\rightarrow \infty$, according to Eq. (\[lim1\]). In addition, using Eq. (\[condi1\]), for all $T>0$ and for all $Q\in \mathbb R$, one has: $$\begin{aligned}
\left| g(Q,T) \right|&\leq& 2\,e^{-Q^4}\,\int \textrm{d}Y\,\left|\Delta_0(Y )\right|\ ,\end{aligned}$$ where the right-hand side is a summable function. Once again, invoking Lebesgue’s continuity theorem leads to: $$\begin{aligned}
\lim_{T\rightarrow\infty}a(T)&=& 0\ . \end{aligned}$$ In summary, we demonstrated that the rescaled solution $f(U,T)$ of Eq. (\[LTFE\_AD\]), given by Eq. (\[Gen\_sol\]) and Eq. (\[Pour\_dev1\]) for any summable initial profile, always converges *uniformly* in time towards the universal self-similar attractor $\phi(U)$ defined in Eq. (\[auxi\]). In other words, we exhibited the intermediate asymptotics of the solutions of the linear capillary-driven thin film equation for flat boundary conditions and we demonstrated that it is simply given by the rescaled Green’s function. We refer again to Fig. \[Fig3\] and Fig. \[Fig4\] for illustration of this result with a gate function and a gaussian function as initial profiles, respectively. The crucial point in these graphs is that the attractor is *identical* for the two summable initial profiles, as summarised in Fig. \[Fig1\].
Remarks
-------
In order to conclude the discussion on the linear case, we enumerate five general remarks below.
- The latter calculations may be extended to all equations of the form: $$\left[\partial_T +\partial_X^{\,2m}\right]\Delta(X,T) =0\ ,$$ with $m\in \mathbb N^*$. Indeed, other than the well known heat equation [@Fourier1822], one can obtain the Green’s function for higher even orders of the spatial derivative and extract analogous conclusions. However, the odd orders being free from dissipation are expected to lead to fundamentally different mathematical solutions.
- According to our primary interest in thin films, we proved the uniform convergence of any summable solution towards the self-similar attractor by explicitly writing the Green’s function of Eq. (\[LTFE\_AD\]). However, this result is more general. In fact, let us consider any diffusive-like linear partial differential equation of two variables for which the Green’s function $G(X,T)$ is self-similar of the form: $$G(X,T)=\frac{1}{T^\beta}\ \phi\left(\frac{X}{T^{\alpha}}\right)\ ,$$ where $\alpha$ and $\beta$ are strictly positive real numbers, and where $\phi$ is a bounded function on $\mathbb R$. Then, the uniform convergence is straightforward to demonstrate, as soon as Eq. (\[Sol\_xt\]) and Eq. (\[cond\_mom\]) remain satisfied, with a summable initial profile.
- Another interesting feature is the following. Let us call $\Delta_{[\Delta_0]}(X,T)$ the solution of Eq. (\[LTFE\_AD\]), for an initial profile $\Delta_0(X)$. Then, the evolution $ \Delta_{[\Delta_0']}(X,T)$ of the first derivative $\Delta_0'$ of the previous initial condition is simply given by $\partial_X \Delta_{[\Delta_0]}(X,T)$. In terms of the self-similar variable introduced in Eq. (\[CDV1\]), one gets: $$\breve{\Delta}_{[\Delta_0']}(U,T)=\frac{1}{T^{1/4}}\,\partial_U \breve{\Delta}_{[\Delta_0]}(U,T)\ .$$ This notably makes the link with our previous work on the particular case of an initial stepped film [@Salez2012]. Indeed, the Dirac distribution being the first derivative of the Heaviside’s distribution, the Green’s function is thus simply given by the derivative of the solution obtained for a stepped initial condition. Generalised to higher order derivatives, this relation naturally becomes: $$\breve{\Delta}_{[\Delta_0^{(n)}]}(U,T)=\left[\frac{1}{T^{1/4}}\,\partial_U \right]^{n} \breve{\Delta}_{[\Delta_0]}(U,T)\ .$$
- One may as well wonder what happens in the particular case of zero algebraic volume[^5], that is when $\mathcal{M}_0=0$. The answer is that there is still an attractive self-similar regime given by the first non-zero derivative $\phi^{(n)}(U)$, under the condition that this quantity is summable.
- At last, we have seen that for any summable initial condition there is long-term self-similarity of the general solution of Eq. (\[LTFE\_AD\]). The question arrises to know whether there exists some specific summable initial conditions that generate solutions that are self-similar *at all positive times*. Interestingly, the solution obtained from the Heaviside initial condition was shown to be self-similar at all positive times for other boundary limits [@Salez2012]. In the present case, let us impose the following constraint at all times $T>0$: $$\label{defsim}
\breve{\Delta}(U,T)=T^{\alpha/4}F(U)\ ,$$ with $U$ as defined in Eq. (\[CDV1\]). By changing variables in Eq. (\[Gen\_sol\]), it is straightforward to see that the initial profile must be homogeneous of degree $\alpha$, meaning that for all real numbers $k$ and $Y$, one has: $$\label{hom}
\Delta_0(kY)=k^{\alpha}\Delta_0(Y)\ .$$ Finally, using Eq. (\[cond\_mom\]) and Eq. (\[hom\]), the algebraic volume $V$ satisfies: $$\begin{aligned}
V&=&\int \textrm{d}X\ \Delta(X,T)\\
&=&T^{(\alpha+1)/4}\int \textrm{d}U\,F(U)\ ,\end{aligned}$$ which implies that $\alpha=-1$, since $V=\mathcal{M}_0$ by volume conservation. Therefore, the only initial profile with finite algrebraic volume that exhibits the self-similarity of Eq. (\[defsim\]) at all positive times is the Dirac distribution. Recalling the remark made after Eq. (\[Sol\_xt\]), this means that the Green’s function is the only summable solution that is self-similar at all positive times, with the definition of Eq. (\[defsim\]).
Extension to the nonlinear equation
===================================
![Normalised numerical solution of the dimensionless nonlinear two-dimensional thin film equation given in Eq. , as a function of the self-similar variable $U$ defined in Eq. , for an initial profile given by an arbitrary function with compact support, as shown in the inset. The solution is plotted for different dimensionless times and compared to the normalised universal asymptotic attractor of the linear case given in Eq. .[]{data-label="Fig5"}](Fig5)
In this last part, we extend the previous results to the nonlinear case through a numerical scheme. The excess profile $\Delta(X,T)$ is still defined by Eq. (\[defdelta\]), but without any restriction on its amplitude. Thus, we consider the full nonlinear partial differential equation given in Eq. . This equation has not been solved analytically yet, but we recently solved it numerically in various geometries [@Salez2012a]. The numerical procedure we used is a finite difference method developed in [@Bertozzi1998; @Zhornitskaya2000]. It ensures capillary energy and entropy dissipation, as required from [@Bernis1990]. In addition to volume conservation, it has been shown that this method ensures positivity of the height profile $H(X,T)$ [@Zhornitskaya2000]. Using this numerical scheme, we verified the existence of an attractive self-similar regime for several *arbitrary* initial profiles, with *compact support* as required from the algorithm. For instance, we plot in Fig. \[Fig5\] the normalised numerical solution of Eq. as a function of the self-similar variable $U$ defined in Eq. (\[CDV1\]), the initial profile being shown in the inset. The solution is plotted for different dimensionless times. We also plot for comparison the normalised function $\phi(U)/\phi(0)$, as given in Eq. . As we see, the latter is thus an attractor for any initial profile with compact support (see Fig. \[Fig1\]). What is remarkable here is that the attractor of the *nonlinear* case is actually equal to the rescaled Green’s function obtained analytically in the *linear* case. Even though this may seem unexpected, it turns out to be quite natural when one realises that as time goes, the initial profile progressively collapses towards the flat film equilibrium shape, thus bringing Eq. closer to its linearised form of Eq. .
Conclusion {#conclusion .unnumbered}
==========
We reported on the intermediate asymptotics of the two-dimensional capillary-driven thin film equation for an arbitrary summable initial profile. First, we derived an analytical solution of the linearised equation. The solution was obtained by seeking the Green’s function, which was found to be given by a combination of generalised hypergeometric functions. As schematised in Fig. \[Fig1\], we then proved that any summable initial condition leads to the uniform convergence in time of the rescaled solution towards a universal self-similar attractor that is proportional to the rescaled Green’s function of the problem; the proportionality factor being equal to the initial algebraic volume. This result appears to be a more general result in diffusive-like processes that are characterised by a self-similar Green’s function with decaying amplitude. At last, we were able to conjecture from compact-support numerical results, as well as to justify on a physical basis, the extension of this convergence behaviour to the nonlinear equation. The important outcome is that the universal self-similar attractor of the nonlinear case appears to be precisely the one of the linear case, that is the rescaled Green’s function.
The recent excellent agreements between thin film theories and experiments with stepped polymer films [@McGraw2011; @McGraw2012; @Salez2012a; @Salez2012], and polymer droplets on identical films [@Cormier2012; @Salez2012a], are very encouraging for the physical relevance of the present analysis. Thus, experimental implementation with viscous nanofilms should be performed in near future. This theoretical work may also be extended to other thin film equations [@Bertozzi1998], such as the one describing flows in Hele-Shaw cells [@Barenblatt1996] and the one governing gravity-driven flows [@Decre2003; @Huppert1982; @Kondic2003]. One could also address coalescence phenomena [@Hernandez2012; @Ristenpart2006] and various diffusive processes, in an identical way. Moreover, it should be feasible to exhibit the necessary conditions acting on a given partial differential equation for showing such a uniform convergence behaviour. Finally, characterising further the conditions for finite-time convergence may be of fundamental importance for the field of intermediate asymptotics, and thus for nonlinear physics in general.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors warmly thank Kari Dalnoki-Veress, Joshua D. McGraw and Oliver Bäumchen for a very stimulating ongoing collaboration. They are also grateful to Ken Sekimoto, Mark Ediger and Alexandre Darmon for interesting discussions. Finally, they thank the École Normale Supérieure of Paris and the Fondation Langlois for financial support.
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[^1]: Along the present article, *summable* means *Lebesgue integrable*, which implies in particular that $\displaystyle\lim_{X\rightarrow\pm\infty}\Delta_0(X)=0$.
[^2]: For comparison, in such a case one gets at $T>0$: $$\begin{aligned}
\breve{\Delta}(U,T)=\frac{\theta_0}{2}\left( 1+\dashint dQ \,\frac{1}{i\pi Q}\,e^{-Q^4}\,e^{iQU} \right)\ ,\nonumber\end{aligned}$$ where the dashed integral represents Cauchy’s principal value and where $\theta_0$ is the amplitude of the step. This solution is self-similar for all $T>0$.
[^3]: $\|f\|_{\infty,x}=\sup\left\{\,\left|f(x,y)\right|,\,x\in \mathbb R\,\right\}. $
[^4]: A function of two variables $f(x,y)$ is defined to be *uniformly convergent* with respect to $y$ if $\displaystyle \lim_{y\rightarrow \infty}\|f\|_{\infty,x} = 0$.
[^5]: One could for instance imagine a *dip* followed by a *bump* of identical shape, as a pathological initial profile.
|
---
abstract: 'An example of the volume and boundary face area of a curved polyhedron for the case of regular spherical and hyperbolic tetrahedron is discussed. An exact formula is explicitly derived as a function of the scalar curvature and the edge length. This work can be used in loop quantum gravity and Regge calculus in the context of a non-vanishing cosmological constant.'
address:
- |
Laboratoire de physique math�matique et subatomique\
Mentouri university, Constantine 1, Algeria
- |
Laboratoire de physique math�matique et subatomique\
Mentouri university, Constantine 1, Algeria
author:
- 'O.Nemoul'
- 'N.Mebarki'
date: 'March 22, 2018'
title: Volume and Boundary Face Area of a Regular Tetrahedron in a Constant Curvature Space
---
Introduction
============
In geometry, the calculation of volume and boundary face area of a curved polyhedron (geodesic polyhedron[^1]) is one of the most difficult problems. In the case of spherical and hyperbolic tetrahedra, a lot of efforts has been made by mathematicians for calculating the volume and boundary face area: the volume formula are discussed by N. Lobachevsky and L. Schlafli in refs [@r1] for an orthoscheme tetrahedron, by G. Martin in ref [@r2] for a regular hyperbolic tetrahedron and by several authors in refs [@r3; @r4; @r5; @r6; @r7; @r8; @r9] for an arbitrary hyperbolic and spherical tetrahedron. All these results are based on the Schlafli differential equation where a unit sectional curvature was taken and they are given by a combination of dilogarithmic or Lobachevsky functions in terms of the dihedral angles. In the present paper, the volume and boundary face area of a regular spherical and hyperbolic tetrahedron are explicitly recalculated in terms of the curvature radius $r=\sqrt{\frac{6}{|R|}}$ and the edge length $a$. We directly perform the integration over the area and volume elements to end up with simple formula for the boundary face area and volume of a regular tetrahedron in a space of a constant scalar curvature $R$. This can be done by using the projection map to the Cayley-Klein-Hilbert coordinates system (CKHcs) which maps a regular geodesic tetrahedron $T(a)$ of an edge length $a$ in the manifold of a constant curvature $R$ to a regular Euclidean tetrahedron $T(a_0)$ of an edge length $a_0$ in the CKHcs. Then, one can express the area and volume measure elements in terms of their Euclidean ones. A comparison between the regular Euclidean, spherical and hyperbolic tetrahedron is studied and their implications are discussed. In physics, a direct application of the volume and boundary face area of a regular tetrahedron is essentially in loop quantum gravity (LQG) and Regge calculus. In LQG, the Euclidean tetrahedron interpretation of a 4-valent intertwiner state was shown in ref [@r10]. The main important feature of the formula which we are looking for is to find another possible correspondence between the 4-valent intertwiner state with a constant curvature regular tetrahedra shapes; this can be achieved by inverting the resulted functions. Thus, one can obtain the scalar curvature measure for a regular tetrahedron shape which allows us to know what kind of space in which the 4-valent intertwiner state can be represented by a regular tetrahedron [@r11]. It is worth mentioning that the idea supporting this new correspondence in the context of LQG with a non-vanishing cosmological constant was initiated in refs [@r11; @r12; @r13; @r14]. In the context of Regge calculus, the use of a constant curvature triangulation of spacetime was suggested in ref [@r15; @r16; @r17] and it can be useful for constructing a quantum gravity version with a non-vanishing cosmological constant. The paper is organized as follows: In section \[Sec2\], the volume and boundary face area of a geodesic polyhedron in general curved space are discussed. In section \[Sec3\], we give general integration formula of the volume and area for constant curvature spaces. In section \[Sec4\], an exact formula for regular spherical and hyperbolic tetrahedra is explicitly derived as a function of the curvature radius and the edge length. Finally, in section \[Sec5\] we draw our conclusions.
Volume and boundary face area of a polyhedron in a general curved space {#Sec2}
=======================================================================
For any n-dimensional Riemannian manifold $M$ equipped with an arbitrary metric $g$ and a coordinates chart $\{U\subset
M,\vec{\tilde{x}}\}$, one has to find another coordinates chart system $\{U\subset M,\vec{x}\}$, such that the straight lines in the second are geodesics of the manifold $M$. In other words, it maps the geodesic curves of the manifold in the first coordinates system$\ $to the straight line in the second one. Such a coordinates system denoted by CKHcs (Cayley-Klein-Hilbert coordinates system)[^2] is very useful to calculate the volume and boundary face area of a geodesic polyhedron (i.e. every geodesic polygons and polyhedrons in the manifold maps to Euclidean polygons and polyhedrons in the CKHcs respectively). Finding such coordinates system is not an easy task for general metric spaces because it depends on the geometry itself and one has to solve a differential equation to find the CKHcs. If we denote by $\varphi $ the coordinates transformation between the first and the CKHcs: $$x^A={\varphi }^A(\vec{\tilde{x}})\ \ \ \ \ \ \
\ \ A=\overline{1.n}\,,\label{eq1}$$ one can define the CKHcs by coordinates transformation that satisfying the following differential equation (See Appendix \[AppA\]): $${\tilde{\nabla}}_V{\tilde{\nabla}}_V{\varphi }^A(\vec{\tilde{x}})=0\,,\label{eq2}$$ where $${\tilde{\nabla}}_VV=0\,,\label{eq3}$$ Eq. holds for any vector field $V$ tangent to geodesic curves and ${\tilde{\nabla}}_V$ stands for the covariant directional derivative along the vector field $V$ in the coordinates system $\{U,\vec{\tilde{x}}\}$. By knowing the metric in the first coordinates system, one can determine the corresponding Christoffel symbols $\widetilde{\mathit{\Gamma}}'s$ and then solve the differential equation to get the ideal frame CKHcs for calculating the volume of a geodesic polyhedron $Pol$ and its boundary face area ${\partial Pol}_f$ in an arbitrary n-dimensional Riemannian space: $$\int_{Pol\subset U\subset M}{{dV}^{Riem}}=\int_{x(Pol)\subset x(U)\subset {\mathbb{R}}^n}{\sqrt{|{det (g(x))\ }|}\ {dV}^{Euc}}\,,\label{eq4}$$ $$\int_{{\partial Pol}_f\subset U\subset M}{{dA}^{Riem}_f}=\int_{x({\partial Pol}_f)\subset x(U)\subset {\mathbb{R}}^n}{\sqrt{{|{det (g(x)|_{{\partial Pol}_f})\ }|}}\ {dA}^{Euc}_f}\,,\label{eq5}$$ where ${dA}^{Euc}_f$ and ${dV}^{Euc}$ are the Euclidean face area and volume measures of a geodesic polyhedron respectively, $g(x)$ is the metric in the CKHcs, $g(x)|_{{\partial Pol}_f}$ is the induced metric in the geodesic surface ${\partial Pol}_f$.
![The Cayley-Klein-Hilbert coordinates system (CKHcs).\[fig1\]](11){width="2.5in"}
Volume and boundary face area of a polyhedron in a 3d- constant curvature space {#Sec3}
===============================================================================
Let $\mathrm{\Sigma }$ be a 3-sphere or 3-hyperbolic metric space. The metric of the $S^3_r$ and $H^3_r$ can be combined in a unified expression and induced from the Euclidean ${Euc}^4$ and the Minkowski ${Mink}^4$ spaces respectively by using a compact form $\epsilon $ such that: $$\epsilon =\left\{ \begin{array}{l}
1\ \ \ \ \ \ \ for\ \ \ \ \ \ \ \ \ \ S^3_r\subset {Euc}^4\\
i\ \ \ \ \ \ \ for\ \ \ \ \ \ \ \ \ \ H^3_r\subset
{Mink}^4\end{array} \right.\,,\label{eq6}$$ Let us consider the cartesian coordinates chart for the two spaces ${Euc}^4$ and ${Mink}^4$ $$\begin{array}{l}
X\ \ :\ \ M\ \ \ \ \ \ \ \ \ \ \longrightarrow \ \ \ \ \ \ \ \ \ \ {\mathbb{R}}^3\times \epsilon \mathbb{R}\\
\ \ \ \ \ \ \ \ \ m\ \ \ \ \ \ \ \ \ \ \ \longmapsto \ \ \ \ \ \ \ \
\ \ X^A(m)=\left(x^1,x^2,x^3,\epsilon x^4\right)
\end{array}
\,,\label{eq7}$$ where $$\epsilon \mathbb{R}=\left\{ \begin{array}{l}
\mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for\ \ \ \ \ \ \ \ \ \ {Euc}^4\ \\
i\mathbb{R}=Im\left(\mathbb{C}\right)\ \ \ \ \ \ \ for\ \ \ \ \ \ \
\ \ \ {Mink}^4 \end{array} \right.\,,\label{eq8}$$ Basically, the metric of the ${Euc}^4$ and ${Mink}^4$ in this coordinates system is written as: $${ds}^2={\delta }_{AB}{dX}^A{dX}^B={\left(dx^1\right)}^2+{\left(dx^2\right)}^2+{\left(dx^3\right)}^2+{\epsilon }^2{\left(dx^4\right)}^2\,,\label{eq9}$$ In the spherical coordinates $\{\vec{\tilde{x}}\}=\{\rho ,\psi
,\theta ,\varphi \}$ one has: $$\left\{ \begin{array}{l}
\rho =\sqrt{{\delta }_{AB}X^AX^B}\\
\psi =\epsilon arctan\left(\frac{\sqrt{{\left(X^1\right)}^2+{\left(X^2\right)}^2+{\left(X^3\right)}^2}}{X^4}\right) \\
\theta ={arctan \left(\frac{\sqrt{{\left(X^1\right)}^2+{\left(X^2\right)}^2}}{X^3}\right)\ } \\
\varphi ={arctan \left(\frac{X^2}{X^1}\right)\ }
\end{array}
\right.\ \ \ \ \ \ \ \ \ \ \ \ \left\{ \begin{array}{l}
X^1=\frac{\rho }{\epsilon }{cos \left(\varphi \right)\ }{sin \left(\theta \right)\ }sin(\epsilon \psi ) \\
X^2=\frac{\rho }{\epsilon }{sin \left(\varphi \right)\ }{sin \left(\theta \right)\ }sin(\epsilon \psi ) \\
X^3=\frac{\rho }{\epsilon }{cos \left(\theta \right)\ }sin(\epsilon \psi ) \\
X^4=\epsilon \ \rho {cos \left(\epsilon \psi \right)\ }
\end{array}
\right.\,,\label{eq10}$$
$${ds}^2={\epsilon }^2{d\rho}^2+{\rho }^2\left[{d\psi }^2+{\epsilon }^2{\mathrm{sin}\mathrm{}}^{\mathrm{2}}(\epsilon \psi )\left({d\theta }^2+{\mathrm{sin}\mathrm{}}^{\mathrm{2}}(\theta ){d\varphi }^2\right)\right]\,,\label{eq11}$$
Now, we define the 3d- metric spaces $S^3_r$ and $H^3_r$ as hyper-surfaces embedded in ${Euc}^4$ and ${Mink}^4$ respectively as: $$X^2={\delta }_{AB}X^AX^B={\left(\epsilon r\right)}^2\,,\label{eq12}$$ where $r$ is a positive real number known as the radius of curvature. Geodesics can be obtained by the intersection of $S^3_r$ (or $H^3_r$) surface with two distinct 3d- hypersurfaces through the centre of the $S^3_r$ (or $H^3_r$): $$\left\{ \begin{array}{l}
{\delta }_{AB}X^AX^B={\left(\epsilon r\right)}^2 \\
a_AX^A=0 \\
b_AX^A=0 \end{array} \right.\,,\label{eq13}$$ Where $a_A$ and $b_A$ are two non-collinear vectors of ${\mathbb{R}}^3\times \epsilon \mathbb{R}$. After dividing Eq. by $cos\left(\epsilon \psi \right)$, the geodesics satisfy: $$\left\{ \begin{array}{l}
a_1{cos \left(\varphi \right)\ }{sin \left(\theta \right)\ }{tan \left(\epsilon \psi \right)\ }+a_2{sin \left(\varphi \right)\ }{sin \left(\theta \right)\ }{tan \left(\epsilon \psi \right)\ }+a_3{cos \left(\theta \right)\ }{tan \left(\epsilon \psi \right)\ }+a_4=0 \\
b_1\ {cos \left(\varphi \right)\ }{sin \left(\theta \right)\ }{tan
\left(\epsilon \psi \right)\ }+b_2{sin \left(\varphi \right)\ }{sin
\left(\theta \right)\ }{tan \left(\epsilon \psi \right)\ }+b_3{cos
\left(\theta \right)\ }{tan \left(\epsilon \psi \right)\ }+b_4=0
\end{array} \right.\,,\label{eq14}$$ where $\psi \neq \frac{\pi }{2}$ is used in the case of the 3-sphere $S^3_r$. Therefore, we can get from the geodesic equations , the coordinates transformation to the CKHcs $\{\vec{x}\}=\{x,y,z\}$ that satisfying the differential equation condition for both spherical and hyperbolic cases:
1. For the spherical case $S^3_r \ (\epsilon =1\Rightarrow R=\frac{6}{r^2})$ , the coordinates transformation to the CKHcs and its inverse read: $$\begin{array}{l}
{\varphi }_{S^3_r}:\tilde{x}(U^{S^3_r}\subset S^3_r)\longrightarrow x(U^{S^3_r}\subset S^3_r) \ \ \ \ \ \ \ \ \ \ \ {\varphi }^{-1}_{S^3_r}:x(U^{S^3_r}\subset S^3_r)\longrightarrow \tilde{x}(U^{S^3_r}\subset S^3_r)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \left(\psi ,\theta ,\varphi
\right)\longmapsto \left(x,y,z\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(x,y,z\right)\longmapsto
\left(\psi ,\theta ,\varphi \right)
\end{array}
\,,\label{eq15}$$ and are defined by $$\left\{ \begin{array}{l}
x=r\ cos\left(\varphi \right)sin\left(\theta \right){tan \left(\psi \right)\ } \\
y=r\ sin\left(\varphi \right)sin\left(\theta \right){tan \left(\psi \right)\ } \\
z=r\ cos\left(\theta \right){tan \left(\psi \right)\ } \end{array}
\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\{
\begin{array}{l}
\psi =arctan\left(\frac{\sqrt{x^2+y^2+z^2}}{r}\right) \\
\theta =arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right) \\
\varphi =arctan\left(\frac{y}{x}\right) \end{array}
\right.\,,\label{eq16}$$ Notice that $U^{S^3_r}\subset
S^3_r$ is the top half 3-sphere divided by the hyper-surface of the equation $\psi =\frac{\pi }{2}$[^3]: $$\tilde{x}(U^{S^3_r})=\{(\psi ,\theta ,\varphi)\mathrel{|\vphantom{(\psi ,\theta ,\varphi) \psi \in [0,\frac{\pi}{2}],\theta \in [0,\pi],\varphi \in [0,2\pi ]}.\kern-\nulldelimiterspace}\psi \in [0,\frac{\pi }{2}],\theta \in [0,\pi],\varphi \in [0,2\pi]\}\,,\label{eq17}$$
2. For the hyperbolic case $S^3_r \ (\epsilon =i\Rightarrow R=\frac{-6}{r^2})$ , the coordinates transformation to the CKHcs and its inverse read: $$\begin{array}{l}
{\varphi }_{H^3_r}:\tilde{x}(U^{H^3_r}\subset H^3_r)\longrightarrow {\left[-r,r\right]}^3 \ \ \ \ \ \ \ \ \ \ \ {\varphi }^{-1}_{H^3_r}:{\left[-r,r\right]}^3\longrightarrow \tilde{x}(U^{H^3_r}\subset H^3_r)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(\psi ,\theta ,\varphi
\right)\longmapsto \left(x,y,z\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \left(x,y,z\right)\longmapsto \left(\psi ,\theta ,\varphi
\right)
\end{array}
\,,\label{eq18}$$ and are defined by $$\left\{ \begin{array}{l}
x=r\ cos\left(\varphi \right)sin\left(\theta \right){tanh \left(\psi \right)\ } \\
y=r\ sin\left(\varphi \right)sin\left(\theta \right){tanh \left(\psi \right)\ } \\
z=r\ cos\left(\theta \right){tanh \left(\psi \right)\ } \end{array}
\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\{
\begin{array}{l}
\psi =arctanh\left(\frac{\sqrt{x^2+y^2+z^2}}{r}\right) \\
\theta \ =arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right) \\
\varphi =arctan\left(\frac{y}{x}\right) \end{array}
\right.\,,\label{eq19}$$ Notice that, in order to get an isomorphism between the two coordinates systems, we have to take the cubic interval ${\left[-r,r\right]}^3$ since ${\mathrm{tanh}
\left(\psi \right)\ }$ is bounded by the interval $\left[-1,1\right]$. Moreover, we have also considered the region $U^{H^3_r}\subset H^3_r$ as the top sheet of the 3d- spherical hyperboloid $H^3_r$.
By using the compact form , one can unify the transformation between the two coordinates charts for both spherical and hyperbolic cases: $$\left\{ \begin{array}{l}
x=\epsilon r\ cos\left(\varphi \right)sin\left(\theta \right){tan \left(\frac{\psi }{\epsilon }\right)\ } \\
y=\epsilon r\ sin\left(\varphi \right)sin\left(\theta \right){tan \left(\frac{\psi }{\epsilon }\right)\ } \\
z=\epsilon r\ cos\left(\theta \right){tan \left(\frac{\psi
}{\epsilon }\right)\ } \end{array} \right.\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \left\{ \begin{array}{l}
\psi =\epsilon \ arctan\left(\frac{\sqrt{x^2+y^2+z^2}}{\epsilon r}\right) \\
\theta =arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right) \\
\varphi =arctan\left(\frac{y}{x}\right) \end{array}
\right.\,,\label{eq20}$$ The metric in the 3-sphere $S^3_r$ and 3-hyperbolic $H^3_r$ spaces is: $${ds}^2=r^2\left[{d\psi }^2+{\epsilon }^2{\mathrm{sin}\mathrm{}}^{\mathrm{2}}(\epsilon \psi )\left({d\theta }^2+{\mathrm{sin}\mathrm{}}^{\mathrm{2}}(\theta ){d\varphi }^2\right)\right]\,,\label{eq21}$$ Using the differential form chain rule, one can write:
& d=dx+dy+dz,&\[eq22\]
& d=dx+dy-dz,&\[eq23\]
& d=dx+dy,&\[eq24\]
Thus, the metric in the CKHcs becomes: $${ds}^2=g_{AB}dx^Adx^B=-{\left(\frac{\sum^3_{A=1}{x^A{dx}^B}}{{\epsilon }^2r^2+{\left|\vec{x}\right|}^2}\right)}^2+\frac{\sum^3_{A=1}{{\left({dx}^A\right)}^2}}{{\epsilon }^2r^2+{\left|\vec{x}\right|}^2}\,,\label{eq25}$$ The components of the metric elements read: $$g_{AB}=\left( \begin{array}{ccc}
\frac{{\epsilon }^2r^2\ \left({\epsilon }^2r^2+y^2+z^2\right)}{{\left({\epsilon }^2r^2+x^2+y^2+z^2\right)}^2} & \frac{-\ {\epsilon }^2\ r^2\ xy}{{\left({\epsilon }^2r^2+x^2+y^2+z^2\right)}^2} & \frac{-\ {\epsilon }^2\ r^2\ xz}{{\left({\epsilon }^2r^2+x^2+y^2+z^2\right)}^2} \\
\frac{-\ {\epsilon }^2\ r^2\ xy}{{\left({\epsilon }^2r^2+x^2+y^2+z^2\right)}^2} & \frac{{\epsilon }^2r^2\ \left({\epsilon }^2r^2+x^2+z^2\right)}{{\left({\epsilon }^2r^2+x^2+y^2+z^2\right)}^2} & \frac{-\ {\epsilon }^2\ r^2\ yz}{{\left({\epsilon }^2r^2+x^2+y^2+z^2\right)}^2} \\
\frac{-\ {\epsilon }^2\ r^2\ xz}{{\left({\epsilon
}^2r^2+x^2+y^2+z^2\right)}^2} & \frac{-\ {\epsilon }^2\ r^2\
yz}{{\left({\epsilon }^2r^2+x^2+y^2+z^2\right)}^2} & \frac{{\epsilon
}^2r^2\ \left({\epsilon }^2r^2+x^2+y^2\right)}{{\left({\epsilon
}^2r^2+x^2+y^2+z^2\right)}^2} \end{array}
\right)\,,\label{eq26}$$ and the Jacobian $J(\vec{x})$
$$J(\vec{x})=\sqrt{\left|{det \left(g(x)\right)\ }\right|}=\frac{r^4\ }{{\left({\epsilon }^2r^2+{\left|\vec{x}\right|}^2\right)}^2}\,,\label{eq27}$$
Finally, we can determine the volume of a geodesic polyhedron $Pol\ $and its boundary face area ${\partial Pol}_f$:
1. For a spherical polyhedron $(R=\frac{6}{r^2})$ $$\int_{{\partial Pol}_f\subset U^{S^3_r}\subset S^3_r}{{dA}^{S^3_r}_f}=\int_{x\left({\partial Pol}_f\right)\subset {\mathbb{R}}^3}{{dA}^{Euc}_f{\sqrt{{|{det (g(x)|_{{\partial Pol}_f}^{S^3_r})\ }|}}}}\,,\label{eq28}$$ $$\int_{Pol\subset U^{S^3_r}\subset S^3_r}{{dV}^{S^3_r}}=\int_{x\left(Pol\right)\subset {\mathbb{R}}^3}{{dV}^{Euc}\frac{r^4\ }{{\left(r^2+{\left|\vec{x}\right|}^2\right)}^2}\ }\,,\label{eq29}$$
2. For a hyperbolic polyhedron $(R=\frac{-6}{r^2})$ $$\int_{{\partial Pol}_f\subset U^{H^3_r}\subset H^3_r}{{dA}^{H^3}}=\int_{x\left({\partial Pol}_f\right)\subset {\mathbb{R}}^3}{{dA}^{Euc}_f{\sqrt{{|{det (g(x)|_{{\partial Pol}_f}^{H^3_r})\ }|}}}}\,,\label{eq30}$$ $$\int_{Pol\subset U^{H^3_r}\subset H^3_r}{{dV}^{H^3_r}}=\int_{x\left(Pol\right)\subset {\mathbb{R}}^3}{{dV}^{Euc}\frac{r^4\ }{{\left(-r^2+{\left|\vec{x}\right|}^2\right)}^2}\ }\,,\label{eq31}$$
The induced Jacobian $\sqrt{{|{det (g(x)|_{{\partial
Pol}_f}^{S^3_r})\ }|}}$ and $\sqrt{{|{det (g(x)|_{{\partial
Pol}_f}^{H^3_r})\ }|}}$ for both spherical and hyperbolic respectively can be determined after restricting the metric in the boundary surface area ${{\partial Pol}_f}$.
Application: Regular tetrahedron in a constant curvature space {#Sec4}
==============================================================
Let $T(a)$ be a regular geodesic tetrahedron with an edge length $a$ embedded in a constant curvature 3d- space $\mathrm{\Sigma }$, and ${\left\{{\vec{A}}_f\right\}}_{f=\overline{1.4}}$ be normal areas vectors of $T(a)$. In what follows, we will calculate the volume of a geodesic regular tetrahedron $T(a)$ and its boundary face area ${\partial T(a)}_f$ in 3d- sphere $S^3_r$ and Hyperbolic $H^3_r$ manifolds: $$A^{\mathit{\Sigma}}_f(r,a)=\int_{x\left({\partial T(a)}_f\right)\subset {\mathbb{R}}^3}{{dA}^{Euc}_f} \sqrt{\left|det(g(x)|_{{\partial T(a)}_f})\right|} \,,\label{eq32}$$ $$V^{\mathit{\Sigma}}\left(r,a\right)=\int_{x\left(T(a)\right)\subset {\mathbb{R}}^3}{{dV}^{Euc}\frac{r^4\ }{{\left({\epsilon }^2r^2+{\left|\vec{x}\right|}^2\right)}^2}}\,,\label{eq33}$$
![A regular tetrahedron $T(a_0)$ in $\mathbb{R}^{3}$ (CKHcs).\[fig2\]](22){width="2.5in"}
The ignorance of how this new coordinates system CKHcs can map an Euclidean length to spherical and hyperbolic length measures, one has to be careful in choosing the location of the tetrahedron $T(a)$. From our choice in Fig. \[fig2\], it obvious to see that the image of a regular geodesic tetrahedron $T(a)$ of an edge length $a$ in the manifold is an Euclidean regular tetrahedron $T(a_0)$ of a different edge length $a_0$ in the CKHcs: $$x\left(T(a)\right)=T(a_0)\,,\label{eq34}$$
Our objective is to have an expression for the starting Euclidean length $a_0$ in terms of the geodesic length $a$. In order to determine how this coordinates system measure the length different from the original one, we have to consider two points $M_1\left(x_1,y_1,z_1\right)$ and $M_2\left(x_2,y_2,z_2\right)$ in the CKHcs where the corresponding geodesic line between them is parameterized by: $$\left\{ \begin{array}{l}
y=\alpha x+\beta \\
z=\gamma x+\delta \end{array} \right.\\,\label{eq35}$$ where $$\alpha =\frac{y_2-y_1}{x_2-x_1}\ \ \ \ \beta =\frac{x_2y_1-x_1y_2}{x_2-x_1}\,,\label{eq36}$$ $$\gamma =\frac{z_2-z_1}{x_2-x_1}\ \ \ \ \delta =\frac{x_2z_1-x_1z_2}{x_2-x_1}\,,\label{eq37}$$
The geodesic length between $M_1$ and $M_2$ is: $$d\left(M_1M_2\right)=\epsilon \ r\ {\left.arctan\left(\frac{\left({\alpha }^2+{\gamma }^2+1\right)x+\alpha \beta +\gamma \delta }{\sqrt{{\epsilon }^2r^2+{\beta }^2+{\delta }^2+\left({\alpha }^2+{\gamma }^2\right){\epsilon }^2r^2+{\alpha }^2{\delta }^2+{\gamma }^2{\beta }^2-2\alpha \beta \gamma \delta }}\right)\right|}^{x_2}_{x_1}\,,\label{eq38}$$ Since $d\left(M_1M_2\right)$ depends strongly on the ending points, a special care has to be done in the location of the Euclidean regular tetrahedron in the CKHcs as it is shown in Fig. \[fig2\]. One can check that: $$a=2\epsilon r\ arctan\left(\frac{1}{2}\frac{a_0}{\sqrt{{\epsilon }^2r^2+\frac{a^2_0}{8}}}\right)\,,\label{eq39}$$ In order to obtain a geodesic edge length $a$, one has to solve Eq. for the unknown $a_0$ and get: $$a_0=\frac{2\ \epsilon \ r\ tan\left(\frac{a}{2\epsilon r}\right)}{\sqrt{\left(1-\frac{1}{2}{tan}^2\left(\frac{a}{2\epsilon r}\right)\right)}}\,,\label{eq40}$$
1. For the spherical case $S^3_r \ (\epsilon =1\Rightarrow R=\frac{6}{r^2})$ , one has: $$a=2r\ arctan\left(\frac{1}{2}\frac{a_0}{\sqrt{r^2+\frac{a^2_0}{8}}}\right)\,,\label{eq41}$$ In this case, one can check that the regular tetrahedron has a maximal edge $a_{max}$ (for $a_0\to \infty $) given by: $$a_{max}=2\ arctan\left(\sqrt{2}\right)\ r\,,\label{eq42}$$
2. For the hyperbolic case $S^3_r \ (\epsilon =i\Rightarrow R=\frac{-6}{r^2})$ , one has: $$a=2r\ arctanh\left(\frac{1}{2}\frac{a_0}{\sqrt{r^2-\frac{a^2_0}{8}}}\right)\,,\label{eq43}$$ Due to the compactness property (see Eq. ) of the coordinates chart, the initial value of the Euclidean length $a_0$ must be bounded $a_0<\frac{2}{3}\sqrt{6}\ r$ . However, $a$ has no upper bound.
Boundary area of a regular tetrahedron in $S^3_r$ and $H^3_r$
-------------------------------------------------------------
The faces area of a geodesic regular tetrahedron of an edge length $a$ are all equal $\left(A^{\mathrm{\Sigma
}}_f\left(r,a\right)=A^{\mathrm{\Sigma }}\left(r,a\right)\ , \
\forall f=\overline{1.4}\right)$ . In fact, the geodesic surface of the $S^3_r$ and $H^3_r$ are portions of the great 2-dimensional spheres $S^2_r$ and hyperbolic $H^2_r$ respectively. Accordingly, we expect to obtain the same area expression of the spherical and hyperbolic trigonometry. Due to the symmetric property of the constant curvature spaces, we restrict ourselves to geodesic triangle face ${\partial T(a)}_f \equiv P_{1}P_{2}P_{3}$ (See Fig. \[fig2\]) in the geodesic surface $z=\frac{-a_{0}}{4}\sqrt{\frac{2}{3}}$ (with $dz=0$). Then the induced Jacobian: $$\sqrt{|det(g(x)|_{P_{1}P_{2}P_{3}})|}=\frac{{\epsilon}^2 r^2 \sqrt{{\epsilon}^2 r^2+\frac{{a_0}^2}{24}}}{\left({\epsilon}^2r^2+x^2+y^2+\frac{{a_0}^2}{24}\right)^{3/2}},$$ The boundary face area is: $$A^{\mathrm{\Sigma }}\left(r,a\right)=\int_{P_{1}P_{2}P_{3}\subset {\mathbb{R}}^3}{{dA}^{Euc}_f}\frac{{\epsilon}^2 r^2 \sqrt{{\epsilon}^2r^2+\frac{{a_0}^2}{24}}}{\left({\epsilon }^2r^2+x^2+y^2+\frac{{a_0}^2}{24}\right)^{3/2}}\,,\label{eq44}$$ with $${dA}^{Euc}_f=\frac{1}{2}\sum^3_{i,j,k=1}{{\epsilon }_{ijk}A^i_fdx^j\wedge dx^k}\,,\label{eq45}$$ where $A^i_f$ is the $i^{th}$ component of the normal area vector ${\vec{A}}_f$. The integral in Eq. is in general very hard to evaluate. To do so, one has to make a series expansion of the Jacobian $J(\vec{x})$ given in with respect to the coordinates variables $\left\{\vec{x}\right\}\ $ and then easily perform the integration over one of the faces $P_{1}P_{2}P_{3}$, we get the following expression: $$\begin{gathered}
A^{\mathrm{\Sigma }}(r,a)=\frac{\sqrt{3}}{4}a^2\{1+\frac{1}{8}{(\frac{a}{\epsilon r})}^2+\frac{1}{60}{(\frac{a}{\epsilon r})}^4+\frac{583}{241920}{(\frac{a}{\epsilon r})}^6\\+\frac{227}{604800}{(\frac{a}{\epsilon r})}^8+\frac{23}{369600}{(\frac{a}{\epsilon r})}^{10}+\frac{1418693}{130767436800}{(\frac{a}{\epsilon r})}^{12}+\mathcal{O}({(\frac{a}{\epsilon r})}^{14})\}\,,\label{eq46}\end{gathered}$$ Using the symmetry of the triangle faces of a regular tetrahedron, the exact formula of the boundary face area reads: $$A^{\mathrm{\Sigma }}\left(r,a\left(a_0\right)\right)=2\int^{\frac{a_0}{2}}_0{dx\int^{-\sqrt{3}x+\frac{\sqrt{3}a_0}{3}}_{\frac{-\sqrt{3}a_0}{6}}{dy}}\ \frac{{\epsilon}^2 r^2 \sqrt{{\epsilon }^2 r^2+\frac{{a_0}^2}{24}}}{\left({\epsilon}^2r^2+x^2+y^2+\frac{{a_0}^2}{24}\right)^{3/2}}\,,\label{eq47}$$ Straightforward but tedious calculations (See Appendix \[AppB\]) give the following analytical expression of the boundary face area $A^{\mathrm{\Sigma }}\left(r,a\right)$ of a regular spherical and hyperbolic tetrahedron with an edge length $a$ in the curved space $\mathrm{\Sigma }$ of a constant curvature $R=\frac{6}{{\epsilon
}^2r^2}$: $$A^{\mathrm{\Sigma }}\left(r,a\right)={\epsilon}^2 r^2\left(3\arccos\left(\frac{\cos(\frac{a}{\epsilon r})}{\cos(\frac{a}{\epsilon r})+1}\right)-\pi\right)\,,\label{eq48}$$ It is easy to check that the expansion of the resulted formula in terms of the $\frac{a}{\epsilon r}$ variable is exactly the one in Eq. and thus ensuring the correctness of the integration.
1. For the spherical case $S^3_r \ (\epsilon =1\Rightarrow
R=\frac{6}{r^2})$ , one has: $$A^{S^3_r}\left(r,a\right)=r^2\left(3\arccos\left(\frac{\cos(\frac{a}{r})}{\cos(\frac{a}{r})+1}\right)-\pi\right)\,,\label{eq49}$$ As it is expected, it is the familiar expression of the regular spherical triangle embedded in the 2-sphere $S^2_r$ where the dihedral angle is defined by $\Theta=\arccos\left(\frac{\cos(\frac{a}{r})}{\cos(\frac{a}{r})+1}\right)$ which is the cosine rule formula for spherical trigonometry. We can check that the boundary area $A^{S^3_r}$ for the maximal edge length $a_{max}$ in Eq. corresponds to an upper bound $A^{S^3_r}_{max}=\pi r^2$. The boundary area of a regular spherical tetrahedron is always greater than the boundary area of a regular Euclidean one.
2. For the hyperbolic case $S^3_r \ (\epsilon =i\Rightarrow
R=\frac{-6}{r^2})$ , one has: $$A^{H^3_r}\left(r,a\right)=r^2\left(\pi-3\arccos\left(\frac{\cosh(\frac{a}{r})}{\cosh(\frac{a}{r})+1}\right)\right)\,,\label{eq50}$$ As it is expected, it is the familiar expression of the regular hyperbolic triangle embedded in the 2-hyperbolic $H^2_r$ where the dihedral angle is defined by $\Theta=\arccos\left(\frac{\cosh(\frac{a}{r})}{\cosh(\frac{a}{r})+1}\right)$ which is the cosine rule formula for hyperbolic trigonometry. Notice that in this case, there is no upper bound and for a given pair $(r,a)$. The boundary area of a regular hyperbolic tetrahedron is always smaller than the boundary area of a regular Euclidean one.
3. For the Euclidean case $Euc^3 \ (R=0)$ , one has: $$A^{{Euc}^3}\left(r,a\right)={\mathop{lim}_{r\to \infty } A^{\mathit{\Sigma}}\left(r,a\right)\ }=\frac{\sqrt{3}}{4}a^2\,,\label{eq51}$$ The Euclidean limit is well-defined.
![: Function surface of the boundary face area for spherical (green), Euclidean (blue) and hyperbolic (red) regular tetrahedra.\[fig3\]](33){width="2.5in"}
Volume of a regular tetrahedron in $S^3_r$ and $H^3_r$
------------------------------------------------------
The volume $V^{\mathrm{\Sigma }}$ of a regular spherical and hyperbolic tetrahedron is: $$V^{\mathrm{\Sigma }}\left(r,a\left(a_0\right)\right)=\int_{T\left(a_0\right)\subset {\mathbb{R}}^3}{{dV}^{Euc}\frac{r^4\ }{{\left({\epsilon }^2r^2+{\left|\vec{x}\right|}^2\right)}^2}}\,,\label{eq52}$$ Since the integration is very hard to deal with, it is better to make again a series expansion of the Jacobian $J(\vec{x})$ given in in terms of the coordinates variables $\left\{\vec{x}\right\}$ and then easily perform the integration to end up with: $$\begin{gathered}
V^{\mathrm{\Sigma }}(r,a)=\frac{\sqrt{2}}{12}a^3\{1+\frac{23}{80}{(\frac{a}{\epsilon r})}^2+\frac{3727}{53760}{(\frac{a}{\epsilon r})}^4+\frac{124627}{7741440}{(\frac{a}{\epsilon r})}^6 \\ +\frac{20283401}{5449973760}{(\frac{a}{\epsilon r})}^8+\frac{14700653069}{17003918131200}{(\frac{a}{\epsilon r})}^{10}+\frac{1651049434189}{8161880702976000}{(\frac{a}{\epsilon r})}^{12}+\mathcal{O}({\frac{a}{r})}^{14})\}\,,\label{eq53}\end{gathered}$$ Using the symmetry of the regular tetrahedron, the exact expression of the volume of a regular spherical and hyperbolic tetrahedron is: $$V^{\mathrm{\Sigma }}\left(r,a\left(a_0\right)\right)=2 \int^{\frac{\sqrt{6}a_0}{4}}_{\frac{-\sqrt{6}a_0}{12}}{dz\ \int^{\frac{\alpha(z)}{2}}_{0}{dx\ \int^{-\sqrt{3}x+\frac{\sqrt{3}\alpha(z)}{3}}_{\frac{-\sqrt{3}\alpha(z)}{6}}{dy\ \frac{r^4}{{\left({\epsilon }^2r^2+{\left|\vec{x}\right|}^2\right)}^2}}}}\,,\label{eq54}$$ where $$\alpha(z)=\frac{-\sqrt{6}}{2}z+\frac{3a_0}{4}$$ Which can be rewritten in the following integral form (See Appendix \[AppC\]) as: $$V^{\mathrm{\Sigma }}\left(r,a\right)=12{\epsilon }^3\ r^3\int^{tan\left(\frac{a}{2\epsilon r}\right)}_0{dt\ \frac{\ t\ arctan\left(t\right)}{\left(3-t^2\right)\sqrt{2-t^2}}}\,,\label{eq55}$$ Notice that this integral has no analytic formula (we can carry the integration by using numerical methods) and can be expressed in terms of some special functions like the dilogarithm ${Li}_2(z)$, the Clausen of order 2 ${Cl}_2\left(\varphi \right)$ or the digamma $\mathrm{\Psi }\left(x\right)$. It is easy to check that the expansion of the resulted formula in terms of the $\frac{a}{\epsilon r}$ variable is exactly the one in Eq. and thus ensuring the correctness of the integration.
1. For the spherical case $S^3_r \ (\epsilon =1\Rightarrow
R=\frac{6}{r^2})$ , one has: $$V^{S^3_r}\left(r,a\right)=12\ r^3\int^{tan\left(\frac{a}{2r}\right)}_0{dt\ \frac{\ t\ arctan\left(t\right)}{\left(3-t^2\right)\sqrt{2-t^2}}}\,,\label{eq56}$$ The volume for a maximal edge length $V^{S^3_r}\left(r,a_{max}\right)$ (as it is expected) ** is half of the 3-dimensional cubic hyperarea of 3-sphere of radius $r$ :$\ $ $$V^{S^3_r}\left(r,a_{max}\right)={\pi }^2r^3=\frac{1}{2}Area\left(S^3_r\subset {\mathbb{R}}^4\right)\,,\label{eq57}$$ Notice that for a given pair $(r,a)$ the volume of a regular spherical tetrahedron is always greater than the regular Euclidean one.
2. For the hyperbolic case $S^3_r \ (\epsilon =i\Rightarrow
R=\frac{-6}{r^2})$ , one has: $$V^{H^3_r}\left(r,a\right)=12\ r^3\int^{tanh\left(\frac{a}{2r}\right)}_0{dt\ \frac{\ t\ arctanh\left(t\right)}{\left(3+t^2\right)\sqrt{2+t^2}}}\,,\label{eq58}$$ has an upper bound : $${\mathop{\mathrm{lim}}_{a\to \infty } V^{H^3_r}\left(r,a\right)\ }=1.0149416064096536250\ r^3\,,\label{eq59}$$ $$\ =Im\left[{Li}_2\left(e^{i\frac{\pi }{3}}\right)\right]r^3=\frac{\sqrt{6}}{3}\left({\mathit{\Psi}}^1\left(\frac{1}{3}\right)-\frac{2}{3}{\pi }^2\right)r^3\ ={Cl}_2\left(\frac{\pi }{3}\right)r^3\ \,,\label{eq60}$$ Notice that for a given pair $(r,a)$ the volume of a regular hyperbolic tetrahedron is always smaller than the regular Euclidean one.
3. For the Euclidean case $Euc^3 \ (R=0)$ , one has: $$V^{{Euc}^3}\left(r,a\right)={\mathop{lim}_{r\to \infty } V^{\mathit{\Sigma}}\left(r,a\right)\ }=\frac{\sqrt{2}}{12}a^3\,,\label{eq61}$$ The Euclidean limit is well-defined.
![Function surface of regular tetrahedron volume for spherical (green), Euclidean (blue) and hyperbolic (red) cases.\[fig4\]](44){width="2.5in"}
The volume-area ratio function
------------------------------
We define the volume-area ratio function ${VRA}^{\mathrm{\Sigma }}$ for a regular geodesic tetrahedron as: $${VRA}^{\mathit{\Sigma}}(r,a)=\frac{V^{\mathit{\Sigma}}\left(r,a\right)}{{\left(A^{\mathit{\Sigma}}\left(r,a\right)\right)}^{\frac{3}{2}}}\,,\label{eq62}$$ It is obvious that the ${VRA}^{\mathrm{\Sigma }}$ for a regular Euclidean tetrahedron is a constant: $${VRA}^{{Euc}^3}={\mathop{lim}_{r\to \infty } VRA(r,a)\ }=\frac{\sqrt{2}}{12{\left(\frac{\sqrt{3}}{4}\right)}^{\frac{3}{2}}}=0.4136\,,\label{eq63}$$
according to the useful inequality $${VRA}^{H^3_r}(r,a)\le {VRA}^{{Euc}^3}(r,a)\le {VRA}^{S^3_r}(r,a)\,,\label{eq64}$$ the ${VRA}^{\mathrm{\Sigma }}$ function allows us to know what kind of geometry inside the regular geodesic tetrahedron: (see Fig. \[fig5\]) $$\left\{ \begin{array}{l}
{VRA}^{\mathit{\Sigma}}(r,a)> 0.4136\ \ \ \ \ \ \ \ \ \ S^3_r\\
{VRA}^{\mathit{\Sigma}}(r,a)= 0.4136\ \ \ \ \ \ \ \ \ \ Euc^3\\
{VRA}^{\mathit{\Sigma}}\left(r,a\right)< 0.4136\ \ \ \ \ \ \ \ \ \ H^3_r\end{array}
\right.\,,\label{eq65}$$
![The volume-area ratio function for spherical (green), Euclidean (blue) and hyperbolic (red) cases.\[fig5\]](55){width="2.5in"}
The volume function in terms of scalar curvature and area
---------------------------------------------------------
From the area formula , one can express the edge length $a$ by: $$a(A,R)=\left(\pi-\arccos\left(\frac{\sin(\frac{-\pi}{6}+\frac{A}{3{\epsilon}^2{r}^2})}{\sin(\frac{-\pi}{6}+\frac{A}{3{\epsilon}^2{r}^2})+1}\right)\right)\epsilon r\,,\label{eq66}$$ substitute it in Eq. to get a volume function in terms of the 3d- Ricci scalar curvature and boundary face area of a regular tetrahedron: $$V^{\mathrm{\Sigma }}=V^{\mathrm{\Sigma }}\left(R,a\left(R,A\right)\right)=V^{\mathrm{\Sigma }}\left(R,A\right)\,,\label{eq68}$$
the volume of a regular geodesic tetrahedron for a fixed boundary area satisfies the following inequality $$For \ any \ \ \ R_1,R_2\in \mathbb{R} \ \ \ if \ \ \ R_1<R_2 \ \ \ then \ \ \
V^{\mathrm{\Sigma }}(R_1,A)<V^{\mathrm{\Sigma
}}(R_2,A)\,,\label{eq69}$$
this results from the fact that the function $V^{\mathrm{\Sigma }}$ increases with respect to $R$ for a fixed area norm $A$ (see Fig. \[fig6\]).
![The volume function in terms of scalar curvature $R$ and area $A$ for spherical (right) and hyperbolic (left) regular tetrahedron.\[fig6\]](66){width="5in"}
Conclusion {#Sec5}
==========
In this paper, we explicitly derived the boundary face area and volume of a regular spherical and hyperbolic tetrahedron in terms of the curvature radius (or the scalar curvature) and the edge length. We have directly performed the integration over the area and volume elements by using the Cayley-Klein-Hilbert coordinates system (CKHcs) to end up with simple formula given in Eqs. (\[eq48\],\[eq55\]). A comparison between the Euclidean, spherical and hyperbolic cases is studied and their implications are discussed. It is shown that the volume function of a regular geodesic tetrahedron for a fixed boundary face area is a strictly increasing in the scalar curvature interval.
Proof of the relation {#AppA}
======================
The geodesics in the CKHcs $\{U\subset M,\vec{x}\}$ are straight lines, one has: $$\ddot{x}^A=0\,,\label{eqA1}$$ The condition $$\Gamma^{A}_{BC}(x)
\dot{x}^B\dot{x}^C=0\,,\label{eqA2}$$ must be hold, which implies: $$\Gamma^{A}_{BC}(x) \frac{{\partial \varphi}^B(\tilde{x})}{{\partial\tilde{x}}^I}\frac{{\partial \varphi}^C(\tilde{x})}{{\partial\tilde{x}}^J}{\dot{\tilde{x}}}^I{\dot{\tilde{x}}}^J=0\,,\label{eqA3}$$ Under the transformation , the Christoffel symbols transform as: $$\Gamma^{A}_{BC}(x)=\frac{{\partial \tilde{x}}^J}{{\partial x}^B}\frac{{\partial \tilde{x}}^K}{{\partial x}^C}\frac{{\partial \varphi}^A}{{\partial
\tilde{x}}^I}{\tilde{\Gamma}}^{I}_{JK}(\tilde{x})-\frac{{\partial
\tilde{x}}^J}{{\partial x}^B}\frac{{\partial \tilde{x}}^K}{{\partial
x}^C}\frac{{\partial^2 \varphi}^A}{{\partial \tilde{x}}^J{\partial
\tilde{x}}^K}\,,\label{eqA4}$$ By substituting it in Eq. , one can obtain the transformation condition Eq. to the ideal CKHcs frame.
Proof of the area formula {#AppB}
=========================
The boundary face area ($P_{1}P_{2}P_{3}$) of a regular spherical and hyperbolic tetrahedron of an edge length $a$ is given by an integral form in Eq. . For simplicity, we drop the triangle face $P_{1}P_{2}P_{3}$ to $\Pi(P_{1}P_{2}P_{3})$ in the $XY$-plane (since the area of a fixed triangle is the same wherever its location inside the constant curvature manifold). In this case, the induced Jacobian can be written as: $$\sqrt{|det(g(x)|_{\Pi(P_{1}P_{2}P_{3})})|}=\frac{{\epsilon}^3 r^3}{({\epsilon}^2r^2+x^2+y^2)^{3/2}},\label{B1}$$ The boundary face area is given by: $$A^{\mathrm{\Sigma }}\left(r,a\right)=2\int^{\frac{a_0}{2}}_0{dx\int^{-\sqrt{3}x+\frac{\sqrt{3}a_0}{3}}_{\frac{-\sqrt{3}a_0}{6}}{dy}}\ \frac{{\epsilon}^3 r^3}{\left({\epsilon }^2r^2+x^2+y^2\right)^{3/2}},\label{B2}$$ where one can check the starting Euclidean length $a_0$ in this case is given by: $$a_0=2\epsilon r \frac{\tan(\frac{a}{2\epsilon r})}{\sqrt{1-\frac{1}{3}\tan(\frac{a}{2\epsilon r})^2}},\label{B3}$$ Performing the Integral over y variable, one get: $$\begin{gathered}
\int^{-\sqrt{3}x+\frac{\sqrt{3}a_0}{3}}_{\frac{-\sqrt{3}a_0}{6}}{dy}\ \frac{{\epsilon}^3 r^3}{\left({\epsilon }^2r^2+x^2+y^2\right)^{3/2}}=\\\frac{{{\epsilon}^3}r^3(-\sqrt{3}x+\frac{\sqrt{3}a_{0}}{3})}{({{\epsilon}^2}r^2+x^2)\sqrt{{{\epsilon}^2}r^2+x^2+(-\sqrt{3}x+\frac{\sqrt{3}a_{0}}{3})^2}}+\frac{{{\epsilon}^3}r^3\frac{\sqrt{3}a_0}{6}}{({{\epsilon}^2}r^2+x^2)\sqrt{{{\epsilon}^2}r^2+x^2+\frac{a_{0}^{2}}{12}}},\label{B4}\end{gathered}$$ Let us preform the second integral over the x variable. By integrating each term separately, one has: $$\begin{gathered}
t_1(x)=\int^{\frac{a_0}{2}}_0{dx \frac{{{\epsilon}^3}r^3(-\sqrt{3}x+\frac{\sqrt{3}a_{0}}{3})}{({{\epsilon}^2}r^2+x^2)\sqrt{{{\epsilon}^2}r^2+x^2+(-\sqrt{3}x+\frac{\sqrt{3}a_{0}}{3})^2}}}={\epsilon}^2r^2\arctan(\frac{F(a_0,r;x)}{G(a_0,r;x)}),\label{B5}\end{gathered}$$ where $$\begin{gathered}
F(a_0,r;x)=-\frac{\sqrt{3}}{3}\sqrt{{\epsilon}^2r^2+4x^2-2xa_0+\frac{a_{0}^{2}}{3}}({\epsilon}^2r^2+\frac{a_{0}^{2}}{9})(-{\epsilon}^2r^2+\frac{a_{0}x}{3})\\-a_{0}(\frac{5{\epsilon}^2r^2}{9}+\frac{a_{0}^{2}}{27})({\epsilon}^2r^2+x^2)+\frac{a_{0}^{4}x}{81}+r^4x+\frac{2a_{0}^{2}{\epsilon}^2r^2x}{9},\label{B6}\end{gathered}$$ and $$\begin{gathered}
G(a_0,r;x)=\frac{\epsilon
r\sqrt{3}}{3}\sqrt{{\epsilon}^2r^2+4x^2-2xa_0+\frac{a_{0}^{2}}{3}}({\epsilon}^2r^2+\frac{a_{0}^{2}}{9})(x+\frac{a_{0}}{3})\\+\frac{2a_{0}^{2}\epsilon
rx^2}{27}+\frac{{\epsilon}^5r^5}{3}-\frac{4a_{0}^{2}{\epsilon}^3r^3}{27}-\frac{a_{0}^{4}{\epsilon}r}{81}+\frac{4{\epsilon}^3r^3x^2}{3},\label{B7}\end{gathered}$$ $$t_2(x)=\int^{\frac{a_0}{2}}_0{dx \frac{{{\epsilon}^3}r^3\frac{\sqrt{3}a_0}{6}}{({{\epsilon}^2}r^2+x^2)\sqrt{{{\epsilon}^2}r^2+x^2+\frac{a_{0}^{2}}{12}}}}={\epsilon}^2r^2\arctan\left(\frac{a_{0}x}{\epsilon r\sqrt{12{{\epsilon}^2}r^2+12x^2+a_{0}^{2}}}\right),\label{B8}$$ Adding the two terms together, we obtain: $$\begin{gathered}
A^{\mathrm{\Sigma }}\left(r,a\right)=2(t_{1}(x)+t_{2}(x))|^{x=a_0/2}_{x=0}=\\2{\epsilon}^2r^2\arctan(\frac{9a_{0}^{2}{\epsilon}r(3a_{0}^{2}-\sqrt{3}a_{0}\sqrt{9{\epsilon}^2r^2+3a_{0}^{2}}+18{\epsilon}^2r^2)}{3a_{0}^{5}-63a_{0}^{3}{\epsilon}^2r^2-216a_{0}r^4+\sqrt{3}\sqrt{9{\epsilon}^2r^2+3a_{0}^{2}}(18a_{0}^{2}{\epsilon}^2r^2+144r^4-a_{0}^{4})})\label{B9}\end{gathered}$$ When we replace $a_0$ given in Eq. , we get the area function formula of Eq. .
Proof of the volume formula {#AppC}
===========================
The volume of a regular spherical and hyperbolic tetrahedron of an edge length $a$ is given by an integral form in Eq. . Using the integration by shell method (taking the sum of parallel triangles of constant $z$). Performing the Integral over the $y$ variable, one get: $$\begin{gathered}
\int^{-\sqrt{3}x+\frac{\sqrt{3}\alpha(z)}{3}}_{\frac{-\sqrt{3}\alpha(z)}{6}}{dy}\frac{r^4\
}{{\left({\epsilon
}^2r^2+x^2+y^2+\frac{{\alpha(z)}^2}{24}\right)}^2}=\\\frac{32\sqrt{3}\
r^4\ \left(-3x+\alpha(z)\right)}{\left(32x^2-16\alpha(z)x+8{\epsilon
}^2r^2+3{\alpha(z)}^2\right)\left(24x^2+24{\epsilon
}^2r^2+{\alpha(z)}^2\right)}+\frac{24\sqrt{6}\ r^4\
arctan\left(\frac{2\sqrt{2}\left(-3x+\alpha(z)\right)}{\sqrt{24x^2+24{\epsilon
}^2r^2+{\alpha(z)}^2}}\right)}{\left(24x^2+24{\epsilon
}^2r^2+{\alpha(z)}^2\right)}\\+\frac{48\sqrt{3}\ r^4\alpha(z)\
}{\left(24x^2+24{\epsilon
}^2r^2+3{\alpha(z)}^2\right)\left(24x^2+24{\epsilon
}^2r^2+{\alpha(z)}^2\right)}+\frac{24\sqrt{6}\ r^4\
arctan\left(\frac{\sqrt{2}\alpha(z)}{\sqrt{24x^2+24{\epsilon
}^2r^2+{\alpha(z)}^2}}\right)}{{\left(24x^2+24{\epsilon
}^2r^2+{\alpha(z)}^2\right)}^{\frac{3}{2}}}\,,\label{eqB1}\end{gathered}$$ Now, let us focus on the second integral over the $x$ variable. By integrating each term separately, one has: $$\begin{gathered}
T_1\ (x)=\int{dx\frac{32\sqrt{3}\ r^4\ \left(-3x+\alpha(z)\right)}{\left(32x^2-16\alpha(z)x+8{\epsilon }^2r^2+3{\alpha(z)}^2\right)\left(24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2\right)}}=\\ \frac{-6\sqrt{3}\ r^4\ ln\left(32x^2-16\alpha(z)x+8{\epsilon }^2r^2+3{\alpha(z)}^2\right)}{72{\epsilon }^2r^2+11{\alpha(z)}^2}-\frac{8\sqrt{3}\ r^4\alpha(z)arctan\left(\frac{8x-2\alpha(z)}{\sqrt{16{\epsilon }^2r^2+2{\alpha(z)}^2}}\right)}{\left(72{\epsilon }^2r^2+11{\alpha(z)}^2\right)\sqrt{16{\epsilon }^2r^2+2{\alpha(z)}^2}}\\+\frac{6\sqrt{3}\ r^4\ r^4\ ln\left(24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2\right)}{72{\epsilon }^2r^2+11{\alpha(z)}^2}+\frac{48\sqrt{3}\ r^4\alpha(z)arctan\left(\frac{12x}{\sqrt{144{\epsilon }^2r^2+6{\alpha(z)}^2}}\right)}{\left(72{\epsilon }^2r^2+11{\alpha(z)}^2\right)\sqrt{144{\epsilon }^2r^2+6{\alpha(z)}^2}}\,,\label{eqB2}\end{gathered}$$ $$\begin{gathered}
T_2\ (x)=\int{dx\frac{24\sqrt{6}\ r^4\ arctan\left(\frac{2\sqrt{2}\left(-3x+\alpha(z)\right)}{\sqrt{24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\left(24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2\right)}}=\\ \frac{48\sqrt{6}\ r^4\sqrt{8{\epsilon }^2r^2+{\alpha(z)}^2}\mathrm{\ arctan}\mathrm{}\left(\frac{\sqrt{2}\left(4x-\alpha(z)\right)}{\sqrt{8{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\left(24{\epsilon }^2r^2+{\alpha(z)}^2\right)\left(72{\epsilon }^2r^2+11{\alpha(z)}^2\right)}-\frac{6\sqrt{3}\ r^4\ ln\left(24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2\right)}{\left(72{\epsilon }^2r^2+11{\alpha(z)}^2\right)}\\ +\frac{6\sqrt{3}\ r^4\ ln\left(96x^2-48\alpha(z)x+24{\epsilon }^2r^2+9{\alpha(z)}^2\right)}{\left(72{\epsilon }^2r^2+11{\alpha(z)}^2\right)}-\frac{24\sqrt{2}\ arctan\left(\frac{\sqrt{24}\ x^2}{\sqrt{24{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\left(72{\epsilon }^2r^2+11{\alpha(z)}^2\right)\sqrt{24{\epsilon }^2r^2+{\alpha(z)}^2}}\\ +\frac{24\sqrt{6}\ r^4\ x\ arctan\left(\frac{2\sqrt{2}\left(-3x+\alpha(z)\right)}{\sqrt{24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\left(24{\epsilon }^2r^2+{\alpha(z)}^2\right)\sqrt{24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2}}\,,\label{eqB3}\end{gathered}$$ $$\begin{gathered}
T_3\ (x)=\int{dx\ \frac{48\sqrt{3}\ r^4\alpha(z)\ }{\left(24x^2+24{\epsilon }^2r^2+3{\alpha(z)}^2\right)\left(24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2\right)}}=\\ \frac{6\sqrt{2}\ r^4\ arctan\left(\frac{2\sqrt{6}\ x}{\sqrt{24{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\alpha(z)\sqrt{24{\epsilon }^2r^2+{\alpha(z)}^2}}-\frac{2\sqrt{6}\ r^4\ arctan\left(\frac{2\sqrt{2}\ x}{\sqrt{8{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\alpha(z)\sqrt{8{\epsilon }^2r^2+{\alpha(z)}^2}}\,,\label{eqB4}\end{gathered}$$ $$\begin{gathered}
T_4\ (x)=\int{dx\ \frac{24\sqrt{6}\ r^4\ arctan\left(\frac{\sqrt{2}\alpha(z)}{\sqrt{24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{{\left(24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2\right)}^{\frac{3}{2}}}}=\\ \frac{\ 6\sqrt{6}\ r^4\sqrt{8{\epsilon }^2r^2+{\alpha(z)}^2}\ arctan\left(\frac{2\sqrt{2}\ x}{\sqrt{8{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\left(24{\epsilon }^2r^2+{\alpha(z)}^2\right)}+\frac{24\sqrt{6}\ r^4x\ arctan\left(\frac{\sqrt{2}\alpha(z)}{\sqrt{24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\left(24{\epsilon }^2r^2+{\alpha(z)}^2\right)\sqrt{24x^2+24{\epsilon }^2r^2+{\alpha(z)}^2}}\\-\frac{6\sqrt{2}\ r^4\ arctan\left(\frac{2\sqrt{6}\ x}{\sqrt{24{\epsilon }^2r^2+{\alpha(z)}^2}}\right)}{\alpha(z)\sqrt{24{\epsilon }^2r^2+{\alpha(z)}^2}}\,,\label{eqB5}\end{gathered}$$ Adding all four terms together, we obtain: $$\begin{array}{c}
2{\left.\left(T_1\ \left(x\right)+T_2\ \left(x\right)+T_3\
\left(x\right)+T_4\
\left(x\right)\right)\right|}^{x=\alpha(z)/2}_{x=0}\\
=\frac{24\sqrt{6}\ r^4\ \alpha(z)\
arctan\left(\frac{\sqrt{2}\alpha(z)}{\sqrt{8{\epsilon
}^2r^2+{\alpha(z)}^2}}\right)}{\left(24{\epsilon
}^2r^2+{\alpha(z)}^2\right)\sqrt{8{\epsilon }^2r^2+{\alpha(z)}^2}}
\end{array}
\,,\label{eqB6}$$ Making the following change of variable in the third integral over $z$: $$t=\frac{\sqrt{2}\alpha(z)}{\sqrt{8\epsilon r^2+{\alpha(z)}^2}}\,,\label{eqC1}$$ When we replace $a_0$ given in Eq. , we get the volume function formula of Eq. .
[0]{}
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[^1]: Geodesic polyhedron is the convex region enclosed by the intersection of geodesic surfaces. A geodesic surface is a surface with vanishing extrinsic curvature and the intersection of two such surfaces is necessarily a geodesic curve.
[^2]: It is usually known as the Klein projection.
[^3]: Knowing that the biggest possible spherical tetrahedron is the half of 3-sphere $S^3_r$.
|
---
abstract: 'Lyapunov exponents measure the average exponential growth rate of typical linear perturbations in a chaotic system, and the inverse of the largest exponent is a measure of the time horizon over which the evolution of the system can be predicted. Here, Lyapunov exponents are determined in forced homogeneous isotropic turbulence for a range of Reynolds numbers. Results show that the maximum exponent increases with Reynolds number faster than the inverse Kolmogorov time scale, suggesting that the instability processes may be acting on length and time scales smaller than Kolmogorov scales. Analysis of the linear disturbance used to compute the Lyapunov exponent, and its instantaneous growth, show that the instabilities do, as expected, act on the smallest eddies, and that at any time, there are many sites of local instabilities.'
author:
- Prakash Mohan
- 'Nicholas Fitzsimmons [^1]'
- 'Robert D. Moser'
bibliography:
- 'paper.bib'
title: Scaling of Lyapunov Exponents in Homogeneous Isotropic Turbulence
---
Introduction
============
One of the defining characteristics of turbulence is that it is unstable, with small perturbations to the velocity growing rapidly. Indeed, turbulent flows in closed domains appear to be chaotic dynamical systems [@keefe1992dimension]. The result is that the evolution of the detailed turbulent fluctuations can only be predicted for a finite time into the future, due to the exponential growth of errors. In a chaotic system, this prediction horizon is inversely proportional to the largest Lyapunov exponent of the system, which is the average exponential growth rate of typical linear perturbations. The maximum Lyapunov exponent $\bar\gamma$ is commonly used to characterize the chaotic nature of a dynamical system [@eckmann1985ergodic]. In a turbulent flow, the maximum Lyapunov exponent is thus a measure of the strength of the instabilities that underlie the turbulence, and its inverse defines the time scale over which the turbulence fluctuations can be meaningfully predicted.
Lyapunov exponents in chaotic fluid flows have been estimated experimentally since the work of Swinney [@wolf1985determining], using indirect methods. In numerical simulations, however, Lyapunov exponents can be determined directly by computing the evolution of linear perturbations. This has been done for weakly turbulent Taylor Couette flow [@vastano1991short] and very low Reynolds number planar Poiseuille flow [@keefe1992dimension]. Remarkably, to the authors’ knowledge, Lyapunov exponents have not been determined for isotropic turbulence, a shortcoming corrected in this paper.
Homogeneous isotropic turbulence is an idealized turbulent flow that has been extensively studied both experimentally [@mydlarski1996onset; @comte1971simple; @comte1966use] and using numerical simulations [@rogallo1981numerical; @vincent1991satial; @jimenez1993structure; @chen1992high]. It is valuable as a model for the small scales of high Reynolds number turbulence away from walls [@durbin2011statistical]. It has been speculated that in isotropic turbulence, the maximum Lyapunov exponent scales with the inverse Kolmogorov time scale [@crisanti1993intermittency], suggesting that the dominant instabilities occur at Kolmogorov length scales as well. If true, then a study of the maximum Lyapunov exponent and the associated instabilities in homogeneous isotropic turbulence will be applicable to a wide range of flows.
This paper focuses on how the maximum Lyapunov exponent and hence the predictability time horizon scale with Reynolds number and computational domain size of a numerically simulated homogeneous isotropic turbulence. The speculation that $\bar\gamma$ should scale as the inverse Kolmogorov time scale $\tau_\eta$ [@crisanti1993intermittency] is in agreement with an estimate from a shell model [@aurell1996growth]. However, this scaling has not been directly tested in direct numerical simulations.
In addition, in the process of computing the maximum Lyapunov exponent in a direct numerical simulation, one necessarily computes the linear disturbance that is most unstable (on average). This can be used in the short-time Lyapunov exponent analysis, as introduced in [@vastano1991short], to characterize the nature of the instabilities. This will be pursued here for isotropic turbulence.
The remainder of this paper includes a brief review of Lyapunov exponents and how they are computed in numerical simulations (section \[sec:lyapunov\]) followed by a description of the direct numerical simulations studied here (section \[sec:dns\]). The results of a scaling study of the Lyapunov exponents are given in section \[sec:scaling\], and a short-time Lyapunov exponent analysis is presented in section \[sec:short\], followed by concluding remarks in section \[sec:conclude\].
Lyapunov Exponent Analysis {#sec:lyapunov}
==========================
Two important characteristics of chaotic dynamical systems for the purposes of the current study are that 1) solutions evolve toward a stable attractor, and 2) solution trajectories on the attractor are unstable so that near-by trajectories diverge exponentially. The rate of this exponential divergence is characterized by the Lyapunov exponents, whose characteristics are recalled briefly here. Further details can be found in [@vastano1991short]. In addition, the use of Lyapunov exponents in the analysis presented in the paper is described.
Evolution of Linear Perturbations
---------------------------------
Consider a solution trajectory $u(t)$ of a chaotic system. The solution will evolve toward an attracting set in phase space (the attractor); in turbulence this corresponds to the solution evolving to a statistically stationary state. Let $u(t_0)$ at some arbitrary starting time $t_0$ be on the attractor, and consider an infinitesimal perturbation $\delta u(t_0)$ of the solution at time $t_0$, and its evolution in time. The Lyapunov exponents describe the growth or decay of the magnitude of $\delta u$. In particular, the multiplicative ergodic theorem [@oseledec1968multiple] implies that the limit $$\bar\gamma=\lim_{t\rightarrow\infty}\frac{1}{t}\log\left(\frac{\|\delta
u(t)\|}{\|\delta u(t_0)\|}\right)$$ exists and $\bar\gamma$ is called a Lyapunov exponent. There is a spectrum of possible Lyapunov exponents, depending on the solution $u(t_0)$ and the perturbation $\delta u(t_0)$ at the starting time. However, for almost all $\delta u(t_0)$, $\bar\gamma=\gamma_1$ the largest Lyapunov exponent, and, due to round-off error and other sources of noise, in practical computations, $\bar\gamma=\gamma_1$ for all $\delta u(t_0)$. Furthermore, the Lyapunov spectrum ($\gamma_1>\gamma_2>\gamma_2>\cdots$) does not depend on $u(t_0)$; it is instead a property of the dynamical system. See the review by Eckmann [@eckmann1985ergodic] for an introduction to the theory.
In addition, in practical computations as discussed above, we expect that in the limit $t\rightarrow \infty$ $$\frac{\delta u(t)}{\|\delta u(t)\|}\rightarrow \overline{\delta
u}(u(t))\qquad\mbox{and}\qquad \frac{1}{\|\delta
u(t)\|}\frac{d\|\delta u(t)\|}{dt}\rightarrow \gamma'(u(t))$$ where $\overline{\delta u}$ and $\gamma'$ depend only on the solution at $t$, and not on the starting conditions $u(t_0)$ and $\delta
u(t_0)$. The perturbation $\overline{\delta u}$ is the disturbance that grows most rapidly in the long run, growing at the average exponential rate $\bar\gamma$. It is defined by the fact that it’s long-time average growth rate forward in time is $\bar\gamma$ and when the evolution is backward in time the long-time average growth rate is $-\bar\gamma$. The short-time Lyapunov exponent $\gamma'$ is simply the instantaneous exponential growth rate of $\overline{\delta u}$.
Because $\gamma'$ and $\overline{\delta u}$ depend only on the solution at the current time, they can be used as diagnostics for the instabilities responsible for a system being chaotic. In particular, when $\gamma'$ is large, the underlying system is particularly unstable, and at that time the Lyapunov disturbance $\overline{\delta u}$ is rapidly growing. Thus by seeking out times when $\gamma'$ is large, and by analyzing the solution $u$ and the Lyapunov disturbance $\overline{\delta u}$ at that time, we can characterize the important instabilities. This is the short-time Lyapunov exponent analysis described by Vastano & Moser [@vastano1991short].
In this paper we will be concerned with the scaling of $\bar\gamma$ with Reynolds number and with the chaotic instabilities revealed by short-time Lyapunov exponent analysis.
Simulations {#sec:dns}
===========
To simulate the base field, we solve the three-dimensional incompressible Navier-Stokes equations on a cube of dimension $L=2\pi$, with periodic boundary conditions, to obtain a computational approximation of homogeneous isotropic turbulence. Turbulence is maintained by introducing a forcing term to the Navier-Stokes equations which only acts at large scales. The forcing formulation is described in section \[sec:negViscos\]. The Navier-Stokes equations are solved using a Fourier-Galerkin spatial discretization with $N$ modes in each direction, and the vorticity formulation of Kim [@kim1987turbulence]. This formulation has the advantage of exactly satisfying the continuity constraint while eliminating the pressure term. A low-storage explicit third-order Runge-Kutta scheme [@spalart1991spectral] is used for time evolution. The simulations are performed using a modified version of the channel flow code PoongBack [@lee2013petascale; @lee2014experiences].
To compute the Lyapunov exponents, we compute the growth rate of a linear perturbation added to the base field. This perturbation satisfies the linearized Navier-Stokes equations: $$\label{lyapNS}
\frac{\partial \delta u_i}{\partial t} + \frac{ \partial}{\partial
x_j}(u_j \delta u_i + \delta u_j u_i) = -\frac{\partial
\delta p}{\partial x_i} + \frac{1}{Re}\nabla^2 \delta u_i$$ $$\partial_i \delta u_i = 0,$$ where $u_i$ is the base field and $\delta u_i$ is the disturbance field. The disturbance equations are solved using the same numerical scheme as the Navier-Stokes equations. Note that the forcing is applied only to the base field and not the perturbation. The implementation of both the base and disturbance field solvers were verified using the method of manufactured solutions.
Forcing {#sec:negViscos}
-------
The goal of the forcing is to inject energy into the large-scale turbulence so that the isotropic turbulence will be stationary. Forcing is applied to Fourier modes with wavenumber magnitudes in a specified range, and is designed to produce a specified rate of energy injection (forcing power), which, when the system is stationary, will be the dissipation rate. By specifying the wavenumber range being forced, forcing power and viscosity, the integral scale, turbulent kinetic energy and Reynolds number can be controlled.
The energy injection is accomplished by the introduction of a forcing term $f_i$ to the Navier Stokes equations: $$\begin{aligned}
\label{nvforceNS}
\frac{\partial u_i}{\partial t} + \frac{\partial u_iu_j}{\partial x_j}= -\frac{\partial p}{\partial x_i} + \frac{1}{Re}\nabla^2 u_i + f_i \\
\partial_iu_i = 0.\end{aligned}$$ Following [@spalart1991spectral], in the Fourier spectral method used here, the Fourier transform of the forcing $\hat f_i$ is specified in terms of the velocity Fourier transform $\hat u_i$ as $$\hat{f}_i(\mathbf{k}) = \alpha |\mathbf{k}|^2 \hat{u}_i(\mathbf{k}).$$ Given that $u_i$ is a Navier-Stokes solution, $f_i$ is guaranteed to be divergence-free. The coefficient $\alpha$ in the above is determined as a function of time so that the forcing power is the target dissipation rate $\epsilon_T$. Since the forcing is applied only to a range of wavenumbers, this yields $$\alpha =
\epsilon_T
\begin{cases}
{\displaystyle
\epsilon_T\left(\sum_{k_{f_{\text{min}}}\leq|\mathbf{k}| \leq
k_{f_{\text{max}}}}
|\mathbf{k}|^2\hat{u}^*_i(\mathbf{k})\hat{u}_i(\mathbf{k})\right)^{-1}}
& k_{f_{\text{min}}}\leq|\mathbf{k}|\leq k_{f_{\text{max}}}\\
0 & \text{otherwise}
\end{cases}$$ where $\cdot^*$ denotes the complex conjugate, and $k_{f_{\text{min}}}$ and $k_{f_{\text{max}}}$ are the bounds on the range of wavenumbers being forced. In the Fourier transform of the Navier-Stokes equations, the viscous term has the same structure as $f_i$, so this forcing can be interpreted as a negative viscosity acting in the specified wavenumber range. The combined forcing and viscous term is then $-(\nu-\alpha)|\mathbf{k}|^2\hat
u_i(\mathbf{k})$. In the numerical solution of the Navier-Stokes equations, this combined term is treated in the same way as the viscous term would be. Note that $f_i$ is just a nonlinear function of $u_i$, so there is no externally imposed stochasticity.
Case $k_{f_{\text{min}}}$ $k_{f_{\text{max}}}$ $N$ $\mathcal{L}$ $Re_\lambda$ $T_{\rm avg}q/\cal{L}$
------ ---------------------- ---------------------- -------- ----- --------------- -------------- ------------------------
1 0 2 0.0235 64 1.43 37.92 455.2
2 0 2 0.0113 96 1.58 58.34 123.8
3 0 2 0.0056 128 1.67 85.68 118.0
4 0 2 0.0038 192 1.70 106.33 51.2
5 0 2 0.0026 256 1.77 130.43 51.3
6 0 2 0.0010 512 1.82 211.76 69.5
7 2 4 0.0093 128 0.71 37.74 277.1
8 4 8 0.0037 256 0.35 37.31 72.1
: Parameters defining the eight direct numerical simulations performed to study Lyapunov exponent scaling. Values of $\mathcal{L}$ are quoted in units in which the domain size is $2\pi$, and averaging times are normalized by eddy turnover time.[]{data-label="tab:cases"}
Simulation Cases
----------------
To investigate the scaling of the maximum Lyapunov exponent $\bar\gamma$ with both Reynolds number and the ratio of the computational domain size $L$ to the integral scale $\mathcal{L}$, eight simulations were performed. These are summarized in table \[tab:cases\]. To study the scaling of $\bar\gamma$ with Reynolds number, six cases where simulated with the same forcing wavenumber range and $\epsilon_T$. This resulted in approximately the same integral scale in each case. The Reynolds number was manipulated by changing the viscosity. To study the potential variation of $\bar\gamma$ with domain size normalized by integral scale, the domain size was kept fixed at $2\pi$ and the integral scale was changed by adjusting the forced wavenumber range, while keeping the Reynolds number approximately fixed. In all cases $k_{\text{max}}\eta>1$, where $k_{\text{max}}$ is the maximum resolved wavenumber, and $\eta$ is the Kolmogorov scale. In a refinement study, this was found to be sufficient to obtain resolution independent values of $\bar\gamma$.
For each case, the simulations were run until the base solution became statistically steady and then the statistics were gathered by time averaging over a period $T_{\rm avg}$ as reported in table \[tab:cases\]. The simulation was confirmed to be stationary by verifying the convergence of the viscous dissipation to $\epsilon_T$ and the statistical convergence rates of $q^2$ and $\bar\gamma$.
Scaling of Lyapunov Exponents {#sec:scaling}
=============================
Of primary concern here is the dependence of the maximum Lyapunov exponent on the Reynolds number and on the domain size. To address this, the maximum Lyapunov exponent $\bar\gamma$, the integral scale ($\mathcal{L}$) and Reynolds number based on the Taylor micro-scale ($Re_\lambda$) are needed, along with their uncertainties. Based on the assumption of isotropy, the latter two were determined to be $\mathcal{L}=0.15q^3/\epsilon$ and $Re_\lambda=q^2\sqrt{5/(3\epsilon\nu)}$ [@pope2001turbulent]. Thus the two statistical quantities that need to be computed from the DNS are $\bar\gamma$ and $q^2$. Both are determined as a time average over averaging time $T_{\rm avg}$ (see table \[tab:cases\]), and the standard deviations $\sigma$ of the uncertainty due to finite averaging time were determined using the technique described by [@Oliver2014]. The values of $\bar\gamma$, $q^2$ and their standard deviations are given in table \[tab:data\]. The standard deviations of the derived quantities $\mathcal{L}$ and $Re_\lambda$ are determined simply as $\sigma_{\mathcal{L}}=(0.225q/\epsilon)\sigma_{q^2}$ and $\sigma_{Re_\lambda}=\sqrt{5/(3\epsilon\nu)}\sigma_{q^2}$, where for $\sigma_{\mathcal{L}}$ it is assumed that $\sigma_{q^2}/q^2\ll 1$. Note that since for each simulation $\epsilon$ and $\nu$ are specified, there is no uncertainty in their values.
case $q^2$ $\sigma_{q^2}$ $\bar\gamma\tau_\eta$ $\sigma_{\bar\gamma}\tau_\eta$
------ ------- ---------------- ----------------------- --------------------------------
1 4.51 0.107 0.0922 0.0038
2 4.80 0.160 0.1075 0.0046
3 4.99 0.075 0.1177 0.0032
4 5.05 0.044 0.1231 0.0040
5 5.19 0.046 0.1304 0.0034
6 5.28 0.084 0.1599 0.0048
7 2.82 0.008 0.0941 0.0019
8 1.76 0.001 0.0945 0.0021
: Values of $q^2$ and the maximum Lyapunov exponent $\bar\gamma$, along with the standard deviation ($\sigma$) of the sampling uncertainty. Values of $q^2$ are quoted in units in which the domain size is $2\pi$ and $\epsilon=1$, and $\bar\gamma$ is normalized by the Kolmogorov time scale $\tau_\eta$.[]{data-label="tab:data"}
The dependence of the maximum Lyapunov exponent in Kolmogorov units on Reynolds number is shown in figure \[fig:dependence\], including uncertainties expressed as the standard deviation. If the hypothesized scaling of the Lyapunov exponent on Kolmogorov time scale were correct, these data would, within their uncertainty, fall along a horizontal line. However, this does not appear to be the case. Indeed, $\bar\gamma\tau_\eta$ appears to be growing with $Re_\lambda$. Also, shown in figure \[fig:dependence\] is the dependence of scaled Lyapunov exponent on domain size at constant Reynolds number. These data do appear to be consistent with the hypothesis that the Lyapunov exponent does not depend on the domain size.
To make these scaling observations quantitative, Bayesian inference is used to infer the coefficients $\alpha$ and $\beta$ in a scaling relationships of the form $$\bar\gamma\tau_\eta=\alpha_1 Re_\lambda^{\beta_1}\quad\mbox{and}\quad
\bar\gamma\tau_\eta=\alpha_2 (\mathcal{L}/L)^{\beta_2},
\label{eq:models}$$ given the data and its uncertainties. These scaling relationships serve as the “model” for the inference. In Bayesian inference for this problem, the joint probability distribution $\pi(\alpha,\beta|{\bf d})$ of the parameters $\alpha$ and $\beta$ conditioned on data ${\bf d}$ (shown in table \[tab:cases\]) is sought. Bayes’ theorem gives this conditional probability as: $$\pi(\alpha,\beta|{\bf d})\propto \pi({\bf
d}|\alpha,\beta)\pi(\alpha,\beta)$$ where $\pi({\bf d}|\alpha,\beta)$ is the likelihood and $\pi(\alpha,\beta)$ is the prior. The likelihood is the joint probability density for the observed quantities evaluated for the observed values of these quantities, as determined by the model with parameters $\alpha$ and $\beta$, and given the uncertainties in the data. The prior represents our prior knowledge about the parameters, independent of the data.
The data are statistical averages obtained from direct numerical simulations. The primary source of uncertainty in such data is statistical sampling error. The central limit theorem implies that in the limit of large samples, the uncertainty associated with sampling error is normally distributed with zero mean. Therefore, to formulate the likelihood, the data are assumed to have Gaussian uncertainty with standard deviations as reported in table \[tab:data\]. The probability distribution for the $i$th observation of the value of $\bar\gamma$ as predicted by the models is thus given by $$\pi(\bar\gamma|\alpha,\beta,x_i)=\frac{1}{\sigma_{\gamma_i}\sqrt{2\pi}}\exp\left[-\frac{(\bar\gamma-\alpha_1x_i^{\beta_1})^2}{2\sigma_{\gamma_i}^2}\right]$$ where $x_i$ is the independent variable ($Re_{\lambda i}$ or $\mathcal{L}_i/L$, depending on which scaling relation is being inferred) of the $i$th observation and $\sigma_{\gamma_i}$ is the standard deviation in $\bar\gamma$ associated with the $i$th observation. However, there are also uncertainties in the values of the independent variables $x$, as determined from the DNS, again with a Gaussian distribution and standard deviation for the $i$th observation of $\sigma_{x_i}$. In this case, the probability distribution of the independent variable $x$ given the observation $x_i$ is $$\pi(x|x_i)=\frac{1}{\sigma_{x_i}\sqrt{2\pi}}\exp\left[-\frac{(x-x_i)^2}{2\sigma_{x_i}^2}\right].$$ The conditional distribution of $\bar\gamma$ given the parameters and the observed independent variable is then given by $$\pi(\bar\gamma|\alpha,\beta,x_i)=\int_x\pi(\gamma|\alpha,\beta,x)\pi(x|x_i)\,
dx.
\label{eq:distgamma}$$ Finally, to obtain the likelihood, (\[eq:distgamma\]) is evaluated at $\bar\gamma=\gamma_i$ and the uncertainties in each observation are assumed to be independent (an excellent assumption), yielding: $$\pi({\bf d}|\alpha,\beta)=\prod_i
\pi(\bar\gamma=\gamma_i|\alpha,\beta,x_i).$$
To inform the prior, we consider the range of time scales in the turbulence. The largest is the eddy turn-over time, which is proportional to $q^2/\epsilon$, and the smallest is the Kolmogorov time scale $\sqrt{\nu/\epsilon}$. The ratio of the turnover to the Kolmogorov times scales as $Re_\lambda$. Therefore, the Lyapunov exponent $\bar\gamma\tau_\eta$ scaling with the turn-over time would imply $\beta=-1$ and scaling with the Kolmogorov scale would imply $\beta=0$. However, theoretical arguments suggest that the Lyapunov exponent scales with the Kolmogorov time scale [@crisanti1993intermittency] ($\beta=0$), and we need to allow for the possibility that this assessment may be in error in either direction. The bounds on the range of plausible values of $\beta$ were therefore extended to $-1\le\beta\le1$, and a uniform distribution over this range was used as a prior for $\beta$. Somewhat arbitrarily, the same range was used for the $\beta$ prior in the domain size scaling relationship. The parameter $\alpha$ is a positive definite scaling parameter, and so following Jaynes [@Jaynes_2003], a Jeffries distribution $\pi(\alpha)\sim1/\alpha$ is used as an (improper) prior. Finally, the priors for $\alpha$ and $\beta$ are independent so $\pi(\alpha,\beta)=\pi(\alpha)\pi(\beta)$.
![Posterior PDFs for $\alpha_1$ and $\beta_1$ in the Reynolds number scaling model (\[eq:models\]). Shown are the marginal distributions of both parameters along with contours of their joint distribution. []{data-label="fig:rescaling"}](Rejoint){width="\linewidth"}
![Posterior PDFs for $\alpha_2$ and $\beta_2$ in the domain size scaling model (\[eq:models\]). Shown are the marginal distributions of both parameters along with contours of their joint distribution.[]{data-label="fig:Bscaling"}](Bsjoint){width="\linewidth"}
Given the likelihood and prior described above, and the data in Table \[tab:data\], samples of the posterior distribution were obtained using a Markov-chain Monte Carlo (MCMC) algorithm [@haario2006dram] as implemented in the QUESO library [@Prudencio2012; @mcdougall2015parallel]. The resulting samples were used to characterize the joint posterior distribution of $\alpha$ and $\beta$ for both the Reynolds number and domain size scaling model, as shown in figures \[fig:rescaling\] and \[fig:Bscaling\]. Notice in these figures that the joint distribution has probability mass concentrated in thin diagonally oriented regions, showing that uncertainty in $\alpha$ and $\beta$ are highly correlated. Indeed, the uncertainty in the $\beta$’s is as large as it is because changes in $\beta$ can be compensated for by changes in $\alpha$ so that the model still fits the data. The MCMC samples were also used to determine the uncertainty in the model predictions for the Lyapunov exponent as a function of Reynolds number and domain size, with the results plotted in figure \[fig:dependence\], along with the data. From this, it is clear that the scaling models as calibrated are consistent with the data and their uncertainty.
The marginal posterior distribution for $\beta$ in the Reynolds number scaling relation shows that the most likely values of $\beta$ are between about $1/4$ and $1/3$, with the possibility that the value is zero essentially precluded. This is remarkable since it suggests instability time scales that will become increasingly faster than Kolmogorov with increasing Reynolds number. The origin of this fast time scale is currently unclear. One possibility to consider is that this fast instability time scale arises as an artifact of the time discretization of the DNS. However the DNS time step in Kolmogorov units $\Delta t/\tau_\eta\sim Re_\lambda^{-1/2}$, so if the Lyapunov exponent were scaling with the DNS time step, $\beta$ would be 1/2, which is also essentially precluded by the posterior distribution. The time discretization thus appears to be an unlikely origin of the observed Reynolds number scaling. This was also verified by running a time refinement study where $\bar\gamma$ was found to be invariant to changing $\Delta t$.
As with the time step, interest in the computational domain size arises because of concern that computational artifacts not impact our Lyapunov exponent analysis. The posterior distribution of $\beta$ in the domain size scaling relationship (figure \[fig:Bscaling\]) shows that $\beta=0$ is highly likely, with the most probable values of $\beta$ ranging from -0.05 to 0.05. If there is an effect of the domain size, the data indicates that it is extremely weak. It therefore appears that the Lyapunov exponent Reynolds number scaling discussed above and the short-time Lyapunov exponent analysis presented in section \[sec:short\] are unaffected by finite domain size effects.
Short-Time Lyapunov Exponent Analysis {#sec:short}
=====================================
As discussed in section \[sec:lyapunov\], both the disturbance field ($\delta u$) used to compute the Lyapunov exponent and its instantaneous exponential growth rate ($\gamma'$) depend only on the instantaneous Navier-Stokes velocity $u$, not on the initial disturbance. In short-time Lyapunov exponent analysis, we study $\gamma'$ and $\delta u$ to learn about the instabilities responsible for the chaotic nature of turbulence.
First, consider the time evolution of the exponential growth rate $\gamma'$, which is shown in figure \[fig:lyapunov-time\] for $Re_\lambda=37$ and 210 (cases 1 and 6 respectively), normalized by $\bar\gamma$. Note that in both cases $\gamma'$ takes large excursions from the mean, of order 3 times the mean value. However, the variations in $\gamma'$ occur on a much shorter time scale and the large excursions seem to occur more often in the high Reynolds number case. The time scale on which $\gamma'$ varies appears to decrease somewhat faster than the Kolmogorov time scale with increasing Reynolds number, as when plotted against $t/\tau_\eta$, $\gamma'$ still varies faster for case 6 (figure \[fig:lyapunov-time-ktime\]). At the same Reynolds number (cases 1 and case 8), the variability of $\gamma'$ decreases sharply with increasing relative computational domain size $L/\mathcal{L}$. The fact that the time scale of the instability, as measured by the Lyapunov exponent, decreases faster than the Kolmogorov time scale suggests that the instability processes are acting at spatial scales near the Kolmogorov scale. In this case, a simulation with a larger domain size relative to intrinsic turbulence length scales would include a larger sample of local unstable turbulent flow features, resulting in smaller variability in $\gamma'$. In comparing case 8 with case 1, the relative volume increases by a factor 64, suggesting that the variability of $\gamma'$ should be about a factor of 8 smaller in case 8 than in case 1, which is indeed consistent with the data.
At the peaks in $\gamma'$, the growth of the disturbance energy is particularly rapid, and the question naturally arises as to what is special about these times. To investigate this, the spatial distribution of the magnitude of the disturbance energy density is visualized in figure \[fig:peak-field\] at three times, just before the beginning of a peak in $\gamma'$, a time half way up that peak and at the peak ($tq/\mathcal{L}=9.58$, 9.85 and 9.89 in figure \[fig:lyapunov-time\]). Notice that before the rapid growth of $\gamma'$ into the peak, the energy in the disturbance field is broadly distributed across the spatial domain. Half way up the peak, the distribution is much more spotty, and finally at the peak, the disturbance energy is primarily focused in a small region, appearing in the lower left corner of figure \[fig:peak-field\](c). The contour levels in these images were chosen so that the contours enclose 60% of the disturbance energy, implying that 60% of the disturbance energy is concentrated in the small feature in the lower left of figure \[fig:peak-field\](c). Another indication of the dominance of the disturbance feature in figure \[fig:peak-field\](c) is that the contour level needed to enclose 60% of the energy is about 2500 times the mean disturbance energy density, while in figure \[fig:peak-field\](a) the contour is only about 15 times the mean. Clearly the growth of the disturbance field in this concentrated area is responsible for the peak in $\gamma'$. However, the spatially local exponential growth rate of the disturbance energy $|\delta
u|^{-2}\,\partial|\delta u|^2/\partial t$ is not particularly large there, large values of this quantity are distributed broadly across the spatial domain. It seems, then, that the large peak in $\gamma'$ is due to a local disturbance that is able to grow over an extended time until it dominates the disturbance energy, so that the disturbance is localized in a region of relatively large growth rate. This is presumably unusual because it requires that the local unstable flow structure responsible for the disturbance growth persists for a long time.
It is of interest to investigate the turbulent flow structures responsible for the large localized disturbance energy at the peak in $\gamma'$. In the region where $\delta u$ is localized, the base field exhibits a pair of co-rotating vortex tubes (figure \[fig:base-vorticity\]). As shown in figure \[fig:slice-vorticity\], the disturbance vorticity is localized on the vortex tubes, with regions of opposite signed disturbance vorticity to one side or the other of each vortex tube. This disturbance, when added to the base field would have the effect of displacing each vortex tube along the line between the positive and negative peaks in the disturbance vorticity associated with each tube. The instability then appears to be one associated with slowing (speeding up) the co-rotation of the vortex tubes while they move away from (toward) each other. Note that the disturbance equations, being linear and homogeneous, are invariant to a sign change, and so the sign of the vortex displacement is indeterminate. Such an instability of co-rotating vortices is reminiscent of the pairing instability in two-dimensional mixing layers.
![Contour of the magnitude of the vorticity of the base field for $Re_\lambda=210$ (case 6) at the peak of $\gamma'$ in the region where the disturbance field is localized (box highlighted in figure \[fig:peak-field\]c) . The contour level is 9.2 times the square root of the mean enstrophy. The vortex tubes are co-rotating, with the direction of rotation indicated by the black arrow.[]{data-label="fig:base-vorticity"}](vortex-tubes-cropped){width="0.6\linewidth"}
![The magnitude of the vorticity (grayscale) and the disturbance vorticity component normal to the plane (contour lines) in a plane perpendicular to and in the middle of the vortex tubes shown in figure \[fig:base-vorticity\]. For the disturbance vorticity, the red and blue contours are of opposite signs.[]{data-label="fig:slice-vorticity"}](1024-vorticity-slice-2){width="0.6\linewidth"}
Conclusions {#sec:conclude}
===========
The results of the scaling study (section \[sec:scaling\]) show definitively that, at least over the Reynolds number range studied, the Lyapunov exponent does not scale like the inverse Kolmogorov time scale, as had been previously suggested [@aurell1996growth]. Instead, $\bar\gamma\tau_\eta$ increases with Reynolds number like $Re_\lambda^\beta$ for $\beta$ in the range from 1/4 to 1/3. Further note that the analysis of Aurell [*et al. *]{}[@aurell1996growth] indicated that a correction for the intermittance of dissipation would yield $\beta < 0$, also inconsistent with the current results. If positive $\beta$ scaling holds to much higher Reynolds numbers, it would be remarkable, as it would mean that there are instability processes that act on time scales shorter than Kolmogorov. However, in the highest Reynolds number ($Re_\lambda=210$) simulation performed here, $\bar\gamma\tau_\eta$ is still only 0.16. It is certainly possible that this Reynolds number dependence of $\bar\gamma\tau_\eta$ is a low Reynolds number effect, caused by insufficient scale separation between the large scales and the scales at which the instabilities act, and that the value will reach a plateau at some much higher Reynolds number. Clearly, this scaling behavior of the maximum Lyapunov exponent is worthy of further study. The current results suggest that the generally accepted and most obvious scaling is not correct, and that, unfortunately, turbulent fluctuations are even less predictable than previously thought.
The short-time analysis described in section \[sec:short\] confirmed that the dominant instabilities in turbulence act on the smallest eddies. Further, at $Re_\lambda=210$, when the instantaneous disturbance growth rate was the largest (about 3 times the mean), the disturbance energy was highly localized, suggesting that it was a particular local instability that was responsible for the rapid growth at that time. However, this was not due to a particularly large local growth rate, as the logarithmic time derivative of the spatially local disturbance energy was equally large in regions spread throughout the domain. It may be that the localized instability we observed is not of particular importance, except that the underlying structure in the turbulent field was especially long-lived. None-the-less, studying it showed that one of the possible instability mechanisms acting in turbulence is reminiscent of pairing instabilities of co-rotating vortices, as in a mixing layer. In this, the short-time Lyapunov analysis pursued here appears to be a valuable tool for the study of the instabilities underlying turbulence.
Acknowledgements {#acknowledgements .unnumbered}
================
Support for the research reported here was provided by the Department of Energy through the Center for Exascale Simulation of Combustion in Turbulence (ExaCT) under subcontract to Sandia National Laboratory, project 1174449, and is gratefully acknowledged. We also wish to thank Dr. Myoungkyu Lee for his assistance with code development for the simulations.
[^1]: Current affiliation: Applied Research Laboratories, the University of Texas at Austin
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abstract: 'Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a normal crossings divisor. We show that there are only finitely many log crystalline ${{\mathbb Z}}_{p^f}$-local systems over $X_K\setminus S_K$ of given rank and with geometrically absolutely irreducible residual representation, up to twisting by a character. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.'
address:
- 'Department of Mathematics, University of Georgia, Athens, GA 30605, USA'
- 'Institut für Mathematik, Universität Mainz, Mainz 55099, Germany'
- 'Institut für Mathematik, Universität Mainz, Mainz 55099, Germany'
author:
- Raju Krishnamoorthy
- Jinbang Yang
- Kang Zuo
title: Finiteness of logarithmic crystalline representations
---
Introduction
============
To state our main theorem, the following setup will be convenient.
\[setup\]Let $p$ be an odd prime, let $f\geq 1$ be a postive integer, and let $k$ be a finite field containing $\mathbb F_{p^f}$. Let $W=W(k)$ be the ring of Witt vectors and set $K=\mathrm{Frac}(W)$. Let $(X,S)/W$ be a smooth proper scheme together with a relative normal crossings divisor over $W$. Denote by $U=X\setminus S$ and by $\mathcal U_K$ the rigid analytification of $U_K=U\times_{\Spec W} \Spec K$.
\[mainthmRep\] Notation as in Setup \[setup\], and fix a positive integer $r\leq p-2$. Then there are only finitely many isomorphism classes of logarithmic crystalline representations $\rho:\pi_1(U_K)\rightarrow \GL_r({{\mathbb Z}}_{p^f})$ whose residual representation is geometrically absolutely irreducible, up to twisting by a character of $\text{Gal}(\bar{K}/K)$.
For clarity, the statement that the residual representation of $\rho$ is geometrically absolutely irreducible means that the composite representation $$\overline{\rho}\colon \pi_1(U_{\bar K})\rightarrow \pi_1(U)\rightarrow \GL_r({{\mathbb Z}}_{p^f})\rightarrow \GL_r(\mathbb{F}_{p^f})\rightarrow \GL_r(\bar{\mathbb F}_p)$$ of the geometric fundamental group is irreducible. Crystalline representations are a $p$-adic analog of polarized variations of Hodge structures. Therefore Theorem \[mainthmRep\] is an arithmetic analog of a theorem of Deligne [@Del87]. See also the very recent work of Litt for a finiteness result in a different spirit [@Li18].
The proof of Theorem \[mainthmRep\] implicitly relies on the work of Abe on a global $p$-adic Langlands correspondence for smooth curves over finite fields, which itself follows the work of Lafforgue on the $l$-adic global Langlands correspondence for smooth curves over finite fields. We now describe the sections of this note.
- In Section \[section:preliminaries\], we explain the preliminary material.
- In Section \[section:proof\], we prove Theorem \[mainthmRep\] by reducing it to a statement about Higgs-de Rham flows.
- In Section \[section:nonexample\], we show that Theorem \[mainthmRep\] is false if $k\cong \overline{\mathbb F}_p$ in Setup \[setup\].
- In Section \[section:uniform\], we speculate on a uniform upper bound of the number of crystalline representations.
Preliminaries {#section:preliminaries}
=============
We briefly describe the main players, with notation as in Setup \[setup\]. Recall that a logarithmic Fontaine-Faltings module over $(X,S)$ is quadruple $(V,\nabla,\Fil,\Phi)$, where
- $V$ is a vector bundle on $X$;
- $\nabla: V\rightarrow V\otimes \Omega_{X/W}^1(\log S)$ is an integrable connection on $X$ with logarithmic poles along $S$;
- $\Fil$ is a filtration of $V$ by subbundles that is locally split and satisfies Griffith transversality; and
- $\Phi$ is a strongly divisible Frobenius structure that is horizontal with respect to $\nabla$.
There are several perspectives on the $\Phi$-structure. The original perspective, proposed by Faltings, is to work locally: one defines a strongly divisible $\Phi$ structure on $(V,\nabla,\Fil)$ over a small affine, whose $p$-adic completion admits a Frobenius lift, and then one glues [@Fal89]. A second view, via nonabelian Hodge theory, only works when the level of $(V,\nabla,\Fil,\Phi)$ is no greater than $p-2$, but it is a global description. The key to this perspective is the $p$-adic inverse Cartier transform defined by Lan-Sheng-Zuo [@LSZ13a]; then $\Phi$ is an isomorphism $$\Phi: C^{-1}(\Gr(V),\Gr(\nabla),V,\nabla,\Fil) \overset{\sim}{\longrightarrow} (V,\nabla).$$ Finally, by forgetting the filtration, a logarithmic Fontaine-Fontaine module yields a logarithmic $F$-crystal in finite, locally free modules on the logarithmic crystalline site of $(X_k,S_k)/W$.
Let $\MF_{[0,a],f}^\nabla((X,S)/W)$ be the category of Fontaine-Faltings modules $\{(V,\nabla,\Fil,\Phi,\iota)\}$ over $(X,S)/W$ with Hodge-Tate weights in $[0,a]$ and endomorphism structure $\iota\colon W(\mathbb{F}_{p^f})\rightarrow \text{End}_{\MF}(V,\nabla,\Fil,\Phi)$. If $a\leq p-2$, then the fundamental work of Fontaine-Lafaille-Faltings constructs a fully faithful functor to the category of $\GL(\mathbb Z_{p^f})$ logarithmic crystalline local systems over $U_K$ and with unipotent local monodromy around $S_K$ [@Fal89].[^1] Let $\FIsoc_{\nilp}^{\log}(X_k)$ be the category of convergent log-$F$-isocrystals on $(X_k,S_k)$ with nilpotent residues around $S_k$. Let $\FIsoc^\dagger(U_k)$ be the category of overconvergent $F$-isocrystals over $U_k$. These are both $\mathbb{Q}_p$-linear Tannakian categories.
Recall that a convergent $F$-isocrystal over $U_k = X_k\setminus S_k$ can be realized as an vector bundle ${{\mathcal E}}$ on ${{\mathcal U}}_K$ equipped with an integrable connection together with an isomorphism $\sigma^*{{\mathcal E}}\simeq{{\mathcal E}}$. By forgetting Hodge filtration, tensoring with $\mathbb Q_p$ and then restricting to $\mathcal U_K$, one obtains a functor $$\psi_1\colon \MF_{[0,a],f}^\nabla((X,S)/W) \rightarrow \FIsoc^{\log}_{\nilp}(X_k)_{\mathbb{Q}_{p^f}}.$$ (The notation on the right hand side refers to the $\mathbb{Q}_{p^f}$-linearization of $\FIsoc^{\log}_{\nilp}(X_k)$.) By a fundamental theorem of Kedlaya, the forgetful functor $$\psi_2\colon \FIsoc^{\log}_{\nilp}(X_k) \rightarrow \FIsoc^\dagger(U_k)$$ is fully faithful [@kedlayasemistableI Proposition 6.3.2]. For the definition of a *tame* $F$-isocrystal, see [@abeesnault 1.2]; in our case, an object $\mathcal{E}\in \FIsoc^{\dagger}(U_k)$ is tame if $\mathcal{E}$ extends to a logarithmic $F$-isocrystal on $(X_k,S_k)$. (In particular, there is no condition on the residues around $S_k$.) The following fundamental theorem follows from work of Abe and Lafforgue and is a consequence of the Langlands correspondence [@abelanglands; @lafforguelanglands].
\[finiteness\] Let $k$ be a finite field of characteristic $p$, let $U_k$ be a smooth curve over $k$. Then there are finitely many isomorphism classes of irreducible tame objects of $\FIsoc^{\dagger}(U_k)_{\overline{\mathbb Q}_p}$ of bounded rank, up to twists by rank 1 objects. As a consequence, if $L/\mathbb{Q}_p$ is a *finite* extension, then there are finitely many isomorphism classes of irreducible tame objects of $\FIsoc^{\dagger}(U_k)_{L}$ with finite order determinant.
The proof {#section:proof}
=========
For any non-negative integer $a\leq p-2$, the category of logarthmic crystalline representations with Hodge-Tate weights in $[0,a]$ is equivalent to the category of logarithmic Fontaine-Faltings modules of level $\leq a$ via Faltings’ $\mathbb{D}$-functor. To prove Theorem \[mainthmRep\], we first use a Lefschetz-style theorem to reduce to the curve case.
\[lemma:lefschetz\]Notation as in Setup \[setup\]. Then there exists a smooth projective relative curve $C\subset X$ over $W$ that intersects $S$ transversely, with the property that $\pi_1(C_K\cap U_K)\rightarrow \pi_1(U_K)$ is surjective. Therefore, to prove Theorem \[mainthmRep\], it suffices to consider the case when $X/W$ has relative dimension 1.
We claim that there exists a smooth ample relative divisor $D\subset X$ over $W$ that intersects $S$ transversely. Indeed, pick some ample line bundle $L$ on $X$; then for all $m\gg 0$, the map $H^0(X,L^m)\rightarrow H^0(X_1,L^m_1)$ is surjective. On the other hand, for $m\gg 0$, the vector space $ H^0(X_1,L^m_1)$ has a section $s_1$ whose zero locus $V(s_1)$ is smooth and intersects $S_1$ transversely by Poonen’s Bertini theorem [@poonenbertini Theorem 1.3]. Take any lift $s\in H^0(X,L^m)$ of $s_1$; then the zero locus $V(s)$ is smooth over $W$ and intersects $S$ transversely. Finally, it is well known that the map on fundamental groups $\pi_1(D_K\cap U_K)\rightarrow \pi_1(U_K)$ is surjective because $D_K\subset X_K$ is ample and $D_K$ intersects $S_K$ transversely (see [@EK15]). Proceed by induction.
Now, as $\pi_1(C_K\cap U_K)\rightarrow \pi_1(U_K)$ is surjective, it follows that to prove Theorem \[mainthmRep\], it suffices to prove it for the pair $(C,S\cap C)$, i.e., we may reduce to the case of curves.
We must now translate the property that the residual representation of $\rho$ is geometrically absolutely irreducible into a property about the objects of the associated Higgs-de Rham flow. This is accomplished by the following lemma.
\[stability\] Notation as in Setup \[setup\] and assume $X/W$ is a smooth curve. Let $\{(V,\nabla,\Fil,\Phi,\iota)_1\}$ be a logarithmic Fontaine-Faltings module over $(X,S)_1/k$ with Hodge-Tate weights in $[0,p-2]$ and endomorphism structure $\iota\colon \mathbb{F}_{p^f}\rightarrow \text{End}_{\MF}((V,\nabla,\Fil,\Phi)_1)$. Suppose the associated crystalline representation $\rho_1\colon \pi_1^{et}(X_K)\rightarrow \GL_r(\mathbb{F}_{p^f})$ is geometrically absolutely irreducible, i.e., the composite representation $$\overline{\rho}\colon \pi_1(X_{\bar K})\rightarrow \GL_r(\bar{\mathbb{F}}_p)$$ is irreducible. Let $$\label{HDF_1} \tiny
\xymatrix{
& (V,\nabla,\Fil)^{(0)}_1 \ar[dr]
&& (V,\nabla,\Fil)^{(1)}_1\ar[dr]
&& \cdots \\
(E,\theta)^{(0)}_1 \ar[ur]
&& (E,\theta)^{(1)}_1 \ar[ur]
&& (E,\theta)^{(2)}_1 \ar[ur]\\
}$$ be the $f$-periodic Higgs-de Rham flow associated to $\{(V,\nabla,\Fil,\Phi,\iota)_1\}$. Then $(V,\nabla)^{(i)}_1$ and $(E,\theta)^{(i)}_1$ are all stable.
When $f=1$, this is [@LSZ13a Corollary 7.3], but for completeness sake we write a proof. We first prove that $(E,\theta)$ is semistable and has vanishing rational Chern classes. This argument does not assume that $X/W$ is a curve.
In the case that $S$ is empty, then as $(E,\theta)^{(0)}_1$ is periodic, it follows immediately from [@LSZ13a Theorem 6.6] that $(E,\theta)^{(0)}_1$ is semistable with vanishing rational Chern classes. The same arguments works in the logarithmic case. Indeed, the Chern classes of all bundles in the flow rationally vanish: the key is that the inverse Cartier is locally a Frobenius pullback, and that the residues of $(V,\nabla)^{(0)}_1$ are nilpotent. In particular, the slope of $(E,\theta)^{(0)}_1$ vanishes. Then, exactly as in [@LSZ13a Theorem 6.6], one shows that a slope $\lambda$ Higgs subsheaf of $(E,\theta)^{(0)}_1$ gives rise to a slope $p\lambda$ Higgs subsheaf of $(E,\theta)^{(1)}_1$. In particular, $(E,\theta)^{(0)}_1$ contains no Higgs subsheaf of positive slope; therefore $(E,\theta)^{(0)}_1$ is semistable with vanishing Chern classes.
Next, note that $(E,\theta)^{(0)}_1$ is stable if and only if $(V,\nabla)^{(0)}_1\cong C^{-1}(E,\theta)^{(0)}_1$ is stable.
We now assume that $X/W$ is a curve. Let us show that $(E,\theta)^{(0)}_1$ is stable (the proof for other terms is precisely analogous). Suppose that $(E,\theta)^{(0)}_1$ is not stable. Then there is a proper Higgs subbundle $(E,\theta)'^{(0)}_1\subset (E,\theta)^{(0)}_1$ of slope $0$; note that $(E,\theta)'^{(0)}_1$ is *automatically semistable*. By running the Higgs-de Rham flow starting with $(E,\theta)'^{(0)}_1$ (using the induced filtration), one obtains a sub Higgs-de Rham flow $$\label{eqn:subflow} \tiny
\xymatrix{
& (V,\nabla,\Fil)'^{(0)}_1 \ar[dr]
&& (V,\nabla,\Fil)'^{(1)}_1\ar[dr]
&& \cdots \\
(E,\theta)'^{(0)}_1 \ar[ur]
&& (E,\theta)'^{(1)}_1 \ar[ur]
&& (E,\theta)'^{(2)}_1 \ar[ur]\\
}$$ The initial bundle $E'^{(0)}_1$ has vanishing rational first Chern class. Therefore the same is true every bundle in Equation \[eqn:subflow\]. There are only finitely many vector subbundles of $E^{(0)}_1$ with vanishing degree because $k$ is a finite field. Therefore Equation \[eqn:subflow\] in fact initiates a *preperiodic Higgs-de Rham flow*. The periodic part of Equation \[eqn:subflow\] forms an $f'$-periodic sub Higgs-de Rham flow of (\[HDF\_1\]). (Note that $f\mid f'$ and $f'$ may be not equal to $f$.) Using the equivalence of logarithmic periodic Higgs-de Rham flows and logarithmic Fontaine-Faltings modules [@LSYZ14 Theorem 1.1], together with the equivalence of logarithmic Fontaine-Faltings modules and torsion crystalline representations [@Fal89 Theorem 2.6\*, p. 41, i], one deduces that the periodic part of Equation \[eqn:subflow\] induces a representation $$\rho'\colon \pi_1(X_{K(\zeta_{p^{f'}-1})}) \rightarrow \mathrm{GL}_{s}(\mathbb F_{p^{f'}})$$ where $s = \rank E'^{(0)}_1$ and $\zeta_{p^{f'}-1}$ a primitive $(p^{f'}-1)$-th root of unity. The inclusion map between these two periodic Higgs de Rham flows induces an inclusion of representations $$\rho'\hookrightarrow \rho\mid_{\pi_1(X_{K(\zeta_{p^{f'}-1})})}$$ In particular one gets a proper sub-representation of $\overline{\rho}$. This contradicts the irreducibility of $\overline{\rho}$.
Notation as in Setup \[setup\]. Let $m_0,\cdots,m_{f-1}\in \mathbb Z$. Define $\mathcal L_{\lambda}(k_0,k_1,\cdots,k_{f-1})$ to be the following flow over one point $\mathrm{Spec}(W)$ $$\tiny
\xymatrix{
& (W,\Fil_0) \ar[dr]
&& (W,\Fil_1) \ar[dr]
&& \cdots \ar[dr] & \\
W \ar[ur]
&& W \ar[ur]
&& W \ar[ur]
&& W \ar@/^20pt/[llllll]^{\varphi = \lambda}\\
&&\\
}$$ where $\Fil_i^{m_i} W= W$ and $\Fil_i^{m_i+1} W= 0$.
The data of $\mathcal L_{\lambda}(m_0,m_1,\cdots,m_{f-1})$ is equivalent to a *rank $1$ filtered $\varphi^f$-module over $W$*, a.k.a. a filtered $\varphi$-module of rank $f$ over $W$ equipped with an endomorphism structure of $\mathbb{Z}_{p^f}$. By the Fontaine-Lafaille correspondence, this corresponds to a crystalline character $\mathrm{Gal}(\overline{K}/K)\rightarrow \mathbb \GL_1(\mathbb Z_{p^f})$.
Therefore, using Lemma Lemma \[lemma:lefschetz\] and \[stability\], to prove Theorem \[mainthmRep\], it suffices to prove the following.
\[mainthmFM\] Notation as in Setup \[setup\], with $X/W$ a curve, and fix $r\leq p-2$. Then there are only finitely many isomorphism classes of $f$-periodic Higgs-de Rham flow over $(X,S)/W$ of rank $r$ such that the reduction modulo $p$ of terms appeared in the flow are stable, up to twisting by a rank $1$ filtered $\varphi^f$-module over $W$.
To prove Theorem \[mainthmFM\], we first record the following lemma, which follows immediately from [@Lan14 Theorem 5.4].
\[fromQtoZ\] Notation as in Setup \[setup\]. Let $(V,\nabla)$ and $(V,\nabla)'$ be logarithmic flat connections over $(X,S)$ that each have stable reduction modulo $p$. If $(V,\nabla)_{\mathbb Q}$ and $(V,\nabla)'_{\mathbb Q}$ are isomorphic on $X_{\mathbb Q}$, then $(V,\nabla)$ and $(V,\nabla)'$ are isomorphic as logarithmic flat connections on $(X,S)$. Moreover, if $f\colon (V,\nabla)\rightarrow (V,\nabla)'$ is a morphism of logarithmic flat connections that is an isomorphism modulo $p$, then $f$ is an isomorphism.
We recall the following lemma, which is due independently to Lan-Sheng-Zuo [@LSZ13a Lemma 7.1] and Langer [@Lan14 Corollary 5.6].
\[lem:LSZ\_Fil\] Let $(Y,D)$ be a smooth projective variety together with a simply normal crossings divisor over an algebraically closed field $k$ and let $(V,\nabla)$ be a logarithmic flat bundle over $(Y,D)$. If there exists a Griffiths transverse filtration $\Fil$ on $(V,\nabla)$ such that the associated graded logarithmic Higgs module $(E,\theta)$ is Higgs stable, then $\Fil$ is unique up to a shift of index.
\[uniqueFil\]Notation as in Setup \[setup\]. Let $\{(V,\nabla,\Fil,\Phi,\iota)\}$ and $\{(V,\nabla,\Fil,\Phi,\iota)'\}$ be two logarithmic Fontaine-Faltings modules over $(X,S)/W$ with Hodge-Tate weights in $[0,p-2]$ and endomorphism structure of $\mathbb{Z}_{p^f}$. Let
$$\label{HDF} \tiny
\xymatrix{
& (V,\nabla,\Fil)^{(0)} \ar[dr]
&& (V,\nabla,\Fil)^{(1)} \ar[dr]
&& \cdots \ar[dr] & \\
(E,\theta)^{(0)} \ar[ur]
&& (E,\theta)^{(1)} \ar[ur]
&& (E,\theta)^{(2)} \ar[ur]
&& (E,\theta)^{(f)} \ar@/^20pt/[llllll]^{\varphi}\\
&&\\
}$$
be the $f$-periodic Higgs-de Rham flow associated to $\{(V,\nabla,\Fil,\Phi,\iota)\}$ and $$\label{HDF'} \tiny
\xymatrix{
& (V,\nabla,\Fil)'^{(0)} \ar[dr]
&& (V,\nabla,\Fil)'^{(1)} \ar[dr]
&& \cdots\ar[dr] & \\
(E,\theta)'^{(0)} \ar[ur]
&& (E,\theta)'^{(1)} \ar[ur]
&& (E,\theta)'^{(2)} \ar[ur]
&& (E,\theta)'^{(f)} \ar@/^20pt/[llllll]^{\varphi'}\\
&&\\
}$$ be the $f$-periodic Higgs-de Rham flow associated to $\{(V,\nabla,\Fil,\Phi,\iota)'\}$. Assume the reduction modulo $p$ of all terms in the two flows are stable and assume there exists a morphism of de Rham bundle $$g^{(0)}\colon (V,\nabla)^{(0)}\rightarrow (V,\nabla)'^{(0)}$$ whose reduction modulo $p$ is nontrivial. Then $g^{(0)}$ is an isomorphism and there exists unique isomorphisms, up to scale, of de Rham bundles $$g^{(i)} \colon (V,\nabla)^{(i)} \rightarrow (V',\nabla')^{(i)}\quad \text{for all } i = 0,1,\cdots,f-1.$$ Moreover, under these isomorphisms the Hodge filtration $\Fil^{(i)}$ coincides with $\Fil'^{(i)}$ after a shift of index and there exists $\lambda\in W^\times$ such that $$\varphi =\lambda\varphi'.$$
After shifting the filtrations, we may assume $$\Fil^0V^{(0)} = V^{(0)}, \quad \Fil^1V^{(0)} \neq V^{(0)}, \quad \Fil'^0V'^{(0)} = V'^{(0)}, \text{ and } \Fil'^1V'^{(0)} \neq V'^{(0)}.$$ By Lemma \[fromQtoZ\], it follows that $g^{(0)}$ is an isomorphism and is unique up to scale. Lemma \[lem:LSZ\_Fil\] implies that $$g^{(0)}(\Fil\mid_{V^{(0)}})\pmod{p} = \Fil'\mid_{V'^{(0)}}\pmod{p}.$$ By following the proof of [@LSZ13a Proposition 7.5] verbatim, one deduces that $g^{(0)}(\Fil\mid_{V^{(0)}}) = \Fil'\mid_{V'^{(0)}}$. Then denote $$g^{(1)}:= C^{-1}\circ \Gr(g^{(0)}).$$ Since $g^{(0)}$ is an isomorphism, $g^{(1)}\colon (V,\nabla)^{(1)} \rightarrow (V,\nabla)'^{(1)}$ is also an isomorphism between de Rham bundles. By the stablility of the modulo $p$ reductions, this isomorphism is unique up to a scale. Inductively on the index $i$, one constructs isomorphisms $g^{(2)},g^{(3)},\cdots$.
That the periodicity maps $\varphi$ and $\varphi'$ differ by an invertible element in $W$ follows from the stability.
\[UniqueUptoTwist\] Notation as in Lemma \[uniqueFil\]. Assume the reduction modulo $p$ of all terms are stable. If the $F$-isocrystals associated to $(V,\nabla,\Fil,\Phi,\iota)$ and $(V,\nabla,\Fil,\Phi,\iota)'$ are isomorphic, then there exists an integer $t$ in $\{0,1,\cdots,f-1\}$ and a filtered $\varphi^f$-module $\mathcal L$ over $W$ of rank 1 such that $$(V,\nabla,\Fil,\Phi,\iota) \cong (V',\nabla',\Fil',\Phi',\sigma^t(\iota'))\otimes \mathcal L$$
First of all, the statement that the underlying $F$-isocrystals are isomorphic is the statement that $(V,\nabla,\Phi)_{\mathbb{Q}}\cong (V',\nabla',\Phi')_{\mathbb Q}$ in $\FIsoc(U_k)$. After shifting the filtration, we may assume $$\Fil^0V = V, \quad \Fil^1V \neq V, \quad \Fil'^0V' = V', \text{ and } \Fil'^1V' \neq V'.$$ Consider the composition maps $$(V,\nabla)^{(0)}_{\mathbb Q} \hookrightarrow (V,\nabla)_{\mathbb Q} \cong (V,\nabla)'_{\mathbb Q} \twoheadrightarrow (V,\nabla)'^{(i)}_{\mathbb Q}.$$ There is at least one index $i$ such that the composition map is not equal to zero. By shifting the index (equivalent to changing the endomorphism structure by a conjugation $\sigma^t$), we may assume $i=0$ and denote the composition map by $$g^{(0)}\colon (V,\nabla)^{(0)}_{\mathbb Q}\rightarrow (V,\nabla)'^{(0)}_{\mathbb Q}$$ Multiplying by a suitable power of $p$, we may assume $$g^{(0)}(V^{(0)}) \subset V'^{(0)} \quad \text{and } g^{(0)}(V^{(0)}) \not\subset pV'^{(0)}.$$ Then $g^{(0)}\pmod{p}$ is a non-trivial morphism between two stable de Rham bundles with the same slope. Thus $g^{(0)}\pmod{p}$ is an isomorphism. By Lemma \[uniqueFil\], $g^{(0)}$ is an isomorphism and there exist isomorphisms $$g^{(i)}\colon (V,\nabla)^{(i)} \rightarrow (V,\nabla)'^{(i)}$$ such that, up to a shift of filtration one has $$g^{(i)}(\Fil\mid_{V^{(i)}}) = \Fil'\mid_{V'^{(i)}}.$$
Now, we identify $(V,\nabla,\Fil)^{(i)}$ and $(V',\nabla',\Fil')^{(i)}$ via $g^{(i)}$. Both $(V,\nabla)^{(0)}$ and $(V',\nabla')^{(0)}$ are crystals on the logarthmic crystalline site of $(X_k,S_k)/W$. We may then consider both $\Phi^f$ and $\Phi'^f$ as morphisms $$(\text{Frob}^*)^f(V,\nabla)^{(0)}\rightarrow (V,\nabla)^{(0)}$$ in the category of logarithmic crystals on $(X_k,S_k)/W$. As stability is an open condition, the logarithmic flat connection $(V,\nabla)^{(0)}_{\mathbb Q}$ is stable. As $\Phi'_{\mathbb Q}$ and $\Phi_{\mathbb{Q}}$ are isomorphisms, it follows that there exists $\lambda\in W^\times$ such that $\Phi = \lambda\Phi'$. Thus there exists an integer $t$ in $\{0,1,\cdots,f-1\}$ and a filtered $\varphi^f$-module $\mathcal L$ over $W$ of rank 1 such that $$(V,\nabla,\Fil,\Phi,\iota) \cong (V',\nabla',\Fil',\Phi',\sigma^t(\iota'))\otimes \mathcal L.$$
Let $HDF$ and $HDF'$ be $f$-periodic Higgs-de Rham flows on $(X,S)$, whose terms modulo $p$ are all stable. Suppose the associated $F$-isocrystals are isomorphic. (By [@kedlayasemistableI Proposition 6.3.2], it does not matter if we consider these as objects of $\FIsoc_{\nilp}^{\log}(X_k)$ or $\FIsoc^{\dagger}(U_k)$.) Then Proposition \[UniqueUptoTwist\] implies that $HDF$ and $HDF'$ differ by a twist. On the other hand, there are only finitely many isomorphism classes of overconvergent $F$-isocrystals on $U_k$, up to twist, by Theorem \[finiteness\]. The result follows.
Theorem \[mainthmRep\] is false over $k=\bar{\mathbb{F}}_p$ {#section:nonexample}
===========================================================
\[section:counterexample\] Let $\lambda\in W$ with $\lambda\not\equiv 0,1\pmod{p}$. Let $(X,S)=(\mathbb P^1,\{0,1,\infty,\lambda\})$. Let $\mathcal M_{\lambda}$ denote the moduli space of semi-stable graded logarithmic Higgs bundles $(E,\theta)$ of rank $2$ and degree $1$ over $(X,S)$. A Higgs bundle in this moduli space may be written as $$(E,\theta)=(\mathcal O\oplus\mathcal O(-1),\, \theta\colon \mathcal O\xrightarrow{\theta}\mathcal{O}(-1)\otimes \Omega^1_X(\log S))\quad (*).$$ We attach a parabolic structure at one of the four points of $D$ with parabolic weight $({1\over 2},\, {1\over 2})$, so that $(E,\theta)$ is parabolically stable and of parabolic degree zero. Let $\mathcal M^{par}_\lambda$ denote the moduli space of those type logarithmic Higgs bundles with a parabolic structure at one of those four cusps. One defines an isomorphism $$\mathcal M_{\lambda}\simeq \mathbb P^1_{W(k)}$$ by sending $(E,\theta)$ to the zero locus $(\theta)_0\in \mathbb P^1$. Consider the self map $\fai=\mathrm{Gr}\circ \mathcal C_{1,2}^{-1}$ on $\mathcal M_{\lambda}\otimes_Wk\simeq \mathbb P^1_{k}$ induced by Higgs-de Rham flow ($\otimes\mathcal O_{\mathbb P^1}((1-p)/2)$). Since $C^{-1}_{1,2}$ is a factor of the composition, $\fai$ factors through the Frobenius map, i.e., there exists a rational function $\psi_{\lambda}\in k(z)$ such that $\fai(z) = \psi_\lambda(z^p)$. In this note we call $\psi_\lambda$ the *Verschiebung part* of the self map $\fai$. The periodic points of the self map $\fai$ are naturally corresponding to the twisted periodic Higgs-de Rham flows on $(X,S)_1$; one forgets the parabolic structure and simply runs a twisted Higgs-de Rham flow [@SYZ17 Section 4].
Conjecturally this self map is related to the muliplication by $p$ map on the elliptic curve $$C_\lambda\colon y^2 = x(x-1)(x-\lambda).$$
\[ConjSYZ\] The following diagram commutes $$\xymatrix{C_\lambda \ar[r]^{[p]} \ar[d]^{\pi}& C_\lambda\ar[d]^{\pi}\\
\mathbb P^1 \ar[r]^{\fai} & \mathbb P^1 \\}$$
Let $(E,\theta)_{n}\in \mathcal M_{\lambda}(W_n)$ be a periodic Higgs bundle over $(X,S)_n$. Consider the self map $\Gr\circ C^{-1}$ on the deformation space $\mathrm{Def}_{(E,\theta)_{n}}(W_{n+1})$. Since $\mathrm{Def}_{(E,\theta)_{n}}(W_{n+1})$ is an $\mathbb A^1$-torsor space, we may identify a self map on $\mathrm{Def}_{(E,\theta)_{n}}(W_{n+1})$ with a self map on $\mathbb A^1$. Under this identification, $\Gr\circ C^{-1}$ is just a polynomial.
\[thm:infinitely\_many\_pgl\] Notation as above. Then
1. There exists exactly $p^{2f}+1$ geometrically absolutely irreducible crystalline $\mathrm{PGL}_2(\mathbb F_{p^f})$-local systems on $(X,S)$ that correspond to twisted $f$-periodic Higgs bundles from $\mathcal M_\lambda$.
2. Let $(E,\theta)_{n}\in \mathcal M_\lambda(W_n(\mathbb F_{q^{h}}))$ be a periodic Higgs bundle over $(X,S)_n$. Then the polynomial associated to the self map $\Gr\circ C^{-1}$ on $\mathrm{Def}_{(E,\theta)_{n}}(W_{n+1})$ is $$\mathbb A^1(\mathbb F_q) \to \mathbb A^1(\mathbb F_q),\quad z\to a\cdot z^p+b,$$ where $a, b \in \mathbb F_q$. Consequently if $a\neq 0$, then by solving the Artin-Schreier equation $az^p-z+b=0$ one obtains $p$ twisted $1$-periodic liftings over $W_{n+1}(\mathbb F_{q^{ph}})$. Moreover the constant $a$ is the derivative of Verschiebung part of $\fai$ at the point associated to $(E,\theta)_{n}\mid_{(X,S)_1}$. In particular the value of $a$ only depends on the reduction modulo $p$ of $(E,\theta)_{n}$.
3. For $p\leq 50$, the Conjecture \[ConjSYZ\] holds. In this case, if the torsion point associated to $(E,\theta)_{n}\mid_{(X,S)_1}$ is not of order $2$, then the coefficient $a\neq 0$ if and only if the associated elliptic curve $C_{\bar{\lambda}}$ is not supersingular.
Let $\pi\colon (X',S')\rightarrow (X,S)$ be the double cover that is ramified at the parabolic point and one other point. Using [@SYZ17 Theorem 4.6] together with Theorem \[thm:infinitely\_many\_pgl\], one obtains the following corollary.
Suppose $p\leq 50$ and the elliptic curve $C_{\bar{\lambda}}$ is not supersingular. Then there exists infinitely many log crystalline $\GL_2(\mathbb{Z}_p)$ local systems on $(X',S')_{\mathbb Q_p^{\text{unr}}}$ whose residual representation is absolutely geometrically irreducible.
We end this section with a final remark.
In view of Theorem \[mainthmRep\] on finiteness if $a\neq0$ then the solutions of the Artin-Schreier equation in $(2)$ must lie in $\mathbb F_{q^{ph}}$ but not in $\mathbb F_{q^h}$ in almost all lifting steps. But, it seems difficult to prove that directly!
Some speculations on a uniform upper bound {#section:uniform}
==========================================
In Section \[section:nonexample\] we made an identification $$M_\lambda \simeq \mathbb P^1$$ by sending $(E,\theta)$ to the zero $\theta_0$ of the Higgs field $\theta$ and let $\pi\colon C_\lambda\rightarrow X$ be the associated double cover of $X=\mathbb P^1$, branched along $S$.
\[varConjSYZ\] A Higgs bundle in $\mathcal M_\lambda$ over a finite unramified extension is twisted $f$-periodic if and only if $\pi^*(\theta_0)$ is a $(p^f\pm 1)$-torsion point in $C_\lambda.$
Conjecture \[varConjSYZ\] is a consequence of Conjecture \[ConjSYZ\]. It imples that the number of $\mathbb P_2(\mathbb Z_{p^f})$-crystalline local systems over $(X,S)$ over finite unramified extensions of $\mathbb Q_p$ is exactly $p^{2f}+1$. Conjecture \[varConjSYZ\] has been checked for the following cases.
1. When we only work modulo $p$ and $p\leq 50$.
2. When $C_{\bar{\lambda}}$ is supersingular and $p\leq 50$.
3. For all $p$, when the torsion point has order $1$, $2$, $3$, $4$ and $6$.
When $C_{\bar{\lambda}}$ is supersingular, any $\text{GL}_2$-crystalline local system corresponding to Higgs bundles in $\mathcal M_\lambda(\mathbb Q_p^{ur})$ automatically descends to a local system over a *finite, unramified* extensions of $\mathbb{Q}_p$. This contrasts with the situation when $C_{\bar{\lambda}}$ is an ordinary elliptic curve; we expect that most $\text{GL}_2$-crystalline local systems corresponding to Higgs bundles in $\mathcal M_\lambda(\mathbb Q_p^{ur})$ over $(X',S')/\mathbb{Q}_p^{ur}$ do not descend to finite unramified extension of $\mathbb{Q}_p$.
We end by posing a conjecture, the first part of which is in the spirit of the Fontaine-Mazur conjecture and the second of which is analogous to a theorem of Litt [@Li18].
Let $(X,S)$ be a log pair over $W(\mathbb F_q)$ with $p^f\mid q$.
1. Let $\mathbb L_1$ and $\mathbb L_2$ be two $\mathbb Z_{p^f}$-crystalline local systems over $(X,S)/W(\mathbb{F}_q)$. If $\mathbb{L}_1\simeq \mathbb{L}_2$ mod $p$, then $\mathbb L_1\simeq\mathbb L_2$.
2. The number of isomorphism classes $\GL_r(\mathbb{Z}_{p^f})$-crystalline local systems over $(X,S)_{W(\mathbb{F}_{q^h})}$, as we let $h$ range through the positive integers, is finite.
A brief discussion on the number of $\GL_2(\mathbb{Z}_p)$-local systems on $\mathbb{P}^1$ minus four points over a finite unramified extension of $\mathbb{Q}_p$ and with eigenvalues -1 around one of the four puncture points.
================================================================================================================================================================================================================================
Maintain notation as in Section \[section:nonexample\]; in particular, $(X,S)=(\mathbb P^1,\{0,1,\infty,\lambda\})$. It seems possible that all of the local systems in Section \[section:nonexample\] could come from families of abelian varieties of Hilbert modular type over $(X,S)/W(\mathbb{F}_q)$. We also conjecture that they correspond to $(p\pm1)$-torsion points on the elliptic curve $C_\lambda/W(\mathbb{F}_q)$.
In fact in a joint paper with Lu-Lv-Sun-Yang-Zuo, we have checked this conjecture for torsion-points of orders 1, 2, 3, 4 and 6, when $K$ is a number field. There exists exactly 26 elliptic curves of $(X,S)/\mathcal{O}_K$ such that the arithmetic local systems attached to those families correspond to arithmetic 1-periodic Higgs bundles from $\mathcal{M}_\lambda/\mathcal{O}_K$ and the zero locus of the Higgs fields are torsion points of order 1, 2, 3, 4 and 6. in $C_\lambda/\mathcal{O}_K$ under the pull back of $\pi.$
Via the techniques of Sections \[section:preliminaries\] and \[section:proof\] together with the theory of $p$-to-$l$ companions (due to Abe), one obtains an inclusion $$\label{eqn:periodic_to_l-adic} (\mathcal{M}_\lambda)^{\text{periodic}}\hookrightarrow \mathcal{M}^{\ell\text{-adic}}_\lambda.$$ Here, $(\mathcal{M}_\lambda)^{\text{periodic}}$ consists of those Higgs bundles in $\mathcal{M}_{\lambda}$ which are periodic (without specifying the periodicity map) and $\mathcal{M}^{\ell\text{-adic}}_\lambda$ is the set of equivalence classses of (geometrically) irreducible systems $\text{GL}_2(\bar {\mathbb Q}_\ell)$ -local systems over $(X_k,S_k)$ with prescribed local monodromy on the cusp points, up to twisting by a character on $\mathbb{F}_q$. Indeed, given a $p$-adically periodic Higgs bundle in $\mathcal M_{\lambda}$, we may forget the filtration to obtain an overconvergent $F$-isocrystal $\mathcal E$ on $U_k$; moreover, this gives an injective map from the set of equivalence classes of such Higgs bundles (without specifying the periodicity map) to the set of isomorphism classes of (overconvergent) $F$-isocrystals up to twisting by a character on $\mathbb{F}_q$ by Proposition \[UniqueUptoTwist\]. Picking a field isomorphism $\sigma\colon \overline{\mathbb Q}_p\rightarrow \overline{\mathbb Q}_l$, we may take the $\sigma$-companion of $\mathcal E$ to obtain a lisse $l$-adic sheaf on $U_k$, whose local monodromy around $S_k$ matches with that of the $F$-isocrystal.
If we choose any of the four points for the parabolic structure, then we may take the associated elliptic curve $\tau: (C_\lambda, S')\to (X,S)$ over $\mathbb{F}_q$ to kill the -1 eigenvalues in the local monodromy. In this way we obtain ${p^2+1}$ $\text{GL}_2(\mathbb{Q}_\ell)$-local systems over $(C_\lambda, S')/\mathbb{F}_q,$ which just corresponds to the $(p\pm1)$-torsion points on $C_\lambda/\mathbb{F}_q.$
Setup as above.
1. Can we intrinsically characterize the image of Equation \[eqn:periodic\_to\_l-adic\]?
2. Is there numerical evidence for the conjecture using the trace formula? More specifically, one may transform the question of counting $\text{GL}_2(\mathbb{Q}_\ell)$ local systems on $(X_1,S_1)$ with perscribed monodromy (in our case, eigenvalues of $-1$ around the parabolic point, with a non-trivial Jordan block and principal unipotent monodromy at the other punctures) to a question about counting certain types of automorphic forms via the Langlands correspondence. Drinfeld, and then Deligne-Flicker have a method to compute such numbers via the trace formula and have fully worked out this number in the case when the sheaves are supposed to have principal unipotent monodromy around each puncture [@drinfeld1982; @deligneflicker; @flicker2015]. Can we see the number $p^{2f}+1$ from the trace formula? Can we see the expected group law on the zeroes of the Higgs field from $(\mathcal{M}_\lambda)^{\text{periodic}}$ via automorphic forms?
We thank Atsushi Shiho for explaining the fact that the functor from log $F$-isocrystals with nilpotent residues to the category of overconvergent $F$-isocrystals is fully faithful. R.K gratefully acknowledges support from NSF Grant No. DMS-1344994 (RTG in Algebra, Algebraic Geometry and Number Theory at the University of Georgia)
[[Ked]{}07]{}
Tomoyuki Abe. Langlands correspondence for isocrystals and the existence of crystalline companions for curves. , 31(4):921–1057, 2018.
Tomoyuki Abe and H[é]{}l[è]{}ne Esnault. A [L]{}efschetz theorem for overconvergent isocrystals with frobenius structure. , 52(5):1243–1264, 2019.
P. [Deligne]{}. , 1987.
Pierre [Deligne]{} and Yuval Z. [Flicker]{}. , 178(3):921–982, 2013.
V. G. [Drinfel’d]{}. , 15:294–295, 1982.
Hélène [Esnault]{} and Lars [Kindler]{}. , 144(12):5071–5080, 2016.
Gerd Faltings. Crystalline cohomology and [$p$]{}-adic [G]{}alois-representations. In [*Algebraic analysis, geometry, and number theory ([B]{}altimore, [MD]{}, 1988)*]{}, pages 25–80. Johns Hopkins Univ. Press, Baltimore, MD, 1989.
Yuval Z. [Flicker]{}. , 137(3):739–763, 2015.
Kiran S. [Kedlaya]{}. , 143(5):1164–1212, 2007.
Laurent Lafforgue. Chtoucas de [D]{}rinfeld et correspondance de [L]{}anglands. , 147(1):1–241, 2002.
Adrian Langer. Semistable modules over [L]{}ie algebroids in positive characteristic. , 19:509–540, 2014.
Daniel Litt. Arithmetic representations of fundamental groups [II]{}: finiteness. , 2018.
Guitang [Lan]{}, Mao [Sheng]{}, Yanhong [Yang]{}, and Kang [Zuo]{}. , 747:63–108, 2019.
Guitang Lan, Mao Sheng, and Kang Zuo. Semistable [H]{}iggs bundles, periodic [H]{}iggs bundles and representations of algebraic fundamental groups. , 21(10):3053–3112, 2019.
Bjorn Poonen. Bertini theorems over finite fields. , 160(3):1099–1127, 2004.
Ruiran Sun, Jinbang Yang, and Kang Zuo. . , 2017.
[^1]: The nilpotence of the residues of the connection may be seen as follows: the Higgs field is nilpotent and the de Rham bundle comes from inverse Cartier transform.
|
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abstract: |
It is often claimed that the security of quantum key distribution (QKD) is guaranteed by the laws of physics. However, this claim is content-free if the underlying theoretical definition of QKD is not actually compatible with the laws of physics. This paper observes that (1) the laws of physics pose serious obstacles to the security of QKD and (2) the same laws are ignored in all QKD “security proofs”.
**Keywords:** security failures, quantum cryptography, quantum key distribution, QKD, side-channel attacks, electromagnetism, gravity, information flow, holographic principle.
author:
- 'Daniel J. Bernstein'
date: '2018.03.12'
title: |
Is the security of quantum cryptography\
guaranteed by the laws of physics?
---
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[^1]
currentlabel[1]{}
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—Hughes, Alde, Dyer, Luther, Morgan, and Schauer, 1995 [@1995/hughes]
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—Shor and Preskill, 2000 [@2000/shor]
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—Christandl, Renner, and Ekert, 2004 [@2004/christandl]
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—Campagna, Chen, Dagdelen, Ding, Fernick, Gisin, Hayford, Jennewein, Lütkenhaus, Mosca, Neill, Pecen, Perlner, Ribordy, Schanck, Stebila, Walenta, Whyte, and Zhang, 2015 [@2015/campagna]
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—ID Quantique, 2016 [@2016/idq-understanding]
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The “provable security” of quantum cryptography
===============================================
The core advertisement for quantum cryptography—in particular, for quantum key distribution, which I’ll focus on—is the claim that its security is guaranteed by the laws of physics. One would expect this claim to be backed by a clearly stated theorem having the following shape:
- “Assume $L$.” Here $L$ is a statement of the laws of physics.
- “Assume $P$.” Here $P$ states physical actions carried out by Alice and Bob.
- “Then $S$.” Here $S$ states a security property: e.g., something about the randomness, from Eve’s perspective, of a key shared between Alice and Bob.
This theorem statement would provide a starting point for security auditors to dive into questions of whether the stated $L$ is actually how the real world works; whether the stated $P$ matches what is actually being sold as “quantum key distribution”; whether the proof of the theorem is correct; and whether the stated $S$ actually includes what the users want.
An auditor who attempts to find this security theorem in the literature will easily find papers claiming to present “security proofs” for several different types of quantum cryptography; see, e.g., [@2004/gottesman]. However, the theorems in these papers never seem to explicitly hypothesize $L$, the laws of physics.
Is this merely a culture gap—when physicists say “Theorem” they implicitly mean a theorem under certain well-known hypotheses? Or is there an important reason that these papers aren’t hypothesizing Newton’s law of gravitation, for example, and Maxwell’s equations for electromagnetism? Or refinements such as general relativity and quantum electrodynamics?
An auditor who dives more deeply into the “security proofs” won’t find Maxwell’s equations used anywhere. This justifies omitting Maxwell’s equations as a hypothesis, but it also strongly suggests that the secret physical actions taken by Alice and Bob don’t involve any electricity or magnetism. How could one possibly prove anything about the effects of electromagnetic actions without invoking the relevant laws of physics? Similarly, Newton’s law isn’t used anywhere, strongly suggesting that the secret physical actions taken by Alice and Bob don’t involve moving any mass around. But then what exactly [*are*]{} the secret actions taken by Alice and Bob?
Abstractions without examples
=============================
A closer look shows that the papers don’t actually say what physical actions Alice and Bob are hypothesized to be carrying out. Sometimes the papers do specify physical details of, e.g., polarized photons being sent from Alice to Bob via Eve in “BB84 QKD”; or entangled pairs of photons being sent from Eve to both Alice and Bob in “device-independent E91 QKD” (see, e.g., [@2009/pironio]); or photons being sent with particular timing in “relativistic QKD” (see, e.g., [@2001/molotkov]); but there are always many other steps taken by Alice and Bob whose physical details are not mentioned anywhere in any of the “security proofs”.
Instead the papers make various abstract hypotheses regarding quantum states. Consider, for example, the following hypothesis from [@2014/masanes]:
> Eve can choose an observable $Z$ and obtains an outcome $E$. We assume that this, together with the public messages exchanged by Alice and Bob, is all information available to her.
Apparently Alice and Bob are supposed to take physical actions that do not interact with Eve’s quantum state except through the public interactions specified by the protocol. This begs the question of what those physical actions are supposed to be.
Obviously there are physical actions that Alice and Bob can take in the real world that might look like QKD but that don’t actually meet the hypotheses of the QKD “security proofs”. In particular, every QKD “security proof” assumes that there are various secret actions by Alice and Bob, actions unobserved by Eve, such as a sizeable fraction of Alice’s choices of polarization bases in the BB84 protocol. If Alice and Bob actually leak these secrets to Eve then these hypotheses are false and the QKD “security proofs” say nothing.
Anyone who claims that the security of QKD is guaranteed by the laws of physics is logically forced to argue that these physical actions by Alice and Bob are not actually QKD: the only physical actions that qualify as QKD are those for which the hypotheses of the theorems are satisfied. On the other hand, many authors insist on using the label “QKD” for actions that are obviously outside the scope of the theorems. Sometimes “security” systems that have already been convincingly demonstrated to fail in the real world (see, e.g., [@2016/huang]) are still sold as “QKD”.
To avoid confusion, let’s use the name “theoretical QKD” to refer to QKD meeting all of the hypotheses of the theorems. Is it clear that theoretical QKD exists within the laws of physics? Are there any examples of physical actions for Alice and Bob that [*do*]{} qualify as theoretical QKD?
Consider the following nightmare scenario for the QKD “security proofs”: The hypotheses of theoretical QKD are actually [*inconsistent with the laws of physics*]{}. For every sequence of physical actions that Alice and Bob can take, the hypotheses turn out to be false. It is vacuous to claim that the security of theoretical QKD is guaranteed by the laws of physics, since the laws of physics imply that theoretical QKD does not exist. What evidence do we have that the QKD “security proofs” are not in this nightmare scenario?
Part of the job of a mathematician or computer scientist stating a theorem is to verify—or at least plausibly conjecture—that there are examples meeting the hypotheses of the theorem. Vacuous lemmas are occasionally stated as intermediate steps inside proofs by contradiction, but these lemmas are also clearly identified as being vacuous, not as saying anything meaningful.
The holographic principle {#holographic}
=========================
Let me focus specifically on the hypothesis that Alice and Bob are taking various actions unobserved by Eve. This is, as I mentioned above, assumed in every QKD “security proof”. Obviously Alice and Bob have no security against Eve if Eve observes all of their secrets, so one cannot avoid making such a hypothesis.
This hypothesis seems to be flatly contradicted by the “holographic principle”. In the words of Brian Greene [@2011/greene page 272]:
> The [*holographic principle*]{} envisions that all we experience may be fully and equivalently described as the comings and goings that take place at a thin and remote locus. It says that if we could understand the laws that govern physics on that distant surface, and the way phenomena there link to experience here, we would grasp all there is to know about reality.
Readers not familiar with the idea that the universe is connected in this way should consider radios as an example of long-distance interaction. Radio communication typically relies on Alice and Bob generating a reasonably strong signal to make the receiver’s job easier, but this cooperation is not necessary; by building a very large array of radio receivers, Eve can pick up a very faint radio signal from ten thousand miles away.
In light of Maxwell’s equations, how can one justify the notion that Eve is unable to observe secret electromagnetic actions by Alice and Bob? Alice and Bob can try to use a Faraday cage to block their signal, but a Faraday cage does not create a truly isolated environment (see, e.g., [@2016/bernstein] and [@2018/guri]); it merely applies some scrambling to the signals emitted from that environment. One could hypothesize that Eve [*is not observing*]{} the actions by Alice and Bob—perhaps Eve is underfunded, or simply lazy—but this is obviously not “absolute security, guaranteed by the fundamental laws of physics”.
It seems that the only way for Alice and Bob to avoid Maxwell’s information leak is to avoid any data flow from secrets to electricity or magnetism. But then how are Alice and Bob supposed to control their photon generators, photon detectors, and other QKD equipment?
More importantly, the holographic principle says that the difficulty here is not limited to electromagnetism. [*Any*]{} physical encoding of information will be visible to a sufficiently resourceful attacker watching signals on a remote screen.
Well-known examples of signal processing (sonar arrays, radar arrays, X-ray tomography, magnetic-resonance imaging, etc.) all seem to fit the following general rule: Eve’s cost for stealing a physically encoded secret grows only polynomially with Eve’s distance from the secret. Alice and Bob might try to hide at some reasonable distance from attackers, and from any attacker-controlled equipment, but this only moderately increases the cost of the attack.
QKD is claimed to have information-theoretic security guaranteed by the laws of physics, not depending on “the conjectured difficulty of computing certain functions”. Even if holographic signal processing actually has superpolynomial cost, the availability of the holographic signal is enough to contradict the QKD security claims. Eve can carry out all necessary computations at her leisure, retroactively breaking QKD without having to interfere with the protocol execution. This directly contradicts the claims in [@2013/unruh] and [@2015/schaffner page 17] that QKD offers “everlasting security”, and the similar claim in [@2015/campagna page 19].
Of course, claimed “laws” of physics do not have the same level of certainty as mathematics. Many past “laws” seem inconsistent with experiment and are no longer believed to be accurate: e.g., Newton’s law of gravitation is merely a first approximation, failing to take relativistic effects into account. One might speculate that (1) the holographic principle is not true in the level of generality suggested in [@1993/hooft], [@1995/susskind], [@2002/bousso], etc.; (2) the leaks of information to Eve are “merely” through electromagnetic waves, gravity, etc.; and (3) there is some way for Alice and Bob to carry out QKD without triggering any of these leaks of information. In other words, it is conceivable that some law of physics not known today will eventually guarantee the security of some form of QKD not specified today. But these unsupported speculations are very far from justifying the claim that QKD provides “absolute security, guaranteed by the fundamental laws of physics”.
A concrete attack against QKD
=============================
Here is an attack to illustrate the principles of Section \[holographic\]. An attacker turns on a large array of radio receivers, leaves it on, and records the resulting data. A tiny budget, on the scale of a million dollars per year, will allow the attacker to store a gigabyte of data[^2] per second.
The victim then begins using a state-of-the-art QKD device. This device relies on the secrecy of non-quantum information encoded inside the equipment as conventional electromagnetic signals (not to be confused with the unclonable quantum information sent via photons). These conventional electromagnetic signals influence the data recorded by the attacker, as per Maxwell’s equations. This influence also depends on other inputs, such as the physical location and shape of objects that have been reflecting radio waves, so the attacker also collects as much information as possible about these inputs by recording data from a similarly large array of cameras, microphones, and other sensors.
The attacker now computes the most likely possibilities for the victim’s QKD secrets, and thus for the resulting secret keys, given the recorded data. This is a mathematical computation exploiting the laws of physics—the equations that determine the influence of the QKD secrets and other inputs upon the data recorded by the attacker.
From an information-theoretic perspective, it seems likely that the recorded data is sufficient to successfully pinpoint the secret keys of all QKD users. There is certainly no guarantee to the contrary. There is an algorithmic challenge of optimizing the computation of these secrets, and perhaps the best algorithm that we can figure out will take a million years to run on future computer equipment, but there is again no guarantee of this.
Now compare this attack to the claim made by Campagna, Chen, Dagdelen, Ding, Fernick, Gisin, Hayford, Jennewein, Lütkenhaus, Mosca, Neill, Pecen, Perlner, Ribordy, Schanck, Stebila, Walenta, Whyte, and Zhang in [@2015/campagna page 19]:
> An important characteristic of quantum key distribution is that any attack (e.g. any attempt to exploit a flaw in an implementation of transmitters or receivers) must be carried out in real time. Contrary to classical cryptographic schemes, in QKD there is no way to save the information for later decryption by more powerful technologies. This greatly reduces the window of opportunity for performing an attack against QKD; the window is much wider for conventional cryptography.
This claim is disproven by the QKD attack explained above:
- The attack does not need to be completed in “real time”.
- The attack does save information for later decryption.
- The attack does take advantage of more powerful technologies available in the future.
The only accurate part of the claim is that the attack must [*begin*]{} before the victim uses QKD—the attacker must plan ahead by collecting and storing data for later analysis—but the comparison in [@2015/campagna] already assumes that the attacker is collecting and storing data for later analysis.[^3] In the real world, ever-increasing amounts of data are being stored from a wide range of collection devices; in other words, the attack has already begun.
Someone who claims that the security of QKD is “guaranteed by the laws of physics” is logically forced to claim that the laws of physics guarantee that this particular attack does not work. When I ask such people why they think that this attack will fail, they typically answer in one of the following ways:
- Speculation: Claim that the electromagnetic signal is too weak for the attacker to pick up at distance $x$.
This might be true in some cases, depending on $x$ and on many more details of what the attacker is doing, but all of this needs analysis—see Section \[countermeasures\]. More to the point, even in situations where this claim is true, this is not the same as security “guaranteed by the laws of physics”.
- Abstraction: Describe today’s electromagnetic devices as mere “implementations” of an abstract concept of theoretical QKD; claim that any exploitable “imperfections” in these “implementations” will be corrected by future “implementations”.
This claim begs the question of what exactly those “implementations” will be: how can theoretical QKD be embedded into the physical world, in light of the holographic principle? More to the point, this claim implicitly admits that the QKD security “guarantee” is not actually from the laws of physics, but rather from a simplified—I say oversimplified—model.
- Marketing: Outline particular examples of next-generation “device-independent” E91 QKD devices; say that it is important to fund development of these devices; and claim that [*these*]{} QKD devices will have security guaranteed by the laws of physics where previous BB84 QKD devices did not.
At this point the claims are not even marginally consistent with the literature, which claims security guarantees for [*both*]{} BB84 and E91.
- Distraction:[^4] Complain that other forms of cryptography are also potentially vulnerable to this type of attack.
This complaint is both correct and irrelevant. What I am disputing is the frequent unjustified claim $Q$ that quantum cryptography has security guaranteed by the laws of physics. Perhaps somewhere there is some other unjustified claim $C$ that some other form of cryptography has security guaranteed by the laws of physics, but it is illogical to use a complaint about $C$ as a justification for $Q$.
I do not have experimental evidence that this particular QKD attack works, but I also see no justification for the claim that the attack will fail.
Previous work
=============
There is a huge literature claiming that the security of theoretical QKD is guaranteed by the laws of physics. I am not aware of previous papers clearly and directly raising the possibility that this claim is vacuous—that theoretical QKD is not actually compatible with the laws of physics.
Plaga in [@2006/plaga] pointed out that speculations about nonlinearity of quantum gravity, if correct, could allow Eve to extract information from the data [*explicitly communicated*]{} in some QKD protocols. What I am saying is quite different: I am focusing on information leaked from the “secret” portions of QKD protocols, and I am relying on core physical phenomena such as electromagnetism.
More relevant is a paper [@2002/rudolph] by Rudolph, which (among other things) questions the “impenetrability/no emanation” hypotheses for theoretical QKD. What I am saying is stronger in three ways:
- Rudolph merely questions the hypotheses, while I am questioning both the hypotheses and the conclusion.
- Regarding details, Rudolph touches upon ways that [*some*]{} secrets could leak, and asks whether these leaks can be limited to “an exponentially small amount of useful information”, while I am pointing to ways that [*all*]{} secrets leak to a sufficiently resourceful attacker.
- Rudolph says that a “black hole lab” would “presumably” satisfy the hypotheses, while I am not drawing any such line. My impression of the consensus of physicists is that black holes are [*not*]{} an exception to the holographic principle. This is important because the extreme case of a “black hole lab” provides some intuition for the best that one can hope for by adding more and more shielding.
Rudolph is saying that secrets are not [*exactly*]{} unobservable, while I am giving reasons to believe that secrets are not even [*approximately*]{} unobservable, and that the answer to Rudolph’s “exponentially small” question is negative. This is important for claims that QKD is improving, eliminating “imperfections”, etc.
Brassard stated in [@2006/brassard] that the original prototype implementation of QKD “was unconditionally secure [*against any eavesdropper who happened to be deaf*]{}! [:-)]{}” (italics and smiley in original). This is still an overstatement of the security provided by the implementation. It is clear that the implementation would also have given up all of its secrets to, e.g., a deaf eavesdropper watching the screen of an oscilloscope attached to a small coil of wire. More importantly, Brassard presented this security failure as being specific to [*one*]{} implementation of QKD, not recognizing the possibility of the laws of physics forcing the same fundamental type of security failure to appear in [*all*]{} forms of QKD.
A series of papers on “quantum hacking”, such as [@2016/huang] by Huang, Sajeed, Chaiwongkhot, Soucarros, Legre, and Makarov, have broken the security of various commercial implementations of QKD. All of these attacks follow the theme of Eve interacting physically (passively or actively) with the “secret” computations by Alice and Bob.
Various earlier attacks are surveyed by Scarani and Kurtsiefer in [@2014/scarani], who generally express skepticism regarding the security of QKD but also claim that “in principle QKD can be made secure”. A careful reading shows that this claim again starts from—and does not attempt to justify—the questionable hypothesis that Alice and Bob are taking various actions unobserved by Eve. This hypothesis is obviously false in various [*examples*]{} of successful QKD attacks summarized in [@2014/scarani]; but Scarani and Kurtsiefer, like Brassard, do not recognize the possibility of the holographic principle allowing Eve to violate the hypothesis for [*all*]{} possible QKD devices.
None of the previous literature has stopped researchers from claiming that the security of QKD is guaranteed by the laws of physics—and dismissing every QKD security failure, every leak of secrets, as a supposedly fixable implementation mistake.
Countermeasures
===============
What should Alice and Bob do if the promise of “security guaranteed by the fundamental laws of physics” is a sham—in particular, if physical effects allow a computationally unlimited attacker to steal secrets from an arbitrary distance?
The obvious answer is to study the [*cost*]{} of Eve’s attack, and to take measures to increase this cost—hopefully beyond anything that Eve can afford. There are several ways to argue that Eve is subject to cost limits, such as the following:
- Perhaps the lifetime of the universe is limited.
- Perhaps, even in an everlasting universe, the cosmological constant is positive, putting a limit on all computations, as explained in [@2000/bousso]. This constant is generally believed to be around $10^{-122}$.
- Perhaps the space aliens who control most of the resources in the universe are happy that Alice and Bob are secretly arranging climate-change protests, and are not willing to help policewoman Eve see these secrets. Compare [@2015/neslen].
If Eve is limited to cost $C$, then Alice and Bob do not need to aim for the unachievable goal of security guaranteed by the laws of physics; Alice and Bob merely need to be secure against all attacks that cost at most $C$.
Earlier I mentioned that Eve’s cost for stealing a physically encoded secret seems to grow only polynomially with Eve’s distance from the secret. However, this polynomial also seems to depend heavily on the choice of encoding mechanism. Alice and Bob can and should choose technologies for encoding their secrets with the goal of making this polynomial as large as possible. For example, one might guess that the following steps help:
- Store and process secrets inside a modern 10nm Intel CPU, rather than using older, less efficient chip technology. Presumably consuming less energy for the same computation will reduce the amount of signal visible to Eve, increasing Eve’s cost for recovering the secrets.
- Hide the secrets by adding shields and by generating extra noise. A Faraday cage leaks information, as mentioned above, but seems to make the information more difficult to intercept and decipher.
- Mask the secrets, for example by encoding bit 0 as a random even-weight 4-bit string and encoding bit 1 as a random odd-weight 4-bit string.
- Apply more sophisticated mathematical encodings, such as “secret-key cryptography” and “public-key cryptography”.
Of course, guessing is not the same as systematically studying the actual impact on attack cost. The premier venue for scientific papers analyzing the costs of attackers learning secrets from (passive and active) physical effects, and analyzing the costs of users defending themselves against these attacks, is the “Cryptographic Hardware and Embedded Systems” (CHES) conference series. CHES has run every year since 1999 and has attracted more than 300 attendees every year since 2009.
I can easily be accused of bias—I’ve served on the CHES program committee every year since 2008 (looking mainly at the costs for users)—but I don’t think anyone can dispute the need for this type of research. Competent attackers take advantage of not merely the information that we declare as public but also all of the information they can see through every available side channel. We have to take the same perspective, taking account of all aspects of how secrets are embedded into the real world, if we want to build information-protection systems that society can afford to use but that the attackers find infeasible to break.
The QKD literature doesn’t try to argue that QKD will produce improvements within the traditional chart of $(x,y)=(\hbox{user
cost},\hbox{attack
cost})$. Instead the QKD literature tries to dodge cost questions by claiming that QKD inhabits a magical realm beyond the top of the chart—security “guaranteed by the laws of physics”. Unfortunately, this claim is not justified anywhere in the literature, and it seems very difficult to justify, in light of what the laws of physics actually say.
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[\[99\]]{}
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[^1]: This work was supported by the European Commission under Contract ICT-645622 PQCRYPTO; by the Netherlands Organisation for Scientific Research (NWO) under grant 639.073.005; and by the U.S. National Science Foundation under grant 1314919. “Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation” (or, obviously, other funding agencies). Permanent ID of this document: [e39346ac8e0f20edb8df3334f8751ac75a600099]{}.
[^2]: This metric presumes that the radio signals are converted from analog to digital before being stored. Perhaps the attack is even more effective if the attacker records the signals directly in some analog form.
[^3]: As a side note on costs, it is not obvious in advance exactly how much data needs to be recorded for this QKD attack to succeed. This uncertainty means that a sensible attacker will opt for an “overkill” approach of recording as much data as he can afford to collect; subsequent analysis will show whether this was enough, and the attacker will then use this experience to optimize subsequent attacks. For comparison, an attacker who records the conventional encoding of all data sent through an Internet router is confidently recording all data that normal users of the router will act upon.
[^4]: Hey, look, a squirrel!
|
---
abstract: 'We prove that the ${\rm PU}(2,1)$ configuration space ${\mathfrak{F}}_4$ of four points in $S^3$ is in bijection with ${\mathfrak{H}}^{\star}\times{\mathbb{R}}_{>0}$, where ${\mathfrak{H}}^\star$ is the hyperbolic Heisenberg group. The latter is a Sasakian manifold and therefore ${\mathfrak{H}}^\star\times{\mathbb{R}}_{>0}$ is Kähler.'
address:
- 'Department of Mathematics and Applied Mathematics, University of Crete, Heraklion Crete 70013, Greece.'
- 'School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, P. R. China.'
author:
- 'Ioannis D. Platis & Li-Jie Sun'
title: 'A Kähler structure for the ${\textrm{PU}}(2, 1)$ configuration space of four points in $S^3$'
---
[^1]
Introduction
============
Moduli spaces of $n$-tuples of points in the boundary of symmetric spaces of rank 1 and of non compact type (that is, hyperbolic spaces) are of great interest. Many times these spaces have remarkable geometric properties and they are interesting on their own. In other cases they appear in the study of deformations spaces of topological surfaces, where they serve as parameter spaces, see for instance [@PP0]. There, Fenchel-Nielsen type coordinates are established for the space of discrete, faithful and totally loxodromic representations of the fundamental group of a closed surface of negative Euler characteristic into ${\rm SU}(2,1)$ (that is, the triple cover of the isometry group ${\rm PU}(2,1)$ of the complex hyperbolic plane ${{\bf H}}^2_{\mathbb{C}}$). Incorporated into these coordinates are specific complex parameters, the [*Korányi-Reimann complex cross-ratios*]{}, see Section \[sec-X\] for details; these parameters are directly related to the configuration space of four points in the boundary of ${{\bf H}}^2_{\mathbb{C}}$. The boundary $\partial{{\bf H}}^2_{\mathbb{C}}$ is in turn identified to $S^3$ or, to the one point compactification of the Heisenberg group. Recall that the Heisenberg group ${\mathfrak{H}}$ is the Lie group with underlying manifold ${\mathbb{C}}\times{\mathbb{R}}$ and multiplication given by $$(z,t)\star(w,s)=\left(z+w,t+s+2\Im(z\overline{w})\right),$$ for every $(z,t),(w,s)\in{\mathbb{C}}\times{\mathbb{R}}$; for details, see Section \[sec:heis\].
By ${\mathfrak{C}}_4$ we shall denote the space of ordered quadruples ${\mathfrak{p}}=(p_1,p_2,p_3,p_4)$ of pairwise distinct points in $S^3$. The group ${\rm PU}(2,1)$ acts diagonally on ${\mathfrak{C}}_4$; we denote the quotient by ${\mathfrak{F}}_4$: this is the [*configuration space of four points in $S^3$*]{} and it has been studied rather extensively, see for instance, [@CG; @FP; @PP1; @Pla-CR]. We consider the following subsets of ${\mathfrak{C}}_4$:
1. The subset ${\mathfrak{C}}^{\mathbb{R}}_4$ comprising quadruples ${\mathfrak{p}}$ such that $p_i$ do not all lie in the same ${\mathbb{C}}$-circle.
2. The subset ${\mathfrak{C}}_4'\subset{\mathfrak{C}}^{\mathbb{R}}_4$ comprising quadruples ${\mathfrak{p}}$ such that $p_2,p_3,p_4$ do not lie in the same ${\mathbb{C}}$-circle.
3. The subset ${\mathfrak{C}}^{\mathbb{C}}_4\subset{\mathfrak{C}}^{\mathbb{R}}_4$ comprising quadruples ${\mathfrak{p}}$ such that $p_1$ and $p_2$ do not lie in the same orbit of the stabiliser of $p_3$ and $p_4$.
In [@FP] it has been proved that:
1. The subset ${\mathfrak{F}}^{\mathbb{R}}_4$ is a 4-dimensional manifold.
2. The subset ${\mathfrak{F}}_4'\subset{\mathfrak{F}}^{\mathbb{R}}_4$ admits a [CR]{} structure of codimension 2.
3. The subset ${\mathfrak{F}}^{\mathbb{C}}_4\subset{\mathfrak{F}}^{\mathbb{R}}_4$ is a 2-dimensional complex manifold, biholomorphic to ${\mathbb{C}}P^1\times({\mathbb{C}}\setminus{\mathbb{R}})$.
All the above results are obtained via the identification of ${\mathfrak{F}}_4$ to the [*cross-ratio variety*]{}, see Section \[sec-X\]. Platis also proved in [@Pla-CR] that ${\mathfrak{F}}^{\mathbb{C}}_4$ admits another complex structure. This complex structure and the complex structure in iii) agree on the the CR structure and they are opposite on its complement in the holomorphic tangent bundle of ${\mathfrak{F}}_4$.
A natural question that arises here, is whether the complex structure of ${\mathfrak{F}}^{\mathbb{C}}_4$ can be also defined on the subset ${\mathfrak{F}}_4'$ which by itself is a CR manifold. To this direction, let ${\mathfrak{p}}=(p_1,p_2,p_3, p_4)\in{\mathfrak{C}}_4'$. We may normalise so that $$p_1=(1,\tan a),\quad p_2=\infty,\quad p_3=(0,0),\quad p_4=(z,t),$$ where $a\in(-\pi/2,\pi/2)$ and $z\neq 0$. Let $r=e^{\tan a}>0$. Then we may define a bijective map ${\mathcal{B}}_0:{\mathfrak{F}}_4'\to{\mathbb{C}}_*\times{\mathbb{R}}_{>0}$ given by ${\mathcal{B}}_0([{\mathfrak{p}}])=(z,t,r)$, see Section \[sec-mainth\]. The set ${\mathbb{C}}_{\ast}\times{\mathbb{R}}$ can be given a structure of a Lie group ${\mathfrak{H}}^{\star}$; we call ${\mathfrak{H}}^{\star}$ the [*hyperbolic Heisenberg group*]{}, see Section \[sec:hhg\]. We can endow ${\mathfrak{H}}^{\star}$ with a contact structure $\omega^{\ast}$, see Proposition \[prop:omegasta\]. With this contact structure, ${\mathfrak{H}}^{\star}$ is a contact open submanifold of the Heisenberg group ${\mathfrak{H}}$ with its natural contact form $\omega=dt+2xdy-2ydx.$ The strictly Levi pseudoconvex CR structure associated to $\omega^{\ast}$ is compatible to the strictly Levi pseudoconvex CR structure of the truncated boundary of complex hyperbolic plane, that is, ${{\bf H}}^{\star}=\partial{{\bf H}}^{2}_{{\mathbb{C}}}\setminus\partial{{\bf H}}^{1}_{{\mathbb{C}}},$ see Section \[sec:trunc\]. Using standard methods of contact Riemannian geometry, we show in Section \[sec:sashhg\] that ${\mathfrak{H}}^{\star}$ has a Sasakian structure (Theorem \[thm:HstarSas\]). This automatically implies that the Kähler cone ${\mathcal{C}}({\mathfrak{H}}^{\star})$ is a Kähler manifold, see Section \[sec:kcon\]. In this manner we obtain our main theorem.
[**Theorem \[thm-main\]**]{} [*${\mathfrak{F}}_4'$ inherits the Kähler structure of $\mathcal C({\mathfrak{H}}^{\star})$*]{}.
It is worth noting that under the identification given by Theorem \[thm-main\], the set ${\mathfrak{F}}'_4$ inherits the structure of the Lie group ${\mathfrak{H}}^{\star}\times{\mathbb{R}}_{>0}.$ The complex structure of ${\mathfrak{F}}_4'$ defined from Theorem \[thm-main\] here and the complex structure defined in the set ${\mathfrak{F}}^{\mathbb{C}}_4$ agree on the subbundle of the CR structure, see Sections \[sec-CReq\] and \[sec-complex\].
This paper is organised as follows: After the preliminaries in Section \[sec:prel\], we study the hyperbolic Heisenberg group ${\mathfrak{H}}^\star$ and show its Sasakian structure in Section \[sec:hhg\]; the Kähler structure of ${\mathcal{C}}({\mathfrak{H}}^\star)$ is studied in Section \[sec:kcon\]. In Section \[sec-conf\] we prove our main result; the rest of the paper is devoted to showing that the structures involved in the Kähler cone ${\mathcal{C}}({\mathfrak{H}}^\star)$, i.e., the CR and the complex structure, are CR compatible with the respective structures that have been established in [@FP].
Preliminaries {#sec:prel}
=============
In this section we review [CR]{}, contact and Sasakian structures (Section \[sec:CRetc\]), the basics about complex hyperbolic plane and its boundary (Section \[sec:chp\]) and finally the Heisenberg group and its Sasakian geometry (Section \[sec:heis\]).
[CR]{}, contact and Sasakian structures {#sec:CRetc}
---------------------------------------
The material of this section is standard. We refer for instance to [@Be], [@Bo-Ga], for further details.
Let $M$ be a $(2p+s)$-dimensional real manifold. A codimension $s$ [CR]{} structure in $M$ is a pair $({\mathcal{H}}, J)$ where ${\mathcal{H}}$ is a $2p$-dimensional smooth subbundle of ${\rm T}(M)$ and $J$ is an almost complex endomorphism of ${\mathcal{H}}$ which is formally integrable: If $X$ and $Y$ are sections of ${\mathcal{H}}$ then the same holds for $\left[X, Y\right]-\left[JX, JY\right], \left[JX, Y\right]+\left[X, JY\right]$ and moreover, $J(\left[X, Y\right]-\left[JX, JY\right])=\left[JX, Y\right]+\left[X, JY\right]$.
If $s=1$, a contact structure on $M$ is a codimension 1 subbundle ${\mathcal{H}}$ of ${\rm T}(M)$ which is completely non-integrable; alternatively, ${\mathcal{H}}$ may be defined as the kernel of a 1-form $\eta$, called the contact form of $M$, such that $\eta\wedge (d\eta)^{p}\neq 0$. The dependence of ${\mathcal{H}}$ on $\eta$ is up to multiplication of $\eta$ by a nowhere vanishing smooth function. By choosing an almost complex structure $J$ defined in ${\mathcal{H}}$ we obtain a [CR]{} structure $({\mathcal{H}}, J)$ of codimension 1 in $M$. The subbundle ${\mathcal{H}}$ is also called the horizontal subbundle of ${\rm T}(M)$. The closed form $d\eta$ endows ${\mathcal{H}}$ with a symplectic structure and we may demand from $J$ to be such that $d\eta(X,JX)>0$ for each $X\in{\mathcal{H}}$; we then say that ${\mathcal{H}}$ is strictly pseudoconvex. The Reeb vector field $\xi$ is the vector field which satisfies $
\eta(\xi)=1$ and $\xi\in\ker(d\eta).
$ By the contact version of Darboux’s Theorem, $\xi$ is unique up to change of coordinates.
Neat examples of contact structures on 3-dimensional manifolds are the strictly pseudoconvex [CR]{} structures on boundaries of domains in ${\mathbb{C}}^2$. Let $D\subset {\mathbb{C}}^2$ be a domain with defining function $\rho:D\to{\mathbb{R}}_{>0}$, $\rho=\rho(z_1,z_2)$. On the boundary $M=\partial D$ we consider the form $d\rho$; if $J$ is the complex structure of ${\mathbb{C}}^2$ we then let $$\eta=-\frac{1}{2}d^c\rho=-\Im(\partial\rho)=-\frac{1}{2}Jd\rho.$$ We thus obtain the [CR]{} structure $({\mathcal{H}}=\ker(\eta),J)$. This is a contact structure if and only if the Levi form $
L=d\eta=i\partial\overline{\partial}\rho
$ is positively oriented.
Let now $(M,\eta,J)$ be a contact manifold with $\dim(M)=2p+1$. The almost complex structure $J$ on ${\mathcal{H}}$ is then extended to an endomorphism $\phi$ of the whole tangent bundle ${\rm T}(M)$ by setting $\phi(\xi)=0$. Subsequently, a canonical Riemannian metric $g$ is defined in $M$ from the relations $$\label{eq:contactmetric}
\eta(X)=g(X,\xi),\quad \frac{1}{2}d\eta(X,Y)=g(\phi X, Y),\quad \phi^2(X)=-X+\eta(X)\xi,$$ for each $X\in {\rm T}(M)$. We then call $(M;\eta,\xi,\phi,g)$ the contact Riemannian structure on $M$ associated to the contact structure $(M;{\mathcal{H}},J)$. If $f:M\to M$ is an automorphism which preserves the contact Riemannian structure, then one may use Eqs. (\[eq:contactmetric\]) to verify straightforwardly that this happens if and only if $f$ is [CR]{}, that is $f_*J=Jf_*$ and also $f^{*}\eta=\eta$.
A contact Riemannian manifold for which the Reeb vector field $\xi$ is Killing (equivalently, $\xi$ is an infinitesimal [CR]{} transformation) is called a K-contact Riemannian manifold.
Consider now the Riemannian cone ${\mathcal{C}}(M)=(M\times{\mathbb{R}}_{>0},\;g_r=dr^2+r^2g)$. We may define an almost complex structure ${\mathbb{J}}$ in ${\mathcal{C}}(M)$ by setting $${\mathbb{J}}X=JX,\quad X\in{\mathcal{H}}(M),\quad {\mathbb{J}}(r\partial_r)=\xi.$$ The fundamental 2-form for ${\mathcal{C}}(M)$ is then the exact form $$\Omega_r=d\left(\frac{r^2}{2}\eta\right)=r\;dr\wedge\eta+\frac{r^2}{2}\;d\eta,$$ and therefore it is closed. We then have that $(M;\eta,\xi,\phi,g)$ is Sasakian if and only if the [*Riemannian cone*]{} $({\mathcal{C}}(M);{\mathbb{J}},g_r,\Omega_r)$ is Kähler. The following proposition is often useful:
\[prop:S-c\] A K-contact Riemannian manifold $M$ is Sasakian if and only if $\xi$ is unit vector field and $$R(X,\xi)Y=g(X,Y)\xi-g(\xi,Y)X,$$ for all vector fields $X,Y$ in $M$. Here $R$ is the Riemannian curvature tensor of $g$.
We wish to further comment on the sub-Riemannian geometry of a contact and a contact Riemannian manifold, respectively. If $(M, {\mathcal{H}}, J)$ is contact, we may first define a Riemannian metric $g_{cc}$ in ${\mathcal{H}}$ (the sub-Riemannian metric); the distance $d_{cc}(p,q)$ between two points $p,q$ of $M$ is given by the infimum of the $g_{cc}$-length of horizontal curves joining $p$ and $q$. By a horizontal curve $\gamma$ we mean a piece-wise smooth curve in $M$ such that $\dot\gamma\in{\mathcal{H}}$. The metric $d_{cc}$ is the Carnot-Carathéodory metric and there are two interesting facts about it: Firstly, the metric topology coincides with the manifold topology and secondly, if $g_{cc}'$ is another sub-Riemannain metric, then $d_{cc}$ and $d_{cc}'$ are bi-Lipschitz equivalent on compact susbsets of $M$. In the case where we construct a contact Riemannian structure $(M;\eta,\xi,\phi,g)$ out of a contact structure $(M;\eta,J)$ as above, the sub-Riemannian metric $g_{cc}$ may be taken as the restriction of $g$ into ${\mathcal{H}}\times{\mathcal{H}}$, i.e., $g=g_{cc}+\eta\otimes\eta$. If $d_g$ is the Riemannian distance corresponding to the Riemannian metric $g$ and $d_{cc}$ is the Carnot-Carathéodory distance corresponding to $g_cc$, then we always have $d_g\le d_{cc}$. It also follows that the group ${\rm Aut}(M)$ of automorphisms of the contact Riemannian structure $g$ is just the group ${\rm Isom}_{cc}(M)$ of isometries of $d_{cc}$. If the contact Riemannian structure is Sasakian, then the group ${\rm Aut}({\mathcal{C}}(M))$ of automorphisms of ${\mathcal{C}}(M)$ is just ${\rm Isom}_{cc}(M)$.
Complex hyperbolic plane {#sec:chp}
------------------------
Let ${\mathbb{C}}^{2, 1}$ be a 3-dimensional ${\mathbb{C}}$-vector space equipped with a Hermitian form of signature $(2, 1)$. For the purpose of our paper we shall work with the Hermitian form given by the matrix $$H=\left(\begin{matrix}
0 &\quad 0 &\quad 1\\
0&\quad1&\quad0\\
1&\quad0&\quad0
\end{matrix}\right).$$ Thus $\langle {\bf z, w}\rangle=\overline{{\bf w}}^{\ast}H{\bf z}=z_1\overline{w_3}+z_2\overline{w_2}+z_3\overline{w_1},$ where ${\bf z}=[z_1, z_2, z_3]^t$ and ${\bf w}=[w_1, w_2, w_3]^t$.
Complex hyperbolic plane ${{\textbf{H}_{\mathbb{C}}^2}}$ is the projectivisation in ${\mathbb{C}}^2$ of negative vectors in ${\mathbb{C}}^{2,1}$, that is, vectors ${{\bf z}}$ such that $\langle{{\bf z}},{{\bf z}}\rangle<0$. The resulting domain, described in coordinates $(z_1,z_2)$ of ${\mathbb{C}}^2$ by $$\label{eq:Sieg}
\rho(z_1,z_2)=2\Re(z_1)+|z_2|^2<0,$$ is the *Siegel domain* model for ${{\textbf{H}_{\mathbb{C}}^2}}$. This is a Kähler manifold, its Kähler metric is the Bergman metric. The boundary $\partial{{\textbf{H}_{\mathbb{C}}^2}}$ of complex hyperbolic space is the projectivisation of null vectors of ${\mathbb{C}}^{2,1}$, that is, vectors ${{\bf z}}$ such that $\langle{{\bf z}},{{\bf z}}\rangle=0$, and is identified with the one point compactification of the boundary of the Siegel domain, that is $S^{3}$. On the other hand, the boundary of the Siegel domain is naturally identified to the 3-dimensional *Heisenberg group* $\mathfrak{H}$. This is the set $\mathbb{C}\times\mathbb{R}$ with the group law $$(\zeta_{1},t_{1})\cdot(\zeta_{2},t_{2})=
(\zeta_{1}+\zeta_{2},t_{1}+t_{2}+2\Im(\zeta_{1}
\overline{\zeta_{2}})).$$ Coordinates for $\mathfrak{H}$ are thus $(z,t)$ and therefore $\partial{{\textbf{H}_{\mathbb{C}}^2}}$ may be viewed as the set comprising [*standard lifts*]{} of points $(z,t)\in\mathfrak{H}$, that is, $$\left[ \begin{array}{ccc}
-|z|^2+it\\
\sqrt{2}\;z\\
1
\end{array} \right],$$ and the point at infinity $\infty$ which shall be $[1,0,0]^{t}$.
Any totally geodesic subspace in ${{\textbf{H}_{\mathbb{C}}^2}}$ is one of the the following types:\
(i) Complex geodesic, which is an isometrically embedded copy of $\textbf{H}^{1}_{\mathbb{C}}$. It has the Poincaré model of hyperbolic geometry with constant curvature -1;\
(ii) Totally real Lagrangian plane, which is an isometrically embedded copy of $\textbf{H}^{2}_{\mathbb{R}}$. It has the Beltrami-Klein projective model with constant curvature -1/4.
The intersection of a complex geodesic $L$ with $\partial {{\textbf{H}_{\mathbb{C}}^2}}$ is called a *$\mathbb{C}$-circle*. Correspondingly, the intersection of a totally real Lagrangian plane with $\partial{{\textbf{H}_{\mathbb{C}}^2}}$ is called an *$\mathbb{R}$-circle*. For more details we refer for instance to [@G].
Heisenberg group {#sec:heis}
----------------
The Heisenberg group ${\mathfrak{H}}$ is a 2-step nilpotent Lie group. Consider the left-invariant vector fields $$\begin{aligned}
X=\frac{\partial}{\partial x}+2y\frac{\partial}{\partial t},\quad Y=\frac{\partial}{\partial y}-2x\frac{\partial}{\partial t},
\quad T=\frac{\partial}{\partial t}.\end{aligned}$$ The vector fields $X,Y,T$ form a basis for the Lie algebra $\mathfrak{h}$ of ${\mathfrak{H}}$; this has a grading $\mathfrak{h} = \mathfrak{v}_1\oplus \mathfrak{v}_2$ with $$\mathfrak{v}_1 = \mathrm{span}_{{\mathbb{R}}}\{X, Y\}\quad \text{and}\quad \mathfrak{v}_2=\mathrm{span}_{{\mathbb{R}}}\{T\}.$$ It is well known that Heisenberg group ${\mathfrak{H}}$ admits a strictly pseudoconvex CR structure with contact form $\omega=d t+ 2 \Im(\bar{z}dz)$, and the Reeb vector field for $\omega$ is $T$. Following the strategy described in Section \[sec:CRetc\], one may define a contact Riemannian structure on $\frak{H}$. The endomorphism $\Phi$ of ${\rm T}({\mathfrak{H}})$ is given by $$\Phi(X)=JX=Y,\quad\Phi(Y)=JY=-X,\quad\Phi(T)=0,$$ and the Riemannian tensor for $g$ may be written in Cartesian coordinates as $$\label{eq:gH}
g=g_{cc}+\omega^2=dx^2+dy^2+(dt+2x\;dy-2y\;dx)^2.$$ It can be then shown that $({\mathfrak{H}};\omega,T,\Phi,g)$ is Sasakian, see for instance [@Bo].
The hyperbolic Heisenberg group and its Sasakian structure {#sec:hhg}
==========================================================
In Section \[sec:hypheis\] we introduce the hyperbolic Heisenberg group ${\mathfrak{H}}^\star$; we then give its left-invariant contact form obtained from the contact structure of the Heisenberg group ${\mathfrak{H}}$ in Section \[sec:con\]. In Section \[sec:trunc\] we study the CR structure of the boundary of the truncated complex hyperbolic plane ${{\bf H}}^{\star}$. We show next that ${\mathfrak{H}}^\star$ has a Sasakian structure in Section \[sec:sashhg\]. Finally we get that: ${\mathcal{C}}({\mathfrak{H}}^\star)={\mathfrak{H}}^\star\times{\mathbb{R}}_{>0}$ is Kähler; also the Sasakian structure of ${\mathfrak{H}}^{\star}$ can be carried to the unit tangent bundle of the hyperbolic plane.
Hyperbolic Heisenberg group {#sec:hypheis}
---------------------------
The hyperbolic Heisenberg group ${\mathfrak{H}}^\star$ is ${\mathbb{C}}_*\times{\mathbb{R}}$ with multiplication rule $$(z,t)\star(w,s)=(zw,t+s|z|^2).$$
One verifies straightforwardly that ${\mathfrak{H}}^\star$ is a non-Abelian group; the unit element of ${\mathfrak{H}}^\star$ is (1,0) and the inverse of an arbitrary $(z,t)\in{\mathfrak{H}}^\star$ is $(1/z,-t/|z|^2)$. The hyperbolic Heisenberg group ${\mathfrak{H}}^\star$ is a Lie group with underlying manifold ${\mathbb{C}}_*\times{\mathbb{R}}$; indeed, the map $$\begin{aligned}
&&
{\mathfrak{H}}^\star\times{\mathfrak{H}}^\star\ni\left((z,t),(w,s)\right)\mapsto (z,t)^{-1}\star (w,s)=\left(\frac{w}{z},\frac{-t+s}{|z|^2}\right)\in{\mathfrak{H}}^\star,\end{aligned}$$ is clearly smooth. We fix a left translation $$F(z,t)=L_{(w,s)}(z,t)=(wz,s+t|w|^2),$$ and we consider the complex matrix $DF$ of the differential $F_{*}$: $$DF=\left[\begin{matrix}
w&0&0\\
0&\overline{w}&0\\
0&0&|w|^2
\end{matrix}\right].$$ The vector fields $$\label{eq:basisc}
Z^{*}=z\frac{\partial}{\partial z},\quad \overline{Z}^{*}=\overline{z}\frac{\partial}{\partial \overline{z}},\quad T^{*}=|z|^2\frac{\partial}{\partial t},$$ form a left-invariant basis for the Lie algebra of ${\mathfrak{H}}^{\star}$ and the corresponding real basis is $$\label{eq:basis}
X^{*}=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y},\quad Y^{*}=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x},\quad T^{*}=(x^2+y^2)\frac{\partial}{\partial t},$$ so that $$Z^{*}=\frac{1}{2}(X^{*}-iY^{*}),\quad \overline{Z}^{*}=\frac{1}{2}(X^{*}+iY^{*}).$$ We have the bracket relations $$\label{eq:nontrivb}
[X^{*},T^{*}]=2T^{*},\quad [X^*,Y^*]=[Y^*,T^*]=0.$$ Also, since ${\textrm{det}}(DF)=|w|^4$ we have that the Haar measure of ${\mathfrak{H}}^{\star}$ is given by $$dm=\frac{dx\wedge dy\wedge dt}{(x^2+y^2)^2}=\frac{i}{2}\cdot\frac{dz\wedge d\overline{z}\wedge dt}{|z|^2}.$$ We consider the natural left-invariant Riemannian metric $g$ in ${\mathfrak{H}}^{\star}$ for which the elements of the left-invariant basis $\{X^{*},Y^{*},T^{*}\}$ are orthonormal. The metric $g$, which may be written in coordinate form as $$g=\frac{dx^2+dy^2}{|z|^2}+\frac{dt^2}{|z|^4},$$ enjoys the property that the sectional curvature is zero for the distinguished planes of $X^{*},Y^{*}$ and $Y^{*},T^{*}$, whereas it is -4 for the planes of $X^{*},T^{*}$. To show this, let $\nabla$ be the Riemannian connection. Using Koszul’s formula $$\label{eq:Koszul}
-2g(Z,\nabla_YX)=g([X,Z],Y)+g([Y,Z],X)+g([X,Y],Z),$$ we find $$\nabla_{T^{*}}X^{*}=-2T^{*},\quad \nabla_{T^{*}}T^{*}=2X^*,$$ and all other covariant derivatives vanish. Thus if $R$ is the Riemannian curvature tensor of $g$, $$\label{eq:Curvten}
R(X,Y)Z=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z,$$ the only non-vanishing sectional curvature is $$K(X^{*},T^{*})=g(R(X^{*},T^{*})X^{*},T^{*})=-4.$$ There are obvious CR structures which may be defined on ${\mathfrak{H}}^{\star}$; for example:
1. By setting $JX^*=T^*$ and $JT^*=-X^*$;
2. By setting $JX^*=Y^*$ and $JY^*=-X^*$;
3. By setting $JY^*=T^*$ and $JT^*=-Y^*$.
But none of them is strongly pseudoconvex; in fact they are all integrable.
To the direction of detecting a strongly pseudoconvex CR structure for ${\mathfrak{H}}^\star$, we define a distinguished basis for the Lie algebra. We consider the vector fields $$\label{eq-XYT}
{{\bf X}}=X^*,\quad {{\bf Y}}=Y^*-2T^*,\quad{{\bf T}}=Y^*.$$ They are left-invariant and form a basis for the tangent space of ${\mathfrak{H}}^{\star}$. The only non-trivial Lie bracket relation between ${{\bf X}},{{\bf Y}}$ and ${{\bf T}}$ is $$[{{\bf X}},{{\bf Y}}]=2({{\bf Y}}-{{\bf T}}).$$ We shall also use the complex vector fields $${{\bf Z}}=\frac{1}{2}({{\bf X}}-i{{\bf Y}}),\quad\overline{{{\bf Z}}}=\frac{1}{2}({{\bf X}}+i{{\bf Y}}).$$ Notice that $${{\bf Z}}=zZ,\quad \overline{{{\bf Z}}}=\overline{z}\overline{Z},$$ where $Z$ and $\overline{Z}$ are the complex left-invariant vector fields of the Heisenberg group ${\mathfrak{H}}$.
Contact structure {#sec:con}
-----------------
There exists a left-invariant contact form for the hyperbolic Heisenberg group ${\mathfrak{H}}^\star$ obtained from the contact structure of the form $\omega$ of the Heisenberg group ${\mathfrak{H}}$. We shall denote the form we are about to define by $\omega^\star$; this form is natural in the sense that it is left-invariant and the Haar measure on ${\mathfrak{H}}^\star$ is obtained by the wedge product of this form by its differential. The following proposition describes this form.
\[prop:omegasta\] We consider the following 1-form in the hyperbolic Heisenberg group ${\mathfrak{H}}^\star$: $$\omega^\star=\frac{\omega}{2|z|^2},$$ where $\omega$ is the restriction to ${\mathbb{C}}_*\times{\mathbb{R}}$ of the contact form $\omega$ of the Heisenberg group ${\mathfrak{H}}$. Then the manifold $({\mathfrak{H}}^\star,\omega^\star)$ is contact. Explicitly:
1. The form $\omega^\star$ of ${\mathfrak{H}}^\star$ is left-invariant.
2. If $dm$ is the Haar measure for ${\mathfrak{H}}^\star$ then $dm=-\omega^\star \wedge d\omega^\star$.
3. The kernel of $\omega^\star$ is $$\ker(\omega^\star)=\langle{{\bf X}},{{\bf Y}}\rangle.$$
4. The Reeb vector field for $\omega^\star$ is ${{\bf T}}$.
5. Let ${\mathcal{H}}^\star=\ker(\omega^\star)$ and consider the almost complex structure $J$ defined on ${\mathcal{H}}$ by $$J{{\bf X}}={{\bf Y}},\quad J{{\bf Y}}=-{{\bf X}}.$$ Then $J$ is compatible with $d\omega^\star$ and moreover, ${\mathfrak{H}}^\star$ is a strictly pseudoconvex [CR]{} structure; that is, $d\omega^\star$ is positively oriented on ${\mathcal{H}}^\star$.
To prove (i) fix an arbitrary $(w,s)\in{\mathfrak{H}}^\star$ and consider the left translation $
F(z,t)=L_{(w,s)}(z,t)=(wz,s+t|w|^2).
$ Then $$\begin{aligned}
&&
F^*(\omega^\star)=\frac{d(s+t|w|^2)+2\Im(\overline{wz}d(wz))}{2|w|^2|z|^2}
=\frac{|w|^2\left(dt+2\Im(\overline{z}dz)\right)}{2|w|^2|z|^2}=\omega^\star.\end{aligned}$$ To prove (ii) we write first $$\omega^*=\frac{dt}{2|z|^2}+d\arg z,$$ therefore $$d\omega^*=2\left(\frac{d(|z|^2)}{2|z|^2}\right)\wedge \left(-\frac{dt}{2|z|^2}\right).$$ We then calculate $$\omega^\star\wedge d\omega^\star=\frac{dt\wedge dx\wedge dy}{|z|^4}=-dm.$$ Condition (iii) is obvious. Now, to prove (iv) we have first that $\omega^\star({{\bf T}})=1$. On the other hand, one verifies straightforwardly that $d\omega^\star({{\bf T}},U)=0$ for all $U\in{\rm T}({\mathfrak{H}}^\star)$.
Finally, for (v) we first observe that the dual basis to $\{{{\bf X}},{{\bf Y}},{{\bf T}}\}$ is $\{\phi^*,\psi^*,\omega^\star\}$ where $$\label{eq:basis-cotr}
\phi^\star=\frac{d(|z|^2)}{2|z|^2},\quad \psi^\star=-\frac{dt}{2|z|^2}.$$ In this basis, $$d\phi^\star=0,\quad d\psi^\star=-2\;\phi^*\wedge\psi^*,\quad d\omega^\star=2\;\phi^\star\wedge\psi^\star.$$ For the symplectic form $d\omega^\star$ we thus have $$d\omega^\star({{\bf X}},{{\bf Y}})=2,\quad d\omega^\star({{\bf X}},{{\bf T}})=d\omega^\star({{\bf Y}},{{\bf T}})=0.$$ This completes the proof.
${\mathfrak{H}}^{\star}$ and ${{\bf H}}^{\star}$ {#sec:trunc}
------------------------------------------------
Consider the set $${{\bf H}}^{\star}=\partial{{\bf H}}^2_{\mathbb{C}}\setminus\partial{{\bf H}}^1_{\mathbb{C}}.$$ The ${{\bf H}}^{\star}$ comprises of points $(z_1,z_2)$ of ${\mathbb{C}}^2$ such that $$\rho^{\star}(z_1,z_2)=\frac{2\Re(z_1)}{|z_2|^2}+1=\frac{\rho(z_1,z_2)}{|z_2|^2}=0,$$ where $\rho$ is the defining function of ${{\bf H}}^2_{\mathbb{C}}$ as in (\[eq:Sieg\]). Let $\Psi:{\mathfrak{H}}^{\star}\to{{\bf H}}^{\star}$ be the bijection given by $$\Psi(z,t)=(-|z|^2+it,\sqrt{2}z).$$
There is a strictly pseudoconvex CR structure on ${{\bf H}}^{\star}$ and the map $\Psi$ is CR. Also, if $\eta^{\star}$ is the corresponding contact form, then $\Psi^{*}\eta^{\star}=\omega^\star$, where $\omega^\star$ is the contact form of the hyperbolic Heisenberg group.
It follows from $\rho^{\star}(z_1,z_2)=\frac{\rho(z_1,z_2)}{|z_2|^2}$ that $$\partial \rho^{\star}=\frac{dz_1+\overline{z_2}dz_2}{|z_2|^2},\qquad
\bar{\partial} \rho^{\star}=\frac{d\overline{z_1}+z_2d\overline{z_2}}{|z_2|^2}.$$ A CR structure is defined by the (1, 0) vector field $Z=-|z_2|^2\frac{\partial}{\partial z_1}+z_2\frac{\partial}{\partial z_2}$, and direct calculation yields to $$\partial\bar{\partial} \rho^{\star}=-\frac{\overline{z_2}}{|z_2|^4}dz_2\wedge d\overline{z_1},$$ which indicates that $$\partial\bar{\partial}{ \rho}^{\star}(Z, \bar{Z})=-\frac{\overline{z_2}}{|z_2|^4}dz_2\wedge d\overline{z_1}\left(-|z_2|^2\frac{\partial}{\partial z_1}+z_2\frac{\partial}{\partial z_2}, -|z_2|^2\frac{\partial}{\partial \overline{z_1}}+\overline{z_2}\frac{\partial}{\partial \overline{z_2}}\right)\\
=1>0.$$ Therefore the Levi form is positively oriented on the CR structure. Now, $$\begin{aligned}
\Psi_{\ast}({{\bf Z}})&=&{{\bf Z}}(-|z|^2+it)\frac{\partial}{\partial z_1}+{{\bf Z}}(-|z|^2-it)\frac{\partial}{\partial \overline{z_1}}
+{{\bf Z}}(\sqrt{2}z)\frac{\partial}{\partial z_2}
+{{\bf Z}}(\sqrt{2}\overline{z})\frac{\partial}{\partial \overline{z_2}}\\
&=&\left(z\frac{\partial}{\partial z}+i|z|^2\frac{\partial}{\partial t}\right)(-|z|^2+it)\frac{\partial}{\partial z_1}+
\left(z\frac{\partial}{\partial z}+i|z|^2\frac{\partial}{\partial t}\right)(\sqrt{2}z)\frac{\partial}{\partial z_2}\\
&=&-|z_2|^2\frac{\partial}{\partial z_1}+z_2\frac{\partial}{\partial z_2}=Z \end{aligned}$$ and also, $$\eta^{\star}=\Im(\partial \rho^{\star})=\frac{dy_1+\Im(\overline{z_2}dz_2)}{|z_2|^2}.$$ The proof is complete.
There is a map $M$ between the hyperbolic Heisenberg group ${\mathfrak{H}}^{\star}$ and the group $\rm{SU}(1, 1)\times \rm{U}(1)$. This is given for each $(z, t)\in{\mathfrak{H}}^{\star}$ by $$M(z, t)=\left(
\left(\begin{matrix}
|z|\quad& i\frac{t}{|z|}\\
0\quad& \frac{1}{|z|}\\
\end{matrix}\right),
e^{i\arg z}
\right).$$ It is clear that $M$ is an immersion and a group homomorphism; one can verify that if $(z, t)\in{\mathfrak{H}}^{\star}$, then $$M((z, t)\star(w, s))=M(z, t)\cdot M(w, s),$$ where $\cdot$ is the multiplication in $\rm{SU}(1, 1)\times \rm{U}(1)$. That is, matrix multiplication in the $\rm{SU}(1, 1)$ component and multiplication of complex numbers in the $\rm{U}(1)$ component.
We consider the $KAN$ decomposition of $\rm{SU}(1, 1)$: $K={\rm SO}(1,1)$, $A=({\mathbb{R}}_{>0},\cdot)$ and $N=({\mathbb{R}},+)$. Then the image of $M$ is exactly $AN\times\rm{U}(1)$. It is well known that $AN$ can be identified to the complex hyperbolic line ${{\bf H}}^1_{\mathbb{C}}$. We have the following.
If ${\mathfrak{H}}^\star$ is the hyperbolic Heisenberg group, then $${\mathfrak{H}}^\star\simeq ({\rm SU}(1,1)\times{\rm U}(1))/{\rm SO}(1,1).$$
In this manner, we have an identification of the Heisenberg group and the unit tangent bundle of the hyperbolic plane $T_1({{\bf H}}^1_{\mathbb{C}})$, which is diffeomorphic to ${{\bf H}}^1_{\mathbb{C}}\times S^1$.
Another bijection $K:$ ${\mathfrak{H}}^\star\to {{\bf H}}^1_{\mathbb{C}}\times S^1$ is given by $$\label{eq:kormap}
K(z,t)=(-|z|^2+it,\arg z).$$ The inverse $K^{-1}:$ ${{\bf H}}^1_{\mathbb{C}}\times S^1\to{\mathfrak{H}}^\star$ is then given by $$\label{eq:kormapinv}
K^{-1}(\zeta, \phi)=(\sqrt{-\Re(\zeta)}e^{i\phi}, \Im(\zeta)),\qquad (\zeta, \phi)\in{{\bf H}}^1_{\mathbb{C}}\times S^1.$$ This map carries the Sasakian structure of ${\mathfrak{H}}^\star$ which we study in the next section, to the usual Sasakian structure of the unit tangent bundle of the hyperbolic plane, see Section \[sec:T1\].
Sasakian structure {#sec:sashhg}
------------------
We start by establishing a contact Riemannian structure on ${\mathfrak{H}}^\star$. Let $\{{{\bf X}},{{\bf Y}},{{\bf T}}\}$ be the basis for the tangent space as in (\[eq-XYT\]) and let also $\{\phi^\star,\psi^\star,\omega^\star\}$ be the dual basis as in (\[eq:basis-cotr\]). By using equations (\[eq:contactmetric\]) we verify straightforwardly that the Riemannian metric which we shall denote by $g^*$ obtained out of the endomorphism of the tangent space (we shall denote by $\Phi^*$, the [CR]{} structure) and the contact form $\omega^\star$ is given by declaring the basis $\{{{\bf X}},{{\bf Y}},{{\bf T}}\}$ orthonormal. Hence, $({\mathfrak{H}}^\star;\omega^\star,{{\bf T}},\Phi^\star,g^\star)$ is a contact Riemannian manifold. The Riemannian tensor is also written as
$$\label{eq-Riem}
g^\star=ds^2=(\phi^\star)^2+(\psi^\star)^2+(\omega^\star)^2=\frac{(d(|z|^2))^2+dt^2}{4|z|^4}+\frac{(dt+2xdy-2ydx)^2}{4|z|^4}.$$
The restriction of $g^\star$ into ${\mathcal{H}}^\star=\{{{\bf X}},{{\bf Y}}\}$ is $$g^\star_{cc}=(\phi^\star)^2+(\psi^\star)^2=\frac{(d(|z|^2))^2+dt^2}{4|z|^4}$$ and defines the Kähler structure on the horizontal tangent bundle ${\mathcal{H}}^\star$. Note that $g^*_{cc}$ which is the pullback from the left half hyperbolic plane of the hyperbolic metric to ${\mathfrak{H}}_{\star}$, via the Korányi map $(z,t)\mapsto -|z|^2=it$ (compare with [@G]). We shall prove:
\[thm:HstarSas\] $({\mathfrak{H}}^\star;\omega^\star,{{\bf T}},\Phi^\star,g^\star)$ is Sasakian.
If $\nabla$ is the Riemannian connection of $g^\star$, then using Koszul’s formula (\[eq:Koszul\]) we have: $$\begin{aligned}
&&
\nabla_{{\bf X}}{{\bf X}}=0,\quad \nabla_{{\bf Y}}{{\bf X}}=-2{{\bf Y}}+{{\bf T}},\quad \nabla_{{\bf T}}{{\bf X}}={{\bf Y}},\\
&&
\nabla_{{\bf X}}{{\bf Y}}=-{{\bf T}},\quad \nabla_{{\bf Y}}{{\bf Y}}=2{{\bf X}},\quad \nabla_{{\bf T}}{{\bf Y}}=-{{\bf X}},\\
&&
\nabla_{{\bf X}}{{\bf T}}={{\bf Y}},\quad \nabla_{{\bf Y}}{{\bf T}}=-{{\bf X}},\quad \nabla_{{\bf T}}{{\bf T}}=0.\end{aligned}$$ Denote by $R$ the curvature tensor. Then by equation (\[eq:Curvten\]) we get that $$\begin{aligned}
\label{eq:CT}
&&\notag
R({{\bf X}},{{\bf Y}}){{\bf X}}=-7{{\bf Y}},\quad
R({{\bf X}}, {{\bf T}}){{\bf X}}={{\bf T}},\quad R({{\bf Y}}, {{\bf T}}){{\bf X}}=0,\\
&&
R({{\bf X}}, {{\bf Y}}){{\bf Y}}={{\bf X}},\quad
R({{\bf X}}, {{\bf T}}){{\bf Y}}=0,\quad R({{\bf Y}}, {{\bf T}}){{\bf Y}}={{\bf T}},\\
&&\notag
R({{\bf X}}, {{\bf Y}}){{\bf T}}=4{{\bf X}},\quad
R({{\bf X}}, {{\bf T}}){{\bf T}}=-{{\bf X}},\quad R({{\bf Y}}, {{\bf T}}){{\bf T}}=-{{\bf Y}}.\end{aligned}$$ To prove that ${\mathfrak{H}}^{\star}$ is $K$-contact, that is, ${{\bf T}}$ is Killing, we show that $$({\mathcal{L}}_{{\bf T}}g^\star)(U,V)=g^\star(\nabla_V{{\bf T}}, U)+g^\star(V,\nabla_U{{\bf T}})=0,$$ for each $U, V\in T({\mathfrak{H}}^\star)$. Letting $U=a{{\bf X}}+b{{\bf Y}}+c{{\bf T}}, V=a'{{\bf X}}+b'{{\bf Y}}+c'{{\bf T}}$ be arbitrary, then we get that $$g^\star(\nabla_V{{\bf T}}, U)+g^\star(V,\nabla_U{{\bf T}})=-ab'+a'b-a'b+ab'=0.$$ Finally, it follows from equations (\[eq:CT\]) that $R(U,{{\bf T}})V=-ac'{{\bf X}}-bc'{{\bf Y}}+(aa'+bb'){{\bf T}}$. Because $g^\star(U, V){{\bf T}}-g^\star({{\bf T}}, V)U=(aa'+bb'+cc'){{\bf T}}-c'(a{{\bf X}}+b{{\bf Y}}+c{{\bf T}})$, we also have $
R(U,{{\bf T}})V=g^\star(U, V){{\bf T}}-g^\star({{\bf T}},V)U
$ for each $U, V\in{\rm T}({\mathfrak{H}}^\star)$ and the theorem is proved.
Using the relation $K(U, V)=g(R(U, V) U, V)$ for sectional curvature of planes spanned by unit vectors $U, V$, we obtain the following:
\[cor:kurv\] The sectional curvatures of distinguished planes are: $$K({{\bf X}},{{\bf Y}})=-7,\quad K({{\bf X}},{{\bf T}})=1,\quad K({{\bf Y}},{{\bf T}})=1.$$
Recall that if $\{X_1,X_2,X_3\}$ is an orthonormal basis of a Riemannian 3-manifold $M$, then the Ricci curvature in the direction of $X_i$ is $${\rm Ric}(X_i)=\frac{1}{2}\sum_{j\neq i}K(X_i,X_j).$$ Moreover, the scalar curvature $K$ is $$K=\frac{1}{3}\sum_{i=1}^3{\rm Ric}(X_i).$$ We obtain straightforwardly:
\[cor:RicH\] The Ricci curvatures ${\rm Ric}(X)$, ${\rm Ric}(Y)$, ${\rm Ric}(T)$, in the directions of $X, Y$ and $T$ respectively are: $$\begin{aligned}
&&
{\rm Ric}(X)=-3,\quad
{\rm Ric}(Y)=-3,\quad
{\rm Ric}(T)=1.\end{aligned}$$ Therefore the scalar curvature is $K=-\frac{5}{3}$.
### From ${\mathfrak{H}}^{\star}$ to $T_1({{{\bf H}}^{1}_{\mathbb{C}}})$ {#sec:T1}
Let $({\mathfrak{H}}^\star;\omega^\star,{{\bf T}},\Phi^\star,g^\star)$ be the Sasakian structure of the hyperbolic Heisenberg group and consider the the Korányi map $K$ and its inverse $K^{-1}$ (Eqs. (\[eq:kormap\]) and (\[eq:kormapinv\]), respectively). Let $(\xi+i\eta, \phi)$ be coordinates for ${{\bf H}}^1_{\mathbb{C}}\times S^1$. Then let $$\omega=(K^{-1})^\ast \omega^{\star},\quad T=K_{*}({{\bf T}}),\quad \Phi=K_{\ast}\circ\Phi^{\star}\circ(K^{-1})_\ast,
\quad g=(K^{-1})^{\ast}g^{\star}.$$ Explicitly, $$\omega=-\frac{d\eta}{2\xi}+d\phi, \quad T=\partial_\phi,\quad
g=\frac{d\xi^2+d\eta^2}{4\xi^2}+\left(d\phi-\frac{d\eta}{2\xi}\right)^2.$$
Then $({{\bf H}}^1_{\mathbb{C}}\times S^1; \omega, T, \Phi, g)$ is a Sasakian structure for the unit tangent bundle of the hyperbolic plane. The Carnot-Carathéodory isometry group is just $\rm{SU}(1, 1)$. [ we refer to Theorem 10 in [@Sa].]{}
The Kähler structure of ${\mathcal{C}}({\mathfrak{H}}^\star)$ {#sec:kcon}
-------------------------------------------------------------
Since $({\mathfrak{H}}^\star;\omega^\star,{{\bf T}},\Phi^\star,g^\star)$ is Sasakian we immediately have for the Riemannian cone ${\mathcal{C}}({\mathfrak{H}}^\star)={\mathfrak{H}}^\star\times_{r^2}{\mathbb{R}}_{>0}$ is Kähler. Here, the complex structure of ${\mathfrak{H}}^\star\times{\mathbb{R}}_{>0}$ is given in terms of the basis $\{{{\bf X}},{{\bf Y}},{{\bf T}},r\partial_r\}$ by $$\label{eq:J-L}
{\mathbb{J}}{{\bf X}}={{\bf Y}},\quad {\mathbb{J}}{{\bf Y}}=-{{\bf X}},\quad {\mathbb{J}}{{\bf T}}=-r\partial_r,\quad {\mathbb{J}}(r\partial_r)={{\bf T}},$$ and the Kähler metric as well as the fundamental 2-form are respectively
$$\begin{aligned}
&&\label{eq:grHC}
g_r^\star=dr^2+r^2\;g^\star,\\
&&\label{eq:OrHC}
\Omega_r^\star=d\left(\frac{r^2\omega^\star}{2}\right)=r\;dr\wedge\omega^\star+\frac{r^2}{2}d\omega^\star=r\;dr\wedge\omega^\star+r^2\;\phi^\star\wedge\psi^\star.\end{aligned}$$
It is clear that $\phi^r=r\phi^\star$, $\psi^r=r\psi^\star$, $\omega^r=r\omega^\star$ and $dr$ form an orthonormal basis for the cotangent space of ${\mathcal{C}}({\mathfrak{H}}^\star)$; in this basis $$\Omega_r^\star=\phi^r\wedge\psi^r+dr\wedge\omega^r.$$ The dual basis is the set $\{{{\bf X}}_r,{{\bf Y}}_r,{{\bf T}}_r,{{\bf S}}_r\}$ where $${{\bf X}}_r=(1/r){{\bf X}},\quad {{\bf Y}}_r=(1/r){{\bf Y}},\quad {{\bf T}}_r=(1/r){{\bf T}},\quad {{\bf S}}_r=\partial/\partial r.$$ The only non-vanishing Lie bracket relations are $$\begin{aligned}
&&
[{{\bf X}}_r,{{\bf Y}}_r]=(2/r)({{\bf Y}}_r-{{\bf T}}_r),\quad[{{\bf X}}_r,{{\bf S}}_r]=(1/r){{\bf X}}_r,\quad
[{{\bf Y}}_r,{{\bf S}}_r]=(1/r){{\bf Y}}_r,\quad[{{\bf T}}_r,{{\bf S}}_r]=(1/r){{\bf T}}_r.\end{aligned}$$
Therefore a basis for the (1, 0) tangent space comprises ${{\bf Z}}_r, {{\bf W}}_r$, where $${{\bf Z}}_r=\frac{1}{2}({{\bf X}}_r-i{{\bf Y}}_r),\qquad {{\bf W}}_r=\frac{1}{2}({{\bf T}}_r+i\partial_r).$$ Accordingly, the (1, 0) cotangent space has a basis comprising $d {{\bf Z}}_r, d{{\bf W}}_r$, where $$d{{\bf Z}}_r=\phi^r+i \psi^r,\qquad d{{\bf W}}_r=\omega^{r}-idr,$$ so that $$\Omega^{r}=\frac{i}{2}\left(d{{\bf Z}}_r\wedge d\overline{{{\bf Z}}_r}+d {{\bf W}}_r\wedge d\overline {{\bf W}}_r\right).$$ Let $\rho: {\mathfrak{C}}({\mathfrak{H}}^{\star})\to {\mathbb{R}}$ be a smooth function. Then the (1, 0) and (0, 1) differentials are given respectively by $$\partial \rho={{\bf Z}}_r(\rho)d{{\bf Z}}_r+{{\bf W}}_r(\rho)d{{\bf W}}_r,$$ $$\bar{\partial}\rho=\overline{{{\bf Z}}_r}(\rho)d\overline{{{\bf Z}}_r}+\overline{{{\bf W}}_r}(\rho)d\overline{{{\bf W}}_r}.$$
In the next proposition we compute the sectional curvatures of distinguished planes.
All sectional curvatures of distinguished planes vanish besides that of the distinguished two planes spanned by ${{\bf X}}_r, {{\bf Y}}_r$ and ${{\bf T}}_r, {{\bf S}}_r$ respectively: $$K_r({{\bf X}}_r, {{\bf Y}}_r)=-8/r^2<0, \qquad K_r({{\bf T}}_r, {{\bf S}}_r)=-1/r^2<0.$$
If $\nabla^r$ is the Riemannian connection, we obtain $$\begin{aligned}
&&
\nabla^r_{{{\bf X}}_r}{{\bf X}}_r=-(1/r){{\bf S}}_r,\quad \nabla^r_{{{\bf Y}}_r}{{\bf X}}_r=(1/r)(-2{{\bf Y}}_r+{{\bf T}}_r),\quad \nabla^r_{{{\bf T}}_r}{{\bf X}}_r=(1/r){{\bf Y}}_r,\quad\nabla^r_{{{\bf S}}_r}{{\bf X}}_r=0,\\
&&
\nabla^r_{{{\bf X}}_r}{{\bf Y}}_r=-(1/r){{\bf T}}_r,\quad \nabla^r_{{{\bf Y}}_r}{{\bf Y}}_r=(1/r)(2{{\bf X}}_r-{{\bf S}}_r),\quad \nabla^r_{{{\bf T}}_r}{{\bf Y}}_r=-(1/r){{\bf X}}_r,\quad\nabla^r_{{{\bf S}}_r}{{\bf Y}}_r=0,\\
&&
\nabla^r_{{{\bf X}}_r}{{\bf T}}_r=(1/r){{\bf Y}}_r,\quad \nabla^r_{{{\bf Y}}_r}{{\bf T}}_r=-(1/r){{\bf X}}_r,\quad \nabla^r_{{{\bf T}}_r}{{\bf T}}_r=-(1/r){{\bf S}}_r,\quad\nabla^r_{{{\bf S}}_r}{{\bf T}}_r=0,\\
&&
\nabla_{{{\bf X}}_r}{{\bf S}}_r=(1/r){{\bf X}}_r,\quad\nabla^r_{{{\bf Y}}_r}{{\bf S}}_r=(1/r){{\bf Y}}_r,\quad\nabla^r_{{{\bf T}}_r}{{\bf S}}_r=(1/r){{\bf T}}_r,\quad\nabla^r_{{{\bf S}}_r}{{\bf S}}_r=0.\end{aligned}$$ Hence for the Riemannian curvature tensor $R^r$ we have $$\begin{aligned}
&&
R^r({{\bf X}}_r,{{\bf Y}}_r){{\bf X}}_r=-(8/r^2){{\bf Y}}_r,\qquad
R^r({{\bf T}}_r,{{\bf S}}_r){{\bf T}}_r=-(1/r^2){{\bf S}}_r.\end{aligned}$$ whereas $$\begin{aligned}
&&
R^r({{\bf X}}_r,{{\bf T}}_r){{\bf X}}_r=0,\quad
R^r({{\bf X}}_r,{{\bf S}}_r){{\bf X}}_r=0,\\
&&
R^r({{\bf Y}}_r,{{\bf T}}_r){{\bf Y}}_r=0,\quad
R^r({{\bf Y}}_r,{{\bf S}}_r){{\bf Y}}_r=0,\quad\end{aligned}$$ The proof follows.
The Ricci curvatures of $g_r$ in the directions of ${{\bf X}}_r,{{\bf Y}}_r,{{\bf T}}_r$ and $d/dr$ are respectively $${\rm Ric}({{\bf X}}_r)={\rm Ric}({{\bf Y}}_r)=-\frac{8}{3r^2},\quad {\rm Ric}({{\bf T}}_r)={\rm Ric}({{\bf S}}_r)=-\frac{1}{3r^2},$$ and the scalar curvature is $$K=-\frac{3}{2r^2}.$$
\[prop:HHHKstar\] The hyperbolic Heisenberg group ${\mathfrak{H}}^\star$ is embedded into ${\mathcal{C}}({\mathfrak{H}}^\star)$ as the hypersurface $r=1$.
Configuration Space of Four Points in $\partial{{\textbf{H}_{\mathbb{C}}^2}}$ {#sec-conf}
=============================================================================
We let ${\mathfrak{C}}_4$ be the set of ordered quadruples of pairwise distinct points in the boundary of the complex hyperbolic plane $\partial{{\textbf{H}_{\mathbb{C}}^2}}$ and we denote by ${\mathfrak{F}}_4$ the configuration space of ${\mathfrak{C}}_4$, that is, the quotient of ${\mathfrak{C}}_4$ with respect to the diagonal action of ${\textrm{PU}}(2, 1)$. There are certain subsets of ${\mathfrak{F}}_4$ with interesting geometrical properties; those properties have been studied in [@FP]:
- The subset ${\mathfrak{F}}_4^{\mathbb{R}}$ comprising orbits of quadruples ${\mathfrak{p}}=(p_1,p_2,p_3,p_4)$ such that not all $p_i$ lie in the same ${\mathbb{C}}$-circle. This is a 4-dimensional real manifold.
- The subset ${\mathfrak{F}}_4'$ comprising orbits of quadruples ${\mathfrak{p}}=(p_1,p_2,p_3,p_4)$ such that $p_2,p_3,p_4$ do not lie in the same ${\mathbb{C}}$-circle. This is a CR manifold of codimension 2, see Section \[sec-CRV4\] below.
- The subset ${\mathfrak{F}}^{\mathbb{C}}_4$ comprising orbits of quadruples ${\mathfrak{p}}=(p_1,p_2,p_3,p_4)$ such that $p_1,p_4$ do not lie in the same orbit of the stabiliser of $p_2,p_3$. This is a 2-dimensional (disconnected) complex manifold, see Section \[sec-complex\].
We prove in Section \[sec-mainth\] our main Theorem \[thm-main\]: the set ${\mathfrak{F}}'_4$ can be endowed with a Kähler structure, namely the Kähler structure of the Kähler cone of the hyperbolic Heisengerg group. We then proceed to examine the relations of the already well-known structures of ${\mathfrak{F}}'_4$ with this new structure. In [@FP], see also [@Pla-CR], the configuration space was identified to the cross-ratio variety ${\mathfrak{X}}$. For clarity, we recall basic facts about ${\mathfrak{X}}$ in Section \[sec-X\]. Here we choose to follow a slightly different route, namely, using the result of Gusevskii-Cunha, [@GC]. In this way, we show in Section \[sec-V4\] that the subset ${\mathfrak{F}}'_4$ is naturally identified to a four dimensional variety inside ${\mathbb{C}}^2\times(-\pi/2,\pi/2)$. We denote this variety by ${\mathfrak{V}}_4$ and we recover the codimension 2 CR structure ${\mathcal{H}}$ of ${\mathfrak{F}}'_4$ in Section \[sec-CRV4\]. The CR structure ${\mathcal{H}}$ is generated by a $(1,0)$ vector field $Z$, see (\[eq-CRV\]). Recall now the subbundle ${\mathcal{H}}^\star$ of $T^{(1,0)}({\mathcal{C}}({\mathfrak{H}}^\star))$ generated by the $(1,0)$-vector field ${{\bf Z}}_r$. In Section \[sec-CReq\] we prove that there is a diffeomorphism from ${\mathcal{C}}({\mathfrak{H}}^\star)$ to ${\mathfrak{V}}_4$ which is CR with respect to ${\mathcal{H}}$ and ${\mathcal{H}}^\star$.
Invariants, Cross-Ratio Variety {#sec-X}
-------------------------------
Recall that the [*Cartan’s angular invariant*]{} $\mathbb{A}({\mathfrak{p}})$ of an ordered triple ${\mathfrak{p}}=(p_1,p_2,p_3)$ of pairwise distinct points in $\partial{{\bf H}}^2_{\mathbb{C}}$ is defined by $$\mathbb{A}({\mathfrak{p}})=\arg\left(-\langle {{\bf p}}_1,{{\bf p}}_2\rangle\langle {{\bf p}}_2,{{\bf p}}_3\rangle\langle {{\bf p}}_3,{{\bf p}}_1\rangle\right)\in[-\pi/2,\pi/2],$$ where ${{\bf p}}_i$ are the lifts of $p_i$ respectively. The definition is independent of the choice of lifts, remains invariant under the action of ${\rm PU}(2,1)$. Cartan’s angular invariant $\mathbb{A}({\mathfrak{p}})$ satisfies the properties (see [@G]): ${\mathbb{A}}({\mathfrak{p}})=\pm\pi/2$ if and only if ${\mathfrak{p}}$ is a triple of points lying in the same ${\mathbb{C}}$-circle and ${\mathbb{A}}({\mathfrak{p}})=0$ if and only if ${\mathfrak{p}}$ is a triple of points lying in the same ${\mathbb{R}}$-circle; moreover, for triples ${\mathfrak{p}}$ and ${\mathfrak{p}}'$ of points not lying in the same ${\mathbb{C}}$-circle, there exists a $g\in{\rm PU}(2,1)$ such that $g({\mathfrak{p}})={\mathfrak{p}}'$ (that is, $g(p_i)=p_i'$, $i=1,2,3$) if and only if ${\mathbb{A}}({\mathfrak{p}})={\mathbb{A}}({\mathfrak{p}}')$.
Given a quadruple ${\mathfrak{p}}=(p_{1},p_{2}, p_{3},p_{4})\in{\mathfrak{C}}_4$, then its [*cross-ratio*]{} is defined by $${\mathbb{X}}({\mathfrak{p}})=\mathbb{X}(p_{1},p_{2}, p_{3},p_{4})=\frac{\langle {{\bf p}}_{4},{{\bf p}}_{2}\rangle\langle {{\bf p}}_{3},{{\bf p}}_{1}\rangle}{\langle {{\bf p}}_{4},{{\bf p}}_{1}\rangle\langle {{\bf p}}_{3},{{\bf p}}_{2}\rangle},$$ where ${{\bf p}}_i$ are lifts of $p_i$. The cross-ratio is independent of the choice of lifts and remains invariant under the action of ${\rm SU}(2,1)$. By permuting the points of the configuration ${\mathfrak{p}}$ we will obtain 24 cross-ratios; all these are functions of the following three cross-ratios: $${\mathbb{X}}_1={\mathbb{X}}(p_1,p_2,p_3,p_4),\quad
{\mathbb{X}}_2={\mathbb{X}}(p_1,p_3,p_2,p_4),\quad
{\mathbb{X}}_3={\mathbb{X}}(p_2,p_3,p_1,p_4).$$ These three cross-ratios satisfy the following two real equations: $$\label{eq-X}
\begin{aligned}
&
|{\mathbb{X}}_2|=|{\mathbb{X}}_1|\cdot|{\mathbb{X}}_3|,\\
&
|{\mathbb{X}}_1+{\mathbb{X}}_2-1|^2=2\Re\left({\mathbb{X}}_1(\overline{{\mathbb{X}}_2}+\overline{{\mathbb{X}}_1}{\mathbb{X}}_3)\right).
\end{aligned}$$ Equations (\[eq-X\]) define the cross-ratio variety $\mathfrak{X}$, see [@FP]. The following identifications hold:
- The subset ${\mathfrak{F}}_4^{\mathbb{R}}$ is identified to the subset ${\mathfrak{X}}\setminus{\mathfrak{X}}_{\mathbb{R}}$ of $\frak{X}$, where ${\mathfrak{X}}_{\mathbb{R}}$ is the subset of ${\mathfrak{X}}$ comprising $({\mathbb{X}}_1,{\mathbb{X}}_2,{\mathbb{X}}_3)$ such that all ${\mathbb{X}}_i$ are real, ${\mathbb{X}}_1+{\mathbb{X}}_2=1$ and ${\mathbb{X}}_3=1-(1/{\mathbb{X}}_1)$.
- The subset ${\mathfrak{F}}_4'$ is identified to the subset ${\mathfrak{X}}\setminus{\mathfrak{X}}'$ of $\frak{X}$, where ${\mathfrak{X}}'$ is the subset of ${\mathfrak{X}}$ comprising $({\mathbb{X}}_1,{\mathbb{X}}_2,{\mathbb{X}}_3)$ such that ${\mathbb{X}}_1+{\mathbb{X}}_2=1$ and ${\mathbb{X}}_3=1-(1/{\mathbb{X}}_1)$.
- The subset ${\mathfrak{F}}^{\mathbb{C}}_4$ is identified to to the subset ${\mathfrak{X}}\setminus{\mathfrak{X}}^{\mathbb{C}}$ of $\frak{X}$, where ${\mathfrak{X}}^{\mathbb{C}}$ comprises of $({\mathbb{X}}_1,{\mathbb{X}}_2,{\mathbb{X}}_3)$ such that ${\mathbb{X}}_3\in{\mathbb{R}}$.
Main Theorem: ${\mathfrak{F}}_4'$ and $\mathcal C({\mathfrak{H}}^{\star})$ {#sec-mainth}
--------------------------------------------------------------------------
In this section, we shall prove our main theorem:
\[thm-main\] There is a bijection $\mathcal B_0: {\mathfrak{F}}_4'\to\mathcal C({\mathfrak{H}}^{\star})$. Therefore ${\mathfrak{F}}_4'$ inherits the Kähler structure of $\mathcal C({\mathfrak{H}}^{\star})$.
Let ${\mathfrak{p}}=(p_1, p_2, p_3, p_4)\in{\mathfrak{C}}_4'$ . We normalise so that $$p_1=(1, \tan a),\quad p_2=\infty, \quad p_3=0, \quad p_4=(z , t),$$ where $a\in(-\pi/2, \pi/2),$ $z\neq 0.$ We then consider $\widetilde{\mathcal B_0}: {\mathfrak{C}}_4'\to\mathcal C({\mathfrak{H}}^{\star})$ given by $$\widetilde{\mathcal B_0}({\mathfrak{p}}) = (z, t, e^{\tan a}).$$ Then the followings hold:
1. If $(z, t, r)\in \mathcal C({\mathfrak{H}}^{\star})$, then there exists a ${\mathfrak{p}}\in{\mathfrak{C}}_4'$ such that $\widetilde{\mathcal B_0}({\mathfrak{p}}) = (z, t, r).$ Indeed we may consider ${\mathfrak{p}}$ with $$p_1=(1, \arctan\log r),\quad p_2=\infty, \quad p_3=0, \quad p_4=(z , t).$$
2. If ${\mathfrak{p}}\in{\mathfrak{C}}_4'$ and $g\in\rm{SU}(2, 1)$, then $\widetilde{\mathcal B_0}({\mathfrak{p}})=\widetilde{\mathcal B_0}(g({\mathfrak{p}})).$
3. If $\widetilde{\mathcal B_0}({\mathfrak{p}})=\widetilde{\mathcal B_0}({{\mathfrak{p}}^{\prime}})$ for ${\mathfrak{p}}, {\mathfrak{p}}'\in{\mathfrak{C}}_4'$, then there exists a $g\in\rm{SU}(2, 1)$ such that ${\mathfrak{p}}'=g({\mathfrak{p}}).$ To see this, we normalise so that $$\begin{aligned}
&p_1=(1, \tan a),\quad\, p_2=\infty, \quad p_3=0, \quad p_4=(z , t),\\
&p_1'=(1, \tan a'),\quad p_2'=\infty, \quad p_3'=0, \quad p_4'=(z' , t').\end{aligned}$$ A $g\in\rm{SU}(2, 1)$ mapping ${\mathfrak{p}}$ to ${\mathfrak{p}}'$ must be of the form $\rm{E}_\lambda\in\rm{SU}(2, 1)$, $\lambda=l+i\theta\in{\mathbb{C}}_{\ast}$, that is, $$E_\lambda(z, t)=(e^{l+i\theta}z, e^{2l}t),$$ since it belongs to $\rm{Stab}(0, \infty).$ It is now clear that $p_1'=p_1,$ and $p_4'=p_4.$
The proof is complete.
${\mathfrak{F}}_4'$ and the variety ${\mathfrak{V}}_4$ {#sec-V4}
------------------------------------------------------
We describe in what follows how the subset ${\mathfrak{F}}_4'$ can be identified to the variety ${\mathfrak{V}}_4$ of ${\mathbb{C}}^2\times(-\pi/2,\pi/2)$ which we will define below. In the first place, we show that if $[{\mathfrak{p}}]\in{\mathfrak{F}}_4'$, ${\mathfrak{p}}=(p_1,p_2,p_3,p_4)$ and $({\mathbb{X}}_1,{\mathbb{X}}_2,{\mathbb{A}})$ correspond to ${\mathfrak{p}}$, that is, ${\mathbb{X}}_1={\mathbb{X}}_1({\mathfrak{p}})$ ${\mathbb{X}}_2={\mathbb{X}}_2({\mathfrak{p}})$ and ${\mathbb{A}}={\mathbb{A}}(p_1,p_2,p_3)$, then $$\label{crv}
|{\mathbb{X}}_1+{\mathbb{X}}_2-1|^2=2\Re\left({\mathbb{X}}_1\overline{{\mathbb{X}}_2}(1+e^{-2i{\mathbb{A}}})\right),$$ where here, ${\mathbb{X}}_1+{\mathbb{X}}_2+1\neq 0$ and $\Re\left({\mathbb{X}}_1\overline{{\mathbb{X}}_2}e^{-i{\mathbb{A}}}\right)>0$.
A variation of this formula is found in [@CG]. For completeness, we prove (\[crv\]) here in a different way; we mention that equation (\[crv\]) follows in [@CG] from the vanishing of the determinant of the Gram matrix of points of ${\mathfrak{p}}$. In our setting we make no use of Gram determinants.
We may normalise so that $p_1=(1,\tan a)$, $p_2=\infty$, $p_3=o$ and $p_4=(z,t)$ with lifts $${{\bf p}}_1=\left[\begin{matrix} -1+i \tan a\\ \sqrt{2}\\1\end{matrix}\right],\quad
{{\bf p}}_2=\left[\begin{matrix} 1\\0\\0\end{matrix}\right],\quad
{{\bf p}}_3=\left[\begin{matrix} 0\\ 0\\1\end{matrix}\right],\quad
{{\bf p}}_{4}=\left[\begin{matrix} u\\\sqrt{2}\;z\\1\end{matrix}\right].$$ Here $a\in(-\pi/2,\pi/2)$ and $u=-|z|^2+it$. One can calculate directly that $$\begin{aligned}
&{\mathbb{X}}_1&=\frac{-1-i\tan a}{u-1-i\tan a+2z},\\
&{\mathbb{X}}_2&=\frac{u}{u-1-i\tan a+2z},\\\end{aligned}$$ from which equation (\[crv\]) follows.
We now define ${\mathfrak{V}}_4$ to be the subset of ${\mathbb{C}}^2\times(-\pi/2,\pi/2)$ comprising $(w_1,w_2,a)$ such that $$|w_1+w_2-1|^2=2\Re\left(w_1\overline{w_2}(1+e^{-2ia})\right),\quad w_1+w_2-1\neq 0,\;\Re\left(w_1\overline{w_2}e^{-ia}\right)>0.$$ The identification of ${\mathfrak{F}}'_4$ to ${\mathfrak{V}}_4$ is by the mapping $B_0:{\mathfrak{F}}_4'\to{\mathfrak{V}}_4$ given for each $[{\mathfrak{p}}]$ by $$B_0([{\mathfrak{p}}])=\left({\mathbb{X}}_1,{\mathbb{X}}_2,{\mathbb{A}}({\mathfrak{p}})\right).$$ Here ${\mathbb{A}}({\mathfrak{p}})={\mathbb{A}}(p_1,p_2,p_3)$. This mapping is bijective. Indeed, given $(w_1,w_2,a)\in{\mathfrak{V}}_4$ we may consider the quadruple ${\mathfrak{p}}$ of points $p_1=(1,\tan a)$, $p_2=\infty$, $p_3=o$ and $p_4=(z,t)$ where $$z=\frac{w_1+w_2-1}{(1+e^{-2ia})w_1},\quad t=-2\Im\left(\frac{w_2}{(1+e^{-2ia})w_1}\right).$$ Straightforward calculation yields to $${\mathbb{X}}_1=w_1,\quad {\mathbb{X}}_2=w_2,\quad {\mathbb{A}}(p)=a.$$ If additionally, for ${\mathfrak{p}}, {\mathfrak{p}}'\in{\mathfrak{C}}_4'$ there exists a $g\in{\rm SU}(2,1)$ such that $g({\mathfrak{p}})=p'$, then clearly ${\mathbb{X}}_i={\mathbb{X}}_i'$, $i=1,2$ and ${\mathbb{A}}({\mathfrak{p}})={\mathbb{A}}({\mathfrak{p}}')$. Finally, if ${\mathfrak{p}}, {\mathfrak{p}}'\in{\mathfrak{C}}_4'$ such that ${\mathbb{X}}_i={\mathbb{X}}_i'$, $i=1,2$ and ${\mathbb{A}}({\mathfrak{p}})={\mathbb{A}}({\mathfrak{p}}')$, then we may normalize so that $$\begin{aligned}
&p_1=(1, \tan a),\quad\, p_2=\infty, \quad p_3=0, \quad p_4=(z , t),\\
&p_1'=(1, \tan a'),\quad p_2'=\infty, \quad p_3'=0, \quad p_4'=(z' , t').\end{aligned}$$ Due to our assertion, we then obtain that $p_1=p_1'$ and $p_4=p_4'$.
We remark that there is also a bijection between varieties ${\mathfrak{X}}'$ and ${\mathfrak{V}}_4$; this is given by $${\mathfrak{V}}_4\ni({\mathbb{X}}_1,{\mathbb{X}}_2,{\mathbb{A}})\mapsto\left({\mathbb{X}}_1,{\mathbb{X}}_2,\frac{{\mathbb{X}}_2}{{\mathbb{X}}_1}e^{2i{\mathbb{A}}}\right)\in {\mathfrak{X}}'.$$
CR structure of ${\mathfrak{F}}_4'$ {#sec-CRV4}
-----------------------------------
We now describe the CR structure of ${\mathfrak{F}}_4'$ as this is obtained by its identification to ${\mathfrak{V}}_4$. For this, we consider the defining function of ${\mathfrak{V}}_4$: $$F(w_1,w_2,a)=|w_1+w_2-1|^2-2\Re(w_1\overline{w_2}(1+e^{-2ia}))=0.
$$ The CR structure is obtained as the kernel of $$\begin{aligned}
\partial F&=&\frac{\partial F}{\partial w_1}dw_1+\frac{\partial F}{\partial w_2}dw_2;
$$ where: $$\begin{aligned}
\frac{\partial F}{\partial w_1}&=&\overline{w_1}-e^{-2ia}\overline{w_2}-1=-\beta,\\
\frac{\partial F}{\partial w_2}&=&\overline{w_2}-e^{2ia}\overline{w_1}-1=\alpha.\end{aligned}$$ Note that $\alpha$ and $\beta$ cannot be simultaneously zero; this would lead to the contradiction $a=\pm\pi/2$. Therefore the codimension 2 CR structure on ${\mathfrak{V}}_4$ is $$\label{eq-CRV}
{\mathcal{H}}=\ker(\partial F)=\left\langle Z=\alpha\frac{\partial}{\partial w_1}+\beta\frac{\partial}{\partial w_2}\right\rangle.$$
CR-equivalence {#sec-CReq}
--------------
We consider ${\mathcal{C}}({\mathfrak{H}}^\star)$ with coordinates $(z,t,r)$ and the variety ${\mathfrak{V}}_4$ as above with coordinates $(w_1,w_2,a)$. We set $u=-|z|^2+it$ and we consider the mapping $G:{\mathcal{C}}({\mathfrak{H}}^\star)\to{\mathfrak{V}}_4$, where $$G(z,t,r)=\left(\frac{-1-i \log r}{u+2z-1-i \log r},\;\frac{u}{u+2z-1-i \log r},\;\arctan \log r\right).$$ This mapping is a diffeomorphic bijection; the inverse $G^{-1}:{\mathfrak{V}}_4\to{\mathcal{C}}({\mathfrak{H}}^\star)$ is given by $$G^{-1}(w_1,w_2,a)=\left(\frac{w_1+w_2-1}{(1+e^{-2ia})w_1},\;
-2\Im \left(\frac{w_2}{(1+e^{-2ia})w_1}\right),\;e^{\tan a} \right).$$ The following proposition displays the equivalence of CR structures for ${\mathfrak{F}}_4'$:
\[prop-CR-H\] We consider ${\mathcal{C}}({\mathfrak{H}}^\star)$ with the CR structure ${\mathcal{H}}^\star=\langle{{\bf Z}}_r\rangle$ and the variety ${\mathfrak{V}}_4$ with the CR structure ${\mathcal{H}}$ as in (\[eq-CRV\]). Then the diffeomorphism $G:{\mathcal{C}}({\mathfrak{H}}^{\star})\to{\mathfrak{V}}_4$ is CR with respect to these structures.
By setting $u=-|z|^2+it$ and $q=-1-i\log r$, we calculate directly $$\begin{aligned}
G_*({{\bf Z}}_r)&=&{{\bf Z}}_r\left(\frac{q}{u+2z+q}\right)\frac{\partial}{\partial w_1}+{{\bf Z}}_r\left(\frac{\overline{q}}{\overline{u}+2\bar{z}+\bar{q}}\right)\frac{\partial}{\partial \overline{w_1}}+\\
& &{{\bf Z}}_r \left(\frac{u}{u+2z+q}\right)\frac{\partial}{\partial w_2}+{{\bf Z}}_r \left(\frac{\bar{u}}{\bar{u}+2\bar{z}+\bar{q}}\right)\frac{\partial}{\partial \overline{w_2}}\\
&& +{{\bf Z}}_r(\arctan\log r)\frac{\partial}{\partial a}.\end{aligned}$$ The second and the fourth term vanish since ${{\bf Z}}_r(\bar{u})={{\bf Z}}_r(\bar{z})=0$. The last term also vanishes because ${{\bf Z}}_r$ does not involve derivation with respect to $r$. On the other hand, since $${{\bf Z}}_r(u)=-\frac{2|z|^2}{r},\quad {{\bf Z}}_r(z)=\frac{z}{r},$$ we have $${{\bf Z}}_r\left(\frac{q}{u+2z+q}\right)=\frac{2z}{r}\cdot\frac{q(\bar{z}-1)}{(u+2z+q)^2},\quad
{{\bf Z}}_r\left(\frac{u}{u+2z+q}\right)=\frac{2z}{r}\cdot\frac{\bar{u}-\bar{z}q}{(u+2z+q)^2}.$$ Hence $$G_*({{\bf Z}}_r)=\frac{2z}{r}\cdot\frac{1}{(u+2z+q)^2}\left(q(\bar{z}-1)\frac{\partial}{\partial w_1}+(\bar{u}-\bar{z}q)\frac{\partial}{\partial w_2}\right).$$ Since $$\begin{aligned}
&&
\frac{2z}{r}\cdot\frac{1}{(u+2z+q)^2}=\frac{1}{2}\cdot e^{-\tan a}w_1(w_1+w_2-1)(1+e^{-2ia}),\\
&&
q(\bar{z}-1)=-\frac{2(\overline{w_2}-e^{2ia}\overline{w_1}-1)}{|1+e^{-2ia}|^2\overline{w_1}},\\
&&\bar{u}-\bar{z}q=\frac{2(\bar{w_1}-\bar{w_2}e^{-2ia}-1)}{|1+e^{-2ia}|^2\overline{w_1}},\end{aligned}$$ we finally obtain $$G_*({{\bf Z}}_r)=kZ,\quad k=-\frac{e^{-\tan a}w_1(w_1+w_2-1)}{(1+e^{2ia})\bar{w_1}},$$ where $Z$ is as in (\[eq-CRV\]). This proves the result.
Comparison of complex structures {#sec-complex}
--------------------------------
For the definition of the complex structure in the subset ${\mathfrak{F}}^{\mathbb{C}}_4$ we refer to [@FP], [@Pla-CR]. In brief, the complex structure is obtained by identifying the set ${\mathfrak{F}}^{\mathbb{C}}_4$ to ${\mathbb{C}}P^1\times{\mathbb{C}}\setminus{\mathbb{R}}$; the map ${\mathcal{B}}_1$ defined in Theorem \[thm-comp\] below is actually the restriction of this identification in ${\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'$. The subset ${\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'$ is the one that will concern us in this section.
\[thm-comp\] Let ${\mathcal{C}}'({\mathfrak{H}}^\star)$ be the (open and disconnected) subset of ${\mathcal{C}}({\mathfrak{H}}^\star)$ at which $$\log r-\frac{t}{|z|^2}\neq 0.$$ There are bijections ${\mathcal{B}}_1:{\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'\to{\mathbb{C}}_*\times({\mathbb{C}}\setminus{\mathbb{R}})$ and ${\mathcal{B}}_2:{\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'\to{\mathcal{C}}'({\mathfrak{H}}^\star)$. Therefore the set ${\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'$ admits two complex structures, namely the one from ${\mathbb{C}}_*\times({\mathbb{C}}\setminus{\mathbb{R}})$ and the other from ${\mathcal{C}}'({\mathfrak{H}}^\star)$. The identity map of ${\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'$ is CR with respect to these two complex structures.
If ${\mathfrak{p}}\in{\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'$, we normalise so that $$p_1=(1,\tan a),\quad p_2=\infty,\quad p_3=(0,0),\quad p_4=(z,t),$$ where $a\neq\pm\pi/2$ and also $\tan a\neq-t/|z|^2$. The map ${\mathcal{B}}_2$ is just the restriction to ${\mathfrak{F}}^{\mathbb{C}}_4\cap{\mathfrak{F}}_4'$ of the map ${\mathcal{B}}_0:{\mathfrak{F}}_4'\to{\mathcal{C}}({\mathfrak{H}}^\star)$. On the other hand, we consider the map ${\mathcal{B}}_1$ given by $$[{\mathfrak{p}}]\mapsto \left(z,\;\frac{|z|^2-it}{1+i\tan a}\right).$$ This map is a bijection: Given $(\zeta,w)\in{\mathbb{C}}_{*}\times({\mathbb{C}}\setminus{\mathbb{R}})$, we set $$z=\zeta,\quad \tan a=\frac{\Re(w)-|\zeta|^2}{\Im(w)},\quad t=\frac{\Re(w)|\zeta|^2-|w|^2}{\Im(w)}.$$ To conclude the proof, let $F={\mathcal{B}}_1\circ {\mathcal{B}}_2^{-1}:{\mathcal{C}}'({\mathfrak{H}}^\star)\to{\mathbb{C}}_*\times({\mathbb{C}}\setminus{\mathbb{R}})$ given by $$F(z,t,r)=\left(z,\;\frac{|z|^2-it}{1+i\log r}\right),\quad (z,t,r)\in{\mathcal{C}}'({\mathfrak{H}}^\star).$$ We will show that $F$ is CR with respect to ${\mathcal{H}}_r^\star=\{{{\bf X}}_r,{{\bf Y}}_r\}$, i.e. $F_{*}({{\bf Z}}_r)\in{\rm T}^{(1,0)}({\mathbb{C}}_*\times({\mathbb{C}}\setminus{\mathbb{R}}))$.
One can know that $F$ is bijective, because $$F^{-1}(\zeta,w)=\left(\zeta,\;\frac{\Re(w)|\zeta|^2-|w|^2}{\Im(w)},\;e^{\frac{\Re(w)-|\zeta|^2}{\Im(w)}}\right),\quad(\zeta,w)\in{\mathbb{C}}_*\times({\mathbb{C}}\setminus{\mathbb{R}}).$$ Thus $F$ is a diffeomorphism and also $$\begin{aligned}
F_*({{\bf Z}}_r)&=&{{\bf Z}}_r(z)\frac{\partial}{\partial\zeta}+
{{\bf Z}}_r(\overline{z})\frac{\partial}{\partial\overline{\zeta}}+
{{\bf Z}}_r\left(\frac{|z|^2-it}{1+i\log r}\right)\frac{\partial}{\partial w}+
{{\bf Z}}_r\left(\frac{|z|^2+it}{1-i\log r}\right)\frac{\partial}{\partial \overline{w}}\\
&=&e^{\frac{|\zeta|^2-\Re(w)}{\Im(w)}}\left(\zeta\frac{\partial}{\partial\zeta}-2i\frac{|\zeta|^2\Im(w)}{\overline{w}-|\zeta|^2}\frac{\partial}{\partial w}\right),\end{aligned}$$ which proves our claim.
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[^1]: [*Acknowledgments.*]{} Part of this work has been carried out while IDP was visiting Hunan University, Changsha, PRC. Hospitality is gratefully appreciated.
|
---
abstract: |
We exactly evaluate the third neighbor correlator $\langle S_j^z S_{j+3}^z \rangle$ and all the possible non-zero correlators $\langle S^{\alpha}_j S^{\beta}_{j+1} S^{\gamma}_{j+2} S^{\delta}_{j+3}
\rangle$ of the spin-1/2 Heisenberg $XXX$ antiferromagnet in the ground state without magnetic field. All the correlators are expressed in terms of certain combinations of logarithm $\ln 2$, the Riemann zeta function $\zeta(3)$, $\zeta(5)$ with rational coefficients. The results accurately coincide with the numerical ones obtained by the density-matrix renormalization group method and the numerical diagonalization.
author:
- Kazumitsu Sakai
- Masahiro Shiroishi
- Yoshihiro Nishiyama
- Minoru Takahashi
date: 'February 27, 2003'
title: 'Third Neighbor Correlators of Spin-1/2 Heisenberg Antiferromagnet'
---
Since Bethe’s pioneering work of the spin-1/2 Heisenberg magnet [@Bethe], $$H=J\sum_{j=1}^{L}\(S_j^xS_{j+1}^x+S_j^yS_{j+1}^y+S_j^zS_{j+1}^z\),
\label{Hami}$$ exact evaluation of the correlation functions has been a long standing problem in mathematical physics. Especially significant are the spin-spin correlators $\bra S_j^z S_{j+k}^z \ket$ (or equivalently $\bra S_j^+ S_{j+k}^-\ket/2$; $S_j^{\pm}=S_j^x\pm i S_j^y$), for which only the first and second neighbor ($k=1,2$) have been calculated so far: $$\begin{aligned}
\bra S_j^{z} S_{j+1}^z\ket&=\frac{1}{12}-\frac{1}{3}\ln2
\simeq-0.14771572685, \label{1st} \db
\bra S_j^{z} S_{j+2}^z\ket&=\frac{1}{12}-\frac{4}{3}\ln2+
\frac{3}{4}\zeta(3)\simeq0.06067976996,
\label{2nd}\end{aligned}$$ where $\zeta(s)$ is the Riemann zeta function and $\bra \cdots \ket$ denotes the ground state expectation value of the antiferromagnetic model ($J>0$). Here we have taken the thermodynamic limit $L\to\infty$. In this letter, we would like to report our new results regarding the third neighbor correlators. Our main result is $$\begin{aligned}
\bra S^z_j & S^z_{j+3}\ket=\frac{1}{2}\bra S_j^+S_{j+3}^-\ket \nn \db
&=\frac{1}{12}-
3\ln 2+\frac{37}{6}\zeta(3)
-\frac{14}{3}\zeta(3)\ln 2
-\frac{3}{2}\zeta(3)^2 \nn \db
&\quad-\frac{125}{24}\zeta(5)+\frac{25}{3}\zeta(5)\ln 2
\simeq -0.05024862726.
\label{3rd}\end{aligned}$$ In addition, we obtain the third neighbor one-particle Green function $\bra c^{\dagger}_j c_{j+3} \ket_{\rm f}$ $$\begin{aligned}
&\bra c^{\dagger}_j c_{j+3} \ket_{\rm f}=
\frac{1}{30}-2\ln 2+\frac{169}{30}\zeta(3)-\frac{10}{3}\zeta(3)\ln 2
\nn \db
&-\frac{6}{5}\zeta(3)^2-\frac{65}{12}\zeta(5)+
\frac{20}{3}\zeta(5)\ln 2
\simeq 0.08228771669,
\label{green}\end{aligned}$$ for the isotropic spinless fermion model corresponding to by the Jordan-Wigner transformation: $$S_k^{-}=\prod_{j=1}^{k-1}(1-2c_j^{\dagger}c_j)
c_k^{\dagger},\quad
S_k^+=\prod_{j=1}^{k-1}(1-2c_j^{\dagger}c_j)c_k.
\label{JW}$$ Here $\bra\dots\ket_{\rm f}$ denotes the expectation value in the half-filled state of the spinless fermion model. Moreover we exactly calculate all the possible non-zero correlators $\langle S^{\alpha}_j
S^{\beta}_{j+1} S^{\gamma}_{j+2}S^{\delta}_{j+3} \rangle$. The result comes from the ground state energy of derived by Hulthén in 1938 [@hult]. The result was obtained by one of the authors in 1977 [@Taka; @Takabook] via the strong coupling expansion for the ground state energy of the half-filled Hubbard model. This result is also reproduced in the framework of the asymptotic Bethe ansatz for an integrable spin chain with variable range exchange [@DitIno]. However, probably due to the complexity of the wave function for these models, no one has succeeded in generalizing the method to obtain the higher neighbor correlators. On the other hand, utilizing the representation theory of the quantum affine algebra $U_q(\widehat{sl_2})$ and the associated vertex operators, in 1992, Jimbo *et al.* derived a universal multiple integral representation of arbitrary correlators for the massive $XXZ$ antiferromagnet [@JMMN; @JMbook]. Their result has been extended to the [@Naka; @KIEU], the massless $XXZ$ [@JM; @KMT] and the $XYZ$ [@Quano] antiferromagnets. However the explicit evaluation, even for the second neighbor correlator , was not achieved for a long time.
In this respect, it is quite remarkable that Boos and Korepin recently devised a general method to evaluate the multiple integral representation especially in the study of the Emptiness Formation Probability (EFP) for the antiferromagnet [@BK1; @BK2]. The EFP, $P(n)$ describes the probability of finding a ferromagnetic string of length $n$ in the antiferromagnetic ground state [@KIEU]. Explicitly it reads $$P(n)=\biggl\bra \prod_{j=1}^n \(S_j^z+\frac{1}{2}\)\biggr\ket.
\label{efp}$$ By reducing the integrand of the multiple integral representation to certain *canonical form*, the EFP for $n=3,4$ [@BK1; @BK2] and $n=5$ [@BKNS] were evaluated by Boos *et al.* (see also recent progress for $n=6$ [@BKS]). Note that $P(2)$ and $P(3)$ in are related to the first and second neighbor correlators as $P(2)=1/4+\bra S_j^zS_{j+1}^z\ket$ and $P(3)=1/8+\bra S_j^zS_{j+1}^z\ket+\bra S_j^zS_{j+2}^z\ket/2$. Here we quote the explicit form of $P(4)$ obtained in [@BK1; @BK2], which is closely related to the third neighbor correlator $\bra S_j^z S_{j+3}^z\ket$. $$\begin{aligned}
P(4)=&\frac{1}{16}+\frac{3}{4}\bra S_j^z S_{j+1}^z\ket+
\frac{1}{2}\bra S_j^z S_{j+2}^z\ket
+\frac{1}{4}\bra S_{j}^zS_{j+3}^z\ket \nn \db
&+\bra S_j^z S_{j+1}^zS_{j+2}^zS_{j+3}^z \ket \nn \db
=&\frac{1}{5}-2\ln2+\frac{173}{60}\zeta(3)-
\frac{11}{6}\zeta(3)\ln2-\frac{51}{80}\zeta(3)^2 \nn \db
&-\frac{55}{24}\zeta(5)+\frac{85}{24}\zeta(5)\ln 2.
\label{p4}\end{aligned}$$ Note that on the antiferromagnetic ground state without magnetic field, all the correlators with an odd number of $S^z$ vanishes. Substituting and into , one finds the relation between the third neighbor correlator $\bra S_{j}^zS_{j+3}^z\ket$ and the four point correlator $\bra S_j^z S_{j+1}^zS_{j+2}^zS_{j+3}^z \ket$. However the exact value of $\bra S_j^{z}S_{j+3}^z\ket$ itself can not be determined solely from $P(4)$. To determine $\bra S_j^z S_{j+3}^z\ket$, we consider the following auxiliary correlator: $$\begin{aligned}
P_{+-+-}^{+-+-}
=&\frac{1}{16}-\frac{3}{4}\bra S_j^z S_{j+1}^z\ket+
\frac{1}{2}\bra S_j^z S_{j+2}^z\ket
-\frac{1}{4}\bra S_{j}^zS_{j+3}^z\ket \nn \db
&+\bra S_j^z S_{j+1}^zS_{j+2}^zS_{j+3}^z\ket.
\label{ap4}\end{aligned}$$ Here and hereafter $P_{\varepsilon_1\varepsilon_{2}
\varepsilon_{3}\varepsilon_{4}}^
{\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{2}
\tilde{\varepsilon}_{3}\tilde{\varepsilon}_{4}}$ (also written as $P_{\varepsilon}^{\tilde{\varepsilon}}$ for simplicity) denotes a correlator of the form $$P_{\varepsilon_1\varepsilon_{2}\varepsilon_{3}\varepsilon_{4}}^
{\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{2}\tilde{\varepsilon}_{3}
\tilde{\varepsilon}_{4}}=\bra
E_1^{\tilde{\varepsilon}_1\varepsilon_1}
E_{2}^{\tilde{\varepsilon}_{2}\varepsilon_{2}}
E_{3}^{\tilde{\varepsilon}_{3}\varepsilon_{3}}
E_{4}^{\tilde{\varepsilon}_{4}\varepsilon_{4}} \ket,
\label{P}$$ where $\varepsilon_j$, $\tilde{\varepsilon}_j=\{+,-\}$ and $E_j^{\varepsilon_j\tilde{\varepsilon}_j}$ is the 2 $\times$ 2 elementary matrix ($E^{\pm \pm}=\pm S^z+1/2$, $E^{- +}=S^{+}$, $E^{+-}=S^-$) acting on $j$th site. In this notation $P(4)=
P_{++++}^{++++}$. Note that because the Hamiltonian has the symmetry under the group $SU(2)$, $P_{\varepsilon}^{\tilde{\varepsilon}}$ possesses a property like $$P_{\varepsilon}^{\tilde{\varepsilon}}
=
P^{\varepsilon}_{\tilde{\varepsilon}}
=
P_{-\varepsilon}^{-\tilde{\varepsilon}}
=
P^{-\varepsilon}_{-\tilde{\varepsilon}}.
\label{psym}$$ As in the case of $P(4)$, the correlators enjoy the multiple integral representation [@JMMN; @JMbook; @Naka; @JM]: $$P_{\varepsilon}^{\tilde{\varepsilon}}
=\prod_{j=1}^4\int_{C}\frac{d \lam_j}{2\pi i}
U(\lam_1,\dots,\lam_4)T(\lam_1,\dots,\lam_4),
\label{mir}$$ where the integration contour $C$ is taken to be a line $[-\infty-i \alpha,\infty-i\alpha]$ $(0<\alpha <1)$. For convenience, we choose $\alpha=1/2$. The integrand $U(\lam_1,\dots,\lam_4)$ is given by $$U(\lam_1,\dots,\lam_4)=\pi^{10}\frac{\prod_{1\le k<j\le 4}
\sh\pi \lam_{jk}}{\prod_{j=1}^{4}\sh^4\pi \lam_j},
\label{ufnc}$$ while $T(\lam_1,\dots,\lam_4)$ depends on the selection of $\varepsilon $ and $\tilde{\varepsilon}$. Here and hereafter we use the notation $\lam_{jk}=\lam_j-\lam_k$ to save space. In particular for the correlator , $T(\lam_1,\dots,\lam_4)$ is given by $$T(\lam_1,\dots,\lam_4)=
\frac{\lam_1(\lam_1+i)^2\lam_2^3 \lam_3(\lam_3+i)^2 \lam_4^3}
{(\lam_{21}-i)\lam_{31}\lam_{41}\lam_{32}\lam_{42}(\lam_{43}-i)}.
\label{tfnc}$$ To calculate the multiple integral , we follow the method by Boos and Korepin [@BK2]. Roughly, their method is described as follows. First Taking carefully into account the property of $U(\lam_1,\dots,\lam_4)$, we modify the integrand $T(\lam_1,\dots,\lam_4)$ such that the integral gives the same result as the original one (“weak equivalence"). In this way it is likely that the integrand $T(\lam_1,\dots,\lam_4)$ can always be reduced to the following form (we call it “canonical form"): $$\begin{aligned}
T_c=&
P_0(\lam_2,\lam_3,\lam_4)+
\frac{P_1(\lam_1,\lam_3,\lam_4)}{\lam_{21}}+
\frac{P_2(\lam_1,\lam_3)}{\lam_{21}\lam_{43}},
\label{canonical}\end{aligned}$$ where $P_0$, $P_1$ and $P_2$ are certain polynomials. Once one derives the canonical form, one can perform the multiple integral by using the Cauchy theorem [@BK2]. The result is written as combinations of the logarithm $\ln 2$, the Riemann zeta function $\zeta(3)$ and $\zeta(5)$ and rational numbers. Consequently, the main part of the calculation for the multiple integral reduces to finding the canonical form . Now we consider the case and show that the Boos–Korepin method is also applicable to our case. Let us introduce the following diagram. $$\graphaa:=\frac{1}{\lam_{jk}},\qquad \graphbb:=\frac{1}{\lam_{jk}-i},$$ where $j>k$. First we expand through partial fractions. The result consists of 24 terms. Taking into account the antisymmetry of the function under transposition of any two variables $\lam_j$ and $\lam_k$ and the symmetry of $T$ under $\lam_1\leftrightarrow\lam_3$, $\lam_2\leftrightarrow\lam_4$, we can reduce the 24 partial fractions to eight ones. Diagrammatically, its denominators are written as $$\begin{aligned}
\grapha\to &\graphi-\graphb+i\left(\graphd+\graphh\right)\nn \\
+2i&\left(\graphf-\graphe-\graphg-\graphc\right).\nn\end{aligned}$$ From a symmetry of the denominator, the second term can further be simplified as $$\begin{aligned}
&-g\graphb\to-\frac{1}{4}\left(g(\lam_1,\lam_3,\lam_2,\lam_4)-
g(\lam_4,\lam_1,\lam_3,\lam_2)\right. \nn \db
&\quad\left.+g(\lam_2,\lam_4,\lam_1,\lam_3)-
g(\lam_3,\lam_2,\lam_4,\lam_1)\right)\graphm \nn \\[4mm]
&\to\frac{1}{2}\lam_1\lam_2\lam_3\lam_4(i\lam_1+i\lam_3+2\lam_1\lam_3)
(i\lam_2+i\lam_4+2\lam_2\lam_4) \nn \\
&\quad\times\graphdd\times\graphee\, , \nn\end{aligned}$$ where $g$ denotes the numerator of . Next using the Cauchy theorem, we shift variables $\lam_j\to\lam_j\pm i$ such that the denominators do not contain $i$. For instance, we have $$\begin{aligned}
g \graphg&=g^{(1)} \graphj+g^{(2)}\graphk \nn \\[4.5mm] \nn
&\to g^{(1)}\graphj+g^{(2)}_{\lam_2\to\lam_2+i}\graphl, \\ \nn\end{aligned}$$ where $$\begin{aligned}
g^{(1)}&=-\lam_1(\lam_1+i)\lam_2^3\lam_3(\lam_3+i)^2\lam_4^3 \nn \\
g^{(2)}&=-\lam_1(\lam_1+i)\lam_2^4\lam_3(\lam_3+i)^2\lam_4^3. \nn\end{aligned}$$ Finally again using the antisymmetric property of , we eliminate the symmetric part with respect to $\lam_j\leftrightarrow\lam_k$. In this way we have obtain the canonical form as $$\begin{aligned}
P_0=&\frac{56}{5}\lam_2\lam_3^2\lam_4^3,\nn \db
P_1=&\frac{27}{10}\lam_4-i\lam_4^2+\frac{33}{5}\lam_3 \lam_4^2+
\frac{4}{5}\lam_4^3+2 i\lam_3\lam_4^3+4\lam_3^2\lam_4^3\nn \db
&+\lam_1(-4i\lam_4+7\lam_4^2-32i\lam_3\lam_4^2-10
i\lam_4^3-12i\lam_3^2\lam_4^3) \nn \db
&+\lam_1^2(4\lam_4-19i\lam_4^2-28\lam_3\lam_4^2-10\lam_4^3 \nn \db
&\qquad\qquad
\qquad\qquad
\qquad\quad-28i\lam_3\lam_4^3-32\lam_3^2\lam_4^3), \nn \db
P_2=&-\frac{3}{10}+\frac{3}{2}i \lam_3+\frac{3}{2}\lam_1\lam_3
-\frac{1}{2}\lam_3^2+4i\lam_1\lam_3^2+6\lam_1^2\lam_3^2. \nn\end{aligned}$$ Subsequently, applying the method developed in [@BK2], we calculate the multiple integral by substituting the above canonical form into . Explicitly, $$\begin{aligned}
&P_{+-+-}^{+-+-}=J_0+J_1+J_2 \nn \db
&J_0=\frac{7}{10}, \qquad
J_1=-\frac{2}{3}+\frac{3}{10}\zeta(3)+\frac{35}{32}\zeta(5),\nn \db
&J_2=-\frac{1}{2}\zeta(3)+\frac{1}{2}\zeta(3)\ln2+\frac{9}{80}\zeta(3)^2
-\frac{25}{32}\zeta(5) \nn \db
&\qquad
-\frac{5}{8}\zeta(5)\ln2,
\label{integral}\end{aligned}$$ where $J_k$ denotes the result of the integration regarding the term $P_k$. We remark that the canonical form is not unique due to the non-uniqueness of partial fraction expansions. Accordingly, the explicit value of each $J_k$ depends on the choice of the canonical form. The final result $J_0+J_1+J_2$, however, is always unique as a matter of course. Combining the result with and , we obtain the third neighbor correlator $\bra S_j^{z}S_{j+3}^z \ket$ and at the same time the correlator $\bra S_{j}^z S_{j+1}^z S_{j+2}^z S_{j+3}^z\ket$ as $$\begin{aligned}
\bra S_{j}^z S_{j+1}^z S_{j+2}^z S_{j+3}^z\ket&=
\frac{1}{80}-\frac{1}{3}\ln 2+\frac{29}{30}\zeta(3)-\frac{2}{3}\zeta(3)\ln 2
\nn \db
-\frac{21}{80}&\zeta(3)^2-\frac{95}{96}\zeta(5)+
\frac{35}{24}\zeta(5)\ln 2.
\label{4body}\end{aligned}$$ Now let us consider the four-point correlators of the form $\bra S_j^{\alpha} S_{j+1}^{\beta} S_{j+2}^{\gamma}
S_{j+3}^{\delta} \ket$ ($\{\alpha,\beta,\gamma,\delta\}\in
\{x,y,z, 0\};\,S_j^0=\openone$). Because the correlators with an odd number of $S^{\{\alpha\}\diagdown 0}$ vanish, the possible non-zero correlators are restricted to the following three types: $\bra S_j^{\alpha}S_{j+1}^{\alpha}S_{j+2}^{\beta}S_{j+3}^{\beta}\ket$, $\bra S_j^{\alpha}S_{j+1}^{\beta}S_{j+2}^{\alpha}S_{j+3}^{\beta} \ket$ and $\bra S_j^{\alpha}S_{j+1}^{\beta}S_{j+2}^{\beta}S_{j+3}^{\alpha}\ket$. Further due to the isotropy of the Hamiltonian , one can find that the independent correlators are written as the following seven ones: $\bra S_{j}^x S_{j+1}^z S_{j+2}^z S_{j+3}^x\ket$, $\bra S_{j}^z S_{j+1}^x S_{j+2}^z S_{j+3}^x\ket$, $\bra S_{j}^z S_{j+1}^z S_{j+2}^x S_{j+3}^x\ket$ and already obtained ones in – and . Then we shall calculate the remaining three correlators here. For convenience we use the operator $S^{\pm}(=S^x\pm i S^y)$ instead of $S^x$. First we consider the correlator $\bra S_j^{+}S_{j+1}^{z}S_{j+2}^z S_{j+3}^{-} \ket
(=2\bra S_j^{x}S_{j+1}^{z}S_{j+2}^z S_{j+3}^{x} \ket)$. From and the property , this correlator is expressed as $\bra S_j^{+}S_{j+1}^{z}S_{j+2}^z S_{j+3}^{-} \ket
=\(P_{-+++}^{+++-}-P_{-+-+}^{++--}\)/2$. Using the relation $\bra S_{j}^{+}S_{j+3}^{-}\ket=
2(P_{-+++}^{+++-}+P_{-+-+}^{++--})$, we obtain $$\bra S_{j}^+ S_{j+1}^z S_{j+2}^z S_{j+3}^-\ket=
P_{-+++}^{+++-}-\frac{1}{2}\bra S_j^z S_{j+3}^z\ket.
\label{p2s1}$$ Here we have used the relation $\bra S_{j}^{+}S_{j+3}^{-}
\ket=2\bra S_{j}^{z}S_{j+3}^{z}\ket$. Similarly the other correlators are given by $$\begin{aligned}
\bra S_{j}^z S_{j+1}^+ S_{j+2}^z S_{j+3}^-\ket&=
P_{+-++}^{+++-}-\frac{1}{2}\bra S_j^z S_{j+2}^z\ket, \nn \db
\bra S_{j}^z S_{j+1}^z S_{j+2}^+ S_{j+3}^-\ket&=
P_{++-+}^{+++-}-\frac{1}{2}\bra S_j^z S_{j+1}^z\ket.
\label{p2s2}\end{aligned}$$ Therefore our goal is to evaluate the auxiliary correlators $P_{-+++}^{+++-}$, $P_{+-++}^{+++-}$ and $P_{++-+}^{+++-}$. They are given if we replace the integrand $T(\lam_1,\dots,\lam_4)$ by $$T^{(l)}=
\frac{(\lam_1+i)^3\lam_2(\lam_2+i)^2(\lam_3+i)\lam_3^2
\lam_4^{4-l}(\lam_4+i)^{l-1}}
{(\lam_{21}-i)(\lam_{31}-i)
(\lam_{32}-i)\lam_{41}\lam_{42}\lam_{43}}.$$ in the multiple integral representation. Here the correlator $P_{+++-}^{-+++}$, $P_{+++-}^{+-++}$ and $P_{+++-}^{++-+}$ correspond to $l=1$, 2 and 3, respectively. Using the procedure similar to the case of $P_{+-+-}^{+-+-}$, one obtains the explicit values of the above auxiliary correlators. As a result, combining the identity and with –, we arrive at $$\begin{aligned}
\bra S_{j}^+ S_{j+1}^z S_{j+2}^z S_{j+3}^-\ket&=
\frac{1}{120}-\frac{1}{2}\ln 2+\frac{169}{120}\zeta(3)\nn \db
-\frac{5}{6}\zeta(3)\ln 2-&\frac{3}{10}\zeta(3)^2-\frac{65}{48}\zeta(5)+
\frac{5}{3}\zeta(5)\ln 2, \label{green2}\db
\bra S_{j}^+ S_{j+1}^z S_{j+2}^- S_{j+3}^z\ket&=
\frac{1}{120}-\frac{1}{3}\ln 2+\frac{77}{60}\zeta(3)\nn \db
-\frac{5}{6}\zeta(3)\ln 2-&\frac{3}{10}\zeta(3)^2-\frac{65}{48}\zeta(5)+
\frac{5}{3}\zeta(5)\ln 2, \label{chiral}\db
\bra S_{j}^+ S_{j+1}^- S_{j+2}^z S_{j+3}^z\ket&=
\frac{1}{120}+\frac{1}{6}\ln 2-\frac{91}{120}\zeta(3)\nn \db
+\frac{1}{3}\zeta(3)\ln 2+\frac{3}{40}&
\zeta(3)^2+\frac{35}{48}\zeta(5)-\frac{5}{12}\zeta(5)\ln 2.\end{aligned}$$ We mention a few remarks of our results. (i) All the above correlators are written as the logarithm $\ln 2$, the Riemann zeta function $\zeta(3)$ and $\zeta(5)$. This agrees with the general conjecture by Boos and Korepin: *arbitrary correlators of the antiferromagnet are described as certain combinations of logarithm $\ln 2$, the Riemann zeta function with odd arguments and rational coefficients.* Especially intriguing is the existence of the non-linear terms such as $\zeta(3)^2$, $\zeta(3)\ln 2$ and $\zeta(5)\ln2$. [(ii)]{} The correlator is interpreted as the third neighbor one-particle Green functions $\bra c^{\dagger}_j
c_{j+3} \ket_{\rm f}/4$ via the Jordan-Wigner transformation . Obviously, the first neighbor one-particle Green function is expressed as $$\bra c_j^{\dagger}c_{j+1}\ket_{\rm f}
=\frac{1}{6}-\frac{2}{3}\ln2\simeq-0.295431453707,$$ which coincides with $\bra S_j^+S_{j+1}^-\ket$. Due to the characteristic $\pi/2$-oscillation: $\bra c^{\dagger}_j c_{k} \ket_{\rm f} \sim A_{jk}
\cos(\pi(k-j+1)/2)$, one finds $\bra c^{\dagger}_j c_{j+2k}\ket_{\rm f}=0$. Therefore the quantity is the first non-trivial exact result of the correlators containing the fermionic nature. [(iii)]{} The difference between and gives the nearest chiral correlator $$\begin{aligned}
\bra (\bm{S}_{j}\times &\bm{S}_{j+1})
\cdot(\bm{S}_{j+2}\times \bm{S}_{j+3})\ket
=3(\bra S_{j}^+ S_{j+1}^z S_{j+2}^- S_{j+3}^z\ \ket
\nn \db
&-\bra S_{j}^+ S_{j+1}^z S_{j+2}^z S_{j+3}^-\ket)
=\frac{1}{2}\ln2-\frac{3}{8}\zeta(3), \nn\end{aligned}$$ which exactly agrees with the one derived from the ground state energy of an integrable two-chain model with four-body interactions [@MT]. To confirm the validity of our formulae, we performed numerical calculations by using the density-matrix renormalization group (DMRG) [@White92; @White93] and numerical diagonalization. As for the DMRG, we followed standard algorithm [@Peschel99]. We have repeated renormalization 500-times. At each renormalization, we kept, at most, 200 relevant states for a (new) block. The numerical diagonalization was performed for the system size $L=24$, 28 and $32$. We extrapolate the data from a fitting function $a_0+a_1/L^2+a_2/L^4$. All our analytical results coincide quite accurately with both numerical ones (TABLE I).
Correlators Exact DMRG Extrap.
------------------------------------------------------- ------------ ------------ ------------
$\bra S_j^z S_{j+3}^z \ket$ -0.0502486 -0.0502426 -0.0502475
$\bra S_{j}^+ S_{j+1}^z S_{j+2}^z S_{j+3}^-\ket$ [^1] 0.0205719 0.0205681 0.0205716
$\bra S_{j}^z S_{j+1}^z S_{j+2}^z S_{j+3}^z\ket$ 0.0307153 0.0307105 0.0307154
$\bra S_{j}^+ S_{j+1}^z S_{j+2}^- S_{j+3}^z\ket$ -0.0141607 -0.0141579 -0.0141606
$\bra S_{j}^+ S_{j+1}^- S_{j+2}^z S_{j+3}^z\ket$ 0.0550194 0.0550108 0.0550198
: Estimates of the correlators by the exact evaluations, DMRG and the extrapolations from the numerical diagonalization for the system size $L=24, 28, 32$.
In closing we would like to comment on generalizations of the present results. The extension to the calculation of higher neighbor correlators $\bra S_j^z S_{j+k}^z\ket_{k\ge 4}$ is of great interest. The fourth neighbor one $\bra S_j^zS_{j+4}^z\ket$, for example, will be calculated by combination of the EFP, $P(5)$ and two independent auxiliary correlators, which can in principle be evaluated. In fact $P(5)$ has been already obtained in [@BKNS]. The computation, however, will be much more complicated. Alternatively, extending the present result to the inhomogeneous case as in [@BKS] and taking into account the property of the quantum Knizhnik-Zamolodchikow equation, we may derive higher neighbor correlators. Using this, eventually we hope to extract the long-distance asymptotics $\bra S_j^{z}S_{j+k}^z
\ket _{k \gg 1}$, which is a crucial problem in conformal field theory [@Affleck; @LT].
The authors are grateful to H.E. Boos and V.E. Korepin for many valuable discussions. KS is supported by the JSPS research fellowships for young scientists. MS and YN are supported by Grant-in-Aid for young scientists No. 14740228 and No. 13740240, respectively. This work is in part supported by Grant-in Aid for the Scientific Research (B) No. 14340099 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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[^1]: $\frac{1}{4}
\bra c_j^{\dagger}c_{j+3}\ket_{\rm f}$
|
---
abstract: 'We propose a multi-dimensional (M-D) sparse Fourier transform inspired by the idea of the Fourier projection-slice theorem, called FPS-SFT. FPS-SFT extracts samples along lines ($1$-dimensional slices from an M-D data cube), which are parameterized by random slopes and offsets. The discrete Fourier transform (DFT) along those lines represents projections of M-D DFT of the M-D data onto those lines. The M-D sinusoids that are contained in the signal can be reconstructed from the DFT along lines with a low sample and computational complexity provided that the signal is sparse in the frequency domain and the lines are appropriately designed. The performance of FPS-SFT is demonstrated both theoretically and numerically. A sparse image reconstruction application is illustrated, which shows the capability of the FPS-SFT in solving practical problems.'
address: |
Department of Electrical and Computer Engineering\
Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA
bibliography:
- 'SFTbib.bib'
title: 'FPS-SFT: A Multi-dimensional Sparse Fourier Transform Based on the Fourier Projection-slice Theorem'
---
Multi-dimensional signal processing, sparse Fourier transform, Fourier projection-slice theorem, sparse image reconstruction
Introduction {#sec:intro}
============
Conventional signal processing methods in radar, sonar, and medical imaging systems usually involve multi-dimensional discrete Fourier transforms (DFT), which can be implemented by the fast Fourier transform (FFT). The sample and computational complexity of the FFT are $O(N)$ and $O(N \log N)$, respectively, where $N$ is the number of samples in the multi-dimensional sample space. Recently, the sparse Fourier transform (SFT) [@hassanieh2012nearly; @ghazi2013sample; @potts2015sparse; @pawar2017ffast] has been proposed, which leverages the sparsity of signals in the frequency domain to reduce the sample and computational cost of the FFT. Different versions of the SFT have been investigated for several applications including a fast Global Positioning System (GPS) receiver, wide-band spectrum sensing, radar signal processing, etc. [@hassanieh2012faster; @hassanieh2014ghz; @wang2017robust; @shi2014light; @hassanieh2015fast].
Multi-dimensional signal processing requires multi-dimensional SFT algorithms. The $2$-dimensional ($2$-D) SFT algorithm proposed in [@ghazi2013sample] achieves sample complexity $O(K)$ and computational complexity of $O(K \log K)$, which are the lower bounds of the complexities of known SFT algorithms to date[@pawar2017ffast]. The reduction of complexity in SFT of [@ghazi2013sample] is achieved by implementing a $2$-D DFT as a series of $1$-dimensional ($1$-D) DFTs, which are applied on a few columns and rows of the input data matrix. The SFT of [@ghazi2013sample] basically extends the $1$-D SFT algorithm of [@hassanieh2012nearly] to two dimensions; such SFT algorithm employs the so-called OFDM-trick to decode the frequencies that are embedded in the phase difference of DFTs of the same signal but with different sample offsets. However, the SFT of [@ghazi2013sample] only applies to the $2$-D cases with equal sample length, $\sqrt{N}$, of the two dimensions; $\sqrt{N}$ is assumed to be a power of $2$. Moreover, to achieve a high success rate of frequency recovery, it assumes that the data is very sparse ($K << \sqrt{N}$) in the $2$-D frequency domain and the frequency locations are distributed uniformly. However, those assumptions are not always valid in practical scenarios.
In this work, we propose FPS-SFT, a new SFT algorithm that uses the basic idea of [@ghazi2013sample] while avoiding the shortcomings of [@ghazi2013sample], and can be generalized to the $D$-dimensional ($D$-D), $D \ge 2$ cases. The FPS-SFT implements a $D$-D DFT via a sequence of $1$-D DFTs, applied on samples of the $D$-D data which are taken along discrete lines; the lines are parametrized with random slopes and offsets. This is different from [@ghazi2013sample], where the lines are restricted along the axis of each dimension, i.e., the rows and the columns. The proposed FPS-SFT can be viewed as a low-complexity, Fourier projection-slice approach for signals that are sparse in the frequency domain. In the Fourier projection-slice theorem [@mersereau1974digital], the Fourier transform of a projection is a slice of the $D$-D Fourier transform along the same line the projection was taken. In FPS-SFT, the $1$-D DFT along a line, which is a $1$-D slice of the $D$-D data is the projection of the $D$-D DFT of the $D$-D data to such line. While the classic Fourier projection-slice based method reconstructs the frequency domain of the signal using interpolation based on frequency-domain slices, the FPS-SFT aims to reconstruct the signal based on DFT of time-domain slices with reduced complexity; this is achieved by leveraging the sparsity of the signal in the frequency domain.
The connection between SFT algorithms and the Fourier projection-slice theorem is also found in [@shi2014light; @hassanieh2015fast], where the SFT algorithms also rely on lines extracted from $D$-D data. The recovery of the frequency locations in those SFT algorithms are based on a voting procedure; specifically, each entry of the DFT along a line is the projection of the $D$-D DFT of the data; the projected DFT values lie in a $D-1$-dimensional hyper-plane, which is orthogonal to the time-domain line. When the entry value of the DFT along a line is significant, each DFT grid in the $D-1$-dimensional hyperplane gains one vote. After applying DFT on a sufficient number of lines with different slopes followed by the voting procedure, the DFT grids with the largest number of votes are recovered as the significant frequencies. When $K$ is moderately large, such method would generate many false frequencies due to that many zero-valued frequency locations also gain large votes stemming from the ambiguity in the voting process. Moreover, the sample and computational complexity of those SFT algorithms do not achieve the lower bounds of the state-of-the-art SFT algorithms[@ghazi2013sample; @pawar2017ffast].
The fundamental difference between the FPS-SFT and the SFT algorithms of [@shi2014light; @hassanieh2015fast] is that the FPS-SFT is inspired by the low-complexity SFT of [@ghazi2013sample], which essentially utilizes phase information to recover the significant frequencies in a progressive manner, i.e., each iteration in the FPS-SFT recovers a subset of significant frequencies, whose contributions are removed in subsequent iterations; this results in a sparser signal. The advantages of the proposed FPS-SFT are summarized as follows. FPS-SFT applies to data of arbitrary dimensions and sizes, which are sparse in the frequency domain. In the $2$-D cases, the FPS-SFT outperforms the SFT of [@ghazi2013sample] significantly when the sparsity of the data reduces. The limitation of the SFT of [@ghazi2013sample] on $K$-sparse signals with large $K$ and uniformly distributed frequencies essentially stems from the fact that the direction of projection in the DFT domain is restricted to be along rows and columns. By randomizing the direction of projection in the DFT domain, achieved by taking DFT along lines with pseudo-random slopes, the FPS-DFT can accommodate signals that contain less sparse, non-uniformly distributed frequencies.
**Notation:** We use lower-case bold letters to denote vectors. $[\cdot]^T$ denotes the transpose of a vector. The $N$-modulo operation is denoted by $[\cdot]_N$. $[S]$ refers to the integer set of $\{0, ... , S-1 \}$. The cardinality of set $\mathbb{S}$ is denoted as $|\mathbb{S}|$. We use $a\perp b$ to denote that $a$ and $b$ are co-prime. The DFT of signal $x$ is denoted by $\hat{x}$.
The FPS-SFT Algorithm {#sec:algorithm}
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We consider the following $2$-D signal model, which is a superposition of $K$ $2$-D complex sinusoids, i.e., $$\label{eq:sigModel2}
x(\mathbf{n}) \triangleq \sum_{(a, \bm{\omega}) \in \mathbb{S}_2} a e^{j \mathbf{n}^T \bm{\omega} },$$ where $\mathbf{n} \triangleq [n_0, n_1]^T \in \mathcal{X}_2 \triangleq [N_0] \times [N_1]$, with $N_0,N_1$ denoting the sample length of the two dimensions, respectively. $(a, \bm{\omega})$ represents a $2$-D sinusoid whose amplitude is $a$ with $a \in \mathbb{C}, a \neq 0$ and frequency is $\bm{\omega} \triangleq [\omega_0, \omega_1]^T$ with $\omega_k =\frac{2 \pi}{N_k} m_k, m_k \in [N_k], k \in \{0,1\}$. The set $\mathbb{S}_2$ with $|\mathbb{S}_2|=K$ includes all the sinusoids. We assume that the signal is sparse in the frequency domain, i.e., $K << N \triangleq N_0 N_1$. The problem we address is the recovery of $\mathbb{S}_2$ from samples of $x(\mathbf{n})$. The generalization to the higher dimension, i.e., $D$-D cases with $D > 2$ is straightforward.
The SFT algorithm of [@ghazi2013sample] {#sec:GHIKPS_SFT}
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According to [@ghazi2013sample], in order to recover the frequency set $\mathbb{S}_2$, $1$-D DFTs are applied on a subset of columns and rows of the data. The $N_0$-point DFT of the $i_{\rm{th}}, i \in [N_1]$ column of the data equals $$\begin{split}
\hat{c}_i (m) &\triangleq \frac{1}{N_0} \sum_{l\in [N_0]} x(l,i) e^{-j \frac{2\pi}{N_0} ml} \\
&= \frac{1}{N_0} \sum_{(a, \bm{\omega}) \in \mathbb{S}_2} \sum_{l\in [N_0]} a e^{j \frac{2\pi}{N_1} m_1 i} e^{j \frac{2\pi}{N_0} l (m_0 - m)}, \; m \in [N_0].
\end{split}$$ For a fixed $m$, $\hat{c}_i (m)$ is the summation of modulated amplitudes of the $2$-D sinusoids, $(a, [2\pi m_0/N_0, 2\pi m_1/N_1]^T) \in \mathbb{S}_2$, whose frequencies lie on line $$\label{eq:column_line}
m_0 - m = 0, m_0 \in [N_1],$$ which is a row in the $N_0\times N_1$-point DFT of (\[eq:sigModel2\]), i.e., $\hat{x}(m_0,m_1), [m_0,m_1]^T \in \mathcal{X}_2$. Thus, $\hat{c}_i (m), m\in [N_0]$, the DFT along a column, can be viewed as the projection of $\hat{x}(m_0,m_1)$ on that column. Similarly, the $N_1$-point DFT applied on a row of (\[eq:sigModel2\]) are projections of columns of $\hat{x}(m_0,m_1)$ on that row. Since the signal is sparse in the frequency domain, if $|\hat{c}_i (m)| \neq 0$, with high probability, there is only one significant frequency laying on line (\[eq:column\_line\]); in such case, we call the frequency bin $m$ to be ‘$1$-sparse’, and $\hat{c}_i (m)$ is reduced to be $\hat{c}_i (m) = \hat{c}_i (m_0) = a e^{j \frac{2\pi}{N_1} m_1 i}$. The amplitude, $a$, can be determined by the $m_0$-th entry of the DFT of the $0$-th column, i.e., $a = \hat{c}_0 (m_0)$, and the other frequency component, $m_1$, is ‘coded’ in the phased difference between the $m_0$-th entries of the DFTs of the $0$-th and the $1$-st columns, which can be decoded by $m_1 = \phi \left( {\hat{c}_1 (m_0)}/{ \hat{c}_0 (m_0)} \right) \frac{N_1}{2 \pi} $, where $\phi(x)$ is the phase of $x$. Note that the $1$-sparsity of the $m_{\rm{th}}$ bin can be effectively tested by comparing $|\hat{c}_0 (m)|$ and $|\hat{c}_1 (m)|$, i.e., $\hat{c}_i (m)$ is $1$-sparse almost for sure when $|\hat{c}_0 (m)| = |\hat{c}_1 (m)|$. Such frequency decoding technique is referred to as OFDM-trick [@hassanieh2012nearly]. The decoded frequencies are removed from the signal, so that the following processing can be applied on a sparser signal, which is likely to generate more $1$-sparse bins in the subsequent processing.
A frequency bin that is not $1$-sparse in column processing might be $1$-sparse in row processing. Also, the removal of frequencies in the column (row) processing may cause bins in the row (column) processing to be $1$-sparse, the SFT of [@ghazi2013sample] runs iteratively and alternatively between columns and rows. The algorithm stops after a finite number of iterations.
The SFT of [@ghazi2013sample] succeeds with high probability only when the frequencies are very sparse; this is due to the ‘deadlock’ structures that exist in the distribution of frequency locations. In a deadlock case, neither a column nor a row DFT contains a $1$-sparse bin. In fact, in many applications, the signal frequency exhibits a block sparsity pattern [@eldar2010block], i.e., the significant frequencies are clustered. In those cases, even when the signal is very sparse, deadlocks are inevitable.
FPS-SFT {#sec:DL_SFT_overview}
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The SFT of [@ghazi2013sample] reduces a $2$-D DFT into $1$-D DFTs of the columns and rows of the input data matrix. The columns and the rows can be viewed as discrete lines of the input data matrix with slopes $\infty$ and $0$, respectively. In this section, by proposing FPS-SFT, we reduce the $2$-D DFT into $1$-D DFTs of the data along discrete lines with random slopes and offsets. The SFT of [@ghazi2013sample] resolves $2$-D frequencies that are projected to $1$-sparse bins of the column and row DFTs, and a deadlock arises when such projections cannot create any $1$-sparse bins. In FPS-SFT, by employing lines with random slopes, the directions of projection are also random, which offers a high probability of creating more $1$-sparse bins and resolving the deadlocks encountered by SFT of [@ghazi2013sample]. This can be illustrated in Fig. \[fig:freqProjection\], where the $4$ $2$-D frequencies in the $8 \times 8$-point DFT domain form a deadlock, as neither a row DFT nor a column DFT creates a $1$-sparse bin. However, the DFT along the diagonal, corresponding to the projection of the $2$-D DFT of data onto the diagonal, produces $4$ $1$-sparse bins, which solves the deadlock.
![Demonstration of $2$-D frequencies projecting onto $1$-D. The projection onto a column or a row causes collisions, while the projection onto the diagonal creates $1$-sparse bins. The colored blocks mark significant frequencies.[]{data-label="fig:freqProjection"}](FreqProjection "fig:") -20pt
FPS-SFT is an iterative algorithm; each iteration returns a subset of recovered $2$-D frequencies. After $T$ iterations, the FPS-SFT returns a frequency set, $\hat{\mathbb{S}}_2$, which is an estimate of $\mathbb{S}_2$ (see (\[eq:sigModel2\])). The frequencies recovered in previous iterations are passed to the next iteration, and their contributions are removed from the signal in order to create a sparser signal.
Within each iteration of FPS-SFT, the signal of (\[eq:sigModel2\]) is sampled along a line with slope $\alpha_1/\alpha_0$ starting at point $(\tau_0,\tau_1)$, with $\bm{\alpha}, \bm{\tau} \in \mathcal{X}_2$, where $\bm{\alpha} \triangleq [\alpha_0, \alpha_1]^T, \bm{\tau} \triangleq [\tau_0,\tau_1]^T$. The sampled signal can be expressed as $$\label{eq:sample_slice}
\begin{split}
&s(\bm{\alpha},\bm{\tau},l) \triangleq x([\alpha_0 l+\tau_0]_{N_0}, [\alpha_1 l+\tau_1]_{N_1})\\ &= \sum_{(a,\bm{\omega})\in \mathbb{S}_2} a e^{j 2\pi \left( \frac {m_0 [\alpha_0 l+\tau_0]_{N_0}}{N_0} + \frac{m_1 [\alpha_1 l+\tau_1]_{N_1}}{N_1} \right)}, l\in [L].
\end{split}$$ Note that such line can wrap around within $x(n_0, n_1)$, and the sampling points along the line are always on the grid of $x(n_0, n_1)$ due to the choice of $\bm{\alpha}, \bm{\tau}$.
On taking an $L$-point DFT on (\[eq:sample\_slice\]), we get $$\label{eq:hs}
\begin{split}
&\hat{s}(\bm{\alpha},\bm{\tau},m) \triangleq \frac{1}{L} \sum_{l \in [L]} s(\bm{\alpha},\bm{\tau}, l) e^{-j 2\pi \frac{l m}{ L}} \\
&= \frac{1}{L} \sum_{(a,\bm{\omega})\in \mathbb{S}_2} a e^{j 2\pi \left(\frac{m_0 \tau_0}{N_0} + \frac{m_1 \tau_1}{N_1} \right)} \sum_{l \in [L]} e^{j2\pi l \left(\frac{m_0 \alpha_0}{N_0} + \frac{m_1 \alpha_1}{N_1} - \frac{m}{L}\right)},
\end{split}$$ where $m \in [L]$.
Let us assume that the line parameters are designed such that the orthogonality condition for frequency projection are satisfied (see Lemma \[le:length\] for details), i.e., for $m \in [L], [m_0,m_1]^T \in \mathcal{X}_2$, $$\label{eq:orthogonal}
\begin{split}
& \hat{f}(m) \triangleq \frac{1}{L} \sum_{l \in [L]} e^{j2\pi l \left(\frac{m_0 \alpha_0}{N_0} + \frac{m_1 \alpha_1}{N_1} - \frac{m}{L}\right)} \in \{0,1\},\\
\end{split}$$ then if $$\label{eq:lines}
\left[\frac{m_0 \alpha_0}{N_0} + \frac{m_1 \alpha_1}{N_1} - \frac{m}{L}\right]_1 = 0, [m_0,m_1]^T \in \mathcal{X}_2,$$ the $m_{\rm{th}}$ entry of (\[eq:hs\]) can be simplified as $$\label{eq:freqSum}
\hat{s}(\bm{\alpha},\bm{\tau},m) = \sum_{(a,\bm{\omega}) \in \mathbb{S}_2} a e^{j 2\pi \left(\frac{m_0 \tau_0}{N_0} + \frac{m_1 \tau_1}{N_1} \right)}.$$
Eqs. (\[eq:freqSum\]) and (\[eq:lines\]) state that each entry of the $L$-point DFT of the data located along a line with slope $\alpha_1/\alpha_0$ represents a projection of the $2$-D DFT values locating along the line of (\[eq:lines\]), which is orthogonal to the time domain line (\[eq:sample\_slice\]). This is closely related to the *Fourier projection-slice theorem*[@mersereau1974digital]. The Fourier projection-slice theorem states that the Fourier transform of a projection is a slice of the Fourier transform of the projected object. While the classical projection is in the time domain and the corresponding slice is in the frequency domain, in the FPS-SFT case, the projection is in the DFT domain and the corresponding slice is in the sample (discrete-time) domain. The important difference between the Fourier projection-slice theorem and FPS-SFT is that while the former reconstructs the frequency domain of the signal via interpolation frequency domain slices, which exhibits high complexity, the latter efficiently recovers the significant frequencies of the signal directly based on the DFT of time-domain $1$-D slices, i.e., samples along random lines. This is achieved by exploring the sparsity nature of the signal in the frequency domain, which is explained in the following.
We apply the assumption that the signal is sparse in the frequency domain; specifically, we assume that $|\mathbb{S}_2| = O(L)$. Then, if $|\hat{s}(\bm{\alpha},\bm{\tau}, m)| \neq 0$, with high probability, the $m_{\rm{th}}$ bin is $1$-sparse, and it holds that $\hat{s}(\bm{\alpha},\bm{\tau}, m) = a e^{j 2\pi \left(\frac{m_0 \tau_0}{N_0} + \frac{m_1 \tau_1}{N_1} \right)}, (a, \bm{\omega}) \in \mathbb{S}_2$. In such case, the $2$-D sinusoid, $(a, \bm{\omega})$, can be ‘decoded’ by three lines of the same slope but different offsets. The offsets for the three lines are designed as $\bm{\tau}, \bm{\tau}_0 \triangleq [[\tau_0+1]_{N_0}, \tau_1]^T, \bm{\tau}_1 \triangleq [\tau_0, [\tau_1+1]_{N_1}]^T$, respectively; such design allows for the frequencies to be decoded independently in each dimension. The sinusoid corresponding to the $1$-sparse bin, $m$, can be decoded as $$\label{eq:decoding}
\begin{split}
&m_0 = \left[ \frac{N_0}{2 \pi} \phi\left( \frac{\hat{s}(\bm{\alpha},\bm{\tau}_0,m) }{\hat{s}(\bm{\alpha},\bm{\tau},m) } \right) \right]_{N_0}, \\
&m_1 = \left[ \frac{N_1}{2 \pi} \phi\left(\frac{\hat{s}(\bm{\alpha},\bm{\tau}_1,m)}{\hat{s}(\bm{\alpha},\bm{\tau},m)} \right) \right]_{N_1},\\
&a = \hat{s}(\bm{\alpha},\bm{\tau},m) e^{-j2\pi(m_0 \tau_0/N_0 + m_1 \tau_1/N_1)}.
\end{split}$$ To recover all the sinusoids in $\mathbb{S}_2$ efficiently, each iteration of FPS-SFT adopts a random choice of line slope (see Lemma \[le:slope\]) and offset. Furthermore, the contribution of the recovered sinusoids in the previous iterations is removed via a *construction-subtraction* approach to creating a sparser signal in the future iterations. Specifically, assuming that for current iteration, the line slope and offset parameters are selected as $\bm{\alpha},\bm{\tau}$, respectively, the recovered sinusoids are projected into $L$ frequency bins to construct the DFT along the line, $\hat{s}_r(\bm{\alpha},\bm{\tau}, m) \triangleq \sum_{(a,\bm{\omega}) \in \mathcal{I}_{m}} a e^{j 2\pi \left(\frac{m_0 \tau_0}{N_0} + \frac{m_1 \tau_1}{N_1} \right)}$, $m \in [L]$, where $\mathcal{I}_{m}, m \in [L]$ represent the subsets of the recovered sinusoids that are related to the constructed DFT along line via projection, i.e., $\mathcal{I}_{m} \triangleq \{(a,\bm{\omega}) : [\frac{m_0 \alpha_0}{N_0} + \frac{m_1 \alpha_1}{N_1} - \frac{m}{L}]_1 = 0, [m_0,m_1]^T \in \mathcal{X}_2 \}, m \in [L]$.
Next, the $L$-point inverse DFT (IDFT) is applied on $\hat{s}_r (\bm{\alpha},\bm{\tau},m), m \in [L]$, from which the line, ${s}_r (\bm{\alpha},\bm{\tau},l), l \in [L]$ due to the previously recovered sinusoids are constructed. Subsequently, those constructed line samples are subtracted from the signal samples of the current iteration. Since the contribution of the recovered sinusoids is removed, the signal appears sparser and thus the recovery of the remaining sinusoids is easier in the future iterations.
Analysis of FPS-SFT
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In this section we provide some lemmas on the design of the lines used in FPS-SFT. Lemma \[le:length\] shows the design of the line length to guarantee orthogonality of projection. Lemma \[le:slope\] provides candidates of line slopes such that, each bin of the DFT along the line corresponds to the same number of frequencies projected to such bin. The uniformity of the projection is likely to create more $1$-sparse bins in the DFT of the lines. The proofs of the lemmas can be found in the Appendices.
\[le:length\] **(Line Length):** Let $L$ be the least common multiple (LCM) of $N_0,N_1$, and $s(\bm{\alpha}, \bm{\tau},l) = x ([\alpha_0 l+\tau_0]_{N_0},[\alpha_1 l+\tau_1]_{N_1})$ with $l \in [L], \bm{\alpha} \triangleq [\alpha_0, \alpha_1]^T, \bm{\tau}\triangleq [\tau_0, \tau_1]^T \in \mathcal{X}_2$ be a discrete line extracted from the signal model of (\[eq:sigModel2\]). Then each entry of the $L$-point DFT of $s(\bm{\alpha}, \bm{\tau},l)$, i.e., $\hat{s}(\bm{\alpha}, \bm{\tau},m), m \in [L]$ is the orthogonal projection of DFT values of the $N_0 \times N_1$-point DFT of (\[eq:sigModel2\]), whose frequencies locate on the discrete line of $[\frac{m_0}{N_0} \alpha_0 + \frac{m_1}{N_1} \alpha_1 - \frac{m }{L}]_1 = 0, [m_0,m_1]^T \in \mathcal{X}_2$. Moreover, $L$ is the minimum length of a line to allow orthogonal projection of DFT values of any frequency location $[m_0, m_1]^T \in \mathcal{X}_2$ with arbitrary choice of $\bm{\alpha} \in \mathcal{X}_2$.
\[le:slope\] **(Line Slope):** Let $s(\bm{\alpha}, \bm{\tau},l) = x([\alpha_0 l+\tau_0]_{N_0},[\alpha_1 l+\tau_1]_{N_1})$ with $l \in [L], L={\mathop{\mathrm{LCM}}\nolimits}(N_0,N_1), \bm{\alpha}\triangleq [\alpha_0, \alpha_1]^T \in \mathcal{A} \subset \mathcal{X}_2, \bm{\tau} \triangleq [\tau_0, \tau_1]^T \in \mathcal{X}_2$ be a discrete line extracted from the signal model of (\[eq:sigModel2\]), where $\mathcal{A} \triangleq \{\bm{\alpha} : \bm{\alpha} \in \mathcal{X}_2, \alpha_0 \perp \alpha_1, \alpha_0 \perp c_1, \alpha_1 \perp c_0\}$ with $c_0 = L/N_0, c_1 = L/N_1$. Let $\hat{s} (\bm{\alpha}, \bm{\tau},m), m \in [L]$ be the $L$-point DFT of $s(\bm{\alpha}, \bm{\tau},l), l \in [L]$. Then each entry of $\hat{s} (\bm{\alpha}, \bm{\tau},m), m \in [L]$ is the projection of DFT values located at $N/L$ different frequency locations in $\mathcal{X}_2$, i.e., $|\mathcal{P}_{m}| = N/L$, where $\mathcal{P}_{m} \triangleq \{[m_0,m_1]^T: [\frac{m_0}{N_0} \alpha_0 + \frac{m_1}{N_1} \alpha_1 - \frac{m }{L}]_1 = 0,[m_0,m_1]^T \in \mathcal{X}_2 \}$. Moreover, $\mathcal{P}_{m} \cap \mathcal{P}_{m'} = \emptyset$ for $m \neq m', m, m' \in [L]$. Thus, the DFT values of $N$ frequency locations in $\mathcal{X}_2$ are uniformly projected into the $L$ frequency bins of $\hat{s} (\bm{\alpha}, \bm{\tau},m), m \in [L]$.
**Complexity analysis:** The FPS-SFT executes $T$ iterations; in the $2$-D case, the samples used in each iteration is $3L$ since $3$ $L$-length lines, with $L={\mathop{\mathrm{LCM}}\nolimits}(N_0,N_1)$ are extracted in order to decode the two frequency components of a $2$-D sinusoid (see (\[eq:decoding\])). Hence, the sample complexity of FPS-SFT is $O(3T L) = O(L)$. The core processing of FPS-SFT is the $L$-point $1$-D DFT, which can be implemented by the FFT with the computational complexity of $O(L \log L)$. The $L$-point IDFT in the construction-subtraction procedure can also be implemented by the FFT. In addition to the FFT, each iteration needs to evaluate $O(K)$ frequencies. Hence the computational complexity of FPS-SFT is $O(L \log L+K)$. Assuming that $K = O (L)$, then the sample and computational complexity can be simplified as $O(K)$ and $O(K \log K)$, respectively, which achieves the lower bounds of the complexity of known state-of-the-art SFT algorithms [@ghazi2013sample; @pawar2017ffast]. **Multi-dimensional extension:** For the $D$-D case with the data cube size of $N_0 \times N_1 \times \cdots N_{D-1}$, the line length can be set as $L = {\mathop{\mathrm{LCM}}\nolimits}(N_0,\cdots,N_{D-1})$; the slope and offset parameters $[\alpha_0, \cdots, \alpha_{D-1}]^T, [\tau_0, \cdots, \tau_{D-1}]^T$ is randomly taken from $\mathcal{X}_D \triangleq [N_0] \times [N_1] \times \cdots [N_{D-1}]$. Each iteration extracts $D+1$ $L$-length lines with a same random slope but different offsets from the $D$-D data cube. The $0$-th line offset is set to be $[\tau_0,\cdots, \tau_{D-1}]^T$, while for the $i_{\rm{th}}$ line with $1 \le i \le D-1$, the offset for the $i_{\rm{th}}$ dimension is set to be $[\tau_i+1]_{N_i}$. With such offset parameters, the frequencies can be decoded independently for each dimension.
Numerical Results
=================
**Comparison to the SFT of [@ghazi2013sample]:** The length of the two dimensions are set to $N_0 = N_1 = 256$. We simulate two scenarios, when frequencies are uniformly distributed and when they are clustered. For the clustered case, we consider clusters of $9$ and $25$ frequencies. When $N_0=N_1$, the line length, $L$, of FPS-SFT equals $N_0$, and each iteration of FPS-SFT uses $3 N_0$ samples. We limit the maximum iterations to $T_{max} = N/(3 L) \approx 85$; this corresponds to roughly $100\%$ samples of the input data. Fig. \[fig:FreqRec\] (a) shows the probability of perfect recovery versus level of sparsity for FPS-SFT and the SFT of [@ghazi2013sample], respectively. When the signal is very sparse, e.g., $K = N_0/2$, the SFT of [@ghazi2013sample] has a high probability for perfect recovery, however, it fails when the sparsity is moderately large, e.g., $K = 2 N_0$. Moreover, the SFT of [@ghazi2013sample] only works for the scenario in which frequencies are distributed uniformly, while it fails when there exists even a single frequency cluster. On the contrary, the FPS-SFT applies to signals with a wide range of sparsity levels. For instance, the success rate of FPS-SFT is approximate $96\%$ when $K = 5 N_0$ and the frequency locations are uniformly distributed, while similar performance is observed for the clustered cases considered. In all cases, the success rates drop to $0$ when $K = 6 N_0$, since we set $T_{max} = 85$. To perfectly reconstruct all frequencies, the FPS-SFT needs to run for roughly $100$ iterations when $K = 6 N_0$. Fig. \[fig:FreqRec\] (b) shows the percentage of samples used by the FPS-SFT for perfect recovery versus different sparsity level for the uniform and clustered cases. The figure shows that the sparser the signal, the fewer samples are required by the FPS-SFT to recover all the frequencies. For example, when $K = N_0$, only $5.9\%$ of the signal samples are required in the uniform-distributed frequency case or the clustered case. The good performance of FPS-SFT arises because the randomized projections can effectively isolate the frequencies into $1$-sparse bins, even when the signal is less sparse ($K$ is large) and the frequencies are clustered.
-10pt
**Sparse image reconstruction:** Due to the duality of the time and frequency, the FPS-SFT is able to reconstruct a signal that is sparse in the time (spatial) domain using the samples in the frequency domain. Here we demonstrate the ability of FPS-SFT to recover images that are sparse in the pixel domain. Such sparse image recovery problem arises in the MRI applications[@lustig2008compressed]. In MRI, samples are directly taken from the frequency domain, from which the images reflecting the inner structure of the examined objects are reconstructed. Fig. \[fig:brain\] (a) shows a $512 \times 576$-pixel brain MRI image [@lustig2008compressed]. This image was sparsified by applying thresholding on the original image. Next, we converted the sparsified images into the frequency domain via a $512 \times 576$-point DFT, on which the $2$-D FPS-SFT was applied to reconstruct the images. Figs. \[fig:brain\] (b), (c) and (d) show that the images with $2.85\%$, $4.48\%$ and $6.61\%$ of non-zero pixels can be perfectly reconstructed by FPS-SFT using $14.0\%$, $23.4\%$, and $70.3\%$ samples in the frequency domain, respectively.
![Image reconstruction. (a) Raw image. (b) $2.9\%$-sparse, $K=8411$. (c) $4.5\%$-sparse, $K=13219$. (d) $6.6\%$-sparse, $K=19506$.[]{data-label="fig:brain"}](brain_raw "fig:") ![Image reconstruction. (a) Raw image. (b) $2.9\%$-sparse, $K=8411$. (c) $4.5\%$-sparse, $K=13219$. (d) $6.6\%$-sparse, $K=19506$.[]{data-label="fig:brain"}](brain_14 "fig:") ![Image reconstruction. (a) Raw image. (b) $2.9\%$-sparse, $K=8411$. (c) $4.5\%$-sparse, $K=13219$. (d) $6.6\%$-sparse, $K=19506$.[]{data-label="fig:brain"}](brain_23 "fig:") ![Image reconstruction. (a) Raw image. (b) $2.9\%$-sparse, $K=8411$. (c) $4.5\%$-sparse, $K=13219$. (d) $6.6\%$-sparse, $K=19506$.[]{data-label="fig:brain"}](brain_70 "fig:")\
(a)(b)(c)(d) -15pt
Conclusion {#sec:conclusion}
==========
We have proposed the FPS-SFT, a low-complexity, multi-dimensional SFT algorithm based on the idea of the Fourier projection-slice theorem. Theoretical and numerical results of FPS-SFT have been provided and an application of FPS-SFT on sparse image reconstruction has been demonstrated.
Collections of Proofs of Lemmas
===============================
Proof of Lemma \[le:length\]
----------------------------
The orthogonality condition derived in (\[eq:orthogonal\]) for $[m_0,m_1]^T, [\alpha_0,\alpha_1]^T \in \mathcal{X}_2, m \in [L]$ is equivalent to $$\label{eq:gline}
\left[ \frac{m_0 \alpha_0}{N_0} + \frac{m_1 \alpha_1}{N_1} - \frac{m}{L} \right]_1 = 0,$$ which can be rewritten as $$\label{eq:orthogonality}
\left[\frac{L}{N_0} m_0 \alpha_0 + \frac{L}{N_1} m_1 \alpha_1 \right]_L = m.$$ It is clear that $L = {\mathop{\mathrm{LCM}}\nolimits}(N_0,N_1)$ satisfies the above orthogonality condition, since $L/N_0, L/N_1$ are integers.
Next, we use contradiction to prove that $L = {\mathop{\mathrm{LCM}}\nolimits}(N_0,N_1)$ is the smallest line length that allows the orthogonal projection for any $[m_0,m_1]^T, [\alpha_0,\alpha_1]^T \in \mathcal{X}_2$.
Assume that $L < {\mathop{\mathrm{LCM}}\nolimits}(N_0,N_1)$, then, the consequence is that at least either $L/N_0$ or $L/N_1$ is not an integer. Without loss of generality, we assume that $\frac{L}{N_0} \notin \mathbb{Z}$, then the right side of (\[eq:orthogonality\]) equals to $[L/N_0]_L \notin [L]$ for $m_0=1,\alpha_0=1, m_1=0$, which is contradictory to the premise that the orthogonality condition holds for any $[m_0,m_1]^T, [\alpha_0,\alpha_1]^T \in \mathcal{X}_2$. Hence $L = {\mathop{\mathrm{LCM}}\nolimits}(N_0,N_1)$ is the smallest line length which allows the orthogonal projection of any frequency to a line with arbitrary slope.
Proof of Lemma \[le:slope\]
---------------------------
This proof is organized as follows. First, by exploring the Bézout’s lemma [@rosen1993elementary], we prove that with the specified line parameters, i.e., $L = {\mathop{\mathrm{LCM}}\nolimits}(N_0, N_1), [\alpha_0, \alpha_1]^T \in \mathcal{A}, [\tau_0, \tau_1]^T \in \mathcal{X}_2$, each entry of the DFT along a line, i.e., $\hat{s}(\bm{\alpha}, \bm{\tau}, m), m \in [L]$ contains at least the projection of the DFT value from one frequency location $(m_0',m_1')$ in $\mathcal{X}_2$, i.e., $|\mathcal{P}_{m}|>0, m \in [L]$. Next, we prove that $|\mathcal{P}_{m}| \ge N/L$, followed by the proof of $\mathcal{P}_{m} \cap \mathcal{P}_{m'} = \emptyset$ for $m \neq m', m, m' \in [L]$, and finally, we conclude that $|\mathcal{P}_{m}| = N/L$.
Let $\alpha_0' = \alpha_0 c_0, \alpha_1' = \alpha_1 c_1$. Since $\alpha_0 \perp \alpha_1, \alpha_0 \perp c_1, \alpha_1 \perp c_0$, and $c_0 \perp c_1$ due to $L = {\mathop{\mathrm{LCM}}\nolimits}(N_0, N_1)$, it is obvious that $\alpha_0' \perp \alpha_1'$. According to the Bézout’s lemma, there exist $m_0, m_1 \in \mathbb{Z}$, such that $$\label{eq:bezout}
\alpha_0' m_0 + \alpha_1' m_1 = 1.$$ By multiplying $m \in [L]$ to the two sides of (\[eq:bezout\]), we get $$\alpha_0' m m_0 + \alpha_1' m m_1 = m ,$$ which, using the Euclidean division, can be written as $$\label{eq:mult_omega}
\alpha'_0 (m_0' + k_0 N_0) + \alpha'_1 (m_1' + k_1 N_1) = m,$$ where $m_0' = [m m_0]_{N_0}, m_1'= [m m_1]_{N_1}$; $k_0, k_1 \in \mathbb{Z}$.
Since that $$\label{eq:L}
[\alpha'_0 k_0 N_0 + \alpha'_1 k_1 N_1]_L = [L(\alpha_0 k_0 + \alpha_1 k_1)]_L = 0,$$ on taking modulo-$L$ of the two sides of Eq. (\[eq:mult\_omega\]), we have $$\label{eq:solve_line}
[\alpha'_0 m_0' + \alpha'_1 m_1']_L = m,$$ which is equivalent to (\[eq:gline\]). It means that there exists a frequency location $[m_0',m_1']^T \in \mathcal{X}_2$, whose DFT value projects to $\hat{s}(\bm{\alpha}, \bm{\tau}, m)$, i.e., $|\mathcal{P}_{m}|>0, m \in [L]$.
Next, let’s explore the solution structure of (\[eq:solve\_line\]). It is easy to see that the frequency locations, $[m_0'+k \alpha'_1, m_1' - k \alpha'_0]^T, k\in \mathbb{Z}$, satisfies (\[eq:solve\_line\]), i.e., $$\label{eq:line_solutions}
[\alpha'_0 (m_0'+k \alpha'_1) + \alpha'_1 (m_1' - k \alpha'_0)]_L = m,$$ which can be written as $$[\alpha'_0 ([m_0'+k \alpha'_1]_{N_0}+k_0 N_0) + \alpha'_1 ([m_1' - k \alpha'_0]_{N_1}+k_1 N_1)]_L = m,$$ where $k_0,k_1 \in \mathbb{Z}$. Again, by substituting (\[eq:L\]), we have $$[\alpha'_0 [m_0'+k \alpha'_1]_{N_0} + \alpha'_1 [m_1' - k \alpha'_0]_{N_1}]_L = m.$$ Hence, the DFT value at frequency locations $\left[[m_0'+k \alpha'_1]_{N_0}, [m_1' - k \alpha'_0]_{N_1}\right]^T \in \mathcal{P}_{m} \subseteq \mathcal{X}_2$, also projects to $\hat{s}(\bm{\alpha}, \bm{\tau}, m)$.
Next, we prove that $|\mathcal{P}_{m}| \ge N/L$. Assume that for $k \neq k'$, there exits two duplicated frequency locations, i.e., $\left[[m_0'+k \alpha'_1]_{N_0}, [m_1' - k \alpha'_0]_{N_1}\right]^T = \left[[m_0'+k' \alpha'_1]_{N_0}, [m_1' - k' \alpha'_0]_{N_1}\right]^T$. It follows that $$[k \alpha'_1]_{N_0} = [k' \alpha'_1]_{N_0}, \; [k \alpha'_0]_{N_1} = [k' \alpha'_0]_{N_1},$$ which can be rewritten as $$k \alpha'_1 = k' \alpha'_1 + k_0 N_0, \; k \alpha'_0 = k' \alpha'_0 + k_1 N_1,$$ where $k_0,k_1 \in \mathbb{Z}$. It is easy to conclude that $k_1/k_0 = \alpha_0/\alpha_1$. Hence we have $$k \alpha'_1 = k' \alpha'_1 + i \alpha_1 N_0, \; k \alpha'_0 = k' \alpha'_0 + i \alpha_0 N_1,$$ where $i \in \mathbb{Z}, i \neq 0$. Hence $$k-k' = i N_0/c_1 = i N/L,$$ which means that the frequency location, $\left[[m_0'+k \alpha'_1]_{N_0}, [m_1' - k \alpha'_0]_{N_1}\right]^T$, repeats every $N/L$ points. In another words, there exist at least $N/L$ frequency locations whose DFT values projecting to $\hat{s}(\bm{\alpha}, \bm{\tau}, m)$, i.e., $|\mathcal{P}_{m}|\ge N/L$.
Next, we prove that $\mathcal{P}_{m} \cap \mathcal{P}_{m'} = \emptyset$ for $m \neq m', m, m' \in [L]$. Assume that $[m_0,m_1]^T \in \mathcal{P}_{m} \cap \mathcal{P}_{m'}$, it can be seen that $$[\alpha'_0 m_0 + \alpha'_1 m_1]_L = m = m',$$ which is contradict with $m \neq m'$. Hence $\mathcal{P}_{m} \cap \mathcal{P}_{m'} = \emptyset$.
Finally, by combing $\mathcal{P}_{m} \cap \mathcal{P}_{m'} = \emptyset$, $m \in [L]$, $|\mathcal{P}_{m}|\ge N/L$ and $|\mathcal{X}_2| = N$, we can conclude that $|\mathcal{P}_{m}| = N/L$. This completes the proof.
|
---
abstract: 'It is shown that the geometry of a class of multisymplectic manifolds, that is, smooth manifolds equipped with a closed nondegenerate form of degree greater than $1$, is characterized by their automorphisms. Such a class is distinguished by a [*local homogeneity*]{} property. Thus, [*locally homogeneous multisymplectic manifolds*]{} extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms and on the study of the local properties of Hamiltonian vector fields on multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.'
author:
- |
A. Ibort[^1]\
\
[A. Echeverría-Enríquez, M. C. Muñoz-Lecanda[^2], N. Román-Roy[^3]]{}\
date:
title: INVARIANT FORMS AND AUTOMORPHISMS OF A CLASS OF MULTISYMPLECTIC MANIFOLDS
---
= 15.5cm = 23cm =-1.5cm =-2cm
\#1
**
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\#1
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3M[Departamento de Matemáticas\
Universidad Carlos III de Madrid\
Avda. Universidad 30,\
E-28911 Leganés, Madrid, SPAIN]{}
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: [*Multisymplectic manifolds, multisymplectic diffeomorphisms, invariant forms, Hamiltonian (multi-) vector fields, graded Lie algebras.*]{}
Introduction and statement of the main results
==============================================
It is well-known that some classical geometrical structures are determined by their automorphism groups, for instance it was shown by Banyaga [@Ba86], [@Ba88], [@Ba97] that the geometric structures defined by a volume or a symplectic form on a differentiable manifold are determined by their automorphism groups, the groups of volume preserving and symplectic diffeomorphisms respectively, i.e., if $(M_i,\alpha_i)$, $i=1,2$ are two paracompact connected smooth manifolds equipped with volume or symplectic forms $\alpha_i$ and $G(M_i,\alpha_i)$ denotes the group of volume preserving or symplectic diffeomorphisms, then if $\Phi\colon G(M_1,\alpha_1) \to G(M_2,\alpha_2)$ is a group isomorphism, there exists (modulo an additional condition in the symplectic case) a unique $C^\infty$-diffeomorphism $\varphi \colon M_1 \to M_2$ such that $\Phi (f)=\varphi \circ f \circ \varphi^{-1}$, for all $f\in G(M_1,\alpha_1)$ and $\varphi^*\alpha_2 = c \, \alpha_1$, with $c$ a constant. In other words, group isomorphisms of automorphism groups of classical structures (symplectic, volume) are inner, in the sense that they correspond to conjugation by (conformal) diffeomorphisms.
An immediate consequence of the previous theorem is that if $(M,\alpha )$ is a manifold with a classical structure (volume or symplectic) and the differential form $\beta$ is an invariant for the group $G(M,\alpha )$, then necessarily $\beta$ has to be a constant multiple of exterior powers of $\alpha$. In other words, the only differential invariants of the groups of classical diffeomorphism are multiples of exterior powers of the defining geometrical structure. The infinitesimal counterpart of this result was already known in the realm of Classical Mechanics. In 1947 Lee Hwa Chung stated a theorem concerning the uniqueness of invariant integral forms (the Poincaré-Cartan integral invariants) under canonical transformations [@Hw-47]. His aim was to use that result in order to characterize canonical transformations in the Hamiltonian formalism of Mechanics; that is, canonical transformations are characterized as those transformations mapping every Hamiltonian system into another Hamiltonian one with respect to the same symplectic structure. Afterwards, this result was discussed geometrically [@LlR-88] and generalized to presymplectic Hamiltonian systems [@GLR-84; @CGIR-85]. The main result there was that in a given a symplectic (resp. presymplectic) manifold, the only differential forms invariant with respect to all Hamiltonian vector fields are multiples of (exterior powers of) the symplectic (resp. presymplectic) form. Since symplectic and presymplectic manifolds represent the phase space of regular and singular Hamiltonian systems respectively, this result allows to identify canonical transformations in the Hamiltonian formalism of Mechanics with the symplectomorphisms and presymplectomorphisms group, on each case.
Returning to the general problem of the relation between geometric structures and their group of automorphisms, it is an open question to determine which geometrical structures are characterized by them. Apart from symplectic and volume, contact structures also fall into this class [@Ba95]. Recent work of Grabowski shows that similar statements hold for Jacobi and Poisson manifolds [@Gr93; @At90]. Our main result shows that a class of multisymplectic manifolds are determined by their automorphisms (finite and infinitesimal). Multisymplectic manifolds are one of the natural generalizations of symplectic manifolds. A multisymplectic manifold of degree $k$ is a smooth manifold $M$ equipped with a closed nondegenerate form $\Omega$ of degree $k\geq 2$ (see [@Ca96a; @Ca96b] for more details on multisymplectic manifolds). In particular multisymplectic manifolds include symplectic and volume manifolds. A diffeomorphism $\varphi$ between two multisymplectic manifolds $(M_i, \Omega_i)$, $i=1,2$, will be called a multisymplectic diffeomorphism if $\varphi^* \Omega_2 = \Omega_1$. The group of multisymplectic diffeomorphisms of a multisymplectic manifold $(M, \Omega )$ will be denoted by $G(M,
\Omega )$. Multisymplectic structures represent distinguished cohomology classes of the manifold $M$ but their origin as a geometrical tool can be traced back to the foundations of the calculus of variations. It is well known that the suitable geometric framework to describe (first order) field theories are certain multisymplectic manifolds (see, for instance [@CCI-91; @EMR-00; @GMS-97; @Go-91b; @Hr-99b; @Ka-98; @MS-99; @Sd-95; @Sa-89] and references quoted therein). The automorphism groups of multisymplectic manifolds play a relevant role in the description of the corresponding system and it is a relevant problem to characterize them in similar terms as in symplectic geometry.
A crucial property for multisymplectic structures is their local homogeneity properties (see Section 4). Then, working with multisymplectic manifolds verifying this property, our main results are:
\[main\] Let $(M_i,\Omega_i)$, $i=1,2$, be two locally homogeneous multisymplectic manifolds and $G(M_i,\Omega_i)$ will denote their corresponding groups of automorphisms. Let $\Phi \colon G(M_1,\Omega_1) \to G(M_2,
\Omega_2)$ be a group isomorphism which is also a homeomorphism when $G(M_i,\Omega_i)$ are endowed with the point-open topology. Then, there exists a $C^\infty$ diffeomorphism $\varphi\colon M_1 \to M_2$, such that $\Phi (f) = \varphi \circ f \circ
\varphi^{-1}$ for all $f\in G(M_1,\Omega_1)$ and the tangent map $\varphi_*$ maps locally Hamiltonian vector fields of $(M_1,\Omega_1)$ into locally Hamiltonian vector fields of $(M_2,\Omega_2)$. In addition, if we assume that $\varphi_*$ maps all infinitesimal automorphisms of $(M_1, \Omega_1)$ into infinitesimal automorphisms of $(M_2, \Omega_2)$ then there is a constant $c$ such that $\varphi^*\Omega_2 = c\,\Omega_1$.
This result generalizes the main theorems in [@Ba86] (Thms. 1 and 2) which are in turn generalizations of a theorem by Takens [@Ta79]. The proof presented here, contrary to the proof in [@Ba86], will not rely on the generalization by Omori [@Om74] of Pursell-Shanks theorem [@Pu54] which do not apply to this situation because of the lack of local normal forms for multisymplectic structures. However we will use the following partial generalization of Lee Hwa Chung theorem.
\[invar\] Let $(M,\Omega)$ be a locally homogeneous multisymplectic manifold of degree $k$, then the only differential forms of degree $k$ invariant under the graded Lie algebra of infinitesimal automorphisms of $\Omega$ are real multiples of $\Omega$.
Local properties of multisymplectic diffeomorphisms of locally homogeneous multisymplectic manifolds will play a crucial role along the discussion. They steam from a localization property for Hamiltonian vector fields that will be discussed in Lemma \[localization\]. These local properties are used to prove another result interesting on its own: the group of multisymplectic diffeomorphisms acts transitively on the underlying manifold. The transitivity of the group of multisymplectic diffeomorphisms relies on the fact that Hamiltonian vector fields span the tangent bundle of the manifold as shown in Lemma \[transitivity\].
The paper is organized as follows: in Section 2 we establish some basic definitions and results, mainly related with the geometry of multisymplectic manifolds. Section 3 is devoted to the definition and basic properties of the graded Lie algebra of the infinitesimal automorphisms of multisymplectic manifolds. In Section 4 the definition and some characteristics of locally homogeneous multisymplectic manifolds is stated, in particular the localization lemma for multisymplectic diffeomorphisms, and the strong local transitivity of the group of multisymplectic diffeomorphisms is proved for locally homogeneous multisymplectic manifolds. In Section 5 we will prove the main results on the structure of differential invariants of locally homogeneous multisymplectic manifolds. Finally, in Section 6, these results are used to characterize the multisymplectic transformations and the proof of the main theorem is completed in Section 7.
All manifolds are real, paracompact, connected and $C^\infty$. All maps are $C^\infty$. Sum over crossed repeated indices is understood.
Notation and basic definitions
==============================
Let $M$ be a $n$-dimensional differentiable manifold. Sections of $\Lambda^m(\Tan M)$ are called $m$-[*multivector fields*]{} in $M$, and we will denote by $\vf^m (M)$ the set of $m$-multivector fields in $M$. Let $\Omega\in\df^k(M)$ be a differentiable $k$-form in $M$ ($k\leq n$). For every $x\in M$, the form $\Omega_x$ establish a correspondence $\hat{\Omega}_m(x)$ between the set of $m$-multivectors $\Lambda^m (\Tan_x M)$ and the $(k-m)$-forms $\Lambda^{k-m} (\Tan_x^* M)$ as $$\begin{array}{ccccl}
\hat\Omega_{m}(x) &\colon&\Lambda^m (\Tan_x M) &
\longrightarrow & \Lambda^{k-m}(\Tan_x^* M)
\\
& & v & \mapsto & \inn(v)\Omega_x
\end{array} .$$ If $v$ is homogeneous, $v = v_1\wedge\ldots\wedge v_m$, then $\inn (v)\Omega_x = \inn
(v_1\wedge\ldots\wedge v_m)\Omega_x = \inn (v_1)\ldots\inn (v_m)\Omega_x$. Thus, an $m$-multivector $X\in\vf^m(M)$ defines a contraction $\inn (X)$ of degree $m$ of the algebra of differential forms in $M$.
The $k$-form $\Omega$ is said to be [*$m$-nondegenerate*]{} (for $1\leq m\leq k-1$) iff, for every $x\in M$ the subspace $\ker\,\hat{\Omega}_{m}(x)$ has minimal dimension. Such subspace will be usually denoted by $\ker^m\Omega_x$. If $\Omega$ is $m$-nondegenerate and $\scriptstyle{\left(\begin{array}{c} n \\ m \end{array}\right) } \leq
\scriptstyle{\left(\begin{array}{c} n \\ k-m \end{array}\right) }$, then $\dim\,(\ker^m\Omega_x) = 0$, but if $\scriptstyle{\left(\begin{array}{c} n \\ m \end{array}\right) } >
\scriptstyle{\left(\begin{array}{c} n \\ k-m \end{array}\right) }$, then $\dim\,(\ker^m\Omega_x) = \scriptstyle{\left(\begin{array}{c} n \\ m
\end{array}\right) } - \scriptstyle{\left(\begin{array}{c} n \\ k-m
\end{array}\right) }$. The form $\Omega$ will be said to be [*strongly nondegenerate*]{} iff it is $m$-nondegenerate for every $m=1,\ldots ,k-1$. Thus, the $m$-nondegeneracy of $\Omega$ implies that the map $\hat{\Omega}_m\colon\Lambda^m(\Tan M) \to \Lambda^{k-m}(\Tan^*M)$ is a bundle monomorphism in the first situation or a bundle epimorphism in the second case. The image of the bundle $\Tan M$ by $\hat{\Omega}_m$ will be denoted by $E_m$. Often, if there is no risk of confusion, we will omit the subindex $m$ and we will denote $\hat{\Omega}_{m}$ simply by $\hat{\Omega}$.
If $X\in\vf^m(M)$, the graded bracket $$[\d , \inn (X)]=\d\inn (X)-(-1)^m\inn (X)\d$$ where $\d$ denotes as usual the exterior differential on $M$, defines a new derivative of degree $m-1$, denoted by $\Lie (X)$. If $X\in\vf^i(M)$, and $X\in\vf^j(M)$ the graded commutator of $\Lie (X)$ and $\Lie (Y)$ is another operation of degree $i + j -2$ of the same type, i.e., there will exists a $(i+j-1)$-multivector denoted by $[X,Y]$ such that, $$[\Lie (X), \Lie (Y)] = \Lie ([X,Y]).$$ The bilinear assignement $X,Y \mapsto [X,Y]$ is called the Schouten-Nijenhuis bracket of $X,Y$.
Let $X$, $Y$ and $Z$ homogeneous multivectors of degrees $i,j,k$ respectively, then the Schouten-Nijenhuis bracket verifies the following properties:
$[X,Y] = -(-1)^{(i+1)(j+1)} [Y,X]$.
$[X, Y\wedge Z] =
[X,Y]\wedge Z + (-1)^{(i+1)(j+1)} Y\wedge [X,Z]$.
$(-1)^{(i+1)(k+1)}[X,[Y ,Z] ] + (-1)^{(j+1)(i+1)}[Y,[Z,X] ]
+(-1)^{(k+1)(j+1)}[Z,[X ,Y] ] = 0 $. The exterior algebra of multivectors has the structure of an odd Poisson algebra, sometimes also called a Schouten algebra. This allows us to define an structure of an odd Poisson graded manifold on $M$ whose sheaf of superfunctions is given by the sheaf of multivector fields and the odd Poisson bracket is the Schouten bracket.
Let $M$ be a $n$-dimensional differentiable manifold and $\Omega\in\df^k(M)$. The couple $(M,\Omega)$ is said to be a [multisymplectic manifold]{} if $\Omega$ is closed and $1$-nondegenerate. The degree $k$ of the form $\Omega$ will be called [*the degree of the multisymplectic manifold*]{}.
Thus, multisymplectic manifolds of degree $k=2$ are the usual symplectic manifolds, and manifolds with a distinguished volume form are multisymplectic manifolds of degree its dimension. Other examples of multisymplectic manifolds are provided by compact semisimple Lie groups equipped with the canonical cohomology 3-class, symplectic 6-dimensional Calabi-Yau manifolds with the canonical 3-class, etc. Notice that there are no multisymplectic manifolds of degrees 1 or $n-1$ because $\ker \Omega$ is nonvanishing in both cases.
Apart from the already cited, another very important class of multisymplectic manifolds are the [*multicotangent bundles*]{}: let $Q$ be a manifold, and $\pi\colon\Lambda^k(\Tan^*Q)\to Q$ the bundle of $k$-forms in $Q$. Then, $\Lambda^k(\Tan^*Q)$ is endowed with a canonical $k$-form $\Theta\in\df^k(\Lambda^k(\Tan^*Q)$ defined as follows: if $\alpha\in\Lambda^k(\Tan^*Q)$, and $\moment{U}{1}{k}\in\Tan_\alpha(\Lambda^k(\Tan^*Q)$, then $$\Theta_\alpha(U_1,\ldots ,U_k):=\inn(\pi_*U_1\wedge\ldots\wedge\pi_* U_k)\alpha$$ If $(x^i,p_{\moment{i}{1}{k}})$ is a system of natural coordinates in $W\subset\Lambda^k(\Tan^*Q)$, then $$\Theta\mid_W=p_{\moment{i}{1}{k}}\d x^{i_1}\wedge\ldots\wedge\d x^{i_k}$$ Therefore, $\Omega:=-\d\Theta\in\df^{k+1}(\Lambda^k(\Tan^*Q)$ is a $1$-nondegenerate form. Such a structure is often called a [*multicotangent bundle*]{}.
Multisymplectic structures of degree $\geq 3$ are abondant. In fact, as it is shown in [@Ma70], if $M$ is a smooth manifold of dimension $\geq 7$, then the space of multisymplectic structures of degree $3\leq k \leq n-3$ is residual. However there is no a local classification of multisymplectic forms, not even in the linear case.
The graded Lie algebra of infinitesimal automorphisms of a multisymplectic manifold
===================================================================================
From now on $(M,\Omega)$ will be a multisymplectic manifold.
A [*multisymplectic diffeomorphism*]{} is a diffeomorphism $\varphi\colon M\to M$ such that $\varphi^*\Omega = \Omega$. A [*locally Hamiltonian vector field*]{} on $(M,\Omega)$ is a vector field $X$ whose flow consists of multisymplectic diffeomorphisms. It is clear that $X$ is a locally Hamiltonian vector field iff $\Lie (X) \Omega = 0$, or equivalently, iff $\inn(X) \Omega$ is a closed $(k-1)$-form. This facts leads to the following generalization:
Let $X\in\vf^m(M)$ ($m\geq 1$).
$X$ is said to be a [Hamiltonian $m$-multivector field]{} iff $\inn (X)\Omega$ is an exact $(k-m)$-form; that is, there exists $\zeta \in\df^{k-m-1}(M)$ such that (X)=\[ham\]
$X$ is said to be a [locally Hamiltonian $m$-multivector field]{} iff $\inn (X)\Omega$ is a closed $(k-m)$-form. In this case, for every point $x\in M$, there is an open neighbourhood $W\subset M$ and $\zeta \in\df^{k-m-1}(U)$ such that $$\inn (X)\Omega = \d\zeta \qquad {\rm (on\ W)}$$
In any case, $\zeta$ is defined modulo closed $(k-m-1)$-forms. The class of $(k-m-1)$-forms defined by $\zeta$ is said to be the [*(local) Hamiltonian*]{} for $X$ and an element $\zeta$ in this class is said to be a [*local Hamiltonian form*]{} for $X$. Conversely, $\zeta \in\df^p(M)$ is said to be a [*Hamiltonian $p$-form*]{} iff there exist a $(k-p-1)$-multivector field $X\in\vf (M)$ such that (\[ham\]) holds.
[**Remarks:**]{}
If $m>k$ the previous definitions are void and all $m$-multivector fields are Hamiltonian.
There are no Hamiltonian forms of degree higher than $k-2$.
Any $m$-multivector field $X$ lying in $\ker^m\Omega$ is a Hamiltonian multivector field with Hamiltonian form the zero form.
Locally Hamiltonian multivector fields $X$ of degree $k-1$ define closed 1-forms $\inn(X) \Omega$, which locally define a smooth function $f$ (up to constants) called the [*local Hamiltonian function*]{} of $X$.
The following Lemma follows immediately from the previous considerations.
\[surjective\] Let $\Omega\in\df^k(M)$ be a closed $m$-nondegenerate form.
For every differentiable form $\zeta\in\df^{m-1}(M)$ such that $\scriptstyle{\left(\begin{array}{c} n \\ m \end{array}\right) } \leq
\scriptstyle{\left(\begin{array}{c} n \\ k-m \end{array}\right) }$, there exists a $(k-m)$-locally Hamiltonian multivector field $X$ possessing it as local Hamiltonian form, i.e. such that $\inn(X)\Omega =\d\zeta$. As a consequence, the differentials of Hamiltonian $(m-1)$-forms of locally Hamiltonian $(k-m)$-multivector fields span locally the $m$-multicotangent bundle of $M$, $\Lambda^m(\Tan^*M)$.
If $\scriptstyle{\left(\begin{array}{c} n \\ m \end{array}\right) } \leq
\scriptstyle{\left(\begin{array}{c} n \\ k-m \end{array}\right) }$ the family of locally Hamiltonian $(k-m)$-multivector fields span locally the $(k-m)$-multitangent bundle of $M$, this is $$\Lambda^{k-m}(\Tan_x M)=\span\set{X_x \mid \Lie (X)\Omega =0 \ , \
X\in\vf^{k-m}(M)}.$$
Let $\Omega$ be $m$-nondegenerate of degree $k$. The map $\hat{\Omega}_{k-m}$ has its rank in the bundle $\Lambda^m(T^*M)$, but $\hat{\Omega}_{k-m}$ is the dual map of $\hat{\Omega}_m$ (up to, perhaps, a minus sign). And as $\hat{\Omega}_m$ is a monomorphism, then $\pm\hat\Omega_m^*=\hat{\Omega}_{k-m}\colon
\Lambda^{k-m}(\Tan M)\to\Lambda^m(\Tan^* M)$ is onto. Then, for every $\zeta\in\df^{m-1}(M)$, $\d\zeta$ defines a section of $\Lambda^m(\Tan^* M)$, hence we can choose a smooth $(k-m)$-multivector field $X$ such that $\hat{\Omega}_{k-m}(x)(X_x)=\d\zeta(x)$, for every $x\in M$.
Taking a family of coordinate functions $x^i$, the same can be done locally for a family of $(m-1)$-forms $x^{i_1}\d x^{i_2}\wedge\ldots\wedge\d x^{i_m}$, showing in this way that the differentials of Hamiltonians $(m-1)$-forms span locally the $m$-multicotangent bundle of $M$.
For every $X\in\vf^{k-m}(M)$, $\inn(X)\Omega\in\df^m(M)$. But, taking into account the above item, for every $x\in M$, there exists a neighbourhood $U\subset M$ such that $\inn(X)\Omega\vert_U=f^i\d\zeta_i$, where $f^i\in \Cinfty (U)$ and $\zeta_i\in\df^{m-1}(M)$ with $\inn(X_i)\Omega\vert_U=\d\zeta_i$ for some locally Hamiltonian $(k-m)$-multivector fields $X_i$. Therefore $X\vert_U=f^iX_i+Z$, with $Z\in\ker^{k-m}\Omega$; that is, $\inn (Z)\Omega=0$, so $Z$ are also locally Hamiltonian $(k-m)$-multivector fields and the proof is finished.
Notice that for $m=1$, if $k\geq 2$, then $n= \scriptstyle{\left(\begin{array}{c} n \\
1 \end{array}\right) } \leq \scriptstyle{\left(\begin{array}{c} n \\k-1
\end{array}\right) }$. Thus if $\Omega$ is $1$-nondegenerate, the above Lemma states that the differentials of Hamiltonian functions of locally Hamiltonian $(k-1)$-multivector fields span locally the cotangent bundle of $M$ and that, on its turn, the family of these $(k-1)$-Hamiltonian multivector fields span locally the $(k-1)$-multitangent bundle of $M$. However the previous Lemma says nothing about the Hamiltonian vector fields. We will analize this question in the following Section.
It is immediate to prove that, as for locally Hamiltonian vector fields in symplectic manifolds, we have the following characterization:
\[lie\_algebra\] A $m$-multivector field $X\in\vf^m(M)$ is a locally Hamiltonian $m$-multivector field if, and only if, $\Lie (X)\Omega =0$. Moreover, if $X$, $Y$ are locally Hamiltonian multivector fields, then $[X,Y]$ is a Hamiltonian multivector field with Hamiltonian form $\inn(X\wedge Y) \Omega$. \[chmf\]
In fact, if $X$, $Y$ are multivector fields of degrees $l,m$ respectively we have, $$\Lie ([X,Y]) \Omega = \Lie (X)\Lie (Y) \Omega - (-1)^{l+m} \Lie (Y) \Lie (X)
\Omega = 0 .$$ Moreover, $\inn([X,Y]) \Omega = \Lie (X)\inn(Y) \Omega -
(-1)^{l+m}\inn(Y)\Lie (X)\Omega = \d (\inn(X)\inn(Y) \Omega )$.
We will denote respectively by $\vf^m_h(M)$ and $\vf^m_{lh}(M)$ the sets of Hamiltonian and locally Hamiltonian $m$-multivector fields in $M$. It is clear by the previous proposition that $\bigoplus_{m\geq 0}\vf^m_{lh}(M)$ is a graded Lie subalgebra of the graded Lie algebra of multivector fields. We will say that a $m$-multivector field is [*characteristic*]{} if it lies in $\ker^m\Omega$. The set of characteristic fields constitute a graded Lie subalgebra of $\bigoplus_{m\geq 0}\vf^m_{lh}(M)$. Moreover, the characteristic multivector fields define a graded ideal of the graded Lie algebra of Hamiltonian multivector fields. We will denote the corresponding quotient graded Lie algebra by ${\cal V}_H^* (M,\Omega)$, and $${\cal V}_H^* (M,\Omega) = \bigoplus_{m\geq 0} {\cal V}_H^m(M,\Omega)
\qquad ,\qquad
{\cal V}_H^m(M,\Omega) = \vf^m_{lh}(M) / \Gamma (\ker\hat{\Omega}_m) .$$ Notice that again if $m>k$, $\ker^m \Omega = \Lambda^m (TM)$, hence ${\cal V}_H^m(M,\Omega) = 0$ and ${\cal V}_H^1(M,\Omega) = \vf_{lh}(M)$. Namely, $${\cal V}_H^* (M,\Omega) = \bigoplus_{m= 0}^k {\cal V}_H^m(M,\Omega) .$$
The Lie algebra ${\cal V}_H^* (M,\Omega)$ will be called the [infinitesimal graded Lie algebra]{} of $(M,\Omega)$ or the [graded Lie algebra of infinitesimal automorphisms]{} of $(M,\Omega )$.
We can translate this structure of graded Lie algebra to the corresponding Hamiltonian forms in a similar way as it is done in symplectic geometry (see [@Ca96a] for more details on this construction).
We will denote by ${\cal H}^p(M)$ the set of Hamiltonian $p$-forms in $M$ and by ${\tilde{\cal H}}^p(M)$ the set of Hamiltonian $p$-forms modulo closed $p$-forms, ${\tilde{\cal H}}^p(M) = {\cal H}^p(M) / Z^p (M)$. The classes in ${\tilde{\cal H}}_h^p(M)$ will be denoted by $\bar{\zeta}$, meaning by that the class containing the Hamiltonian $p$-form $\zeta$. Let ${\tilde{\cal H}}^*(M) := \oplus_{p\geq 0}{\tilde{\cal H}}^p(M)$. We can then define a graded Lie bracket on ${\tilde{\cal H}}^*(M)$.
Let $\bar\xi\in{\cal H}^p(M)$, $\bar\zeta\in{\cal H}^m(M)$ and $X_{\xi}\in\vf^{k-p-1}_h(M)$, $Y_{\zeta}\in\vf^{k-m-1}_h(M)$ their corresponding Hamiltonian multivector fields modulo $\Gamma
(\ker\hat{\Omega}_*)$. The bracket of these Hamiltonian classes (related to the multisymplectic structure $\Omega$) is the $(p+m-k+2)$-Hamiltonian class $\{\bar\xi ,\bar\zeta\}$ containing the form, $$\{\xi ,\zeta\}:= \Omega (X_{\xi},Y_{\zeta})=
\inn (Y_{\zeta})\inn (X_{\xi})\Omega=
\inn (Y_{\zeta})\d\xi=
(-1)^{(k-p-1)(k-m-1)}\inn (X_{\xi})\d\zeta$$
It is an easy exercise to check that $\{\bar\xi ,\bar\zeta\}$ is well defined. In the same way that in the symplectic case the Poisson bracket is closely related to the Lie bracket, now we have:
Let $X_{\xi}\in\vf^p_h(M)$, $Y_{\zeta}\in\vf^m_h(M)$ Hamiltonian multivector fields and $\bar\xi\in{\tilde{\cal H}}^{k-p-1}(M)$, $\bar\zeta\in{\tilde{\cal
H}}^{k-m-1}(M)$ the corresponding Hamiltonian classes. Then the Schouten-Nijenhuis bracket $[X_{\xi},Y_{\zeta}]$ is a Hamiltonian $(p+m-1)$-multivector field whose Hamiltonian $(k-p-m-2)$-form is $\{\zeta ,\xi\}$; that is, $$X_{\{\zeta ,\xi\}}=[X_{\xi},Y_{\zeta}]$$
By definition $$\inn (X_{\{\zeta ,\xi\}})\Omega =\d\{\zeta ,\xi\}$$ On the other hand, because of Prop. \[lie\_algebra\] (\[X\_,Y\_\])= (X\_)(Y\_)= {,} Thus $$\inn (X_{\{\zeta ,\xi\}})\Omega =
\inn ([X_{\xi},Y_{\zeta}])\Omega$$ and therefore $X_{\{\zeta ,\xi\}} =[X_{\xi},Y_{\zeta}]$.
As a consequence of this we have:
$({\tilde{\cal H}}^*(M),\{\ ,\ \})$ is a graded Lie algebra whose grading is defined by $|\bar \eta | = k- p -1$ if $\eta$ is a $p$-Hamiltonian form.
[**Remarks:**]{}
- The center of the graded Lie algebra $({\tilde{\cal H}}^*(M),\{\ ,\ \})$ is a graded Lie subalgebra, whose elements will be called Casimirs. We must point it out that there are no Casimirs of degree 0, i.e., functions commuting with anything, because if this were the case, then there will be a function $S$ such that $$\{ S, \eta \} = 0 ,$$ for all Hamiltonian forms $\eta$. In particular, $S$ will commute with $(k-1)$-Hamiltonian forms, but this implies that $X(S) = 0$, for all Hamiltonian vector fields. But this is clearly impossible, because Hamiltonian vector fields span the tangent bundle by Lemma \[transitivity\].
- The graded Lie algebra ${\cal V}_H^* (M,\Omega)$ possesses as elements of degree zero the Lie algebra of locally Hamiltonian vector fields on $(M,\Omega)$ which is the Lie algebra of the ILH-group [@Om74] of smooth multisymplectic diffeomorphisms. This suggests the possibility of embracing in a single structure of supergroup both smooth multisymplectic diffeomorphisms and infinitesimal automorphisms of a multisymplectic manifold $(M,\Omega)$. This can certainly be done extending to the graded setting some of the techniques used to deal with ILH-Lie groups.
Locally homogeneous multisymplectic manifolds. The group of multisympletic diffeomorphisms
==========================================================================================
As it was mentioned earlier, in general, it is not true that the locally Hamiltonian vector fields in a multisymplectic manifold span the tangent bundle of this manifold. However, there are a simple property, that was already mentioned in the introduction, that implies it (among other things).
Let $M$ be a differential manifold. Consider a point $x\in M$, and a compact set $K$ such that $x\in\stackrel{\circ}{K}$.
A [local Liouville vector field]{} at $x$ is a vector field $\Delta^x$ which verifies that ${\rm supp}\,\Delta^x:=\overline{\{ y\in M\, | \, \Delta^x(y)\not= 0\}}$ is contained in $K$, and there exists a diffeomorphism $\varphi\colon\overbrace{{\rm supp}\,\Delta^x}^{\circ}\to\Real^n$ such that $\varphi_*\Delta^x=\Delta$, where =x\^i is the standard [*Liouville*]{} or [*dilation*]{} vector field in $\Real^n$.
A differential form $\Omega\in\df^k(M)$ is said to be [locally homogeneous]{} at $x$ if there exists a local Liouville vector field $\Delta^x$ at $x$, such that (\^x)=c,c\[lochomog\] $\Omega$ is [locally homogeneous]{} if it is locally homogeneous for all $x\in M$.
A couple $(M,\Omega)$, where $M$ is a manifold and $\Omega\in\df^k(M)$ is locally homogeneous is said to be a [locally homogeneous manifold]{}.
Notice that the last property implies that, for every point $x\in M$, there exists a neighbourhood $B$ of $x$, with a local Liouville vector field, such that $\overbrace{{\rm supp}\,\Delta^x}^{\circ}=B$.
Clearly, symplectic an volume forms are locally homogeneous. It is important to remark that, apart from these ones, multicotangent bundles are [*locally homogeneous multisymplectic manifolds*]{}. In fact, let $\alpha_0\in\Lambda^k(\Tan^*Q)$, and consider local natural coordinates $(x^i,p_{\moment{i}{1}{k}})$ in a small neighbourhood of $\alpha_0$. We define r\^2:=\_i(x\^i)\^2+\_i(p\_)\^2 , and $\lambda\equiv\lambda(r^2)$ a bump fuction. Then consider the local Liouville vector field $\Delta_\lambda:=\lambda\Delta$. Therefore, a straighforward calculation shows that, for the natural multisymplectic form $\Omega\in\df^{k+1}(\Lambda^k(\Tan^*Q))$, $$\Lie(\Delta_\lambda)\Omega=[\lambda+2\lambda'(r^2)]\Omega$$
Nevertheless, not all the multisymplectic manifolds are locally homogeneous. A simple example of such situation is provided by $(\Real^7,\Omega)$, where $\Omega\in\df^3(\Real^7)$ is given by &=& x\^1x\^2x\^3+x\^1x\^4x\^5+ x\^1x\^6x\^7\
& & +x\^2x\^4x\^6-x\^2x\^5x\^7- x\^3x\^4x\^7-x\^3x\^5x\^6 This form is analyzed in [@Br87], and it is shown that the group of its multisymplectic diffeomorphisms is $G_2$. As it is seen in [@Ib2000], this fact contradicts the local homogeneity of $(\Real^7,\Omega)$.
At this point, we can show that Hamiltonian vector fields in locally homogeneous multisymplectic manifolds can be localized. This property plays a crucial role in the discussion to follow.
\[localization\] Let $X$ be a locally Hamiltonian vector field on a locally homogeneous multisymplectic manifold $(M, \Omega )$. Let $x$ be a point in $M$, then there exist neighborhoods $V,U$ of $x$ such that $V\subset \bar{V} \subset U$, $\bar{V}$ compact, and a locally Hamiltonian vector field $X^\prime$ such that $X^\prime$ coincides with $X$ in $V$ and $X^\prime$ vanishes outside of $U$.
Let $X$ be a locally Hamiltonian vector field on $(M,\Omega )$, i.e., $i (X) \Omega = \eta $ with $\eta$ a closed $(k-1)$-form. We shall follow a deformation technique used to show that certain forms are isomorphic [@Mo62] in a similar way as it is applied to prove Poincaré’s lemma. Let $x$ be a point in $M$. We shall choose a contractible neighborhood $U$ of $x$ (if necessary we can shrink it to be contained in a coordinate chart). Let $\rho_t$ be a smooth isotopy defining a strong deformation retraction from $U$ to $x$, i.e., $\rho_0 = id$, and $\rho_1$ maps $U$ onto $x$. Let $\Delta_t$ be the time-dependent vector field whose flow is given by $\rho_t$, i.e., $$\frac{\d}{\d t} \rho_t = \Delta_t
\circ \rho_t .$$ Then, $$\frac{\d}{\d t}(\rho_t^* \eta ) =
\rho^*_t (\Lie (\Delta_t)\eta,$$ hence, \[moser\] \_1\^\* -\_0\^\* = \_0\^1 (\_t\^\* ) t= \_0\^1 \^\*\_t ((\_t) ) t = \_0\^1 \_t\^\* ((\_t) ) t , thus, $$\eta = - \d \int_0^1 \rho_t^* (\inn(\Delta_t) \eta) \d t ,$$ and in $U$, $\eta = \d\zeta$ with = - \_0\^1 \_t\^\* ((\_t)) t .
We will localize the vector field $X$ by using a bump function $\lambda$ centered at $x$, i.e., we shall choose $\lambda$ such that $V= \supp\, \lambda$ will be a compact set $K$ contained in $U$. Unfortunately the vector field $\lambda X$ is not locally Hamiltonian, hence we will proceed modifying the Hamiltonian form $\zeta$ of $X$ instead. As $M$ is locally homogeneous at $x$, we can define a new vector field $\Delta_t^\prime$ by scaling the vector field $\Delta_t$ by $\lambda$, i.e., $$\Delta_t^\prime = \lambda \Delta_t .$$ We will denote the flow of $\Delta_t^\prime$ by $\rho_t^\prime$. It is clear that the subbundle $E_1$ is invariant by the flow $\rho_t^\prime$ of $\Delta_t^\prime$. In fact, if $\zeta\in E_1$, then there is $v$ such that $\inn(v)\Omega=\zeta$. Thus, $\rho_t^{\prime *}\zeta=\rho_t^{\prime *}(\inn(v)\Omega)=
\inn(\rho_{t*}^\prime v)(\rho_t^{\prime *}\Omega)\circ\rho_t^\prime$. But, as a consequence of (\[lochomog\]), we have that $\rho_t^{\prime *}\Omega=f_t\Omega$, for some $f_t$. Hence $\rho_t^{\prime *}\zeta\in E_1$. Moreover, we can choose the function $\lambda$ such that $\Delta_t (\lambda ) = r (t)$ and, \[rho\] r (t) = {
[l]{} 0 0 t 1/3\
r(t) r(t) > 0 1/3 < t < 2/3\
1 2/3 t 1
. . hence, as the flow $(\rho_t^\prime)^*$ leaves invariant the subbundle $E_1 = \hat{\Omega}_1 (\Tan M)$, we have that $(\rho_t^\prime)^* \eta \in E_1$ for all $0\leq t \leq 1$. Again, repeating the computation leading to eq. (\[moser\]), and using the vector field $\Delta_t^\prime$ instead, we get, \[eta\_prime\] (\_1\^)\^\* - (\_0\^)\^\* = \_0\^1 (\_t\^)\^\* ((\_t) ) t . As in the undeformed situation ($\lambda =1$), $\rho_0^\prime ={\rm Id}$, but $\rho_1^\prime$ is not a retraction of $U$ onto $x$. The $(k-1)$-form \^= - \_0\^1 (\_t\^)\^\* ((\_t))t is in $E_1$, because both, $(\rho_1^\prime)^* \eta$ and $(\rho_0^\prime)^* \eta$, are in $E_1$, thus there exists a vector field $X^\prime$ such that $$\inn(X^\prime ) \Omega = \eta^\prime .$$ The form $\eta^\prime$ is closed by construction, hence $X^\prime$ is locally Hamiltonian.
Moreover, if $y$ is a point lying in the set $\lambda^{-1}(1) \subset V$ then, $\rho_s^\prime (y) = \rho_s (y)$. Consequently, from eq. (\[eta\_prime\]), $\eta^\prime (y) =
\eta (y)$ and $X^\prime (y) = X(y)$. If $y$ on the contrary lies outside the compact set $K$, we have $\rho_t^\prime (y) = y$ for all $t$ because $\lambda$ vanishes there, thus $\Delta_t^\prime$ vanishes and the flow is the identity. Then $\eta^\prime (y) = 0$ and $X^\prime (y) = 0$.
A far reaching consequence of the localization lemma is the transitivity of the group of multisymplectic diffeomorphisms. We will first proof the following result.
\[transitivity\] Let $(M,\Omega)$ be a locally homogeneous multisymplectic manifold. Then the family of locally Hamiltonian vector fields span locally the tangent bundle of $M$, this is \[span\_ham\] \_x M = .
We will work locally. Let $U$ be a contractible open neighborhood of a given point $x\in M$. We can shrink $U$ to be contained in a coordinate chart with coordinates $x^i$. The tensor bundles of $M$ restricted to $U$ are trivial. In particular the subbundle $E_1$ restricted to $U$ is trivial. Let $v\in \Tan_x M$ an arbitrary tangent vector. Let $\nu =
\hat{\Omega}_1(x) v \in E_1 \subset\Lambda^{k-1} (\Tan_x^* M)$. Consider a vector field $X$ on $U$ such that $X(x) = v$. Then $\inn(X) \Omega = \eta$ and the $(k-1)$-form $\eta$ is not closed in general. We shall consider as in Lemma \[localization\] a strong deformation retraction $\rho_t$ and the corresponding vector field $\Delta_t$. Now on one hand we have, $$\int_0^1 \frac{\d}{\d s} (\rho_s^* \eta )\,\, \d s = -\eta ,$$ and on the other hand, \[intes\] \_0\^1 (\_s\^\* ) s = \_0\^1 \_s\^\* ((\_s) ) s + \_0\^1 \_s\^\* ((\_s) ) s, but, because, $\d\eta = \d\inn(X) \Omega = \Lie (X) \Omega$, then $$\inn(\Delta_t) \d\eta = \inn(\Delta_t)\Lie (X) \Omega =
\inn([\Delta_t, X])\Omega + \Lie (X) \inn(\Delta_t)\Omega .$$ Thus, returning to eq. (\[intes\]), we obtain, $$-\eta = \d \int_0^1 \rho_s^* \inn(\Delta_s) \eta\, \d s + \int_0^1 \rho_s^*
(\inn([\Delta_s, X]) \Omega + \Lie (X) \inn(\Delta_s)\Omega ) \, \d s .$$ Choosing the vector field $X$ such that its flow leaves $E_1$ invariant the second term on the r.h.s. of the previous equation will be in $E_1$, hence the first term will be in $E_1$ too. Let us define $\eta' = -
\d \left(\int_0^1 \rho_s^* \inn(\Delta_s) \eta\, \d s \right)$, and let us denote by $X'$ the hamilonian vector field on $U$ defined by $$\inn(X') \Omega = \eta' .$$ Evaluating $\eta'$ at $x$ we find that $\eta' (x) = \eta (x)$, hence $X' (x) =v$. Then, we localize the vector field $X'$ in such a way that the closure of its support is compact and is contained in $U$. Then, we can extend this vector field trivially to all $M$ and this extension is locally Hamiltonian. Finally the value of this vector field at $x$ is precisely $v$.
We shall recall that a group of diffeomorphisms $G$ is said to act $r$-transitively on $M$ if for any pair of collections $\{ x_1, \ldots, x_r \}$, $\{ y_1, \ldots, y_r \}$ of distinct points of $M$, there exists a diffeomorphism $\phi
\in G$ such that $\phi (x_i ) = y_i$. If the group $G$ acts transitively for all $r$, then it is said to act $\omega$-transitively or transitively for short. The transitivity of a group of diffeomorphisms can be reduced to a local problem because (strong) local transitivity implies transitivity. More precisely, we will say that the group of diffeomorphisms $G$ is strongly locally transitive on $M$ if for each $x\in M$ and a neighborhood $U$ of $x$, there are neighborhoods $V$ and $W$ of $x$ with $\bar{V} \subset W \subset \bar{W} \subset
U$, $\bar{W}$ compact, such that for any $y\in V$ there is a smooth isotopy $\phi_t$ on $G$ joining $\phi$ with the identity, $\phi_1 = \phi$, $\phi_0 = id$, such that $\phi_1 (x) = y$ and $\phi_t$ leaves fixed every point outside $\bar{W}$. Then, if $G$ is strongly locally transitive on $M$, then $G$ acts transitively on $M$ [@Bo69].
\[trans\] The group of multisymplectic diffeomorphisms $G(M,\Omega )$ of a locally homogeneous multisymplectic manifold is strongly locally transitive on $M$.
By Lemma \[transitivity\] we can construct a local basis of the tangent bundle in the neighborhood of a given point $x$ made of locally Hamiltonian vector fields $X_i$. Using Lemma \[localization\] we can localize the vector fields $X_i$ in such a way that the localized Hamiltonian vector fields $X_i^\prime$ will have common supports. We will denote this common support by $V$ and we can assume that it will be contained in a compact subset contained in $U$. But the vector fields $X_i^\prime$ will generate the module of vector fields inside the support $V$, hence the flows of local vectors fields cover the same set as the flows of local Hamiltonian vector fields, but the group of diffeomorphisms is locally strongly transitive and the same will happen for the group of multisymplectic diffeomorphisms.
\[cor\_trans\] The group of multisymplectic diffeomorphisms $G(M,\Omega)$ of a locally homogeneous multisymplectic manifold $(M,\Omega)$ acts transitively on $M$.
The conclusion follows from the results in [@Bo69] and Thm. \[trans\].
Invariant differential forms
============================
In order to prove the main statement in this section, we will establish first two lemmas:
Let $(M,\Omega )$ be a multisymplectic manifold of degree $k$ and $\alpha\in\df^p(M)$ (with $p\geq k-1$) a differential form which is invariant under the set of locally Hamiltonian $(k-1)$-multivector fields, that is, $\Lie (X)\alpha =0$, for every $X\in\vf^{k-1}_{lh}(M)$. Then:
For every $X,Y\in\vf^{k-1}_{lh}(M)$, (X)(Y)+(Y)(X)=0 \[clave1\]
In particular, for every $X\in\vf^{k-1}_{lh}(M)$ with $\inn (X)\Omega =0$ (that is, $X\in\ker^{k-1}\Omega$), then (X)=0 \[clave2\] \[basico1\]
Since $\alpha$ is invariant under $\vf_{lh}^{k-1}(M)$, for every $X\in\vf_{lh}^{k-1}(M)$ we have $\Lie (X)\alpha = 0$, then, (X)=(-1)\^[k-1]{}(X)\[uno\] Let $X,Y\in\vf_{lh}^{k-1}(M)$, for every $x\in M$ there exists an open neighbourhood of it, $U\subset M$, and $f,g\in \Cinfty (U)$ such that $\inn (X)\Omega\feble{U}\d f$ and $\inn (Y)\Omega\feble{U}\d g$ (from now on we will write $X\vert_U\equiv X_f$ and $Y\vert_U\equiv X_g$). Then, consider the locally Hamiltonian vector field $X_h\in\vf_{lh}^{k-1}(U)$ whose expression in $U$ is $X_h\feble{U}fX_g+gX_f$; its Hamiltonian function in $U$ is $h=fg\in\Cinfty (U)$ since $$\inn (X_h)\Omega \feble{U}
\inn (fX_g+gX_f)\Omega =f\inn (X_g)\Omega +g\inn (X_f)\Omega =
f\d g+g\d f = \d h$$ Hence $$\inn (X_h)\alpha\feble{U} f\inn (X_g)\alpha +g\inn (X_f)\alpha$$ and then $$\d\inn (X_h)\alpha\feble{U}\d f\wedge\inn (X_g)\alpha+f\d\inn (X_g)\alpha+
\d g\wedge\inn (X_f)\alpha+g\d\inn (X_f)\alpha$$ But, taking into account (\[uno\]), $$\d\inn (X_h)\alpha\feble{U}
f((-1)^{k-1}\inn (X_g)\d\alpha )+g((-1)^{k-1}\inn (X_f)\d\alpha )=
f\d\inn (X_g)\alpha+g\d\inn (X_f)\alpha$$ and comparing both results we conclude f(X\_g)+g(X\_f)0 \[dos\] which is the local expression of equation (\[clave1\]).
Taking $X\in\ker^{k-1}\Omega$ in (\[clave1\]), the equation (\[dos\]) gives $\inn (Y) \Omega \wedge \inn (X)\Omega =0$ for every $Y$. But because of Lemma \[surjective\], this implies that, $$\d g\wedge\inn (X)\alpha\feble{U}0$$ for every $g\in \Cinfty (U)$; hence, $\inn(X)\alpha =0$, for every $X\in\ker^{k-1}\Omega$.
Let $(M,\Omega )$ be a multisymplectic manifold of degree $k$ and $\alpha\in\df^p(M)$ a differential form which is invariant under the set of locally Hamiltonian $(k-1)$-multivector fields. Then:
If $p=k-1$ then $\alpha =0$.
If $p=k$, there exists a unique $\alpha'\in\Cinfty(M)$ such that $$\inn (X)\alpha =\alpha'\inn (X)\Omega$$ for every $X\in\vf^{k-1}_{lh}(M)$. \[basico2\]
The starting point is the equality (\[clave1\]). Taking $X=Y\not\in\ker^{k-1}\Omega$ (if $X\in\ker^{k-1}\Omega$ then $\inn (X)\alpha=0$ by hypothesis), we obtain (X)(X)=0 \[tres\] for every $X\in\vf^{k-1}_{lh}(M)$. Therefore we have:
If $p=k-1$ then $\inn (X)\alpha\in \Cinfty (M)$ and, according to the first item of Lemma \[surjective\] (for $1$-nondegenerate forms), (\[tres\]), together with (\[clave2\]), leads to the result $\inn (X)\alpha
=0$, for every $X\in\vf_{lh}^{k-1}(M)$. But, taking into account the item 2 of Lemma \[surjective\] (for $1$-nondegenerate forms), this holds also for every $X\in\vf^{k-1}(M)$ and we must conclude that $\alpha =0$.
If $p=k$ and $\inn (X)\alpha =0$ for all $X$, then, $\alpha = 0$. Thus, let us assume that $\inn (X)\Omega \neq 0$ for some $X$, then the solution of eq. (\[tres\]) is (X)=\_X’(X)\[cuatro\] and it is important to point out that the equation (\[clave2\]) for $\alpha$ implies that the function $\alpha'_X\in\Cinfty(M)$ is the same for every $X,X'\in\vf_{lh}^{k-1}(M)$ such that $\inn (X)\Omega=\inn (X')\Omega$.
Now, returning to equation (\[clave1\]) we obtain the relation $$\inn (Y)\Omega\wedge\inn (X)\Omega (\alpha_X'-\alpha_Y')=0$$ But $\alpha_X',\alpha_Y'\in \Cinfty (M)$ are the unique solution of the respective equations (\[tres\]) for each $X,Y\in\vf_{lh}^{k-1}(M)$; then we have the following options:
If $\inn (Y)\Omega\wedge\inn (X)\Omega\not= 0$ then $\alpha_X'=\alpha_Y'$.
If $\inn (Y)\Omega\wedge\inn (X)\Omega=0$ then $X=fY+Z$, where $f\in\Cinfty (M)$ and $Z\in\ker^{k-1}\Omega$. Therefore:
If $X\in\ker^{k-1}\Omega$ then $Y\in\ker^{k-1}\Omega$. Therefore, taking into account the item 2 of lemma \[basico1\], the corresponding equations (\[cuatro\]) for $X$ and $Y$ are identities and, thus, $\alpha_X'$ and $\alpha_Y'$ are arbitrary functions which we can take to be equal.
If $X\not\in\ker^{k-1}\Omega$ then $Y\not\in\ker^{k-1}\Omega$. Therefore, taking into account the item 2 of lemma \[basico1\], we have (X)&=& (fY+Z)=f(Y)\
&=& f\_Y’(Y)=\_Y’(fY+Z)=\_Y’(X)which, comparing with (\[cuatro\]), gives $\alpha_X'=\alpha_Y'$: In any case $\alpha_X'=\alpha_Y'$ and, as a consequence, the function $\alpha'$ solution of (\[tres\]) is the same for every $X\in\vf_{lh}^{k-1}(M)$.
At this point we can state and prove the following fundamental result:
\[pre\_invar\] Let $(M,\Omega )$ be a locally homogeneous multisymplectic manifold and $\alpha\in\df^p(M)$, with $p=k-1,k$ a differential form which is invariant by the set of locally Hamiltonian $(k-1)$-multivector fields and the set of locally Hamiltonian vector fields; that is, $\Lie (X)\alpha =0$ and $\Lie (Z)\alpha =0$, for every $X\in\vf^{k-1}_{lh}(M)$ and $Z\in\vf_{lh}(M)$. Then we have:
If $p=k$ then $\alpha =c\, \Omega$, with $c\in\Real$.
If $p=k-1$ then $\alpha =0$.
First suposse that $p=k$.
For every $X\in\vf_{lh}^{k-1}(M)$, according to Lemma \[basico2\] (item 2), we have $$\inn (X)\alpha=\alpha'\inn (X)\Omega=\inn (X)(\alpha'\Omega)$$ where $\alpha'\in\Cinfty (M)$ is the same function for every $X\in\vf_{lh}^{k-1}(M)$. But, taking into account the item 2 of Lemma \[surjective\] (for 1-nondegenerate forms), the above equality holds for every $X\in\vf^{k-1}(M)$ and thus $\alpha=\alpha'\Omega$.
Then, for every $Z\in\vf_{lh}(M)$, by hypothesis $$0=\Lie (Z)\alpha = (\Lie (Z)\alpha' ) \Omega .$$ Hence $\Lie (Z)\alpha' = 0$, but because of Lemma \[transitivity\], locally Hamiltonian vector fields span the tangent space, thus $\alpha'=c$ (constant). So $$\inn (X)\alpha = c \inn (X)\Omega = \inn (X)(c\Omega )$$ and, taking into account the item 2 of Lemma \[surjective\] (for $1$-nondegenerate forms) again, this relation holds also for every $X\in\vf^{k-1}(M)$, therefrom we have to conclude that $\alpha =c\Omega$.
If $p=k-1$ the result follows straightforwardly from the first item of the lemma \[basico2\].
- From Thm. \[pre\_invar\] it follows immediately Thm. \[invar\].
- Another immediate consequence of this theorem is that, if $\alpha\in\df^k(M)$ is a differential form invariant by the sets of locally Hamiltonian $(k-1)$-multivector fields and locally Hamiltonian vector fields, then it is invariant also by the set of locally Hamiltonian $m$-multivector fields, for $1<m<k-1$.
- As it is evident, if $k=2$ we have proved (partially) the classical [*Lee Hwa Chung’s theorem*]{} for symplectic manifolds.
Characterization of multisymplectic transformations
===================================================
Now we are going to use the theorems above in order to give several characterization of multisymplectic transformations in the same way as Lee Hwa Chung’s theorem allows to characterize symplectomorphisms in the symplectic case [@Hw-47; @LlR-88; @GLR-84].
A vector field $X$ on a multisymplectic manifold $(M,\Omega )$ will be said to be a [*conformal Hamiltonian vector field*]{} iff there exists a function $\sigma$ such that \[loc\_conf\] (X) = . It is immediate to check that, if $\Omega^r \neq 0$, $r>1$, then $\sigma$ must be constant. Then:
A diffeomorphism $\varphi \colon M_1 \to M_2$ between the multisymplectic manifolds $(M_i, \Omega_i)$, $i=1,2$, is said to be a [special conformal multisymplectic diffeomorphism]{} iff there exists $c\in \R$, such that $\varphi^*\Omega_2 = c \Omega_1$. The constant factor $c$ will be called the [scale]{} or [valence]{} of $\varphi$.
Therefore we will prove:
\[cgm\] Let $(M_i,\Omega_i)$, $i=1,2$, be two locally homogeneous multisymplectic manifolds. A diffeomorphism $\varphi\colon M_1\to M_2$ is a special conformal multisymplectic diffeomorphism if, and only if, the differential map $\varphi_*\colon\vf (M_1)\to\vf (M_2)$ induces an isomorphism between the graded Lie algebras ${\cal V}_H^* (M_1,\Omega_1)$, ${\cal V}_H^* (M_2,\Omega_2)$. Then we will have that $$\varphi_*X_{\xi}=\frac{1}{c}X_{\varphi^{*-1}\xi}.$$ In addition, if $X_1\in\vf_h^m(M_1)$ is any Hamiltonian (resp. locally Hamiltonian) multivector field with $\xi_1\in\df^{k-m-1}(M_1)$ a Hamiltonian form for it (resp. locally Hamiltonian in some $U_1\subset M_1$), and $\varphi_*X_1 = X_2\in\vf_{lh}^m(M_2)$ with Hamiltonian form $\xi_2\in\df^{k-m-1}(M_2)$ (resp. locally Hamiltonian in $\varphi (U_1)=U_2\subset M_2$); then c\_1=\^\*\_2+\[rfh\]where $\eta\in\df^{k-m-1}(M_1)$ is a closed form. In other words, $\varphi^*$ induces an isomorphism between classes of Hamiltonian forms.
Taking into account proposition \[chmf\] we have:
($\Longleftarrow$)For every $X_1\in\vf_h^m(M_1)$ (resp. $X_1\in\vf_{lh}^m(M_1)$) we have that $\varphi_ *X_1=X_2\in\vf_h^m(M_2)$ (resp. $X_2\in\vf_{lh}^m(M_2)$). In any case $\Lie (X_2)\Omega_2=0$, then we obtain $$0=\varphi^*\Lie (X_2)\Omega_2=\Lie (\varphi_*^{-1}X_2)\varphi^*\Omega_2=
\Lie (X_1)\varphi^*\Omega_2$$ therefore, by theorem \[pre\_invar\], we have that $\varphi^*\Omega_2=c\Omega_1$.
($\Longrightarrow$)Conversely, for every $X_{\xi_1}\in\vf_h^m(M_1)$ we have that $\inn (X_{\xi_1})\Omega_1-\d\xi_1=0$. Then, since $\varphi^*\Omega_2=c\Omega_1$, we obtain 0=\^[\*-1]{}((X\_[\_1]{})\_1-\_1)&=& (\_\*X\_[\_1]{})\^[\*-1]{}\_1-\^[\*-1]{}\_1= (\_\*X\_[\_1]{})\_2-\^[\*-1]{}\_1\
&& (\_\*X\_[\_1]{})\_2- (\^[\*-1]{}\_1)=0 so, $\varphi_*X_{\xi_1}=X_{\xi_2}\in\vf_h^m(M_2)$ and its Hamiltonian form $\xi_2\in\df^{k-m-1}(M_2)$ is related with $\xi_1$ by Eq. (\[rfh\]).
In an analogous way, using $\varphi^{-1}$, we would prove that $\varphi_*^{-1}X_2\in\vf_h^m(M_1)$, for every $X_2\in\vf_h^m(M_2)$.
The proof for locally Hamiltonian multivector fields is obtained in the same way, working locally on $U_1\subset M_1$ and $U_2=\varphi (U_1)\subset M_2$.
As a consequence of the previous theorem there is another characterization of conformal multisymplectomorphisms.
Let $(M_i,\Omega_i)$, $i=1,2$, be two locally homogeneous multisymplectic manifolds. A diffeomorphism $\varphi\colon M_1\to M_2$ is a special conformal multisymplectic diffeomorphism if, and only if, for every $U_2\subset M_2$ and for every $\xi_2\in\df^p(U_2)$ and $\zeta_2\in\df^m(U_2)$ ($p,m<k-1$ ), we have \^\*{\_2,\_2} ={\^\*\_2,\^\*\_2} \[pp\]
Let $X_{\xi_2}\in\vf^{k-p-1}_h(M_2)$ and $Y_{\zeta_2}\in\vf^{k-m-1}_h(M_2)$ be Hamiltonian multivector fields having $\xi_2$ and $\zeta_2$ as Hamiltonian forms in $U_2$.
($\Longrightarrow$)We have \^\*{\_2,\_2} =\^\*(Y\_[\_2]{})\_2= (\_\*\^[-1]{}Y\_[\_2]{})\^\*\_2 \[ecu\] but, if $\varphi$ is a conformal multisymplectomorphism (of valence $c$), according to Thm. \[cgm\], $\varphi_*^{-1}Y_{\zeta_2}\in\vf_h^m(M_1)$ and $$\inn(\varphi_*^{-1}Y_{\zeta_2})\Omega_1=\frac{1}{c}\d\varphi^*\zeta_2$$ that is, $\varphi_*^{-1}Y_{\zeta_2} = \frac{1}{c}Y_{\varphi^*\zeta_2}$. Therefore, because of eq. (\[ecu\]) we conclude $$\varphi^*\{\xi_2,\zeta_2\} =\inn (\varphi_*^{-1}Y_{\zeta_2})\varphi^*\d\xi_2=
\frac{1}{c}\inn (Y_{\varphi^*\zeta_2})\varphi^*\d\xi_2=
\frac{1}{c}\{\varphi^*\xi_2,\varphi^*\zeta_2\}$$
($\Longleftarrow$)Assuming that eq. (\[pp\]) holds and using again the definition of Poisson bracket we can write it as $$\varphi^*\inn (Y_{\zeta_2})\d\xi_2=
\inn (\varphi_*^{-1}Y_{\zeta_2})\varphi^*\d\xi_2=
\frac{1}{c}\inn (Y_{\varphi^*\zeta_2})\varphi^*\d\xi_2$$ for every $\xi_2$. Hence we conclude that $$\varphi_*^{-1}Y_{\zeta_2}=\frac{1}{c}Y_{\varphi^*\zeta_2}\in\vf_h^m(M_1)$$ for every $Y_{\zeta_2}\in\vf_h^m(M_2)$ and because of Thm. \[cgm\] again, $\varphi$ is a special conformal multisymplectomorphism.
Proof of the main Theorem
=========================
We will prove now theorem \[main\]:
Let $\Phi$ be a group isomorphism from $G(M_1,\Omega_1)$ to $G(M_2,\Omega_2)$ which is in addition a homeomorphism if we endow $G(M_i,\Omega_i)$ with the point-open topology. Then, Corollary \[cor\_trans\] implies that the group $G(M_i,\Omega_i)$ acts transitively on $M_i$, $i=1,2$, hence by the main theorem in [@We54] there exists a bijective map from $M_1$ to $M_2$ such that $\Phi (f) = \varphi \circ f \circ
\varphi^{-1}$. Moreover the map $\varphi$ is a conformal multisymplectic diffeomorphism if it verifies the conditions in Thm. \[main\] as the following argument shows.
$\varphi$ is a homeomorphism.
Let ${\cal A} (M)$ be the class of fixed subsets of $G(M,\Omega)$, i.e., $${\cal A} (M) = \set{ \Fix (f) \mid f \in G(M,\Omega ) }, ~~~~~ \Fix (f) =
\set{x\in M \mid f(x) = x} .$$ Let ${\cal B} (M)$ be the class of complements of elements of ${\cal A} (M)$, this is $${\cal B} (M) = \set{ B = M - A \mid A \in {\cal A} (M) } ,$$ hence, ${\cal B} (M)$ is a class of open subsets of $M$. If $B\in {\cal B}
(M)$ we can construct a multisymplectic diffeomorphism $g$ such that $B$ is the interior of $\supp (g)$. In fact, for any point $x\in M$ and a neighborhood $U$ of $x$, it follows from Lemma \[localization\] that there exists $B\in {\cal B}
(M)$ such that $x\in B\subset U$. Thus, ${\cal B} (M)$ is a basis for the topology of $M$. Moreover, if $f\in
G(M_1,\Omega_1)$, then $\Fix (\varphi \circ f \circ \varphi^{-1} ) = \varphi
(\Fix (f))$ and if $g\in G(M_2, \Omega_2)$, then $\Fix (\varphi^{-1} \circ g \circ \varphi ) =
\varphi^{-1} (\Fix (g))$. Hence, $\varphi$, $\varphi^{-1}$ take basic open sets (in ${\cal B} (M)$) into basic open sets, thus they are both continuous, i.e., $\varphi$ is a homeomorphism.
$\varphi$ is a smooth diffeomorphism.
To proof this we will adapt the proof in [@Ta79] and [@Ba86] to our setting. To prove that $\varphi$, and $\varphi^{-1}$ are $C^\infty$ it is enough to show that $h\circ \varphi \in
C^\infty (M_1)$ for all $h\in C^\infty (M_2)$ and $k\circ \varphi^{-1} \in
C^\infty (M_2)$ for all $k\in C^\infty (M_1)$.
Let $x\in M_1$ and $U$ an open neighborhood of $x$ which is the domain of a local coordinate chart $\psi \colon U \to \R^n$. According to Lemma \[transitivity\], there exist Hamiltonian vector fields $X_i$, with compact supports on $U$, which are a local basis for the vector fields on an open neighborhood of $x$ contained in $U$. Let $\phi_t^i$ the 1-parameter group of diffeomorphisms generated by $X_i$. Let now $X$ be any locally Hamiltonian vector field on $M_1$. We will localize it on a neighborhood of $x$ in such way that its compact support will be contained in $U$. We will denote the localized vector field again by $X$. Let $\phi_t$ be 1-parameter group of multisymplectic diffeomorphisms generated by $X$ (which exists because $X$ is complete). For each $t$, $\Psi_t := \Phi (\phi_t) =
\varphi
\circ \phi_t \circ \varphi^{-1}$ is a $C^\infty$ multisymplectic diffeomorphism. The evaluation map, $$\begin{array}{ccccl}\Psi &\colon&\R \times M_2 &
\longrightarrow & M_2 \\ & & (t,x) & \mapsto & \Phi_t (x) = \Phi (\phi_t) (x) =
\varphi \circ \phi_t \circ \varphi^{-1} (x) \end{array}$$ is continuous. Moreover $\Psi_0 = {\rm Id}$ and $\Psi_{t+s} = \Psi_t \circ \Psi_s$. Therefore the map $\Psi$ is a continuous action of $\R$ on $M_2$ by $C^\infty$ diffeomorphisms. By Montgomery-Zippin theorem, since $\R$ is a Lie group, this action is $C^\infty$, i.e, $\Psi$ is smooth in both variables $t$ and $x$. Therefore, the 1-parameter group of multisymplectic diffeomorphisms $\Psi_t$ has an infinitesimal generator, i.e., a $C^\infty$ locally Hamiltonian vector field $X_\Psi$ such that, $$\frac{\d}{\d t} \Psi_t = X_\Psi \circ \Psi_t .$$ Given $h\in C^\infty (M_2)$, its directional derivative $X_\Psi (h)$ is a $C^\infty$ function. For any $x\in M_1$ we have, $$X_\Psi (h) (\varphi (x)) = \left. \frac{\d}{\d t} h(\Psi_t(\varphi
(x))\right|_{t=0} = \left. \frac{\d}{\d t} (h\circ\varphi) (\phi_t (x))
\right|_{t=0} .$$ Therefore if $X$ is any of the Hamiltonian vector fields $X_i$ above, for all $y$ in a small neighborhood of $x$, the preceding formula gives $$X_i(h\circ\varphi )(y)=
\left. \frac{\d}{\d t} (h\circ \varphi) (\phi_t^i (y)) \right|_{t=0} =
(X_i)_\Psi
(h) (\varphi (y))$$ This formula shows that $h\circ \varphi$ is a $C^1$-map and that for any locally Hamiltonian vector field $X$, \[transf\] (X\_(h)) = X (h) . To compute higher partial derivatives, we just iterate this formula using the vector fields $X_i$, for instance, $$(X_j)_\Psi ((X_i)_\Psi (h))\circ \varphi = X_j (X_i (h\circ \varphi ))) ,$$ Since the Hamiltonian vector fields $X_i$ are a local basis for the vector fields on an open neighborhood of $x$, we have proved that $h\circ \varphi \in C^\infty (M_1)$.
$\varphi_*$ maps locally Hamiltonian vector fields into locally Hamiltonian vector fields.
Equation (\[transf\]) shows that $X_\Psi = \varphi_* X$ and because $\Psi_t$ is a flow of multisymplectic diffeomorphisms, then $\varphi_*X$ is another locally Hamiltonian vector field. Thus, $\varphi_*$ maps every locally Hamiltonian vector field into a locally Hamiltonian vector field.
$\varphi$ is a special conformal multisymplectic diffeomorphism.
We finally show that, with the additional hypothesis stated in Thm. 1, then $\varphi^* \Omega_2 = c \Omega_1$.
In fact, if in addition we assume that the tangent map $\varphi_*$ maps all infinitesimal automorphisms of $(M_1, \Omega_1)$ into infinitesimal automorphisms of $(M_2, \Omega_2)$, then as a consequence of Thm. \[cgm\], we have that $\varphi^*\Omega_2 = c\, \Omega_1$.
It is important to point out that this conclusion cannot be reached unless this new hypothesys is assumed, since the starting set of assumptions allows us to prove only that $\varphi_*$ maps locally Hamiltonian vector fields into locally Hamiltonian vector fields; but this result cannot be extended to Hamiltonian $m$-multivector fields, with $m>1$.
Conclusions and outlook
=======================
We have shown that locally homogeneous multisymplectic forms are characterized by their automorphisms (finite and infinitesimal). As it was pointed in the introduction it is remarkable that it is not known a Darboux-type theorem for multisymplectic manifolds, although a class of multisymplectic manifolds with a local structure defined by Darboux type coordinates has been characterized [@Ca96b]. This has forced us to use a proof that does not rely on normal forms. The statement in Theorem \[main\] can be made slighty more restrictive assuming that we are given a bijective map $\varphi \colon M_1 \to M_2$ such that it sends elements $f\in G(M_1, \Omega_1)$ to elements $\varphi \circ f \circ\varphi^{-1} \in G(M_2, \Omega_2)$, then along the proof of the theorem in Section 6 we show that $\varphi$ is $C^\infty$. The generalization we present here uses the transitivity of the group of multisymplectic diffeomorphisms and is a simple consequence of theorems by Wechsler [@We54] and Boothby [@Bo69]. However we do not know yet if the continuity assumption for $\Phi$ can be dropped and replaced by weaker conditions like in the symplectic and contact cases. To answer these questions it would be necessary to describe the algebraic structure of the graded Lie algebra of infinitesimal automorphisms of the geometric structure as in the symplectic and volume cases [@Ba78]. A necessary first step in this direction will be describing the extension of Calabi’s invariants to the multisymplectic setting.
We will like to stress that in the analysis of multisymplectic structures beyond the symplectic and volume manifolds, it is necessary to consider not only vector fields, but the graded Lie algebra of infinitesimal automorphisms of arbitrary order. Only the Lie subalgebra of derivations of degree zero is related to the group of diffeomorphisms, however derivations of all degrees are needed to characterize the invariants. It is also meaningful the fact (that was pointed out in Section 4) that there are “exceptional” multisymplectic geometries (i.e., multisymplectic forms) whose group of automorphisms is finite-dimensional.
Finally, we want to remark that Theorem \[pre\_invar\] (which plays a relevant role in this work) is just a partial geometric generalization for multisymplectic manifolds of Lee Hwa Chung’s theorem. A complete generalization would have to characterize invariant forms of every degree (work in this direction is in progress). Our guess is that, in order to achive this, additional hypothesis must be considered, namely: strong nondegeneracy of the multisymplectic form and invariance by locally Hamiltonian multivector fields of every order. Nevertheless, it is important to point out that the hypothesis that we have assumed here (1-nondegeneracy, local homogeneity and invariance by locally Hamiltonian vector fields and locally Hamiltonian $(k-1)$-multivector fields) have been sufficient for our aim. This is a relevant fact since, as an example, in the jet bundle description of classical field theories (the regular case), the Lagrangian and Hamiltonian multisymplectic forms are just 1-nondegenerate [@CCI-91; @EMR-00; @GMS-97; @Go-91b; @Hr-99b; @Ka-98; @MS-99; @Sd-95; @Sa-89], and in the analysis of the field equations, only locally Hamiltonian $(k-1)$-multivector fields are relevant [@EMR-97; @EMR-99].
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors want to thank F. Cantrijn and G. Bor for their useful remarks and comments during the preparation of this work.
The author AI wishes to ackwnoledge the partial financial support provided by CICYT under the programmes PB95-0401 and PB98-0861 NATO collaborative research grant 940195 and the CRM for the hospitatility and facilities provided by the Institute were part of this work was done.
The authors AEE, MCML, NRR wish to thank the financial support of CICYT programmes TAP94-0552-C03-02 and PB98-0821.
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|
---
author:
- 'Soumya D. Sanyal'
title: Bruns
---
“ ****”\
**Winfried Bruns**\
**Journal of Algebra **39**, 429-439 (1976)**\
A translation into English:\
**‘Every’ finite free resolution is a free resolution of an ideal generated by three elements.**\
Translation by: Soumya D. Sanyal [^1] $^,$ [^2]\
June 29, 2010\
Present draft: December 24, 2010\
Introduction
============
In \[4\] and \[7\], Burch and Kohn proved that when $R$ is a commutative Noetherian ring, and $n$ a natural number that is the homological dimension [^3] of a torsionless $R$-module, then there exists an ideal in $R$ generated by at most 3 elements, whose homological dimension is precisely $n$. Buchsbaum and Eisenbud \[3, pg. 135, Conjecture; see also Theorem 7.2\] have further conjectured that ‘every’ finite free resolution is a free resolution of such an ideal. We prove the following generalisation of their conjecture:\
Let:\
be a projective resolution of $M$, where $F_m$ is a free $R$-module and $M$ is an m-torsionless $R$-module (or equivalently, $M$ is an m-th syzygy module) having a well defined rank. Let $r$ be the rank of the submodule Im$(f_{m+1})$. Then there exist homomorphisms:\
$$c: F_m \rightarrow R^{r+m}; f_m: R^{r+m} \rightarrow R^{2m-1}; f_j: R^{2j+1} \rightarrow R^{2j-1}, j=1, \dots , m-1$$
such that if $f'_{m+1}:=c \circ f_{m+1}$, the sequence:\
is exact. (Theorem 3). When $m=2$ and the $F_k$, $k=2, \dots n$ are free modules, we obtain the conjecture of Buchsbaum and Eisenbud.\
Our result also shows that any natural number that is the homological dimension of an $m$-torsionless $R$-module is the homological dimension of an $m$-torsionless $R$-module of rank $m$ generated by at most $2m+1$ elements.\
The maps $f_i$ and $c$ above are constructed in Theorem 2, which states that if firstly $M$ is an $m$-torsionless $R$-module of rank $r>m$ and of finite homological dimension, and if secondly $g: R^n \rightarrow M$ is an epimorphism, then there exists a basis element $x$ of $R^n$ such that $M/{Rg(x)}$ is $m$-torsionless and of rank $r-1$. This result requires that $g(x)$ can be included as part of a basis for the free $R_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ for all prime ideals $\mathfrak{p}$ of $R$ having the property that depth$(R_{\mathfrak{p}}) \leq m$. Buchsbaum and Eisenbud have shown the existence of such elements $x \in R^n$, assuming certain hypotheses on the Krull dimension of the ring $R/\mathfrak{p}$. We are able to prove a similar statement, in which we replace their hypotheses on the Krull dimension of $R/\mathfrak{p}$ with conditions on the difference between $m$ and the depth of $R_{\mathfrak{p}}$ (Theorem 1).\
One may replace the condition on the homological dimension of $M$ in the previous paragraph with a condition on the regularity of the rings $R_{\mathfrak{p}}$ whose depth is at most $m$ and still have the result hold. For rings satisfying this condition, we are able to generalize a theorem of Bourbaki \[2, pg. 76, Theorème 6\], which states that any $m$-torsionless $R$-module of rank at least $m$ is an extension of an $m$-torsionless $R$-module of rank $m$ by a free $R$-module (Corollary 2 to Theorem 2).\
In our paper, $R$ will throughout denote a commutative Noetherian ring and all $R$-modules $M, N, \dots$ shall be finitely generated. We denote by $Q(R)$ the total quotient ring of $R$. If $M \otimes Q(R)$ is a free $Q(R)$-module, then the rank of $M$ (denoted rank $M$) is defined to be the number of elements in a basis of $M \otimes Q(R)$. We remind the reader that if any two modules in a short exact sequence have well defined ranks, then so does the third (\[9, chap 6\]). We denote the homological dimension of $M$ over $R$ by dh($M$), and the homological codimension (depth) by codh($M$). When $R$ is a local ring we define the (well-defined) number of elements in a minimal generating set of $M$ by $\mu(M)$. Ass $M$ denotes the (finite) set of associated primes of $M$. We denote the grade of an ideal $\mathfrak{a}$ (with respect to elements in $R$) by grad($\mathfrak{a}$); thus grad($\mathfrak{a}) = $min$ \{$codh$ (R_{\mathfrak{p}}) \mid \mathfrak{p} \supset \mathfrak{a}\}$. For notational convenience, we shall define $\mathfrak{C}_n$ to be the set of prime ideals $\mathfrak{p}$ of $R$ such that codh$(R_{\mathfrak{p}}) \leq n$.\
We briefly explain the notion of ‘$m$-torsionless’ for a module. Suppose $F_1 \rightarrow F_0 \rightarrow M \rightarrow 0$ is a presentation of a module $M$, and let $D(M)$ be the cokernel of the dual of the homomorphism $F_1 \rightarrow F_0$ [^4]. The Ext$^i_R(D(M),R)$ modules for $i \geq 1$ do not depend on the choice of presentation of $M$. Thus we may say that $M$ is $m$-torsionless provided that Ext$^i_R(D(M),R)=0$ for $i=1, \dots m$ \[1\]. The nomenclature derives from the fact that Ext$^1_R(D(M),R)$ and Ext$^2_R(D(M),R)$ are the kernel and the cokernel respectively of the natural homomorphism of $M$ into its bidual. Hence $M$ is $1$-torsionless (resp. $2$-torsionless) if $M$ is torsionless (resp. reflexive) in the usual sense. If dh$(M) < \infty$ there are the following equivalent \[1, Theorem 4.25\] characterisations of $m$-torsionlessness:\
1. $(a_m)$ Every R-regular sequence of at most $m$ elements is $M$-regular (thus if $m = 1$ then $M$ is torsion-free).
2. $(b_m)$ codh$(M_{\mathfrak{p}}) \geq$ min$(m,$ codh$(R_{\mathfrak{p}}))$ for every prime ideal $\mathfrak{p}$ of $R$.
3. $(s_m)$ $M$ is an $m$-th syzygy module of a projective resolution.
4. $(t_m)$ $M$ is $m$-torsionless.
When the localization $R_{\mathfrak{p}}$ is a Gorenstein ring for all prime ideals $\mathfrak{p} \in \mathfrak{C}_{m-1}$, the above equivalences also hold even if dh$(M)$ is not finite \[6, Theorem 4.6\]. We call rings $R$ satisfying this property on its localisations $m$-Gorenstein. In this paper we will only consider $m$-Gorenstein rings; in fact, the rings $R_{\mathfrak{p}}: \mathfrak{p} \in \mathfrak{C}_{m-1}$, shall be regular local rings.\
In formulating Theorem 1, we have used the term ‘$s$-basic’. A submodule $M'$ of $M$ is called $s$-basic in $M$ at $\mathfrak{p}$ if $\mu(M/M')_{\mathfrak{p}} \leq \mu(M_{\mathfrak{p}}) - s$. An element $x \in M$ is called *basic* at $\mathfrak{p}$ if $Rx$ is 1-basic. In the proof of Theorem 1 we consider the set of prime ideals of $R$ such that $\mu(M_{\mathfrak{p}})>n$. This set is a variety in Spec$(R)$: if $I_n(M)$ is the sum of colon ideals $(N:M)$ where $N$ is generated by at most $n$ elements, then $\mu(M_{\mathfrak{p}})>n$ if and only if $\mathfrak{p} \supset I_n(M)$.\
On the existence of basic elements
==================================
In this section we shall prove Theorem 1. The theorem asserts, under certain conditions, the existence of elements $y \in M$ that are basic at all $\mathfrak{p} \in \mathfrak{C}_m$. The result will allow us to pass from an $m$-torsionless $R$-module $M$ to an $m$-torsionless $R$-module $M/{Ry}$. Theorem 1 is largely analogous to \[5, Theorem A\]; we replace their condition on the Krull dimension of $R/{\mathfrak{p}}$ by a different condition on $\mathfrak{p}$ and an additional condition on $M$. (Remark 1 following the proof of Theorem 1 indicates a special case of this result).\
Let $n \geq 0$ be a natural number and $M$ be an $R$-module such that for all prime ideals $\mathfrak{p}, \mathfrak{q} \in {\mathfrak{C}}_{n}$ we have that $(R_{\mathfrak{p}}) >$ $(R_{\mathfrak{q}})$ implies $\mu(M_{\mathfrak{p}}) \geq \mu(M_{\mathfrak{q}})$. Then:\
1. If $\mu(M_{\mathfrak{p}})>n$ for every prime ideal $\mathfrak{p} \in \mathfrak{C}_n$, then there exists some $x \in M$ that is basic in $M$ at all $\mathfrak{p} \in \mathfrak{C}_n$.
2. Let $x_1, \dots , x_k \in M$, for $k \geq 1$ and define $M' := \Sigma_{i=1}^k{Rx_i}$. If $M'$ is $(k,n+1-$$(R_{\mathfrak{p}}))$-basic in $M$ at all $\mathfrak{p} \in \mathfrak{C}_n$, then there exists $x' \in \Sigma_{i=2}^k{Rx_i}$ such that $x_1+x'$ is basic in $M$ at all $\mathfrak{p} \in \mathfrak{C}_n$.
We largely follow the proof of \[5, Theorem A\], although we use the following lemma instead of \[5, Lemma 1\]:\
Let $\mathfrak{a}$ be an ideal of $R$. Then there exist only finitely many prime ideals $\mathfrak{p} \supset \mathfrak{a}$ such that $(R_{\mathfrak{p}})=$ $(\mathfrak{a})$.
The case $\mathfrak{a}=R$ is trivial. So suppose that $\mathfrak{a} \lneq R$ and let $x_1, \dots, x_n$, where $n=$ grad$({\mathfrak{a}})$, be an $R$-regular sequence contained in $\mathfrak{a}$. Then if $\mathfrak{p} \supset \mathfrak{a}$, $x_1, \dots, x_n$ is also an $R_{\mathfrak{p}}$-regular sequence; thus when codh$(R_{\mathfrak{p}})=n$, we have:\
codh$((R/{Rx_1+ \dots + Rx_n})_{\mathfrak{p}}) = \hspace{1mm} $codh$(R_{\mathfrak{p}}/({R_{\mathfrak{p}}x_1+ \dots + R_{\mathfrak{p}}x_n})) = 0.$\
Thus $\mathfrak{p} \in $Ass $(R/{(Rx_1+\dots + Rx_n)})$. But there are only finitely many primes in Ass $(R/{Rx_1+\dots + Rx_n})$.
**Proof of Theorem 1.** (1) follows from (2) by taking $x_1, \dots x_k$ to be members of a system of generators of $M$. We will prove (2) by induction on $k$. The case $k=1$ is trivial. Suppose then that $k>1$ and let $a_1, \dots ,a_{k-1}$ be given such that $M'' = R(x_1+a_1x_k) + \Sigma_{i=2}^{k-1}{R(x_i+a_ix_k)}$ is min$(k-1,n+1-$codh$(R_{\mathfrak{p}}))$-basic at all $\mathfrak{p} \in \mathfrak{C}_n$.\
We begin by proving that there are only finitely many primes $\mathfrak{p} \in \mathfrak{C}_n$ at which $M'$ is not min$(k,n+2-$codh$(R_{\mathfrak{p}}))$-basic, and hence only finitely many $\mathfrak{p} \in \mathfrak{C}_n$ such that\
$\mu(M/M')_{\mathfrak{p}}>\mu(M_{\mathfrak{p}})-$min$(k,n+2-$codh$(R_{\mathfrak{p}}))$.\
We note that it is enough to prove this for fixed $t:=$codh$(R_{\mathfrak{p}})$ and fixed $s:=\mu(M_{\mathfrak{p}})-$min$(l,n+2-t)$, since $s$ and $t$ will take only finitely many values. For any $\mathfrak{q} \in \mathfrak{C}_n$ with codh$(R_{\mathfrak{q}})<t$, we obtain the following bound\
$\mu((M/M')_{\mathfrak{q}}) \leq \mu(M_{\mathfrak{q}})-$min$(k,n+1-$codh$(R_{\mathfrak{q}})) \leq \mu(M_{\mathfrak{q}})-$min$(k,n+2-t) \leq s,$\
since for every $\mathfrak{p} \in \mathfrak{C}_n$ such that codh$(R_{\mathfrak{p}})=t$, we have $\mu(M_{\mathfrak{q}})\leq \mu(M_{\mathfrak{p}})$ by hypothesis. Thus $\mathfrak{q}$ does not contain $I_s(M/M')$, whence grad$(I_s(M/M')) \geq t$. In the case that grad$(I_s(M/M'))>t$, there are no prime ideals $\mathfrak{p}$ with codh$(R_{\mathfrak{p}})=t$ that contain $I_s(M/M')$, and in the case that grad$(I_s(M/M'))=t$ there are only finitely many such ideals, as follows from the previous lemma.\
Now let $E$ be the finite set of primes $\mathfrak{p} \in \mathfrak{C}_n$ at which $M'$ is not min$(k,n+2-$codh$(R_{\mathfrak{p}}))$-basic in $M$. By \[5, Lemma 3\] there exist $a_1, \dots , a_{k-1} \in R$ such that $M'' = R(x_1+a_1x_k)+\Sigma_{i=2}^{k-1}{R(x_i+a_ix_k)}$ is min$(k-1,n+1-$codh$(R_{\mathfrak{p}}))$-basic at all $\mathfrak{p} \in E$. Note that this latter condition also holds at all $\mathfrak{p} \in \mathfrak{C}_n \setminus E$.\
1. The statement and proof of Theorem 1 are valid more generally (for instance, as in \[5, Theorem A\]), provided that $R$ is a commutative Noetherian ring, $A$ a (not necessarily commutative) $R$-algebra that is finitely generated as an $R$-module, and $M$ a finitely generated $A$-module. For given $A$-modules $N$ the definitions of $\mu(N_{\mathfrak{p}})$ and $I_s(N)$ correspond to $\mu(A_{\mathfrak{p}},N_{\mathfrak{p}})$ and $I_s(A,N)$ respectively, as in \[5\]. However, we may not relax the condition that $R$ be Noetherian. Furthermore, Part 2 of Theorem 1 (which is analogous to \[5, Theorem A(iib)\]) can be generalised in the required manner by assuming $(a,x_1)$ is basic in $A \oplus M$ at all $\mathfrak{p} \in \mathfrak{C}_n$ for every $a \in A$, thus obtaining basic elements of the form $x_1 + ax', x' \in \Sigma_{i=2}^{k}{Ax_i}$.
2. Several conclusions follow immediately from Theorem 1, each analogous to the colloraries of \[5, Theorem A\]. We give the following example, corresponding to a theorem of Serre, that indicates how to formulate these conclusions along the lines of \[5\]: if $P$ is a projective $R$-module with a well defined rank greater than max $\{$codh$(R_{\mathfrak{p}}) \mid \mathfrak{p} \in $Spec$(R) \}$, then $P$ splits as a direct sum of rank $1$ free modules. (For a proof, see \[5, Corollary 1\]; the hypotheses of Theorem 1 are satisfied since $P$ has a well defined rank).\
The reduction of ranks of $m$-torsionless modules
=================================================
In this section we show that we may ‘replace’, in an appropriate sense, an $m$-torsionless $R$-module of rank greater than $m$ by an $m$-torsionless $R$-module of rank $m$.\
Let $M$ be an $m$-torsionless $R$-module of rank $r \geq m$, and suppose that either $(M) < \infty$ or $R_{\mathfrak{p}}$ is a regular local ring for all $\mathfrak{p} \in \mathfrak{C}_m$. Suppose further that $g: R^n \rightarrow M$ is an epimorphism, that $e_1, \dots , e_n$ constitute a basis for $R^n$ and $s := n-r+m+1$. Then there exist elements $x_s, \dots , x_n \in R^n$ satisfying the following conditions:\
1. $e_1, \dots , e_{s-1}, x_s, \dots , x_n$ is a basis for $R^n$.
2. $g(x_s), \dots , g(x_n)$ are linearly independent elements of $M$.
3. $M' := M/{\Sigma_{i=s}^{n}{Rg(x_i)}}$ is $m$-torsionless and of rank $m$.
4. The natural epimorphism $p: R^n \rightarrow F := R^n/{\Sigma_{i=s}^{n}{Rx_i}}$ gives an isomorphism between $(g)$ and the kernel of the induced epimorphism $g': F \rightarrow M'$.
We proceed by induction on $r$. If $r=m$, there is nothing to prove. If $r>m$, it is enough to show the existence of an element $x \in R^n$ satisfying the following properties:\
1. $e_1, \dots , e_{n-1}, x$ is a basis for $R^n$,
2. $g(x)$ is linearly independent,
3. $M'' := M/{Rg(x)}$ is $m$-torsionless and of rank $r-1$,
4. the canonical projection $p' := R^n \rightarrow R^n/{Rx}$ gives an isomorphism between Ker$(g)$ and the kernel of the induced epimorphism $g'' := R^n/{Rx} \rightarrow M''$.
The inequality $r>m$ implies $\mu(M_{\mathfrak{p}})>m$ for all prime ideals $\mathfrak{p}$ in $R$. Our hypotheses on $M$ and $R$ imply that dh$(M_{\mathfrak{p}})< \infty$ for all $\mathfrak{p} \in \mathfrak{C}_m$ (in fact, dh$(M_{\mathfrak{p}})=0$), since if $\mathfrak{p} \in \mathfrak{C}_m$, one obtains by the characterization ($b_m$) of $m$-torsionless $R$-modules in the introduction that codh$(M_{\mathfrak{p}})=$ codh$(R_{\mathfrak{p}})$. Hence $M_{\mathfrak{p}}$ is free for every $\mathfrak{p} \in \mathfrak{C}_m$. As $M$ has rank $r$, we have that $\mu(M_{\mathfrak{p}})=\mu(M_{\mathfrak{q}})=r$ for all $\mathfrak{p}, \mathfrak{q} \in \mathfrak{C}_m$. The elements $g(e_1), \dots g(e_n)$ generate $M$, and so by Theorem 1, (2), there exist $a_1, \dots , a_{n-1} \in R$ such that if $x := e_n + \Sigma_{i=1}^{n-1}{a_ie_i}$, then $g(x)$ is not in $\mathfrak{p}M_{\mathfrak{p}}$, at all $\mathfrak{p} \in \mathfrak{C}_{m}$.\
It is clear that for this choice of $x$, condition (1’) is satisfied. Since $g(x)$ can be included as part of a basis of the free $R_{{\mathfrak{p}}}$-module $M_{{\mathfrak{p}}}$ for any ${\mathfrak{p}} \in {\mathfrak{C}}_{m}$ (and in particular, for any ${\mathfrak{p}} \in $Ass $R)$, we see that $M''_{{\mathfrak{p}}}$ is a free $R_{{\mathfrak{p}}}$-module for these primes ${\mathfrak{p}}$ and that $g(x)$ is a linearly independent element of $M$. Thus (2’) follows.\
If ${\mathfrak{p}} \notin {\mathfrak{C}}_m$ (i.e., codh$(R_{{\mathfrak{p}}}) > m$), it follows from the exact sequence\
that codh$(R_{{\mathfrak{p}}}g(x))=$ codh$(R_{{\mathfrak{p}}})>m$, codh$(M_{{\mathfrak{p}}}) \geq m$ and codh$(M''_{{\mathfrak{p}}}) \geq m$. Altogether, this gives codh$(M''_{{\mathfrak{p}}}) \geq $ min$(m,$ codh$(R_{{\mathfrak{p}}}))$ for all primes ${\mathfrak{p}}$ in $R$. Since either $M''$ has finite homological dimension or $R$ is $m$-Gorenstein, it follows that $M''$ is $m$-torsionless. From (2’) it follows that rank$(M'')=r-1$.\
It only remains to verify (4’). It is elementary to show that $p'$ maps the kernel of $g$ onto the kernel of $g''$. But rank(Ker$(g))=$ rank$($Ker$(g''))$, whence Ker$(p'\mid_{Ker(g)})$ is a torsion-free $R$-module of rank $0$, and thus equal to the zero module.\
We may relax the hypotheses of Theorem 2 while ensuring that Theorem 1 is still applicable to get the following statement:\
Suppose $M$ is an $R$-module such that $M_{{\mathfrak{p}}}$ is free for ${\mathfrak{p}} \in {\mathfrak{C}}_m$ and such that $\mu(M_{{\mathfrak{p}}}) \geq \mu(M_{{\mathfrak{q}}})$ whenever codh$(R_{{\mathfrak{p}}}) >$ codh$(R_{{\mathfrak{q}}})$ for ${\mathfrak{p}}, {\mathfrak{q}} \in {\mathfrak{C}}_{m}$. Further, let
$r:=$ min$\{\mu(M_{{\mathfrak{q}}}) \mid {\mathfrak{q}} \in $Ass $R \} \geq m,$\
and $g, e_1, \dots , e_n, s$ be as in Theorem 2. Then there exist elements $x_s, \dots, x_n \in R^n$ satisfying conditions (1), (2) and (4) of Theorem 2 and such that the following condition (in place of (3)) holds: $(M/(\Sigma_{i=s}^{n}{Rg(x_i)}))_{{\mathfrak{p}}}$ is free for ${\mathfrak{p}} \in {\mathfrak{C}}_m$ and there is ${\mathfrak{q}} \in $Ass $R$ with $$\langle M/(\Sigma_{i=s}^{n}{Rg(x_i)}) \rangle _{{\mathfrak{q}}} =m.$$
(The proof is similar to that of Theorem 2 and proceeds by induction on $r$, in which the conclusion of Theorem 1 serves as the inductive step. One requires only a slightly more subtle argument for (4): an $R$-homomorphism $f: N \rightarrow N'$, where $N$ is torsionfree, is injective if and only if for all ${\mathfrak{q}} \in $Ass$ R$, $f \otimes_R R_{{\mathfrak{q}}}$ is injective.)\
The following corollary shows that under the slightly weaker hypotheses of Theorem 2 on $M$ (respectively $R$), the projective modules in an exact sequence:\
can be chosen in the following prescribed manner:\
Let $M$ be an $m$-torsionless $R$-module of well defined rank $r$, where $m \geq 2$. Let either $(M)< \infty$ or $R_{{\mathfrak{p}}}$ be a regular local ring for all ${\mathfrak{p}} \in {\mathfrak{C}}_{m-1}$. Then there exists an exact sequence\
In particular, $M$ admits an embedding $f: M \rightarrow R^{r+m-1}$ such that $(f)$ is $(m-1)$-torsionless and $M$ is the $(m-1)$-th syzygy module of an ideal of $R$. When $m>2$ this ideal is generated by three elements, and when $m=2$ it is generated by $r+1$ elements.\
Projective modules in the above exact sequence that satisfy property $(s_m)$ and are $m$-torsionless can without loss of generality be assumed to be free, so that there exists an exact sequence\
where $F$ is free and $N$ is $(m-1)$-torsionless. $N$ has a well defined rank and is of finite homological dimension if $M$ is. Applying Theorem 2 to the epimorphism $F \rightarrow N$ gives in the case that rank$(F) \geq r+m-1$ an exact sequence $$0 \rightarrow M \rightarrow F' \rightarrow N' \rightarrow 0,$$ where $N'$ is $(m-1)$-torsionless and of rank $m-1$, whence $F'$ is isomorphic to $R^{r+m-1}$. If rank$(F) < r+m-1$ one obtains this exact sequence by adding to $F$ a free direct summand of the appropriate rank.
The observation just made about $M$ yields for $N'$ an exact sequence
$$0 \rightarrow N' \rightarrow R^{2m-3} \rightarrow N'' \rightarrow 0,$$
such that $N''$ is $(m-2)$-torsionless, etc.\
A theorem of Bourbaki \[2, pg. 76, Theorème 6\] is obtained as a special case of Theorem 2. Bourbaki proves the special case $m=1$ in the following corollary (under the additional assumption that $R$ does not contain any zero-divisors).\
Let the localizations $R_{{\mathfrak{p}}}$ be regular local rings for ${\mathfrak{p}} \in {\mathfrak{C}}_m$ and let $M$ be an $m$-torsionless $R$-module having a well-defined rank $\geq m$. Then there exists a free submodule $F$ of $M$, such that $M/F$ is $m$-torsionless and of rank $m$.
On the extension of certain projective resolutions to resolutions of ideals generated by three elements
=======================================================================================================
We are now prepared to give an easy proof of the following result that was highlighted in the introduction:\
Let $M$ be an $m$-torsionless $R$-module having a well-defined rank, and let
be a projective resolution of $M$, with $n>m \geq 1$, $F_m$ a free $R$-module and $r:= $$(f_{m+1}))$. Then there exist homomorphisms $c: F_m \rightarrow R^{r+m}, f_m: R^{r+m} \rightarrow R^{2m-1}, f_j: R^{2j+1} \rightarrow R^{2j-1}, j=1, \dots , m-1,$ such that if $f'_{m+1}:=c \circ f_{m+1}$, the sequence:\
is exact.
If $u:= rank(F_m) \leq r+m$, we may choose $c$ to be the embedding $F_m \rightarrow F_m \oplus R^{r+m-u}$. Then\
is exact, Coker$(f'_{m+1})$ is $m$-torsionless and of rank $m$.\
If $u>r+m$, then by Theorem 2 there exist basis elements $x_s, \dots, x_u, s:= r+m+1$ of $F_m$ such that $M':= M/(Rg(x_s)+ \dots + Rg(x_u))$ is $m$-torsionless and of rank $m$. Further, the natural epimorphism $c: F_m \rightarrow F_m/(Rx_s + Rx_u) \cong R^{r+m}$ induces an exact sequence: $0 \rightarrow$ Im$(f_{m+1}) \cong$ Coker$(f_{m+2}) \rightarrow R^{r+m} \rightarrow M' \rightarrow 0$. Thus $f'_{m+1}$ has the required propery in this case as well. Finally, Corollary 1 to Theorem 2 gives the homomorphisms $f_i, i=1, \dots, m$.\
1. When the localizations $R_{{\mathfrak{p}}}$ are regular local rings, Theorem 3 applies to infinite resolutions also. (See Theorem 2 and Corollary 1.)\
2. It would suffice to prove Theorem 3 in the case that the modules $F_{n-1}, \dots, F_{m}$ are free and $F_n$ is projective and of well-defined rank, since $M$ has such a resolution. Conversely, if $M$ has such a resolution, then $M$ automatically has a well-defined rank.\
3. If all the $F_i$ are free modules, then $M$ is $m$-torsionless if and only if grade$(I(f_j)) \geq j$ for all $j= m+1, \dots, n$. (The following definition of $I(f_j)$ is found in \[3\]: $I(f_j)$ is the (rank(Coker$(f_j)))$-th Fitting ideal of Coker$(f_j)$). This follows immediately from the characterization of $m$-torsionless $R$-modules of finite homological dimension ($b_m$) in the introduction.\
4. In Theorem 2 (and its corollaries) $m$ is always a lower bound for the rank of the constructed $m$-torsionless $R$-modules. We conjecture that this is not a consequence of our method of proof, but is in fact because there are no non-projective $m$-torsionless $R$-modules of finite homological dimension and of rank $< m$. This is always true if either $m \leq 2$ or dh$(M) \leq 1$. To see this it is enough to consider the case when $R$ is local. The case $m=1$ is trivial. A $1$-torsionless, or even more generally, a $2$-torsionless $R$-module of rank $1$ is isomorphic to an ideal ${\mathfrak{a}}$ of $R$, which under the given hypotheses has a finite free resolution and has grad$({\mathfrak{a}}) \geq 1$. But then ${\mathfrak{a}}$ is isomorphic to an ideal ${\mathfrak{a'}}$ with grade$({\mathfrak{a'}}) \geq 2$ (\[8, Corollary 5.6\] or \[3, Corollary 5.2\]). The ideal ${\mathfrak{a'}}$ is not $2$-torsionless, unless ${\mathfrak{a'}} = R$. When dh$(M) \leq 1$, our claim follows from known estimates on the degree of determinantal ideals: the degree of a (rank$(M))$-th Fitting ideal of $M$ is, for $M$ free, less than or equal to rank$(M)+1$ (see also \[10, Theorem 2\]).\
5. [\[3, Theorem 8.1\]]{} derives a particular case of Theorem 3, where $n=3$, $m=2$, rank$(F_2)=$ rank$(F_1)+2$. The proof given there uses the structure theory of free resolutions developed in \[3\].\
The following corollary says that every natural number that is the homological dimension of an $m$-torsionless $R$-module is also the homological dimension of a ($2m+1$)-generated $m$-torsionless $R$-module of rank $m$. The special case $m=1$ was proved using entirely different methods by Burch \[4\] and Kohn \[7\].\
Let $s, m \geq 1$ be natural numbers such that there exists an $R$-module $N$ with $(N)=s+m$. Then there is an $m$-torsionless $R$-module $M$ with $(M)=s$, having rank $m$, and generated by at most $2m+1$ elements.
Let $n:=s+m$. Since dh$(N)=n$, there exists an $R$-regular sequence $x_1, \dots, x_n$. When $s=1$, we let $M=R^n/R(x_1, \dots x_n)$. If $s>1$, consider the following section of the Koszul complex associated to $(x_1, \dots, x_n)$:\
Coker$(f_{m+2})$ is $(m+1)$-torsionless and has a well-defined rank; let $r:=$ rank(Im$(f_{m+2}))$. By Theorem 3, there are homomorphisms $f'_{m+2}, f_{m+1}$ such that the complex
is exact and Coker$(f_{m+1})$ is $m$-torsionless. It is clear that the claims pertaining to the rank and the number of generators of $M$ hold. When $s=2$, we have $f'_{m+2}=f_n$ (indeed, rank(Coker$(f_n))=m+1$) and when $s>2$, $f_n$ is independent of the construction of $f'_{m+2}$. Since $f_n$ does not split, we have dh$(M)=s$.\
If the conjecture made in Remark 4 to Theorem 3 holds, then there is an $m$-torsionless $R$-module of finite homological dimension (in fact of homological dimension $\leq 1$), generated by less than $2m+1$ elements. [^5] (It is sufficient, again, to consider only local rings $R$; then $M$ has a well-defined rank whenever dh$(M) < \infty$.)
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G. Scheja and U. Storch, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, *Math. Ann.* **197** (1972), 137-170.
U. Vetter, Zu einem Satz von G. Trautmann über den Rang gewisser kohärenter analytischer Moduln, *Arch. Math.* **XXIV** (1973), 158-161.
[^1]: Soumya Deepta Sanyal, 28 Math. Sci. Bldg., Columbia, MO 65211; [email protected]\
[^2]: Translator’s note: The translator gratefully acknowledges support from the AMS, the NSF and the AMS Mathematics Research Communities program. The translation was begun during the 2010 AMS MRC in Commutative Algebra.\
The translator thanks H. Srinivasan for helpful comments on the translation, A. Heinecke for proofreading the translation and C. Chiasson for assistance with the translation of Remark 4 to Theorem 3.\
The translator thanks W. Bruns for helpful comments and for permitting the distribution of this translation.\
[^3]: Translator’s note: The homological dimension of a module is now more commonly known as its projective dimension.
[^4]: Translator’s note: this is the ‘transpose’ of $M$ in the sense of Auslander, as remarked in “Linear Free Resolutions and Minimal Multiplicity”, by D. Eisenbud and S. Goto; Journal of Algebra **88**, 89-133 (1984).
[^5]: *Addendum & Correction*. D. Eisenbud has informed us that the conjecture stated in Remark 4 to Theorem 3 was first made by P. Hackman in a currently unpublished article “Exterior Powers and Homology”. The referee requested that we make a reference to the article “Tout ideal premier d’un anneau noethérien est associé à un ideal engendré par trois éléments” by T. Gulliksen (*C. R. Acad. Sci. Paris Ser. A* **271** (1970), 1207-1208). The main result of the paper stated in its title follows immediately from the corollary to Theorem 3.
|
---
abstract: 'An analytical model of the electrostatic force between the tip of a non-contact Atomic Force Microscope (nc-AFM) and the (001) surface of an ionic crystal is reported. The model is able to account for the atomic contrast of the local contact potential difference (CPD) observed while nc-AFM-based Kelvin Probe Force Microscopy (KPFM) experiments. With the goal in mind to put in evidence this short-range electrostatic force, the Madelung potential arising at the surface of the ionic crystal is primarily derived. The expression of the force which is deduced can be split into two major contributions: the first stands for the coupling between the microscopic structure of the tip apex and the capacitor formed between the tip, the ionic crystal and the counter-electrode; the second term depicts the influence of the Madelung surface potential on the mesoscopic part of the tip, independently from its microscopic structure. The former has the lateral periodicity of the Madelung surface potential whereas the latter only acts as a static component, which shifts the total force. These short-range electrostatic forces are in the range of ten pico-Newtons. Beyond the dielectric properties of the crystal, a major effect which is responsible for the atomic contrast of the KPFM signal is the ionic polarization of the sample due to the influence of the tip/counter-electrode capacitor. When explicitly considering the crystal polarization, an analytical expression of the bias voltage to be applied on the tip to compensate for the local CPD, *i.e.* to cancel the short-range electrostatic force, is derived. The compensated CPD has the lateral periodicity of the Madelung surface potential. However, the strong dependence on the tip geometry, the applied modulation voltage as well as the tip-sample distance, which can even lead to an overestimation of the real surface potential, makes quantitative KPFM measurements of the local CPD extremely difficult.'
author:
- 'Franck Bocquet$^{1, 2}$, Laurent Nony$^{1, 2, }$[^1], and Christian Loppacher$^{1, 2}$'
- Thilo Glatzel
title: 'Analytical Approach to the Local Contact Potential Difference on (001) Ionic Surfaces: Implications for Kelvin Probe Force Microscopy'
---
Published in Phys. Rev. B **78**, 035410 (2008)
Introduction
============
Electrostatic forces play a key role in non-contact Atomic Force Microscopy (nc-AFM), not only in the imaging process [@sadewasser03a] but also for the investigation of the surface electronic properties. Electronic properties such as the work function and surface charges can be acquired by Kelvin Probe Force Microscopy (KPFM) [@weaver91a; @nonnenmacher91a] simultaneously to imaging topography by nc-AFM. In KPFM, a feedback is used to apply a voltage between the tip and the sample in order to minimize the electrostatic tip-sample interaction. For metals, this voltage is equal to the contact potential difference (CPD), *i.e.* the bias voltage to be applied between the tip and the surface to align their fermi levels. It is connected to the difference between the work functions of the two surfaces, and thereby to their local electronic properties, according to:
$$\label{EQU_LOCAL_CPD}
\Delta \phi=\phi_{tip}-\phi_{sample}=qV_{cpd},$$
$q$ being the elementary electrical charge: $q=1.6
\times 10^{-19}$ C.
Nowadays, two KPFM-based techniques provide facilities to map the spatial variations of the CPD on the nanometer scale, namely Frequency- [@kitamura98a] or Amplitude-Modulation-KPFM [@kikukawa96a; @sommerhalter99a; @glatzel03a] (FM- or AM-KPFM, respectively). These methods were primarily applied to metallic and semiconducting surfaces to study the distribution of dopants in semiconductors [@kikukawa95a], or the adsorption of organic molecules (for an overview see ref.\[\]). In a few experiments, even molecular [@sasahara01a] or atomic [@sugawara99a; @kitamura00a; @okamoto03a] contrast has been reported. The extension of the technique to insulating surfaces, was performed more recently, as demonstrated by the results reported on thin ionic films on metals [@krok04a; @loppacher04a], or on the contribution of bulk defects to the surface charge state of ionic crystals [@barth06a; @barth07a].
In this work, atomic corrugation of the CPD signal is reported for the first time on the (001) surface of a bulk ionic crystal of KBr. For that purpose, KPFM experiments were carried out in ultrahigh vacuum with a base pressure below $10^{-10}$ mbar using a home built non-contact atomic force microscope operated at room temperature [@howald93a]. A highly doped silicon cantilever with a resonance frequency $f_0\approx 160$ kHz, and a spring constant $k\approx 21$ N.m$^{-1}$ was used. The typical oscillation amplitude of the fundamental bending resonance was $\approx 10$ nm. The cantilever was annealed (30 min @ 120$^\circ$) and gently sputtered with Ar$^+$ ions (1-2 min @ 680 eV). The KBr crystal was cleaved in ultrahigh vacuum along the (001)-plane and subsequently annealed at 120$^\circ$ during an hour. The KPFM signal was detected using the AM mode, as described in detail in Ref. . While these experiments, atomic-scale contrast was as well visible in the topography channel (data not shown). The CPD measurements are reported in figs.\[FIG\_EXP\]a and b. In fig.\[FIG\_EXP\]a, the image exhibits atomic features, the measured period of which is 0.63 nm, which is visible in the joint cross section. This value matches to a good agreement the lattice constant of KBr, 0.66 nm. The vertical contrast yields about 100 mV around an average value of -3.9 V, the origin of which will be discussed in section \[SEC\_KPFM\]. A striking aspect of those results is the robustness of the imaging process in terms of stability and reproducibility upon the tips used. These results suggest an intrinsic imaging process relying on the microscopic origin of the contact potential arising at the sample surface. In this case, the CPD rather turns into *local CPD*, consistently with the concept of *local work function* which has been introduced by Wandelt on metals[@wandelt97a]. By “intrinsic imaging process", it is meant that the contrast can be accounted for with a tip consisting of a single material, namely a metal. Thus, the atomic contrast neither relies on adsorbed nor on unstable species at the tip apex, as often reported for topographic[@hoffmann04a] or dissipation[@hoffmann07a] imaging by nc-AFM.
![a- Experimental image showing the atomic contrast of the compensated CPD measured on a (001) surface of KBr in ultra-high vacuum by means of AM-KPFM. The vertical contrast ranges from -3.95 to -3.85 V from dark to white spots. The dashed line depicts the cross section shown in b-. The dotted square depicts the area corresponding to the ball model shown in fig.\[FIG\_GEOM\]a. b- Corresponding cross section.[]{data-label="FIG_EXP"}](FIG01.eps "fig:"){width="\columnwidth"}\
In this work, in order to understand the local CPD contrast formation, an electrostatic model is proposed that allows us to derive an expression of the short-range electrostatic force occurring between a biased metallic tip of a nc-AFM microscope and the surface of a bulk dielectric. On the contrary to more refined numerical methods proposed in the literature for almost similar systems[@kantorovich_SciFi; @lyuksyutov04a], the analytical development is restricted to a simple tip geometry and a classical, continuous electrostatic approach. Notwithstanding, the model allows us to define a general frame, able to put in relation the surface electrostatic properties with the imaging process yielding atomic contrast of the CPD on ionic surfaces. Obviously, the main results presented here remain qualitatively correct for more complex tip geometries, although numerical methods are then required to get quantitative numbers.
The motivations for that work are twofold. On the one hand, to our knowledge, a compact modelization of the short-range electrostatic forces responsible for the atomic contrast in KPFM on a bulk ionic crystal is still lacking. On the other hand, when evaluating KPFM experiments, the relative complexity of FM- or AM-KPFM experimental setups, both including four electronic controllers, makes the interpretation of the experimental images, and primarily CPD images, tedious, especially when dealing with atomic resolution. Several groups analyzed the KPFM imaging mechanism in order to evaluate their data in terms of quantitative values and lateral resolution[@jacobs98a; @colchero01a; @mcmurray02a; @sadewasser03b; @rosenwaks04a; @takahashi04a; @palacios05a; @zerweck05a; @leendertz06a; @zerweck07a], a few of them also compared AM- and FM-KPFM in terms of evaluating the force and its gradient, respectively[@glatzel03a; @zerweck05a]. The fact that many of the above mentioned calculations point out that KPFM results, especially for nano-objects, show a strong distance dependence, clearly points out that for a careful analysis, it is not sufficient to only calculate the electrostatic tip-sample interaction. It is rather important to perform simulations including all imaging mechanisms and in particular also the distance control in order to exclude artifacts due to the feedback circuits. In order to explore the origin of the CPD atomic contrast, we are aiming to closely mimic a real KPFM setup by means of an earlier developed nc-AFM simulator[@nony06a]. In the present case of an ionic surface, it is required to consider a large slab of ions, virtually infinite, to describe properly the electrostatic interaction. This is hardly feasible by means of *ab initio* calculations which fail to describe tip-surface systems involving a too big number of atoms. Therefore, prior to simulating the CPD contrast on ionic surfaces by means of our simulator, which is the scope of a future work, we have to find an analytical expression for the electrostatic tip-sample interaction.
The following section details the boundary-value electrostatic problem leading to the expression of the force (section \[SEC\_FORCE\]). In section \[SEC\_DISCUSSION\], the analytical expression of the local CPD is derived and discussed to be put in relation with the experimental observations. The implications for KPFM experiments are discussed as well.
Electrostatic model {#SEC_MODEL}
===================
![a- KBr lattice with a fcc structure corresponding to the dotted square shown in fig.\[FIG\_EXP\]a. The white spots of the experimental image have been placed on top of anions arbitrarily. b- Scheme of the KPFM experiment defining the electrostatic boundary-value problem to be solved. The metallic tip is biased with respect to a metallic counter-electrode placed a few millimeters far from it owing to the thickness of the ionic crystal. The bias voltage polarizes the crystal, which results, at the surface, in a modulation of the positions of the ions. c- and d- Schemes of the splitting of the original electrostatic boundary-value problem schemed in b- defining problems A and B, respectively.[]{data-label="FIG_GEOM"}](FIG02.eps "fig:"){width="\columnwidth"}\
The geometry of the problem of FM- or AM-KPFM experiments applied to bulk insulating materials is reported in fig.\[FIG\_GEOM\]b. The dielectric sample is an alkali halide crystal like NaCl, KBr, KCl with a fcc structure and a lattice constant $a$ (*cf.* fig.\[FIG\_GEOM\]a). In the area where the tip is, the model assumes that the crystal carries neither net charge, nor local dipole. Its thickness $h_d$ is much larger than all other distances of the problem, typically a few millimeters. Below the surface, the crystal is treated as a continuous dielectric medium with a dielectric permittivity $\epsilon_d$. At the surface, the atomic corrugation of the crystal is described by a single layer of alternate point charges arranged with a fcc structure perpendicular to the \[001\] direction. The layer extends infinitely in the plane direction. The motivation for such a rationalization of the problem will be justified in section (\[SEC\_SIGMA\_MU\]).
The crystal lies on a metallic sample holder (hereafter referred to as the counter-electrode), with respect to which the tip is biased. The counter-electrode is a planar and perfect conductor. The tip is also assumed to be a perfect conductor which is biased at $V_b$. In order to preserve the analyticity of the model, the electrostatic boundary-value problem is restricted to a tip with a very simple apex geometry, namely: a hemispherical mesoscopic part (radius $R \simeq 5$ nm) on top of which is half-embedded a small spherical asperity (radius $R_a \ll R$). The contributions of the cantilever and of the macroscopic part of the tip to which the apex is connected to are assumed to be negligible. This issue will be justified in section \[SEC\_FORCE\_M\]. The tip-surface distance is denoted $z_\mu$ (typically a few Å).
The electric field $\overrightarrow{E}$ produced locally between the tip, the dielectric, and the counter-electrode polarizes the ionic crystal, which acquires a macroscopic polarization $\overrightarrow{P}=n_v\overrightarrow{p_l}$ oriented along $\overrightarrow{E}$. In the former equation, $n_v$ is the number of polarizable species *per* unit volume and $\overrightarrow{p_l}$, the local dipolar moment *per* unit cell. In KPFM, the bias voltage $V_b$ is modulated at frequencies which do not influence the electronic polarization of the ions. The major part of the crystal polarization has rather an ionic character, *i.e.* a net displacement in opposite directions of the ions due to their charge $\pm q$ with respect to their equilibrium positions, $\pm \delta/2$, with $\delta = p_l/q$, $q$ being the elementary electrical charge. The polarization effect occurs as well at the crystal surface, where the positions of the ions become modulated perpendicularly to the surface plane, *i.e.* $\delta \rightarrow \delta^\bot$, as sketched in figs.\[FIG\_GEOM\]b and d. $\delta^\bot$ is proportional to the local electric field $\overrightarrow{E_l}$ at any ionic site and to the total polarizability of the dielectric restricted, in our approach, to the ionic polarizability $\alpha_i$ (ref.\[\]). Usually, $\overrightarrow{E_l}$ differs from the external electric field $\overrightarrow{E}$ due to the biased tip because $\overrightarrow{E_l}$ explicitly depends on the polarization of the dielectric. The Lorentz’s model links $\overrightarrow{E_l}$ and $\overrightarrow{E}$ (ref.\[\]), thus, $\overrightarrow{p_l}$ is written:
$$\label{EQU_POLARIZATION}
\overrightarrow{p_l}=\alpha_i\epsilon_0
\overrightarrow{E_l}=\chi_d\overrightarrow{E}=q\delta^\bot
\frac{\overrightarrow{E}}{E},$$
with $\chi_d=\alpha_i\epsilon_0/(1-n_v\alpha_i/3)$, the dielectric susceptibility of the sample. In the former equation, it is important to notice that $\overrightarrow{p_l}$ depends on $\overrightarrow{E}$ and hence on the bias voltage $V_b$. Consequently, this is also true for $\delta^\bot$. For the sake of discussions, the bias dependence will henceforth be explicitly outlined $\delta^\bot \rightarrow \delta^\bot(V_b)$. Despite $\delta^\bot(V_b)$ cannot be estimated at this point, it is crucial to keep in mind that the sample surface is polarized by the influence of the bias since this is a key aspect of the origin of the CPD atomic contrast.
Our approach of the electrostatic boundary-value problem relies on an *ad hoc* assumption. The tip, being a perfect conductor, develops a surface charge density $\sigma$, the origin of which is split into two main contributions $\sigma=\sigma_m+\sigma_\mu$, namely:
- a charge density $\sigma_m$ due to the capacitor formed between the biased tip and the counter-electrode with the dielectric in between (*cf.* fig.\[FIG\_GEOM\]c). Owing to the distance between the electrodes, *i.e.* roughly the dielectric thickness $h_d$, $\sigma_m$ has a mesoscopic character. It is not influenced by the local structure of the tip apex, but rather by its overall shape.
- a charge density $\sigma_{\mu}$ originating from the Madelung potential that expands at the crystal surface. When the tip is at a distance $z_\mu$ where the potential is effective, typically a few Ångströms, then it develops, in addition to $\sigma_m$, a surface charge density $\sigma_\mu$ (*cf.* fig.\[FIG\_GEOM\]d). $\sigma_{\mu}$ has a microscopic character and must strongly depend on the local structure of the tip apex and on $z_\mu$.
Despite the simple tip geometry that has been assumed, the calculation of the electrostatic force acting on it due to the combined influence of the capacitive coupling and of the Madelung surface potential has no exact analytical solution. Nevertheless, one can build up an approximate solution to the boundary-value problem relying on the superposition principle. For that purpose, the problem is split up into two boundary-value sub-problems schemed in figs.\[FIG\_GEOM\]c and d:
- problem $A$: the tip biased at $V_b$ in front of a dielectric continuous medium (height $h_d$, dielectric permittivity $\epsilon_d$) held on an infinite planar conductor, the counter-electrode (fig.\[FIG\_GEOM\]c). The local structure of the dielectric is not supposed to influence the tip. This is the description of the “capacitive" problem, the solution of which provides the surface charge density $\sigma_m$.
- problem $B$: the tip now biased at 0 V with an infinite plane of alternate point charges located at the same distance than the surface of the dielectric in problem A, *i.e.* $z_\mu$. The layer of point charges is polarized under the electric field that occurs in problem A (fig.\[FIG\_GEOM\]d). This describes the “microscopic" problem. The solution provides the surface charge density $\sigma_\mu$.
Besides, in order to carry out the calculations more easily, it is convenient to distinguish the two following geometrical areas on the tip (*cf.* fig.\[FIG\_GEOM\]d), namely: the asperity, area (1), *i.e.* a half-sphere with radius $R_a$ and the mesoscopic tip apex around it, area (2), a sphere with radius $R
\gg R_a$ (typically $R/R_a \geq 50$). These two areas do not overlap, but the continuity between them is ensured. The vertical force acting on the tip[@jackson] is written in terms of $\sigma_m$ and $\sigma_\mu$ as described above:
$$F=\int_\text{tip}\frac{\left(\sigma_m+\sigma_{\mu}
\right)^2}{2\epsilon_0}\widehat{n}.\widehat{u}_zd\mathcal{S} =
F_{m}+F_{m\mu}+F_{\mu}$$
$\widehat{n}$ and $\widehat{u}_z$ are the normal to the tip surface and the unitary vector along the vertical $z$ axis of the problem, respectively. Doing so, we only focus on the vertical resultant of the force acting onto the tip. The above expression can be expanded into three components: a purely capacitive part, $F_{m}$ originating from the tip/dielectric/counter-electrode capacitor; a coupling part, $F_{m\mu}$, which can be interpreted in terms of the resulting force of all the elementary forces due to the electric field $\sigma_m/\epsilon_0$ onto each elementary charge $\sigma_{\mu} d\mathcal{S}$ produced on the tip by the influence of the Madelung potential of the crystal, $V_s$; and a purely microscopic part, $F_\mu$, standing for the short-range electrostatic force due to $V_s$.
Problem A: estimation of $\sigma_m$ {#SEC_PROBLEM_A}
-----------------------------------
Although the boundary-value problem of a planar conductor biased with respect to another one with an incomplete dielectric layer in between yields an expression of the surface charge density, the problem with the sphere does not. However, one can argue that the expression of $\sigma_m$ must be a combination between a configuration in which there is no dielectric medium in the sphere/counter-electrode interface and an opposite one, where the interface is completely filled with it. One can therefore postulate an effective dielectric permittivity $\widetilde{\epsilon}_d=K\epsilon_d$, where $K~(<1)$ is a constant to be set. Owing to the fact that the mesoscopic part of the tip apex, referred to as area (2), is located at a distance $z_m=z_\mu+h_d \gg R$ from the counter-electrode, the analytical expression of $\sigma_m^{(2)}$, explicitly calculated in refs.\[\], asymptotically trends towards the surface charge density of an isolated, biased sphere [@jackson]:
$$\label{EQU_SIGMA_M2}
\sigma_m^{(2)} \overset{z_m \gg R}{=}
\frac{\widetilde{\epsilon}_d\epsilon_0 V_b}{R}$$
To get the surface charge density on area (1), $\sigma_m^{(1)}$, we seek the potential $V_m^{(1)}$ which equals $V_b$ over the asperity. For that purpose, a spherical coordinate system $(r,\theta,\varphi)$ centered on the asperity is chosen. The problem having an azimuthal symmetry, the sought potential can be expanded in Legendre polynomials [@jackson]:
$$V_m^{(1)}(r,\theta)=\sum_{n=0}^{\infty}\left( \alpha_n r^n +
\frac{\beta_n}{r^{n+1}} \right) P_n(\cos \theta)$$
At large distance from the asperity, $r \gg R_a$, the potential must be similar to the one of a sphere with radius $R$ biased at $V_b$, namely [@jackson]:
$$V_m^{(1)}(r,\theta) \overset{R>r\gg
R_a}{=}V_b\sum_{n=0}^{\infty}\left(\frac{r}{R}\right)^n P_n(\cos
\theta)$$
Hence, the coefficients $\alpha_n$ of the expansion are known. The coefficients $\beta_n$ are deduced from the property of orthogonality of the Legendre polynomials at the boundary condition $V(r=R_a)=V_b$. The potential is finally written:
$$\label{EQU_POTENTIAL_Vm1}
V_m^{(1)}(r,\theta)=V_b\left\{ 1+ \sum_{n=1}^\infty
\left[\left(\frac{r}{R}\right)^n-\frac{R_a^{2n+1}}{R^n r^{n+1}}
\right]P_n(\cos \theta) \right\}$$
The former equation rigorously describes the potential of a system apex/asperity where the junction point between the two spheres is smoothed and not singular, as sketched in fig.\[FIG\_GEOM\]d. Nevertheless, for $R \gg R_a$, the influence of the singular area is negligible. Therefore, equ.\[EQU\_POTENTIAL\_Vm1\] is a good approximation to the boundary-value problem. The normal derivation along the surface of the asperity yields the expression of $\sigma_m^{(1)}$:
$$\label{EQU_SIGMA_M1}
\sigma_m^{(1)}= -\frac{\widetilde{\epsilon}_d\epsilon_0
V_b}{R}\sum_{n=1}^{\infty}(2n+1)\left(\frac{R_a}{R}\right)^{n-1}P_n(\cos
\theta)$$
Owing to the condition $R \gg R_a$, the sum can be restricted to the first term. Therefore:
$$\sigma_m^{(1)}= -\frac{3\widetilde{\epsilon}_d\epsilon_0
V_b}{R}\cos \theta$$
Thus, despite $\sigma_m^{(1)}$ develops on the asperity, its strength is governed by the radius of the mesoscopic part of the tip apex, $R$ and not by the local radius of curvature of the asperity, $R_a$.
Problem B: estimation of $\sigma_{\mu}$ {#SEC_SIGMA_MU}
---------------------------------------
The calculation of the surface charge density $\sigma_{\mu}$ on the mesoscopic sphere+asperity is more tedious, primarily because it relies on the estimation of the Madelung potential of the ionic crystal, $V_s$. The boundary-value problem is now restricted to the determination of the surface charge density arising on a metallic tip at zero potential under the influence of an infinite planar slab of point charges. Again, the solution of such a problem has no straightforward analytical solution. However, we can again use the condition $R \gg R_a$, as depicted, to some extend, in fig.\[FIG\_GEOM\]d. Consequently, the mesoscopic part of the apex can be assumed as equivalent to an infinite planar conductor, at least in a small area along the sides of the asperity (light grey area in fig.\[FIG\_GEOM\]d). The new boundary-value problem defined by an infinite planar conductor influencing another infinite planar conductor at zero potential carrying a hemispherical bump with a radius $R_a$, now yields a solution [@durand]. The method of the image charges is used to solve it. The first set of image charges ensuring a zero-potential value on the counter-electrode[@Note_Nony07f] produces an electric field which influences the tip. But as a matter of fact, this contribution can be neglected because the distance between the tip and the counter-electrode is on the millimeter range and the image charges originate from the Madelung potential, which is known to decay exponentially fast[@watson81a; @Note_Nony07c] (*cf.* also hereafter). The second set of image charges[@Note_Nony07g] is quasi-punctual and located at the center of the asperity. Hence, owing to the simplified geometry of the electrode, the problem is reduced to a sphere with radius $R_a$ at zero potential in the influence of two infinite planes, *i.e.* the slab and its image, which are anti-symmetrically spaced with respect to the sphere. This procedure ensures a zero-potential on the approximated plane within which the asperity is embedded. Again, it is more convenient to use a spherical coordinate system centered on the asperity. Then, the surface charge density $\sigma_{\mu}=\sigma_{\mu}^{(1)}+\sigma_{\mu}^{(2)}$ is derived by normal derivation along areas (1) and (2), namely:
$$\label{EQU_SIGMA_MU}
\sigma_{\mu}=-\left.\underbrace{\epsilon_0 \frac{\partial
V_\mu(r,\theta,\varphi)}{\partial
r}}_\text{(1)=asperity}\right|_{r=R_a}-\left.\underbrace{\epsilon_0\frac{\partial
V_\mu(r,\theta,\varphi)}{r\partial \theta}}_\text{(2)=planar
area}\right|_{\theta=\pi/2},$$
The potential $V_\mu$ is derived from the Madelung potential of the ionic crystal, previously referred to as $V_s$, by the method of the image charges and the Kelvin transform (influence on a sphere biased at zero potential)[@durand], namely:
$$\label{EQU_KELVIN_TRANSFORM}
\begin{array}{l}
V_\mu(r,\theta,\varphi)=\left.\left\{V_s(r,\theta,\varphi)-\frac{R_a}{r}V_s\left(\frac{R_a^2}{r},\theta,\varphi
\right)\right\}\right|_\text{slab}-\\
\left.\left\{V_s(r,\pi-\theta,\varphi)-\frac{R_a}{r}V_s\left(\frac{R_a^2}{r},\pi-\theta,\varphi
\right)\right\}\right|_\text{image slab}\end{array}$$
The former equation fulfills the boundary condition $V_\mu(r=R_a)=0$ everywhere along the surface asperity or along the surface of the local planar area around it.
$V_s$ can be estimated on the base of the work by Watson *et al.* [@watson81a]. When considering an infinite planar slab of point charges, the authors state that the potential, so-called Madelung surface potential, reaches its asymptotic value in a very short distance normal to the slab. Consequently, the ions within the crystal at a distance only one lattice constant from the surface have Madelung potentials which are indistinguishable from those of the bulk. In other words, *the tip will mainly be influenced by the surface potential and not by the one arising from the bulk part of the ionic crystal.* This is why a single, infinite, layer of point charges is enough to describe the influence of the Madelung surface potential on the tip, which motivates our initial assumption. The potential is written:
$$\label{EQU_MADELUNG_POT}
V_s\left(\overrightarrow{\rho},z_\mu\right)=\frac{1}{4\pi
\epsilon_0}\left(\frac{2\pi}{a'^2}\sum_{\overrightarrow{G}}q(
\overrightarrow{G})e^{i\overrightarrow{G}.\overrightarrow{\rho}}e^{-Gz_\mu}
\right),$$
where $\overrightarrow{\rho}=x\widehat{i}+y\widehat{j}$ is the polar vector of any ion of the surface slab in an orthogonal basis $(O,\widehat{i},\widehat{j})$, $O$ being the projection of the center of the asperity on the surface, $\widehat{i}$ and $\widehat{j}$ the unitary vectors of the fcc unit cell. The summation is performed over the vectors $\overrightarrow{G}$ of the reciprocal lattice of an arbitrarily defined unit cell and $a'$ is proportional to the lattice constant $a$ of the fcc unit cell. $q(\overrightarrow{G})$ is a structure factor:
$$\label{EQU_STRUCTURE_FACTOR}
q(\overrightarrow{G})=\frac{1}{G}\sum_k q_k
e^{i\overrightarrow{G}.\overrightarrow{\delta}_k^\|}e^{G
\delta_k^\bot}$$
It is summed over the ions within the defined unit cell. The k$^{th}$ ion carries an electrical charge $q_k$. Its planar and perpendicular coordinates from the origin of the cell are given by the two vectors $\overrightarrow{\delta}_k^\|$ and $\overrightarrow{\delta}_k^\bot$. The latter reflects the polarization effect felt by the ion within the unit cell, previously referred to as $\delta^\bot(V_b)$. In equ.\[EQU\_STRUCTURE\_FACTOR\], it is assumed that the polarization of the ions extends all over the (001) surface plane. In any case, it must extend over an much larger area than the tip asperity. This assumption is consistent with the electric field produced by area (2), which is constant within an area roughly scaling as the mesoscopic tip radius $R$. $V_s$ is calculated from the unit cell defined in fig.\[FIG\_GEOM\]a (light grey). It consists of 4 anions and a cation weighting for a fourth and one, respectively. The vectors of the direct lattice are $\overrightarrow{\alpha}=a'\widehat{i}$ and $\overrightarrow{\beta}=a'\widehat{j}$, where $a'=a\sqrt{2}/2$. Owing to the exponential decay of the potential with $z_\mu$, visible in equ.\[EQU\_MADELUNG\_POT\], the calculation of the structure factor can be restricted to the first four reciprocal vectors, namely: $\overrightarrow{G}_{i \text{ or
} j}^{\pm}=\pm 2\pi /a'(\widehat{i} \text{ or } \widehat{j}$). The calculation yields:
$$\label{EQU_MADELUNG_POT_XPLICIT}
V_s(x,y,z_\mu)= -\frac{q}{\pi
\epsilon_0a'}\cosh[\widetilde{\delta}^\bot(V_b)]\widetilde{\chi}(x,y)
e^{-\frac{2\pi}{a'}z_\mu}$$
with: $\widetilde{\delta}^\bot(V_b)=\frac{2\pi}{a'}\delta^\bot(V_b)$ and $\widetilde{\chi}(x,y)=\cos\left[\frac{2\pi}{a'}(x-x_0)\right]+\cos
\left[\frac{2\pi}{a'}(y-y_0) \right]$, a spatial modulation term. $x_0$ and $y_0$ are the $x$ and $y$ coordinates of the center of the asperity projected onto the unit cell. Setting $x_0=y_0=0$ locates the asperity and therefore the tip on top of an anion, the reference ion within the defined unit cell. The above expression exhibits the expected exponential decaying behavior as a function of $z_\mu$. The potential is reported in fig.\[FIG\_POT\] for $a=0.66$ nm, $\delta^\bot=11$ pm and $z_\mu=4$ Å. The value of $\delta^\bot$ will be justified in section \[SEC\_DISCUSSION\].
![(Color online). a- Madelung surface potential calculated from equ.\[EQU\_MADELUNG\_POT\_XPLICIT\] for $a=0.66$ nm, $\delta^\bot=11$ pm and $z_\mu=4$ Å. The vertical contrast ranges from $-100$ (blued spots) to $+100$ mV (reddish spots). The unit cell depicted with a dotted line is centered on a cation. b- Distance dependence of the potential on top of an anion (dotted curve) and on top of a cation (continuous curve) showing the exponential decay of the potential. c- Cross section along the dotted line shown in a-.[]{data-label="FIG_POT"}](FIG03.eps "fig:"){width="\columnwidth"}\
The expression of $\sigma_{\mu}=\sigma_{\mu}^{(1)}+\sigma_{\mu}^{(2)}$ can now be derived from equ.\[EQU\_SIGMA\_MU\]. The calculation of $\sigma_{\mu}^{(1)}$ yields:
$$\label{EQU_SIGMA_MU1}
\begin{array}{r}
\sigma_{\mu}^{(1)}=\frac{q}{
a'^2}\cosh[\widetilde{\delta}^\bot(V_b)]\left\{
\widetilde{\chi}(R_a,\theta,\varphi)\mathcal{F}^{(1)}(\theta)-\right.\\
\left.\widetilde{\zeta}(R_a,\theta,\varphi)\mathcal{G}^{(1)}(\theta)\right\}e^{-\frac{2\pi}{a'}(z_\mu+R_a)}\end{array}$$
with $\widetilde{\chi}(r,\theta,\varphi)$, the expression of $\widetilde{\chi}(x,y)$ in the spherical coordinate system centered on the asperity and $\widetilde{\zeta}(r,\theta,\varphi)$, the spherical expression of another spatial modulation term given by: $\widetilde{\zeta}(x,y)=(x\sin\left[\frac{2\pi}{a'}(x-x_0)\right]+y\sin
\left[\frac{2\pi}{a'}(y-y_0) \right])/\sqrt{x^2+y^2}$. The functions $\mathcal{F}^{(1)}$ and $\mathcal{G}^{(1)}$ are written:
$$\mathcal{F}^{(1)}(\theta)=\frac{2a'}{\pi
R_a}\sinh\left(\frac{\eta_\theta}{2}\right)-
8\cos\theta\cosh\left(\frac{\eta_\theta}{2}\right)$$
and:
$$\mathcal{G}^{(1)}(\theta)=8\sin\theta\sinh\left(\frac{\eta_\theta}{2}\right)$$
with: $\eta_\theta= 4\pi R_a\cos\theta/a'$ (see also the appendix). Thus, it was necessary to assume area (2) as an infinite plane in order to derive the expression of $\sigma_{\mu}^{(1)}$ by means of the method of the images. In such a case, the expression of $\sigma_\mu^{(2)}$ is rigorously derived from equ.\[EQU\_SIGMA\_MU\] and is explicitly given in ref.\[\]. For $x$ and $y$ positions large compared to $R_a$ however, this description does not fit with the geometry of the tip apex defined in problem B. The “infinite plane" must actually be shrunk down to a spatially limited area around the asperity, as sketched in fig.\[FIG\_GEOM\]d. This is made possible when assuming that any planar area is the asymptotic limit of a sphere with large radius compared to its extension. The former statement is fulfilled by the condition $R \gg R_a$. The surface charge density of a sphere with radius $R$ under the influence of the surface potential $V_s$ is also derivable from the Kelvin transform. Here, the spherical coordinate system is centered on the sphere with radius $R$. $\sigma_{\mu}^{(2)}$ is written:
$$\label{EQU_SIGMA_MU2}
\begin{array}{r}
\sigma_{\mu}^{(2)}=\frac{q}{a'^2}\cosh[\widetilde{\delta}^\bot(V_b)]\left\{
\mathcal{F}^{(2)}(\theta)\widetilde{\chi}(R,\theta,\varphi)-\right.\\
\left.\mathcal{G}^{(2)}(\theta)
\widetilde{\zeta}(R,\theta,\varphi)\right\}e^{-\frac{2\pi}{a'}(z_\mu+R_a)}\end{array}$$
The functions $\mathcal{F}^{(2)}$ and $\mathcal{G}^{(2)}$ are written:
$$\mathcal{F}^{(2)}(\theta)=\left(\frac{a'}{\pi R}-4\cos \theta
\right)e^{-\frac{2\pi}{a'}R(\cos \theta+1)}$$
and:
$$\mathcal{G}^{(2)}(\theta)=4\sin \theta e^{-\frac{2\pi}{a'}R(\cos
\theta+1)}$$
Since $\mathcal{F}^{(2)}(\theta)$ and $\mathcal{G}^{(2)}(\theta)$ decrease exponentially fast as one moves away from the foremost position of the tip apex, it can readily be verified that equ.\[EQU\_SIGMA\_MU2\] and the expression given in ref.\[\] do fit for $x$ or $y$ $\in [R_a;3R_a]$. Thus, even in the vicinity of the asperity, expression \[EQU\_SIGMA\_MU2\] can be used instead of ref.\[\]. Therefore, the most part of the contribution of the sphere to $\sigma_\mu^{(2)}$ is restricted to a small area that can be assumed as locally planar.
The graph of the projection of $\sigma_\mu$ on the tip apex (areas (1) and (2)) is reported in fig.\[FIG\_SIGMA\_MU\] on top of an anion (positive charge density on the asperity) at a distance $z_\mu=4$ Å from the surface and for $R=5$ nm and $R_a=1$ Å. The oscillations of $\sigma_\mu$ at the surface of the tip due to the image charges of the crystal are readily visible, although their amplitude decreases exponentially fast along the sides of the asperity. The relevant oscillations in area (2) spread out in a circular area with only two unit cells radius. Owing to the exponential decay of the Madelung surface potential farther away from the asperity, the tip surface is too far from the crystal surface to produce relevant image charges. On top of the asperity, the surface charge density reaches about $2\times
10^{-2}$ C.m$^{-2}$. For a half-sphere with radius $R_a=1$ Å and with $\delta^\bot=11$ pm, this results in a local charge carried by the tip of about $0.8\times 10^{-2}~q$.
![(Color online) a- Perspective view of the projection of the microscopic surface charge density $\sigma_\mu$ on the tip calculated on top of an anion from equs.\[EQU\_SIGMA\_MU1\] and \[EQU\_SIGMA\_MU2\] with $R_a=1$ Å and $R=5$ nm. $\sigma_\mu$ is strongly increased on top of the asperity. Then, owing to the exponential decay of the Madelung surface potential, the surface charge density strongly decreases, which makes the contribution of the mesoscopic apex weak. b- Top view of $\sigma_\mu$. The oscillations due to the image charges of the ions at the surface of the crystal are well visible. The most part of the attenuation of $\sigma_\mu$ occurs within a single unit cell. Two unit cells apart from the asperity, $\sigma_\mu$ is almost zero. c-. Cross section along the dotted white line shown in b.[]{data-label="FIG_SIGMA_MU"}](FIG04.eps "fig:"){width="\columnwidth"}\
Estimation of the force {#SEC_FORCE}
=======================
Estimation of $F_m$ {#SEC_FORCE_M}
-------------------
Owing to the geometry of the problem and since $\sigma_m$ is equivalent to the surface charge density of an isolated, conducting sphere, we can now fully evaluate the purely capacitive component $F_{m}$ to the total force. The force effectively acting on the mesoscopic part of the tip apex can be derived from the image charge of the sphere $-4\pi R^2 \sigma_m$ placed at a symmetric position with respect to the counter-electrode, *i.e.* at a distance $2h_d$ from the mesoscopic apex:
$$F_{m}=-\frac{(4\pi R^2 \sigma_m)^2}{4\pi\epsilon_0
(2h_d)^2}=-\frac{\pi
R^2}{h_d^2}\widetilde{\epsilon}_d^2\epsilon_0V_b^2$$
With $\epsilon_d \simeq 4.87$ (ref.\[\]) and $K=0.9$ (arbitrarily), then $\widetilde{\epsilon}_d=K\epsilon_d=4.38$. For a typical ac modulation of the bias of about 0.5 V and considering $R\simeq
5$ nm and $h_d= 5$ mm, the former equation gives an estimation for the mesoscopic component of the electrostatic force: $F_{m}\simeq
8 \times 10^{-11}$ pN. As suspected, this contribution is negligible compared to the other two components[@Note_Nony07d]. Therefore the expression of the total force simplifies to:
$$F=F_{m\mu}+F_{\mu}=\int_\text{tip}{\frac{\sigma_m\sigma_{\mu}}{\epsilon_0}d\mathcal{S}}+\int_\text{tip}{\frac{\sigma_{\mu}^2}{2\epsilon_0}d\mathcal{S}}$$
The reason why the tip has been restricted to its apex is now clear. Regarding the former capacitive force, owing to the distance between the tip and the counter-electrode, the contribution of a macroscopic body in addition to the mesoscopic apex would not change notably the total force. Regarding the other force components, $F_{m\mu}$ and $F_{\mu}$, owing to the exponential decay of the Madelung surface potential, the influence of the macroscopic body of the tip is expected to be negligible as well.
Estimation of $F_\mu$ {#SEC_FORCE_MU}
---------------------
Let us first focus on $F_{\mu}$. The integral must be performed over the asperity and the mesoscopic sphere around it, areas (1) and (2), respectively. It is recalled that the two areas do not overlap each together. For each of them, the spherical coordinate system must be centered on the corresponding sphere. Thus:
$$\begin{array}{r}
F_{\mu}=F_{\mu}^{(1)}+F_{\mu}^{(2)}=\frac{R_a^2}{2\epsilon_0}\int_{\frac{\pi}{2}}^\pi\int_0^{2\pi}\cos
\theta \sin \theta {\sigma_\mu^{(1)}}^2 d\theta d\varphi+\\
\frac{R^2}{2\epsilon_0}\int_{\frac{\pi}{2}}^{\theta_M}\int_0^{2\pi}\cos
\theta \sin \theta {\sigma_\mu^{(2)}}^2 d\theta
d\varphi\end{array}$$
The radial coordinates are fixed to $r=R_a$ and $r=R$ in areas (1) and (2), respectively. The polar integration in area (1) is performed with $\theta \in [\pi/2;\pi]$, whereas in area (2), it is performed with $\theta \in [\pi/2;\theta_M]$, where $\theta_M=\pi-\arcsin(R_a/R)$, which ensures the continuity from (1) to (2). Regarding area (2), the choice of the beginning angle of the interval ($\pi/2$) does not influence notably the result of the integration. In other words, owing to the exponential decay of $\sigma_\mu^{(2)}$, the exact shape of the mesoscopic part of the tip apex far from the asperity is not relevant. This justifies *a posteriori* the choice, although simple, of a spherical geometry. Both integrations over the azimuthal angle must be performed over $\varphi \in [0; 2\pi]$. Note that in each of the former integrals, the term $\cos \theta$ stands for the vertical projection of the force, as stated initially. Although the integration over the azimuthal angle yield an analytical result, the integration over $\theta$ does not, which requires to evaluate some integrals numerically. The expression of $F_{\mu}^{(1)}$ is written:
$$\label{EQU_F_MIC_1}
\begin{array}{r}
F_{\mu}^{(1)}=\frac{q^2R_a^2}{2\epsilon_0
a'^4}\cosh^2[\widetilde{\delta}^\bot(V_b)]e^{-\frac{4\pi}{a'}(z_\mu+R_a)}\left\{
A^{(1)}+\right.\\
\left.B^{(1)}\left[\cos\left(\widetilde{x}_0\right)+\cos\left(\widetilde{y}_0\right)\right]+2C^{(1)}\cos\left(\widetilde{x}_0\right)\cos\left(\widetilde{y}_0\right)\right\}\end{array}$$
with: $\widetilde{x}_0=2\pi x_0/a'$ and $\widetilde{y}_0=2\pi y_0/a'$, the reduced coordinates of the tip above the crystal surface. The integral forms of coefficients $A^{(1)}$, $B^{(1)}$ and $C^{(1)}$ are reported in the appendix as functions of $R_a$ and $a'$. Taking a typical lattice constant for alkali halides $a'=a\sqrt{2}/2 \simeq 0.45$ nm and assuming $R_a
\simeq 1$ Å, we get:
$$A^{(1)} \simeq -130 \hspace{0.5cm} B^{(1)} \simeq -70
\hspace{0.5cm} C^{(1)} \simeq A^{(1)} \simeq -130$$
Thus, $F_\mu^{(1)}$ explicitly depends on the spatial modulation of the surface potential. Note also the exponential decay with the distance, actually faster than the distance dependence of the Madelung surface potential and also the doubling spatial period term.
Similar integration on area (2) yields:
$$\begin{array}{r}
F_{\mu}^{(2)}=\frac{q^2R^2}{2\epsilon_0
a'^4}\cosh^2[\widetilde{\delta}^\bot(V_b)]e^{-\frac{4\pi}{a'}(z_\mu+R_a)}\left\{
A^{(2)}+\right.\\
\left.B^{(2)}\left[\cos\left(\widetilde{x}_0\right)+\cos\left(\widetilde{y}_0\right)\right]+2C^{(2)}\cos\left(\widetilde{x}_0\right)\cos\left(\widetilde{y}_0\right)\right\}\end{array}$$
The integral forms of coefficients $A^{(2)}$, $B^{(2)}$ and $C^{(2)}$ are derived from those of coefficients $A^{(1)}$, $B^{(1)}$ and $C^{(1)}$, by replacing $R_a$, $\mathcal{F}^{(1)}(\theta)$ and $\mathcal{G}^{(1)}(\theta)$ with $R$, $\mathcal{F}^{(2)}(\theta)$ and $\mathcal{G}^{(2)}(\theta)$, respectively. The integration is now performed with $\theta \in
[\pi/2;\theta_M]$. With similar parameters than before and setting $R=5$ nm, we now get:
$$A^{(2)} \simeq -8 \hspace{0.5cm} B^{(2)} \simeq C^{(2)} \simeq 0$$
The striking discrepancy between $A^{(2)}$, $B^{(2)}$ and $C^{(2)}$ is due to the combined contribution of the Bessel functions and the exponential decay of the functions $\mathcal{F}^{(2)}(\theta)$ and $\mathcal{G}^{(2)}(\theta)$ occurring in the integral forms of the coefficients $B^{(2)}$ and $C^{(2)}$. Hence, the force simplifies to:
$$\label{EQU_F_MIC_2}
F_{\mu}^{(2)}=\frac{q^2R^2}{2\epsilon_0
a'^4}\cosh^2[\widetilde{\delta}^\bot(V_b)]e^{-\frac{4\pi}{a'}(z_\mu+R_a)}A^{(2)}$$
The spatial modulation of the potential does not influence the mesoscopic part of tip while scanning the surface. This contribution acts as a static shift to the total electrostatic force, similarly as the Van der Waals long-range interaction for the short-range chemical interactions which are responsible for the topographic atomic contrast in nc-AFM.
Estimation of $F_{m\mu}$ {#SEC_FORCE_MMU}
------------------------
The geometrical splitting in terms of areas (1) and (2) used for the estimation of $F_{\mu}$, can equivalently be applied to $F_{m\mu}$. Thus:
$$\begin{array}{ll}
F_{m\mu}&=F_{m\mu}^{(1)}+F_{m\mu}^{(2)}=
\\& \frac{R_a^2}{\epsilon_0}\int_{\frac{\pi}{2}}^\pi\int_0^{2\pi}\cos \theta \sin \theta
{\sigma_m^{(1)} \sigma_\mu^{(1)}} d\theta d\varphi+\\ &
\frac{R^2}{\epsilon_0}\int_{\frac{\pi}{2}}^{\theta_M}\int_0^{2\pi}\cos
\theta \sin \theta \sigma_m^{(2)}\sigma_\mu^{(2)} d\theta
d\varphi\end{array}$$
$\sigma_m^{(1)}$ and $\sigma_m^{(2)}$ are the surface charge densities on areas (1) and (2) within the frame of problem A. The calculation of $F_{m\mu}^{(1)}$ yields:
$$\label{EQU_F_MICMES_1}
\begin{array}{r}
F_{m\mu}^{(1)}=\frac{3\widetilde{\epsilon}_dqR_a^2}{a'^2R}V_b \cosh[\widetilde{\delta}^\bot(V_b)] e^{-\frac{2\pi}{a'}(z_\mu+R_a)}\times \\
D^{(1)}\left[\cos\left(\widetilde{x}_0\right)+\cos\left(\widetilde{y}_0\right)\right]
\end{array}$$
The integral form of $D^{(1)}$ is reported in the appendix. The numerical integration results in $D^{(1)} \simeq
-15$. Similar integration over area (2) gives a similar expression of $F_{m\mu}^{(2)}$ by replacing indexes (1) with indexes (2) and $R_a$ with $R$. The integral form of the coefficient $D^{(2)}$ is similar to the one of $D^{(1)}$, except that the term $\cos^2
\theta$ is replaced by $\cos \theta$. The numerical integration of $D^{(2)}$ yields almost zero. This was expected since the integration is performed over the oscillations of the charge density $\sigma_\mu^{(2)}$ due to the image charges of the surface potential (*cf.* fig.\[FIG\_SIGMA\_MU\]). On the opposite, this was not observed in $F_\mu^{(2)}$ because the integration was performed on the square of $\sigma_\mu^{(2)}$. Therefore $F_{m\mu}$ is finally written:
$$F_{m\mu}=F_{m\mu}^{(1)}$$
Hence, the former equation states that the coupling of the force to the bias $V_b$ actually occurs only by means of the microscopic surface charge density at the foremost part of the tip, *i.e.* on the asperity. Owing to the integration over the mesoscopic part of the tip apex and the subsequent cancellation due to the oscillations of $\sigma_\mu^{(2)}$, no relevant coupling between the bias voltage and the mesoscopic part of the tip apex can occur. This aspect strongly suggests that the effect we are reporting is mainly controlled by the foremost structure of the tip. The effect is expected to be much enhanced in case of tips with sharper geometries, particularly those with apexes including atomically sharp edges.
The expression for the vertical contribution of the total force acting on the tip due to the combined influence of the capacitive coupling and of the Madelung surface potential is finally written: $F=F_{m\mu}+F_\mu=F_{m\mu}^{(1)}+F_{\mu}^{(1)}+F_{\mu}^{(2)}$, with $F_{m\mu}^{(1)}$ given by equ.\[EQU\_F\_MICMES\_1\], $F_{\mu}^{(1)}$ by equ.\[EQU\_F\_MIC\_1\] and $F_{\mu}^{(2)}$ by equ.\[EQU\_F\_MIC\_2\]. The graph of the total force and of its components is reported in fig.\[FIG\_FORCE\] for similar parameters than previously, namely: $a=0.66$ nm, $\delta^\bot=11$ pm and $z_\mu=4$ Å. A typical value of bias has been set, $V_b=+1$ V. Thus, it is visible that the term $F_\mu^{(1)}$ is negligible compared to others. This is due to the prefactor $R_a^2/a'^4$ *vs.* $R^2/a'^4$ for $F_\mu^{(2)}$. The total electrostatic force $F$ finally simplifies to:
$$\label{EQU_TOTAL_FORCE}
F=F_{m\mu}^{(1)}+F_{\mu}^{(2)}$$
In fig.\[FIG\_FORCE\], the force reaches an average value of about 9 pN (absolute value) and a corrugation of about 2 pN (peak to peak).
![(Color online). a- Total electrostatic force over a unit cell calculated from $F=F_{m\mu}^{(1)}+F_{\mu}^{(1)}+F_{\mu}^{(2)}$ for $a=0.66$ nm, $\delta^\bot=11$ pm, $z_\mu=4$ Å and $V_b=+1$ V. The unit cell depicted with a dotted line is centered on a cation. The vertical contrast ranges from -10 (blued spots) to -8 pN (reddish spots). The force is more repulsive on top of cations (central ion) than on top of anions, consistently with the bias polarity. b- $z_\mu$ dependence of the electrostatic force on top of an anion (dotted curve) and on top of a cation (continuous curve). c- Cross section along the dotted line shown in a- showing the total force (thick continuous line), $F_{m\mu}^{(1)}$ (dashed line), $F_{\mu}^{(1)}$ (dotted line) and $F_{\mu}^{(2)}$ (greyed line). $F_\mu^{(1)}$ is negligible compared to the others.[]{data-label="FIG_FORCE"}](FIG05.eps "fig:"){width="\columnwidth"}\
IMPLICATIONS FOR KPFM {#SEC_DISCUSSION}
=====================
Estimation of $\delta^\bot(V_b)$
--------------------------------
The net displacement $\delta^\bot(V_b)$ of the topmost ionic layer induced by the polarization can be estimated out of the electric field $\overrightarrow{E}$ between the tip and the surface (*cf.* equ.\[EQU\_POLARIZATION\]). Section \[SEC\_PROBLEM\_A\] has shown that the field induced by the bias voltage was essentially controlled by the mesoscopic radius of the tip, $R$, and not by the foremost asperity. Therefore, owing to equs.\[EQU\_SIGMA\_M2\] or \[EQU\_SIGMA\_M1\]: $E \simeq V_b/R$. The expression of $\delta^\bot(V_b)$ can now be deduced using equ.\[EQU\_POLARIZATION\]:
$$\label{EQU_DELTA_PERP}
\delta^\bot(V_b)=\frac{\chi_d}{q}E=V_b\frac{\chi_d}{qR}$$
An order of magnitude for $\delta^\bot(V_b)$ can now be calculated as follows: with $V_b \simeq 1$ V, $R=5$ nm, $\alpha_i=70 \times 10^{-30}$ m$^3$ (ref.\[\]) and $n_v=8\sqrt{2}/a^3\simeq 40 \times 10^{27}$ m$^{-3}$ (number of polarizable ionic species *per* volume unit in a fcc crystal of KBr with a lattice constant $a=0.66$ nm), we get $E=2\times 10^8$ V.m$^{-1}$, $\chi_d \simeq 9 \times
10^{-39}$ F.m$^2$ and therefore $\delta^\bot \simeq 11$ pm.
Detected signal in KPFM: connection with the local CPD {#SEC_KPFM}
------------------------------------------------------
When performing KPFM experiments, the bias voltage $V_b$ is modulated with a frequency $f_k$ and may as well include a static component to compensate for the long-range electrostatic forces. This is the reason why the average value of the experimental CPD image shown in fig.\[FIG\_EXP\]a reaches -3.9 V. It is not rare that, on ionic surfaces, many volts are required to compensate for the long-range electrostatic forces due to trapped charges while the cleavage of the cristal [@barth06a]. Thus:
$$\label{EQU_Vb}
V_b=V_{dc}+V_{ac}\sin(2\pi f_k t)$$
The electrostatic force is thus triggered at $f_k$ and then detected as an additional low- or high- frequency component when doing FM- or AM-KPFM, respectively. In both techniques, a proper dc bias voltage produced by an external controller, hereafter referred to as $V_{dc}^{(c)}$, is applied between the tip and the counter-electrode to cancel the modulated component at $f_k$, *i.e.* the oscillation amplitude of the second bending eigenmode of the cantilever in AM-KPFM, or the one of the frequency shift in FM-KPFM. When applied to the tip (*i.e.* with the counter-electrode grounded), this dc bias is the opposite of the local CPD defined in equ.\[EQU\_LOCAL\_CPD\]: $V_{dc}^{(c)}=-V_\text{cpd}$.
In order to stick to the AM-KPFM experiments (fig.\[FIG\_EXP\]), it is necessary to estimate the amplitude of the second eigenmode of the cantilever and then derive a condition on the dc value of the bias ultimately able to nullify it. On the one hand, it is therefore mandatory to check carefully all the occurrences of the modulated component of the bias voltage in the expression of the force. This includes explicit dependencies, such as those due to the polarization, but also implicit ones, as discussed hereafter. On the other hand, an expression of the oscillation amplitude of the mode modulated at $f_k$ must be derived.
We first address the problem of explicit and implicit bias dependencies in the expression of the electrostatic force. Owing to the explicit $V_b$ dependence in $\delta^\bot$, which has been kept throughout the description of the model, it can be seen that the polarization effect is mainly included in the $\cosh$ function of $F_{m\mu}^{(1)}$ and $F_{\mu}^{(2)}$ through a linear and a quadratic dependence, respectively (*cf.* equs.\[EQU\_F\_MICMES\_1\] and \[EQU\_F\_MIC\_2\]). Furthermore, since $\delta^\bot$ is small compared to $a'$, $\cosh[\widetilde{\delta}^\bot(V_b)]$ can be expanded in series. To first order:
$$\cosh[\widetilde{\delta}^\bot(V_b)] \simeq 1+
\widetilde{\delta}^\bot(V_b)^2=1+\frac{4\pi^2}{a'^2}\delta^\bot(V_b)^2=1+(\chi'_dV_b)^2$$
with $\chi'_d=2\pi\chi_d/(a'qR)$. Replacing this expansion in the expressions of the components of the force and keeping the linear and quadratic terms in $V_b$ yields, with compact notations:
$$F_{m\mu}^{(1)}=\widetilde{\epsilon}_dK_{m\mu}^{(1)}\Phi_{m\mu}^{(1)}\frac{qV_b}{R}$$
and:
$$F_{\mu}^{(2)}=K_{\mu}^{(2)}\Phi_{\mu}^{(2)}[1+2(\chi'_dV_b)^2]\frac{q^2}{\epsilon_0
a'^2},$$
where $K_{m\mu}^{(1)}$ and $K_{\mu}^{(2)}$ are two dimensionless coefficients standing for geometrical factors of areas (1) and (2), respectively:
$$K_{m\mu}^{(1)}=\frac{3R_a^2}{a'^2}D^{(1)} \hspace{0.5cm}
\text{and} \hspace{0.5cm} K_{\mu}^{(2)}=\frac{R^2}{2a'^2}A^{(2)}$$
$\Phi_{m\mu}^{(1)}$, $\Phi_{\mu}^{(2)}$ are also two dimensionless coefficients carrying the spatial dependence of each force component:
$$\label{EQU_PHI_MMU1}
\Phi_{m\mu}^{(1)}=e^{-\frac{2\pi}{a'}(z_\mu+R_a)}\left[\cos\left(\widetilde{x}_0\right)+\cos\left(\widetilde{y}_0\right)\right]$$
and:
$$\Phi_{\mu}^{(2)}=e^{-\frac{4\pi}{a'}(z_\mu+R_a)}$$
Implicit $V_b$ dependencies are now discussed. In the two former equations, particular attention must be paid to $z_\mu$. So far, this parameter was defined as the tip-surface distance and set to an arbitrary, constant, value. However, when dealing with AM-KPFM, $z_\mu$ is not static but actually coupled to the bias and to the oscillation amplitude of the fundamental eigenmode of the cantilever. In the following, for the sake of clarity, the variables related to the fundamental eigenmode of the cantilever will be denoted with index “0" and those of the second eigenmode with index “1". Thus, let $z_0$, $z_1$ and $D$ be the instantaneous position of the fundamental eigenmode of the cantilever, the instantaneous position of the second eigenmode of the cantilever and the distance between the surface and the equilibrium position of the cantilever at rest, respectively. Therefore, $z_\mu(t)=D-z_0(t)-z_1(t)$. Hence, if $V_b$ has the form given in equ.\[EQU\_Vb\], one can postulate, to first order: $z_1(t)=A_1\sin(2\pi f_k
t + \varphi_1)$. $A_1$ and $\varphi_1$ stand for the oscillation amplitude of this mode and its phase lag with respect to the electrostatic actuation, respectively. Their exact expressions are not easily derivable, but it must be noticed that $A_1$ must be connected to the amplitude of the modulation, namely $V_{ac}$. When $f_k$ accurately matches the actual resonance frequency of the second eigenmode[@Note_Nony07i], then $\varphi_1=-\pi/2$. $z_0$ is experimentally driven by the control electronics of the microscope at the actual resonance frequency of the fundamental mode of the cantilever, $f_0$. It is known that it has an almost harmonic behavior of the form: $z_0(t)=A_0\sin(2\pi f_0t-\pi/2)$. Thus, $\Phi_{m\mu}^{(1)}$ and $\Phi_{\mu}^{(2)}$ have a component at $f_0$ which is further modulated by the dynamics of the second eigenmode, electrostatically actuated at $f_k$.
We can now propose a self-consistent approximated solution to the equation of motion for $z_1(t)$ and thus derive the expression of the oscillation amplitude $A_1$. This equation has the standard form:
$$\label{EQU_DIFF_Z1}
\ddot{ z_1 }(t) +\frac{\omega_1}{Q_1}\dot{z_1}(t) + \omega_1^2
z_1(t)=\frac{F_{ext}}{m_1}+\frac{F_{m\mu}^{(1)}+F_\mu^{(2)}}{m_1},$$
where $F_{ext}$ is an external force oscillating at $f_0$ which controls the dynamics of $z_0(t)$. Let us assume: i- $A_1 \ll A_0$, *i.e.* the amplitude of the mode is much smaller than the one of the fundamental mode, ii- $A_1 \ll a'$, where $a'=a\sqrt{2}/2$, $a$ being the lattice constant of the crystal and iii- that the dynamics of $z_1(t)$ is mainly influenced by components at $f_k$. Assumptions i- and ii- are not too strong, since the experimental estimations of $A_1$ yield a few tens of picometers. Assumption i- implies that the dynamics of the fundamental mode is not much influenced by the one of the second eigenmode. Hence, the solution of the equation of motion of $z_0(t)$ has indeed the form postulated above. Assumption ii- allows us to linearize $\Phi_{m\mu}^{(1)}$ and $\Phi_{\mu}^{(2)}$ with respect to $z_1(t)$. Finally, assumption iii-, which is consistent with the postulated solution for $z_1(t)$, $z_1(t)=A_1\sin(2\pi f_k t+\varphi_1)$, simplifies further the above equation of motion. Now, owing to the former assumptions, equ.\[EQU\_DIFF\_Z1\] can be solved by injecting the postulated expressions of both eigenmodes and keeping only the terms oscillating at $f_k$. For that purpose, the exponential term wherein $z_0(t)$ occurs must be expanded in Fourier series. Then, the only possibility to preserve terms at $f_k$ is to keep the lone static component of the Fourier expansion, hereafter referred to as $a_0$ (expansion of $\Phi_{m\mu}^{(1)}$) and $b_0$ (exansion of $\Phi_\mu^{(2)}$). After linearization, $\Phi_{m\mu}^{(1)}$ and $\Phi_{\mu}^{(2)}$ can finally be written as:
$$\label{EQU_TOT_PHIMMU1}
\begin{array}{r}
\Phi_{m\mu}^{(1)}\overbrace{\simeq}^\text{i-, ii-}
\left[1-\frac{2\pi}{a'}z_1(t)
\right]e^{-\frac{2\pi}{a'}(D-A_0+R_a)}\times
\\
\{a_0+\underbrace{\sum_{n=1}^{\infty}a_n\cos(2\pi n f_0
t)}_{\text{neglected, owing to
iii-}}\}\left[\cos\left(\widetilde{x}_0\right)+\cos\left(\widetilde{y}_0\right)\right]
\end{array}$$
and:
$$\label{EQU_TOT_PHIMU2}
\Phi_\mu^{(2)}\simeq \left[1-\frac{4\pi}{a'}z_1(t)
\right]e^{-\frac{4\pi}{a'}(D-A_0+R_a)}
\{b_0+\underbrace{\sum_{n=1}^{\infty}b_n\cos(2\pi n f_0
t)}_{\text{neglected, owing to iii-}}\}$$
With:
$$\begin{array}{l}
a_n=e^{-\frac{2\pi}{a'}A_0}I\left[n,{\frac{2\pi}{a'}A_0}\right]\\
b_n=e^{-\frac{4\pi}{a'}A_0}I\left[n,{\frac{4\pi}{a'}A_0}\right]
\end{array}$$
$I$ is the modified Bessel function of the first kind. When replacing equs.\[EQU\_TOT\_PHIMMU1\] and \[EQU\_TOT\_PHIMU2\] with $z_1(t)=A_1\sin(2\pi f_kt+\varphi_1)$ in equ.\[EQU\_DIFF\_Z1\], it is possible to derive an expression for $A_1$. The condition on $V_{dc}^{(c)}$ to match $A_1=0$ is finally written:
$$\label{EQU_COMPENSATED_CPD}
V_{dc}^{(c)}=-\frac{{\widetilde{\epsilon}}_d \epsilon_0 a'^2 }{4R
q
{\chi'_d}^2}\frac{a_0}{b_0}\frac{K_{m\mu}^{(1)}}{K_\mu^{(2)}}e^{\frac{2\pi}{a'}
(D+R_a-A_0)}[\cos(\widetilde{x_0})+\cos(\widetilde{y_0})]$$
The graph of $V_{dc}^{(c)}$ is reported in fig.\[FIG\_COMPENSATED\_CPD\] for $a=0.66$ nm, $\delta^\bot=11$ pm, $A_0=5$ nm (hence $a_0=0.0487$ and $b_0=0.0344$) and $(D-A_0)=3.5$ Å. The value of $A_0$ has been chosen consistently with the experimental conditions. On the contrary, the tip-surface distance has been chosen arbitrary, but however in a range where the atomic contrast is usually experimentally achieved. In fig.\[FIG\_COMPENSATED\_CPD\], the lateral periodicity of the underlying lattice is readily visible, but surprisingly, the potential scales between -0.6 to +0.6 V from an anionic to a cationic site, respectively. At similar height, this is three times larger than the Madelung surface potential (*cf.* fig.\[FIG\_POT\]b). The comparison with the experimental results is more severe since the theoretical prediction is one order of magnitude larger. At this point, it is recalled that the strong tip geometry dependence of the problem makes a straightforward comparison between the theoretical prediction (equ.\[EQU\_COMPENSATED\_CPD\]) and the experimental results difficult, since our analytical expression of $V_{dc}^{(c)}=-V_{cpd}$ relies on a somewhat unrealistic tip.
Experimental implications for AM-KPFM
-------------------------------------
The figure and the above formula show that the bias voltage to be applied on the tip to compensate for the electrostatic force is governed by three main factors:
- the dielectric properties of the sample such as its dielectric permittivity and lattice constant.
- a subtle balance between mesoscopic and microscopic geometric factors of the tip.
- a lateral periodicity similar to the Madelung surface potential of the crystal.
A straightforward consequence is that the atomic corrugation of the CPD reported experimentally might stand for the spatial fluctuations of the Madelung surface potential, however with an amplitude that depends on the surface polarization and hence on the applied ac voltage. It is also important to notice that the CPD compensation is proportional to $K_{m\mu}^{(1)}$, *i.e.* to the asperity size. The former being also the source of the coupling between the tip/dieletric/counter-electrode capacitor and the Madelung surface potential, *i.e.* the source of the KPFM signal, the atomic contrast of the CPD is therefore closely connected with the geometry of the very foremost part of the tip. This is consistent with the short-range character of the interaction. But, on the other hand, the explicit dependence with geometric factors of the tip, unambiguously proves that quantitative measurements of the local CPD are unlikely to be performed in KPFM, unless the tip geometry be accurately known, which is practically never true.
Furthermore, although equ.\[EQU\_COMPENSATED\_CPD\] explicitly exhibit a distance dependence, consistently with the experimental observations in the above mentioned references, an increase of the compensated CPD as a function of the distance is nevertheless surprising. The residual exponential dependence originates from $\Phi_{\mu}^{(2)}$, *i.e.* from the influence of the Madelung surface potential on the mesoscopic tip apex. As discussed above, the CPD compensation being partly governed by the asperity, a tip-surface distance increase $\Delta z_\mu$ produces a decrease of the related force $F_{m\mu}^{(1)}$ proportional to $\exp{(-2\pi
\Delta z_\mu/a')}$ (*cf.* $\Phi_{m\mu}^{(1)}$, equs.\[EQU\_PHI\_MMU1\] or \[EQU\_TOT\_PHIMMU1\]). This abrupt change is compensated by an equivalent exponential increase of the compensated CPD. This process can obviously not occur at any tip-surface distance. Prior to being cancelled, the $f_k$ component must stand for a measurable signal. Therefore the above discussion stands within a narrow range of distances from the surface, typically a few times the asperity radius.
Nevertheless, an increase of the measured local CPD as a function of the distance is still expected to occur if the distance dependence in $\Phi_\mu^{(2)}$ decays faster than the one in $\Phi_{m\mu}^{(1)}$. In other words $V_{dc}^{(c)}$ must increase with the distance as soon as the distance dependence of the force induced by the influence of the Madelung surface potential on the mesoscopic part of the tip decays faster than the one of the force induced on the asperity due to the capacitive coupling with the image charges of the surface.
Let us finally point out that such a distance dependence of the CPD might make the experimental achievement of the atomic contrast easier, which strengthens the argument of an intrinsic imaging process of the local CPD. Indeed, no major tip and/or surface distortion is expected to occur in an equivalent range of distance ($\geq 3$ Å, ref.\[\]). In that case, instabilities due to adsorbed and/or mobile atomic or ionic species at the tip apex are less likely to occur, which makes the imaging process robust, as experimentally observed.
To conclude, the analytical approach, although restricted to a tip with a basic geometry, remains helpful, primarily because it provides an expression of the short-range electrostatic force that can be connected to the nc-AFM-KPFM simulator. There are obvious limitations to our approach, the most important one being the use of classical, continuous electrostatics to treat the angstrom-size nanoasperity. This obviously must break down at a certain point, and be replaced by a proper quantum mechanical treatment of the problem. In the near future, the electrostatic model should be extended to a bit more complex systems such as local dipoles, charges or defects at the surface and at steps of ionic crystals.
![(Color online). a- $V_{dc}^{(c)}$ bias voltage required to compensate the local CPD calculated from equ.\[EQU\_COMPENSATED\_CPD\] for $a=0.66$ nm, $\delta^{\bot}=11$ pm, $A_0=5$ nm and $z_\mu=3.5$ Å. The vertical contrast ranges from -0.6 (blued spots) to +0.6 V (reddish spots). The unit cell depicted with a dotted line is centered on a cation. b- Distance dependence of the potential on top of an anion (dashed curve) and on top of a cation (continuous curve). c- Cross section along the dotted line in a-.[]{data-label="FIG_COMPENSATED_CPD"}](FIG06.eps "fig:"){width="\columnwidth"}\
CONCLUSION
==========
The aim of this work was to provide a consistent approach to describe the short-range electrostatic force between the tip of an nc-AFM and the (001) surface of a perfect ionic crystal. In order to develop an analytical expression for the total electrostatic force, the tip has been restricted to a simple geometry and the influence of the sample has been described by means of its Madelung surface potential. In such a way, an analytical solution for the total electrostatic force was found within the boundary-value problem assuming a thick dielectric sample and an infinite top-layer of ionic surface charges.
Two major contributions to the electrostatic force can be extracted: the first stands for a coupling term between the microscopic structure of the tip apex and the capacitor formed between the tip, the dielectric ionic crystal and the counter-electrode due to the bias voltage $V_b$; the second term depicts the influence of the fluctuations of the Madelung surface potential arising at the surface of the ionic crystal on the mesoscopic part of the tip, independently from its microscopic structure. The former has the lateral periodicity of the Madelung surface potential whereas the latter only acts as a static component which shifts the total force.
Beyond the dielectric properties of the crystal, which are explicitly included in the model, the ionic polarization of the sample due to the influence of the bias voltage applied to the tip/counter-electrode capacitor is mainly responsible for the atomic contrast of the KPFM signal. Typical orders of magnitude give a net displacement of the ions of about $\pm$10 pm from their equilibrium positions. Note that this displacement only occurs if a tip-sample bias is applied (ac or dc), which is always the case in KPFM experiments.
A detailed analysis of the bias voltage required to compensate for the electrostatic force shows that the compensated CPD has the lateral periodicity of the Madelung surface potential. However, there is a strong dependence on the tip geometry, the applied modulation voltage as well as the tip-sample distance, which can even lead to an overestimation of the real surface potential.
For a quantitative evaluation of KPFM results, it is thus essential to account for all the parameters of the experiment, among which the tip shape. The analytical expression developed in this work provides an alternative tool to elucidate the contrast formation in KPFM on ionic crystals, and in combination with the nc-AFM simulator it might enable us to interpret our results more accurately.
APPENDIX {#appendix .unnumbered}
========
After integration of $F_{\mu}^{(1)}$ over the azimuthal angle $\varphi \in[0;2\pi]$ , we have:
$$\begin{array}{r}
F_{\mu}^{(1)}=\frac{q^2R_a^2}{2\epsilon_0
a'^4}\cosh^2[\widetilde{\delta}^\bot(V_b)]e^{-\frac{4\pi}{a'}(z_\mu+R_a)}\left\{
A^{(1)}+\right.\\
\left.B^{(1)}\left[\cos\left(\widetilde{x}_0\right)+\cos\left(\widetilde{y}_0\right)\right]+2C^{(1)}\cos\left(\widetilde{x}_0\right)\cos\left(\widetilde{y}_0\right)\right\}\end{array}$$
where:
$$A^{(1)}= \pi\int_{\frac{\pi}{2}}^{\pi}\cos\theta \sin \theta
[2\mathcal{F}^{{(1)}^2}(\theta)+\mathcal{G}^{{(1)}^2}(\theta)]d\theta$$
$$\begin{array}{r}
B^{(1)}=\pi\int_{\frac{\pi}{2}}^{\pi}\cos\theta \sin \theta
\left\{\left(\mathcal{F}^{{(1)}^2}(\theta)-\frac{\mathcal{G}^{{(1)}^2}(\theta)}{2}\right)J_0\left(\eta_\theta \right) -\right.\\
\left.2\mathcal{F}^{(1)}(\theta)\mathcal{G}^{(1)}(\theta)J_1\left(\eta_\theta
\right)+\frac{\mathcal{G}^{{(1)}^2}(\theta)}{2}J_2\left(\eta_\theta
\right) \right\}d\theta\end{array}$$
and:
$$\begin{array}{r}
C^{(1)}=\pi\int_{\frac{\pi}{2}}^{\pi}\cos\theta \sin \theta
\left\{2\mathcal{F}^{{(1)}^2}(\theta)J_0\left(\eta'_\theta \right) -\right.\\
\left.2\sqrt{2}\mathcal{F}^{(1)}(\theta)\mathcal{G}^{(1)}(\theta)J_1\left(\eta'_\theta \right)+\right.\\
\left.\mathcal{G}^{{(1)}^2}(\theta)J_2\left(\eta'_\theta \right)
\right\}d\theta\end{array}$$
with: $\eta_\theta=4\pi R_a\sin \theta /a'$ and $\eta'_\theta=2\sqrt{2}\pi R_a\sin \theta /a'$. $J_0$, $J_1$ and $J_2$ are the Bessel functions of the first kind.
Regarding $F_{m\mu}^{(1)}$, the integration over the azimuthal angle $\varphi \in[0;2\pi]$ yields:
$$\begin{array}{r}
F_{m\mu}^{(1)}=\frac{3\widetilde{\epsilon}_dqR_a^2}{a'^2R}\cosh[\delta_\bot(V_b)]V_b D^{(1)}\left[\cos\left(\frac{2\pi}{a'}x_0\right)+\right.\\
\left.\cos\left(\frac{2\pi}{a'}y_0\right)\right]
\end{array}$$
where:
$$\begin{array}{r}
D^{(1)}=-2\pi\int_{\frac{\pi}{2}}^\pi \cos^2 \theta \sin \theta
\left\{\mathcal{F}^{(1)}(\theta)J_0\left(\frac{\eta_\theta}{2}\right)-\right.\\
\left.\mathcal{G}^{(1)}(\theta)J_1\left(\frac{\eta_\theta}{2}\right)
\right\} d\theta \end{array}$$
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The ionic polarizability $\alpha_i$ is connected to the dielectric permittivity $\epsilon_d$ by Claussius-Mossotti’s equation.
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This is also true because it has been assumed that the ionic crystal carried no net charge. Therefore there is no long-range electrostatic interaction.
The latter is now obtained by anti-symmetrical transform of the layer of ionic charges into the asperity.
The expression of $\sigma_\mu^{(2)}(r>R_a,\theta=\pi/2, \varphi )$ for an infinite planar geometry is: $\sigma_\mu^{(2)}=\frac{q}{a'^2}\cosh[\delta^\bot(V_b)]\{4\widetilde{\chi}(R%
,\pi/2,\varphi)-4\frac{R^3}{r^3}\widetilde{\chi}(R^2/r,\pi/2,\varphi)\}$.
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As a matter of fact, $h_d$ being very large, it can readily be verified that this result is always true whatever is the exact shape of the mesoscopic part of the tip apex.
It must be noticed that the resonance frequency of that mode might be shifted owing to the influence of the overall attractive force, *i.e.* including Van der Waals, short-range and electrostatic components, similarly to the resonance frequency of the fundamental mode.
In ref.\[\], the foremost unstable atom of the tip which is responsible for the contrast in dissipation is triggerred at a distance of about 3 Å from the surface. Above that threshold, no instability is observed.
[^1]: To whom correspondence should be addressed; E-mail: [email protected].
|
---
author:
- 'Jonathan Tammo Siebert, Florian Dittrich, Friederike Schmid, Kurt Binder, Thomas Speck, and Peter Virnau'
bibliography:
- 'references.bib'
title: |
[Supplementary Information]{}\
Critical behavior of active Brownian particles
---
Model
=====
The dynamics of active Brownian Particles are governed by the coupled equations of motion $$\dot{\mathbf r}_k = -\nabla_kU + \frac{{\mathrm{Pe}}}{d_{\mathrm{BH}}/\sigma}\left(
\begin{array}{c}
\cos{\varphi}_k \\ \sin{\varphi}_k
\end{array}\right) + \sqrt{2}\mathbf R_k$$ and $$\dot{{\varphi}}_k = \sqrt{2 D_{\mathrm{r}}} T_k$$ with Peclet number ${\mathrm{Pe}}=(3v_0)/(d_{\mathrm{BH}}D_{\mathrm{r}})$ and independent and normal distributed Gaussian noises $\mathbf R_k$ and $T_k$. The interactions $$U(\{\mathbf{r}_i\}) = \sum_{i<j}u_\text{WCA}(|\mathbf{r}_i-\mathbf{r}_j|)$$ are modeled via the strongly repulsive Weeks-Chandler-Anderson pair potential [@Weeks:1971] $$u_{\text{WCA}}(r) = \begin{cases}4\epsilon \left(r^{-12}-r^{-6}+\frac{1}{4}\right) & r < 2^{1/6} \\
0 & r \ge 2^{1/6}.
\end{cases}
\label{eq:potential}$$ As units of length and time we choose $\sigma$ and $\sigma^2/D$ with bare diffusion coefficient $D$, respectively. The interaction strength $\epsilon=100$ is chosen to only allow for small overlaps. The resulting effective diameter [@Barker:1967] is $d_{\mathrm{BH}}\approx 1.10688$. The rotational diffusion constant is given by $D_{\mathrm{r}}=3/d_{\mathrm{BH}}^2$. The equations are then integrated by an Euler scheme except for the data on the larger boxes that were used to determine the static structure factor, which were integrated using a higher order predictor-corrector scheme [@Kloeden:1999].
Improved block-distribution method
==================================
As the overexpression of interfaces seems to be the main problem in the original scheme, we try to include only subboxes that do not contain an interface. By simulating an elongated box with side length ratio $1:3$, we force the system into a slab geometry, exploiting the stability of interfaces in finite systems. We then place four subboxes with side lengths $\ell$ in the system. To ensure a proper sampling of both phases but also the exclusion of the interface region we place two subboxes aligned parallel to the slab in the middle of each, the dense and the dilute region. The dense boxes are positioned around the center of mass in $x$ direction [@Bai:2008]. The dilute boxes are then shifted by $3\ell$.
As a proof of concept, we test this method using the example of the two dimensional Ising model. In contrast to the original approach for which the curves corresponding to different system sizes do not cross at all (Fig. 1 a) of the manuscript), the new method is indeed able to predict the critical point with remarkable accuracy. Fig. 1 b) of the manuscript shows $Q_{\ell}(T)$ with subbox magnetization $m$ for subbox sizes of $\ell = 8, 10, 12, 15$. Below the critical point, i.e. in the phase separated region, the $Q_{\ell}(T)$-curves are ordered going from large values for large subboxes to small values for small subboxes. At the critical temperature (which is indicated by the dashed vertical line) the curves cross and at even higher temperatures, i.e., in the homogeneous region, they invert their order. This shows that our new method indeed allows to accurately predict the critical temperature in the two dimensional Ising model, circumventing the main problems of the original block-magnetization-distribution method. Excluding the interface region, the measured distribution is much closer to the true grand canonical distribution. Furthermore, simulating different sized systems for the different subbox sizes allows to eliminate the additional scaling variable of box length over subbox length occurring in the original block-distribution method.
In addition to the determination of the critical point, data from the analysis can be used to estimate critical exponents (cf. Fig. 3) of the manuscript). Fig. \[fig:exponentsIsing\] demonstrates that the analysis is working nicely in the case of the 2D Ising model, for which the slopes in the log-log-plot reproduce the exponents well within the uncertainties of the data.
![a) Log-log-plot of the susceptibility *vs.* the subbox size. The slope corresponds to ${\gamma}/\nu$ (cf. Fig. 3b) of the manuscript). The dashed line shows corresponds to the literature value, fitting perfectly to the data’s slope. b) Log-log-plot of $Q_{\ell}$’s slope against the subbox size. The derivative $\mathrm{d}Q_{\ell}/\mathrm{d}\tau\big|_{\tau\simeq0}$ is determined by fitting a linear function to $Q_{\ell}(\tau)$ in the critical region. The slope of the curve in the log-log-plot corresponds to $1/\nu$ [@sm] (cf. Fig. 3c) of the manuscript). 2D-Ising ($\nu=1$) universality is shown for reference as dashed lines, fitting to the data’s slope within its uncertainties.[]{data-label="fig:exponentsIsing"}](exponentsIsing)
Qualitative justification for crossing of cumulants
===================================================
A crucial aspect of this subsystem method is that in the regime of the ordered phase two of the four $\ell \times \ell$ subboxes are centered in the middle of the “liquid” domain, which has the linear dimension $(2\ell) \times (3\ell)$ (Fig. 1 of the manuscript). This constraint must be maintained not only at low temperatures but also in the critical region, noting that (in the grandcanonical ensemble) the order parameter distribution in $d=2$ dimensions still has a pronounced double-peak structure, throughout the critical region, and the two peaks merge in a single peak above $T_c$ when the correlation length $\xi$ has become much smaller than the linear dimension $2\ell$ of the system. For this reason, it is useful to choose the coordinate system such that the origin coincides with the center of mass of the system (the Ising model is then considered as lattice gas, particles have mass unity). The geometry then looks as shown in Figure 1 a) of the manuscript.
For systems in the Ising universality class, critical correlations grow isotropically. Thus, the growth of the correlation length $\xi$ is limited by the smaller linear dimension $2\ell$ in the chosen geometry. The average distance of the liquid-vapor interfaces from the boundaries of the measurement boxes is $\ell$, and when $\xi$ is distinctly smaller than $2\ell$, the typical length scale $w$ of interfacial fluctuations in the $d=2$ Ising model is, from capillary wave theory and exact solutions [@Abraham:1986; @Binder:1995]
$$\label{eq1}
w \propto \sqrt{2\ell\xi}$$
where the prefactor is of the order unity. Eq. (\[eq1\]) implies that for $\xi \ll 2\ell$ the “measurements” of the density in the subboxes cannot be affected by interfacial fluctuations, and it is reasonable to assume that the fluctuations in the liquid subboxes are independent from the fluctuations in the vapor subboxes. Of course, the fluctuations in the two liquid subboxes are not independent of each other, because they interact across their boundaries in $y$ directions twice (each subbox has the same upper and lower neighbor subbox, because of the periodic boundary conditions in $y$-direction).
However, in $x$-direction there are intervening $2\ell \times 2\ell$ regions on both sides of the subboxes (half filled by liquid regions, half by vapor) which act as “particle reservoirs” for the particle number fluctuations in the “measurement boxes”. This situation is not identical to the grandcanonical ensemble of statistical mechanics, but qualitatively similar (in the grandcanonical ensemble, one considers a subbox which can exchange particles with an infinitely large reservoir at the same average density [@Landau:1958a]). Thus, when $2\ell \gg \xi$, one can in principle divide each $\ell \times \ell$ subbox into many weakly interacting subsystems to conclude that the distribution of density in the subbox must be (approximately) Gaussian, due to the law of large number [@Landau:1958a; @Binder:1981; @Rovere:1993; @Roman:1997a; @Roman:1998; @Sengupta:2000]
$$\label{eq2}
p^{\rm liqiud}_{\rm subbox} (\rho) \propto \exp \{-(\rho-\rho^{\rm coex}_\ell)^2 \, \ell^2/(2 k_BT \chi^\ell_{\rm eff}) \}\text{,}$$
$$\label{eq3}
p^{\rm vapor}_{\rm subbox} (\rho) \propto \exp \{-(\rho-\rho^{\rm coex}_v)^2 \, \ell^2/(2 k_BT \chi^\upsilon_{\rm eff}) \}\text{.}$$
Taking the average of all 4 subboxes yields the desired double-peak distribution, analogous to the grandcanonical ensemble.
For a lattice gas, we have a symmetry for the “susceptibilities” $\chi^\ell_{\rm eff}= \chi^\upsilon_{\rm eff}$, while no such symmetry is expected for off-lattice fluids, of course. However, it can be asserted that for the chosen geometry these effective susceptibilities typically will be smaller than their counterparts in the grandcanonical ensemble, but still of the same order of magnitude. The fact that they are smaller can be concluded from the expectation that the constraints of conserved total density in the system (we work here at an average density $\rho=\rho_{\rm crit}$, $\rho_{\rm crit}=(\rho^{\rm coex}_v+ \rho^{\rm coex}_\ell)/2=1/2$ in the lattice gas) removes some fluctuations, which still are possible in the grandcanonical ensemble. The expectation that $\chi^\ell_{\rm eff}$, $\chi^v_{\rm eff}$ are of the same order as their grandcanonical counterpart $\chi^\ell$, $\chi^v$ can be justified from the explicit computation of these quantities for $\ell \times \ell$ subsystems of $L \times L$ homogeneous systems in the canonical ensemble [@Roman:1997a; @Roman:1998; @Sengupta:2000] for various fluids. Due to the presence of two interfaces in our system, however, one cannot take over any of the results in the literature [@Roman:1997a; @Roman:1998; @Sengupta:2000] to the present case quantitatively.
When Eqs. (\[eq2\], \[eq3\]) hold, it follows immediately [@Binder:1981] that the cumulant of the density distribution (with respect to the average density $\rho_{\rm crit}$) converges to 1 for $T<T_c$ when $\ell \rightarrow \infty$, i.e. with $\Delta \rho=\rho-\rho_{\rm crit}$ we have $$\label{eq5}
Q_{\ell} =\frac{\langle (\Delta \rho)^2 \rangle^2_{\ell}}{\langle (\Delta \rho)^4 \rangle_{\ell}} \quad {\underset{\ell \rightarrow \infty}{\longrightarrow}} 1 \quad .$$ Note that this result does not hold for subboxes which contain interfaces, and hence when one fails to exclude those [@Rovere:1993] the finite size analysis of the density distribution no longer is straightforward.
Of course, when one wishes to study critical phenomena, one is not only interested in the region $T < T_c$, but one wishes to pass through the critical region well up into the region of the weakly correlated disordered phase. For $T \gg T_c$, when one still fixes the boxes which were related to the liquid (for $T \leq T_c$) at the origin, there will nevertheless be no longer any significant difference in the density distribution of any of the boxes; the $6 \ell \times 2\ell$ system has density inhomogeneities only on scales $\xi\ll 2\ell$ at $T \gg T_c$). So averaging over all 4 subboxes again follows a Gaussian distribution, but now centered at the average density $\rho_{\rm crit}$,
$$\label{eq6}
p_{\rm subbox} (\rho) \propto \exp \{-(\rho-\rho_{\rm crit})^2 \ell^2/(k_BT \chi_{\rm eff})\}$$
and hence in this region the cumulant $Q_{\ell} \rightarrow 1/3$ as $\ell \rightarrow \infty.$ Of course, for the region near $T_c$ one can postulate (which includes additional assumptions about scaling relations at this point) the same finite size scaling hypothesis as proposed in [@Rovere:1993], again referring to an average over all 4 subboxes\
$$\label{eq7}
p_{\rm subbox} (\rho)=\ell^{\beta/\nu} \tilde{p}\{(\rho-\rho_{\rm crit}) \ell^{\beta/\nu}, \, \ell^{1/\nu} \tau \} \quad,$$ where $\tau=1-T/T_c$, $\beta$ and $\nu$ are the critical exponents of the Ising model, and $\tilde{p}$ is a scaling function, similar - but not identical - to the scaling function that applies in the grand-canonical ensemble.
Eq. (\[eq7\]) is essentially identical to the proposal given in [@Rovere:1993], for subsystems of a canonical ensemble. However, in this paper an average over all subsystems contained in the total system was taken, not conceiving that one needs to distinguish between subsystems for the “measurements” and subsystems containing the interfaces, and moreover acting as particle reservoirs. Thus, although Eq. (\[eq7\]) was proposed earlier [@Rovere:1993], this proposal really referred to a physically different situation, and also its usefulness could NOT be demonstrated previously, for systems in the canonical ensemble of statistical mechanics.
Note that Eq. (\[eq7\]) leads to the standard expressions for the moments, for instance $$\label{eq8}
\langle (\Delta \rho)^{2k} \rangle= \ell^{-2k \beta/\nu} f_{2k} (\ell^{1/\nu} \tau)$$ where $f_{2k}$ is a scaling function whose explicit form is not needed, and hence $$\label{eq9}
Q_{\ell}=\tilde{Q}(\ell^{1/\nu} \tau) \quad,$$ in the regime where $\ell \gg 1$ and $\xi \gg 1$, and $\ell$, $\xi$ are about of the same order ($\xi$ would become infinite for $\tau=0$, for a macroscopic system, of course). Near $\tau=0$ the scaling function $\tilde{Q}$ can be expanded as a Taylor series\
$$\label{eq10}
Q_{\ell} =\tilde{Q} (0) + \tilde{Q}' \ell^{1/\nu} \tau + \cdots ;$$ the constant $\tilde{Q}(0)$ is similar (but not identical) to the corresponding constant of the grand canonical ensemble. Note that from Eqs. (\[eq8\]), (\[eq10\]) both exponents $\beta/\nu$ and $1/\nu$ can be estimated, since
$$\label{eq11}
\langle (\Delta \rho)^2 \rangle \propto \ell^{-2 \beta/\nu} \quad, \quad d Q_{\ell}/d\tau\big|_{\tau=0} \, \propto \ell^{1/\nu} \quad .$$
Eqs. (\[eq11\]) are identical to their counterparts in the grandcanonical ensemble, but for subsystems taken from a canonical ensemble their practical usefulness has not been shown earlier.
Results of the original subsystem method
========================================
![Original subsystem method:a) Dependence of $Q_{\ell}$ in subsystems of different sizes on the overall density at a propulsion strength in the homogeneous region ${\mathrm{Pe}}=33.21$. The curves exhibit maxima at increasing densities for increasing subsystem size. As suggested in [@Luijten:2002; @Kim:2003; @Kim:2003a] we evaluate $Q_{\ell}$ for all subsystems along their locus of maximum value (as indicated by the crosses).b) Crossing of $Q_{\ell,\mathrm{max}}({\mathrm{Pe}})$ when going from the homogeneous to the phase separated region. Generally, the results agree with those of our modified subsystem method presented in the manuscript. Note, however, a more pronounced spread of intersections starting at ${\mathrm{Pe}}\approx 34$ for small subsystems and going up to ${\mathrm{Pe}}\approx 41$ for the larger subsystems.[]{data-label="fig:oldMethod"}](oldMethod.pdf)
We also compared our estimate for the critical point of ABPs to the results of the original subsystem method [@Binder:1981; @Binder:1987; @Rovere:1988; @Rovere:1990; @Rovere:1993], in which a quadratic box of side length $L$ is subdivided into $N^2$ subsystems by a regular grid of spacing $\ell=L/N$. Each of these strongly correlated subsystems is then treated as a quasi-grandcanonical system which allows to determine $Q_{\ell}$. In contrast to the modified method used in our manuscript, the medium density averaged over all subsystems is fixed. Measuring $Q_{\ell}$ at different overall densities yields maxima as seen in Figure \[fig:oldMethod\] a) for a system in the homogeneous region. Analyzing $Q_{\ell}$ along their loci of maximum value $Q_{\ell,\mathrm{max}}({\mathrm{Pe}})$ [@Luijten:2002; @Kim:2003; @Kim:2003a] yields intersections similar to those of the modified method presented in the manuscript. Due to a more pronounced spread of intersections, the uncertainty is much larger though. As shown in Figure \[fig:oldMethod\] b) these intersection move from ${\mathrm{Pe}}\approx 34$ up to ${\mathrm{Pe}}\approx 41$ starting with smaller and going to larger subsystems. Possible causes for this could be the over expression of interfaces due to the phase separation as well as the additional scaling variable $N$, which were already mentioned in the manuscript. Note that, as is common in subsystem distribution methods [@Binder:1981; @Binder:1987; @Rovere:1988; @Rovere:1990; @Rovere:1993; @Watanabe:2012; @Trefz:2017], only a small range of subsystem sizes can be used and the selection process is somewhat empirical. The critical density can be estimated by the position of the locus of large subsystems at the critical propulsion strength. This estimate is rather rough, as the resolution of density points used is low. Despite the comparably large uncertainties and the other shortcomings of the method, its results corroborate the estimate of the critical point in the manuscript in both propulsion strength and density.
Static structure factor
=======================
We also studied the low-$q$ limit of the static structure factor measured in a quadratic box of side length $L=130$ at a constant particle number corresponding to $\phi\simeq\phi{_\text{cr}}$. For sufficiently large systems it is expected to follow a Lorentzian [@Fisher:1964] (see also Ref. [@Fily:2012] for $\eta=0$): $$S(q)=\frac{S_0}{1+(\xi q)^{2-\eta}}\text{,}
\label{eq:lorentzian}$$ that can be fitted to the data to get an estimate of the correlation length $\xi$. In this case $S_0$ is a free fit parameter, $\eta$ is the (constant) anomalous dimension that determines the power law scaling $S(q)\propto q^{-2+\eta}$ for intermediary $q$ values. Note, though, that this power law is only valid for $\frac{2\pi}{\ell}\ll q \ll\frac{2\pi}{D}$, where $D$ is the typical distance between nearest neighbor particles at the chosen density. In addition one must be in the regime $\frac{2\pi}{\xi}\ll q$. Figure \[fig:differentEta\]a)-c) show the structure factor data, comparing it to power law slopes indicated by dashed lines. By naively fitting the low-$q$ limit of the data to Eq. (\[eq:lorentzian\]) assuming an anomalous dimension corresponding to the maximum slope, one finds that only $\eta<0$ is able to reproduce the measured $S(q)$ reasonably well. However, a negative anomalous dimension is not physical, as it would imply a divergence of order parameter auto-correlations with increasing distance.
It turns out that an naive extraction of anomalous dimensions and correlation lengths from the static structure factor in this finite system with constant particle number is not successful. It seems that the system size $L=130a$ studied in this work is still not sufficient to allow a reasonable extraction of the power law behavior $S(q)\propto q^{-2+\eta}$ for intermediary $q$ values. Instead, neglecting important finite size effects is giving unphysical results.
|
---
abstract: 'We present first results from electronic Multi-Element Remotely Linked Interferometer Network (e-MERLIN) and electronic European VLBI Network (e-EVN) observations of a small sample of ultra-steep spectrum radio sources, defined as those sources with a spectral index $\alpha < -1.4$ between 74MHz and 325MHz, which are unresolved on arcsecond scales. Such sources are currently poorly understood and a number of theories as to their origin have been proposed in the literature. The new observations described here have resulted in the first detection of two of these sources at milliarcsecond scales and show that a significant fraction of ultra-steep spectrum sources may have compact structures which can only be studied at the high resolution available with very long baseline interferometry (VLBI).'
author:
- |
M. K. Argo$^{1}$, Z. Paragi$^{2}$, H. Röttgering$^{3}$, H.-R. Klöckner$^{4}$, G. Miley$^{3}$, M. Mahmud$^{5}$\
$^{1}$ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA, Dwingeloo, The Netherlands\
$^{2}$Joint Institute for VLBI in Europe (JIVE), Postbus 2, 7990 AA, Dwingeloo, The Netherlands\
$^{3}$Leiden Observatory, Universiteit Leiden, Niels Bohrweg 2, NL-2333 CA Leiden, The Netherlands\
$^{4}$Max-Planck-Institute für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany\
$^{5}$Formerly at JIVE\
bibliography:
- 'refs.bib'
date: Revision January 2013
title: 'Probing the nature of compact ultra-steep spectrum radio sources with the e-EVN and e-MERLIN'
---
\[firstpage\]
radio continuum: galaxies – galaxies: active
Introduction
============
Radio sources with an ultra steep radio spectrum (USS; defined here as having $\alpha < -1.4$ where S$_{\nu} \propto \nu^{\alpha}$) are often associated with extreme phenomena in the universe. USS sources with a classical FRII type radio morphology (highly collimated jets and well-defined lobes with prominent hotspots) are, for the most part, associated with very distant galaxies ($2 < z < 5.2$; @breuck00 [@rott97]). Being very massive and often located in proto-clusters, these sources can be used to study the origin and evolution of massive galaxies and clusters. Recent high-redshift results include (i) the finding that massive radio galaxies with masses of around $10^{12}$M$_{\odot}$ exist at up to $z \sim 5$ [@seymour07], and (ii) the discovery of proto-clusters up to $z \sim 4$ with total masses as expected for progenitors of nearby clusters (a few $10^{14}$M$_{\odot}$; @venemans07). USS sources with mostly diffuse radio emission are virtually always associated with close-by clusters up to $z \sim 0.5$ [@weeren09] and the emitting regions can have spectacular sizes up of to a few Mpc. The most likely explanation for this class of object is radio emitting plasma tracing shocks in merging clusters. The study of these two types of USS source started in the late seventies with studies of their morphology, and follow-up work with large optical and X-ray telescopes subsequently gave an understanding of their enormous importance for studies of galaxy and cluster evolution.
A third class of USS has not yet received much attention: those which are unresolved on arcsecond scales. The aim of the observations presented here is to take a first step towards determining the nature of these objects. There are a great number of possible scenarios for the origin of such compact USS objects including (i) radio galaxies located at or near the epoch of re-ionisation (it has been suggested that objects with the steepest spectral index correspond to the highest redshift objects, e.g. @krolik91), (ii) young obscured radio galaxies with the early phase perhaps accompanied by a starburst (@reuland04 inferred star formation rates of $\sim$1000M$_{\odot}$/year), (iii) steep spectrum core quasars with the steepening perhaps caused by cooling of the plasma via scattering of infrared photons from the torus by the synchrotron-emitting electrons, and (iv) pulsars (@breuck00 found a clear over density of sources with $\alpha<-1.6$ near the Galactic plane). Long baseline radio interferometric observations may give very important clues about the nature of these objects. Star formation activity in galaxies is expected to produce low brightness temperature emission ($\sim10^4$ K, see @alexandroff12 and references therein) and diffuse morphology. Synchrotron emission from radio lobes on kiloparsec scales can be very well imaged with e-MERLIN, but will still be mostly resolved out on milliarcsecond scales. Finally, steep spectrum quasars and pulsars will be compact when probed with VLBI; the potential Galactic origin of the source can then be verified by multi-epoch proper motion measurements. This letter describes e-MERLIN and e-EVN observations of a sample of compact ultra-steep spectrum sources, aimed at attempting to investigate their morphologies and narrow down the possibilities. Section \[SecObs\] describes the sample selection and observations, the results are discussed in section \[SecResults\] and in section \[SecDiscussion\] we draw some conclusions.
\[USSsample\]
Name —– RA —– —– Dec —– $\alpha^{\rm 74\,MHz}_{\rm 325\,MHz}$ ${S^{\rm NVSS}_{\rm 1400\,MHz}} \over {\rm S}^{\rm FIRST}_{\rm 1400\,MHz}$ $S^{\rm VLSS}_{\rm 74\,MHz}$ $S^{\rm WENSS}_{\rm 325\,MHz}$ $S^{\rm NVSS}_{\rm 1400\,MHz}$ $S^{\rm FIRST}_{\rm 1400\,MHz}$
---------------- -------------- -------------- --------------------------------------- ---------------------------------------------------------------------------- ------------------------------ -------------------------------- -------------------------------- ---------------------------------
J072212+291042 07:22:12.603 +29:10:41.51 $-$1.45 1.1 650 76 11 10
J082916+383453 08:29:17.351 +38:34:52.62 $-$1.44 0.9 940 112 11 12
J130612+514407 13:06:12.176 +51:44:06.93 $-$1.58 1.1 1640 159 27 25
J131655+483200 13:16:55.642 +48:32:00.13 $-$1.52 1.1 2390 252 67 61
J151229+470245 15:12:29.174 +47:02:45.59 $-$0.50 1.0 1130 543 217 221
Sample selection and observations {#SecObs}
=================================
The initial sample of USS sources was found by correlating the 74MHz VLSS catalogue [@cohen07] against the 325MHz WENSS database [@rengelink97] and selecting the small fraction of VLSS sources with a sufficiently steep spectrum. A sub-sample of compact sources was then selected by choosing objects which were both unresolved in the FIRST survey [@becker95], and had a flux ratio between the NVSS [@condon98] and FIRST catalogues of $0.8 < R < 1.2$, ensuring that the 1.4GHz emission is indeed compact on scales of $\sim$1arcsecond. The FIRST and NVSS maps were also visually inspected to ensure that the sources were unresolved and that there were no nearby objects which could have led to ambiguities in the measured fluxes. The resulting sample contained 17 sources. A sub-sample of five sources was selected for an exploratory VLBI survey, all of these sources had NVSS flux densities larger than 10mJy (see Table \[USSsample\]).
e-EVN and WSRT observations
---------------------------
The five sources (J072212+291042, J082916+383453, J130612+514407, J131655+483200 and J151229+470245) were observed with the EVN at 1.6 GHz in e-VLBI mode [@szomoru08] over two 10-hour runs carried out on 10$^{\rm th}$ and 11$^{\rm th}$ June 2010 (programmes EP070A and B; PI Röttgering). Ten stations participated in the experiment: Effelsberg, Medicina, Onsala, Torun, Westerbork (12 antennas tied array), Lovell, Cambridge, Darnhall, Knockin and Sheshan. The target sources were phase-referenced to nearby calibrators; the typical on-source time was 2.5 hours. The total aggregate bitrate per telescope was 1024Mb/s, except for the MERLIN antennas which participated with a lower rate of 128 Mbps. Both LCP and RCP polarizations were observed with 2-bit sampling. The data were processed using standard procedures following the EVN Data Analysis Guide[^1].
During the e-EVN observations, synthesis array data were also recorded at Westerbork and were independently processed in AIPS. The data were recorded using 8 bands of 20MHz, each split into 64 channels. For amplitude calibration we used 3C286. There was significant interference during the observation resulting in the loss of the first two subbands.
Out of the five objects from the sample which were observed with the e-EVN, J072212+291042, J130612+514407 and J151229+470245 were detected. Note that initially J130612+514407 was not detected because of a 12.8-arcsecond error in the pointing position; a clear detection of the source was obtained only after this error was discovered and a wide-field image was made. Table \[detections\] summarises the detections of these sources. Since, for the sources which were detected, the e-EVN observations recovered only $\sim$20% of the VLA flux at the same frequency, e-MERLIN commissioning observations were requested in order to investigate their structures on intermediate scales. Comments on each of these sources are given in the next section.
e-MERLIN observations
---------------------
e-MERLIN observations of two of the EVN-detected sources were requested in order to investigate their structures on intermediate scales. The e-MERLIN observations were carried out in March 2011, during the commissioning period. At this point in the commissioning phase, five stations of the array were available with the new C-band (4- to 6-GHz) system: Mk2, Tabley, Darnhall, Knockin and Defford. Observations were carried out at 6.6GHz using the new capabilities of the upgraded e-MERLIN array[^2] and used four adjacent bands of 128MHz, each split into 512 channels, starting at 6.64GHz with a central frequency of 6.89GHz. In both cases 3C286 was used as the primary flux calibrator and OQ208 was used as the bandpass calibrator. Both sources were observed for $\sim$4.5 hours on 26th March and $\sim$10 hours on March 30th 2011. J151229+470245 used the source 1500+478 for phase referencing, J072212+291042 used 0736+299 as the phase calibrator. Of the total observing time, only $\sim$9 hours on each source were ultimately usable due to a variety of commissioning issues. The data were flagged spectrally, averaged in frequency, fringe fitted and then calibrated using standard methods for phase referencing experiments.
Results {#SecResults}
=======
J072212+291042
--------------
With a flux density of 10mJy in FIRST and 11mJy in the NVSS (Table \[detections\]), J072212+291042 is the weakest of the three VLBI detections. The e-EVN observations showed a $\sim$2mJy source with no clear evidence of structure on VLBI scales, consistent with a point source model, albeit at significantly lower flux density than that detected by the VLA and Westerbork: only $\sim$30% of the Westerbork flux is recovered by the e-EVN at the same frequency indicating the presence of significant structure on intermediate scales.
The source is not detected with e-MERLIN at 6.9GHz to a 3$\sigma$ limit of 0.12mJy/beam. Comparing the 3-$\sigma$ limit at 6.9GHz with the flux detected by the e-EVN at 1.4GHz gives a limit on the spectral index of $\alpha < -1.4$, showing that, in this case, the spectral index remains steep at higher frequencies. A search of the catalogues in the NASA/IPAC Extragalactic Database (NED)[^3] shows that J072212+291042 has no known counterparts in other wavebands. The compact nature of this object suggests that the most likely explanation is either a steep spectrum quasar core or a Galactic pulsar. Although this source has a low galactic latitude (see Table \[detections\]) a search of pulsar catalogues around this location shows no currently known pulsar or pulsar candidate at these coordinates (Eatough, private communication).
J130612+514407
--------------
Of the detected sources, J130612+514407 has the steepest spectral index at low frequencies. Without an observation at 6.8GHz, however, it is impossible to tell if the spectrum remains steep at higher frequencies - the positional offset in the e-EVN observations was only discovered after the e-MERLIN observations had taken place. The e-EVN data show an unresolved point source with an integrated flux density of 7.1mJy, recovering $\sim30\%$ of the emission detected by the Westerbork array during the same observation, implying structure on intermediate scales at 1.4GHz. However, the primary beam of the Westerbork tied-array is significantly smaller than the 12.8" offset of the source position from the pointing centre of the e-EVN observation; imaging the source without including the Westerbork array results in a peak flux density of 18.5 mJy and an integrated intensity of 21.6 mJy, recovering almost all of the WSRT-only flux, although it should be noted that these measurements are somewhat uncertain due to the large positional offset.
Unlike J072212+291042, J130612+514407 has an optical counterpart in SDSS, J130612.15+514407.0, a diffuse-looking galaxy with an SDSS spectrum giving a redshift of 0.2773. The source also has a counterpart in the JVAS/CLASS sample with an integrated flux density at 8.4GHz of 16.2mJy. If the source does not vary, then this results in a spectral index of $-$0.31 between 1.4 (NVSS) and 8.4GHz making it slightly steep, but certainly not ultra-steep at these frequencies. However, the CLASS observations were carried out with the VLA in its largest A configuration resulting in an angular resolution of 220 milli-arcseconds (mas), very different to the beam size of the NVSS survey, making this spectral index measurement unreliable. This object also has counterparts in 2MASS and the ROSAT X-ray catalogue, and is classified as the brightest cluster galaxy in [@koester07], making the quasar core scenario the most likely.
J151229+470245
--------------
In contrast, J151229+470245 is clearly resolved in both the e-EVN and e-MERLIN images (see Fig. \[fig1512\]). The WSRT-only data show an unresolved source, as expected from the NVSS/FIRST images, with a peak flux of 192mJy/beam and an integrated flux density of 203mJy. This is within 8% of the flux recorded in the FIRST and NVSS catalogues. The e-EVN observation shows a source elongated in a NE-SW direction, with a peak brightness to total flux density ratio of 1:4.
The map obtained with e-MERLIN at 6.9GHz (Fig. \[fig1512\]) shows a very different structure. The source is clearly resolved and appears to show a core-jet morphology. The e-EVN component is coincident with the brightest feature observed with e-MERLIN. A two-component fit was carried out on the e-MERLIN map using the AIPS task [jmfit]{}: a relatively compact (compared to the e-MERLIN beam), bright Gaussian with a peak of 39.5mJy/beam, an integrated flux of 66.1mJy and a size of 98$\times$73 mas at a position angle of 71 degrees, and a more extended Gaussian “jet" component with a fitted peak brightness of 4.14mJy/beam, an integrated flux density of 28.3mJy with a size of 0.2$\times$0.12 arcseconds at a position angle of 145 degrees.
The nature of this source is rather puzzling. Interestingly, while the e-MERLIN map (taken on its own) appears to show a classic core-jet morphology, the e-EVN (“core") component is very resolved suggesting that it is not a simple AGN core. Unless there is very strong scatter broadening of the target, the observed structure is more consistent with lobe emission detected on 10-mas scales. The observed steep spectrum, low brightness temperature and no variability (between VLA and WSRT epochs) all support this scenario. Therefore we may have identified a “naked jet-lobe” source without strong AGN core emission. The absence of a compact, self-absorbed core is a sign of a Type 2 AGN, when the jet is viewed at a large angle compared to our line of sight. It is intriguing though, why such an unbeamed object would have a prominent one-sided jet morphology. An alternative interpretation is that we are observing a two-sided jet-lobe with a very weak core within the structure.
On inspection of the WENSS map, we found that J151229+470245 is a complex source at the resolution of Westerbork at 325MHz. An examination of the map in comparison with the survey database, the corresponding fields in both NVSS and FIRST, and the new Westerbork data obtained during the e-EVN observations, determined that the fluxes listed in the WENSS catalogue do not refer to the source components at the listed coordinates. The calibrated fits file for the WENSS field was retrieved and Gaussian components were fitted to the three sources in the region. This showed that the source in which we are interested actually had an integrated flux of 543mJy, not 123mJy as listed in the WENSS catalogue. This gives the source a spectral index of -0.50 between 74 and 325MHz, making it steep spectrum, but not ultra-steep. The spectrum is slightly steeper between WENSS and FIRST/NVSS with $\alpha$ = -0.62 between 325MHz and 1.4GHz. Therefore, based on its spectrum and the observed VLBI and e-MERLIN radio structure we may classify J151229+470245 as a peculiar compact steep spectrum radio source (CSS).
This source is also an example of the class of radio sources known as Infrared Faint Radio Sources (IFRS), those with clear radio emission but no corresponding infrared detection [e.g. @norris06]. Such sources are difficult to identify due to their lack of counterparts outside the radio regime and, as yet, no IFRS source has a measured redshift, but they are thought to be high redshift quasars due to their generally compact nature, mostly steep spectral indexes and high brightness temperatures. While some of the known IFRS sources are extended on arcsecond scales with the VLA, and a small number of such sources have previously been detected at VLBI-resolution (e.g. @norris07 [@middelberg08]), this appears to be the only example so far which is resolved on VLBI scales.
Name RA Dec ($l$,$b$) $S^{\rm WSRT}_{1670 \rm MHz}$ $S^{\rm{e\textrm{-}EVN}}_{1670 \rm MHz}$ $S^{\rm{e\textrm{-}MERLIN}}_{6890 \rm MHz}$ $\alpha^{1400\,\rm MHz}_{1670 \, \rm MHz}$ $S^{\rm{e\textrm{-}EVN}}_{1670 \rm MHz} \over S^{\rm WSRT}_{1670 \rm MHz}$
---------------- ---------------- ---------------- ----------- ------------------------------- ------------------------------------------ --------------------------------------------- --------------------------------------------- ----------------------------------------------------------------------------
J072212+291042 07:22:12.60906 +29:10:41.6648 189, 19 7.6 2.1 $<$0.12 $-1.82$ 0.28
J130612+514407 13:06:12.16262 +51:44:06.9627 118, 65 23.1 21.6 $-$ $-1.28$ 0.94
J151229+470245 15:12:29.19905 +47:02:45.3837 78, 56 203 42.0 94.0 $-0.51$ 0.21
![\[fig1512\]e-MERLIN map of J151229+470245 at 6.9GHz (blue contours) together with the e-EVN map at 1.4GHz (red contours). Contours are plotted at -1,1,2,4,8,16,32,64,128 $\times$ 0.23mJy/beam for the e-MERLIN data and -1,1,2,4,8,16 $\times$ 0.36mJy/beam for the e-EVN data. The boxes in the lower left indicate the size of the restoring beam for each image.](1512_MERLIN+EVNkntr.ps){width="8cm"}
Discussion and conclusions {#SecDiscussion}
==========================
Of the five sources in the sample, one, J151229+470245, is not an ultra steep spectrum source. Of the remaining four, two are undetected with VLBI, two are detected but are unresolved on milli-arcsecond scales. These new high resolution data are not sufficient to uniquely determine the nature of these sources, however they do provide valuable information on the variety of sources one may find in a larger USS sample. As one may expect, a significant fraction of these sources are completely resolved out since steep spectrum emission is often related to very extended lobes in radio galaxies. A detection rate of fifty per cent, however, indicates that very high resolution studies of a moderate fraction of USS sources will be possible.
Both of our USS source detections, J072212+291042 and J130612+514407, are consistent with a point source on VLBI scales. Note that in these short, exploratory observations with limited $uv$-coverage (especially on the long baselines) we cannot adequately constrain the source sizes, but they do appear more compact than our beam of 32$\times$26 mas. The emission in these two cases is, therefore, consistent with either a Galactic pulsar or a very steep spectrum quasar core. The former hypothesis will be easily tested by further VLBI observations that can potentially show the proper motion of the pulsar in our Galaxy.
Since there are generally no detections of these sources in any other wavelength regime and no redshift information is available, they are very difficult to classify. These new radio observations provide fresh information and vital clues in the search for an understanding of their nature.
J072212+291042 is compact on VLBI scales but the e-EVN observations presented here recover less than 20% of the flux seen at the same frequency on VLA scales, clearly indicating the existence of structure on intermediate spatial scales to which neither of these arrays are sensitive. e-MERLIN observations at 6.9GHz fail to detect the source to a 3-$\sigma$ limit of 120$\mu$Jy/beam, confirming the steep spectrum extends to higher frequencies. Observations with e-MERLIN at 1.4GHz would probe this intermediate-scale emission, but this band was not available at the time when the commissioning observations described here were obtained. The fact that only $\sim$20 per cent of the FIRST/NVSS flux is recovered on VLBI scales strongly suggests the existence of structure on intermediate scales for these sources. A study by [@kloeckner09] of a sample of 11 high-redshift obscured quasars also found similar evidence for structures on intermediate scales, although the VLBI-recovered fluxes there ranged from 30 to 100 per cent of the low-resolution measurements.
J130612+514407 is also compact on VLBI scales with significantly less flux lost when imaged with long baselines. Unlike J072212+291042, however, this source does have counterparts in other wavebands which suggest the source is associated with a galaxy at a redshift of 0.2773.
While J151229+470245 appears to be a compact steep spectrum object, rather than an ultra steep spectrum source, it is still compact with a steep spectrum at low frequencies and interesting structure at higher resolutions. VLBI observations of J151229+470245 only recover 20% of the NVSS flux. In sharp contrast to J072212+291042 and J130612+514407 however, the object is clearly resolved on both e-MERLIN and VLBI scales, showing different structures in the two maps. While the e-MERLIN map shows what appears to be a classic core-jet morphology, the “core" is clearly resolved in the e-EVN map. Observations with e-MERLIN at L-band (1.4GHz) would again allow us to probe the morphology of the missing flux.
While firm conclusions regarding the nature of these sources remain elusive, the observations and unexpected results presented here do show that compact ultra-steep spectrum sources are a diverse class of object whose nature can be probed with VLBI.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Ralph Eatough for useful discussions. e-MERLIN is a National Facility operated by the University of Manchester at Jodrell Bank Observatory on behalf of STFC. We thank the e-MERLIN operations team for data taken during commissioning time. The EVN is a joint facility of European, Chinese, South African and other radio astronomy institutes funded by their national research councils. e-VLBI research infrastructure in Europe is supported by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement RI-261525 NEXPReS. The WSRT is operated by ASTRON (Netherlands Institute for Radio Astronomy) with support from the Netherlands foundation for Scientific Research. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23. This research also made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
\[lastpage\]
[^1]: http:/$\!$/www.evlbi.org/user\_guide/guide/userguide.html
[^2]: See http://www.e-merlin.ac.uk/tech/ for the current capabilities of the e-MERLIN array.
[^3]: http://ned.ipac.caltech.edu/
|
---
author:
- 'Artur Palha, Pedro Pinto Rebelo and Marc Gerritsma'
bibliography:
- './AdvectionICOSHAM2012Submit\_reviewed.bib'
title: Mimetic Spectral Element Advection
---
INTRODUCTION {#Section::Introduction}
============
Consider the classical advection problem for a scalar function in conservation form, $$\frac{\partial \rho}{\partial t} + \nabla\cdot\left(\vec{v}\rho\right) = 0,
\label{eq::scalar_advection_diffusion}$$ where $\vec{v}$ is a prescribed uniformly Lipschitz continuous vector field and $\rho$ the advected scalar function. The method presented in this work is based on the approximation of the differential operators with the focus on the spatial discretization and a time-stepping scheme that distinguishes between quantities evaluated at time instants and quantities evaluated over time intervals. The mimetic framework, presented in [@kreeft2011mimetic], showed that using the differential geometric approach for the representation of physical laws clarifies the underlying structures. One clearly identifies to what kind of geometrical object a certain physical quantity is associated and this determines how its discretization must be done (e.g.: evaluation at points, integration over lines, surfaces or volumes). Additionally, a well defined, metric free, representation of differential operators is obtained, together with their metric dependent Hilbert adjoints. For these reasons, the authors followed this approach for the advection equation. It is known, see [@abraham_diff_geom pp. 317], that is a particular case of the generalized advection equation which can be written in terms of differential geometry as, $$\frac{\partial{\alpha^{(k)}}}{\partial t} + \mathcal{L}_{\vec{v}}\, {\alpha^{(k)}} = 0 \label{eq::testCases_advection_diffusion}.$$ The advection operator, $\mathcal{L}_{\vec{v}}$, is the Lie derivative for the prescribed velocity field $\vec{v}$ and the advected quantity is given by the $k$-differential form ${\alpha^{(k)}}$. Depending on the index $k$ the quantity $\alpha^{(k)}$ can represent scalar, vector and higher dimensional quantities.
DIFFERENTIAL GEOMETRY {#Section::DifferentialGeometry}
=====================
In this section a brief introduction to differential geometry is given. For a more detailed introduction the reader is directed to [@abraham_diff_geom]. Given an $n$-dimensional smooth orientable manifold $\Omega$ it is possible to define in each point a tangent vector space $E$ of dimension $n$. The space of smooth vector fields on a manifold is the space, $\Gamma$, of smooth assignments of elements of $E$ to each point of the manifold. We denote by $\Lambda^{k}$, $k$ an integer $0 \leq k \leq n$, the space of differentiable $k$-forms, i.e. all smooth $k$-linear, antisymmetric maps ${\omega^{(k)}}: E \times \cdots \times E \rightarrow {\mathbb{R}}$, at every point of the manifold. We recall the wedge product $\wedge : \Lambda^{k} \times \Lambda^{l} \rightarrow \Lambda^{k+l}$ for $k+l \leq n$ with the property that ${\alpha^{(k)}} \wedge {\beta^{(l)}} = (-1)^{kl} {\beta^{(l)}} \wedge {\alpha^{(k)}}$. The inner product ${\left( \cdot, \cdot\right)}$ on $E$ induces at each point of the manifold an inner product ${\left( \cdot, \cdot\right)}$ on $\Lambda^{1}$. In turn, this can be extended to a local inner product on $\Lambda^{k}$, [@morita::geometry_diff_forms pp. 149]. The local inner product gives rise to a unique metric operator, Hodge-$\star$, $\star: \Lambda^{k} \rightarrow \Lambda^{n-k}$, defined by ${\alpha^{(k)}} \wedge \star {\beta^{(k)}} = {\left({\alpha^{(k)}},{\beta^{(k)}}\right)_{}}\omega^{(n)}$, where $\omega^{(n)}=\star 1$ is the standard volume form. By integration, one can define an inner product on $\Omega$ as ${\left( \cdot, \cdot\right)}_{L^{2}}:=\int_{\Omega}{\left( \cdot, \cdot\right)} \omega^{(n)}$. The exterior derivative ${\mathrm{d}}: \Lambda^{k} \rightarrow \Lambda^{k+1}$ satisfies the following rule, ${\mathrm{d}}\left({\alpha^{(k)}} \wedge {\beta^{(l)}} \right) = {\mathrm{d}}{\alpha^{(k)}} \wedge {\beta^{(l)}} +(-1)^k {\alpha^{(k)}} \wedge {\mathrm{d}}{\beta^{(l)}}$ and by definition ${\mathrm{d}}\alpha^{(n)} = 0$. The flat operator, $\flat$, is a mapping $\flat:\Gamma\mapsto\Lambda^{1}$.
The Lie derivative along a tangent vector field, $\vec{v}$, is denoted by $\mathcal{L}_{\vec{v}}$ and represents the advection operator in differential geometry. It is a mapping $\mathcal{L}_{\vec{v}}:\Lambda^{k}\mapsto\Lambda^{k}$. From Cartan’s homotopy formula the Lie derivative can be written as $$\mathcal{L}_{\vec{v}}\, {\alpha^{(k)}} ():= {\mathrm{d}}\iota_{\vec{v}}{\alpha^{(k)}} + \iota_{\vec{v}} {\mathrm{d}}{\alpha^{(k)}}\;,$$
where the interior product of a tangent vector field, $\vec{v}$, with a $k$-form, ${\alpha^{(k)}}$, is a mapping $\iota_{\vec{v}}{\alpha^{(k)}}: \Lambda^{k} \rightarrow \Lambda^{k-1}$ given by: $$\iota_{\vec{v}}{\alpha^{(k)}}(\vec{X}_{2},\cdots,\vec{X}_{k}) := {\alpha^{(k)}}\left(\vec{v},\vec{X}_{2}, \cdots,\vec{X}_{k} \right),\quad \forall \vec{X}_{i}\in\Gamma\quad\mathrm{and}\quad\iota_{\vec{v}}{\alpha^{(0)}}=0,\quad\forall\vec{v}\in\Gamma\;.$$
The interior product is the adjoint of the wedge product, made explicit by: $${\left( \iota_{\vec{v}}\,{\alpha^{(k)}}, {\beta^{(k-1)}}\right)}_{L^{2}\Lambda^{k-1}} = {\left( {\alpha^{(k)}}, \vec{v}^{\flat}\wedge{\beta^{(k-1)}}\right)}_{L^{2}\Lambda^{k}},\quad\forall{\beta^{(k-1)}}\in\Lambda^{k-1}\label{eq::adjoint_interior_product}$$ where $\vec{v}^{\,\flat}={\nu^{(1)}}\in\Lambda^{1}$ and ${\alpha^{(k)}}_{h}\in \Lambda^{k}$.
The relevance of this adjoint relation between the interior product and the wedge product lies in the fact that it shows how a physical quantity represented by an interior product with a vector field can be represented by its dual differential 1-form.
For a volume form ${\rho^{(n)}}$ the Lie derivative is simply $\mathcal{L}_{\vec{v}}\, {\rho^{(n)}} = {\mathrm{d}}\iota_{\vec{v}}{\rho^{(n)}}$ and for a $0$-form, $\mathcal{L}_{\vec{v}}\, {\alpha^{(0)}} = \iota_{\vec{v}} {\mathrm{d}}{\alpha^{(0)}}$.
MIMETIC DISCRETIZATION {#Section::Discretization}
======================
In this section a brief introduction to the discretization of physical quantities and to the discretization of the exterior derivative is presented. For a more detailed presentation the reader is directed to [@kreeft2011mimetic; @palhaLaplaceDualGrid; @gerritsma::geometricBasis]
Consider a three dimensional domain $\Omega$ and an associated grid consisting of a collection of points, $\tau_{(0),i}$, line segments connecting the points, $\tau_{(1),i}$, surfaces bounded by these line segments, $\tau_{(2),i}$, and volumes bounded by these surfaces, $\tau_{(3),i}$.
Let $\Lambda^{k}$ be the space of smooth differentiable $k$-forms. Additionally, let the finite dimensional space of differentiable forms be defined as $\Lambda^{k}_{h}=\mathrm{span}(\{{\epsilon_{i}^{(k)}}\}), \,\, i=1,\cdots,\dim(\Lambda^{k}_{h})$, where ${\epsilon_{i}^{(k)}}\in \Lambda^{k}$ are basis $k$-forms. Under these conditions it is possible, see [@kreeft2011mimetic; @palhaLaplaceDualGrid], to define a projection operator $\pi_{h}$ which projects elements of $\Lambda^{k}$ onto elements of $\Lambda^{k}_{h}$ which satisfies: $$\pi_{h}{\mathrm{d}}= {\mathrm{d}}\pi_{h}\;. \label{eq:commuting_projection}$$ It is possible to write: $$\pi_{h} {\alpha^{(k)}} = {\alpha_{h}^{(k)}}=\sum_{i}\alpha_{i}{\epsilon_{i}^{(k)}}\;,$$ where $$\alpha_{i} = \int_{\tau_{(k),i}}{\alpha^{(k)}} \quad\mathrm{and}\quad \int_{\tau_{(k),i}}{\epsilon_{j}^{(k)}} = \delta_{ij},\quad k=0,1,\cdots,n\;.$$
A set of basis functions yielding a projection operator $\pi_{h}$ that satisfies can be constructed using piecewise polynomial expansions on the quadrilateral elements using tensor products. Thus, it suffices to derive the basis forms in one dimension on a reference interval and generalize them in $n$ dimensions.
In one dimension take a 0-form, $\alpha^{(0)} \in {\Lambda^{0}\left(Q_{ref}\right)}$, where $Q_{ref} := \left[ -1,1 \right]$. Define on $Q_{ref}$ a cell complex $D$ of order $p$ consisting of $(p+1)$ nodes $\tau_{(0),i}=\xi_{i}$ with $i=0,\cdots,p$, where $-1 \leq \xi_{0} < \cdots < \xi_{i}< \cdots \xi_{p}\leq 1$ are the Gauss-Lobatto quadrature nodes, and $p$ edges, $\tau_{(1),i} = \left[ \xi_{i-1},\xi_{i} \right]$ with $i=1,\cdots,p$. The projection operator $\pi_{h}$ reads: $$\begin{aligned}
{\pi_{h}}\alpha^{(0)} \left( \xi \right) = \sum_{i=0}^{p} \alpha_{i} \epsilon^{(0)}_{i}(\xi)\;,\end{aligned}$$ where $\epsilon^{(0)}_{i}(\xi) = l_{i} \left( \xi \right)$ are the $p^{th}$ order *Lagrange polynomials* and $\alpha_{i}=\alpha^{0}(\xi_{i})$. Similarly in one dimension for the projection of 1-forms Gerritsma [@gerritsma::edge_basis] and Robidoux [@robidoux-polynomial] derived 1-form polynomials called *edge polynomials*, $\epsilon^{(1)}_{i} \in {\Lambda^{1}_{h}\left(Q_{ref}\right)}$, $$\begin{aligned}
\epsilon^{(1)}_{i} (\xi) = e_{i} \left( \xi \right) {\mathrm{d}}\xi, \quad \text{with} \quad e_{i}(\xi) = - \sum_{k=0}^{i-1} \frac{d l_{k}}{d \xi}\;.\end{aligned}$$ Note that in this way we have: $$\begin{aligned}
\int_{\xi_{j-1}}^{\xi_{j}}\epsilon^{(1)}_{i} = \int_{\xi_{j-1}}^{\xi_{j}} e_{i} \left( \xi \right){\mathrm{d}}\xi = \delta_{ij}\;.\end{aligned}$$
Moreover, the exterior derivative of the basis 0-forms is given by: $${\mathrm{d}}{\epsilon_{i}^{(0)}} = \frac{{\mathrm{d}}l_{i}}{{\mathrm{d}}\xi}\,{\mathrm{d}}\xi = -\sum_{k=0}^{i-1}\frac{{\mathrm{d}}l_{k}}{{\mathrm{d}}\xi}\,{\mathrm{d}}\xi - \left(-\sum_{k=0}^{i}\frac{{\mathrm{d}}l_{k}}{{\mathrm{d}}\xi}\,{\mathrm{d}}\xi\right) = {\epsilon_{i}^{(1)}} -{\epsilon_{i+1}^{(1)}},\quad i=1,\cdots,p-1\;.$$ In this way, the exterior derivative of a discrete 0-form can be written as: $${\mathrm{d}}{\alpha_{h}^{(0)}} = {\mathrm{d}}\sum_{i=0}^{p}\alpha_{i}{\epsilon_{i}^{(0)}} = \sum_{i=0,j=1}^{p}{\mathsf{E}^{(1,0)}}_{ij}\alpha_{j}{\epsilon_{i}^{(1)}}\;, \label{incidence_d}$$ where, ${\mathsf{E}^{(1,0)}}_{ij}$ is the incidence matrix containing only the values, 0, 1 and -1, see [@kreeft2011mimetic; @palhaLaplaceDualGrid; @gerritsma::geometricBasis] for more details. This idea can be extended to higher dimensions, giving rise to $k$-incidence matrices, ${\mathsf{E}^{(k+1,k)}}_{ij}$, which represent the discrete exterior derivative on discrete $k$-forms, see [@palhaLaplaceDualGrid; @gerritsma::geometricBasis].
Mimetic spectral advection: an application to 1D advection {#Section::Numerical}
==========================================================
In this section we want to illustrate how to discretize the advection equation. Take the Lie advection of a 1-form, $$\frac{\partial{\rho^{(1)}}}{\partial t} + {\mathrm{d}}\iota_{\vec{v}}{\rho^{(1)}} = 0 \Leftrightarrow\left\{
\begin{array}{l}
\frac{\partial{\rho^{(1)}}}{\partial t} = -{\mathrm{d}}{\varsigma^{(0)}}\\
\\
\iota_{\vec{v}}\,{\rho^{(1)}} = {\varsigma^{(0)}}
\end{array}\;.
\right. \label{eq::system_advection}$$ Here ${\rho^{(1)}}$ is the advected quantity, say mass density, and ${\varsigma^{(0)}}$ represents the instantaneous fluxes of the advected quantity under the vector field $\vec{v}$, which are discretized in space as, $${\varsigma_{h}^{(0)}} = \sum_{i=0}^{p}\varsigma_{i}(t){\epsilon_{i}^{(0)}}\quad\mathrm{and}\quad{\rho_{h}^{(1)}} = \sum_{i=1}^{p}\rho_{i}(t){\epsilon_{i}^{(1)}}\;. \label{eq::discretization_variables}$$ For the sake of clarity in the method presentation we first introduce the time treatment, then the interior product discretization and finally their combination for a numerical solution of the advection problem.
Time integration
----------------
The time integrator used for solving the time evolution part of the advection equation is the canonical mimetic one, an arbitrary order symplectic operator derived in [@palha::phd], which is connected to canonical Gauss collocation integrators. Take an ordinary differential equation of the unknown function $y(t)$: $$\frac{{\mathrm{d}}y}{{\mathrm{d}}t} = h(y,t), \quad t\in I\subset\mathbb{R}\;.\label{eq:time_ode}$$ Discretizing $y(t)$ as $y_{h} = \sum_{k=0}^{p}y^{k}l_{k}(t)$, one gets: $$\frac{{\mathrm{d}}y_{h}}{{\mathrm{d}}t} = \sum_{k=0}^{p} y^{k}\frac{{\mathrm{d}}l_{k}(t)}{{\mathrm{d}}t} = \sum_{k=1}^{p}(y^{k} - y^{k-1})\,e_{k}(t)\;,$$ where the superscript $k$ denotes the time level.
The approximated solution, $y_{h}(t)$, is a polynomial of order $p$ determined by means of $(p+1)$ degrees of freedom such as its values at the Gauss-Lobatto nodes, red dots in [Figure \[fig:mimetic\_integrator\]]{}. On the other hand, $\frac{{\mathrm{d}}y_{h}}{{\mathrm{d}}t}$ is a polynomial of order $(p-1)$ defined by only $p$ degrees of freedom. One can set these degrees of freedom to be the values of the derivative in one point inside each of the $p$ intervals $[t^{k}, t^{k+1}]$. A choice that results in a symplectic integrator of order $2p$ is to select these points as the Gauss nodes of order $(p-1)$, the blue nodes of [Figure \[fig:mimetic\_integrator\]]{}. Notice that along the trajectory these nodes will not show the usual Gauss-Lobatto and Gauss distribution patterns, since in general the velocity field is not constant. In this way the discrete integrator becomes:
$$\sum_{k=1}^{p}(y^{k} - y^{k-1})\,e_{k}(\tilde{t}^{q}) = h\left(\sum_{k=0}^{p}y^{k} l^{k}(\tilde{t}^{q}),\tilde{t}^{q}\right),\quad q=1,2,\cdots,p\;,
\label{eq::TestCases_primal_dual_integrator}$$
with $\tilde{t}^{j}$ the $p$ nodes of a Gauss quadrature formula. The fact that the instants in time, $t^{k}$, where the $y^{k}_{i}$ are defined alternate with the instants in time, $\tilde{t}^{j}$, where the $h_{i}$ are evaluated (see [Figure \[fig:mimetic\_integrator\]]{}), corresponds to a staggering in time. This staggering also appears in leap-frog methods and in the implicit midpoint rule, for instance.
![Geometric interpretation of the solution of as given by : $(t,y^{(0)}(t))$. In red the Gauss-Lobatto nodes where the trajectory is discretized. In blue, the Gauss nodes where its derivative is discretized. The flow field, represented by arrows, is tangent to the curve at the Gauss nodes. That is, the derivative of the approximate trajectory is exactly equal to the flow field at the Gauss nodes.[]{data-label="fig:mimetic_integrator"}](./mimetic_ode_example-crop.pdf){width="40.00000%"}
The first equation in using the discretization and can be written as: $$\dfrac{\sum_{i}{\mathrm{d}}\rho_{i}(t){\epsilon_{i}^{(1)}}}{{\mathrm{d}}t} = -\sum_{il}{\mathsf{E}^{(1,0)}}_{il}\varsigma_{l}(t){\epsilon_{i}^{(1)}}\quad\Rightarrow\quad\dfrac{{\mathrm{d}}\rho_{i}(t)}{{\mathrm{d}}t} = -\sum_{l}{\mathsf{E}^{(1,0)}}_{il}\varsigma_{l}(t)\;.
\label{eq::system_advection_discretization_03}$$ This equation has a similar form as , but now as a system of equations, therefore one can apply the mimetic integrator, yielding: $$\sum_{k} (\rho_{i}^{k+1} - \rho_{i}^{k})\,e_{k}(\tilde{t}^{q}) = -\sum_{l}{\mathsf{E}^{(1,0)}}_{il}\varsigma_{l}^{q}\;.
\label{eq::system_advection_discretization_11}$$ Recall that $\rho_{i}^{k}$ is the discrete degree of freedom of the advected quantity at the $t^{k}$ instants of time associated to Gauss-Lobatto nodes and $\varsigma_{l}^{q}$ is the discrete degree of freedom of the fluxes of the advected quantity at the $\tilde{t}^{q}$ instants of time associated to the Gauss nodes, just as stated for the systems of ordinary differential equations.
Interior product
----------------
The discretization of the interior product is done using , in the following way:
\[def::discrete\_interior\_product\] In one dimension, the discrete interior product $\iota_{\vec{v},h}:\Lambda^{1}_{h}{\rightarrow}\Lambda^{0}_{h}$ is such that: $${\left( \iota_{\vec{v},h}\,{\alpha^{(1)}}_{h}, {\epsilon^{(0)}}_{i}\right)}_{L^{2}} = {\left( {\alpha^{(1)}}_{h}, \vec{v}^{\flat}\wedge{\epsilon^{(0)}}_{i}\right)}_{L^{2}},\quad\forall{\epsilon_{i}^{(0)}}\in\Lambda^{0}_{h}\label{eq::discrete_interior_product}$$ where $\vec{v}^{\,\flat}={\nu^{(1)}}\in \Lambda^{1}$ and ${\alpha^{(1)}}_{h}\in\Lambda^{1}_{h}$.
In this way one satisfies the duality pairing between the interior product and the wedge product in the discrete setting. Partitioning the domain $\Omega$ in a spectral element cell complex one can apply the discretization of the interior product in each spectral element, obtaining: $$\sum_{i}\rho_{i}(t){\left( {\epsilon_{i}^{(1)}}, \nu^{(1)}\wedge{\epsilon_{j}^{(0)}}\right)}_{L^{2}} = \sum_{i}\varsigma_{i}(t){\left( {\epsilon_{i}^{(0)}}, {\epsilon_{j}^{(0)}}\right)}_{L^{2}}\;,\quad\forall{\epsilon_{j}^{(0)}}\in \Lambda^{0}_{h}\;.
\label{eq::discrete_lie_derivative}$$
Putting things together: advection
----------------------------------
The complete discrete systems becomes: $$\begin{cases}
\sum_{k} (\rho_{i}^{k+1} - \rho_{i}^{k})\tilde{e}_{k}(\tilde{t}^{q}) = -\sum_{l}{\mathsf{E}^{(1,0)}}_{il}\varsigma_{l}^{q}\\
\\
\sum_{i,k}\rho_{i}^{k}{\epsilon_{k}^{(0)}}(\tilde{t}^{q}){\left( {\epsilon_{i}^{(1)}}, \star\nu^{(0)}\wedge{\epsilon_{j}^{(1)}}\right)}_{L^{2}\Lambda^{1}(\Omega_{m})} = \sum_{i}\varsigma_{i}^{q}{\left( {\epsilon_{i}^{(0)}}, {\epsilon_{j}^{(0)}}\right)}_{L^{2}\Lambda^{0}(\Omega_{m})},\,\forall{\epsilon_{j}^{(0)}}\in\Lambda^{0}(\Omega_{m})
\end{cases}
\label{eq::system_advection_discretization_02}$$
Numerical results
=================
This approach was applied to the two dimensional solution of an advected sine wave and a sine bell in a constant velocity field $\vec{v}=\vec{e}_{x}$: $\rho^{(2)}(x,y) = \sin(\pi x)\sin(\pi y){\mathrm{d}}x{\mathrm{d}}y$ (sine wave) and $\rho^{(2)}(x,y) = \sin(2\pi x)\sin(2\pi y){\mathrm{d}}x{\mathrm{d}}y$ if $(x,y) \in[0,0.5]\times[0,0.5]$ and $\rho^{(2)}(x,y) = 0$ in $(x,y) \in \mathbb{R}^{2}\backslash[0,0.5]\times[0,0.5]$ (sine bell), on a domain with periodic boundary conditions.
In [Figure \[fig:advection\_time\_error\_p3-12\]]{} the error in time of the numerical solution of for a mesh of $4\times4$ elements with a $\Delta t = 0.1s $ and various polynomial orders in space, $p$, and time, $p_{t}$, is presented. The initial error, due to the discretization, is conserved, as long as the time integration is sufficiently accurate.
In [Figure \[fig:advection\_space\_error\_convergence\]]{}, the $h$- and $p$-convergence plots are shown for different values of the order of the time integration scheme, $p_{t}$, and $\Delta t=0.1s$. It is possible to see that the method presents algebraic $h$-convergence rates of order $(p+1)$ as long as the time integration error does not dominate the spatial one. The method shows a spectral $p$-convergence as soon as the time integration is accurate enough.
In [Figure \[fig:advection\_numerical\_dispersion\]]{} the error on the velocity is presented as a function of the advected sine wave frequency. This figure shows that the numerical method introduces an artificial dispersion if the time scheme is not accurate enough.
![Error in velocity as a function of the frequency of the advected sine wave: numerical dispersion. $p=10$, $\Delta t=0.1s$ and $n=4\times 4$ elements.[]{data-label="fig:advection_numerical_dispersion"}](./frequencyResponse-crop.pdf){width="40.00000%"}
Another fundamental aspect is the conservation of the advected quantity. [Figure \[fig:advection\_mass\_error\_sine\_bell\]]{} shows the mass error in time, that is: $\int_{\Omega}{\rho^{(2)}}_{t}-\int_{\Omega}{\rho^{(2)}}_{t_{0}}$. The error goes from the zero machine in the first $10^{3}$ time steps while thereafter it steadily increases. Notice that even after $2\times10^{4}$ time steps the error is still below $10^{-12}$.
![Sum of the local errors of the advected 2-form for a sine bell in a velocity field $\vec{v}=\vec{e}_{x}$, with $50\times 50$ elements of order $p=0$ (blue) and $4\times 4$ elements of order $p=9$ (red), $\Delta t = 0.01$ and $p_{t}=2$.[]{data-label="fig:advection_mass_error_sine_bell"}](./massConservationLongTime_p_0_p_9_pt_2_dt_001_ne_50x50_ne_4x4-crop.pdf){width="40.00000%"}
In [Figure \[fig:advection\_rudman\_vortex\]]{} one can see the advection of a sine wave of frequency $\omega=\pi$ in a Rudman vortex for 100 time steps after which the direction of the flow is reversed and the calculation is continued for another 100 time steps, with $4\times 4$ curved elements of order $p=9$, $\Delta t=0.1s$ and time integration of order $p_{t}=2$, on a distorted mesh. The mimetic advection enables one to recover the initial solution, thus demonstrating that the integration method is reversible.
The authors would like to thank the valuable comments of both reviewers and the funding received by FCT - Foundation for science and technology Portugal through SRF/BD/36093/2007 and SFRH/BD/79866/2011.
|
---
author:
- 'Sotaro <span style="font-variant:small-caps;">Sasaki</span> Hiroaki <span style="font-variant:small-caps;">Ikeda</span> and Kosaku <span style="font-variant:small-caps;">Yamada</span>'
title: |
Perturbation Theory for a Repulsive Hubbard Model\
in Quasi-One-Dimensional Superconductors
---
Superconductivity in quasi-one-dimensional conductors has been studied as an important phenomenon. Today, some quasi-one-dimensional superconductors, such as (TMTSF)$_2$X [@rf:Ishiguro; @rf:Jerome] and Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$ [@rf:Uehara], have been discovered, and their superconductivity has been investigated. Recently, a superconducting transition in $\beta$-Na$_{0.33}$V$_2$O$_5$, which has a quasi-one-dimensional lattice structure similar to Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$, has been discovered [@rf:Yamauchi], and this phenomenon has attracted our attention. The transition temperature is $\Tc\simeq 8\,{\rm K}$ under high pressures of approximately $8\,$GPa. At ambient pressure, this material shows quasi-one-dimensional metallic behavior in an electric resistivity experiment at high temperature [@rf:Yamada], and encounters a charge-ordered transition at $T_{\rm CO}\simeq 135\,$K. [@rf:Yamada] Furthermore, below $T_{\rm N}\simeq 25\,{\rm K}$, a antiferromagnetic ordered phase appears in the charge-ordered phase. [@rf:Ueda] Under high pressures of approximately 8$\,$GPa, the charge-ordered phase abruptly vanishes, accompanied by the superconducting transition [@rf:Yamauchi]. It is not clear under high pressure whether the antiferromagnetic phase in the charge-ordered phase survives or not. However, the existence of the antiferromagnetic phase at ambient pressure suggests that the electron correlation is important.
Such an electron correlation effect leads to unconventional superconductivity rather than conventional s-wave superconductivity induced by the electron-phonon coupling [@rf:Yanase]. Such investigations have already been reported in the quasi-one-dimensional superconductors, (TMTSF)$_2$X and Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$. The superconductivity in (TMTSF)$_2$X has been investigated using the fluctuation-exchange approximation (FLEX) [@rf:Kino] and the third-order perturbation theory (TOPT) [@rf:Nomura]. Both theoretical calculations suggest that a d-wave like spin-singlet state is the most stable. Also, in Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$, the FLEX calculation for the trellis lattice indicated a d-wave like spin-singlet state [@rf:Kontani]. However, experimentally, in both materials, the Knight shift does not change above and below $\Tc$, and the spin-triplet state is indicated [@rf:Lee; @rf:Fujiwara], although it is still confusing. Thus, intensive investigations on quasi-one-dimensional superconductors should be carried out. In this letter, we investigate in detail unconventional superconductivity in a quasi-one-dimensional Hubbard model, taking up the superconductivity in $\beta$-Na$_{0.33}$V$_2$O$_5$.
Now, let us consider the lattice structure and the band structure in $\beta$-Na$_{0.33}$V$_2$O$_5$. This material has three types of vanadium site, V$1$, V$2$ and V$3$. V$1$ is on the VO$_6$ zigzag chain, V$2$ is on the VO$_6$ ladder chain and V$3$ is on the VO$_5$ zigzag chain. At ambient pressure, NMR experiments indicated that V3 does not have any conduction electrons at low temperatures [@rf:Itoh]. Assuming that conduction electrons at V3 are also empty under high pressures, we can consider the lattice structure with the V network shown in Fig. \[fig:lattice\](a), which is important for electric conductivity. This structure is different from the trellis lattice only in the V$1$ zigzag chain in the middle of Fig. \[fig:lattice\](a).
![(a) Schematic figure of the lattice used in this calculation. $t_i(1\leq i \leq 7)$ is the hopping integral. The region enclosed by rectangle is primitive cell. The primitive cell topologycally composes triangular lattice. (b) The Fermi surface for $n=0.90$, $t_0=1.0$. Since the lattice is topologycally triangular lattice, the Brillouin zone is hexagonal. []{data-label="fig:lattice"}](lattice.eps){width="8cm"}
The unit cell is a thick-line rectangle, and contains four V sites. Since there is no information on the band structure of $\beta$-Na$_{0.33}$V$_2$O$_5$, we discuss a simple tight-binding model with an s orbital on each V site. In this case, we consider $7$ types of hopping integrals $(t_1\sim t_7)$ displayed in Fig. \[fig:lattice\](a). By numerically diagonalizing the energy matrix, we obtain four bands, since there are four orbitals in the unit cell. Among them, we only use the lowest energy band. In $\beta$-Na$_{0.33}$V$_2$O$_5$, there is one electron per unit cell, if we simply count the valence electrons. Under the ideal condition that all electrons occupy only the lowest energy band, it becomes half-filled. Therefore, we deal with the electron number density $n$ as a parameter less than the half-filled state. If the superconductivity of this material is caused by electron correlation, electron correlation must be strong. For electron correlation to be strong, a high-density band is required. Therefore, the electrons should mainly occupy the lowest energy band. From these viewpoints, we use the single-band model. If there is no high-density band, we have to consider a mechanism other than electron correlation. As a typical set of parameters, we use $t_1=1.0$, $t_2=t_0$, $t_3=t_4=t_5=0.3t_0$ and $t_6=t_7=0.2t_0$. Here, $t_0$ is a measure of one-dimensionality, and with decreasing $t_0$, the Fermi surface becomes more one-dimensional. We assume that the results of calculations mainly depend on the one-dimensionality of the Fermi surface. Actually, when we change the ratio of the transfer integrals $t_i$ maintaing the one-dimensionality of the Fermi surface, the results of the calculations are almost unchanged. Therefore, we can study the dependence of the form of the Fermi surface using the parameter $t_0$. In Fig. \[fig:lattice\](b), we show a typical quasi-one-dimensional Fermi surface for $t_0=1.0$ and the electron number density $n=0.90$. This Fermi surface possesses a less nesting property, and is different from the band structures with an almost perfect nesting property discussed so far in the quasi-one-dimensional model calculations. Here, we investigate in detail superconductivity in such a situation. We consider the quasi-one-dimensional single-band Hubbard model with the lowest band $\varepsilon(k)$ discussed above, $$\begin{split}
H&=\sum_{k,\sigma}\varepsilon(k)c^{\dagger}_{k\sigma}
c_{k\sigma}\\
&+\frac{U}{2N}\sum_{k_i}\sum_{\sigma\neq\sigma'}
c^{\dagger}_{k_1\sigma}c^{\dagger}_{k_2\sigma'}c_{k_3\sigma'}
c_{k_4\sigma}\delta_{k_1+k_2,k_3+k_4}.
\end{split}$$ We treat this model using the third-order perturbation expansion. Hereafter, in order to obtain a moderate transition temperature $\Tc$, we set $U=5.0$, which is almost equal to the bandwidth. The third-order perturbation theory in the strongly correlated region has been justified by higher-order calculations of pairing interactions. [@rf:Nomura2] We can apply the perturbation theory for the appropriate values of $U$ to obtain the reliable value of $T_{\rm c}$. We apply the third-order perturbation theory with respect to $U$ to our model. Diagrams of the normal self-energy are shown in Fig. \[fig:selfenergy\].
![The diagrams of Normal self-energy. The solid lines represent Green’s function $G_0(k)$ and the broken lines represent the Coulomb repulsion $U$. []{data-label="fig:selfenergy"}](selfenergy.eps){width="8cm"}
The normal self-energy is given by $$\begin{split}
\Sigma_{\rm N}(k)&= \frac{T}{N}\sum_{k'} [U^2 \chi_0(k-k') G_0(k') \\
& +U^3 \left( \chi_0^2(k-k')+\phi_0^2(k+k') \right) G_0(k')],
\end{split}$$ where $$\begin{split}
&G_0(k)=\frac{1}{i\omega_n-\varepsilon (\ve{k})+\mu}, \\
&\chi_0(q)=-\frac{T}{N}\sum_{k} G_0(k)G_0(q+k), \\
&\phi_0(q)=-\frac{T}{N}\sum_{k}G_0(k)G_0(q-k).
\end{split}$$ Here, $G_0(k)$ with the short notation $k=(\ve{k},\omega_n)$ represents the bare Green’s function. Since the first-order normal self-energy is constant, it can be included by the chemical potential $\mu$. The dressed Green’s function $G(k)$ is given by $$\begin{split}
G(k)=\frac{1}{i\omega_n-\varepsilon(\ve{k})-\Sigma_{\rm N}(k)+\mu+\delta\mu}.
\end{split}$$ Here, the chemical potential $\mu$ and the chemical potential shift $\delta\mu$ are determined so as to fix the electron number density $n$, $$\begin{split}
n=2\frac{T}{N}\sum_k G_0(k)=2\frac{T}{N}\sum_k G(k).
\end{split}$$
We also expand the effective pairing interaction up to the third order with respect to $U$. For the spin-singlet state, the effective pairing interaction is given by $$\begin{split}
V^{\rm Singlet}(k;k')=V_{\rm RPA}^{\rm Singlet}(k;k')
+V_{\rm Vertex}^{\rm Singlet}(k;k'),
\end{split}$$ where $$\begin{split}
V_{\rm RPA}^{\rm Singlet}(k;k')=U+U^2\chi_0(k-k')+2U^3\chi_0^2(k-k'),
\end{split}$$ and $$\begin{split}
&V_{\rm Vertex}^{\rm Singlet}(k;k')=2(T/N)\Re\Big [\sum_{k_1}G_0(k_1) \\
&\times(\chi_0(k+k_1)-\phi_0(k+k_1))G_0(k+k_1-k')U^3\Big ].
\end{split}$$
For the spin-triplet state, $$\begin{split}
V^{\rm Triplet}(k;k')=V_{\rm RPA}^{\rm Triplet}(k;k')
+V_{\rm Vertex}^{\rm Triplet}(k;k'),
\end{split}$$ where $$\begin{split}
V_{\rm RPA}^{\rm Triplet}(k;k')=-U^2\chi_0(k-k'),
\end{split}$$ and $$\begin{split}
&V_{\rm Vertex}^{\rm Triplet}(k;k')=2(T/N)\Re\Big [\sum_{k_1}G_0(k_1)\\
&\times (\chi_0(k+k_1)+\phi_0(k+k_1))G_0(k+k_1-k')U^3\Big ].
\end{split}$$ Here, $V_{\rm RPA}^{\rm Singlet(Triplet)}(k,k')$ is called the RPA terms and $V_{\rm Vertex}^{\rm Singlet(Triplet)}(k,k')$ is called the vertex corrections. Near the transition point, the anomalous self-energy $\Delta(k)$ satisfies the linearized Eliashberg equation, $$\begin{split}
\lambda_{\rm max}\Delta(k)=-\frac{T}{N}\sum_{k'}V(k;k')|G(k')|^2\Delta(k'),
\end{split}$$ where, $V(k;k)$ is $V^{\rm Singlet}(k;k')$ or $V^{\rm Triplet}(k;k')$, and $\lambda_{\rm max}$ is the largest positive eigenvalue. Then, the temperature at $\lambda_{\rm max}=1$ corresponds to $T_{\rm c}$. By estimating $\lambda_{\rm max}$, we can determine which type of pairing symmetry is stable. For numerical calculations, we take 128 $\times$ 128 $\ve{k}$-meshes for twice space of the first Brillouin zone and 2048 Matsubara frequencies.
![Calculated maximum eigenvalues $\lambda_{\rm max}$ for spin-singlet (or spin-triplet) state. The line with white squares (circles) is the result for the spin-singlet (spin-triplet) state obtained using the third-order perturbation theory. The line with the black squares (circles) is the result for the spin-singlet (spin-triplet) state without the pairing interaction due to the third-order terms. The parameters are $n=0.90$ and $t_0=1.0$.[]{data-label="fig:temp"}](temp2.eps){width="8cm"}
In Fig. \[fig:temp\], we show the results for $\lambda_{\rm max}$ in the case with $n=0.90$ and $t_0=1.0$. With decreasing temperature, $\lambda_{\rm max}$ increases. The spin-singlet state and the spin-triplet state possess almost the same transition temperature, $\Tc\simeq 0.004$. If we assume that the bandwidth $W \simeq 5$ corresponds to $1\,{\rm eV}$, then $\Tc\simeq 8\,{\rm K}$ is obtained in accordance with the experimental value for $\beta$-Na$_{0.33}$V$_2$O$_5$. In Fig. \[fig:temp\], we also show the results for $\lambda_{\rm max}$ obtained without the pairing interaction due to the third-order terms. For the spin-triplet state, we see that the vertex corrections are important for stabilizing the spin-triplet state from the comparison.
![(a) Contour plots of $\chi_0(\ve{q},0)$. The peaks exist near $\ve{q}=(\pi,\pi)$ and $\ve{q}=(0,\pi)$ (b) Contour plots of $\phi_0(\ve{q},0)$. The peaks exist at $\ve{q}=(0,0)$. The parameters are $n=0.90$, $t_0=1.0$ and $T=0.004$. []{data-label="fig:chiphi"}](chiphi.eps){width="8cm"}
In Fig. \[fig:chiphi\], we show the momentum dependence of $\chi_0(\ve{q},0)$ and $\phi_0(\ve{q},0)$ in the case with $n=0.90$, $t_0=1.0$ and $T=0.004$. Since the Fermi surface has a nesting property, $\chi_0(\ve{q},0)$ has peaks near $\ve{q}=(\pi , \pi)$ and $\ve{q}=(0 , \pi)$. Since the Brillouin zone is hexagonal, $(\pi,\pi)$ is equivalent to $(0,\pi)$. At the half-filled state, $\chi_0(\ve{q},0)$ has peaks beside $\ve{q}=(\pi,\pi)$ and $\ve{q}=(0,\pi)$. If the electron number density $n$ is shifted from the half-filled state, then the peaks are shifted from $\ve{q}=(\pi,\pi)$ and $\ve{q}=(0,\pi)$ . On the other hand, $\phi_0(\ve{q},0)$ has a peak at $\ve{q}=(0 , 0)$ .
![(a) Contour plot of the anomalous self-energy. $\Delta(k)=0$ for the spin-singlet state has d-wave like momentum dependence. (b) Contour plot of the anomalous self-energy. $\Delta(k)=0$ for the spin-triplet state has p-wave like momentum dependence. The parameters are $n=0.90$, $t_0=1.0$ and $T=0.01$.[]{data-label="fig:gapsgapt"}](gapsgapt.eps){width="8cm"}
In Fig. \[fig:gapsgapt\], we show the contour plots of the anomalous self-energy in the case of $n=0.90$, $t_0=1.0$ and $T=0.01$. For the spin-singlet state, the momentum dependence of the anomalous self-energy on the Fermi surface is a d-wave like state with node. For the spin-triplet state, the momentum dependence of the anomalous self-energy is p-wave like, and is a fully gapped state. For the spin-singlet state, the RPA terms are dominant in the effective interaction terms. In this case, we can easily understand the gap structure in Fig. \[fig:gapsgapt\](a) from the structure in the Eliashberg equation as follows. The peak structures of $\chi_0(\ve{q},0)$ in Fig. \[fig:chiphi\](a) originate from the nesting property, between around point A and around point B, or around point B’ on the Fermi surfaces in Fig. \[fig:gapsgapt\](a). In order to obtain a positive value of $\lambda_{\rm max}$, it is favorable that signs of $\Delta(k)$ around points B and B’ are different from its sign around point A. The structure of $\Delta(k)$ in Fig. \[fig:gapsgapt\](a) just becomes so.
![$\lambda_{\rm max}$ as a function of $n$. The parameters are $T=0.01$ and $t_0=1.0$. []{data-label="fig:n"}](n.eps){width="8cm"}
![$\lambda_{\rm max}$ as a function of $t_0$. The parameters are $
n=0.90$ and $T=0.01$. []{data-label="fig:t0"}](t0.eps){width="8cm"}
In Fig. \[fig:n\], $n$ dependence of $\lambda_{\rm max}$ is shown. If $n$ is near the half-filled state, the spin-singlet and the spin-triplet states have low values of $\lambda_{\rm max}$. With decreasing $n$, $\lambda_{\rm max}$ increases, and the spin-singlet and spin-triplet states yield nearly the same $\lambda_{\rm max}$ at $n\simeq 0.90$. Moreover, with decreasing $n$ from $n=0.90$, the values of $\lambda_{\rm max}$ for the spin-triplet state become larger than those for the spin-singlet state. This indicates that the spin-triplet state may be realized far from the half-filled state. We can easily understand why the values of $\lambda_{\rm max}$ are suppressed for the spin-singlet and spin-triplet states at around the half-filled state. For spin-singlet state, at the half-filled state, $\chi_0(\ve{q},0)$ has a peak beside $\ve{q}=(0,\pi)$. Considering the structure of the Eliashberg equation, in order to obtain a large positive value of $\lambda_{\rm max}$, it is not favorable that signs of $\Delta(k)$ around point A are the same as its sign around point C on the Fermi surface in Fig. \[fig:gapsgapt\](a). Therefore, the values of $\lambda_{\rm max}$ are strongly suppressed around the half-filled state by the conflicting peaks of $\chi_0(\ve{q},0)$. On the other hand, if the electron number density is far from the half-filled state, the peak of $\chi_0(\ve{q},0)$ is far from $\ve{q}=(0,\pi)$. Therefore, the suppression of the values of $\lambda_{\rm max}$ becomes weak. For the spin-triplet state, when the Fermi surface has perfect particle-hole symmetry, the vertex corrections are perfectly canceled out, and at approximately the half-filled state, owing to approximate particle-hole symmetry, vertex corrections are approximately canceled out and the values of $\lambda_{\rm max}$ are suppressed. Here, the normal self-energy term corresponding to Fig. \[fig:selfenergy\](c) make the mass enhancement factor small. When the Fermi surface have perfect particle-hole symmetry, the terms corresponding to Figs. \[fig:selfenergy\](b) and \[fig:selfenergy\](c) are perfectly canceled out. However, for $n\leq 0.82$, particle-hole symmetry deteriorates, and the mass enhancement factor is much smaller than unity. Therefore, reliable numerical calculation cannot be obtained in the range of $n\leq 0.82$.
In Fig. \[fig:t0\] we show the $t_0$ dependence of $\lambda_{\rm max}$. With decreasing $t_0$, the spin-triplet state becomes dominant, and with increasing $t_0$, the spin-singlet state becomes dominant. If $t_0$ is small, $\chi_0(\ve{q},0)$ have the character of one-dimensionality. In this case, the values of $\lambda_{\rm max}$ are suppressed by the conflict of the peaks of $\chi_0(\ve{q},0)$ like the above case.
In conclusion, we have investigated pairing symmetry and the transition temperature on the basis of a quasi-one-dimensional Hubbard model. We have solved the Eliashberg equation using the third-order perturbation theory with respect to the on-site repulsion $U$. We find that if $n$ is shifted from the half-filled state, the transitions into unconventional superconductivity is expected. If one-dimensionality is weak, a spin-singlet pairing is more stable than a spin-triplet one. In contrast, if one-dimensionality is strong and $n$ is far from the half-filled state, a spin-triplet pairing is more stable than a spin-singlet one. Thus, we suggest the possibility of unconventional superconductivity in $\beta$-Na$_{0.33}$V$_2$O$_5$ caused by the on-site Coulomb repulsion. Numerical calculation in this work was carried out at the Yukawa Institute Computer Facility.
[99]{} For review, T. Ishiguro, K. Yamaji and G. Saito: [*Organic Superconductors*]{} (Springer-Verlag, Heiderberg, 1998) D. Jérome and H. J. Schulz: Adv. Phys. [**31**]{} (1982) 299. M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Môri and K. Kinoshita: J. Phys. Soc. Jpn. [**65**]{} (1996) 2764. T. Yamauchi, Y. Ueda and N. Môri: Phys. Rev. Lett. [**89**]{} (2002) 057002. H. Yamada and Y.Ueda: J. Phys. Soc. Jpn. [**68**]{} (1999) 2735. Y. Ueda: J. Phys. Soc. Jpn. [**69**]{} (2000) Suppl. B. pp. 149. Y. Yanase, T. Jujo, T. Nomura, H. Ikeda, T. Hotta, K. Yamada: Phys. Rep. [**387**]{} (2003) 1. H. Kino and H. Kontani: J. Phys. Soc. Jpn. [**68**]{} (1999) 1481. T. Nomura and K. Yamada: J. Phys. Soc. Jpn. [**70**]{} (2001) 2694. H. Kontani and K. Ueda: Phys. Rev. Lett. [**80**]{} (1998) 5619. I. J. Lee [*et al*]{}: Phys. Rev. B [**68**]{} (2003) 092510. N. Fujiwara, N. Môri, Y. Uwatoko, T. Matsumoto, N. Motoyama and S. Uchida: Phys. Rev. Lett. [**90**]{} (2003) 137001. M. Itoh, N. Akimoto, H, Yamada, M. Isobe and Y, Ueda: J. Phys. Soc. Jpn. [**69**]{} (2000) Suppl. B. pp. 155. T.Nomura and K. Yamada: J. Phys. Soc. Jpn. [**72**]{} (2003) 2053.
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---
abstract: 'The web plays an important role in people’s social lives since the emergence of Web 2.0. It facilitates the interaction between users, gives them the possibility to freely interact, share and collaborate through social networks, online communities forums, blogs, wikis and other online collaborative media. However, an other side of the web is negatively taken such as posting inflammatory messages. Thus, when dealing with the online communities forums, the managers seek to always enhance the performance of such platforms. In fact, to keep the serenity and prohibit the disturbance of the normal atmosphere, managers always try to novice users against these malicious persons by posting such message (DO NOT FEED TROLLS). But, this kind of warning is not enough to reduce this phenomenon. In this context we propose a new approach for detecting malicious people also called ’Trolls’ in order to allow community managers to take their ability to post online. To be more realistic, our proposal is defined within an uncertain framework. Based on the assumption consisting on the trolls’ integration in the successful discussion threads, we try to detect the presence of such malicious users. Indeed, this method is based on a conflict measure of the belief function theory applied between the different messages of the thread. In order to show the feasibility and the result of our approach, we test it in different simulated data.'
author:
- 'Imen Ouled Dlala, Dorra Attiaoui, Arnaud Martin, and Boutheina Ben Yaghlane [^1]'
title: Trolls Identification within an Uncertain Framework
---
[Shell : Bare Demo of IEEEtran.cls for Journals]{}
Q&AC, trolls, belief function theory, conflict measure.
Introduction
============
way we look for, and acquire information has shifted greatly into to instant, easy and low cost process. In fact, thanks to the Internet one can make a research in any given topic, get a huge amount of information by a simple click. Although, for some problems it is difficult to get satisfactory answers by searching directly on a traditional search engine. Instead, we prefer to find someone who has expertise or experience. In order to have the best answer, one of the tools that has widened the scope of information exchange is Question Answering Communities (Q&AC). These systems allow everyone to contribute as much as they can on a given community. Unfortunately, not all messages can be considered as reliable: some users claim themselves as experts, and other people post messages without any utility for the one who is seeking for answers. Thus, the managers of these communities seek to always enhance the performance of such platforms. Although, the increase of the useless messages can be attributed to the presence of trolls. The term of trolling has been defined in several works within different communities, including \[2\], \[6\] and \[17\]. These Malicious people intend to insidiously mislead the subject of the discussion in order to provoke controversy and disrupt the discussion. They aim to make normal users fall into their traps by deviating them from the main topic of the discussion. In fact, the only way to deal with a troll is to ignore him, or detect his presence in order to notify him or take away his ability to post online. Thus, some other works tried to detect not just their characteristics but also their presence in order to avoid them. To address this problem Cambria et al. \[14\] proposed a technique based on semantic computing to automatically detect and check web trolls. This work aims to prevent the malicious people from emotionally hurting other users or communities within the same social network. In another work Ortega et al. \[15\] proposed a method to classify users in a social network regarding to their trustworthiness. The goal of their method is to detect trolls from the other users by preventing such malicious users to gain high reputation in the network. Patxi et al. \[16\] dealt with Trolling users on twitter social network. These studies were explored in different social networks within certain framework.
When dealing with real-world applications, the massive amounts of data are inseparably connected with imperfection. In fact, this kind of data can be imprecise and/or uncertain or even missing. Different theories have emerged to deal with this kind of data such as fuzzy set theory \[21\], possibility theory \[22\] and belief function theory \[1\]. Thus, to be closer to reality and to obtain more relevant results, we propose a new method dealing with uncertain data. This method aims to detect trolls in Q&AC using the framework of belief function.
The paper proceeds as follows: in Section 2, we introduce the Q&AC and briefly review related works. In section 3, we present the necessary background regarding the different concepts of the belief functions theory. We define the different steps of our proposal based on a conflict measure in section 4. Finally, we present the feasibility of the proposed method on an illustrated example.
Q&AC: Quick overview
====================
In this section we introduce some concepts related the Q&AC. First we will start by presenting the main actors in these forums, then a little overview on sources identification and finally the levels of uncertainty we can face in Q&AC.
Users within Q&AC
-----------------
Users are considered as the main actors within Question Answering Communities. We can define different types such as experts, trolls and learners.
- **Reliable user / Expert**: a person who is very knowledgeable about or skilful in a particular area.
- **Troll**: a person who seeks to disturb the serenity of the concerned community. His purpose is to create controversial debates by multiplying irrelevant messages that we keep unanswered.
- **Learner**: a normal user of the Question Answering Community, trying to gain information and expertise.
Sources Identification within Q&AC
----------------------------------
Several researches have been exploring this field, trying to evaluate sources of information in Q&AC. Such as Bouguessa et al. \[7\] who proposed a model to identify authoritative users based on the number of best answers provided by them. A best answer is selected either by the asker or by other users via a voting procedure. In \[12\], the author focused on the selection of questions a user would choose for answering. Based on these studies, experts prefer answering questions where they have a higher chance of making a valuable contribution. Recently in \[13\], the authors proposed a framework for evaluating both the reliability and the expertise of an information provider. Considering some cognitive and behavioral criteria of the users, they were able to establish a trust system. Using a response matrix summarizing the interactions between peers of persons, each one is capable of estimating and providing an opinion. Using the subjective logic to aggregate these evaluations, they provided later a global reliability and expertise value for each user within Q&AC.
Uncertainty within Q&AC
-----------------------
When dealing with information provided by humans, we are facing several levels of uncertainty. Gjergji et al. proposed three levels for Q&AC \[11\], the first one is related to the extraction and integration of uncertainty, the second deals with information sources uncertainty and finally the inherent knowledge related to the information itself. In our case, we are more interested in the evaluation of the sources and the part of uncertainty related to them. The main issue in these communities is that we are facing users that we do not always have an *apriori* knowledge about them. We ignore every thing about the sources’ credibility, reliability, relevance, objectivity and expertise. In this context, we will exploit all the mathematical background and large panel of sepcificities provided by the theory of belief functions to help us considering this problem in an uncertain point of view.
Theory of Belief Functions {#sec:greetings}
==========================
This section recalls the necessary background related to the belief function theory It has been developed by Dempster in his work on upper and lower probabilities \[1\]. Based on that, he was able to represent more precisely the observed data. A belief function must take into consideration all the possible events on which a source can describe a belief. Based on that, we can define the frame of discernment.
Frame of discernment
--------------------
It is a finite set of disjoint elements noted $\Omega$ where $\Omega= \{\omega_{1},...,\omega_{n}\}$. This theory allows us to affect a mass on a set of hypotheses not only a singleton like in the probabilistic theory. Thus, we are able to represent ignorance, imprecision...
Basic belief assignment $(bba)$
-------------------------------
A $bba$ is defined on the set of all subsets of $\Omega$, named power set and noted $2^{\Omega}$. It affects a real value from $ [0, 1]$ to every subset of $ 2^{\Omega}$ reflecting sources amount of belief on this subset. A $bba$ $m$ verifies:
$$\begin{aligned}
\sum_{X\subseteq \Omega} m(X) = 1.\end{aligned}$$
We consider any positive elementary mass $m(X)>0$ as a focal element such that X belongs to $2^\Omega$.
Combination rules
-----------------
Many combination rules have been proposed taking in consideration the nature of the sources.
### Dempster’s combination rule
The first one was proposed by Dempster in 1967 \[1\] which is a conjunctive normalized combination rule also called the orthogonal sum. Given two mass functions $m_{1}$ and $m_{2}$, for all $X \in 2^{\Omega}$, $X\neq \emptyset$, the Dempster’s rule is defined by: $$\begin{aligned}
m_{D}(X)=m_1\cap m_2(X)= \frac{1}{1-k} \sum_{Y_{1} \cap Y_{2}=X} m_{1}(Y_{1})m_{2}(Y_{2})\end{aligned}$$
where $k=\sum_{Y_{1}\cap Y_{2}=\emptyset} m_{1}(Y_{1})m_{2}(Y_{2})$ is the inconsistency of the fusion (or of the combination) can also be called the conflict or global conflict. $(1-k)$ is the normalization factor of the combination in a closed world.
### The conjunctive combination rule
In order to consider the issues of the open world, the conjunctive combination rule was introduced by Smets \[9\]. Considering two mass functions $m_{1}$ and $m_{2}$, for all $X \in 2^{\Omega}$ $m_{conj}$ is defined by:
$$\begin{aligned}
m_{conj}(X)= \sum_{Y_{1}\cap Y_{2}=X} m_{1}(Y_{1})m_{2}(Y_{2})\end{aligned}$$
### The disjunctive combination rule
First introduced by Dubois and Prade 1986 \[18\], the induced results of two bbas $m_{1}$ and $m_{2}$ is defined as follows: $$\begin{aligned}
\forall X \subseteq \Omega m_{disj}(X)= \sum_{Y_{1}\cup Y_{2}=X} m_{1}(Y_{1}) m_{2}(Y_{2})\end{aligned}$$
The disjunctive combination rule can be used when one of the sources is reliable or when we have no knowledge about their reliability.
Inclusion as a conflict measure for belief functions
====================================================
Recently Martin in \[3\] used a degree of inclusion as involved in the measurement made in order to determine the conflict during the combination of two belief functions. He presented an index of inclusion having binary values where: $$\label{}
Inc(X_{1},Y_{2}) = \left\{
\begin{array}{ll}
\ 1, \mbox{if} X_{1} \subseteq Y_{2} \\
\ 0, \mbox{otherwise}
\end{array}
\right.$$
With $X_{1}$, $Y_{2}$ being respectively the focal elements of $m_{1}$ and $m_{2}$. This index is then used to measure the degree of inclusion of the two mass functions and defined as:
$$\begin{aligned}
d_{inc}= \frac{1}{|F_{1}| |F_{2}|} \sum_{X_{1}\in F_{1}} \sum_{Y_{2}\in F_{2}} Inc(X_{1},Y_{2})
\end{aligned}$$
Where $|F_{1}|$ and $|F_{2}|$ are the number of focal elements of $m_1$ and $m_2$. He define the degree of inclusion of $m_1$ and $m_2$: $\sigma_{inc}(m_{1},m_{2})$ as follows: $$\begin{aligned}
\sigma_{inc}(m_{1},m_{2})= max(d_{inc}(m_{1},m_{2}),d_{inc}(m_{2},m_{1}))
\end{aligned}$$
Where $d_{inc}$ is the degree of inclusion of $m_{1}$ in $m_{2}$ and inversely. This inclusion is used as a conflict measure for two mass functions, using it like presented: $$\begin{aligned}
Conf(m_{1},m_{2})=(1-\sigma_{inc}(m_{1},m_{2})d(m_{1},m_{2}))
\end{aligned}$$ where $d(m_{1},m_{2})$, is the distance of Jousselme \[10\]: $$\begin{aligned}
d(m_{1},m_{2})=\sqrt{\frac{1}{2}(m_{1}-m_{2})^{T} \underline{\underline{D}} (m_{1}-m_{2})}
\end{aligned}$$ where $\underline{\underline{D}}$ is a metric based on the measure of Jaccard: $$\label{}
D(A,B) = \left\{
\begin{array}{ll}
\ 1,if A=B=\emptyset \\
\frac{|A \cap B|}{|A \cup B|}, \forall A,B \in 2^\Omega
\end{array}
\right. \\$$
Trolls Identification based in a conflict measure
=================================================
Based on the assumption that consists of the trolls’ integration in the successful discussion threads, we propose a new method for detecting malicious people in online communities forums. This approach is defined within the framework of belief functions. Indeed, it is based on a conflict measure of this theory applied between the different messages of the thread. We can summarize our proposed method in three major steps that will be discussed in depth in the following.
Users’ messages
---------------
Hardarker proposed primary characteristics of a troll \[2\] (Aggression, Deception, Disruption, Success). In 2014, Buckels et al. \[6\] specified the behavioral characteristics of a troll. They described them as persons having sadism, psychopathy and machiavilism. To them, trolling is a “deceptive, destructive or disruptive manner in social media”.\
In the context of this work, to distinguish between the troll and the other users, we tried to manually extracted the characteristics of their responses from the answers and comments in different forums. Based on these characteristics, the content of the messages can be: Off-topic, senseless or controversy. Using these characteristics, we have defined the frame of discernment that can characterize a message in a forum: $$\begin{aligned}
\Omega_{msg}= \left\{Off-topic, Senseless,1,\ldots,N \right\}\end{aligned}$$
- Senseless: how much the response is empty of meaning?
- Off-topic: How irrelative the answer can be?
- $[1..N]$ : number of topics where, $[1..N] \backslash i$ with $i$ being the relevant topic, and $[1..N] \backslash i$ are the controversy topics posted by a troll.
During this step, we assume that a method of analysis expresses a piece of evidence concerning the nature of each message. This method aims to analyze the messages relative to the posted question or topic.
Users’ conflict
---------------
Detecting irrelevant messages does not only means that this user is a troll. Thus, it is not only the content of the messages that can characterize the trolls. We can find a victim user that responds to a message posted by a troll. Besides, the subject of the discussion can change gradually. In fact, to distinguish between trolls and other users in a community, we need to quantify how a given user is in conflict with the rest of all the other users. Thus, we will base our approach on measuring the conflict between the messages of each person posting answers. The list of notations is shown in table \[not\].
[|c|p[5.5cm]{}|]{} **Notations** & **Description**\
$\textbf{U}$ & Users\
$\textbf{N}$ & Number of users\
$\textbf{NP}$ & Number of all the previous messages\
$\bf{NP_j}$ & Number of the previous messages of a user $U_{j}$\
$\bf{N_{i}}$ & Number of all messages posted by a user $U_{i}$\
$\bf{N_{j}}$ & Number of all messages posted by a user $U_{j}$\
$\bf{m_{k}}$ & $k^{th}$ message of a user $U_{i}$\
$\bf{m_{s}}$ & $s^{th}$ message of a user $U_{j}$\
$\bf{Rank(m)}$ & Rank of the message m\
$\bf{Tab1}$ & Contains in each time the conflict of a message relative to each user\
$\bf{Tab2}$ & Contains in each time the number of the previous messages of a message\
$\bf{Tab3}$ & Contains the total conflict of each user\
$\bf{Conf_t}$ & Contains the sum of conflict of each user\
\[not\]
Using the inclusion as a conflict measure for belief functions, for each user $U_i$ we will measure:
- **$Conf_{msg_{/U}}$**: measures the conflict between the $k^{th}$ message posted by $U_{i}$ and the messages that were posted before it by each other users $U_{j}$.
$ Conf_{msg_{/U}}(m_{k}(U_{i}),m(U_{j}))=$
$$\!\!\! \frac{1}{NP_j}\sum_{s=1}^{NP_j} Conf(m_{k}(U_{i}),m_{s}(U_{j})),
(i\neq j) \nonumber$$
- **$Conf_{msg}$**: measures the conflict between the $k^{th}$ message posted by $U_{i}$ and the all messages that were posted before it by all the other users $U$ based on a weighted mean. This measure takes into account the number of messages posted by every user in order to determine the level of conflict especially between a troll and an expert.
$Conf_{msg}(m_{k}(U_{i}),m(U))= $
$$\sum_{j=1}^{N} \frac{NP_j}{NP} Conf_{msg_{/U}}(m_{k}(U_{i}),m(U_{j}))$$
- **$Conf_{user}$**: measures the global conflict of the user $U_{i}$
$$Conf_{user}= \frac{1}{N_i} \sum_{k=1}^{N_i} Conf_{msg}(m_{k}(U_{i}),m(U)) \\
\label{eq}$$
The value of the total conflict of a user can be risen when this user launches into an interminable debate with a troll. In this case, this victim user becomes in his turn a troll. Thus, the managers have to control the behavior of the users in many discussion threads.
Users’ clustering
-----------------
The last step consists on the classification of the users according to their conflict results into two groups. Therefore, to make decision we base our approach on an unsupervised classification method using the k means algorithm.
It was introduced by McQueen \[19\] and implemented in its current forms by Forgy \[20\]. The Kmeans algorithm aims to construct from the objects of the training set K partitions (clusters) concentrated and isolated from each other. In our case, we will devise the users into two partitions: K= 2. Since the value of the troll ’s conflict is bigger than the conflict of the other users:
- Trolls belong to the group having the biggest value of center.
- The other users belong to the group having the least value of center.
Experimentation
===============
To illustrate the comportment of our proposed method, we have tested it in different simulated data. In this section, we will present two different examples.
Example 1
---------
As we presented our method, it has three main steps. Indeed, we will present the results of each step:
### Users’ messages
Our assumption consists on the integration of the trolls in the successful discussion threads. From this point of view, we simulate the data of analyze of messages as depicted in Figure \[sim\]. In fact, in this example we will try to detect a troll in a group of 4 users. In this scenario, the discussion thread contains 16 messages posted by different users and among whom three messages are published by a troll. In this example, our frame of discernment is composed by 4 elements: Relevant=$X_1$, off-topic=$X_2$, senseless=$X_3$, controversy-topic=$X_4$. As shown in Figure 2 each row presents: the owner of the message, the order of the message in the discussion thread and the mass function of this message (as mention in section \[sec:greetings\]. B each bba must be equal to 1).
In this example, the first message of the troll ($U_4$) is controversy: $m(X_4)=0.9210$. His second message is empty of meaning: $m(X_3)=0.9716$. His third message is controversy: $m(X_4)=0.8387$.
![Simulation Results[]{data-label="sim"}](mydata22.png){height="5.5in" width="3.6in"}
### Users’ conflict
Based on the method of the inclusion and applying our algorithm, we will present the total conflict of each user of our example in Table \[table2\]: $U_4$ has the biggest value of conflict. The total conflicts of users $U_1$ and U$_2$ is small relative to the total conflict of $U_4$ despite the fact that they responded to the first message of the troll by posting each one a controversy message. this result can be explained by the answers provided by these two users who have published relevant messages. $U_3$ has a small value of conflict, he published three relevant messages where in his first message $m(X_1)=0.9732$, in his second message $m(X_1)=0.7782$, and his third message $m(X_1)=0.9632$.
[|l|l|l|l|l|]{} & $\bf{U_1}$&$\bf{U_2}$&$\bf{U_3}$&$\bf{U_4}$\
$\bf{Conf_{user}}$ & 0.0610&0.0639&$0.0489$&0.2030\
\[table2\]
### Users’ clustering
Applying the $K$-means algorithm to the different values of total conflict of all users we obtained two clusters.
- Trolls$=\{ U_4\}$
- Other users$=\{ U_1, U_2, U_3\}$
Our proposal provides us a correct classification of the users. This result shows the feasibility of our proposed method.
Example 2
---------
For this simulation we will assume that we are dealing with 8 users, among them two trolls. The discussion thread contains 31 messages. The result of the total conflict of each user expressed in equation \[eq\] is illustrated in figure \[fig\_sim2\].
The first troll $U_4$ published 2 controversy messages and the second troll $U_8$ published 3 messages: The two first ones are off-topic, and the last one is controversy.\
- $U_1$ posted 3 relevant messages and 2 controversy messages to respond to the first troll.\
- $U_2$ posted 7 relevant messages and 2 controversy messages to respond to the first troll.\
- $U_3$ posted 4 relevant messages and one off-topic message to respond to the second troll.\
- $U_5$ posted one relevant message.\
- $U_6$ published 3 relevant messages.\
- $U_7$ published 2 relevant messages.
The total conflict of the troll $U_4$ is bigger relatively to the other troll $U_8$ because he published his posts after a big number of reliable messages provided by the other users. So, this situation created a higher value of a conflict. Applying the Kmeans algorithm our method provides us a correct classification:
- Trolls$=\{ U_4, U_8\}$
- Other users$=\{ U_1, U_2, U_3, U_5, U_6, U_7\}$
The users $U_1$, $U_2$ and $U_3$ are not classified among the trolls in spite of their posts that can be categorized as trolls’ messages. This result is explained by the fact that they have other relevant messages.
![The steps of detecting trolls[]{data-label="fig_sim2"}](bar2.png){width="6.5cm"}
Conclusion
==========
We proposed in this paper a new method for detecting ’Trolls’ in Q&AC. Relying on this approach managers, can control the behavior of the users in many discussion threads in order to notify them to stop trolling. Our work is defined within an uncertain framework. It is based on a conflict measure in the belief function theory applied between the messages of the different users of the thread. First of all, this method aims to analyze the messages relative to the posted question or topic. But detecting irrelevant message is not enough to judge if this user is a troll or not. Thus, not only the content of the messages that can characterize the trolls but also their behaviors. Next, using the results of this analysis we measured the conflict between the different users. Finally, after calculating the conflict of each user we applied the kmeans method in order to distinguish trolls from the other users. Indeed, we have classified the users according to their conflict results into two clusters. This method was tested in different simulated data to check its feasibility. Since our proposed method for detecting malicious users dealt only with one discussion thread, we aim to extend this approach to detect trolls inside the community.
[1]{}
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[Imen Ouled Dlala]{} Univeristy of Tunis, LARODEC laboratory, High Institut of Management, Cite Bouchoucha, le Bardo, TUNISIA; E-mail: [email protected]
[Dorra Attiaoui]{} Univeristy of Tunis, LARODEC laboratory, High Institut of Management, Cite Bouchoucha, le Bardo, TUNISIA; University of Rennes 1, IRISA, Rue E. Branly, 22300 Lannion, FRANCE; E-mail: [email protected]
[Arnaud Martin]{} University of Rennes 1, IRISA, Rue E. Branly, 22300 Lannion, FRANCE; E-mail: [email protected]
[Boutheina Ben Yaghlane]{} University of Carthage, IHEC Carthage, Carthage Presidence, TUNISIA, LARODEC laboratory; E-mail: [email protected]
[^1]: Manuscript received.
|
---
author:
- |
Elise van der Pol\
UvA-Bosch Deltalab\
University of Amsterdam\
`[email protected]`\
Daniel E. Worrall\
Philips Lab\
University of Amsterdam\
`[email protected]`\
Herke van Hoof\
UvA-Bosch Deltalab\
University of Amsterdam\
`[email protected]` Frans A. Oliehoek\
Department of Intelligent Systems\
Delft University of Technology\
`[email protected]` Max Welling\
UvA-Bosch Deltalab\
University of Amsterdam\
`[email protected]`
bibliography:
- 'main.bib'
title: |
MDP Homomorphic Networks:\
Group Symmetries in Reinforcement Learning
---
Introduction
============
Background
==========
Method
======
Experiments
===========
Related Work
============
Conclusion
==========
Broader Impact
==============
The Symmetrizer
===============
In this section we prove three properties of the symmetrizer: the symmetric property ($S(\bW)\in \eqspace$ for all $\bW\in \fullspace$ ), the fixing property ($\bW \in \eqspace \implies S(\bW) = \bW$) , and the idempotence property ($S(S(\bW)) = S(\bW)$ for all $\bW \in \fullspace$).
#### The Symmetric Property
Here we show that the symmetrizer $S$ maps matrices $\bW \in \fullspace$ to equivariant matrices $S(\bW) \in \eqspace$. For this, we show that a symmetrized weight matrix $S(\bW)$ from Equation \[eq:symmetrizer\] satisfies the equivariance constraint of Equation \[eq:equivariant-layer\].
We begin by recalling the equivariance constraint $$\begin{aligned}
\ztransout \bW \bz = \bW \ztransin \bz, \qquad \text{for all } g \in G, \bz \in \mathbb{R}^{D_\text{in} + 1}. \label{eq:equivariant-layer-copy}\end{aligned}$$ Now note that we can drop the dependence on $\bz$, since this equation is true for all $\bz$. At the same time, we left-multiply both sides of this equation by $\ztransout^{-1}$, which is possible because group representations are invertible. This results in the following set of equations $$\begin{aligned}
\bW = \ztransoutinv \bW \ztransin, \qquad \text{for all } g \in G.\end{aligned}$$ Any $\bW$ satisfying this equation satisfies Equation \[eq:equivariant-layer-copy\] and is thus a member of $\eqspace$. To show that $S(\bW)$ is a member of $\eqspace$, we thus would need show that $S(\bW) = \ztransoutinv S(\bW) \ztransin$ for all $\bW \in \fullspace$ and $g\in G$. This can be shown as follows: $$\begin{aligned}
\bK_g^{-1} S(\bW) \bL_g &= \bK_g^{-1} \left ( \frac{1}{|G|}\sum_{h \in G} \bK_h^{-1} \bW \bL_h \right ) \bL_g && \text{substitute } S(\bW) = \ztransoutinv S(\bW) \ztransin\\
&= \frac{1}{|G|}\sum_{h \in G} \bK_g^{-1} \bK_h^{-1} \bW \bL_h \bL_g \\
&= \frac{1}{|G|}\sum_{h \in G} \bK_{hg}^{-1} \bW \bL_{hg} && \text{representation definition: } \bL_h\bL_h = \bL_{hg} \\
&= \frac{1}{|G|}\sum_{g'g^{-1} \in G} \bK_{g'}^{-1} \bW \bL_{g'} && \text{change of variables } g' = hg, h = g'g^{-1} \\
&= \frac{1}{|G|}\sum_{g' \in Gg} \bK_{g'}^{-1} \bW \bL_{g'} && g'g^{-1} \in G \iff g' \in Gg \\
&= \frac{1}{|G|}\sum_{g' \in G} \bK_{g'}^{-1} \bW \bL_{g'} && G = Gg \\
&= S(\bW) && \text{definition of symmetrizer}.\end{aligned}$$ Thus we see that $S(\bW)$ satisfies the equivariance constraint, which implies that $S(\bW) \in \eqspace$.
#### The Fixing Property
For the symmetrizer to be useful, we need to make sure that its range covers the equivariant subspace $\eqspace$, and not just a subset of it; that is, we need to show that $$\begin{aligned}
\eqspace = \{S(\bW) \in \eqspace | \bW \in \fullspace \}.\end{aligned}$$ We show this by picking a matrix $\bW \in \eqspace$ and showing that $\bW \in \eqspace \implies S(\bW) = \bW$.
We begin by assuming that $\bW \in \eqspace$, then $$\begin{aligned}
S(\bW) &= \frac{1}{|G|} \sum_{g \in G} \bK_g^{-1} \bW \bL_g && \text{definition} \\
&= \frac{1}{|G|} \sum_{g \in G} \bK_g^{-1} \bK_g \bW && \bW \in \eqspace \iff \bK_g \bW = \bW \bL_g,\,\forall g \in G \\
&= \frac{1}{|G|} \sum_{g \in G} \bW \\
&= \bW\end{aligned}$$ This means that the symmetrizer leaves the equivariant subspace invariant. In fact, the statement we just showed is stronger in saying that each point in the equivariant subspace is unaltered by the symmetrizer. In the language of group theory we say that subspace $\eqspace$ *is fixed under* $G$. Since $S: \fullspace \to \eqspace$ and there exist matrices $\bW$ such that for every $\bW \in \eqspace$, $S(\bW) = \bW$, we have shown that $$\begin{aligned}
\eqspace = \{S(\bW) \in \eqspace | \bW \in \fullspace \}.\end{aligned}$$
#### The Idempotence Property
Here we show that the symmetrizer $S(\bW)$ from Equation \[eq:symmetrizer\] is idempotent, $S(S(\bW))$.
Recall the definition of the symmetrizer $$\begin{aligned}
S(\bW) = \frac{1}{|G|}\sum_{g \in G} \bK_g^{-1} \bW \bL_g.\end{aligned}$$ Now let’s expand $S(S(\bW))$: $$\begin{aligned}
S(S(\bW)) &= S \left ( \frac{1}{|G|}\sum_{h \in G} \bK_h^{-1} \bW \bL_h \right ) \\
&= \frac{1}{|G|}\sum_{g \in G} \bK_g^{-1} \left ( \frac{1}{|G|}\sum_{h \in G} \bK_h^{-1} \bW \bL_h \right ) \bL_g \\
&= \frac{1}{|G|}\sum_{g \in G} \left ( \frac{1}{|G|}\sum_{h \in G} \bK_g^{-1} \bK_h^{-1} \bW \bL_h \bL_g \right ) && \text{linearity of sum} \\
&= \frac{1}{|G|}\sum_{g \in G} \left ( \frac{1}{|G|}\sum_{h \in G} \bK_{hg}^{-1} \bW \bL_{hg} \right ) && \text{definition of group representations}\\
&= \frac{1}{|G|}\sum_{g \in G} \left ( \frac{1}{|G|}\sum_{g'g^{-1} \in G} \bK_{g'}^{-1} \bW \bL_{g'} \right ) && \text{change of variables } g' = hg \\
&= \frac{1}{|G|}\sum_{g \in G} \left ( \frac{1}{|G|}\sum_{g' \in Gg} \bK_{g'}^{-1} \bW \bL_{g'} \right ) && g'g^{-1} \in G \iff g' \in G g \\
&= \frac{1}{|G|}\sum_{g \in G} \left ( \frac{1}{|G|}\sum_{g' \in G} \bK_{g'}^{-1} \bW \bL_{g'} \right ) && Gg = G \\
&= \frac{1}{|G|}\sum_{g' \in G} \bK_{g'}^{-1} \bW \bL_{g'} && \text{sum over constant} \\
&= S(\bW)\end{aligned}$$ Thus we see that $S(\bW)$ satisfies the equivariance constraint, which implies that $S(\bW) \in \eqspace$.
Experimental Settings
=====================
Designing representations
-------------------------
In the main text we presented a method to construct a space of intertwiners $\eqspace$ using the symmetrizer. This relies on us already having chosen specific representations/transformation operators for the input, the output, and for every intermediate layer of the MDP homomorphic networks. While for the input space (state space) and output space (policy space), these transformation operators are easy to define, *it is an open question how to design a transformation operator for the intermediate layers* of our networks. Here we give some rules of thumb that we used, followed by the specific transformation operators we used in our experiments.
For each experiment we first identified the group $G$ of transformations. In every case, this was a finite group of size $|G|$, where the size is the number of elements in the group (number of distinct transformation operators). For example, a simple flip group as in Pong has two elements, so $|G|=2$. Note that the group size $|G|$ does not necessarily equal the size of the transformation operators, whose size is determined by the dimensionality of the input/activation layer/policy.
#### MLP-structured networks
For MLP-structured networks (CartPole), typically the activations have shape `[batch_size, num_channels]`. Instead we used a shape of `[batch_size, num_channels, representation_size]`, where for the intermediate layers `representation_size=|G|+1` (we have a +1 because of the bias). The transformation operators we then apply to the activations is the set of permutations for group size $|G|$ appended with a 1 on the diagonal for the bias, acting on this last ‘representation dimension’. Thus a forward pass of a layer is computed as $$\begin{aligned}
\textbf{y}_{b,c_\text{out},r_\text{out}} = \sum_{c_\text{in}=1}^{\texttt{num\_channels}} \sum_{r_\text{in}=1}^{\texttt{|G|+1}} \bz_{b,c_\text{in},r_\text{in}} \bW_{c_\text{out},r_\text{out},c_\text{in},r_\text{in}}\end{aligned}$$ where $$\begin{aligned}
\bW_{c_\text{out},r_\text{out},c_\text{in},r_\text{in}} = \sum_{i=1}^{\text{rank}(\eqspace)} c_{i,c_\text{out},c_\text{in}} \textbf{V}_{i, r_\text{out},r_\text{in}}.\end{aligned}$$
#### CNN-structured networks
For CNN-structured networks (Pong and Grid World), typically the activations have shape `[batch_size, num_channels, height, width]`. Instead we used a shape of `[batch_size, num_channels, representation_size, height, width]`, where for the intermediate layers `representation_size=|G|+1`. The transformation operators we apply to the input of the layer is a spatial transformation on the `height, width` dimensions and a permutation on the `representation` dimension. This is because in the intermediate layers of the network the activations do not only transform in space, but also along the representation dimensions of the tensor. The transformation operators we apply to the output of the layer is just a permutation on the `representation` dimension. Thus a forward pass of a layer is computed as $$\begin{aligned}
\textbf{y}_{b,c_\text{out},r_\text{out},h_\text{out},w_\text{out}} = \sum_{c_\text{in}=1}^{\texttt{num\_channels}} \sum_{r_\text{in}=1}^{\texttt{|G|+1}}
\sum_{h_\text{in},w_\text{in}} \bz_{b,c_\text{in},r_\text{in},h_\text{out}+h_\text{in},w_\text{out}+w_\text{in}} \bW_{c_\text{out},r_\text{out},c_\text{in},r_\text{in},h_\text{in},w_\text{in}}\end{aligned}$$ where $$\begin{aligned}
\bW_{c_\text{out},r_\text{out},c_\text{in},r_\text{in},h_\text{in},w_\text{in}} = \sum_{i=1}^{\text{rank}(\eqspace)} c_{i,c_\text{out},c_\text{in}} \textbf{V}_{i, r_\text{out},r_\text{in},h_\text{in},w_\text{in}}.\end{aligned}$$
Cartpole-v1
-----------
#### Group Representations
For states: $$\begin{aligned}
\bL_{g_e} =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix},
\bL_{g_1} =
\begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{pmatrix}\end{aligned}$$ For intermediate layers and policies: $$\begin{aligned}
\bK^\pi_{g_e} =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},
\bK^\pi_{g_1} =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}\end{aligned}$$ For values we require an invariant rather than equivariant output. This invariance is implemented by defining the output representations to be $|G|$ identity matrices of the desired output dimensionality. For predicting state values we required a 1-dimensional output, and we thus used $|G|$ 1-dimensional identity matrices, i.e. for value output $V$: $$\begin{aligned}
\bK^V_{g_e} =
\begin{pmatrix}
1
\end{pmatrix},
\bK^V_{g_1} =
\begin{pmatrix}
1
\end{pmatrix}\end{aligned}$$
#### Hyperparameters
For both the basis networks and the MLP, we used Xavier initialization. We trained PPO using ADAM on 16 parallel environments and fine-tuned over the learning rates $\{0.01, 0.05, 0.001, 0.005, 0.0001, 0.0003, 0.0005\}$ by running 25 random seeds for each setting, and report the best curve. The final learning rates used are shown in Table \[tab:cartpole\_lrs\]. Other hyperparameters were defaults in RLPYT [@stooke2019rlpyt], except that we turn off learning rate decay.
Equivariant Nullspace Random MLP
------------- ----------- -------- -------
0.01 0.005 0.001 0.001
: Final learning rates used in CartPole-v1 experiments.[]{data-label="tab:cartpole_lrs"}
#### Architecture
\
Basis networks:
BasisLinear(repr_in=4, channels_in=1, repr_out=2, channels_out=64)
ReLU()
BasisLinear(repr_in=2, channels_in=64, repr_out=2, channels_out=64)
ReLU()
BasisLinear(repr_in=2, channels_in=64, repr_out=2, channels_out=1)
BasisLinear(repr_in=2, channels_in=64, repr_out=1, channels_out=1)
First MLP variant:
Linear(channels_in=1, channels_out=64)
ReLU()
Linear(channels_in=64, channels_out=128)
ReLU()
Linear(channels_in=128, channels_out=1)
Linear(channels_in=128, channels_out=1)
Second MLP variant:
Linear(channels_in=1, channels_out=128)
ReLU()
Linear(channels_in=128, channels_out=128)
ReLU()
Linear(channels_in=128, channels_out=1)
Linear(channels_in=128, channels_out=1)
GridWorld
---------
#### Group Representations
For states we use `numpy.rot90`. The stack of weights is rolled.
For the intermediate representations: $$\begin{aligned}
\bL_{g_e} =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix},
\bL_{g_1} =
\begin{pmatrix}
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0
\end{pmatrix},
\bL_{g_2} =
\begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0
\end{pmatrix},
\bL_{g_3} =
\begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0
\end{pmatrix}\end{aligned}$$ For the policies: $$\begin{aligned}
\bK^\pi_{g_e} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix},
\bK^\pi_{g_1} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0
\end{pmatrix},
\bK^\pi_{g_2} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1\\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0
\end{pmatrix},
\bK^\pi_{g_3} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0
\end{pmatrix}\end{aligned}$$ For the values: $$\begin{aligned}
\bK^V_{g_e} =
\begin{pmatrix}
1
\end{pmatrix},
\bK^V_{g_1} =
\begin{pmatrix}
1
\end{pmatrix},
\bK^V_{g_2} =
\begin{pmatrix}
1
\end{pmatrix},
\bK^V_{g_3} =
\begin{pmatrix}
1
\end{pmatrix}\end{aligned}$$
#### Hyperparameters
For both the basis networks and the CNN, we used He initialization. We trained A2C using ADAM on 16 parallel environments and fine-tuned over the learning rates $\{0.00001, 0.00003, 0.0001, 0.0003, 0.001, 0.003\}$ on 20 random seeds for each setting, and reporting the best curve. The final learning rates used are shown in Table \[tab:grid\_lrs\]. Other hyperparameters were defaults in RLPYT [@stooke2019rlpyt].
Equivariant Nullspace Random MLP
------------- ----------- -------- -------
0.001 0.003 0.001 0.003
: Final learning rates used in grid world experiments.[]{data-label="tab:grid_lrs"}
#### Architecture
\
Basis networks:
BasisConv2d(repr_in=1, channels_in=1, repr_out=4, channels_out=$\lfloor{\frac{16}{\sqrt{4}}} \rfloor$,
filter_size=(7, 7), stride=2, padding=0)
ReLU()
BasisConv2d(repr_in=4, channels_in=$\lfloor{\frac{16}{\sqrt{4}}} \rfloor$, repr_out=4, channels_out=$\lfloor{\frac{32}{\sqrt{4}}} \rfloor$,
filter_size=(5, 5), stride=1, padding=0)
ReLU()
GlobalMaxPool()
BasisLinear(repr_in=4, channels_in=$\lfloor{\frac{32}{\sqrt{4}}} \rfloor$, repr_out=4, channels_out=$\lfloor{\frac{512}{\sqrt{4}}} \rfloor$)
ReLU()
BasisLinear(repr_in=4, channels_in=$\lfloor{\frac{512}{\sqrt{4}}} \rfloor$, repr_out=5, channels_out=1)
BasisLinear(repr_in=4, channels_in=$\lfloor{\frac{512}{\sqrt{4}}} \rfloor$, repr_out=1, channels_out=1)
CNN:
Conv2d(channels_in=1, channels_out=$16$,
filter_size=(7, 7), stride=2, padding=0)
ReLU()
Conv2d(channels_in=$16$,channels_out=$32$,
filter_size=(5, 5), stride=1, padding=0)
ReLU()
GlobalMaxPool()
Linear(channels_in=$32$, channels_out=$512$)
ReLU()
Linear(channels_in=$512$, channels_out=5)
Linear(channels_in=$512$, channels_out=1)
Pong
----
#### Group Representations
For the states we use `numpy`’s indexing to flip the input, i.e.\
`w = w[..., ::-1, :]`, then the permutation on the `representation` dimension of the weights is a `numpy.roll`, since the group is cyclic.
For the intermediate layers: $$\begin{aligned}
\bL_{g_e} =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},
\bL_{g_1} =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}\end{aligned}$$
#### Hyperparameters
For both the basis networks and the CNN, we used He initialization. We trained A2C using ADAM on 4 parallel environments and fine-tuned over the learning rates $\{0.0001, 0.0002, 0.0003\}$ on 15 random seeds for each setting, and reporting the best curve. The learning rates to fine-tune over were selected to be close to where the baseline performed well in preliminary experiments. The final learning rates used are shown in Table \[tab:pong\_lrs\]. Other hyperparameters were defaults in RLPYT [@stooke2019rlpyt].
Equivariant Nullspace Random MLP
------------- ----------- -------- --------
0.0002 0.0002 0.0002 0.0001
: Learning rates used in Pong experiments.[]{data-label="tab:pong_lrs"}
#### Architecture
\
Basis Networks:
BasisConv2d(repr_in=1, channels_in=4, repr_out=2, channels_out=$\lfloor{\frac{16}{\sqrt{2}}} \rfloor$,
filter_size=(8, 8), stride=4, padding=0)
ReLU()
BasisConv2d(repr_in=2, channels_in=$\lfloor{\frac{16}{\sqrt{2}}} \rfloor$, repr_out=2, channels_out=$\lfloor{\frac{32}{\sqrt{2}}} \rfloor$,
filter_size=(5, 5), stride=2, padding=0)
ReLU()
Linear(channels_in=$2816$, channels_out=$\lfloor{\frac{512}{\sqrt{2}}} \rfloor$)
ReLU()
Linear(channels_in=$\lfloor{\frac{512}{\sqrt{2}}} \rfloor$, channels_out=6)
Linear(channels_in=$\lfloor{\frac{512}{\sqrt{2}}} \rfloor$, channels_out=1)
CNN:
Conv2d(channels_in=4, channels_out=$16$, filter_size=(8, 8), stride=4, padding=0)
ReLU()
Conv2d(channels_in=$16$,channels_out=$32$, filter_size=(5, 5), stride=2, padding=0)
ReLU()
Linear(channels_in=$2048$, channels_out=$512$)
ReLU()
Linear(channels_in=$512$, channels_out=6)
Linear(channels_in=$512$, channels_out=1)
Cartpole-v1 Deeper Network Results
==================================
We show the effect of training a deeper network – 4 layers instead of 2 – for CartPole-v1 in Figure \[exp:cartpole\_deeper\]. The performance of the regular depth networks in Figure \[exp:cartpole\] and the deeper networks in Figure \[exp:cartpole\_deeper\] is comparable, except that for the regular MLP, the variance is much higher when using deeper networks.
[0.49]{} ![<span style="font-variant:small-caps;">Cartpole</span>: Trained with PPO, all networks fine-tuned over 7 learning rates. 25%, 50% and 75% quantiles over 25 random seeds shown. a) Equivariant, random, and nullspace bases. b) Equivariant basis, and two MLPs with different degrees of freedom. []{data-label="exp:cartpole_deeper"}](figures/cartpole_versus_bases_deeper.pdf "fig:"){width="\textwidth"}
[0.49]{} ![<span style="font-variant:small-caps;">Cartpole</span>: Trained with PPO, all networks fine-tuned over 7 learning rates. 25%, 50% and 75% quantiles over 25 random seeds shown. a) Equivariant, random, and nullspace bases. b) Equivariant basis, and two MLPs with different degrees of freedom. []{data-label="exp:cartpole_deeper"}](figures/cartpole_versus_mlp_deeper.pdf "fig:"){width="\textwidth"}
Bellman Equations
=================
$$\begin{aligned}
V^\pi(s) &= \sum_{\a \in \A} \pi(s, a) \left [ \R(\s, \a) + \gamma \sum_{\s' \in \S} \T(\s, \a, \s') V^\pi(\s') \right ] \label{eq:bellman-equation}\\
Q^\pi(\s, \a) &= \R(\s, \a) + \gamma \sum_{\s' \in \S} \T(\s, \a, \s') V^\pi(\s').\end{aligned}$$
|
---
abstract: 'We demonstrate that a certain class of low scale supersymmetric “Nelson-Barr” type models can solve the strong and supersymmetric CP problems while at the same time generating sufficient weak CP violation in the $K^{0}-\bar{K}^{0}$ system. In order to prevent one-loop corrections to $\bar{\theta}$ which violate bounds coming from the neutron electric dipole moment (EDM), one needs a scheme for the soft supersymmetry breaking parameters which can naturally give sufficient squark degeneracies and proportionality of trilinear soft supersymmetry-breaking parameters to Yukawa couplings. We show that a gauge-mediated supersymmetry breaking sector can provide the needed degeneracy and proportionality, though that proves to be a problem for generic Nelson-Barr models. The workable model we consider here has the Nelson-Barr mass texture enforced by a gauge symmetry; one also expects a new U(1) gauge superfield with mass in the TeV range. The resulting model is predictive. We predict a measureable neutron EDM and the existence of extra vector-like quark superfields which can be discovered at the CERN Large Hadron Collider. Because the $3\times 3$ Cabbibo-Kobayashi-Maskawa matrix is approximately real, the model also predicts a flat unitarity triangle and the absence of substantial CP violation in the $B$ system at future $B$ factories. We discuss the general issues pertaining to the construction of such a workable model and how they lead to the successful strategy. A detailed renormalization group study is then used to establish the feasibility of the model considered.'
address: |
$^a$ Institute of Field Physics, Department of Physics and Astronomy,\
University of North Carolina, Chapel Hill, NC 27599-3255\
$^b$ Department of Physics and Astronomy,\
University of Rochester, Rochester NY 14627-0171 [^1]
author:
- '**Otto C. W. Kong$^{a,b}$ and Brian D. Wright$^a$ [^2]**'
title: 'A Solution to the Strong CP Problem with Gauge-Mediated Supersymmetry Breaking'
---
Introduction
============
The strong CP problem is without question one of the most important problems faced by the Standard Model (SM). Its origin lies in the necessity of adding the so-called $\theta$ term to the effective QCD Lagrangian due to the contribution of instantons present in the topologically nontrivial QCD vacuum[@instanton]: $${\mathcal L}_{{\mathrm eff}} = \frac{\theta\alpha_s}{8\pi}
F^A_{\mu\nu}\tilde{F}^{A\mu\nu}~,\label{leffqcd}$$ where the dual field strength is given by $\tilde{F}_{\mu\nu} = \frac{1}{2}
\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$. Through the anomaly in the axial ${\mathrm U}(1)$ current of QCD, chiral U(1) transformations lead to shifts in $\theta$, leaving the physical combination $\bar{\theta} = \theta - {\mathrm arg\, det} M_q$, where $M_q$ is the quark mass matrix. Since ${\mathcal L}_{\mathrm eff}$ clearly violates CP, it gives a strong interaction contribution to the neutron electric dipole moment[@nedm] and leads to the experimental contraint $$\bar{\theta} < 10^{-9}~.$$ The real problem therefore is one of naturalness or fine-tuning: Why is $\bar{\theta}$ so incredibly small?
There are currently three notable classes of possible solutions to this problem: (1) vanishing up quark mass, (2) the axion[@visax; @invisax] or (3) CP conservation and subsequent spontaneous breaking. The first and simplest possibility appears to be disfavored by current algebra relations between pseudoscalar meson masses[@Weinberg], but is still controversial (see [*e.g.*]{} Ref. [@KapMan]). Of these, the most popular is the invisible axion alternative[@invisax]. Here one introduces a global chiral ${\mathrm U}(1)_{PQ}$ (Peccei-Quinn[@PQ]) symmetry which is spontaneously broken at a high energy scale $f$ and explicitly broken by instantons. The $\theta$ parameter is replaced by a dynamical field — a pseudo Goldstone boson of the ${\mathrm U}(1)_{PQ}$ — whose potential dynamically relaxes $\bar{\theta}$ to zero. The advantage of this scheme is that it is simple, generic and has observable consequences both in terrestrial experiments and in astrophysics and cosmology. Astrophysical constraints from axion-induced cooling during stellar evolution[@Raffelt] and effects on the neutrino signal from supernova 1987A[@Turner] give a lower bound, $f \agt 10^{10}$ GeV, while a cosmological upper bound of $\sim 10^{12}$ GeV is given by the contribution to the universal energy density of the vacuum energy associated with ${\mathrm U}(1)_{PQ}$ breaking as the axion vacuum expectation value relaxes to zero[@axcosmo]. On the aesthetic side, one may complain that we are merely replacing the $\bar{\theta}$ fine-tuning problem with another: the smallness of the ratio of the weak scale to the ${\mathrm U}(1)_{PQ}$ breaking scale $\sim 10^{-(8{\mathrm -}10)}$. Another possible problem is the dependence of the solution on a global symmetry, generally not preserved by gravity, so not likely to appear from a more fundamental theory. This appears to be a significant problem[@aPl], at least in Einstein gravity. However, it has been argued that gravitational violations of global symmetries may be suppressed in certain extensions, including string theories[@KLLS], where at least the universal dilaton-axion is always present. The axion alternative also does not provide an explanation of weak CP violation. Here one must assume the Kobayashi-Maskawa (KM) origin of CP violating phases. Of course the ultimate test is to detect actually an axion[@axexpts].
In this paper we will focus on the third alternative. That is, we will assume that the fundamental theory of nature preserves CP and that at sub-Planck energies it is spontaneously broken. Indeed there is evidence that CP has its origin as a gauge symmetry remnant of superstring theories[@DLM]. In this manner the smallness of $\bar{\theta}$ reflects the existence of an underlying symmetry. Such models were first constructed in the context of Grand Unified theories (GUTs) by Nelson and refined by Barr[@NB] and incorporate extra heavy quarks which mix with the observed quarks. Then, relying on specific symmetries, one can obtain a texture of the full quark mass matrices which guarantee the tree-level vanishing of $\bar{\theta}$ after the spontaneous CP violation (SCPV). After integrating out the heavy fields, the low energy quark mass matrices contain the usual KM phase. The generic difficulty with these models comes from the need to ensure that large contributions to $\bar{\theta}$ do not arise at higher loops, while at the same time having sufficient weak CP violation from the KM phase. Thus while the SCPV approach is conceptually rather simple, it is not so generic and requires careful model building.
Given some of the tantalizing hints of low energy supersymmetry and the plausible gauge origin of CP symmetry, it is worthwhile to attempt to construct a supersymmetric (SUSY) model with a Nelson-Barr type mechanism for solving the strong CP problem. The SCPV feature then also resolves the so-called SUSY phases problem. The latter problem was originally described in the context of a minimal supergravity origin of the soft SUSY breaking terms[@PW], and is usually worse in a general SUSY breaking scenario. There are two extra phases in the universal soft mass parameters, beyond $\delta_{KM}$ and $\bar{\theta}$, which give CP violating effects in the low energy effective theory. These can be written as effective phases in the coefficients $A$ and $B$ of the trilinear and bilinear soft SUSY breaking scalar terms, respectively, given by $$\phi_A = {\mathrm arg}(A M_{1/2}^\ast)\qquad
\phi_B = {\mathrm arg}(B M_{1/2}^\ast)~,\label{susyphases}$$ where $M_{1/2}$ is the universal gaugino mass. The problem is that from 1-loop diagrams involving squarks, these phases must be fine-tuned to order $10^{-2}$ – $10^{-3}$ to satisfy the limit on the neutron electric dipole moment unless all the superpartners are “heavy”, $\sim 1$ TeV. With CP spontaneously broken in a sector independent of SUSY breaking, these phases would be naturally zero at first order.
Attempts to realize the Nelson-Barr mechanism in SUSY models[@BMS] have, however, run up against a formidable difficulty: There generically exist potentially large 1-loop contributions to $\bar{\theta}$ in these models[@DKL]. The dangerous diagrams are shown in Fig.1, where now in the supersymmetric case $\bar{\theta}$ also gets contributions from the argument of the gluino mass (Fig.1b): $$\bar{\theta} = \theta - {\mathrm arg\, det} M_q - 3\, {\mathrm arg} M_g~.
\label{thetabarsusy}$$ As discussed at length in Ref. [@DKL], one requires an exceptionally high degree of proportionality of the soft SUSY breaking trilinear scalar couplings to their associated Yukawa couplings as well as degeneracy among the soft squark mass terms for each charge and color sector, if these contributions are to be sufficiently suppressed. This is equivalent to the statement that when the quark and squark mass matrices are diagonalized by the same set of unitary matrices, no phase can appear in the diagrams of Fig.1. The degree of proportionality and degeneracy required among the soft SUSY breaking parameters is very difficult to maintain due to the effects of renormalization.
So, is SCPV doomed to be disfavored as a solution to the strong CP problem in supersymmetric models? We will argue that models with a specifically modified Nelson-Barr mechanism together with the recently popular gauge-mediated SUSY breaking (GMSB) scenario[@GMSB; @MMM; @fx; @Bor; @rev] can overcome the difficulty. The GMSB scenario ensures that the soft masses at the intrinsic SUSY breaking scale $M_{mess}$ are proportional and degenerate, while renormalization effects that violate these conditions are reduced by having to run soft masses and couplings from the messenger scale $M_{mess} \simeq 10 - 100$ TeV instead of the reduced Planck mass $\simeq 2\times 10^{18}$ GeV. In the minimal version of such GMSB models[@MMM; @Bor], the $A$- and $B$-terms are zero at $M_{mess}$. This tends to give additional suppression of the dangerous contributions to $\bar{\theta}$. However, large third generation Yukawa couplings can still lead to significant violations of proportionality and degeneracy and a detailed numerical analysis of the situation is necessary to determine if the supersymmetric Nelson-Barr type models are viable solutions to the strong CP problem. We will answer this in the affirmative with what to our knowledge is the only full renormalization group (RG) analysis of such models in the literature.
The models to be discussed here have CP spontaneously broken at low energies (of order a TeV), and a source of weak CP violation distinct from that in the Standard Model, namely the exchange of a new U(1) gauge boson, the symmetry of which enforces the Nelson-Barr texture. The KM phase is very small. This makes it possible to account for the smallness of $\bar{\theta}$ without making weak CP violation inconsistent with observations. In the SUSY-GUT Nelson-Barr models, even with gauge-mediated SUSY breaking, one predicts a too large $\bar{\theta}$ if the experimental requirement that $\delta_{KM}
\sim {\mathcal O}(1)$ is imposed. Moreover, one has a richer and quite distinct phenomenology. The extra quarks and other fields needed to construct the Nelson-Barr texture and break CP will be within the reach of future accelerators. A non-supersymmetric version of the type of model is the aspon model[@aspon; @FN; @asponB]. The situation with SUSY incorporated is first discussed in Ref.[@FK] where its advantage over the generic SUSY Nelson-Barr type models is highlighted. Another possible advantage for such low scale models would arise if it were somehow possible to imbed the sector responsible for the Nelson-Barr texture in the SUSY breaking and messenger sectors. An immediate disadvantage of this approach is that by breaking CP at low energies, one introduces a serious domain wall problem[@domwall]. However, this can be solved via a period of inflation just above the weak scale. Indeed, in the SUSY context, some authors have argued that this type of inflation can be natural[@weakinfl] and desirable for other reasons ([*e.g.*]{} as a solution to the cosmological moduli problem).
This article is organized as follows: In Section 2, the considerations leading to viable models for solving the strong and supersymmetric CP problems are discussed. We also note some intriguing alternatives worthy of further consideration. We give in Section 3 a detailed summary of the dangerous 1-loop contributions to $\bar{\theta}$ and their dependence on proportionality and squark-degeneracy violating mass insertions. The question of the detailed structure of the spontaneous CP breaking part of the superpotential is taken up in Section 4. We emphasize its importance in determining certain dangerous contributions to $\bar{\theta}$ and present a minimal example we use in further analysis. The renormalization group analysis used to estimate the $\bar{\theta}$ contributions is described in Section 5 and the full numerical results presented. Some remarks on related questions of interest are presented in the conclusion.
Model-Building Considerations
=============================
We assume full CP symmetry in the visible sector, including the soft SUSY breaking terms, down to energies where spontaneous CP breaking occurs. From Eq.(\[thetabarsusy\]) we see that there should be no tree-level phases in the quark mass determinant, nor in the SUSY breaking gluino mass. The latter holds by assumption and the former is obtained through a Nelson-Barr texture. To obtain the texture, we introduce an extra heavy right-handed down quark superfield $\bar{D}$ together with its mirror $D$ coupling to the ordinary down quarks via the superpotential[^3] $$W_d = Y_d^{ij}Q_j\bar{d}_i H_d + \mu_{{\scriptscriptstyle D}} D\bar{D} +
\gamma^{ia}D\chi_a \bar{d}_i~,\label{Wd}$$ where the VEVs of the scalar components of $\chi_a$ contain a relative phase, thus breaking CP. The details of the superpotential accomplishing this are postponed to Section 4, and here it is sufficient to note that at least two $\chi$ fields are necessary, since one phase can always be absorbed by a field redefinition of the extra quarks. After CP and ${\rm SU}(2)_W\times {\rm U}(1)_Y$ breaking, we have the down sector fermion mass matrix: $$m_q = \left( \begin{array}{cc}
m_d & x \mu_{{\scriptscriptstyle D}} \mbox{\boldmath $a$} \\
0 & \mu_{{\scriptscriptstyle D}}
\end{array}\right)~,\label{mf}$$ where $m_d$ is the usual $3\times 3$ down sector mass matrix and [$a$]{} is a complex $3$-vector with components $a^i = \frac{1}{x \mu_{{\scriptscriptstyle D}}}
\gamma^{ia}\langle\chi_a\rangle$, and the real parameter $x$ is defined such that [$a$]{} is normalized to 1, [*i.e.*]{} [$a^{\dag} a$]{}$=1$. The magnitude of mixing between the ordinary quarks and the extra singlet is characterized by $x$. Clearly the determinant of $m_q$ is real and at energies below $\mu_{{\scriptscriptstyle D}}$ the low energy effective theory has a KM phase of at most order $x$. Without some additional source for weak CP violation, this must be ${\mathcal O}(1)$ and as we shall see this in turn makes the suppression of 1-loop contributions to $\bar{\theta}$ problematic.[^4] For this reason, we shall assume that CP is broken at relatively low scales with a nonstandard mechanism for weak CP violation.
The specific form of the mass matrix in Eq.(\[mf\]) can be enforced by a variety of symmetries, though by the non-renormalization theorems, the Nelson-Barr texture is not upset by renormalization of terms in the superpotential. In the aspon scenario, as discussed in Ref.[@FN], the $D$, $\bar{D}$ and $\chi_a$ can be given charges $1,-1$ and $1$, respectively, under a new gauged U(1) symmetry. The major source of weak CP violation, in the $K-\bar{K}$ system for instance, then comes from exchange of the new U(1) gauge boson (aspon) which becomes massive at the scale where CP is spontaneously broken. This places an upper bound on the mass scale of CP breaking of ${\mathcal O}({\rm TeV})$. More important in our SUSY version, this allows the parameter $x$ to be small, [*e.g.*]{} $x^2\sim 10^{-5}$, which contributes significantly in suppression $\bar{\theta}$ from loop corrections. Note that we will need at least some extra mirror partners for $\chi_a$ superfields to cancel the gauge anomaly introduced by their fermionic components.
If the Nelson-Barr texture is obtained from a discrete/global symmetry, one must rely on superbox diagrams involving gluino and chargino exchange to generate $\epsilon_K$; a scenario recently re-analyzed in the context of the minimal supersymmetric Standard Model (MSSM)[@epK]. However, in present setting, this proves very difficult to do. This is unfortunate since it is easier to construct unifiable models in this latter case.[^5]
When SUSY is broken we generate nonzero gaugino masses and soft scalar bilinear and trilinear couplings,[^6] including the following terms relevant for the down-sector analysis \[a complete description is given in Eqs.(\[mssmsoft\],\[extrasoft\])\]: $$\begin{aligned}
V_{\text{soft}}^d & = & \hat{Q}^{\dagger}\tilde{m}_Q^2 \hat{Q}
+ \hat{\bar{d}} \tilde{m}_{\bar{d}}^2 \hat{\bar{d}}^{\dagger}
+ \hat{D}^{\dagger}\tilde{m}_D^2 \hat{D}
+ \hat{\bar{D}} \tilde{m}_{\bar{D}}^2 \hat{\bar{D}}^{\dagger}
+ \chi_a^{} \tilde{m}_{\chi ab}^2 \chi_b^{\dagger}
\nonumber \\
&&{} + \hat{\bar{d}} h_d \hat{Q} H_d
+ \hat{\bar{d}}_i h_{\gamma}^{ia} \hat{D} \chi_a
+ B_{{\scriptscriptstyle D}} \mu_{{\scriptscriptstyle D}}\hat{\bar{D}} \hat{D}
+ \text{h.c.} \label{vdsoft}\end{aligned}$$ The general form of the down squark mass matrix can be written as, $${\mathcal M}^2_d = \left( \begin{array}{cc}
{\mathcal M}_{RR}^2 & {\mathcal M}_{RL}^2 \\
{\mathcal M}_{RL}^{2 \dagger} & {\mathcal M}_{LL}^2
\end{array} \right)~, \label{8sqm}$$ where $$\begin{aligned}
{\mathcal M}_{LL}^2 & = & \left( \begin{array}{cc}
\tilde{m}_d^2 + m_d^{\dag}m_d & x \mu_{{\scriptscriptstyle D}}
m_d^{\small \text{T}} \mbox{\boldmath $a$} \\
x \mu_{{\scriptscriptstyle D}} \mbox{\boldmath $a^\dagger$} m_d &
\mu_{{\scriptscriptstyle D}}^2 ( 1 + x^2) + \tilde{m}_D^2
\end{array}\right)~,\label{mll} \\
%
{\mathcal M}_{RR}^2 & = & \left( \begin{array}{cc}
\tilde{m}_{\bar{d}}^2 + m_d m_d^{\text{T}} +
x^2 \mu_{{\scriptscriptstyle D}}^2 \mbox{\boldmath $a a^\dagger$} &
x \mu_{{\scriptscriptstyle D}}^2 \mbox{\boldmath $a$} \\
x \mu_{{\scriptscriptstyle D}}^2 \mbox{\boldmath $a^\dagger$} &
\mu_{{\scriptscriptstyle D}}^2 + \tilde{m}_{\bar{D}}^2
\end{array}\right)~,\label{mrr} \\
%
{\mathcal M}_{RL}^2 & = & \left( \begin{array}{cc}
h_d v_d + m_d \mu_{{\scriptscriptstyle H}} \tan\!\beta
& M_5^2 \mbox{\boldmath $b$} \\
0 & B_{{\scriptscriptstyle D}} \mu_{{\scriptscriptstyle D}}
\end{array}\right)~.\label{mrl}\end{aligned}$$ In the expression for ${\mathcal M}_{RL}^2$ we have used $$M_5^2 b^i = h_{\gamma}^{ia}\langle\chi_a\rangle - \gamma^{ia}
\langle F_{\chi_a}\rangle~,\label{mrloffdiag}$$ where [$b$]{} is normalized to 1 and $F_{\chi_a}$ is the $F$-term for the $\chi_a$ field, which depends on the specific form of the soft SUSY breaking mass terms related to the spontaneous CP breaking part of the superpotential. The form of these squark mass matrices will be critical in the calculation of $\bar{\theta}$ in the next section. In particular, the $\langle F_{\chi_a}\rangle$’s bear complex phases independent of those in the $\langle\chi_a\rangle$’s in a generic setting, and hence constitute a major source of trouble.
We have implicitly assumed in the above that the sectors responsible for the Nelson-Barr texture and CP breaking are disjoint from those involved in the intrinsic breaking of supersymmetry. Since the successful example model we focus on has gauge-mediated SUSY breaking at a relatively low scale $M_{mess}$, it is [*a priori*]{} possible that all or part of the extra field content and symmetries required for the Nelson-Barr mechanism is contained in the SUSY breaking hidden and messenger sectors. Although this is an intriguing possibility, we have not been able to construct viable models of this type thus far. Actually, there appears to be an intrinsic incompatibility between the role a field takes in CP violation and the one it takes in SUSY breaking, as far as constraining $\bar{\theta}$ is concerned. Recall that in GMSB, the scalar particles get their soft SUSY breaking masses from the gauge interactions they share with the messenger sector particles which see SUSY breaking directly. The squark masses in each sector, for instance, would then be degenerate, as the process is flavor blind. The extra singlet $\bar{D}$ introduced here may easily upset the situation. Naively, the best strategy is to make the GMSB also blind to the aspon U(1). We will see below that this happens to have a even more important merit — it guarantees the suppression of the very dangerous $\left\langle F_{\chi_a}\right\rangle$’s. In other words, hiding the CP-breaking sector from SUSY breaking helps to suppress the SUSY loop contributions to $\bar{\theta}$. This is the less ambitious strategy we have taken in the model analyzed in detail below.
Calculation of 1-Loop $\bar{\theta}$ Constraints
================================================
Here we review and extend the work in Refs.[@DKL; @FK] to compute the 1-loop contributions to $\bar{\theta}$ of Figs.1a and 1b, using the mass-insertion approximation[@mia]. These results are generally valid for any type of low energy SCPV model, with or without the U(1)$_A$. The scale, $\mu_{\scriptscriptstyle D}$, and hence the characteristic scale of the SCPV, here is chosen near or below $\tilde{m}_{sq}$, the average squark mass. This is more or less dictated by the aspon scenario of weak CP[@FN]. The $D$ and $\bar{D}$ superfields are handled on the same footing as the other quark superfields. The analysis is basically the same as that given in Ref.[@FK] except here we pay full attention to the explicit phase factors and family indices, and also treat the $M_5^2$ term, as given by Eq.(\[mrloffdiag\]), in full detail. These turn out to be very important in understanding how the scenario can provide a feasible solution. In taking the large $\mu_{\scriptscriptstyle D}$ limit, which corresponds to situation discussed in Ref.[@DKL], one has to be careful in handling the loop momentum integrals properly. The latter are however not explicitly given in this paper, though they are included in our numerical computations.
The 1-loop contribution is given by $$\begin{aligned}
\delta\bar{\theta} &=& \text{Im Tr}\,
m_{\scriptscriptstyle F}^{-1} \delta\! m_{\scriptscriptstyle F}
+ 3 M_g^{-1} \delta\!M_g \nonumber \\
&=& \frac{\alpha_s}{4\pi} \sum_{i,I} {\rm Im}[Z^{iI*}Z^{(i+4)I}]
{\mathcal M}^2_{d{\scriptscriptstyle I}}
\left( \frac{M_g}{m_{\scriptscriptstyle F\!i}}
\frac{8/3}{M_g^2-{\mathcal M}^2_{d{\scriptscriptstyle I}}}
\ln \frac{{\mathcal M}^2_{d{\scriptscriptstyle I}} }{M_g^2}
+ \frac{m_{\scriptscriptstyle F\!i}}{M_g}
\frac{3}{{\mathcal M}^2_{d{\scriptscriptstyle I}}
- m_{\scriptscriptstyle F\!i}^2}
\ln \frac{{\mathcal M}^2_{d{\scriptscriptstyle I}} }{ m_{\scriptscriptstyle F\!i}^2}
\right) \; ,\end{aligned}$$ where $m_{\scriptscriptstyle F\!i}$ runs over the four eigenvalues of the quark mass matrix \[cf. Eq.(\[mf\])\] and ${\mathcal M}^2_{d{\scriptscriptstyle I}}$ over the eight eigenvalues for the squarks \[cf. Eq.(\[8sqm\])\], all in the down-sector; $Z^{IJ}$ is the unitary rotation that diagonalizes the squark mass matrix in the quark mass eigenstate basis. This full formula, while it can be used in the numerical calculations once all the quantities involved are known, hides its physics content behind the $Z$-matrix elements. In the limit of exact degeneracy and proportionality, the latter is just the identity matrix and $\bar{\theta}$ is zero. Otherwise, the mass-insertion approximation, as discussed below, is more illustrative.
We first assume an approximate degeneracy and that the diagonal blocks in ${\mathcal M}_d^2$ dominate over the off-diagonal block ${\mathcal M}^2_{RL}$ and write $$\begin{aligned}
\tilde{m}^2_{\bar{d}} = \bar{m}^2_{\bar{d}} \times 1\!\!\!1
+ \delta\tilde{m}^2_{\bar{d}} \; ,
\quad \tilde{m}^2_{\bar{\scriptstyle D}} = \bar{m}^2_{\bar{d}}
+ \delta\tilde{m}^2_{\bar{\scriptstyle D}} \; , \nonumber \\
\tilde{m}^2_{d} = \bar{m}^2_{d} \times 1\!\!\!1
+ \delta\tilde{m}^2_{d} \; ,
\quad \tilde{m}^2_{{\scriptstyle D}} = \bar{m}^2_{d}
+ \delta\tilde{m}^2_{{\scriptstyle D}} \; .\end{aligned}$$ The squarks are then treated as scalars of masses $\bar{m}^2_{\bar{d}}$ and $\bar{m}^2_{d}$ with the $\delta\tilde{m}^2_{..}$ and ${\mathcal M}^2_{RL}$ treated as admissible mass-insertions is the loop-diagrams Figs.1a and 1b. Explicit forms of the matrices needed to diagonalize $m_{\scriptstyle F}$ are useful. Expressions up to order $x^2$ are available in the literature[@x2]. To parametrize the effect of proportionality violation among the three families, we write $$h_d = \bar{A}_d Y_d + \delta\!A_d \; .$$ The situation for the related parameter in the $d$-$D$ mixings is more complicated. Recall that $M_5^2 b^i = h_{\gamma}^{ia}\langle\chi_a\rangle - \gamma^{ia}
\langle F_{\chi_a}\rangle$ \[Eq.(\[mrloffdiag\])\]. It has been emphasized in Ref.[@FK] that the $F$-terms being small is paramount to the success of any model of the Nelson-Barr type. These terms are dangerous because in general one has no reason to expect these $F$-terms to obey even an approximate proportionality (to the $x \mu_{\scriptscriptstyle D} a^i$ terms). On the contrary, contributions of the other part to $\bar{\theta}$ can be interpreted as a proportionality violation among the $\gamma^{ia}$’s by writing $$h^{ia}_{\gamma} = \bar{A}_{\gamma} \gamma^{ia}
+ \delta\!A_{\gamma}^{ia}\; ;$$ the term proportional to $\bar{A}_{\gamma}$ does not contribute. We further introduce the simplified notation: $$\delta\!A_{\gamma} c^i = \frac{1}{x \mu_{\scriptscriptstyle D}} h^{ia}_{\gamma}
\left\langle \chi_a \right\rangle - \bar{A}_{\gamma} a^i~, \label{dagam}$$ where complex vector [$c$]{} is normalized to 1. Hence, we have $$M_5^2 b^i = \bar{A}_{\gamma}( x \mu_{\scriptscriptstyle D} a^i )
+ \delta\!A_{\gamma} ( x \mu_{\scriptscriptstyle D} c^i ) - \gamma^{ia}
\left\langle F_{\chi_a}\right\rangle~. \label{mfiveb}$$
In terms of the above notation, the list of major contributions to $\bar{\theta}$ is given in Tables 1a and 1b. The $\bar{\theta}$ contributions involving $M_5^2$ are complicated. To make it easier to see the effects of the different parts, we list some of those terms in tables before and after the above mentioned splitting. For example, entry 1 in the Table 1a is split into two parts: the first part is a proportionality violation effect involving $\delta\!A_{\gamma}$ and Im($a^*_ic^i$) (both are suppressed in our model), the second is the $F$-term contribution ($\gamma^{ia} \left\langle F_{\chi_a}\right\rangle$), where the relevant complex phase is taken to be ${\cal O}(1)$. One other notable feature among the $\bar{\theta}$ contributions is the combination $M_5^2 b^i - x \mu_{\scriptscriptstyle D} B_{\scriptscriptstyle D} a^i$, as shown in entry 9 of Table 1b. When the $M_5^2 b^i$ term is split as above, the second term actually can be combined with the first term in Eq.(\[mfiveb\]) to give $x \mu_{\scriptscriptstyle D} (\bar{A}_{\gamma} -
B_{\scriptscriptstyle D}) a^i$, which can be interpreted as a proportionality violation among the corresponding trilinear and bilinear terms. The other parts involve $\delta\!A_{\gamma}$ and $\gamma^{ia} \left\langle F_{\chi_a}\right\rangle$, as explicitly shown in the table. All other entries with a $M_5^2$ can be split and interpreted in the same way. We will see in the final result that the $F$-term contribution [*is the most dangerous*]{}.
The Spontaneous CP Violation Sector
===================================
Spontaneous breaking of the U(1)$_A$ symmetry is the only source of CP violation in our model. This CP violation effect feeds directly into the $x \mu_{\scriptscriptstyle D}$ and $M_5^2$ terms in the quark and squark mass matrices, with complex phase vectors [$a$]{} and [$b$]{}, respectively. To implement the mechanism, we need a sector of U(1)$_A$-charged SM singlet superfield with a superpotential that not only gives rise to the complex $\left\langle \chi_a \right\rangle$’s, but also gives us a good control on the dangerous $\left\langle F_{\chi_a} \right\rangle$’s. Soft SUSY breaking terms should also be taken into consideration, when determining the true scalar potential. The $F$-terms, of course, characterize SUSY breaking. We consider the scenario in which the messengers communicating SUSY breaking to the visible sector are U(1)$_A$-blind, [*i.e.*]{} they do not carry any U(1)$_A$ charges; furthermore, they are not directly coupled to the CP-breaking sector. The superfields of the latter are then hidden from SUSY breaking.
We have to consider at least five superfields, two $\bar{\chi}$’s of conjugate U(1)$_A$ charges to the $\chi_a$’s and a singlet $\aleph$, in order to have both gauge anomaly cancellation and a possible CP violating vacuum solution[@FK; @4h]. We consider the superpotential[^7] $$W_{\chi} = \bar{\chi}_a \mu_{\chi}^{ab} \chi_{b}
+ \aleph \bar{\chi}_a \lambda^{ab} \chi_b
+ \lambda_{{\scriptscriptstyle \aleph}} \aleph^3
+ \mu_{{\scriptscriptstyle \aleph}} \aleph^2 \label{wchi} \; .$$ The five $F$-flat conditions yield four independent equations, which, together with the $D$-flat condition, give a unique vacuum solution. The solution is CP violating for most of the parameter space. Hence, neglecting the soft SUSY breaking terms, we have a SUSY preserving vacuum that breaks CP.
The GMSB scenario we considered allows the unwanted soft SUSY breaking terms of the sector to be zero at $M_{mess}$. They are, however, generated through RG evolution, as discussed in the next section. With their nonvanishing values taken into consideration, the scalar potential is then given by $$V_{\chi} = D_{\chi}^2 + F^{}_{\chi_a} F_{\chi_a}^*
+ F^{}_{\bar{\chi}_a} F_{\bar{\chi}_a}^* + F^{}_{\scriptscriptstyle \aleph}
F_{\scriptscriptstyle \aleph}^* + V_{s\chi}$$ where $$\begin{aligned}
V_{s\chi} &=&
\bar{\chi}_a B_{\chi}^{ab} \chi_b
+ \bar{\chi}_a h_{\lambda}^{ab} \chi_b \aleph
+ h_{\scriptscriptstyle \aleph} \aleph^3
+ B_{{\scriptscriptstyle \aleph}}\aleph^2 \nonumber \\
& &
+ \bar{\chi}^{\dagger}_a \tilde{m}_{\bar{\chi} ab}^2
\bar{\chi}_b^{}
+ \chi_a^{} \tilde{m}_{\chi ab}^2 \chi_b^{\dagger}
+ \aleph^{\dagger} \tilde{m}_{\scriptscriptstyle \aleph}^2 \aleph \; .\end{aligned}$$ Solving for the potential minimum to determine the $\left\langle F_{\chi_a}\right\rangle$ values is not tractable, as $W_{\chi}$ and $V_{s\chi}$ involves a large number of parameters which are not otherwise constrained, apart from yielding a CP violating solution. However, one can easily obtain a reasonable order of magnitude estimate of the shifts in the $\left\langle F_{\chi_a}\right\rangle$’s as a result of including the small $V_{s\chi}$ terms. For example, the equation $$\frac{\partial V_{\chi}}{\partial \bar{\chi}_a}
= 2D_{\chi}\frac{\partial D_{\chi}}{\partial \bar{\chi}_a}
- ( \mu_{\chi}^{ab} + \aleph \lambda^{ab} ) F_{\chi_b}
+ F_{\scriptscriptstyle \aleph} \frac{\partial
F_{\scriptscriptstyle \aleph}^*}{\partial \bar{\chi}_a}
+ B_{\chi}^{ab} \chi_b + h_{\lambda}^{ab} \chi_b \aleph
+ \bar{\chi}^{\dagger}_b \tilde{m}_{\bar{\chi} ba}^2 =0$$ suggests that $\left\langle F_{\chi}\right\rangle$ (here we drop all indices and phases) is given by the magnitude of $$B_{\chi} \quad
\text{or} \quad \tilde{m}_{\bar{\chi}}^2 \quad\
\text{or} \quad
h_{\lambda} \left\langle \chi \right\rangle / \lambda \; .\label{Fest}$$ An alternative way to estimate the $\left\langle F_{\chi_a}\right\rangle$’s is given by the SUSY breaking diagrams shown in Fig.2. Here Figs.2a and 2b give exactly the same results as the first two terms listed above. Figure 2c, however, gives the estimate $\left\langle F_{\chi}\right\rangle \sim h_{\lambda}\left\langle \chi
\right\rangle \lambda / 16\pi^2$. For perturbative values of the $\lambda$ coupling, this is of course smaller than the third estimate in Eq.(\[Fest\]), hence we neglect it. A similar diagram, Fig.2d, also suggests a contribution $\sim h_{\gamma}\left\langle \chi \right\rangle \gamma / 16\pi^2$, though the $\gamma$ dependence of the $\left\langle F_{\chi} \right\rangle$’s is implicitly incorporated into the generation of the $V_{s\chi}$ terms through RG running. We will use all these in our numerical estimates to determine whether the $F$-term is sufficiently small that its contributions to $\bar{\theta}$, listed in Tables 1 and 2, are under control. Finally, we emphasize again that the complex phases in the $\left\langle F_{\chi_a}\right\rangle$’s are not related to those of the $\left\langle \chi_a \right\rangle$’s directly.
Renormalization Group Analysis
==============================
As pointed out in the introduction, we need a full theory for the soft SUSY breaking parameters to see if the $\bar{\theta}$ constraints can be satisfied and the GMSB scenario may provide the only viable possibility. In particular, we use here only the minimal version of such a theory [@MMM]. This version has a few special merits: it provides practically a one-parameter model of soft SUSY breaking, radiative breaking of electroweak symmetry is naturally implemented and, within the MSSM framework, it has been studied with extensive renormalization group analysis and shown to be compatible with all known experimental constraints[@Bor]. From our perspective of solving the strong CP problem by augmenting the ${\mathrm{U(1)}}_A$ sector, it actually represents a relatively demanding setting among GMSB models, where a large $\tan\!\beta$ allows all the Yukawa couplings of the third family to have substantial effects on the RG-runnings. A smaller $\tan\!\beta$ in general would only make it easier to satisfy the $\bar{\theta}$ constraints.
We will refrain from elaborating extensively on the details of the GMSB model or the RG-analysis itself. For more specific details on finding the correct electroweak symmetry breaking vacuum and meeting other experimental constraints in the minimal GMSB model, readers are referred to Ref.[@Bor]. [*Our interest here is in adapting the machinery to our extended model at a level of sufficient sophistication to calculate $\bar{\theta}$ to 1-loop and establish our solution to the strong CP problem.*]{}
We use 1-loop renormalization group equations (RGE’s) with naive step thresholds between $M_Z = M_{SUSY}$ and $M_{mess}$. The RG-improved tree level Higgs potential is considered in finding the electroweak-symmetry breaking solution. The RGE’s for the extra content of the model are also implemented at the 1-loop level; relevant formulae are in Appendix A. The ${\mathrm{U(1)}}_A$, and hence CP symmetry, breaking is imposed by hand. The idea is to study the general situation independent of the details of the SCPV sector, as the latter is to a certain extent more flexible and less constrained. It is important to note that the extra superfield content in the model is partially decoupled from the MSSM part, with the only direct coupling being gauge couplings of $D$ and $\bar{D}$, and the small Yukawa couplings $\gamma^{ia}$. The computation concerning the SCPV sector, as well as the RG analysis can certainly be made more sophisticated, however, we consider our treatment sufficient for our purpose. In the sample analysis for which numerical results are presented in this paper in detail (Appendix B and the last column of Table 1), the values of the various $\gamma^{ia}$ Yukawa couplings are generated randomly in the range $0.005 - 0.01$. The latter is chosen to target an $x$-value of around $0.01$. The value of $\mu_{\scriptscriptstyle D}$ is fixed at $500$ GeV; $M_{mess} \sim \Lambda$ at $50$ TeV. For the soft SUSY breaking parameters from GMSB, all $A$- and $B$- terms are taken to be zero at $M_{mess}\equiv X$. The scalar soft masses from GMSB are given by $${\tilde{m}^2(X)} = \frac{\Lambda^2}{8\pi} \left\{
C_3 \,\alpha_3^2(X) + C_2 \,\alpha_2^2(X)
+ \frac{3}{5} Y^2 \,\alpha_1^2(X)
\right\}\,f(y) \; ,
\label{bcscal}$$ where $C_3 = 4/3, 0$ for triplets and singlets of $SU(3)_C$, $C_2 =3/4,0$ for doublets and singlets of $SU(2)_L$; $Y =Q-T_3$ is the hypercharge. The function $f(y)$, derived in Ref.[@fx], is simply set to 1. Note that the above formula is independent of the ${\mathrm{U(1)}}_A$ charge; a SM singlet with or without ${\mathrm{U(1)}}_A$ charge, such as the $\chi_a$ and $\aleph$ scalars, has no initial soft mass. Gaugino masses are likewise given by the MSSM formula, omitted here. The new ${\mathrm{U(1)}}_A$ gaugino (aspino) has no tree-level SUSY breaking mass. The gauge coupling $g_A$ is taken to be around $g_{em}$. The SUSY-breaking aspino mass $M_A$ then remains vanishingly small even after finite loop effects and RG evolution are taken into account.
After the electroweak symmetry breaking solution is obtained, various $\bar{\theta}$ contributions are calculated through the mass-insertion approximation to order $x^2$. To impose the ${\mathrm{U(1)}}_A$ symmetry breaking, we set, in the sample analysis, $|\left\langle \chi_1 \right\rangle |^2 + |\left\langle \chi_2
\right\rangle |^2 \simeq \mu_{\scriptscriptstyle D}^2$ and choose random values for the VEVs and their complex phases within the constraint. Effects of higher order in $x$ are checked to be insignificant.
Values of parameters in $W_{\chi}$ are needed for the RG-runnings of particularly the SCPV sector soft SUSY breaking parameters, discussed in the previous section. To simplify the situation, we input all these mass parameters as $ \mu_{{\scriptscriptstyle D}}$ and all dimensionless couplings as random numbers in the range $0.1-0.8$, for the sample calculation. This oversimplification certainly begs the question of consistency of the vacuum solution for this sector, or the whole model. However, in the small $x$ domain of interest, the influence of the extra ingredients on the values of the other MSSM parameters is insignificant, as to be expected. The only practical effect of those parameters is in the RG evolution of the related soft terms which we needed to estimate the $\left\langle F_{\chi} \right\rangle$’s. We have checked, for instance, that the particular input values used in the sample run reported here does lead to generic magnitudes of the latter.
Appendix B contains a collection of some of the numerical results, while those for the $\bar{\theta}$ contributions, [*without*]{} the $\left\langle F_{\chi}
\right\rangle$’s are listed in the last column of Table 1. Estimates of the $\left\langle F_{\chi} \right\rangle$, following the discussion in the previous section, and their contribution to $\bar{\theta}$ are given in Table 2 (second column). The latter can be easily checked using the $\left\langle F_{\chi} \right\rangle$ value and the listing of $M_5^2$-terms in Table 1. We also list in Table 2 results from a number of different runs with different values of the $\gamma$’s (reflected by the $x$-value obtained) and $\lambda$’s. The former, which can have a significant effect on the various MSSM parameters, are restricted by $x^2 \sim 10^{-3}$ – $10^{-5}$ from the weak CP considerations. Our results indicate that only a relatively large value of $x$ can upset the strong CP solution, by first driving $\left\langle F_{\chi} \right\rangle$ too large (see column 4 of Table 2). One should be cautious in using this result quantitatively, as our $\left\langle F_{\chi} \right\rangle$ estimates are meant to be conservative upper bounds. However, the result is certainly illustrative of the importance of the $\left\langle F_{\chi}\right\rangle$ in estimating $\bar{\theta}$. With $x$ restricted to the workable range, the basic features of the RGE solutions are quite stable. This is true even with a relatively large variation of the $\lambda$’s, as illustrated by column 5 and 6 of Table 2. Note that though the $h_\lambda \left\langle \chi \right\rangle / \lambda$ term may have an explicit dependence on $\lambda$, its numerical value does not have a large variation with $\lambda$ as one might naively expect. This is, like the approximate proportionality of a general $A$-term, a natural result of the RG equations.
All in all, the $F$-term contributions to $\bar{\theta}$ dominate, and the overall $\bar{\theta}$ value is comfortably within the required bound for the major region of the parameter space of our model under consideration, hence solving the strong CP problem.
Conclusions
===========
To recapitulate, we discussed a complete spontaneous CP violation model with gauge-mediated supersymmetry breaking and why this type of model is particularly favored over a generic supersymmetric SCPV model in solving the strong CP problem. Results from numerical RGE studies are used to explicitly establish the feasibility of the approach. The treatment of parameters in the SCPV sector is admittedly oversimplified. The correlations between $x$ and $\mu_{\scriptscriptstyle D}$, and between the various mass parameters at the $\mu_{\scriptscriptstyle D}$ scale and the values of the various $\lambda$ coupling, for instance, are neglected. However, it is easy to see from our discussion that such details are not going to change the essential features of our results, though they would determine explicitly the specific “large-$x$" region of the parameter space that could be ruled out. The model has a rich spectrum of new particles at the $\mu_{\scriptscriptstyle D}$ or SCPV scale. Until such experimental data become available, a detailed study of the parameter space may not be feasible.
While the weak CP aspects of this model have been analyzed in the non-supersymmetric setting as in Ref. [@FN], SUSY particles could lead to new contributions through super-box and penguin diagrams. These contributions are in general subdominant, as are the Standard Model box diagrams.
Our model predicts a measureable neutron EDM, which could be close to the present experimental bound for $x >.01$. The model also has an extra pair of vector-like quark superfields, a new gauge boson, and a number of neutral fermions and scalars with no direct couplings to the Standard Model gauge bosons, all with masses around the TeV scale. This scale is dictated by the weak CP phenomenology. Hence it offers a rich spectrum of new particles to be discovered at the CERN Large Hadron Collider. Because the $3\times 3$ Cabbibo-Kobayashi-Maskawa matrix is approximately real, the model also predicts a flat unitarity triangle and the absence of substantial CP violation in the $B$ system at future $B$ factories[@asponB]. Moreover, there will be a lack of any substantial CP violating effects in the up-quark sector.
The authors want to thank P.H. Frampton for being a constant source of support and encouragement. O.K. is in debt to colleagues in Rochester, where the major part of this manuscript was finished. Special thanks go to M. Bisset for reading the manuscript. B.W. thanks P. Sikivie for discussions. The authors were supported in part by the U.S. Department of Energy under Grant DE-FG05-85ER-40219, Task B; O.K. was also supported in part by the U.S. Department of Energy under Grant DE-FG02-91ER-40685.\
Renormalization Group Equations
===============================
Here we collect the modified one loop MSSM renormalization group equations (RGE’s) to account for the extra vector-like chiral superfields and the new Yukawa couplings in our model. In many cases we give only the extra contributions and refer the interested reader to Ref. [@MV] with whom we share conventions. The complete two-loop renormalization group equations for a softly broken supersymmetric theory can be found in Refs.[@MV; @RGE].
The complete superpotential can be written as $$\begin{aligned}
W & = & \bar{u}Y_u Q H_u + \bar{d} Y_d Q H_d + \bar{e} Y_e L H_d +
\mu_{{\scriptscriptstyle H}} H_u H_d + \bar{d}_i \gamma^{ia} D \chi_a
+ \mu_{{\scriptscriptstyle D}} \bar{D} D\nonumber\\
&&{} + \bar{\chi}_a \mu_{\chi}^{ab} \chi_{b}
+ \aleph \bar{\chi}_a \lambda^{ab} \chi_b
+ \lambda_{{\scriptscriptstyle \aleph}} \aleph^3
+ \mu_{{\scriptscriptstyle \aleph}} \aleph^2
\label{superpot}\end{aligned}$$ where family indices are implicit except in the new Yukawa coupling and $a,b = 1,2$. The soft supersymmetry breaking Lagrangian can be written as $\mathcal{L}_{\text{soft}} = \mathcal{L}_{\text{soft}}^
{\scriptscriptstyle{\text{MSSM}}} + \mathcal{L}_{\text{soft}}^{\text{extra}}$, where $$\begin{aligned}
- \mathcal{L}_{\text{soft}}^
{\scriptscriptstyle{\text{MSSM}}} & = & \hat{\bar{u}} h_u \hat{Q} H_u
+ \hat{\bar{d}} h_d \hat{Q} H_d + \hat{\bar{e}} h_e \hat{L} H_d
+ B_{{\scriptscriptstyle H}} \mu_{{\scriptscriptstyle H}}H_u H_d
+ \text{h.c.} \nonumber\\
&&{} + \hat{Q}^{\dagger}\tilde{m}_Q^2 \hat{Q}
+ \hat{L}^{\dagger}\tilde{m}_L^2 \hat{L}
+ \hat{\bar{u}} \tilde{m}_{\bar{u}}^2 \hat{\bar{u}}^{\dagger}
+ \hat{\bar{d}} \tilde{m}_{\bar{d}}^2 \hat{\bar{d}}^{\dagger}
+ \hat{\bar{e}} \tilde{m}_{\bar{e}}^2 \hat{\bar{e}}^{\dagger}\label{mssmsoft}\\
&&{} + m^2_{H_u} H_u^{\dagger} H^{}_u
+ m^2_{H_d} H_d^{\dagger} H^{}_d~, \nonumber\end{aligned}$$ and $$\begin{aligned}
- \mathcal{L}_{\text{soft}}^
{\text{extra}} & = & \hat{\bar{d}}_i h_{\gamma}^{ia} \hat{D} \chi_a
+ B_{{\scriptscriptstyle D}} \mu_{{\scriptscriptstyle D}}\hat{\bar{D}} \hat{D}
+ \bar{\chi} B_{\chi}^{ab} \chi
+ \bar{\chi}_a h_{\lambda}^{ab} \chi_b \aleph
+ h_{\scriptscriptstyle \aleph} \aleph^3
+ B_{\scriptscriptstyle \aleph} \aleph^2 + \text{h.c.}
\nonumber\\
&&{} + \hat{\bar{D}} \tilde{m}_{\bar{\scriptscriptstyle D}}^2
\hat{\bar{D}}^{\dagger}
+ \hat{D}^{\dagger}\tilde{m}_{\scriptscriptstyle D}^2 \hat{D}
+ \bar{\chi}^{\dagger}_a \tilde{m}_{\bar{\chi} ab}^2
\bar{\chi}_b^{}
+ \chi_a^{} \tilde{m}_{\chi ab}^2 \chi_b^{\dagger}
+ \aleph^{\dagger} \tilde{m}_{\scriptscriptstyle \aleph}^2 \aleph~.
\label{extrasoft}\end{aligned}$$ Note that $B_{\chi}$ and $B_{{\scriptscriptstyle \aleph}}$ are defined in a different way from $B_{{\scriptscriptstyle D}}$ and $B_{{\scriptscriptstyle H}}$; the former have dimension (mass)$^2$ and are analogs of $B_{{\scriptscriptstyle D}} \mu_{{\scriptscriptstyle D}}$ and $B_{{\scriptscriptstyle H}} \mu_{{\scriptscriptstyle H}}$. Also, $B_{\chi}$ is a $2\times 2$ matrix. Finally we have supersymmetry breaking gaugino masses $M_a (a = 1,2,3)$ for the MSSM and a possible gaugino mass $M_A$ under ${\mathrm{U(1)}}_A$.
We give the MSSM one loop RGE’s for these interactions below. The gauge couplings are computed to two loops and are given by $$\frac{dg_a}{dt} = \frac{g_a^3}{16\pi^2} B^{(1)}_a + \frac{g_a^3}{(16\pi^2)^2}
\left(\sum_{b=1}^{3} B^{(2)}_{ab} g_b^2 - \sum_{x=u,d,e,\gamma} C^x_a
{\mathrm{Tr}}(Y_x^{\dagger} Y_x^{})\right)~,\label{gaugerge}$$ where $B^{(1)} = (\frac{33}{5} + \frac{2}{5}N_D + \frac{3}{5}N_L,
1 + N_L, -3 + N_D)$, $$B^{(2)} = \left(\begin{array}{ccc}
\frac{199}{25} + \frac{8}{75}N_D
+ \frac{9}{25}N_L & \frac{27}{5} + \frac{9}{5}N_L & \frac{88}{5}
+ \frac{32}{15}N_D \\
\frac{9}{5} + \frac{3}{5}N_L & 25 + 7 N_L & 24 \\
\frac{11}{5} + \frac{4}{15}N_D & 9 & 14 + \frac{34}{3}N_D \end{array} \right)~,$$ and $$C^{u,d,e,\gamma} = \left( \begin{array}{cccc}
\frac{26}{5} & \frac{14}{5} & \frac{18}{5} & \frac{4}{5}\\
6 & 6 & 2 & 0 \\
4 & 4 & 0 & 2
\end{array} \right)~.$$ In the above we have allowed for the possibility of $N_D$ heavy vector-like pairs of charge $\frac{1}{3}$ color triplets \[SU(2) singlets\] and $N_L$ such pairs of hypercharge $-1$ SU(2) doublets (color singlets). These include both the extra mirror pairs $D + \bar{D}$ and $L + \bar{L}$ which can interact directly with MSSM fields, but also extra mirror pairs originating in the messenger sector at higher scales. We have for simplicity omitted the possible Yukawa interactions of the extra mirror lepton doublets and the two loop ${\mathrm{U(1)}}_A$ contributions. In the actual computations, we use $N_D=1$, $N_L=0$. At one loop we also have $$16\pi^2\frac{dg_A}{dt} = 10 g_A^3~,\label{garge}$$ when the ${\mathrm{U(1)}}_A$ is present.
Using the above definitions, the two loop gaugino mass equations are $$\begin{aligned}
\frac{dM_a}{dt} & = & \frac{2g_a^2}{16\pi^2} B^{(1)}_a M_a
+ \frac{2g_a^3}{(16\pi^2)^2} \Big[ \sum_{b=1}^{3} B^{(2)}_{ab} g_b^2
\left(M_a + M_b\right) \nonumber\\
&&{} + \sum_{u,d,e,\gamma} C^x_a
\left({\mathrm{Tr}}(Y^{\dagger}_x h_x^{})
- M_a {\mathrm{Tr}}(Y^{\dagger}_x Y_x^{})
\right)\Big]~.\label{mgauginorge}\end{aligned}$$ At one loop we have $$16\pi^2\frac{dM_A}{dt} = 20 g_A^2 M_A~.\label{marge}$$
The running down quark Yukawa matrix is modified by the presence of the $D$-$\bar{d}$ couplings $\gamma^{ia}$ which are treated as $3\times 2$ matrices below. We have for the up and down one loop Yukawas: $$\begin{aligned}
16\pi^2 \frac{dY_d}{dt} & = & Y_d\left(
{\mathrm{Tr}}\left(3Y_d^{\dagger}Y_d^{} + Y_e^{\dagger}Y_e^{}\right) +
3Y_d^{\dagger}Y_d^{} + Y_u^{\dagger}Y_u^{} - \frac{16}{3}g_3^2 - 3 g_2^2
-\frac{7}{15}g_1^2\right) + \gamma\gamma^{\dagger}Y_d~,\label{ydrge}\\
%
16\pi^2 \frac{dY_u}{dt} & = & Y_u\left(
3{\mathrm{Tr}}\left(Y_u^{\dagger}Y_u^{}\right) +
3Y_u^{\dagger}Y_u^{} + Y_d^{\dagger}Y_d^{} - \frac{16}{3}g_3^2 - 3 g_2^2
-\frac{13}{15}g_1^2\right)~,\label{yurge}\end{aligned}$$ and for $\gamma$: $$16\pi^2\frac{d\gamma}{dt} = \gamma\left(
{\mathrm{Tr}}\left(\gamma^{\dagger}\gamma\right) +
2\gamma^{\dagger}\gamma + \lambda^{\dagger}\lambda
- \frac{16}{3}g_3^2 -\frac{4}{15}g_1^2 -
4 g_A^2\right) + Y_d^{} Y_d^{\dagger}\gamma~,\label{gammarge}$$ where for $\gamma$ we have included the effect of a possible extra ${\mathrm{U(1)}}_A$ as described in the text.
The running supersymmetric $\mu_{{\scriptscriptstyle D}}$ parameter is given by $$16\pi^2\frac{d\mu_{{\scriptscriptstyle D}}}{dt} =
\mu_{{\scriptscriptstyle D}}\left({\mathrm{Tr}}\left(\gamma^{\dagger}
\gamma\right) - \frac{16}{3}g_3^2 - \frac{4}{15}g_1^2 -
4g_A^2\right)~.\label{mudrge}$$ The equation for the corresponding soft mass parameter is $$16\pi^2\frac{dB_{{\scriptscriptstyle D}}}{dt} =
2 {\mathrm{Tr}}\left(\gamma^{\dagger}
h_{\gamma}\right) + \frac{32}{3}g_3^2M_3 + \frac{8}{15}g_1^2M_1
+ 8g_A^2 M_A~.\label{bdrge}$$ Similar parameters for the SCPV sector have $$16\pi^2\frac{d\mu_{{\scriptscriptstyle \chi}}}{dt} =
\mu_{{\scriptscriptstyle \chi}}\lambda^{\dagger}\lambda
+ \lambda\lambda^{\dagger} \mu_{{\scriptscriptstyle \chi}}
+3 \mu_{{\scriptscriptstyle \chi}}\gamma^{\dagger}\gamma
-4g_A^2 \mu_{{\scriptscriptstyle \chi}}~,$$ $$\begin{aligned}
16\pi^2\frac{dB_{{\scriptscriptstyle \chi}}}{dt} &=&
B_{{\scriptscriptstyle \chi}} \left(\lambda^{\dagger} \lambda
+ 3 \gamma^{\dagger} \gamma \right)
+ \lambda \lambda^{\dagger} B_{{\scriptscriptstyle \chi}}
+ \lambda\left[{\mathrm{Tr}}\left(\lambda^{\dagger}
B_{{\scriptscriptstyle \chi}}\right) + \lambda_{\aleph}
B_{{\scriptscriptstyle \aleph}}\right]
\nonumber \\
& &
+ 2 \mu_{{\scriptscriptstyle \chi}}\lambda^{\dagger} h_{\lambda}
+ 2 h_{\lambda}\lambda^{\dagger} \mu_{{\scriptscriptstyle \chi}}
+ 6 \mu_{{\scriptscriptstyle \chi}}\gamma^{\dagger} h_{\gamma}
- 4\left( B_{{\scriptscriptstyle \chi}} - 2 \mu_{{\scriptscriptstyle \chi}}
M_A\right) g_A^2~,\end{aligned}$$ and $$16\pi^2\frac{d\mu_{{\scriptscriptstyle \aleph}}}{dt} =
\mu_{{\scriptscriptstyle \aleph}}
\left[\lambda_{{\scriptscriptstyle \aleph}}^2
+2 {\mathrm{Tr}}\left(\lambda^{\dagger}\lambda\right)\right]~,$$ $$\begin{aligned}
16\pi^2\frac{dB_{{\scriptscriptstyle \aleph}}}{dt} &=&
B_{{\scriptscriptstyle \aleph}} \left[ \lambda_{{\scriptscriptstyle \aleph}}^2
+ 2 {\mathrm{Tr}}\left(\lambda^{\dagger}\lambda\right)\right]
+ \lambda_{\scriptscriptstyle \aleph}
\left[\lambda_{\scriptscriptstyle \aleph} B_{{\scriptscriptstyle \aleph}}
+2 {\mathrm{Tr}}\left(\lambda^{\dagger}B_{{\scriptscriptstyle \chi}}\right)\right]
\nonumber \\
& &
+ 2 \mu_{{\scriptscriptstyle \aleph}} \left[\lambda_{{\scriptscriptstyle \aleph}}
h_{{\scriptscriptstyle \aleph}}
+2 {\mathrm{Tr}}\left(\lambda^{\dagger}h_{\lambda}\right)\right]~.\end{aligned}$$ There are also RGE’s for the extra Yukawa couplings: $$\begin{aligned}
16\pi^2\frac{d\lambda}{dt} & = & \lambda\left(
{\mathrm{Tr}}\left(\lambda^{\dagger}\lambda\right)
+ \frac{1}{2} \lambda_{{\scriptscriptstyle \aleph}}^2
+ 2 \lambda^{\dagger}\lambda
+ 3 \gamma^{\dagger}\gamma
-4g_A^2 \right)~, \\
16\pi^2\frac{d\lambda_{{\scriptscriptstyle \aleph}}}{dt} & = &
\lambda_{{\scriptscriptstyle \aleph}}
\left[\lambda_{{\scriptscriptstyle \aleph}}^2 +
2 {\mathrm{Tr}}\left(\lambda^{\dagger}\lambda\right) \right]~.\end{aligned}$$ Note that $\mu_{{\scriptscriptstyle \chi}}~, B_{{\scriptscriptstyle \chi}}~,
\lambda$ and $h_{\lambda}$ are all $2\times 2$ matrices.
The relevant soft supersymmetry breaking trilinear coupling RGE’s are given by $$\begin{aligned}
16\pi^2\frac{dh_d}{dt} & = & h_d\left(
{\mathrm{Tr}}\left(3Y_d^{\dagger}Y_d^{} + Y_e^{\dagger}Y_e^{}\right) + 5
Y_d^{\dagger}Y_d^{} + Y_u^{\dagger}Y_u^{} - \frac{16}{3}g_3^2 - 3 g_2^2
-\frac{7}{15}g_1^2\right) + \gamma\gamma^{\dagger}h_d\nonumber\\
&&{} + Y_d\Bigl({\mathrm{Tr}}\left(6Y_d^{\dagger}h_d^{}
+ 2Y_e^{\dagger}h_e^{}\right) + 4Y_d^{\dagger}h_d^{}
+ 2Y_u^{\dagger}h_u^{}\nonumber\\
&&{} + \frac{32}{3}g_3^2M_3 + 6 g_2^2M_2
+ \frac{14}{15}g_1^2M_1\Bigr) +
2 h_{\gamma}\gamma^{\dagger}Y_d~,\label{hdrge}\\
%
16\pi^2\frac{dh_u}{dt} & = & h_u\left(
3{\mathrm{Tr}}\left(Y_u^{\dagger}Y_u^{}\right) + 5Y_u^{\dagger}Y_u^{}
+ Y_d^{\dagger}Y_d^{} - \frac{16}{3}g_3^2 - 3 g_2^2
-\frac{13}{15}g_1^2\right) \nonumber\\
&&{} + Y_u\Bigl(6{\mathrm{Tr}}\left(Y_u^{\dagger}h_u^{}\right)
+ 4Y_u^{\dagger}h_u^{}
+ 2Y_d^{\dagger}h_d^{}\nonumber\\
&&{} + \frac{32}{3}g_3^2M_3 + 6 g_2^2M_2
+ \frac{26}{15}g_1^2M_1\Bigr)~,\label{hurge}\\
%
16\pi^2\frac{dh_{\gamma}}{dt} & = & h_{\gamma}\left(
{\mathrm{Tr}}\left(\gamma^{\dagger}\gamma\right)
+ 3\gamma^{\dagger}\gamma + \lambda^{\dagger} \lambda
- \frac{16}{3}g_3^2
- \frac{4}{15}g_1^2 - 4 g_A^2\right)
+ Y_d^{} Y_d^{\dagger}h_{\gamma}\nonumber\\
&&{} + \gamma\left(2{\mathrm{Tr}}\left(\gamma^{\dagger}h_{\gamma}\right)
+ 3\gamma^{\dagger}h_{\gamma} + \lambda^{\dagger} h_{\lambda}
+ \frac{32}{3}g_3^2M_3
+ \frac{8}{15}g_1^2M_1 + 8 g_A^2 M_A\right) \nonumber \\
&&{} +2 h_d^{} Y_d^{\dagger}\gamma~, \label{hgammarge} \\
%
16\pi^2\frac{dh_{\lambda}}{dt} & = & h_{\lambda}\left(
{\mathrm{Tr}}\left(\lambda^{\dagger}\lambda\right)
+ 3\gamma^{\dagger}\gamma + 3 \lambda^{\dagger} \lambda
+ \frac{1}{2} \lambda_{{\scriptscriptstyle \aleph}}^2
- 4 g_A^2\right) \nonumber\\
&&{} + \lambda\left(2{\mathrm{Tr}}\left(\lambda^{\dagger}h_{\lambda}\right)
+ 6\gamma^{\dagger}h_{\gamma} + 3 \lambda^{\dagger} h_{\lambda}
+ \lambda_{{\scriptscriptstyle \aleph}} h_{{\scriptscriptstyle \aleph}}
+ 8 g_A^2 M_A\right) ~,\\
%
16\pi^2\frac{dh_{{\scriptscriptstyle \aleph}}}{dt} & = &
h_{{\scriptscriptstyle \aleph}}\left(\frac{9}{2}
\lambda_{{\scriptscriptstyle \aleph}}^2
+ 3 {\mathrm{Tr}}\left(\lambda^{\dagger}\lambda\right)\right)
+ 6 \lambda_{{\scriptscriptstyle \aleph}}
{\mathrm{Tr}}\left(\lambda^{\dagger}h_{\lambda}\right)\; .\end{aligned}$$
Finally we give the modifications of the MSSM soft hermitian quadratic mass parameters (see Ref.[@MV] for a complete description). All of the MSSM RGE’s have the following change in a D-term contribution: the factor $\mathcal{S}$ defined in Eq.(4.27) of [@MV] is now given by $${\mathcal{S}} = m_{H_u}^2 - m_{H_d}^2 -
\tilde{m}_D^2 + \tilde{m}_{\bar{D}}^2
+ {\mathrm{Tr}}\left(\tilde{m}_Q^2 - \tilde{m}_L^2 -
2\tilde{m}_{\bar{u}}^2 + \tilde{m}_{\bar{d}}^2 +
\tilde{m}_{\bar{e}}^2\right)~.\label{bigs}$$ Besides this modification, the equation for $\tilde{m}_{\bar{d}}^2$ has the only nontrivial change due to the coupling $\gamma$: $$\begin{aligned}
16\pi^2\frac{d\tilde{m}_{\bar{d}}^2}{dt} & = &
\left(2\tilde{m}_{\bar{d}}^2 + 4m_{H_d}^2\right)Y_d^{} Y_d^{\dagger}
+ 4 Y_d^{} \tilde{m}_{Q}^2 Y_d^{\dagger}
+ 2 Y_d^{} Y_d^{\dagger}\tilde{m}_{\bar{d}}^2
+ \gamma\gamma^{\dagger}\tilde{m}_{\bar{d}}^2
+ \tilde{m}_{\bar{d}}^2\gamma\gamma^{\dagger}\nonumber\\
&&{} + 2 \tilde{m}_D^2 \gamma\gamma^{\dagger}
+ 2 \gamma\tilde{m}_{\chi}^2\gamma^{\dagger}
+ 4h_d^{} h_d^{\dagger} + 2h_{\gamma}^{}h_{\gamma}^{\dagger}\nonumber\\
&&{} - \frac{32}{3}g_3^2\left|M_3\right|^2
- \frac{8}{15}g_1^2\left|M_1\right|^2
+ \frac{2}{5}g_1^2{\mathcal{S}}~,\label{msqdbarrge}\end{aligned}$$ where $\tilde{m}_{\chi}^2$ is a $2\times 2$ matrix. The equations for the new soft squark masses are $$\begin{aligned}
16\pi^2\frac{d\tilde{m}_{D}^2}{dt} & = & 2
{\mathrm{Tr}}\left(\tilde{m}_{D}^2\gamma^{\dagger}\gamma
+ \gamma\tilde{m}_{\chi}^2\gamma^{\dagger}
+ \gamma^{\dagger}\tilde{m}_{\bar{d}}^2\gamma
+ h_{\gamma}^{\dagger}h_{\gamma}^{}\right)\nonumber\\
&&{} - \frac{32}{3}g_3^2\left|M_3\right|^2
- \frac{8}{15}g_1^2\left|M_1\right|^2
- 8 g_A^2\left|M_A\right|^2 - \frac{2}{5}g_1^2{\mathcal{S}}
+ 2 g_A^2{\mathcal{S_A}}~,\label{msqbigdrge}\\
%
16\pi^2\frac{d\tilde{m}_{\bar{D}}^2}{dt} & = &
- \frac{32}{3}g_3^2\left|M_3\right|^2
- \frac{8}{15}g_1^2\left|M_1\right|^2
- 8 g_A^2\left|M_A\right|^2 + \frac{2}{5}g_1^2{\mathcal{S}}
- 2 g_A^2{\mathcal{S}}_A~,\label{msqbigdbarrge}\\
%
16\pi^2\frac{d\tilde{m}_{\chi}^2}{dt} & = &
3\gamma^{\dagger}\gamma \tilde{m}_{\chi}^2
+ 3\tilde{m}_{\chi}^2\gamma^{\dagger}\gamma
+ 6\gamma^{\dagger}\gamma \tilde{m}_{D}^2
+ 6\gamma^{\dagger}\tilde{m}_{\bar{d}}^2\gamma
+ 6 h_{\gamma}^{\dagger} h_{\gamma}^{}\label{msqchirge}\\
&&{} + \lambda^{\dagger}\lambda \tilde{m}_{\chi}^2
+ \tilde{m}_{\chi}^2\lambda^{\dagger}\lambda
+ 2\lambda^{\dagger}\lambda \tilde{m}_{{\scriptscriptstyle \aleph}}^2
+ 2\lambda^{\dagger}\tilde{m}_{\bar{\chi}}^2\lambda
+ 2 h_{\lambda}^{\dagger} h_{\lambda}^{}
- 8 g_A^2\left|M_A\right|^2
- 2 g_A^2{\mathcal{S}}_A~,\nonumber \\
%
16\pi^2\frac{d\tilde{m}_{\bar{\chi}}^2}{dt} & = &
\tilde{m}_{\bar{\chi}}^2 \lambda \lambda^{\dagger}
+ \lambda \lambda^{\dagger} \tilde{m}_{\bar{\chi}}^2
+ 2\lambda \lambda^{\dagger} \tilde{m}_{{\scriptscriptstyle \aleph}}^2
+ 2\lambda \tilde{m}_{\chi}^2 \lambda^{\dagger}
+ 2 h_{\lambda}^{}h_{\lambda}^{\dagger}
- 8 g_A^2\left|M_A\right|^2
+ 2 g_A^2{\mathcal{S}}_A~, \\
%
16\pi^2\frac{d\tilde{m}_{{\scriptscriptstyle \aleph}}^2}{dt} & = &
\left( 3\lambda_{{\scriptscriptstyle \aleph}}^2 +
2 {\mathrm{Tr}}\lambda^{\dagger} \lambda \right)
\tilde{m}_{{\scriptscriptstyle \aleph}}^2
+ 2 {\mathrm{Tr}}\left(\lambda \tilde{m}_{\chi}^2 \lambda^{\dagger}\right)
+ 2 {\mathrm{Tr}}\left(\lambda^{\dagger} \tilde{m}_{\bar{\chi}}^2 \lambda\right)
\nonumber \\
&&{}+ h_{{\scriptscriptstyle \aleph}}^2
+ 2 {\mathrm{Tr}} h_{\lambda}^{\dagger}h_{\lambda}^{}~,\end{aligned}$$ where ${\mathcal{S}}_A = 3\tilde{m}_{D}^2 - 3\tilde{m}_{\bar{D}}^2
+ {\mathrm{Tr}}(\tilde{m}_{\bar{\chi}}^2 - \tilde{m}_{\chi}^2)$.
Some numerical results
======================
We collect here some of the numerical results in our sample calculation. Recall that we use $M_{mess}=50$ TeV, $\mu_{\scriptscriptstyle D}=
\mu_{\scriptscriptstyle \chi}=500$ GeV, and random values of $\gamma$’s and $\lambda$’s in the ranges $.005 - .01$ and $ .1 -.8$ respectively. As inputs we also used the values $M_t = 175$ GeV and $\alpha_s = 0.12$.
The electroweak-symmetry breaking solution is obtained with $$\tan\!\beta = 43.18\; , \quad \mu_H = -370.5{\rm\ GeV}\; ,
\quad B_H = 3.938 {\rm\ GeV}\; .$$ In the following, we concentrate on the down-sector as it is the only one of relevance to the understanding the $\bar{\theta}$ value. Average values of the squared left- and right-handed squark masses are $5.337\times 10^5$ GeV$^2$ and $5.031\times 10^5$ GeV$^2$, respectively, and $\bar{A}_d$ is $-243.5$ GeV. The lack of proportionality of the $A_d$-terms is given by $$\delta\!A/\tilde{m}_{sq} =
\left( \begin{array}{ccc}
0. & 2.12\times 10^{-8} & 8.45\times 10^{-7}\\
1.18\times 10^{-9} & 6.39\times 10^{-7} & 4.02\times 10^{-7}\\
-2.37\times 10^{-7} & 1.73\times 10^{-6} & 1.63\times 10^{-2}
\end{array}\right) \; ,$$ while degeneracy violations are given by $$\delta \tilde{m}_{d}^2/\tilde{m}_{sq}^2 =
\left( \begin{array}{ccc}
0. & 2.54\times 10^{-5} & -6.13\times 10^{-4}\\
2.54\times 10^{-5} & -2.53\times 10^{-4} & 4.42\times 10^{-3}\\
-6.13\times 10^{-4} & 4.42\times 10^{-3} & -0.172
\end{array}\right) \; ,$$ and $$\delta \tilde{m}_{\bar{d}}^2/\tilde{m}_{sq}^2 =
\left( \begin{array}{ccc}
0. & -8.22\times 10^{-6} & -1.12\times 10^{-5}\\
-8.22\times 10^{-6} & -1.59\times 10^{-4} & -1.17\times 10^{-5}\\
-1.12\times 10^{-5} & -1.17\times 10^{-5} & -0.135
\end{array}\right) \; ,$$ where $\tilde{m}_{sq}^2$ is the average squark mass. In addition, we have $$\begin{aligned}
\delta \tilde{m}_{\bar{D}}^2/\tilde{m}_{sq}^2 &=& 1.65\times 10^{-5}\; , \\
\delta \tilde{m}_D^2/\tilde{m}_{sq}^2 &=& -5.64\times 10^{-2}\; .\end{aligned}$$ The proportional part of the $A$-terms for the $\gamma$ Yukawas is very close to the $B_D$ value, given by $$\bar{A}_{\gamma} \sim B_D = -248.1\, {\rm GeV}\; ,$$ with a difference of only $4.27\times 10^{-2}$ GeV. Some other quantities of interest are : $$M_g (\equiv M_3) = 478.1 {\rm\ GeV}\; , \quad
x = 1.213\times 10^{-2} \; ;$$ and, as defined by Eq.(\[dagam\]), $$\delta\! A_{\gamma} = 4.782\times 10^{-4} {\rm GeV}\; ,
\quad {\rm Im}(\mbox{\boldmath $a^{\dag}c$})
= 5.531\times 10^{-3} \; .$$
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[**Table Captions.**]{}\
Table 1: Analysis of the major 1-loop $\bar{\theta}$ contributions and numerical results from the sample run. Table 1a contains contributions from gluino mass corrections; Table 1b contains those from quark mass corrections. Entry 1 of Table 1a and entries 9, and 10 of Table 1b are shown together with explicit splittings of $M_5^2$ according to Eq.(3.6) below the first lines. Numerical results given in the last column do not include the $\left\langle F_{\chi}\right\rangle$ term contributions, but otherwise are complete, [*i.e.*]{} they include all other numerical factors from color indices summation, momentum loop integrals, and full summation over family indices ($i,j,k$) so that the full $\bar{\theta}$ value without the $F$-term contributions, apart from some unlisted subdominating terms, is given by the sum of all the entries.
Table 2: Estimates of the $\left\langle F_{\chi}\right\rangle$ term and its contribution to $\bar{\theta}$, for our sample run and a few runs with different $\gamma$ and $\lambda$ inputs ($\mu_{\scriptscriptstyle D}$ and $\mu_{\chi}$’s are all set at $500$ GeV, $M_{mess}$ at $50$ TeV). Note that the entries $B_{\chi}$, $\tilde{m}^2_{\bar{\chi}}$, $h_{\lambda} \left\langle \chi\right\rangle / \lambda$, and $h_{\gamma} \left\langle \chi\right\rangle \gamma / 16\pi^2$ are our $\left\langle F_{\chi}\right\rangle$ estimates, as discussed; all these are quantities of dimension (mass)$^2$ in units of GeV$^2$ (not shown explicitly). The $\left\langle F_{\chi}\right\rangle$ estimates and its contributions to $\bar{\theta}$ are meant to be upper bounds. Overall $\bar{\theta}$ contributions from gluino and quark mass corrections without the $F$-term are also listed.
[**Figure Captions.**]{}\
Fig.1 : 1-loop mass-correction diagrams leading to $\bar{\theta}$ contributions. (a) 1-loop quark mass; (b) 1-loop gluino mass.
Fig.2 : Diagrams giving estimates of $\left\langle F_{\chi} \right\rangle$ magnitudes. Note that $\left\langle \chi \right\rangle$, $\left\langle \bar{\chi} \right\rangle$, $\mu_{\chi}$ and all propagator masses in the diagrams can be taken as around the same scale, namely $\mu_{\scriptscriptstyle D}$; a SUSY breaking vertex or mass insertion is required in each case, as shown.
Table 1: Analysis of the major 1-loop $\bar{\theta}$ contributions and numerical results from the sample run. Table 1a contains contributions from gluino mass corrections; Table 1b contains those from quark mass corrections. Entry 1 of Table 1a and entries 9, and 10 of Table 1b are shown together with explicit splittings of $M_5^2$ according to Eq.(3.6) below the first lines. Numerical results given in the last column do not include the $\left\langle F_{\chi}\right\rangle$ term contributions, but otherwise are complete, [*i.e.*]{} they include all other numerical factors from color indices summation, momentum loop integrals, and full summation over family indices ($i,j,k$) so that the full $\bar{\theta}$ value without the $F$-term contributions, apart from some unlisted subdominating terms, is given by the sum of all the entries.\
Table 1a: Gluino mass correction contributions\
No.
------ ----------------------------- --------------------------------------------------------- ---------------------------------------------------------------------- ------------------------------ -- ---------------------------------------------------------------- ---------------------------------------------------------- --------------------------------------------------------------------------------- --------------------------------------------- ------------------------- -------------------------
(1) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_i b^i)$ $-5.17\times 10^{-15}$
– $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\!A_{\gamma}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_i c^i)$ - -
– $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{\tilde{m}_{sq}}{M_g}$ (cf. table 2)
(2) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{\tilde{m}_{\scriptscriptstyle A}^2 m_i^2}{\tilde{m}_{sq}^4}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_i b^i)$ $-1.53\times 10^{-14}$
(3) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d m_i}{\tilde{m}_{sq}^2}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_j a^i)$ $1.53\times 10^{-18}$
(4) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{\tilde{m}_{\scriptscriptstyle A} m_i^2}{\tilde{m}_{sq}^3}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! \tilde{m}_{\bar{d}}^{2jk}}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_j a^i)$ $1.09\times 10^{-16}$
(5) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! \tilde{m}_{\bar{d}}^{2ij}}{\tilde{m}_{sq}^2}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_j b^i)$ $1.27\times 10^{-14}$
(6) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d m_i}{\tilde{m}_{sq}^2}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! \tilde{m}_{\bar{d}}^{2jk}}{\tilde{m}_{sq}^2}$ $\frac{\delta\! A^{ki}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_j a^i)$ $2.39\times 10^{-19}$
(7) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d m_i}{\tilde{m}_{sq}^2}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! \tilde{m}_{d}^{2ki}}{\tilde{m}_{sq}^2}$ $\frac{\delta\! A^{jk}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_j a^i)$ $-5.99\times 10^{-18}$
(8) $\frac{\alpha_s}{4\pi}$ $\frac{v_d m_i}{\tilde{m}_{sq}^2}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ $\frac{(M_5^2)^2}{\tilde{m}_{sq}^4}$ ${\rm Im}(b^{*}_j b^i)$ $8.43\times 10^{-20}$
(9) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{v_d m_i \tilde{m}_{\scriptscriptstyle A}}{\tilde{m}_{sq}^3}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_j a^i)$ $1.14\times 10^{-18}$
(10) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{v_d^2}{\tilde{m}_{sq}^2}$ $\frac{\tilde{m}_{sq}}{M_g}$ $\frac{\delta\! A^{jk}}{\tilde{m}_{sq}} \frac{\delta\! A^{ik}}{\tilde{m}_{sq}}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_j b^i)$ $5.26\times 10^{-22}$
Table 1b: Quark mass correction contributions\
No.
------ ----------------------------- --------------------------------------------------------- -------------------------------------------------------------------------------------- ------------------------------ -- ------------------------------------------------------------------------------- ---------------------------------------------------------- ------------------------------------------ --------------------------------------------------------------------- ----------------------------------------------------- -------------------------
(1) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_i b^i)$ $-1.52\times 10^{-15}$
(2) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{\tilde{m}_{\scriptscriptstyle A} B_{\scriptscriptstyle D}}{\tilde{m}_{sq}^2}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_i b^i)$ $3.45\times 10^{-15}$
(3) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d m_i}{\tilde{m}_{sq}^2}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_j a^i)$ $3.10\times 10^{-19}$
(4) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{m_i}{m_j}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{\bar{d}}^{2ij}}{\tilde{m}_{sq}^2}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_i b^j)$ $2.02\times 10^{-14}$
(5) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d}{m_k}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{\bar{d}}^{2jk}}{\tilde{m}_{sq}^2}$ $\frac{\delta\! A^{ik}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_i a^j)$ $2.00\times 10^{-15}$
(6) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d}{m_j}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{\bar{\scriptscriptstyle D}}^2}{\tilde{m}_{sq}^2}$ $\frac{\delta\! A^{ij}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_i a^j)$ $8.00\times 10^{-18}$
(7) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{m_i \tilde{m}_{\scriptscriptstyle A}}{m_k \tilde{m}_{sq}}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{\bar{d}}^{2kj}}{\tilde{m}_{sq}^2}$ $\frac{\delta\! \tilde{m}_{d}^{2ik}}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_i a^j)$ $-7.49\times 10^{-13}$
(8) $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{m_j \tilde{m}_{\scriptscriptstyle A}}{m_i \tilde{m}_{sq}}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{\bar{\scriptscriptstyle D}}^2}{\tilde{m}_{sq}^2}$ $\frac{\delta\! \tilde{m}_{d}^{2ji}}{\tilde{m}_{sq}^2}$ ${\rm Im}(a^{*}_i a^j)$ $2.26\times 10^{-13}$
(9) $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{m_j}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{d}^{2ji}}{\tilde{m}_{sq}^2}$ $-5.74\times 10^{-15}$
– $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{m_j}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{d}^{2ji}}{\tilde{m}_{sq}^2}$ $\frac{\bar{A}_\gamma - B_{\scriptscriptstyle D} }{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_j a^i)$ - -
– $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{m_j}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{d}^{2ji}}{\tilde{m}_{sq}^2}$ $\frac{\delta\! A_\gamma}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_j c^i)$ - -
– $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{m_j}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! \tilde{m}_{d}^{2ji}}{\tilde{m}_{sq}^2}$ (cf. table 2)
(10) $\frac{\alpha_s}{4\pi} $ $\frac{v_d}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ $\frac{(M_5^2)^2}{\tilde{m}_{sq}^4}$ ${\rm Im}(b^{*}_j b^i)$ $1.78\times 10^{-16}$
– $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d \bar{A}_{\gamma}^2}{m_i \tilde{m}_{sq}^2}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ ${\rm Im}(a^{*}_j a^i)$ - -
– $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ $\frac{\delta\! A_{\gamma}^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(c^{*}_j c^i)$ - -
– $\frac{\alpha_s}{4\pi} x^2$ $\frac{\mu_{\scriptscriptstyle D}^2}{\tilde{m}_{sq}^2}$ $\frac{v_d \bar{A}_{\gamma}}{m_i \tilde{m}_{sq}}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ $\frac{\delta\! A_\gamma}{\tilde{m}_{sq}}$ ${\rm Im}(c^{*}_j a^i+a^{*}_j c^i)$ - -
– $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{v_d \bar{A}_{\gamma}}{m_i \tilde{m}_{sq}}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ (cf. table 2)
– $\frac{\alpha_s}{4\pi} x$ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{v_d}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ $\frac{\delta\! A_\gamma}{\tilde{m}_{sq}}$ $\frac{\gamma^{ia} |F_{\chi_a}|}{\tilde{m}_{sq}^2}$ (cf. table 2)
– $\frac{\alpha_s}{4\pi} $ $\frac{v_d}{m_i}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ (cf. table 2)
(11) $\frac{\alpha_s}{4\pi} x $ $\frac{\mu_{\scriptscriptstyle D}}{\tilde{m}_{sq}}$ $\frac{v_d B_{\scriptscriptstyle D}}{m_i \tilde{m}_{sq}}$ $\frac{M_g}{\tilde{m}_{sq}}$ $\frac{\delta\! A^{ji}}{\tilde{m}_{sq}}$ $\frac{M_5^2}{\tilde{m}_{sq}^2}$ ${\rm Im}(b^{*}_j a^i)$ $3.08\times 10^{-21}$
Table 2: Estimates of the $\left\langle F_{\chi}\right\rangle$ term and its contribution to $\bar{\theta}$, for our sample run and a few runs with different $\gamma$ and $\lambda$ inputs ($\mu_{\scriptscriptstyle D}$ and $\mu_{\chi}$’s are all set at $500$ GeV, $M_{mess}$ at $50$ TeV). Note that the entries $B_{\chi}$, $\tilde{m}^2_{\bar{\chi}}$, $h_{\lambda} \left\langle \chi\right\rangle / \lambda$, and $h_{\gamma} \left\langle \chi\right\rangle \gamma / 16\pi^2$ are our $\left\langle F_{\chi}\right\rangle$ estimates, as discussed; all these are quantities of dimension (mass)$^2$ in units of GeV$^2$ (not shown explicitly). The $\left\langle F_{\chi}\right\rangle$ estimates and its contributions to $\bar{\theta}$ are meant to be upper bounds. Overall $\bar{\theta}$ contributions from gluino and quark mass corrections without the $F$-term are also listed.\
No. 1 2 (sample) 3 4 5 6
-------------------------------------------------------------------- ------------ ------------ ------------ ------------ --------------- ------------
$x$ $.0081$ $.012$ $.02$ $.068$ $.0086$ $.0077$
$\lambda$ $ .23-.44$ $ .18-.68$ $ .53-.77$ $ .27-.61$ $.0038-.0094$ $ .75-1.3$
$B_{\chi}$ $1.5$ $3.4$ $9.5$ $101$ $1.9$ $1.5$
$\tilde{m}_{\bar{\chi}}^2 $ $3.9$ $3.5$ $.24$ $18$ $4.7$ $2.5$
$h_{\lambda} \left\langle \chi \right\rangle / \lambda $ $2.3$ $1.0$ $11$ $7.9$ $4.6$ $1.8$
$\frac{h_{\gamma}\left\langle \chi \right\rangle \gamma}{16\pi^2}$ $.033$ $.054$ $.16$ $1.9$ $.036$ $.032$
$\left\langle F_{\chi} \right\rangle$ estimate $10$ $10$ $25$ $130$ $10$ $10$
$\frac{\gamma^{ia} |F_{\chi_a}|}{\tilde{m}_{sq}^2}$ estimate $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-5}$ $10^{-7}$ $10^{-7}$
$\longrightarrow \; \; \bar{\theta}$ $10^{-11}$ $10^{-11}$ $10^{-11}$ $10^{-8}$ $10^{-11}$ $10^{-11}$
$\bar{\theta}$ (quark) $10^{-14}$ $10^{-13}$ $10^{-11}$ $10^{-9}$ $10^{-13}$ $10^{-13}$
$\bar{\theta}$ (gluino) $10^{-15}$ $10^{-15}$ $10^{-14}$ $10^{-11}$ $10^{-15}$ $10^{-15}$
[^1]: Present address; E-mail: [email protected]
[^2]: Current E-mail address: [email protected]
[^3]: Other type of phenomenological features from an extra vector-like quark singlet have been studied by various authors in a different context. See for example Ref.[@singlet], and references therein.
[^4]: This is exactly the reason why the small $x$ scenario is discarded in the analysis of Ref.[@DKL].
[^5]: To construct a unifiable model in the discrete symmetry case, we need to add a pair of heavy lepton doublets coupling in a superpotential analogous to Eq.(\[Wd\]). The same thing can be done to make the aspon model GUT-compatible, but unifying the extra U(1) with the other gauge interactions is not possible.
[^6]: Note that the trilinear couplings, $h_d$ and $h^{ia}_{\gamma}$, are also commonly written as products of the $A$-parameters and the corresponding Yukawa couplings, [*e.g.*]{} elements of the $h_d$ matrix correspond to $A_d^{ij}Y_d^{ij}$ (no sum).
[^7]: In Ref.[@FK], a $W_{\chi}$ without the $\mu_{\chi}$-terms is suggested. While that could have a CP violating vacuum with SUSY preserved, the situation is not as general and natural as the one considered here, and would have to rely on a linear $\aleph$ term to fix the symmetry breaking scale.
|
---
abstract: |
We investigate the nature of the physical structures giving rise to damped [Ly$\alpha$ ]{}absorption systems (DLAS) at high redshift. In particular, we examine the suggestion that rapidly rotating large disks are the only viable explanation for the characteristic observed asymmetric profiles of low ionization absorption lines. Using hydrodynamic simulations of galaxy formation in a cosmological context we demonstrate that irregular protogalactic clumps can reproduce the observed velocity width distribution and asymmetries of the absorption profiles equally well. The velocity broadening in the simulated clumps is due to a mixture of rotation, random motions, infall and merging. The observed velocity width correlates with the virial velocity of the dark matter halo of the forming protogalactic clump ($\Delta v \approx 0.6\; v_{\rm
vir}$ for the median values, with a large scatter of order a factor two between different lines-of-sight). The typical virial velocity of the halos required to give rise to the DLAS population is about $100 {\, {\rm km \, s}^{-1} }$ and most standard hierarchical structure formation scenarios can easily account even for the largest observed velocity widths. We conclude that the evidence that DLAS at high redshift are related to large rapidly rotating disks with $v_{\rm circ} \ga 200 {\, {\rm km \, s}^{-1} }$ is not compelling.
author:
- 'Martin G. Haehnelt, Matthias Steinmetz, Michael Rauch'
title: |
Damped [Ly$\alpha$ ]{}absorbers at high redshift —\
large disks or galactic building blocks?
---
1.cm
1.cm
[email protected], [email protected], [email protected]
Introduction
============
Damped [Ly$\alpha$ ]{}absorption systems (DLAS) have often been interpreted as large high-redshift progenitors of present-day spirals which have evolved little apart from forming stars (Wolfe 1988; Lanzetta et al. 1991; Wolfe 1995; Wolfe et al. 1995; Lanzetta, Wolfe & Turnshek 1995). A number of observational results have been quoted as being in support of this hypothesis (see section 6 below). Most recently, Prochaska & Wolfe (1997) have investigated a variety of idealised models for the spatial distribution and kinematics of the absorbing gas to test whether they could produce the absorption line profiles of low ionisation ionic species (LIS) associated with DLAS. Of those models they investigated, only the one in which the lines-of-sight (LOS) intersect rapidly rotating thick galactic disks can explain both the large velocity spreads (up to 200kms$^{-1}$) and the characteristic asymmetries of the observed LIS absorption profiles. In particular, they find that if the embed their disk model within a CDM structure formation scenario, the result is inconsistent with the observed velocity widths. In this paper we demonstrate that the inconsistency with galaxy formation models within hierarchical cosmogonies (e.g. Kauffmann 1996) disappears if the gas is modeled with a more realistic spatial distribution and kinematic structure.
For this purpose we use numerical simulations of galaxy formation in a CDM cosmogony including gas dynamics and realistic initial conditions. These exhibit a complex relationship between high column density absorption features and the underlying dark matter distribution (Katz et al. 1996; Haehnelt, Steinmetz & Rauch 1996, paper I; Rauch, Haehnelt & Steinmetz 1997, paper II; Gardner et al. 1997a/b). Agglomerations of neutral hydrogen with central column densities larger than $10^{20}{{\, \rm cm} }^{-2}$ and with the masses of dwarf galaxies do occur commonly in these simulations. These objects form by gravitational collapse in CDM potential wells. Subsequent cooling produces an optically thick, mostly neutral phase in the inner ten to twenty kpc. We have already demonstrated that the large number of these objects and their clustering and merging into larger units can explain many observed features of metal absorption systems, for example the ionization and thermal state of the gas, and the observed multi-component structure of the absorption line profiles (paper I&II). In these models the high rate of incidence of damped [Ly$\alpha$ ]{}systems is a result of the high abundance of protogalactic clumps (PGC) which are the progenitors of large present-day galaxies. This must be contrasted with the popular picture of DLAS where a population of very large disks evolves without merging to form present-day spirals.
Prochaska & Wolfe (1997) have highlighted two crucial questions which a hierarchical structure formation model must be able to address satisfactorily:
- How do the observed asymmetries of the absorption line complexes arise?
- Can absorption by groups of PGCs in a hierarchical universe reproduce the observed velocity width distribution?
Below we will investigate the velocity width and shape of LIS absorption profiles using artificial spectra for lines-of-sight through numerically simulated regions of ongoing galaxy formation. We then examine the underlying physical conditions responsible for the kinematic structure of these systems. We further investigate the connection between the velocity width of the absorption systems and the depth of the forming potential well and assess the problem of accommodating the observed velocity width within standard hierarchical cosmogonies. Finally we discuss our results and some other, observational clues to the nature of DLAS, and draw conclusions.
Numerical simulations of damped [Ly$\alpha$ ]{}systems
======================================================
The hydrodynamical simulations
------------------------------
Spatial regions of the universe selected to contain one or a few normal galaxies at redshift zero are simulated with the hydrodynamic GRAPE-SPH code (Steinmetz 1996) in the framework of a standard CDM cosmogony ($\sigma_{8} = 0.67$, $H_{0} = 50{\, {\rm km \, s}^{-1} }{\, {\rm Mpc} }^{-1}$, $\Omega_{\rm b} = 0.05$). Temperature, density and peculiar velocity arrays along LOS through the simulated boxes are used to produce artificial absorption spectra. For a detailed description of the properties of the simulations and the resulting absorption features see Steinmetz (1996), Navarro & Steinmetz (1997), and papers I&II. The strategy of simulating small regions of ongoing galaxy formation preselected from a large dark matter simulation allows us to achieve a spatial resolution of 1kpc and a mass resolution of $5\times 10^6\,{\mbox{M$_\odot$}}$ (in gas). This high resolution — about a factor ten higher than that in most other cosmological hydro-simulations — is crucial to resolve the rich substructure within the damped region induced by the frequent merging of protogalactic clumps in hierarchical structure formation scenarios. Despite this, the resolution is still not sufficient to account for a possible clumping on sub-kpc scales due to thermal instabilities (Mo 1994; Mo & Miralda-Escudé 1996). Furthermore energy and momentum feedback due to star formation are not included. Both effects are likely to produce additional substructure in physical as well as in velocity space.
The region of neutral hydrogen — self-shielding
-----------------------------------------------
To study the kinematic structure of damped systems we extend our previous work to LOS passing through regions of collapsed dark matter halos with integrated column densities exceeding $2 \times 10^{20}$ cm$^{-2}$. The main problem which then arises is the treatment of the self-shielding of the dense gas against radiation beyond the Lyman edge. With the current generation of computers it is not yet possible to run cosmological hydro-simulations which solve the full radiative transfer equations. We therefore have adopted a simple scheme to mimic the effect of self-shielding which is motivated by the tight correlation between column density and density predicted by the numerical simulations (Miralda-Escude et al 1996; paper II). A column density of $10^{17}{{\, \rm cm} }^{-2}$, above which self-shielding becomes important, occurs for LOS with an absorption-weighted density of about $10^{-3} {{\, \rm cm} }^{-3}$ to $10^{-2} {{\, \rm cm} }^{-3}$. This is easy to understand by looking at the photoionization equilibrium equation for a highly ionized optically thin homogeneous slab of hydrogen. The column density of neutral hydrogen then scales as $N_{\rm HI} \propto n_{\rm H}^{2}\, D\,J_{912}^{-1}$, where $D$ is the thickness of the slab and $J_{912}$ is the flux of the UV background at the Lyman edge. The hydrogen density at the onset of self-shielding in the central plane can be written as, $$n_{\rm shield} \sim
3 \times 10^{-3} \;
\left ( \frac{D}{10 {\, {\rm kpc} }} \right )^{-0.5} \;
\left (\frac{J_{912}}{0.3 \times 10^{21}
{{\, \rm erg} }{{\, \rm cm} }^{-2}\sec^{-1}{{\, \rm Hz} }^{-1}{{\, \rm sr} }^{-1}}
\right )^{0.5}
{{\, \rm cm} }^{-3},$$ where the spectral shape proposed by Haardt & Madau (1996) was used to transform the flux at the Lyman edge $J_{912}$ into a photoionization rate. The geometry of collapsed regions in the numerical simulations is certainly more complicated than that of a slab, but $D$ = 10 kpc is a typical scale. To be on the safe side we assumed that all the gas above a density threshold of $10^{-2}{{\, \rm cm} }^{-2}$ is self-shielded. This probably underestimates the size of the self-shielding region.
Profiles of low ionization ionic species
----------------------------------------
In observed DLAS hydrogen is predominantly neutral due to the self-shielding of the gas while the other atomic species attain low ionization states (Viegas 1995). Optically thin transitions of LIS like , and are therefore generally considered as suitable tracers of the the kinematic and density structure of the neutral gas (Wolfe 1995). We have chosen the 1808 absorption feature for our investigation. Silicon was assumed to be predominantly in the first ionization state within the self-shielding region, $$\begin{aligned}
\nonumber \frac{[{\mbox{Si{\scriptsize II}}}]}{\rm [Si]} = 1, &n_{\rm H} >10^{-2} {{\, \rm cm} }^{-3}, & \\
\nonumber &&\\
\nonumber \frac{[{\mbox{Si{\scriptsize II}}}]}{\rm [Si]} = 0, &{\rm otherwise}.& \\ \end{aligned}$$ A homogeneous silicon abundance of \[Si/H\] = $-1$ was assumed for the self-shielding region. Artificial spectra were made to resemble typical Keck data obtainable within a few hours from a 16-17th magnitude QSO (S/N= 50 per 0.043 Å pixel, FWHM = 8km/s). To make contact with our previous work we will also show the corresponding absorption line profiles. For these a homogeneous carbon abundance of \[C/H\] = $-1.5$ was assumed. The fraction was calculated using the photoionization code CLOUDY (Ferland 1993) as described in papers I&II.
A gallery of simulated damped [Ly$\alpha$ ]{}absorbers
------------------------------------------------------
Figs 1-5 present some typical examples of simulated damped [Ly$\alpha$ ]{}absorption systems. The [*bottom left*]{} panel shows the absorption spectrum for the 1808 and 1548 transitions, the [*top left*]{} and [*bottom right*]{} show the total hydrogen density and peculiar velocity along the LOS, while the [*top right*]{} panel shows density and velocity fields in a thin slice containing the LOS. The coordinates are proper distance, and the projection is such that the LOS along which the absorption spectrum is determined lies along the z-axis. The wavelength axis of the spectra is in ${\, {\rm km \, s}^{-1} }$. Velocities are relative to the center-of-mass velocity of a sphere with 30 kpc radius. The assumed density threshold for self-shielding is indicated by the dashed line in the density profile and by the thick contour in the slice of the density field.
Fig. 1 shows a typical example of two merging PGCs. The density profile along the line of sight has two peaks within the self-shielding region. On larger scales the gas is flowing in from the left and the right, but in the self-shielded region the flow is rather quiescent with a velocity gradient of only about $30 {\, {\rm km \, s}^{-1} }$. This results in a double-peaked 1808 absorption profile with about $30{\, {\rm km \, s}^{-1} }$ velocity difference between the peaks.
Fig. 2 shows another example of merging PGCs. This time the density profile along the line of sight has three rather marginal peaks within the self-shielding region. The velocity profile is smooth but shows large gradients due to an eddy-like motion in the shock produced by the infalling clump. The corresponding 1808 absorption profile is complex and extends over $200 {\, {\rm km \, s}^{-1} }$, showing a prominent leading edge. The hump in the velocity profile at 5kpc and the three density peaks show all up as individual absorption features.
Fig. 3 shows a similar third example of merging PGCs. The peculiar velocity profile shows the same large gradients but is much more chaotic than that in Fig. 2. The peaks of the density profiles and the features of the velocity profile in the self-shielding region can again be identified in the 1808 absorption profile which also shows a prominent leading edge. One should note here the strong difference between the 1808 profile and the 1548 profile. The latter arises from absorption by the spatially separated warm gas surrounding the self-shielding region.
Fig. 4 and 5 show two of the rather rare cases where there is a large rotational component in the motion of the gas. For these “rotating” PGCs we generally find rather smooth density and velocity profiles in the self-shielding region. In most cases these result in single-peaked 1808 absorption profiles with mild asymmetries or extended wings to one side as in Fig. 4. The latter are often difficult to detect unless the main peak is already saturated. The best example of a leading edge profile produced by rotation which we could find is shown in Fig. 5.
Orientation effects
-------------------
As discussed above, the detailed structure of the absorption profile of the LIS regions depends strongly on the substructure in physical and velocity space. This makes the absorption profiles very sensitive to the orientation of the LOS. In order to demonstrate this we plot in Fig. 6 ten different randomly oriented LOS, each giving rise to damped [Ly$\alpha$ ]{}absorption, in the vicinity of the PGC shown in Fig. 1. The nature of the absorption profiles varies from a single symmetric peak to double and multiple peaks. The rapid changes in the details of the velocity profile along the LOS are due to the rather chaotic velocity field of the merging PGC, and are the main reason for the dramatic changes in the LIS absorption profile. It is also clearly seen that the absorption profile of the higher ionization ion varies independently of . This is due to the fact that arises mainly from the warmer gas outside the self-shielding region.
Distribution of velocity widths and shapes — Observed [*vs*]{} simulated
========================================================================
Prochaska & Wolfe (1997) have introduced four parameter to characterize LIS absorption profiles. In order to facilitate comparison with their observed sample of DLAS (a sample of 17 DLAS obtained by Sargent et al. with the Keck HIRES instrument) we have applied their selection criteria and characterized the simulated LIS profiles in exactly the same manner. The 1808 absorption profiles were transformed into an apparent optical depth profile which was then smoothed over a range of 9 pixels. The first parameter — the velocity width $\Delta v$ of the LIS region — is defined as the velocity interval which contributes the central 90 percent to the optical depth weighted velocity integral, $$\tau_{\rm tot} = \int{ \tau {\,{\rm d}}v.}$$ The mean velocity $v_{\rm mean}$ is defined as the midpoint of the velocity width interval (setting $v$ = 0 at the left edge) while the median velocity $v_{\rm med}$ bisects the integral in equation (2) performed over the velocity width interval. The three shape parameters designed to detect asymmetries in the absorption complexes are defined as follows, $$f_{\rm edg} = \frac{|v_{\rm pk} - v_{\rm mean} |}{(\Delta v/2)},$$ $$f_{\rm mm} = \frac{|v_{\rm median} - v_{\rm mean} |}{(\Delta v/2)},$$ $$f_{\rm 2pk} = \pm \frac{|v_{\rm 2pk} - v_{\rm mean} |}{(\Delta v/2)},$$ where $v_{\rm pk}$ and $v_{\rm 2pk}$ are the velocity of the highest and second highest significant peak in the smoothed apparent optical depth profile. For the 2-peak test the plus (minus) sign holds if the velocity of the second peak falls (falls not) between the velocity of the first peak and the mean velocity. In the case of single peaks $f_{\rm 2pk}$ is set equal to $f_{\rm
edg}$. To avoid saturation effects and to ensure sufficient signal-to-noise only absorption profiles with $$0.1 \le \frac{I_{\rm pk}}{I_{0}}\le 0.6$$ were considered, where $I_{\rm pk}$ and $I_{0}$ are the intensity of the strongest peak and the continuum respectively.
A sample of 640 simulated damped [Ly$\alpha$ ]{}absorption systems with column densities above $2\times 10^{20} {{\, \rm cm} }^{-2}$ and satisfying the criterion in equation (6) was assembled by choosing random LOS in the vicinity of 40 protogalactic clumps identified with a friends-of-friends group-finder. For the discussion of velocity widths a minimum threshold of $\Delta v$ $>$ 30 kms$^{-1}$ was imposed on both the observed and simulated velocity widths to avoid incompleteness effects. The redshift of the simulated DLAS is $z$ = 2.1, the median redshift of the observed sample.
[cccccc]{} &0.69 & 0.98& 0.23 & 0.28&\
Fig. 7 summarizes the comparison between the observed and simulated samples. The top panels show the differential distributions of the velocity width, the edge-leading, the mean-median, and the 2-peak parameter. The observed sample is still small and we therefore had to use rather large bins for the observed data. The absolute numbers of the observed sample are plotted while the simulated sample is normalized accordingly. The error bars indicate the Poisson errors and the width of the bin respectively. The velocity width is plotted relative to the median value of the observed sample and relative to the median value of subsamples of 16 LOS around each PGC in the simulated sample. This allows us to test the relative velocity distribution independent of the cosmological model chosen. We come back to the absolute velocity width of the LIS region in section 5. The bottom panels show the corresponding cumulative distribution. The agreement between observed and simulated spectra is within the expected statistical errors. The KS test values for the cumulative distribution are given in Table 1. For none of the parameters is the KS test probability smaller than 20%. We would, however, like to caution against using KS tests to discriminate between different models. Small KS probabilities can be very misleading if a very special representation of a general class of models is chosen. We have e.g. varied our density threshold for the self-shielding region, the redshift of the sample and the metallicity of the gas and found significant changes in the distribution of the parameter. In some cases they further improved the agreement, and in some cases they led to KS probabilities as small as a tenth of a percent for one or two of the parameters. We believe that a significantly larger sample and a careful assessment of the selection effects are necessary in order to draw strong conclusions from the detailed distributions of the shape parameters introduced by Prochaska & Wolfe.
Physical conditions giving rise to damped Lyman $\alpha$ systems
================================================================
In the previous section we have shown that LOS passing the vicinity of PGCs can give rise to DLAS with LIS absorption profiles which reproduce the characteristic velocity width distribution and asymmetries of observed DLAS. It remains to be seen which underlying physical conditions are giving rise to these features. In hierarchical structure formation scenarios the “progenitor” of a present-day galaxy consists of several PGCs often moving along filamentary structures to merge into larger objects. We found the turbulent gas flows and inhomogeneous density structures related to the merging of two or more clumps to be the main reason for the occurrence of multiple LIS absorption systems with large velocity widths. Rotational motions of the gas play only a minor role for these absorption profiles, as does the velocity broadening due to the Hubble expansion between aligned clumps which is important for higher ionization species like (see papers I&II). The latter is easily understandable given the small cross section of the LIS region.
Properties of the absorbing protogalactic clumps
------------------------------------------------
We have systematically investigated the physical properties of our sample of 40 PGCs in order to understand what kind of motions are reflected in the LIS line profiles and how their velocity width is related to the depth of the potential wells in which they are embedded. For this purpose we determined the following quantities for the absorbers: total velocity dispersion of the large scale motions of the gas in the self-shielding region, velocity dispersion due to radial motions of the gas, overall rotational velocity of the gas and virial velocity of the dark matter halo. The virial velocity is defined as $v_{\rm vir} = \sqrt{GM/r}$ in a sphere overdense by a factor 200 compared to the mean cosmic density. One should note here that the DM halo(s) are not necessarily virialized during the merger of two PGCs.
In Fig. 8a the median value of the velocity width for 16 randomly orientated LOS around each PGC is plotted [*vs*]{} the virial velocity of the DM halo. There is a strong correlation indicating that the velocity width reflects the depth of the potential reasonably well even though there is considerable scatter. The velocity width of the LIS absorption region is typically 60 percent of the virial velocity of the DM halo. The solid line shows the least-square fit.
Fig. 8b shows the relation between rotational velocity and velocity width. There seems to be no correlation. The rotational velocity is generally too small to account for the observed LIS velocity width. In Fig. 8c the relative contribution of radial motions and rotation to the total velocity dispersion of the gas is shown. The contributions of rotation and radial motions (mainly infall and merging) range between 0 and 70 percent. As expected these are anti-correlated. The contribution of additional random motions is generally between 30 and 70 percent.
How do asymmetric and leading edge profiles arise?
--------------------------------------------------
The main motivation for interpreting observed absorption profiles as a signature of rotation are their leading edges,[*i.e.,*]{} the strongest absorption feature often occurs at one of the edges of the profile. As demonstrated by Prochaska & Wolfe (1997) such profiles occur naturally in a thick disk model with an exponential density and an isothermal velocity profile. Fig. 5 shows such an example. There is, however, an equally simple and plausible explanation for such leading edges in a scenario of merging protogalactic clumps. In the case of two merging clumps the strongest absorption feature will generally be caused by the high density central region of the clump which is closest to the LOS. The dense regions of merging PGC will move faster than their surroundings because of the smaller deceleration by the (density-dependent) ram pressure. The strongest absorption feature then occurs naturally at the edge of the absorption profile. Smaller features are produced by density fluctuations in stripped material behind or by shocked material in front of the dense region. This situation is illustrated schematically in Fig. 9. Except in the rare case where the LOS passes both dense regions symmetrically this is an intrinsically asymmetric configuration in velocity space. This is the principal reason for the asymmetric LIS absorption profiles in our models. We caution, however, against over-interpretation of leading edge profiles. One should keep in mind that in the case of three randomly ordered components of varying strength the probability that the strongest component is at one edge is 2/3 and in the case of four components it is still 1/2.
Absolute velocity widths and cosmological models.
=================================================
In section 3 we have shown that the width distribution of the simulated LIS profiles relative to its median value is consistent with that observed, and in section 4.1 we demonstrated that the width of the profiles is correlated with the the virial velocity of the associated DM halo. Fig. 10a shows the complete velocity width distribution of the 640 DLAS in our sample. The median value is about 60 percent of the virial velocity of the associated DM halo, $${\rm median}(\Delta v) \approx 0.6 \times v_{\rm vir}.$$ This value depends somewhat on the assumed density threshold for self-shielding, the assumed metallicity, the selection criterion of the PGC and the redshift of the sample. Varying these parameter we found the ratio of velocity width to virial velocity to vary between 0.5 and 0.75. One should note here that the virial velocity which we infer from a given velocity width is a factor 1.5 to 2.5 times smaller than in the rotating disk model.
It is difficult to assure that the sample of PGCs picked from our numerical simulations is fully representative of the simulated cosmological model. Furthermore it is very CPU time-consuming to simulate a large number of different cosmogonies. The exact velocity width distribution will depend on the distribution of virial velocities in a chosen cosmological model weighted by the cross-section for damped absorption, $$p(\Delta v,N_{\rm HI}>N_{\rm damp}, v_{\rm vir}) =
p(\Delta v | N_{\rm HI}> N_{\rm damp},v_{vir})\times
p(N_{\rm HI}> N_{\rm damp} | v_{vir})\times
p(v_{vir}).$$ We take the following approach to calculate the distribution of absolute velocity widths. The third factor in equation (8) the relative number of halos with different virial velocities is calculated using the Press-Schechter formalism (Press & Schechter 1974). The cross section for damped absorption is assumed to scale linearly with mass, $p(N_{\rm HI}> N_{\rm damp}|v_{vir})\propto
M \propto v_{\rm vir}^{3}$. This is the simplest possible scaling suggested by the constant column density threshold defining a DLAS. A good estimate for the first factor is obtainable from our numerical simulations. We found that $p(\Delta v | N_{\rm HI}>N_{\rm damp}, v_{\rm vir})$ depends mainly on the ratio of $\Delta v/v_{\rm vir}$ and only weakly on the virial velocity of the dark matter halo itself. We therefore used the velocity width distribution of all 640 DLAS shown in Fig. 10a. The distribution of absolute velocity widths is obtained by integrating over virial velocity, $$p(\Delta v,N_{\rm HI}>N_{\rm damp}) =
\int_{v_{\rm min}} ^{\infty}
{p(\Delta v , N_{\rm HI}> N_{\rm damp},v_{vir}) {\,{\rm d}}v_{\rm vir}}.$$ The result is shown is shown in Fig. 10b for a standard CDM model at $z= 2.1$ with three different values of $\sigma_{8}$ and a minimum virial velocity of 30 ${\, {\rm km \, s}^{-1} }$. Even the largest observed velocity width, $\sim 200 {\, {\rm km \, s}^{-1} }$, can be accommodated in a CDM model with $\sigma_{8}$ as low as 0.5 to 0.6. Most of the currently favored variants of hierarchical galaxy formation should therefore have no serious problems in account for the observed velocity width distribution. Constraints on different hierarchical scenarios by the overall incidence rate for damped absorption have been discussed extensively by other authors (see also section 6).
Damped [Ly$\alpha$ ]{}absorber — Large disks or protogalactic clumps?
=====================================================================
Current hydrodynamical simulations, including those presented here, are undoubtedly unable to model all the details of the spatial distribution and kinematics of the gas in the innermost regions of collapsed dark matter halos. We believe, however, that our simulations already catch many of the significant features and as dicussed above they should underestimate rather than overestimate the amount of structure in the density and velocity field. The simulations therefore clearly cast serious doubt on the claim that only objects as massive as present-day sprials can produce the velocity widths of the observed LIS profiles, and that rotation is the only possible interpretation for the shape of the profiles.
Hierarchical structure formation models can explain many other features of the cosmic matter distribution seen in absorption, e.g., the rate of incidence, column density distribution, Doppler parameters, ionization state, sizes, and the opacity of Lyman $\alpha$ forest clouds, and the abundance, kinematics, temperatures, and ionization conditions of heavy element absorbers (Cen et al. 1994; Hernquist et al. 1996; Petitjean, Mücket & Kates 1995; Miralda-Escudé et al. 1996; Zhang et al. 1997; Croft et al. 1997; Rauch et al. 1997, Bi & Davidsen 1997; Hellsten et al. 1997; paper I&II). The fact that the same models can account for the essential features of DLAS (Katz et al. 1996; Gardner et al. 1997a/b; paper I&II) makes absorption by protogalactic clumps an even more attractive explanation for DLAS. However, some of the observed properties of absorption systems are not unique to hierarchical models and we do not consider the rapidly rotating, large disk hypothesis to be ruled out by our results. Here we briefly comment on other arguments which have been put forward in discussing the nature of DLAS, mostly in favor of DLAS as large, protogalactic disks:
\(1) The high column densities:
Large column densities of DLAS are indeed reminiscent of present-day disks (e.g. Wolfe 1988). However, simple analytical estimates and the various simulations quoted above show that these column densities can equally well be produced in gas-rich protogalactic clumps with masses expected in typical hierarchical structure formation models.
\(2) The large impact parameter:
Large separations between the absorber and detected emission attributed to associated starlight are taken as indicative of extended, massive objects. However, very few examples are currently known (Møller & Warren 1995; Warren & Møller 1995, Djorgovski 1996). It is not yet clear whether the inferred sizes actually contradict the predictions by hierarchical structure formation scenarios (Mo, Mao & White 1997). This will depend crucially on the detailed gas distribution in the outskirts of protogalactic clumps. Moreover, there are several uncertainties. We do not know yet how emission and absorption properties are related. As demonstrated in section 2 and papers I&II the region responsible for DLAS often contains several protogalactic clumps within a few tens of kpc. Also, the few DLAS identified in emission may only be the tip of the iceberg, at the upper end of the mass and size distribution.
\(3) The continuity of $\Omega_b$:
$\Omega_b$ in the phase of high-redshift DLAS is roughly similar to the baryon content in the stellar component of present-day spirals (Wolfe 1986; Lanzetta et al. 1995; Storrie-Lombardi 1996). This has been interpreted as continuity in baryon content between DLAS and present-day galaxies. If correct this continuity may mean that the gas constituting the stars observed today has already cooled and settled into collapsed objects at these redshifts. The coincidence contains, however, no information on the size distribution of the collapsed objects. Hierarchical models have been shown to reproduce the observed $\Omega_b$ in DLAS and its evolution with redshift (Kauffmann & Charlot 1994; Ma & Bertschinger 1994; Mo & Miralda-Escudé 1994; Klypin et al. 1995; Gardner et al. 1997a/b; Ma et al. 1997).
\(4) The high rate of incidence:
The rate of incidence only depends on the product of space density of the absorbers times their cross section for damped absorption. It gives no constraints on the size of individual absorbers. Hierarchical models have also been shown to reproduce the observed rate of incidence of DLAS and its evolution with redshift (Kauffmann 1996; Baugh et al. 1997).
\(5) The alignment of the edge of the absorption profile with the redshift of observed emission:
For two of the DLAS where emission has been observed it is possible to determine the relative position of emission and absorption in redshift space (Lu, Sargent & Barlow 1997). In the first case the emission falls on one edge of the LIS absorption profile in absorption while the strongest absorption feature lies on the opposite edge. In the second case the emission redshift lies roughly at the centre of the absorption profile. If the LIS absorption profiles showing the leading-edge signature were solely due to rotation, if, in addition, the center of emission coincided with the center of the disk and if finally there were no optical depth effects, then the emission redshift should indeed occur preferentially at the edge of the LIS absorption profile opposite to the edge coinciding with the strongest absorption feature. The situation is, however, similar if the leading edges are due to merging/collision of PGCs. The emission of a stellar continuum would most likely originate in one of the central regions of the merging clumps and should therefore also coincide with one of the two edges of the absorption profile. However, we see no reason why that should happen preferentially opposite to the edge with the strongest absorption feature. Thus, a large number of cases like that reported by Lu et al. would indeed argue for rotation dominating the dynamics.
For completeness, we mention a counterargument against massive disks that has received some attention in the past, the issue of the metallicities in DLAS. The metal abundances (\[Fe/H\]) in DLAS at high z are much lower than expected for local spiral disks (Pettini et al. 1994; Lanzetta, Wolfe, and Turnshek (1995); Lu et al. (1996); Prochaska & Wolfe 1996), which has led to suggestions that DLAS show abundance patterns of dwarf galaxies or galactic halos. In the CDM picture, this inference is correct.
Conclusions
===========
We have used hydrodynamical simulations of galaxy formation in a cosmological context to study the line profiles of low ionization species associated with damped [Ly$\alpha$ ]{}absorption systems. Observed velocity widths and asymmetries of the line profiles of the low ionization species are well reproduced by a mixture of rotation, random motions, infall and merging of protogalactic clumps. The asymmetries are mainly caused by random sampling of irregular density and velocity fields of individual halos and by intrinsically asymmetric configurations arising when two or more clumps collide. We show why leading-edge asymmetries occur naturally in the latter case; the dense central regions of the clumps move faster than surrounding less dense material.
We have further shown that the presence of non-circular motions reduces the depth of the potential well necessary to produce a given velocity width compared to a model where the absorption is solely due to rotation. The reduction is typically a factor about 2. The observed velocity width can therefore be explained by gas moving into and within (forming) dark matter halos with typical virial velocities of about $100{\, {\rm km \, s}^{-1} }$. Velocity width and virial velocity are linearly correlated but the scatter is large; there are outliers with large velocity width due to an occasional alignment of clumps with well separated dark matter halos.
Our final conclusion is that asymmetric profiles of the kind observed are not necessarily the signature of rotation and that there is no problem of accommodating the observed velocity widths within standard hierarchical cosmogonies.
Acknowledgments
===============
We thank Simon White for a careful reading of the manuscript, and Limin Lu, Hojun Mo, Jason Prochaska, and Art Wolfe for very helpful discussions. MR is grateful to NASA for support through grant HF-01075.01-94A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support by NATO grant CRG 950752 and the “Sonderforschungsbereich 375-95 für Astro-Teilchenphysik der Deutschen Forschungsgemeinschaft” is also gratefully acknowledged.
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abstract: 'We give an example of two $n\times n$ chess positions, $A$ and $B$, such that (1) there is a sequence $\sigma$ of legal chess moves leading from $A$ to $B$; (2) the length of $\sigma$ cannot be less than $\exp \Theta(n)$.'
address: 'National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia'
author:
- Yaroslav Shitov
title: 'Chess God’s number grows exponentially'
---
Introduction
============
This note presents a contribution in the complexity theory of *puzzles*, or *one-player games*, which can be considered directed graphs with vertices called *positions* and arcs called *moves*. A player is given a pair of positions and needs to transform one to the other using a sequence of moves. Well known examples of puzzles include Rubik’s cube, Fifteen game, computer simulations like Atomix and Sokoban, and other games. An important invariant of a puzzle is its *diameter*, that is, the greatest possible distance between a pair of positions, with distance being the length of a shortest sequence of moves transforming one to the other. This invariant, hard to be calculated in general [@15], is sometimes referred to as *God’s number*. For instance, the Rubik’s cube God’s number has recently been found by an extensive computer search [@Rubik].
If the diameter of a certain puzzle has an upper bound polynomial in its size, the player can decide whether a solution exists in non-deterministic polynomial time just by executing every possible sequence of moves. This is indeed the case for $n\times n$ generalizations of Fifteen game [@15], Rubik’s cube [@Rubik2], and some other puzzles. On the other hand, the Sokoban God’s number has no polynomial upper bound [@SB], and this game turns out to be PSPACE-complete. The order of growth is still unknown for diameters of several classical puzzles, and this note aims to treat the problem for $n\times n$ chess. A related problem has been mentioned in 1981 by Fraenkel and Lichtenstein [@Chess], who noted that *reachability* of a given position from another one may be not quite infeasible. Chow gives the following formulation of this problem.
[@MO]\[probexp\] Does there exist an infinite sequence $(A_n,B_n)$ of pairs of chess positions on an $n\times n$ board such that the minimum number of legal moves required to get from $A_n$ to $B_n$ is exponential in $n$?
Our note gives a positive solution of this problem, and we use the following notation throughout. As said above, a *puzzle* is a directed graph with vertices called *positions* and arcs called *moves*. We say that a sequence $\sigma=\sigma_1\ldots\sigma_n$ (where $\sigma_i$ is a move *from* $\pi_i$ *to* $\rho_i$) is *legal* if $\rho_i=\pi_{i+1}$ holds for every $i$. We call $\pi_1$ the *initial* position and $\rho_n$ and the *resulting* position of $\sigma$. Moreover, we say that $\sigma$ is a *repetition* if $\sigma$ is legal, $n=2$, and $\pi_1=\rho_2$.
An auxiliary game
=================
We need to introduce a new game in order to give a solution for Problem \[probexp\]. Consider a graph $G$ which is a union of a cycle of length $3m$ and $m$ triangles; we denote the $j$th vertex of $i$th triangle by $(i,j)$ with $i\in\{1,\ldots,m\}$ and $j\in\{1,2,3\}$. The vertices of the large cycle are labeled $(0,t)$ with $t\in\mathbb{Z}/3m\mathbb{Z}$. The game is played with two types of pieces, one *chip* and $3m$ *switches*. We assume that every vertex of $G$ is either empty, or occupied by the chip, or occupied by exactly one switch. (In particular, it is not possible for a vertex to be occupied by more than one switch at the same time, or by the chip and a switch at the same time.) Also, we assume that either $(i,j)$ or $(0,3i+j-3)$ is occupied by a switch. The game consists in completing a legal sequence of *single moves* each of which belongs to one of the following categories.
*Swap.* If the chip is located in one of the vertices $(i,j)$ and $(0,3i+j-3)$, it can be swapped with a switch from the other of these vertices.
*Jump.* If the chip is located on a triangle, it can jump (that is, can be moved) to any vacant vertex of the same triangle. If $(0,j)$ is occupied by the chip, it can jump to $(0,j+1)$ or $(0,j-1)$ if any of these vertices is vacant.
In initial position $P$, switches occupy vertices $(i,1)$, $(i,3)$, $(0,3i-1)$; the chip is located initially in arbitrary vertex on the large cycle. Figure \[figinitaux\] represents the starting position with $m=3$ with dark squares being switches and *C* a chip.
![Initial position of auxiliary game.[]{data-label="figinitaux"}](start.png)
Consider a move sequence $\sigma$ resulting in a position $P'$. Denote by $\#(\sigma,i_0,j_1,j_2)$ the number of single moves from $\sigma$ that are jumps from $(i_0,j_1)$ to $(i_0,j_2)$. (In particular, $\#(\sigma,0,j_1,j_2)$ may be nonzero only if $j_1-j_2$ is either $1$ or $-1$.) Define the *flow* $\mathcal{F}(\sigma,i_0,j_0)$ through $(i_0,j_0)$ as the half of $$\#(\sigma,i_0,j_0-1,j_0)+\#(\sigma,i_0,j_0,j_0+1)-
\#(\sigma,i_0,j_0+1,j_0)-\#(\sigma,i_0,j_0,j_0-1).$$
\[lemflow\] If $\sigma$ is legal and $P=P'$, then $\mathcal{F}(\sigma,i,2)=\mathcal{F}(\sigma,0,3i-1)$.
Focus the attention on those single moves from $\sigma$ which involve vertices $(i,2)$ and $(0,3i-1)$, and those which change the positions of switches acting between $(i,j)$ and $(0,3i+j-3)$; we call these moves *effective*, and all other moves *ineffective*. By $\sigma'$ we denote the subsequence of $\sigma$ consisting of all effective moves; note that $\sigma'$ need not be legal. By definition of flow, $\sigma$ and $\sigma'$ have the same flows through $(i,2)$ and the same flows through $(0,3i-1)$. The combinatorial analysis shows that any effective move leads from a position indexed $t$ on Figure \[fig12\] to the position indexed either $t+1$ or $t-1$ (such a move is called a *direct type $t$* move in the former case and a *reversed type $t$* move in the latter case).
![Positions corresponding to effective moves.[]{data-label="fig12"}](pos1-3.png "fig:") ![Positions corresponding to effective moves.[]{data-label="fig12"}](pos4-6.png "fig:") ![Positions corresponding to effective moves.[]{data-label="fig12"}](pos7-9.png "fig:") ![Positions corresponding to effective moves.[]{data-label="fig12"}](pos10-12.png "fig:")
Note that a legal sequence of ineffective moves cannot lead from a position indexed $t_1$ to a position indexed $t_2$ unless $t_1=t_2$ or $\{t_1,t_2\}=\{1,12\}$. Therefore, a move of direct type $t$ (modulo $11$) can be followed in $\sigma'$ by a type $t+1$ move only; similarly, a move of reversed type $t$ can be followed by a type $t-1$ move only. Since a direct type $t$ move and a reversed type $t+1$ move form a sequence with zero flow, we can remove such a pair from $\sigma'$ without changing its flow. Therefore, it suffices to consider the case when $\sigma'$ is a sequence of $11$ consecutive direct type $1,\ldots,11$ moves (or inverse type $12,\ldots,2$ moves). In this case, the quantities $\mathcal{F}(\sigma',i,2)$ and $\mathcal{F}(\sigma',0,3i-1)$ equal $-1$ (or $1$, respectively).
Let us complete the legal sequence of effective moves leading from 1 to 12, then move the chip from $(0,3i-2)$ to $(0,3i-3)$. Then, we can do these moves again for switches between $(i-1,j)$ and $(0,3(i-2)+j)$ instead of those between $(i,j)$ and $(0,3i+j-3)$. By induction, we construct a legal sequence $\tau$ of moves which leads to the starting position and satisfies $\mathcal{F}(\tau,i,2)=1$, for any $i$. Completing the sequence $\tau$ several times still leaves us in the starting position, and we have $\mathcal{F}(\tau^k,i,2)=k$.
The chess positions
===================
Define a *short bishop* graph on the $n\times n$ chessboard by declaring a pair of squares adjacent if they have the same color and one of them can be reached from the other by a single king move. A subset of chessboard is called a *bishop cycle* if the short bishop graph it induces is a cycle. Removing two non-adjacent squares splits a bishop cycle into two connected components which we call *segments*. We say that a pair of squares is *touching* if they have different colors and one of them can be reached from the other by a single king move.
Now we are ready to provide a pair of positions $P$ and $P'$ on the $n\times n$ chessboard such that (1) there is a legal sequence $\sigma$ of chess moves leading from $A$ to $B$; (2) the length of $\sigma$ cannot be less than some function growing exponentially in $n$. The positions $A$ and $B$ will use the same multiset of pieces, and the pawns in $A$ will be located in the same positions as in $B$. Therefore, $\sigma$ can contain no captures and no pawn movements. Also, removing any repetition contained in $\sigma$, we get a legal sequence of smaller length with the same starting and initial positions; therefore, we can assume that $\sigma$ contains no repetitions.
The only pieces actually used in our positions are pawns, bishops, and rooks. These pieces are located in a region $R$ of the board bounded by a pawn chain. Since the pawns are untouchable, this chain forbids the pieces to leave $R$. We can add $O(1)$ more ranks and files to $R$ and locate there a pair of kings, in order to make our positions satisfy the rules of chess. These settings allow us to consider a chess-like game played in region $R$ with standard chess rules except that (1) White and Black do not have to alternate their moves, (2) captures, pawn moves, and repetitions are forbidden, and (3) there are no kings.
In order to describe the position within $R$, denote $p_1=7$ and let $p_{i+1}$ be the smallest prime exceeding $p_i$. Let $R$ contain $m$ disconnected dark-squared bishop cycles of lengths $2(p_1+1),\ldots,2(p_m+1)$ and one light-squared bishop cycle; all other squares in $R$ are occupied by pawns. There are $3m$ pairwise non-adjacent squares on light cycle which we call *switch squares* and identify with vertices $(0,t)$ from $G$; we assume that one of the segments between $(0,t)$ and $(0,t+1)$ contains no switch squares. On $i$th dark cycle, there will be also three non-adjacent *switch squares* identified with vertices $(i,1)$, $(i,2)$, $(i,3)$ from $G$. We say that a square $x$ lies *between* $(i_0,j_0)$ and $(i_0,j_0+1)$ if $x$ belongs to that segment with ends $(i_0,j_0)$ and $(i_0,j_0+1)$ which contains no switch squares. We assume that the $(i,j)$ and $(0,3i-3+j)$ switch squares are touching, and no other pair of squares on cycles is touching. In the starting position, the switch squares $(i,1)$, $(i,3)$, $(0,3i-1)$ are occupied by rooks; one of the squares on the light cycle is vacant, and all the other squares are occupied by bishops. One of the bishops on every dark cycle is white, and all the other pieces are black. (In fact, there is no need to specify the colors of pieces outside the dark cycles.) It is easy to construct such a position in the case when $p_m<cn$, for some sufficiently small absolute constant $c$. An example, which corresponds to the case $p_1=7$, $p_2=11$, $p_3=13$, is provided on Figure \[figchess\]; the dotted squares correspond to pawns, which are untouchable in our model.
![Position $P$. A swap of marked bishops leads to $P'$.[]{data-label="figchess"}](pic42.png)
Now we explain how is the constructed position related to the game described in previous section. Note that every rook can only move between the $(i,j)$ and $(0,3i-3+j)$ switch squares. Since repetitions are not allowed, we see that moving a bishop located between switch squares $(i_0,j_0)$ and $(i_0,j_0+1)$ to the switch square $(i_0,j_0+1)$ is to be followed by the sequence of consecutive bishop moves eventually leaving $(i_0,j_0)$ vacant. Therefore, playing chess in the constructed position is the same as playing the game from previous section as we identify rooks with switches and the vacant square, which is unique in region $R$, with chip.
Now let $\sigma$ be any legal sequence of moves such that the starting and resulting positions coincide, up to color of pieces. We can note that the flow $\mathcal{F}(\sigma,0,j)$ is equal to the number of times a bishop located between switch squares $(0,j)$ and $(0,j+1)$ moved through $(0,j)$ to a position between $(0,j)$ and $(0,j-1)$ minus the number of times bishops moved in opposite direction. In particular, it is now clear that the flow $\mathcal{F}(\sigma,0,t)$ does not depend on $t$. Similarly, since there are one white and $2p_i-1$ black bishops moving in $i$th cycle, the remainder of $\mathcal{F}(\sigma,i,2)$ modulo $2p_i$ determines the position of white bishop. By Lemma \[lemflow\], the flow $\mathcal{F}(\sigma,i,2)$ is independent of $i$, but it can take any value as $\sigma$ varies.
Now consider the starting position $P$ and construct a new position $P'$ by swapping the positions of the white bishop from the first dark cycle and a black bishop located a distance two apart from this white bishop. (In particular, $P'$ can be obtained from the position on Figure \[figchess\] by swapping the bishops contained in ovals; in general, we say that two bishops from the same dark cycle lie a *distance two apart* if they define a segment containing exactly one bishop.) Then, we conclude by the Chinese remainder theorem that there is a sequence $\sigma'$ leading from $P$ to $P'$. Then, $\sigma'$ is such that $\mathcal{F}(\sigma',i,2)$ divides $2p_i$ if and only if $i\neq1$. Now we see that $\mathcal{F}(\sigma',i,2)$ is non-zero and divides $\prod_{i=2}^mp_i$; the latter quantity is exponential in $n$ since the product of primes not exceeding $n$ is $\exp(n+o(n))$.
Concluding remarks
==================
Problem \[probexp\] has also been studied for some less natural generalizations of chess, see a discussion in [@MO]. In particular, there was an attempt of solving it for a chess-like game with $O(n)$ kings none of which can be leaved under attack. Also, there is a negative solution for Problem \[probexp\] if a game is assumed to terminate after fifty moves without captures and pawn moves; this generalization, however, seems rather artificial. In fact, the rules of $8\times8$ chess do not say that a game terminates immediately after fifty consecutive moves of this kind: The players are allowed to play on if neither of them wants a draw [@ChessLaws]. Finally, note that some authors [@Chess] consider an upper bound for number of pieces growing as a fractional power of board size. In order to make our positions satisfy this restriction, we can add sufficiently many empty ranks and files to them, getting a lower bound for the diameter of $n\times n$ chess exponential in a fractional power of $n$.
I would like to thank Timothy Y. Chow for interesting discussion and many helpful suggestions on presentation of the result.
[0]{} T. Chow, Do there exist chess positions that require exponentially many moves to reach? MathOverflow. `http://mathoverflow.net/q/27944/56203`. E. D. Demaine, M. L. Demaine, S. Eisenstat, A. Lubiw, A. Winslow, Algorithms for solving Rubik’s cubes, Algorithms–ESA, Springer Berlin Heidelberg (2011) 689–700. A. S. Fraenkel, D. Lichtenstein, Computing a perfect strategy for $n\times n$ chess requires time exponential in $n$, J. Combin. Theory Ser. A. 31 (1981) 199–214. R. A. Hearn, E. D. Demaine, PSPACE-Completeness of Sliding-Block Puzzles and Other Problems through the Nondeterministic Constraint Logic Model of Computation, Theoretical Comp. Sci. 343(1) (2005), 72–96. T. Just (ed.), U.S. Chess Federation’s Official Rules of Chess, 6th Ed., USCF, 2014. D. Ratner, M. Warmuth, Finding a shortest solution for the $N\times N$ extension of the $15$-puzzle is intractable, J. Symb. Comp. 10 (1990) 111–137. T. Rokicki, H. Kociemba, M. Davidson, J. Dethridge, The Diameter of the Rubik’s Cube Group Is Twenty, SIAM J. Discrete Math., 27(2) (2013), 1082–1105.
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abstract: 'In this work, we investigate numerical solutions of the two-dimensional shallow water wave using a fully nonlinear Green-Naghdi model with an improved dispersive effect. For the purpose of numerics, the Green-Naghdi model is rewritten into a formulation coupling a pseudo-conservative system and a set of pseudo-elliptic equations. Since the pseudo-conservative system is no longer hyperbolic and its Riemann problem can only be approximately solved, we consider the utilization of the central discontinuous Galerkin method which possesses an important feature of needlessness of Riemann solvers. Meanwhile, the stationary elliptic part will be solved using the finite element method. Both the well-balanced and the positivity-preserving features which are highly desirable in the simulation of the shallow water wave will be embedded into the proposed numerical scheme. The accuracy and efficiency of the numerical model and method will be illustrated through numerical tests.'
address:
- 'School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan, 611731, P.R. China '
- 'College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China'
author:
- Maojun Li
- Liwei Xu
- Yongping Cheng
title: 'A CDG-FE method for the two-dimensional Green-Naghdi model with the enhanced dispersive property'
---
Enhanced dispersive property, Green-Naghdi model, Central discontinuous Galerkin method, Finite element method, Positivity-preserving property, Well-balanced scheme
Introduction {#sec:1}
============
In an incompressible and inviscid fluid, the propagation of surface waves is governed by the Euler equation with nonlinear boundary conditions at the free surface and the bottom. In its full generality, this problem is very complicated to be solved, both mathematically and numerically. Usually, simplified models have been derived to describe the behavior of the solution in some physical specific regimes, such as the nonlinear shallow water equations, the Korteweg-de Vries equations, the Boussinesq type models and the Green-Naghdi models, and to name a few.
The nonlinear shallow water equations (also called Saint-Venant equations) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid. They can model the propagation of strongly nonlinear waves up to breaking and run-up in near-shore zones. However, they fail to properly describe wave propagation in deep water or wave shoaling because they do not incorporate frequency dispersion. The Boussinesq systems carry weak frequency dispersion but are typically restricted to small amplitude waves with relatively weak nonlinearity. Extensions of the shallow water equations that incorporate frequency dispersion can be traced back to Serre ([@Serre1953]) who derived a one-dimensional (1D) system of equations for fully nonlinear weakly dispersive waves over the flat bottom in 1953. In 1976, Green and Naghdi ([@Green1976]) presented the two-dimensional (2D) counterpart of these equations for wave propagation over variable bottom topography. The Green-Naghdi model is a class of fully nonlinear weakly dispersive shallow water wave equations, including dispersive effects and supporting traveling solitary wave solutions. Therefore, it can simulate the long-time propagation of solitary waves with relatively large amplitude.
Due to the significance of the Green-Naghdi model, there has been increasing interest in the numerical simulation of the Green-Naghdi model in the past decade. A fourth-order compact finite volume scheme was proposed for solving the fully nonlinear and weakly dispersive Boussinesq type equations ([@Cien2006; @Cien2007]). A hybrid numerical method using a Godunov type scheme was presented to solve the Green-Naghdi model over the flat bottom ([@MGH10]). A pseudo-spectral algorithm was developed for the solution of the rotating Green-Naghdi shallow water equations ([@Pearce2010]). A hybrid finite volume and finite difference splitting approach was presented for numerical simulation of the fully nonlinear and weakly dispersive Green-Naghdi model ([@Bonneton2011; @Bonneton2011-1; @Chazel2011; @Lannes2015]). In particular, numerical investigations on a dispersive-effect-improved Green-Naghdi model have been reported in [@Chazel2011; @Lannes2015]. A well-balanced central discontinuous Galerkin (CDG) method coupling with the finite element (FE) method ([@LiM2014]) was employed to solve the 1D fully nonlinear weakly dispersive Green-Naghdi model over varying topography. Recent works of other kinds of discontinuous Galerkin (DG) methods for the Green-Naghdi model include [@Duran2015; @Panda2014].
The DG method is a class of high order finite element methods, which was originally introduced in 1973 by Reed and Hill ([@Reed1973]) for the neutron transport equation. Thereafter, it has achieved great progress in a series of pioneer papers ([@Cock1989; @Cock1989-1; @Cock1990; @Cock1998]). The DG method has its own advantages in dealing with numerical solutions for many problems in sciences and engineering, consisting of being very flexible to achieve high order of accuracy and handle complicated geometry and boundary conditions, and to name a few. As a variant of the DG method, the CDG method is also one of popular high order numerical methods defined on overlapping meshes, which was originally introduced for hyperbolic conservation laws ([@Liu2007]), and then for diffusion equations ([@Liu2008]). By evolving two sets of numerical solutions defined on overlapping meshes which provide more information on numerical solutions, the CDG method does not rely on any exact or approximate Riemann solver at element interfaces as in the simulation of the DG method. The CDG method has been successfully applied to solve various partial differential equations, such as the Hamilton-Jacobi equations ([@LiF2010]), the ideal MHD equations ([@LiF2011; @LiF2012]), the Euler equations ([@LiM2016]) and the shallow water equations ([@LiM2014; @LiM2017]). Recently, a reconstructed CDG method has been developed in [@Dong2017] for the improvement of computational efficiency, and two kinds of CDG methods defined on unstructured overlapping meshes have also been presented in [@Xu2016; @LiM2019] for the treatment of complex computational domains.
In this paper, we first derive a formulation describing the propagation of 2D fully nonlinear weakly dispersive water waves, using the same approach as in [@Su1969] where the authors derived the 1D equation over the flat bottom. Based on this formulation, we derive a Green-Naghdi model with an enhanced dispersive effect, and then carry out a linear dispersion analysis for the corresponding linear equation to show the improvement on dispersive effects. This improvement on computational modeling is of great importance during the procedure of seeking more accurate numerical solutions of shallow water waves. In simulation of the derived 2D Green-Naghdi model, we usually encounter several difficulties: dealing with the mixed spatial and temporal derivative terms in the flux gradient and the source term, particularly in the dispersive-effect-improved model to be considered in the current work; maintaining the nonnegativity of the water depth and preserving the still-water stationary solution; computational efficiency and accuracy in 2D simulation. Concerning the first challenge, we reformulate the model into a hybrid system of the pseudo-conservative form with a nonhomogeneous source term and the pseudo-elliptic equation ([@MGH10; @LiM2014]). Since the resulting dynamic equations are no longer hyperbolic, we utilize the CDG method ([@Liu2007]) as the base scheme which has an important feature of being free of Riemann solvers. Meanwhile, we employ the FE method to solve the elliptic part, and point out that there is no need for the CDG-FE hybrid method to make special treatments on the continuity and discontinuity due to the fact of two copies of solutions on overlapping meshes. Secondly, the reason for naming the pseudo-elliptic equation in our numerical model lies in the fact that the computed variable of water depth appears in the coefficients of the second-order derivative term and may be trivial in the dry area. On the other hand, it explains that the nonnegativity of computational water depth is extremely important in our numerical model and numerical scheme, and this issue has not been solved in the previous 1D work ([@LiM2014]) yet. In this work, we design a CDG method for the Green-Naghdi model not only maintaining the well-balanced property for the stationary solution but also preserving the nonnegativity for the solution at the dry area. In addition to these issues on numerical modeling and schemes, computational efficiency ([@Xu2009; @Lannes2015]) is also significantly important for 2D water wave simulations. Since the CDG method is a local scheme similar to the DG method, the computation is implemented element by element at each time step and is thus friendly to the parallel execution. The computation load of solving the FE equation is dominant in our computation, and its fast solver will be considered in the future work.
The remainder of the paper is organized as follows. In section 2, we present a fully nonlinear strongly dispersive water wave model in 2D domain. In section 3, we propose a family of high order schemes, coupling positivity-preserving well-balanced CDG methods and continuous FE methods. In section 4, we perform a series of numerical experiments to demonstrate the well-balanced property, positivity-preserving property, high order accuracy as well as the capability of the Green-Naghdi model to describe the propagation of strongly nonlinear and dispersive waves. Some concluding remarks are given in section 5.
Mathematical models
===================
Green-Naghdi model {#sec:2.1}
------------------
In this paper, we consider the following 2D Green-Naghdi model $$\label{GN-NFB-2D}
\left\{\begin{array}{lclcl}
h_t+(hu)_x+(hv)_y=0,\\
(hu)_t+\left( hu^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3 \Phi + \frac{1}{2} h^2 \Psi \right)_x + \left( huv \right)_y =-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_x,\\
(hv)_t+\left(huv\right)_x+\left( hv^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3\Phi+\frac{1}{2}h^2\Psi\right)_y=-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_y,
\end{array} \right.$$ where $$\begin{aligned}
\Phi &=& - u_{xt}-u u_{xx}+ u_x^2 - v_{yt}-v v_{yy}+ v_y^2 - uv_{xy}- u_{xy}v+ 2 u_x v_y, \label{phi2D} \\
\Psi &=& b_x u_{t}+b_x u u_{x}+b_{xx} u^2 + b_y v_{t} + b_y v v_y +b_{yy} v^2 + b_y u v_x + b_x u_y v + 2b_{xy}uv. \label{psi2D}\end{aligned}$$ The derivation of the model is shown in Appendix.
Green-Naghdi model with enhanced dispersion effect {#sec:2.2}
--------------------------------------------------
We can observe from the last two equations in that ([@Chazel2011]) $$\begin{aligned}
u_t &=& -g(h+b)_x-uu_x-u_yv+\text{higher order terms}\\
&\simeq& \alpha u_t+(1-\alpha)(-g(h+b)_x-uu_x-u_yv),\end{aligned}$$ and $$\begin{aligned}
v_t &=& -g(h+b)_y-vv_y-uv_x+\text{higher order terms}\\
&\simeq& \alpha v_t+(1-\alpha)(-g(h+b)_y-vv_y-uv_x).\end{aligned}$$ Replacing $u_t$ and $v_t$ with $\alpha u_t+(\alpha-1)(uu_x+u_yv+g(h+b)_x)$ and $\alpha v_t+(\alpha-1)(vv_y+uv_x+g(h+b)_y)$ in and , respectively, we obtain a modified Green-Naghdi model as follows $$\label{GN-NFB-2Da}
\left\{\begin{array}{lclcl}
h_t+(hu)_x+(hv)_y=0,\\
(hu)_t+\left( hu^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3 \Phi + \frac{1}{2} h^2 \Psi \right)_x + \left( huv \right)_y =-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_x,\\
(hv)_t+\left(huv\right)_x+\left( hv^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3\Phi+\frac{1}{2}h^2\Psi\right)_y=-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_y.
\end{array} \right.$$ with $$\begin{aligned}
\Phi &=& - \alpha u_{tx}-(\alpha-2)u_x^2-\alpha uu_{xx}-\alpha u_{xy}v-2(\alpha-1)u_yv_x-(\alpha-1)g(h+b)_{xx} \nonumber \\
& & -\alpha v_{ty}-(\alpha-2)v_y^2-\alpha vv_{yy}-\alpha uv_{xy}-(\alpha-1)g(h+b)_{yy}+2u_xv_y, \label{phi2Da}\\
\Psi &=& \alpha b_xu_t+\alpha b_xuu_x+\alpha b_xu_yv+(\alpha-1)gb_x(h+b)_x+ \alpha b_yv_t \nonumber\\
& & +\alpha b_yvv_y+\alpha b_yuv_x+(\alpha-1)gb_y(h+b)_y+b_{xx}u^2+b_{yy}v^2+2b_{xy}uv. \label{psi2Da}\end{aligned}$$
It is apparent that the standard Green-Naghdi model corresponds to a particular case of the modified Green-Naghdi model with $\alpha=1$.
Linear dispersive analysis {#sec:2.3}
--------------------------
We will carry out a linear dispersion analysis for the modified Green-Naghdi model given in the previous section. We first linearize the model for the flat-bottom case ($b=\text{constant}$) with the trivial solution $h = h_0 > 0$ ($h_0$ is a constant), $u = u_0 = 0$ and $v = v_0 = 0$, i.e. looking for solutions of the form
$$h=h_0+\tilde{h}, \ \ u=\tilde{u}, \ \ v=\tilde{v},$$
where $(\tilde{h},\tilde{u},\tilde{v})$ are small perturbations, and retain only linear terms. As a result, we have $$\left\{ {\begin{array}{*{20}{l}}
\tilde{h}_t + h_0\tilde{u}_x + h_0\tilde{v}_y = 0~,\\
\tilde{u}_t + g\tilde{h}_x - \frac{1}{3}h_0^2 \left(\alpha \tilde{u}_{txx} + \alpha \tilde{v}_{tyx} + (\alpha-1)\tilde{h}_{xxx} + (\alpha-1)\tilde{h}_{yyx}\right) = 0~, \\
\tilde{v}_t + g\tilde{h}_y - \frac{1}{3}h_0^2 \left(\alpha \tilde{u}_{txy} + \alpha \tilde{v}_{tyy} + (\alpha-1)\tilde{h}_{xxy} + (\alpha-1)\tilde{h}_{yyy}\right) = 0~.
\end{array}} \right.$$ Then we look for perturbations as plane waves taking the form $$\left( \begin{array}{l}
\tilde{h}\\
\tilde{u}\\
\tilde{v}
\end{array} \right) = \left( \begin{array}{l}
\hat{h}\\
\hat{u}\\
\hat{v}
\end{array} \right){e^{i(k_1x+k_2y - \omega t)}},$$ where $k_1$ and $k_2$ are the wave numbers, and $\omega$ is the frequency of the perturbations. Without loss of generality, $k_1$ and $k_2$ are taken to be real and positive. We end up with the linear system $$\left( {\begin{array}{*{20}{c}}
-\omega & h_0k_1 & h_0k_2\\
gk_1 + \frac{\alpha-1}{3}gh_0^2k_1(k_1^2+k_2^2) & -\omega(1 + \frac{\alpha}{3}h_0^2 k_1^2 ) & -\omega \frac{\alpha}{3}h_0^2 k_1 k_2 \\
gk_2 + \frac{\alpha-1}{3}gh_0^2k_2(k_1^2+k_2^2) & -\omega \frac{\alpha}{3}h_0^2 k_1 k_2 & -\omega(1 + \frac{\alpha}{3}h_0^2 k_2^2 )
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\hat{h}\\
\hat{u}\\
\hat{v}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
0\\
0\\
0
\end{array}} \right),$$ and it has nontrivial solutions provided that the determinant of the coefficient matrix is zero, i.e. $$\label{dispersion relation}
\omega = {\omega _ \pm } = \pm |k| \sqrt {g{h_0}(1 + \frac{{\alpha - 1}}{3}h_0^2{|k|^2}){{(1 + \frac{\alpha }{3}h_0^2{|k|^2})}^{ - 1}}}$$ with $|k|=\sqrt{k_1^2+k_2^2}$. This gives the linear dispersion relation for the modified Green-Naghdi equations in the flat-bottom case with respect to the trivial solution. In Figure \[Fig:Dispersion\], we compare the linear dispersion relation ($\alpha=1.0$ and $\alpha=1.159$) and that of the full water wave problem in finite depth, $$\omega = {\omega _ \pm } = \pm \sqrt {g |k| \tanh ({h_0}|k|)},$$ with $g = 1$, $h_0 = 1$. It can be observed from this figure that compared with the case $\alpha=1.0$, the linear dispersion relation of the modified model with $\alpha=1.159$ shows a better agreement with that of the full water wave problem.
![Comparison between the linear dispersion relation of the Green-Naghdi equations with $\alpha=1.0$ (blue line with dots) and $\alpha=1.159$ (red line with circles) and the exact linear dispersion relation of the full water wave problem (black solid line), with $g = 1$, $h_0 = 1$.[]{data-label="Fig:Dispersion"}](Dispersion.eps){height="8cm" width="12cm"}
Numerical methods for the Green-Naghdi models {#sec:3}
=============================================
The first difficulty in designing numerical schemes for the Green-Naghdi model comes from the appearance of mixed spatial and temporal derivatives in the equations. To tackle this, we make a reformulation from the original equations through introducing auxiliary variables, and the procedure will be given in Section \[sec:3.1\]. This new model will be adopted for the 2D shallow water wave simulations. In addition, in the case of a variable bottom, these equations admit still-water stationary solutions for the system which are given by $$\label{Eq:SSW2}
u=0~,\qquad v=0~,\qquad h+b = \mbox{constant}~.$$ Numerical errors in discretization will produce the spurious oscillations which lead to a wrong set of solutions. Another major difficulty in the simulation of shallow water waves has to do with the appearance of dry areas. In particular, this case happens frequently in so many important applications involving rapidly moving interfaces between wet and dry areas, for instance, the wave run-up on beaches or over man-made structures, dam break and tsunami. If no special care is taken, non-physical phenomena may arise and it will lead to a breakdown of computation. Following the work in [@Xing2010; @LiM2017], we design in Section \[sec:3.2\] a high order positivity-preserving well-balanced CDG-FE method for the 2D simulation of the modified Green-Naghdi model in Section \[sec:3.1\].
Reformulation of modified Green-Naghdi model {#sec:3.1}
--------------------------------------------
In order to remove the mixed spatial and temporal derivatives in the equations, we introduce two new unknown variables $P$ and $Q$ which satisfy: $$\begin{aligned}
\label{Eq:hP}
hP &=& -\left(\frac{\alpha}{3}h^3 u_x + \frac{\alpha}{3}h^3 v_y - \frac{\alpha}{2}h^2 v b_y \right)_x-\left(\frac{\alpha}{2}h^2 v b_x \right)_y \nonumber \\
& & + h \Big(1+\alpha h_xb_x+\frac{\alpha}{2}h b_{xx} +\alpha b_x^2 \Big)u+h \Big(\alpha h_yb_x+\frac{\alpha}{2}h b_{xy} +\alpha b_x b_y \Big)v~,\end{aligned}$$ and $$\begin{aligned}
\label{Eq:hQ}
hQ &=& -\left(\frac{\alpha}{3}h^3 u_x + \frac{\alpha}{3}h^3 v_y - \frac{\alpha}{2}h^2ub_x \right)_y-\left(\frac{\alpha}{2}h^2 u b_y \right)_x \nonumber \\
& & + h \Big(\alpha h_xb_y+\frac{\alpha}{2}h b_{xy} +\alpha b_x b_y \Big)u + h \Big(1+\alpha h_yb_y+\frac{\alpha}{2}h b_{yy} +\alpha b_y^2 \Big)v~.\end{aligned}$$ Then the system can be reformulated into a balance law $$\label{GN-NFB-2D-law}
\bU_t+\mathbf{F}(\bU,u,v;b)_x+\mathbf{G}(\bU,u,v;b)_y=\mathbf{S}(\bU,u,v;b),$$ where $\mathbf{U}=(h,hP,hQ)^\top$ is the unknown vector, $$\begin{aligned}
\label{GN-NFB-2D-Flux1}
\mathbf{F}(\bU,u,v; b)&=&\left(hu, hPu +hQv +\frac{1}{2}gh^2 -\alpha huvb_xb_y+\frac{1-\alpha}{2}h^2(u^2b_{xx}+v^2b_{yy})\right.\nonumber \\
& & \left.-hv^2(1+\alpha b_y^2)-(\frac{4\alpha-2}{3}h^3u_x^2+\frac{6\alpha-2}{3}h^3u_xv_y+\frac{4\alpha-2}{3}h^3v_y^2)\right.\nonumber \\
& & \left. +\alpha h^2u(u_x+v_y)b_x+\frac{3}{2}\alpha h^2v( u_x+v_y)b_y -\frac{2}{3}(\alpha-1)h^3u_yv_x +(1-\alpha)h^2uvb_{xy}\right. \nonumber \\
& & \left. -\frac{\alpha-1}{3}gh^3((h+b)_{xx}+(h+b)_{yy})+\frac{\alpha-1}{2}gh^2(b_x(h+b)_x+b_y(h+b)_y), \right.\nonumber \\
& & \left. huv(1+\alpha b_y^2)+\alpha hu^2b_xb_y-\frac{\alpha}{2}h^2u(u_x+v_y)b_y \right)^\top \nonumber \\\end{aligned}$$ and $$\begin{aligned}
\label{GN-NFB-2D-Flux2}
\mathbf{G}(\bU,u,v; b)&=&\left(hv, huv(1+\alpha b_x^2)+\alpha hv^2b_xb_y-\frac{\alpha}{2}h^2v(u_x+v_y)b_x, hPu +hQv \right. \nonumber \\
& & \left. +\frac{1}{2}gh^2 -\alpha huvb_xb_y+\frac{1-\alpha}{2}h^2(u^2b_{xx}+v^2b_{yy})-hu^2(1+\alpha b_x^2)\right. \nonumber \\
& & \left. - (\frac{4\alpha-2}{3}h^3u_x^2+\frac{6\alpha-2}{3}h^3u_xv_y+\frac{4\alpha-2}{3}h^3v_y^2)+\frac{3}{2}\alpha h^2u (u_x +v_y)b_x \right. \nonumber \\
& & \left. + \alpha h^2v (u_x + v_y )b_y-\frac{2}{3}(\alpha-1)h^3u_yv_x +(1-\alpha)h^2uvb_{xy}\right.\nonumber \\
& & \left. -\frac{\alpha-1}{3}gh^3((h+b)_{xx}+(h+b)_{yy})+\frac{\alpha-1}{2}gh^2(b_x(h+b)_x+b_y(h+b)_y)\right)^\top \nonumber \\\end{aligned}$$ are the flux terms, and $$\begin{aligned}
\label{GN-NFB-2D-source}
\bS(\bU,u,v; b)&=&\left(0, -ghb_x-\frac{\alpha}{2}h^2 u (u_x+v_y) b_{xx} -\frac{\alpha}{2}h^2 v (u_x+v_y) b_{xy}+(2\alpha-1)h u^2 b_x b_{xx} \right. \nonumber \\
& & \left. +huv((3\alpha-2)b_xb_{xy}+\alpha b_{xx}b_y)+(\alpha-1)h^2(u_x^2+u_xv_y+u_yv_x+v_y^2)b_x \right. \nonumber \\
& & \left. + \alpha hv^2b_{xy}b_y +(\alpha-1)hv^2b_xb_{yy} +\frac{\alpha-1}{2}gh^2((h+b)_{xx}+(h+b)_{yy})b_x \right. \nonumber \\
& & \left. -(\alpha-1)gh(b_x^2(h+b)_x+b_xb_y(h+b)_y), \right. \nonumber \\
& & \left. -ghb_y-\frac{\alpha}{2}h^2 u (u_x+v_y) b_{xy}-\frac{\alpha}{2}h^2 v (u_x+v_y) b_{yy} +(2\alpha-1)h v^2 b_y b_{yy}\right. \nonumber \\
& & \left. +huv((3\alpha-2)b_{xy}b_y+\alpha b_xb_{yy})+(\alpha-1)h^2(u_x^2+u_xv_y+u_yv_x+v_y^2)b_y \right. \nonumber \\
& & \left. +\alpha hu^2b_xb_{xy}+(\alpha-1)hu^2b_{xx}b_y +\frac{\alpha-1}{2}gh^2((h+b)_{xx}+(h+b)_{yy})b_y \right. \nonumber \\
& & \left. -(\alpha-1)gh(b_xb_y(h+b)_x+b_y^2(h+b)_y) \right)^\top \nonumber \\\end{aligned}$$ is the source term.
Based on the new equations, the solution of the modified Green-Naghdi model amounts to finding the unknowns ${(h, hP, hQ)^\top}$ based on and the unknowns ${(u,v)^\top}$ from -. We remark that the linear dispersive relationship of the reformulated Green-Naghdi model - is the same as the one of .
CDG-FE methods {#sec:3.2}
--------------
In this section, we develop the numerical method for the solution of the equations -. Let $\mathcal{T}^C=\{C_{ij}, \forall i, j \}$ and $\mathcal{T}^D=\{D_{ij}, \forall i, j \}$ define two overlapping meshes for the computational domain $\Omega=[x_{\min}, x_{\max}] \times [y_{\min}, y_{\max}]$, with $C_{ij}=[x_{i-\frac{1}{2}}, x_{i+\frac{1}{2}}] \times [y_{j-\frac{1}{2}}, y_{j+\frac{1}{2}}]$, $D_{ij}=[x_{i-1}, x_{i}] \times [y_{j-1}, y_{j}]$, $x_{i+\frac{1}{2}}=\frac{1}{2}(x_{i}+x_{i+1})$ and $y_{j+\frac{1}{2}}=\frac{1}{2}(y_{j}+y_{j+1})$, where $\{x_i\}_i$ and $\{y_j\}_j$ are partitions of $[x_{\min}, x_{\max}]$ and $[y_{\min}, y_{\max}]$ respectively. Associated with each mesh, we define the following discrete spaces $$\begin{aligned}
\mathcal {V}^C&=&\mathcal {V}^{C,k}=\{{\bf{v}}:{\bf{v}}|_{C_{ij}}\in [P^k(C_{ij})]^3~, \forall\, i, j \}~,\\
\mathcal {V}^D&=&\mathcal {V}^{D,k}=\{{\bf{v}}:{\bf{v}}|_{D_{ij}}\in [P^k(D_{ij})]^3~, \,
\forall \, i, j \}~.\end{aligned}$$ To approximate $u$ and $v$, we define two continuous finite element spaces $$\begin{aligned}
\mathcal {W}^C&=&\mathcal{W}^{C,k}=\{w:w|_{C_{ij}}\in P^k({C_{ij}})~, \forall\, i,j \text{ and $w$ is continuous} \}~,\\
\mathcal {W}^D&=&\mathcal W^{D,k}=\{w:w|_{D_{ij}}\in P^k({D_{ij}})~, \,
\forall \,i,j \text{ and $w$ is continuous} \}~.\end{aligned}$$ We only present the schemes with the forward Euler method for time discretization. High-order time discretizations will be discussed in Section \[sec:3.3\]. The proposed methods evolve two copies of numerical solution, which are assumed to be available at $t=t_n$, denoted by $\bU^{n,\star}=(h^{n,\star}, (hP)^{n,\star}, (hQ)^{n,\star})^\top \in \mathcal{V}^\star$, and we want to find the solutions at $t=t_{n+1}=t_n+\Delta t_n$. Only the procedure to update $\bU^{n+1,C}$ will be described. We project the bottom topography function $b$ into $P^k(C_{ij})$ on $C_{ij}$ (resp. into $P^k(D_{ij})$ on $D_{ij}$) in the $L^2$ sense, and obtain an approximation $b^{C}$ (resp. $b^{D}$) throughout the computational domain.
### Standard CDG-FE method {#sec:3.2.1}
We first apply to the standard CDG methods of Liu et al. ([@Liu2007]) for space discretization and the forward Euler method for time discretization. That is, we look for $\bU^{n+1,C}=(h^{n+1,C}, (hP)^{n+1,C}, (hQ)^{n+1,C})^\top\in\mathcal{V}^{C,k}$ such that for any $\bV\in\mathcal{V}^{C,k}|_{C_{ij}}$ with any $i$ and $j$, $$\begin{aligned}
\label{Comp-SW-2D-1}
\int_{C_{ij}} \bU^{n+1,C}\cdot \bV dxdy &=& \int_{C_{ij}}\left( \theta \bU^{n,D} + (1-\theta) \bU^{n,C}\right)\cdot \bV dxdy \nonumber\\
&+&\Delta t_n \int_{C_{ij}} \left[ \bF(\bU^{n,D},u^{n,D},v^{n,D};b^D)\cdot \bV_x + \bG(\bU^{n,D},u^{n,D},v^{n,D};b^D)\cdot \bV_y \right] dxdy \nonumber\\
&-&\Delta t_n \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}\left[\bF(\bU^{n,D}(x_{i+\frac{1}{2}},y),u^{n,D}(x_{i+\frac{1}{2}},y),
v^{n,D}(x_{i+\frac{1}{2}},y);b^D(x_{i+\frac{1}{2}},y)) \cdot \bV(x^-_{i+\frac{1}{2}},y)\right. \nonumber\\
&-&\left.\bF(\bU^{n,D}(x_{i-\frac{1}{2}},y),u^{n,D}(x_{i-\frac{1}{2}},y),v^{n,D}(x_{i-\frac{1}{2}},y);
b^D(x_{i-\frac{1}{2}},y)) \cdot \bV(x^+_{i-\frac{1}{2}},y) \right] dy \nonumber\\
&-&\Delta t_n \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\left[\bG(\bU^{n,D}(x,y_{j+\frac{1}{2}}),u^{n,D}(x,y_{j+\frac{1}{2}}),
v^{n,D}(x,y_{j+\frac{1}{2}});b^D(x,y_{j+\frac{1}{2}})) \cdot \bV(x,y^-_{j+\frac{1}{2}})\right. \nonumber\\
&-&\left.\bG(\bU^{n,D}(x,y_{j-\frac{1}{2}}),u^{n,D}(x,y_{j-\frac{1}{2}}),v^{n,D}(x,y_{j-\frac{1}{2}});
b^D(x,y_{j-\frac{1}{2}})) \cdot \bV(x,y^+_{j-\frac{1}{2}}) \right] dx \nonumber\\
&+& \Delta t_n\int_{C_{ij}}\bS(\bU^{n,D},u^{n,D},v^{n,D}; b^{D}) \cdot \bV dxdy~.\end{aligned}$$ Here $\theta=\Delta t_n/\tau \in [0, 1]$ with $\tau$ being the maximal time step allowed by the CFL restriction ([@Liu2007]). In general, this numerical scheme does not necessarily maintain the still-water stationary solution and preserve the non-negativity of water depth.
Once $\mathbf{U}^{n+1,C}$ is available, we can obtain $u^{n+1,C}$ and $v^{n+1,C}$by applying a continuous finite element method to and : look for $u^{n+1,C}, v^{n+1,C} \in \widetilde{W}^{C,k}$ such that for any $\widehat{u}, \widehat{v} \in \widehat{W}^{C,k}$, $$\begin{aligned}
\label{FE1}
& &\int_\Omega \left(\frac{\alpha}{3}(h^{n+1,C})^3(u^{n+1,C})_x+\frac{\alpha}{3}(h^{n+1,C})^3(v^{n+1,C})_y
-\frac{\alpha}{2}(h^{n+1,C})^2v^{n+1,C}(b^C)_y \right)\widehat{u}_xdxdy \nonumber\\
&+&\int_\Omega \left(\frac{\alpha}{2}(h^{n+1,C})^2v^{n+1,C}(b^C)_x \right)\widehat{u}_ydxdy
+\int_\Omega f(h^{n+1,C},u^{n+1,C},v^{n+1,C};b^C)\widehat{u}dxdy \nonumber\\
&=&\int_\Omega (hP)^{n+1,C}\widehat{u}dxdy,\end{aligned}$$ $$\begin{aligned}
\label{FE2}
& &\int_\Omega \left(\frac{\alpha}{3}(h^{n+1,C})^3(u^{n+1,C})_x+\frac{\alpha}{3}(h^{n+1,C})^3(v^{n+1,C})_y
-\frac{\alpha}{2}(h^{n+1,C})^2u^{n+1,C}(b^C)_x \right)\widehat{v}_ydxdy \nonumber\\
&+&\int_\Omega \left(\frac{\alpha}{2}(h^{n+1,C})^2u^{n+1,C}(b^C)_y \right)\widehat{v}_xdxdy
+\int_\Omega g(h^{n+1,C},u^{n+1,C},v^{n+1,C};b^C)\widehat{v}dxdy \nonumber\\
&=&\int_\Omega (hQ)^{n+1,C}\widehat{v}dxdy,\end{aligned}$$ where $$\begin{aligned}
f(h^{n+1,C},u^{n+1,C},v^{n+1,C};b^C)&=&h^{n+1,C}\left (1+\alpha(h^{n+1,C})_x(b^C)_x+\frac{\alpha}{2}h^{n+1,C}(b^C)_{xx}+\alpha(b^C)_{x}^2 \right)u^{n+1,C} \nonumber\\
&+&h^{n+1,C}\left (\alpha(h^{n+1,C})_y(b^C)_x+\frac{\alpha}{2}h^{n+1,C}(b^C)_{xy}+\alpha (b^C)_x (b^C)_y \right)v^{n+1,C}, \nonumber \\\end{aligned}$$ $$\begin{aligned}
g(h^{n+1,C},u^{n+1,C},v^{n+1,C};b^C)&=&h^{n+1,C}\left (\alpha(h^{n+1,C})_x(b^C)_y+\frac{\alpha}{2}h^{n+1,C}(b^C)_{xy}+\alpha (b^C)_x (b^C)_y \right)u^{n+1,C}, \nonumber \\
&+&h^{n+1,C}\left (1+\alpha(h^{n+1,C})_y(b^C)_y+\frac{\alpha}{2}h^{n+1,C}(b^C)_{yy}+\alpha(b^C)_{y}^2 \right)v^{n+1,C}, \nonumber \\\end{aligned}$$ and $\widetilde{W}^{C,k}$, $\widehat{W}^{C,k}$ are variants of $\mathcal{W}^{C,k}$ with consideration of the boundary conditions ([@LiM2014]).
When the bottom is flat ($b=\mbox{constant}$), the unique solvability of this FE method can be obtained in a straightforward manner if $h \geq h_0 >0$ and $\alpha \geq \alpha_0>0$. When the bottom is not flat, the unique solvability of this FE scheme is more difficult to be determined. Even though $h \geq h_0 >0$ and $\alpha \geq \alpha_0>0$, a constraint condition for the $b$ as in the 1D case in [@LiM2014] is needed for the unique solvability of the FE method. Due to the complexity of the 2D case, the condition can not be obtained explicitly. However, except for the tests in Sections 4.2(Case B) and 4.3 which involve a dry area or a near dry area, the FE system of the other numerical applications in Section 4 (satisfy $h \geq h_0 >0$ and $\alpha \geq \alpha_0>0$) was found to be uniquely solvable in numerics.
For the cases in Sections 4.2(Case B) and 4.3 which involve a dry area or a near dry area, the FE equation is ill-conditioned, so we shall evaluate the velocity $u$ using an approximation technique. We solve the FE equation on the cells with $h^{n+1,C} \geq h_0 >0$, while for other cells, we omit the high order terms in - and get $$\begin{aligned}
\label{Eq:hP-1}
hP & \simeq & h \Big(1 +\alpha b_x^2 \Big)u+\alpha h b_x b_y v~,\\
hQ & \simeq & \alpha h b_x b_y u + h \Big(1+\alpha b_y^2 \Big)v~.\end{aligned}$$ Then the velocity $u$ and $v$ can be evaluated by $$\begin{aligned}
\label{Eq:hP-2}
u & \simeq & \frac{1+\alpha b_y^2}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \frac{hP}{h} - \frac{\alpha b_x b_y}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \frac{hQ}{h},\\ \label{Eq:hP-3}
v & \simeq & \frac{1+\alpha b_x^2}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \frac{hQ}{h} - \frac{\alpha b_x b_y}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \frac{hP}{h},\end{aligned}$$ To deal with the singularity in -, we employ the regularization technique presented in [@Kurganov2007] and thus we have: $$\begin{aligned}
\label{Eq:hP-4}
u & \simeq & \frac{1+\alpha b_y^2}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \widetilde{P} - \frac{\alpha b_x b_y}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \widetilde{Q} ,\\ \label{Eq:hP-5}
v & \simeq & \frac{1+\alpha b_x^2}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \widetilde{Q} - \frac{\alpha b_x b_y}{1 +\alpha b_x^2 + \alpha b_y^2} \cdot \widetilde{P} ,\end{aligned}$$ where $$\begin{aligned}
\label{Eq:hP-6}
\widetilde{P}=\frac{\sqrt{2}h (hP)}{\sqrt{h^4+\max(h^4,\varepsilon)}}, \widetilde{Q}=\frac{\sqrt{2}h (hQ)}{\sqrt{h^4+\max(h^4,\varepsilon)}}\end{aligned}$$ with $\varepsilon=\min((\Delta x)^{4},(\Delta y)^{4})$. In the computation, $h_0=\max((\Delta x)^{k+1},(\Delta y)^{k+1})$ for $k=1,2$.
### Well-balanced CDG-FE method {#sec:3.2.2}
In this subsection, we propose a family of high order well-balanced CDG-FE schemes for the model -, which exactly preserves the still-water steady state solution . The well-balance property can be achieved by adding some terms to the scheme , $$\begin{aligned}
\label{Comp-WB-SW-2D-0}
& &\int_{C_{ij}} \bU_h^{n+1,C}\cdot \bV dxdy = \mbox{right-hand side of \eqref{Comp-SW-2D-1}} + \btU(b_h^{C}, b_h^{D},\bV)\nonumber\\
&+& \Delta t_n \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} \left[\btS_1(\bU_h^{n,D}; b_h^{D}(x_i^+,y))-\btS_1(\bU_h^{n,D}; b_h^{D}(x_i^-,y))\right] \cdot \bV(x_i,y) dy \nonumber\\
&+&\Delta t_n \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \left[\btS_2(\bU_h^{n,D}; b_h^{D}(x,y_j^+))-\btS_2(\bU_h^{n,D}; b_h^{D}(x,y_j^-))\right] \cdot \bV(x,y_j) dx~,\nonumber\end{aligned}$$ where the correction terms are given by $$\label{Solution-Correction-1}
\btU(b_h^{C}, b_h^{D}, \bV)= \theta\int_{C_{ij}}\left( b_h^{D}-b_h^D, 0, 0 \right)^\top\cdot \bV dxdy~,$$ $$\label{Source-Correction-1}
\btS_1(\bU_h^{n,D}; b_h^{D})=\left(0, \frac{g}{2}(b_h^D)^2-\gamma^{n,D}_{ij} g b_h^D, 0 \right)^\top~,$$ $$\label{Source-Correction-2}
\btS_2(\bU_h^{n,D}; b_h^{D})=\left(0, 0, \frac{g}{2}(b_h^D)^2-\gamma^{n,D}_{ij} g b_h^D\right)^\top~.$$ Here, $\gamma^{n,D}_{ij}$ is a special constant which represents the average value of the water surface $\eta_h^{n,D}=h_h^{n,D}+b_h^{D}$ in the element $C_{ij}$. Particularly, we can take $$\begin{array}{lclcl}
\gamma^{n,D}_{ij} &=& \frac{1}{4}\left[\eta_h^{n,D}(x_{i+\frac{1}{2}},y_{j+\frac{1}{2}}) +\eta_h^{n,D}(x_{i-\frac{1}{2}},y_{j+\frac{1}{2}})\right. \\
&+&\left. \eta_h^{n,D}(x_{i+\frac{1}{2}},y_{j-\frac{1}{2}}) + \eta_h^{n,D}(x_{i-\frac{1}{2}}, y_{j-\frac{1}{2}})\right]~.
\end{array}$$ With the following decomposition of the source term $$\begin{aligned}
\label{decom-2D}
\bS(\bU_h^{n,D},u_h^{n,D},v_h^{n,D}; b_h^{D})&=& \left(0,-g(\eta_h^{n,D}-b_h^D)(b_h^D)_x, -g(\eta_h^{n,D}-b_h^D)(b_h^D)_y \right)^{\top} + \mbox{the remaining items} \nonumber\\
&=& \left(0,\frac{g}{2}(b_h^D)^2-g\gamma_{ij}^{n,D}b_h^D, 0 \right)_x^{\top} + \left(0, 0, \frac{g}{2}(b_h^D)^2-g\gamma_{ij}^{n,D}b_h^D \right)_y^{\top} \nonumber\\
&-& \left(0, g(\eta_h^{n,D}-\gamma_{ij}^{n,D})(b_h^D)_x, g(\eta_h^{n,D}-\gamma_{ij}^{n,D})(b_h^D)_y \right)^{\top} \nonumber\\
&+& \mbox{the remaining items},\end{aligned}$$ the scheme can be rewritten as $$\begin{aligned}
\label{Comp-WB-SW-2D-1}
& &\int_{C_{ij}} \bU_h^{n+1,C}\cdot \bV dxdy \nonumber\\
&=& \int_{C_{ij}}\left( \theta \bU_h^{n,D} + (1-\theta) \bU_h^{n,C}\right)\cdot \bV dxdy + \btU(b_h^{C}, b_h^{D}, \bV)\nonumber\\
&+&\Delta t_n \int_{C_{ij}} \left[ \bF(\bU_h^{n,D},u_h^{n,D},v_h^{n,D};b_h^D)\cdot \bV_x + \bG(\bU_h^{n,D},u_h^{n,D},v_h^{n,D};b_h^D)\cdot \bV_y \right] dxdy \nonumber\\
&-&\Delta t_n \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}\left[\bF(\bU_h^{n,D}(x_{i+\frac{1}{2}},y),u_h^{n,D}(x_{i+\frac{1}{2}},y),
v_h^{n,D}(x_{i+\frac{1}{2}},y);b_h^D(x_{i+\frac{1}{2}},y)) \cdot \bV(x^-_{i+\frac{1}{2}},y)\right. \nonumber\\
&-&\left.\bF(\bU_h^{n,D}(x_{i-\frac{1}{2}},y),u_h^{n,D}(x_{i-\frac{1}{2}},y),v_h^{n,D}(x_{i-\frac{1}{2}},y);
b_h^D(x_{i-\frac{1}{2}},y)) \cdot \bV(x^+_{i-\frac{1}{2}},y) \right] dy \nonumber\\
&-&\Delta t_n \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\left[\bG(\bU_h^{n,D}(x,y_{j+\frac{1}{2}}),u_h^{n,D}(x,y_{j+\frac{1}{2}}),
v_h^{n,D}(x,y_{j+\frac{1}{2}});b_h^D(x,y_{j+\frac{1}{2}})) \cdot \bV(x,y^-_{j+\frac{1}{2}})\right. \nonumber\\
&-&\left.\bG(\bU_h^{n,D}(x,y_{j-\frac{1}{2}}),u_h^{n,D}(x,y_{j-\frac{1}{2}}),v_h^{n,D}(x,y_{j-\frac{1}{2}});
b_h^D(x,y_{j-\frac{1}{2}})) \cdot \bV(x,y^+_{j-\frac{1}{2}}) \right] dx \nonumber\\
&-& \Delta t_n \int_{C_{ij}}\left(0, g(\eta_h^{n,D}-\gamma_{ij}^{n,D})(b_h^D)_x, g(\eta_h^{n,D}-\gamma_{ij}^{n,D})(b_h^D)_y \right)^{\top} \cdot \bV dxdy \nonumber\\
&-& \Delta t_n \int_{C_{ij}}\left(0, \frac{g}{2}(b_h^D)^2-g\gamma_{ij}^{n,D})b_h^D,0 \right)^{\top} \cdot \bV_x dxdy \nonumber\\
&-& \Delta t_n \int_{C_{ij}}\left(0, 0, \frac{g}{2}(b_h^D)^2-g\gamma_{ij}^{n,D})b_h^D \right)^{\top} \cdot \bV_y dxdy \nonumber\\
&+& \Delta t_n \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} \left(0, \frac{g}{2}(b_h^D(x_{i+\frac{1}{2}},y))^2-g\gamma_{ij}^{n,D})b_h^D(x_{i+\frac{1}{2}},y) ,0 \right)^{\top} \cdot \bV(x_{i+\frac{1}{2}}^-,y) dy \nonumber\\
&-& \Delta t_n \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} \left(0, \frac{g}{2}(b_h^D(x_{i-\frac{1}{2}},y))^2-g\gamma_{ij}^{n,D})b_h^D(x_{i-\frac{1}{2}},y) ,0 \right)^{\top} \cdot \bV(x_{i-\frac{1}{2}}^+,y) dy \nonumber\\
&+& \Delta t_n \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \left(0 ,0, \frac{g}{2}(b_h^D(x,y_{j+\frac{1}{2}}))^2-g\gamma_{ij}^{n,D})b_h^D(x,y_{j+\frac{1}{2}}) \right)^{\top} \cdot \bV(x,y_{j+\frac{1}{2}}^-) dx \nonumber\\
&-& \Delta t_n \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \left(0,0,\frac{g}{2}(b_h^D(x,y_{j-\frac{1}{2}}))^2-g\gamma_{ij}^{n,D})b_h^D(x,y_{j-\frac{1}{2}}) \right)^{\top} \cdot \bV(x,y_{j-\frac{1}{2}}^+) dx \nonumber\\
&+& \mbox{the remaining items}~.\end{aligned}$$
\[Th3.1\] The numerical scheme, defined in , and and their counterparts for $\bU_h^{n+1,D}$, $u_h^{n+1,D}$ and $v_h^{n+1,D}$ , to solve the 2D Green-Naghdi model - is well-balanced, in the sense that it preserves the still-water stationary solution (\[Eq:SSW2\]).
The proof can be easily obtained by the mathematical induction with time step $n$ as the Proposition in [@LiM2014].
### Positivity-preserving CDG-FE method {#sec:3.2.3}
We now discuss the positivity-preserving CDG-FE method for -. Firstly, let $\hat L_i^{1,x}=\{ \hat x_i^{1,\beta}, \beta=1,2,...,{\hat N} \}$ and $\hat L_i^{2,x}=\{\hat x_i^{2, \beta}, \beta=1,2,...,{\hat N}\}$ be the Legendre Gauss-Lobatto quadrature points on $[x_{i-\frac{1}{2}}, x_{i}]$ and $[x_{i}, x_{i+\frac{1}{2}}]$ respectively, while $\hat L_j^{1,y}=\{ \hat y_j^{1,\beta}, \beta=1,2,...,{\hat N} \}$ and $\hat L_j^{2,y}=\{\hat y_j^{2, \beta}, \beta=1,2,...,{\hat N}\}$ represent the Legendre Gauss-Lobatto quadrature points on $[y_{j-\frac{1}{2}}, y_{j}]$ and $[y_{j}, y_{j+\frac{1}{2}}]$ respectively, $\forall i, j$. The corresponding quadrature weights on the reference element $[-\frac{1}{2}, \frac{1}{2}]$ are $\hat \omega_{\beta}, \beta =1, 2, ... , {\hat N}$, and $\hat N$ is chosen such that $2 \hat N-3 \geq k$. In addition, let $L_i^{1,x}=\{ x_i^{1,\alpha}, \alpha=1,2,...,N \}$ and $L_i^{2,x}=\{x_i^{2, \alpha}, \alpha=1,2,...,N\}$ denote the Gaussian quadrature points on $[x_{i-\frac{1}{2}}, x_{i}]$ and $[x_{i}, x_{i+\frac{1}{2}}]$ respectively, while $L_j^{1,y}=\{ y_j^{1,\alpha}, \alpha=1,2,...,N \}$ and $L_j^{2,y}=\{y_j^{2, \alpha}, \alpha=1,2,...,N\}$ represent the Gaussian quadrature points on $[y_{j-\frac{1}{2}}, y_{j}]$ and $[y_{j}, y_{j+\frac{1}{2}}]$ respectively. The corresponding quadrature weights $\omega_{\alpha}, \alpha=1, 2, ... , N$ are distributed on the interval $[-\frac{1}{2}, \frac{1}{2}]$ and $N$ is chosen such that the Gaussian quadrature is exact for the integration of univariate polynomials of degree $2k+1$. Define $L_{i,j}^{l,m}=(L_i^{l,x}\otimes \hat L_j^{m,y})\cup (\hat L_i^{l,x}\otimes L_j^{m,y})$ with $l,m=1,2$. By taking the test function $\bV=(\frac{1}{\Delta x\Delta y}, 0, 0)^\top$, we get the equation satisfied by the cell average of $h^{n,C}$, $$\begin{aligned}
\label{gene-cell-2D}
\bar h_{ij}^{n+1,C}&=& (1-\theta) \bar h_{ij}^{n,C} + \frac{\theta}{\Delta x \Delta y} \int_{C_{ij}}h^{n,D}dxdy \nonumber\\
&-&\frac{\Delta t_n}{\Delta x \Delta y} \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}\left[h^{n,D}(x_{i+\frac{1}{2}},y)u^{n,D}(x_{i+\frac{1}{2}},y)-
h^{n,D}(x_{i-\frac{1}{2}},y)u^{n,D}(x_{i-\frac{1}{2}},y)\right] dy \nonumber\\
&-&\frac{\Delta t_n}{\Delta x \Delta y} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\left[h^{n,D}(x,y_{j+\frac{1}{2}})v^{n,D}(x,y_{j+\frac{1}{2}})-
h^{n,D}(x,y_{j-\frac{1}{2}})v^{n,D}(x,y_{j-\frac{1}{2}})\right] dx~,\end{aligned}$$ where $\bar h_{ij}^{n,C}$ denotes the cell average of the CDG solution $h^C$ on $C_{ij}$ at time $t_n$.
In the numerical implementation, the definite integrals in the intervals $[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}]$ and $[y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}]$ are evaluated by applying the Gaussian quadrature rule described above to each half of the interval (also see Section 3.2 in [@Cheng2012]), the scheme becomes $$\begin{aligned}
\label{cell1-2D-1}
\bar{h}_{ij}^{n+1,C}
&=& (1-\theta) \bar{h}_{ij}^{n,C} + \frac{\theta}{\Delta x\Delta y}\int_{C_{ij}} h^{n,D} dxdy\nonumber\\
&-&\frac{\Delta t_n}{2\Delta x}\sum_{m=1}^{2}\sum_{\alpha=1}^{N}\omega_{\alpha}\left[ h^{n,D}\left(\hat x_i^{2,\hat N},y_j^{m,\alpha}\right)u^{n,D}\left(\hat x_i^{2,\hat N},y_j^{m,\alpha}\right)\right. \nonumber\\
&-&\left.h^{n,D}\left(\hat x_i^{1,1},y_j^{m,\alpha}\right)u^{n,D}\left(\hat x_i^{1,1},y_j^{m,\alpha}\right) \right] \nonumber\\
&-&\frac{\Delta t_n}{2\Delta y}\sum_{l=1}^{2}\sum_{\alpha=1}^{N}\omega_{\alpha}\left[ h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{2,\hat N}\right)v^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{2,\hat N}\right)\right. \nonumber\\
&-&\left.h^{n,D}\left(x_i^{l,\alpha},\hat y_j^{1,1}\right)v^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{1,1}\right) \right],\end{aligned}$$ herein, we used $\hat x_i^{1,1}=x_{i-\frac{1}{2}}, \hat x_i^{2,\hat N}=x_{i+\frac{1}{2}}, \hat y_j^{1,1}=y_{j-\frac{1}{2}}, \hat y_j^{2,\hat N}=y_{j+\frac{1}{2}}$. Now we have the following result.
\[Th3.2\] For any given $n\ge 0$, we assume $\bar h_{ij}^{n,C}\ge 0$ and $\bar h_{ij}^{n,D}\ge 0$, $\forall i, j$. Consider the scheme in and its counterpart for $\bar h_{ij}^{n+1,D}$, if $h^C(x,y,t_n)\ge 0$ and $h^D(x,y,t_n)\ge 0$, $\forall (x,y) \in L_{i,j}^{l,m}, \forall i,j$ with $l,m=1,2$, then $\bar h_{ij}^{n+1, C} \ge 0$ and $\bar h_{ij}^{n+1, D} \ge 0$, $\forall i,j$, under the CFL condition $$\label{gene-CFL-con-2d}
\lambda_x a_x +\lambda_y a_y \leq \frac{1}{4} \theta \hat \omega_1~,$$ where $\lambda_x=\Delta t_n/\Delta x$, $\lambda_y=\Delta t_n/\Delta y$, $a_x=\max(\|u^{n,C}\|_\infty, \|u^{n,D}\|_\infty)$, $a_y=\max(\|v_h^{n,C}\|_\infty, \|v_h^{n,D}\|_\infty)$.
Since the numerical solution $h^{n,D}$ is a piecewise polynomial with degree $k$, the integral of $h^{n,D}$ in cell $C_{ij}$ in is exactly evaluated in our numerical implementation. However, in order to discuss the non-negativity of the cell average $\bar{h}_{ij}^{n+1,C}$. We equivalently evaluate the integral using a combination of the Gauss quadrature rule and the Legendre Gauss-Lobatto quadrature rule as follows: $$\begin{aligned}
\label{cell1-2D-2}
\frac{\theta}{\Delta x\Delta y}\int_{C_{ij}} h^{n,D} dxdy &=& \frac{\theta}{2\Delta x\Delta y}\int_{C_{ij}} h^{n,D} dxdy + \frac{\theta}{2\Delta x\Delta y}\int_{C_{ij}} h^{n,D} dxdy \nonumber\\
&=& \frac{\theta}{8}\sum_{l,m=1}^{2}\sum_{\beta=1}^{\hat N} \sum_{\alpha=1}^{N} \hat {\omega}_{\beta} \omega_{\alpha} h^{n,D}\left(\hat x_i^{l,\beta},y_j^{m,\alpha}\right) + \frac{\theta}{8}\sum_{l,m=1}^{2}\sum_{\alpha=1}^{N} \sum_{\beta=1}^{\hat N} \omega_{\alpha} \hat{\omega}_{\beta} h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{m,\beta}\right) \nonumber\\\end{aligned}$$ Plugging into , one obtains $$\begin{aligned}
\label{cell1-2D-3}
\bar{h}_{ij}^{n+1,C}
&=& (1-\theta) \bar{h}_{ij}^{n,C} \nonumber\\
&+&\sum_{m=1}^{2}\sum_{\alpha=1}^{N}\omega_{\alpha}\left[ \frac{\theta}{8} \hat {\omega}_{1}-\frac{\Delta t_n}{2\Delta x} u^{n,D}\left(\hat x_i^{2,\hat N},y_j^{m,\alpha}\right) \right]h^{n,D}\left(\hat x_i^{2,\hat N},y_j^{m,\alpha}\right) \nonumber\\
&+&\sum_{m=1}^{2}\sum_{\alpha=1}^{N}\omega_{\alpha}\left[ \frac{\theta}{8} \hat {\omega}_{1} + \frac{\Delta t_n}{2\Delta x} u^{n,D}\left(\hat x_i^{1,1},y_j^{m,\alpha}\right) \right]h^{n,D}\left(\hat x_i^{1,1},y_j^{m,\alpha}\right) \nonumber\\
&+&\sum_{l=1}^{2}\sum_{\alpha=1}^{N}\omega_{\alpha}\left[ \frac{\theta}{8} \hat {\omega}_{1}-\frac{\Delta t_n}{2\Delta y} v^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{2,\hat N}\right) \right] h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{2,\hat N}\right) \nonumber\\
&+&\sum_{l=1}^{2}\sum_{\alpha=1}^{N}\omega_{\alpha}\left[ \frac{\theta}{8} \hat {\omega}_{1} + \frac{\Delta t_n}{2\Delta y} v^{n,D}\left(x_i^{l,\alpha},\hat y_j^{1,1}\right) \right]h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{1,1}\right) \nonumber\\
&+&\frac{\theta}{8}\sum_{l,m=1}^{2}\sum_{\beta=2}^{\hat N - 1} \sum_{\alpha=1}^{N} \hat {\omega}_{\beta} \omega_{\alpha} h^{n,D}\left(\hat x_i^{l,\beta},y_j^{m,\alpha}\right) \nonumber\\
&+&\frac{\theta}{8}\sum_{m=1}^{2} \sum_{\alpha=1}^{N} \omega_{\alpha} \hat {\omega}_{1} \left( h^{n,D}\left(\hat x_i^{2,1},y_j^{m,\alpha}\right) + h^{n,D}\left(\hat x_i^{1,\hat N},y_j^{m,\alpha}\right) \right) \nonumber\\
&+&\frac{\theta}{8}\sum_{l,m=1}^{2}\sum_{\alpha=1}^{N} \sum_{\beta=2}^{\hat N - 1} \omega_{\alpha} \hat{\omega}_{\beta} h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{m,\beta}\right) \nonumber\\
&+&\frac{\theta}{8}\sum_{l=1}^{2}\sum_{\alpha=1}^{N} \omega_{\alpha} \hat{\omega}_{1} \left( h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{2,1}\right) + h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{1,\hat N}\right) \right).\end{aligned}$$ Here, we used $\hat{\omega}_{\hat N}=\hat{\omega}_{1}$. A few observations can be made. Firstly, $\bar{h}_{ij}^{n+1,C}$ is a linear combination of $\bar{h}_{ij}^{n,C}$, $h^{n,D}\left(x_i^{l,\alpha}, \hat y_j^{m,\beta}\right)$ and $h^{n,D}\left(\hat x_i^{l,\beta},y_j^{m,\alpha}\right)$, $l,m=1,2,\alpha=1,2,...,N,\beta==1,2,...,\hat N$, which are all non-negative according to the conditions in this Proposition. Secondly, the CFL condition and $\theta \in [0,1]$ imply that all coefficients in the linear combination are non-negative. Therefore $\bar{h}_{ij}^{n+1,C} \geq 0 $, $\forall i,j$. Similarly, one can show $\bar{h}_{ij}^{n+1,D} \geq 0 $, $\forall i,j$.
Although the line to prove Proposition \[Th3.2\] is similar to the one in [@LiM2017], there is an explicit difference in the proof. In [@LiM2017], the water depth $h$ and the momentum $hu$ may be not consistent due to the numerical error, namely, $hu\neq0$ if $h \equiv 0$. This issue hampers the proof of the non-negativity of the water depth. To overcome the issue, a numerical technique has been employed to modify the momentum to be consistent with the water depth. While according to the numerical scheme in the present paper, the velocity is always equal to zero as the water depth is equal to zero. Therefore, the proof for Proposition \[Th3.2\] is more natural and simpler than the one in [@LiM2017].
Next, we give the positivity-preserving limiters which modify the CDG solution polynomials $h^{n,C}$ and $h^{n,D}$ into $\tilde h^{n,C}$ and $\tilde h^{n,D}$ which satisfy the sufficient condition given in Proposition \[Th3.2\]. In fact, the limiters are the same as in [@Zhang2010; @Zhang2010-1; @Xing2010], as long as the notation $K$ and $\hat L_K$ are re-defined as follows: On the primal mesh, $K$ denotes a mesh element $C_{ij}$ and $\hat L_K$ represents the set of relevant quadrature points in $K$, namely $\hat L_K=\cup_{l,m=1}^2 L_{i,j}^{l,m}$. On the dual mesh, $K$ denotes a mesh element $D_{ij}$ and $\hat L_K$ represents the set of relevant quadrature points in $K$, namely $\hat L_K=L_{i,j}^{1,1} \cup L_{i,j-1}^{1,2}\cup L_{i-1,j}^{2,1}\cup L_{i-1,j-1}^{2,2}$. Following [@Xing2010], the positivity-preserving limiter is given as follows: On each mesh element $K$, we modify the water depth $h^{n,\star} $ into $\tilde h^{n,\star}=\alpha_K(h^{n,\star}-\bar h^{n,\star})+ \bar h^{n,\star}$, with $\alpha_K=\min_{x \in \hat L_K} \left\{1, |\bar h^{n,\star} /(\bar h^{n,\star}- h^{n,\star}(x)) | \right\}$ and $\star=C,D$.
### Positivity-preserving well-balanced CDG-FE method {#sec:3.2.4}
Finally, we study the positivity-preserving and well-balanced CDG-FE method for the 2D Green-Naghdi model -. We start with the well-balanced CDG method satisfied by the cell average of the numerical solution $h^C$, which is obtained by taking the test function $\bV=(\frac{1}{\Delta x \Delta y}, 0, 0)^\top$ in the scheme (\[Comp-WB-SW-2D-1\]), $$\begin{aligned}
\label{WB-gene-cell-2D}
\bar h_{ij}^{n+1,C}&=& (1-\theta) \bar h_{ij}^{n,C} + \frac{\theta}{\Delta x \Delta y} \int_{C_{ij}}h^{n,D}dxdy \nonumber\\
&-&\frac{\Delta t_n}{\Delta x \Delta y} \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}\left[h^{n,D}(x_{i+\frac{1}{2}},y)u^{n,D}(x_{i+\frac{1}{2}},y)-
h^{n,D}(x_{i-\frac{1}{2}},y)u^{n,D}(x_{i-\frac{1}{2}},y)\right] dy \nonumber\\
&-&\frac{\Delta t_n}{\Delta x \Delta y} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\left[h^{n,D}(x,y_{j+\frac{1}{2}})v^{n,D}(x,y_{j+\frac{1}{2}})-
h^{n,D}(x,y_{j-\frac{1}{2}})v^{n,D}(x,y_{j-\frac{1}{2}})\right] dx \nonumber\\
&+& \theta\left( \frac{1}{\Delta x \Delta y}\int_{C_{ij}}b^{D}\cdot dxdy - \bar b_{ij}^{C} \right)~,\end{aligned}$$ where $\bar b_{ij}^C$ (resp. $\bar b_{ij}^D$) is the cell average of the bottom topography $b^C$ (resp. $b^D$) on the element $C_{ij}$ (resp. $D_{ij}$).
\[Th3.3\] For any given $n\ge 0$, we assume $\bar h_{ij}^{n,C}\ge 0$ and $\bar h_{ij}^{n,D}\ge 0$, $\forall i, j$. Consider the scheme in and its counterpart for $\bar h_{ij}^{n+1,D}$, if $$\label{bot-con-2d}
\bar b_{ij}^C=\frac{1}{\Delta x \Delta y} \int_{C_{ij}} b^{D} dxdy~, \qquad \bar b_{ij}^D= \frac{1}{\Delta x \Delta y} \int_{D_{ij}} b^{C} dxdy~, \quad \forall i, j~,$$ and $h^C(x,y,t_n)\ge 0$, $h^D(x,y,t_n)\ge 0$, $\forall (x,y) \in L_{i,j}^{l,m}$, $\forall i, j$ with $l,m=1,2$, then $\bar h_{ij}^{n+1, C}\ge 0$ and $\bar h_{ij}^{n+1, D} \ge 0$, $\forall i, j$, under the CFL condition $$\label{CFL_con_2D}
\lambda_x a_x + \lambda_y a_y \leq \frac{1}{4}\theta \hat \omega_1~.$$
The proof is a direct result of Proposition \[Th3.2\] and condition .
To enforce the sufficient conditions in Proposition \[Th3.3\], we also need to modify the approximations to the bottom and use the positivity-preserving limiter in Section \[sec:3.2.3\]. The new approximations to the bottom topography $b(x,y)$, still denoted by $b^C$ and $b^D$, are obtained by solving a constrained minimization problem similar to the one in [@LiM2017] with the Lagrange multiplier method.
High-order time discretizations and nonlinear limiters {#sec:3.3}
------------------------------------------------------
To achieve better accuracy in time, the strong stability preserving (SSP) high-order time discretizations ([@Gottlieb2001]) will be used in the numerical simulations . Such discretizations can be written as a convex combination of the forward Euler method, and therefore the resulting SSP schemes are also well-balanced and positivity-preserving. In this paper, we use the third order TVD Runge-Kutta method for the time discretization.
When the CDG method is applied to nonlinear problems, nonlinear limiters are often needed to prevent numerical instabilities. In this work, we use the total variation bounded (TVB) minmod slope limiter with parameter $M=10$ ([@Cock1998]) in a componentwise way as it is needed. This limiter is applied to $(h+b, hP, hQ)^\top$ and it is used prior to application of the positivity-preserving limiter.
Numerical examples
==================
Accuracy test {#sec:4.1}
-------------
In this example, we test the convergence rate of the proposed CDG-FE method by varying the mesh size. The Green-Naghdi model with $b=0$ has an exact solution given by ([@Su1969]) $$\label{Eq:so}
\left\{\begin{array}{lcl}
h(x,y,t)=h_1+(h_2-h_1)\mbox{sech}^2\left(\frac{x-Dt}{2}\sqrt{\frac{3(h_2-h_1)}{h_2h_1^2}} \right)\\
u(x,y,t)=D\left(1-\frac{h_1}{h(x,t)} \right)~,
v(x,y,t)=0,
\end{array}\right .$$ where $h_1$ is the typical water depth, $h_2$ corresponds to the solitary wave crest and $D=\sqrt{gh_2}$ is the wave speed. In this test, we employ a solitary wave with $h_1=1$ and $h_2=2.25$ in which is initially located at $x=0$ and propagating in the positive $x$-direction. The computational domain is $[-30, 50]\times[-1,1]$ and the final time is $1$. An outgoing boundary condition is used in the $x$-direction and a periodic boundary condition is used in the $y$-direction. We use regular meshes with $\Delta x=\Delta y=1,0.5,0.25,0.125$. The time step is $\Delta t =0.1 \Delta x$. We present $L^2 $ errors and orders of accuracy for $h$ and $u$ in Table \[table:smooth1:1\]. The results show that the CDG-FE method is $(k+1)$st order accurate for $P^k$ with $k=1,2$ and therefore it is optimal with respect to the approximation properties of the discrete spaces.
------------------ -- ------------- ------- -- ------------- ------- -- ------------- ------- -- ------------- -------
\[accuracytest\] $h$ $u$
$\Delta x$ $P^1$ $P^2$ $P^1$ $P^2$
$L^2$ error Order $L^2$ error Order $L^2$ error Order $L^2$ error Order
1 2.28E-01 — 7.80E-02 — 5.16E-01 — 1.05E-01 —
0.5 6.00E-02 1.93 9.94E-03 2.97 1.27E-02 2.03 1.47E-02 2.84
0.25 1.53E-02 1.97 1.27E-03 2.97 2.94E-02 2.11 1.84E-03 2.99
0.125 3.53E-03 2.12 1.64E-04 2.96 7.03E-03 2.07 2.29E-04 3.01
------------------ -- ------------- ------- -- ------------- ------- -- ------------- ------- -- ------------- -------
: $L^2$ errors and orders of accuracy of $(h,u)$.
\[table:smooth1:1\]
Stationary solution {#sec:4.2}
-------------------
In this test, we validate the well-balanced feature and the positivity-preserving property of the proposed method as applied to continuous and discontinuous variable bottoms. The initial conditions are $$\label{SS-Initial2D}
u(x,y,0)=0~, \qquad v(x,y,0)=0~, \qquad h(x,y,0)+b(x,y)=0.50001~,$$ and the continuous bottom profile (Case A) is defined by $$b(x,y)=\left\{\begin{array}{lclclcl}
0.2~, & r\leq 0.3~,\\
0.5-r~, & 0.3\leq r\leq 0.5~,\\
0~, & \mbox{otherwise}~,
\end{array}\right.$$ with $r=\sqrt{x^2+y^2}$, while the discontinuous bottom profile (Case B) is given by $$b(x,y)=\left\{\begin{array}{lcl}
0.5~, & -0.5\leq x,y\leq 0.5~,\\
0~, & \mbox{otherwise}~.
\end{array}\right .$$ We choose $[-1,1]\times[-1,1]$ as the computational domain, divided into $20\times20$ elements, and use outgoing boundary conditions. We compute the solution up to $t=10$ by the well-balanced CDG methods. Notice that there exists a near dry area for the second case.
For these cases, the standard CDG-FE method usually fails to preserve the still-water stationary solution exactly. Especially for the second case, the standard CDG-FE method will produce negative water depth due to the numerical oscillation, and thus the computation will blow down. To demonstrate that the positivity-preserving well-balanced CDG-FE scheme indeed preserves the still-water stationary solution exactly (i.e., up to machine precision), we perform the computation in both single and double precision. The corresponding $L^2$ errors on the water surface $h+b$ and velocity $(u,v)$ are listed in Table \[table:stationary\_solution2D\] for both topographies. We see that the errors have orders of a magnitude consistent with the machine single and double precision, and thus the numerical results verify the well-balanced property and the positivity-preserving property.
------ -- ----------- -- ---------- -- ---------- -- ---------- --
Case precision $h+b$ $u$ $v$
A single 5.96E-08 8.86E-10 9.73E-10
A double 2.56E-16 2.81E-16 2.23E-16
B single 5.96E-08 4.21E-09 2.45E-09
B double 4.93E-16 5.32E-16 2.33E-16
------ -- ----------- -- ---------- -- ---------- -- ---------- --
: $L^{\infty}$ errors on $(h+b,u,v)$ for the stationary solution at $t=10$.
\[table:stationary\_solution2D\]
Solitary wave overtopping a seawall {#sec:4.3}
-----------------------------------
In this test, we consider the simulation of a solitary wave overtopping a seawall, which has been studied experimentally in [@Hsiao2010] and numerically in [@Lannes2015]. This example is also used to investigate the validity of our positivity-preserving well-balanced scheme. The initial solitary wave ($h_1=0.2,h_2=0.27$ in ) and the bottom topography including the seawall are shown in Figure \[Fig:seawall\] (along $y=0$). The computational domain is $[-5,20]\times[-0.2,0.2]$ discretized into $500 \times 8$ uniform elements. The outgoing boundary condition is used in the $x$-direction and the periodic boundary condition is used in the $y$-direction. For the standard CDG-FE method, negative water depth was generated during the simulation at the dry or near dry areas, this causes inaccurate velocity which is used as the boundary condition when solving the elliptic equations, and then the numerical solution blows up within a few dozen time steps. For the positivity-preserving well-balanced CDG-FE method, however, the water depth remains non-negative during the entire simulation. The numerical surface profiles (along $y=0$) at time $t=5, 7.5, 12.5, 20$ are plotted in Figure \[Fig:seawall-profiles\]. The time series of the wave elevation at several positions of gauges ($x=5.9, 7.6, 9.644, 10.462, 10.732$ and $11.12$, $y=0$) are illustrated in Figure \[Fig:seawall-gages\]. The time origin has been shifted in order to compare with results reported in [@Hsiao2010]. Both models with $\alpha=1$ and $\alpha=1.159$ give almost the same results. Our numerical results are similar to experimental data and the numerical results in [@Hsiao2010].
![The sketch of the topography and the initial wave for solitary wave overtopping a seawall.[]{data-label="Fig:seawall"}](seawall_bottom.eps){height="6.0cm" width="12.0cm"}
![Water surface at several times ($t=5, 7.5, 12.5, 20$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-profiles"}](seawall_profiles_1.eps "fig:"){height="4.5cm" width="8cm"} ![Water surface at several times ($t=5, 7.5, 12.5, 20$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-profiles"}](seawall_profiles_2.eps "fig:"){height="4.5cm" width="8cm"}\
![Water surface at several times ($t=5, 7.5, 12.5, 20$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-profiles"}](seawall_profiles_3.eps "fig:"){height="4.5cm" width="8cm"} ![Water surface at several times ($t=5, 7.5, 12.5, 20$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-profiles"}](seawall_profiles_4.eps "fig:"){height="4.5cm" width="8cm"}
![The time series of the free surface elevation due to waves interacting against seawall at several gages ($x=5.9, 7.6, 9.644, 10.462, 10.732, 11.12$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-gages"}](seawall_gages_1.eps "fig:"){height="2.6cm" width="8.0cm"} ![The time series of the free surface elevation due to waves interacting against seawall at several gages ($x=5.9, 7.6, 9.644, 10.462, 10.732, 11.12$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-gages"}](seawall_gages_2.eps "fig:"){height="2.6cm" width="8.0cm"}\
![The time series of the free surface elevation due to waves interacting against seawall at several gages ($x=5.9, 7.6, 9.644, 10.462, 10.732, 11.12$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-gages"}](seawall_gages_3.eps "fig:"){height="2.6cm" width="8.0cm"} ![The time series of the free surface elevation due to waves interacting against seawall at several gages ($x=5.9, 7.6, 9.644, 10.462, 10.732, 11.12$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-gages"}](seawall_gages_4.eps "fig:"){height="2.6cm" width="8.0cm"}\
![The time series of the free surface elevation due to waves interacting against seawall at several gages ($x=5.9, 7.6, 9.644, 10.462, 10.732, 11.12$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-gages"}](seawall_gages_6.eps "fig:"){height="2.6cm" width="8.0cm"} ![The time series of the free surface elevation due to waves interacting against seawall at several gages ($x=5.9, 7.6, 9.644, 10.462, 10.732, 11.12$, from left to right, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:seawall-gages"}](seawall_gages_7.eps "fig:"){height="2.6cm" width="8.0cm"}
Periodic waves propagation over a submerged bar {#sec:4.4}
-----------------------------------------------
In this example, we investigate the robustness of the modified Green-Naghdi model. We first consider the propagation of periodic Stokes waves over a submerged bar with plane slopes. The bottom variation is specified by $$\label{Eq:HarmonicB}
b(x,y)=\left\{\begin{array}{lcl}
-0.4+0.05(x-6), & 6\le x\le 12 \\
-0.1, & 12 \le x \le 14\\
-0.1 - 0.1(x-14), & 14 \le x \le 17\\
-0.4, & \mbox{elsewhere}~,
\end{array}\right.$$ and is also exhibited in Figure \[Fig:harmonics\_bottom\] (along $y=0$) in which we also label the positions of $10$ gauges used in [@Ding1994].
As shown in experimental work ([@Ding1994]), regular waves break up into higher-frequency free waves as they propagate past a submerged bar. As the waves travel up the front slope of the bar, higher harmonics are generated due to nonlinear interactions, causing the waves to steepen. These harmonics are then released as free waves on the downslope, producing an irregular pattern behind the bar. This experiment is particularly difficult to simulate because it includes nonlinear interactions and requires accurate propagation of waves in both deep and shallow water over a wide range of depths. Therefore it has often been used as a discriminating test case for nonlinear models of surface wave propagation over variable bottom ([@Chazel2011; @Ding1994; @Guyenne2007]).
In the simulation, the computational domain is $[0, 25]\times[-0.2, 0.2]$, divided into $500 \times 8$ uniform cells. At initial time, $h+b=0$ and $u=v=0$ in the computational domain. The incident wave (entering from the left) is a third-order Stokes wave ([@Fenton1985]) given by $$\begin{aligned}
\label{harmonics-incident}
\eta(x,t)&=&a_0 \cos\left( 2 \pi \left( \frac{x}{\lambda} - \frac{t}{T_0} \right) \right)
+ \frac{\pi a_0^2}{\lambda} \cos\left( 4 \pi \left( \frac{x}{\lambda} - \frac{t}{T_0} \right) \right) \nonumber \\
&-& \frac{\pi^2 a_0^3}{2 \lambda^2} \left[ \cos\left( 2 \pi \left( \frac{x}{\lambda} - \frac{t}{T_0} \right) \right) - \cos\left( 6 \pi \left( \frac{x}{\lambda} - \frac{t}{T_0} \right) \right) \right]~,\end{aligned}$$ where $T_0$, $a_0$ and $\lambda$ denote the wave period, amplitude and wavelength, respectively. We choose $(T_0, a_0, \lambda) = (2.02, 0.01, 3.73)$ corresponding to one of the experiments in [@Ding1994]. An absorbing boundary condition is applied at the right boundary and a periodic boundary condition is used at the upper and bottom boundaries.
Figure \[Fig:harmonics6\] depicts the time histories of the water surface at the first $6$ gauges ($x=2$, $4$, $10.5$, $12.5$, $13.5$ and $14.5$, $y=0$) and Figure \[Fig:harmonics4\] depicts the time histories of the water surface at the last $4$ gauges ($x=15.7$, $17.3$, $19$ and $21$, $y=0$). The time origin has been shifted in order that the numerical results match the measurements for the first gauge at $x = 2, y=0$. We compare three sets of data: experimental data ([@Ding1994]), numerical solutions from the original Green-Naghdi model ($\alpha=1$), numerical solutions from the improved Green-Naghdi model ($\alpha=1.159$). It can be seen from Figure \[Fig:harmonics6\] that before the crest of the bar ($x \le 14.5$), the numerical results from both models ($\alpha=1$ and $\alpha=1.159$) match well with each other and compare well with experimental data. However, it can be seen from Figure \[Fig:harmonics4\] that, compared with the experimental data, discrepancies in amplitude and phase can be observed for gauges beyond the crest of the bar ($x \ge 15.7$) for the numerical solutions with $\alpha=1$, these discrepancies should be attributed to the weakly dispersive character of the original Green-Naghdi model, while for the modified Green-Naghdi model, the numerical solutions match well with the experimental data.
![Experimental set-up and locations of the wave gauges as used in [@Ding1994].[]{data-label="Fig:harmonics_bottom"}](harmonics_bottom.eps){height="6.0cm" width="12.0cm"}
![Time series of surface elevations for waves passing over a submerged bar at $x=2$, $4$, $10.5$, $12.5$, $13.5$ and $14.5$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics6"}](Harmonic_gauges_1.eps "fig:"){height="3.5cm" width="8.0cm"} ![Time series of surface elevations for waves passing over a submerged bar at $x=2$, $4$, $10.5$, $12.5$, $13.5$ and $14.5$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics6"}](Harmonic_gauges_2.eps "fig:"){height="3.5cm" width="8.0cm"}\
![Time series of surface elevations for waves passing over a submerged bar at $x=2$, $4$, $10.5$, $12.5$, $13.5$ and $14.5$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics6"}](Harmonic_gauges_3.eps "fig:"){height="3.5cm" width="8.0cm"} ![Time series of surface elevations for waves passing over a submerged bar at $x=2$, $4$, $10.5$, $12.5$, $13.5$ and $14.5$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics6"}](Harmonic_gauges_4.eps "fig:"){height="3.5cm" width="8.0cm"}\
![Time series of surface elevations for waves passing over a submerged bar at $x=2$, $4$, $10.5$, $12.5$, $13.5$ and $14.5$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics6"}](Harmonic_gauges_5.eps "fig:"){height="3.5cm" width="8.0cm"} ![Time series of surface elevations for waves passing over a submerged bar at $x=2$, $4$, $10.5$, $12.5$, $13.5$ and $14.5$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics6"}](Harmonic_gauges_6.eps "fig:"){height="3.5cm" width="8.0cm"}
![Time series of surface elevations for waves passing over a submerged bar at $x=15.7$, $17.3$, $19$ and $21$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics4"}](Harmonic_gauges_7.eps "fig:"){height="3.5cm" width="8.0cm"} ![Time series of surface elevations for waves passing over a submerged bar at $x=15.7$, $17.3$, $19$ and $21$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics4"}](Harmonic_gauges_8.eps "fig:"){height="3.5cm" width="8.0cm"}\
![Time series of surface elevations for waves passing over a submerged bar at $x=15.7$, $17.3$, $19$ and $21$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics4"}](Harmonic_gauges_9.eps "fig:"){height="3.5cm" width="8.0cm"} ![Time series of surface elevations for waves passing over a submerged bar at $x=15.7$, $17.3$, $19$ and $21$ (from left to right, from top to bottom). Circles: experimental data ([@Ding1994]), green solid line: numerical results with $\alpha=1$, blue solid line: numerical results with $\alpha=1.159$.[]{data-label="Fig:harmonics4"}](Harmonic_gauges_10.eps "fig:"){height="3.5cm" width="8.0cm"}
In this test, we further study the dispersive effect of the modified Green-Naghdi model. We consider a periodic wave propagation over a submerged bar with elliptic slope (see Figure \[Fig:elliptic-slope\]), which is also given by $$\label{Eq:elliptic-slope}
b(x,y)=\left\{\begin{array}{lcl}
-0.1, & r < \frac{47}{576} , \\
1.2 \sqrt{1-r}-1.25 , & \frac{47}{576} \le r \le \frac{287}{576},\\
-0.4, & \mbox{elsewhere}~,
\end{array}\right.$$ with $r=\frac{(x-12.5)^2}{100}+\frac{y^2}{16}$.
![The bottom topography for periodic wave propagation over a submerged bar with elliptic slope.[]{data-label="Fig:elliptic-slope"}](bottom_elliptic_slope.eps){height="6.0cm" width="12.0cm"}
The computational domain is $[0,25]\times[-1,1]$, which is discretized into $125\times20$ uniform elements. At initial time, $h+b=0$ and $u=v=0$ in the computational domain. The incident wave (entering from the left) is a third-order Stokes wave given in with $(T_0, a_0, \lambda) = (3, 0.01, 3.73)$. Solid wall boundary conditions are used at the top and bottom boundaries, an absorbing boundary condition is used at right boundary. The numerical free surfaces at $t=30$ for both models ($\alpha=1$ and $\alpha=1.159$) are shown in Figure \[Fig:Surface-elliptic-slope\]. Overall, both numerical surfaces compare well with each other. To observe the discrepancies more clearly, the time series of numerical surfaces at several positions ($A_1(8,0)$, $A_2(9,0.5)$, $A_3(21,0)$, $A_4(18,0)$, $A_5(19,0.5)$ and $A_6(22,0.5)$) are shown in Figure \[Fig:Surface-elliptic-slope-time\]. It can be seen that both numerical surfaces compare well with each other for gauges before the crest of the bar ($x \le 9$). However, the discrepancies in amplitude and phase can be observed for gauges beyond the crest of the bar ($x \ge 18$). These discrepancies again should be attributed to the dispersive character of the models.
![Numerical surface at $t=30$ for periodic wave propagation over a submerged bar with elliptic slope. Top: numerical results with $\alpha=1$; bottom: numerical results with $\alpha=1.159$. []{data-label="Fig:Surface-elliptic-slope"}](Surface_elliptic_slope.eps){height="9.5cm" width="12.0cm"}
![Time series of numerical surface at six gages ($A_1(8,0)$, $A_2(9,0.5)$, $A_3(21,0)$, $A_4(18,0)$, $A_5(19,0.5)$ and $A_6(22,0.5)$, from left to right, from top to bottom) for periodic wave propagation over a submerged bar with elliptic slope. Blue lines: numerical results with $\alpha=1$; red dashed lines: numerical results with $\alpha=1.159$. []{data-label="Fig:Surface-elliptic-slope-time"}](surface_elliptic_slope_time_1.eps "fig:"){height="3.1cm" width="8.0cm"} ![Time series of numerical surface at six gages ($A_1(8,0)$, $A_2(9,0.5)$, $A_3(21,0)$, $A_4(18,0)$, $A_5(19,0.5)$ and $A_6(22,0.5)$, from left to right, from top to bottom) for periodic wave propagation over a submerged bar with elliptic slope. Blue lines: numerical results with $\alpha=1$; red dashed lines: numerical results with $\alpha=1.159$. []{data-label="Fig:Surface-elliptic-slope-time"}](surface_elliptic_slope_time_2.eps "fig:"){height="3.1cm" width="8.0cm"}\
![Time series of numerical surface at six gages ($A_1(8,0)$, $A_2(9,0.5)$, $A_3(21,0)$, $A_4(18,0)$, $A_5(19,0.5)$ and $A_6(22,0.5)$, from left to right, from top to bottom) for periodic wave propagation over a submerged bar with elliptic slope. Blue lines: numerical results with $\alpha=1$; red dashed lines: numerical results with $\alpha=1.159$. []{data-label="Fig:Surface-elliptic-slope-time"}](surface_elliptic_slope_time_3.eps "fig:"){height="3.1cm" width="8.0cm"} ![Time series of numerical surface at six gages ($A_1(8,0)$, $A_2(9,0.5)$, $A_3(21,0)$, $A_4(18,0)$, $A_5(19,0.5)$ and $A_6(22,0.5)$, from left to right, from top to bottom) for periodic wave propagation over a submerged bar with elliptic slope. Blue lines: numerical results with $\alpha=1$; red dashed lines: numerical results with $\alpha=1.159$. []{data-label="Fig:Surface-elliptic-slope-time"}](surface_elliptic_slope_time_4.eps "fig:"){height="3.1cm" width="8.0cm"}\
![Time series of numerical surface at six gages ($A_1(8,0)$, $A_2(9,0.5)$, $A_3(21,0)$, $A_4(18,0)$, $A_5(19,0.5)$ and $A_6(22,0.5)$, from left to right, from top to bottom) for periodic wave propagation over a submerged bar with elliptic slope. Blue lines: numerical results with $\alpha=1$; red dashed lines: numerical results with $\alpha=1.159$. []{data-label="Fig:Surface-elliptic-slope-time"}](surface_elliptic_slope_time_5.eps "fig:"){height="3.1cm" width="8.0cm"} ![Time series of numerical surface at six gages ($A_1(8,0)$, $A_2(9,0.5)$, $A_3(21,0)$, $A_4(18,0)$, $A_5(19,0.5)$ and $A_6(22,0.5)$, from left to right, from top to bottom) for periodic wave propagation over a submerged bar with elliptic slope. Blue lines: numerical results with $\alpha=1$; red dashed lines: numerical results with $\alpha=1.159$. []{data-label="Fig:Surface-elliptic-slope-time"}](surface_elliptic_slope_time_6.eps "fig:"){height="3.1cm" width="8.0cm"}
Solitary wave propagation over a composite beach {#sec:4.5}
------------------------------------------------
To further investigate the robustness of the modified Green-Naghdi model, we simulate the propagation of solitary waves over a composite beach, which consists of three piece-wise linear segments, terminated with a vertical wall on the left. The slopes of the topography are defined as follows ([@Lannes2015]): $$\label{Eq:compo-beach}
s(x,y)=\left\{\begin{array}{lcl}
0, & x\leq 15.04, \\
1/53, & 15.04 \le x \leq 19.4,\\
1/150, & 19.4 \le x \leq 22.33,\\
1/13, & 22.33 \le x \leq 23.23.
\end{array}\right.$$
We consider the propagation of a solitary wave: $h_1=0.22,h_2=1.73h_1$. The initial solitary waves, produced by using , are located at $x=0$ and propagate to the right. The computational domain is $[-5,23.23]\times[-0.2,0.2]$ discretized into $500 \times 8$ uniform elements. The bottom along with $y=0$ and the initial solitary wave is shown in Figure \[Fig:composite\_bottom\]. The outgoing boundary condition is used at the left boundary, the reflective boundary condition is employed at the right boundary and the periodic boundary condition is used at the upper and bottom boundaries. We observe the propagation over the beach, reflection on the vertical wall before traveling back to the left boundary. The time series of the wave elevation at several positions of gauges ($x=15.04, 19.4, 22.33$, $y=0$) are shown in Figure \[Fig:composite-beach-gages\]. Overall, both sets of the numerical results obtained from the Green-Naghdi models ($\alpha=1$ and $\alpha=1.159$) compare well together before the wave encounters the vertical wall. However, discrepancies in amplitude and phase can be observed after the wave reflects on the vertical wall. Especially, discrepancies are clearer when the positions of gauges are far away from the vertical wall. Our numerical solution with $\alpha=1.159$ matches well with the one reported in [@Lannes2015].
![The initial solitary wave and the bottom topography for solitary wave propagation over a composite beach.[]{data-label="Fig:composite_bottom"}](composite_bottom.eps){height="6.0cm" width="10.0cm"}
![The time series of the wave elevation at several gages ($x=15.04, 19.4, 22.33$, $y=0$, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:composite-beach-gages"}](composite-beach-gages-0d7-1.eps "fig:"){height="3.5cm" width="14.0cm"}\
![The time series of the wave elevation at several gages ($x=15.04, 19.4, 22.33$, $y=0$, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:composite-beach-gages"}](composite-beach-gages-0d7-2.eps "fig:"){height="3.5cm" width="14.0cm"}\
![The time series of the wave elevation at several gages ($x=15.04, 19.4, 22.33$, $y=0$, from top to bottom). Blue line: numerical results with $\alpha=1$; Red dashed line: numerical results with $\alpha=1.159$.[]{data-label="Fig:composite-beach-gages"}](composite-beach-gages-0d7-3.eps "fig:"){height="3.5cm" width="14.0cm"}
Conclusions
===========
In this work, we derive a Green-Naghdi model with enhanced dispersive property and then develop a family of high order positivity-preserving and well-balanced numerical methods for its numerical solutions. These methods are based on the reformulation of the original system into a pseudo-conservation law coupled with an elliptic system. Numerical experiments are presented to demonstrate the accuracy and achievement of expected properties for the proposed schemes, and the capability of the Green-Naghdi equations to model a wide range of shallow water wave phenomena. Application of the proposed schemes to Green-Naghdi models taking into account the irrotational effect and the Coriolis effect due to the Earth’s rotation, and proposal of fast solvers for the elliptic part of the model will be envisioned for our future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
Maojun Li is partially supported by NSFC (Grant Nos. 11501062, 11701055, 11871139). Liwei Xu is partially supported by a Key Project of the Major Research Plan of NSFC (Grant No. 91630205) and a NSFC (Grant No. 11771068).
Appendix {#appendix .unnumbered}
========
Here, we derive the fully nonlinear and weakly dispersive shallow water equations over the non-flat bottom in 2D space along the same lines in [@Su1969] where the authors derived the one-dimensional equations on the flat bottom. Let $\Omega(t) $ be the domain in $R^3$ occupied by the water at time $t$. The propagation of the water is described by the fully nonlinear Euler equations $$\begin{aligned}
\tilde u_t + \tilde u \tilde u_x + \tilde v \tilde u_y + \tilde w \tilde u_z &=& -p_x ~, \label{Euler1}\\
\tilde v_t + \tilde u \tilde v_x + \tilde v \tilde v_y + \tilde w \tilde v_z &=& -p_y ~, \label{Euler2}\\
\tilde w_t + \tilde u \tilde w_x + \tilde v \tilde w_y + \tilde w \tilde w_z &=& -p_z-g ~, \label{Euler3}\end{aligned}$$ and the continuity equation $$\label{cont-equa}
\tilde u_x + \tilde v_y + \tilde w_z = 0 ~,$$ where $(\tilde u, \tilde v, \tilde w)$ denotes the velocity of the water, $p$ is the pressure and $g$ is the gravitational constant. The subscript $t$ denotes the partial derivative with respect to the time variable, $x$, $y$, and $z$ denote the partial derivatives with respect to the space variables. We assume that the density is taken as one. The boundary conditions are given by
(BC1) the kinematic condition at the free surface $$\label{boun-cond-1}
\tilde{w}^{(s)} = h_t + \tilde{u}^{(s)} (h+b)_x + \tilde{v}^{(s)} (h+b)_y~,$$
(BC2) the impermeability of the bottom $$\label{boun-cond-2}
\tilde{w}^{(b)} = \tilde{u}^{(b)} b_x + \tilde{v}^{(b)} b_y~,$$ where $h(x,y,t)$ denotes the depth of the water and $b(x,y)$ is the bottom topography. The superscript $(s)$ and $(b)$ denote the quantities evaluated at the free surface and the bottom, respectively.
Let $u(x,y,t)$ and $v(x,y,t)$ denote the vertically averaged horizontal velocity in the x- and y-directions, respectively, and be defined by $$\begin{aligned}
u(x,y,t)&=&\frac{1}{h(x,y,t)}\int_{b}^{h+b}\tilde{u}(x,y,z,t)dz \label{aver-velo-u},\\
v(x,y,t)&=&\frac{1}{h(x,y,t)}\int_{b}^{h+b}\tilde{v}(x,y,z,t)dz. \label{aver-velo-v}\end{aligned}$$
Along the same lines as in [@Su1969], we want to find a set of equations which governs the evolution of the water depth $h(x,y,t)$, the average velocity $u(x,y,t)$ and $v(x,y,t)$ under the following assumptions:
(A1) we assume that the vertical movement of a particle is small compared with the horizontal movement, that is, we use the shallow water hypothesis, so that we can write $$\label{assu-1}
\tilde u(x,y,z,t) \simeq u(x,y,t) \mbox{ and } \tilde v(x,y,z,t) \simeq v(x,y,t),$$
(A2) the dynamic condition at the free surface is assumed to be $$\label{assu-2}
p^{(s)} \simeq \mbox{constant}.$$
Integrating (\[cont-equa\]) with respect to $z$ gives $$\label{tilde-w}
\tilde{w}=\tilde{w}^{(b)}-\int_{b}^{z} \tilde{u}_x dz - \int_{b}^{z} \tilde{v}_y dz=-\left( \int_{b}^{z} \tilde{u} dz \right)_x - \left(\int_{b}^{z} \tilde{v} dz\right)_y~.$$ Here, we have used . Equation also implies $$\label{tilde-ws}
\tilde{w}^{(s)}=\tilde{w}^{(b)}-\int_{b}^{h+b} \tilde{u}_x dz - \int_{b}^{h+b} \tilde{v}_y dz.$$
Multiplying by $h$ and taking the derivative with respect to $x$, multiplying by $h$ and taking the derivative with respect to $y$, and then adding them together, one arrives at $$\begin{aligned}
\label{hux-hvy}
(hu)_x+(hv)_y
&=&\left( \int_{b}^{h+b}\tilde{u}dz \right)_x + \left( \int_{b}^{h+b}\tilde{v}dz \right)_y \nonumber \\
&=& \int_{b}^{h+b}\tilde{u}_xdz + \tilde{u}^{(s)}(h+b)_x - \tilde{u}^{(b)} b_x \nonumber \\
&+& \int_{b}^{h+b}\tilde{v}_ydz + \tilde{v}^{(s)}(h+b)_y - \tilde{v}^{(b)} b_y.\end{aligned}$$
Utilizing , and , one gets from $$\label{GN2D-1}
h_t+(hu)_x+(hv)_y=0.$$ This equation gives the evolution of $h$ provided we know the evolution of $hu$ and $hv$.
Integrating with respect to $z$ from $z=b$ to $z=h+b$, performing an integration by parts for the fourth term and utilizing , and , one obtains $$\label{hut1}
(hu)_t+\left( \int_{b}^{h+b}(\tilde{u}^2+p)dz\right)_x+\left( \int_{b}^{h+b}(\tilde{u} \tilde{v})dz\right)_y=p^{(s)}(h+b)_x-p^{(b)}b_x.$$ Similarly, one has from $$\label{hvt1}
(hv)_t+\left( \int_{b}^{h+b}(\tilde{u} \tilde{v})dz\right)_x+\left( \int_{b}^{h+b}(\tilde{v}^2+p)dz\right)_y=p^{(s)}(h+b)_y-p^{(b)}b_y.$$
Integrating with respect to $z$ from $z$ to $h+b$ yields $$\label{pres1}
p = \int_{z}^{h+b}( \tilde w_t + \tilde u \tilde w_x + \tilde v \tilde w_y + \tilde w \tilde w_z)dz + p^{(s)} + g(h+b-z),$$ thus one gets $$\begin{aligned}
\label{aver-p}
\int_{b}^{h+b}pdz &=&\int_{b}^{h+b} \left( \int_{z}^{h+b}( \tilde w_t + \tilde u \tilde w_x + \tilde v \tilde w_y + \tilde w \tilde w_z)dz + p^{(s)} + g(h+b-z) \right)dz \nonumber \\
&=& \int_{b}^{h+b} (z-b)( \tilde w_t + \tilde u \tilde w_x + \tilde v \tilde w_y + \tilde w \tilde w_z ) dz + p^{(s)}h + \frac{1}{2}gh^2 \nonumber \\
&=& -\int_{b}^{h+b} (z-b)\left( \left( \int_{b}^{z} \tilde{u} dz \right)_{xt} + \left(\int_{b}^{z} \tilde{v} dz\right)_{yt} \right)dz \nonumber \\
&-& \int_{b}^{h+b} (z-b)\tilde{u} \left( \left( \int_{b}^{z} \tilde{u} dz \right)_{xx} + \left(\int_{b}^{z} \tilde{v} dz\right)_{yx} \right)dz \nonumber \\
&-& \int_{b}^{h+b} (z-b)\tilde{v} \left( \left( \int_{b}^{z} \tilde{u} dz \right)_{xy} + \left(\int_{b}^{z} \tilde{v} dz\right)_{yy} \right)dz \nonumber \\
&+& \int_{b}^{h+b} (z-b)(\tilde{u}_x+\tilde{v}_y) \left( \left( \int_{b}^{z} \tilde{u} dz \right)_{x} + \left(\int_{b}^{z} \tilde{v} dz\right)_{y} \right)dz \nonumber \\
&+& p^{(s)}h + \frac{1}{2}gh^2~.\end{aligned}$$ Here, we have used and .
Now, substituting into and , respectively, and using the assumptions -, one obtains $$\label{GN2D-2}
(hu)_t+\left( hu^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3 \Phi + \frac{1}{2} h^2 \Psi \right)_x + \left( huv \right)_y =-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_x$$ and $$\label{GN2D-3}
(hv)_t+\left(huv\right)_x+\left( hv^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3\Phi+\frac{1}{2}h^2\Psi\right)_y=-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_y$$ where $$\begin{aligned}
\Phi &=& - u_{xt}-u u_{xx}+ u_x^2 - v_{yt}-v v_{yy}+ v_y^2 - uv_{xy}- u_{xy}v+ 2 u_x v_y, \label{phi2D-1} \\
\Psi &=& b_x u_{t}+b_x u u_{x}+b_{xx} u^2 + b_y v_{t} + b_y v v_y +b_{yy} v^2 + b_y u v_x + b_x u_y v + 2b_{xy}uv. \label{psi2D-1}\end{aligned}$$
Therefore, we get a set of equations by combining , and $$\label{GN-NFB-2D-1}
\left\{\begin{array}{lclcl}
h_t+(hu)_x+(hv)_y=0,\\
(hu)_t+\left( hu^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3 \Phi + \frac{1}{2} h^2 \Psi \right)_x + \left( huv \right)_y =-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_x,\\
(hv)_t+\left(huv\right)_x+\left( hv^2+ \frac{1}{2}gh^2+\frac{1}{3}h^3\Phi+\frac{1}{2}h^2\Psi\right)_y=-\left(gh+\frac{1}{2}h^2\Phi+h\Psi\right)b_y.
\end{array} \right.$$
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abstract: 'Angle resolved photoemission spectroscopy line shapes measured for quasi-one-dimensional Li$_{0.9}$Mo$_6$O$_{17}$ samples grown by a temperature gradient flux technique are found to show Luttinger liquid behavior, consistent with all previous data by us and other workers obtained from samples grown by the electrolyte reduction technique. This result eliminates the sample growth method as a possible origin of considerable differences in photoemission data reported in previous studies of Li$_{0.9}$Mo$_6$O$_{17}$.'
author:
- 'G.-H. Gweon$^{\dagger}$'
- 'S.-K. Mo'
- 'J. W. Allen'
- 'J. He'
- 'R. Jin'
- 'D. Mandrus'
- 'H. Höchst'
title: 'Luttinger liquid ARPES spectra from samples of Li$_{0.9}$Mo$_6$O$_{17}$ grown by the temperature gradient flux technique'
---
Li$_{0.9}$Mo$_6$O$_{17}$, also known as the Li purple bronze, is a quasi-one-dimensional metal which displays metallic T-linear resistivity and temperature independent magnetic susceptibility for temperatures down to $T_X \approx 24$ K, where a phase transition of unknown origin is signaled by a very weak anomaly in the specific heat [@Schlenker1985]. As $T$ decreases below $T_X$, the resistivity increases. However the d.c. magnetic susceptibility is unchanged below $T_X$ [@Schlenker1985; @matsuda86], implying no single particle gap opening, and infrared optical studies [@degiorgi-purple-optics] below $T_X$ also show no gap opening down to $1$ meV. Consistent with this evidence for the lack of a single particle gap, repeated x-ray diffraction studies [@pouget-Li-struct] show no charge density wave or spin density wave.
The various transport and spectroscopy studies of this fascinating material have been made on samples prepared by two methods, an electrolyte reduction technique [@Schlenker1985] and a temperature gradient flux technique [@greenblatt84]. Angle resolved photoemission spectroscopy (ARPES) is the only measurement for which any major inconsistency in data obtained from samples prepared by the two different methods has been reported, and the inconsistency is very serious. In particular an extensive set of ARPES data from two groups [@grioni; @denlinger; @gweon1; @gweon2; @gweon3; @allen; @gweon4] obtained on electrolyte reduction samples show non-Fermi liquid ARPES line shapes consistent[@denlinger; @gweon2; @gweon3; @allen; @gweon4] with Luttinger liquid (LL) behavior and no low temperature Fermi energy ($E_F$) gap, whereas ARPES data reported [@xue; @smith_reply; @smith] for temperature-gradient-flux grown samples show Fermi liquid (FL) line shapes, a large low temperature $E_F$ gap and an additional feature inconsistent with the known band structure of the material. These differences between the two ARPES data sets are summarized in Ref. \[\]. The LL line shapes have been verified repeatedly in subsequent studies [@gweon1; @gweon2; @gweon3; @allen; @gweon4] of samples prepared with the electrolyte reduction technique. Nonetheless it has been a lingering possibility that FL line shapes and a large low temperature gap could perhaps be characteristic of temperature gradient flux grown samples. This Brief Report dispels that possibility by reporting ARPES spectra for temperature gradient flux grown samples that are in full agreement with the line shapes obtained for electrolyte reduction samples.
The spectra reported here were obtained on the PGM beamline at the Wisconsin Synchrotron Radiation Center. Photons of energy 30 eV were used to excite photoelectrons whose kinetic energies and angles were analyzed with a Scienta SES 200 analyzer. Measurement on a freshly prepared Au surface was used to determine the position of $E_F$ in the spectra and the overall energy resolution of 21 meV due to both the monochromator and the analyzer. The angle resolution was set at $\pm 0.1^\mathrm{o}$, better than that $\pm 0.25^\mathrm{o}$ in our earlier work [@gweon1] and exactly the same as used in previous ARPES studies [@xue; @smith_reply; @smith] of temperature gradient flux samples. The sample surface was obtained by cleaving [*in situ*]{} and the data were taken at a sample temperature of 200 K, much higher than the transition temperature 24 K.
For the endstation in place at the time of taking the data reported here, the angular dispersion direction of the SES 200 analyzer was vertical. ARPES symmetry analysis of the data obtained shows that the one dimensional $\Gamma$–Y chain axis direction was (unintentionally) oriented at an angle of 13$^\mathrm{o}$ to the vertical. Nonetheless we will refer to this geometry as the “vertical geometry” from here on in the paper. Due to this small angular offset, the dispersions in this data set are slightly different from those that we obtained previously along the $\Gamma$–Y axis, as documented carefully in discussing Fig. 2 below. We have repeated the measurement in exactly the same geometry as that of our previous experiments [@denlinger; @gweon1; @gweon2; @gweon3; @allen; @gweon4], i.e. one in which both the one dimensional chain axis and the angular dispersion direction of the analyzer are horizontal and well aligned, and found dispersions essentially identical to those of the previous [@denlinger; @gweon1; @gweon2; @gweon3; @allen; @gweon4] data. We will refer to this geometry as the “horizontal geometry” below. Despite the small angular offset, we present here the data taken in the vertical geometry because (1) these data happen to show the $E_F$-crossing line shapes most clearly among all of our data sets, by virtue of having fortuitously the maximum intensity of the band crossing $E_F$ relative to the intensities of the bands that do not cross $E_F$, and (2) the differences in dispersions have been verified to arise from the small offset and are in any case so slight as to be insignificant for the central thrust of this paper. Both of these points are elaborated below. Another advantage of the vertical geometry setup was that it allowed acquisition of intensity maps like the one presented below (Fig. 1) for many parallel one dimensional paths crossing the FS. Thereby we could verify that the LL behavior holds for such paths anywhere in the Brillouin zone, regardless of the exact location of the momentum space cut across the Fermi surface, so that Fermi liquid behavior does not occur for some very specific cut, as reported previously [@xue; @smith_reply] for temperature gradient flux samples.
Fig. 1 shows ARPES spectra taken in the vertical geometry on a sample grown by the temperature gradient method. Panels (a) and (c) summarize the overall electronic structure with k labels denoting the k values projected onto the $\Gamma$-Y axis. We label the bands A,B,C in the order of decreasing binding energy at $\Gamma$. As we will discuss in connection with Fig. 2 below, the overall band structure revealed by the data is consistent with the data in the literature [@grioni; @denlinger; @gweon1; @gweon2; @gweon3; @allen; @gweon4] obtained on samples grown by the electrolyte reduction method, as well as with band theory [@whangbo]. Furthermore, the LL line shapes observed previously [@denlinger; @gweon1; @gweon2; @gweon3; @allen; @gweon4] are not just confirmed, but actually better observed due to the enhanced strength of band C relative to that of bands A,B in the present data. For example, we can now clearly observe that the spectral weight of band C shows a back-bending behavior after the peak has crossed the Fermi level (darker curve), one of the key signatures of the LL line shape. In panel (b) we show the k-sum of the ARPES data. As found previously, the resulting line shape is far from the Fermi edge line shape expected of a FL and instead is described much better by a LL with $\alpha > 0.5$, where $\alpha$ is the so-called the anomalous dimension of the LL.
Panel (a) of Fig. 2 summarizes the overall band structure determined from the present data (open circles) and compares it with that from our previous result (diamonds) taken on a sample grown by the electrolyte reduction technique. The small differences arising from the slightly different k-paths can be seen. For example, in the new data band B becomes almost non-dispersive when peak C crosses $E_F$ while this occurs for larger k values for the data perfectly along $\Gamma$-Y. As shown in panel (b) band theory predicts bands A, B, and C essentially as observed, and also a fourth band D. Bands A and B do not cross $E_F$ and C and D become degenerate and cross $E_F$ together. All four bands have been observed [@denlinger; @gweon1; @gweon2] for various k-paths, although band D is typically very weak, just a slight shoulder on the leading edge of peak C, and is clearly seen only for a particular k-path [@denlinger; @gweon3] where it appears as a main peak. In the vertical geometry data, band D is nearly undetectable (see Fig. 1 (c)) but was observed very weakly in the horizontal geometry data, consistent with previous results. For completeness, we mark the approximate position of band D thus found for the present sample as a gray region. This position is similar to that found for previous samples along the same, i.e. the $\Gamma$-Y, direction as well as along the special k-path [@denlinger; @gweon3] where D is strong.
![Agreement of the overall band structures obtained on the two samples grown with different methods. (a) Momentum-energy dispersion relations as extracted by taking the peak positions of energy distribution curves (e.g. in Fig. 1 (c)). The data plotted in circles correspond to the data of the current sample (Fig. 1) grown by the temperature gradient flux method and the data plotted in diamonds correspond to the data reported in Ref. \[\] for sample grown by the electrolyte reduction method. See text for discussion of small differences visible, arising from slightly different k-paths. The approximate position of the band D for both samples is indicated as a gray region. (b) Extended Hückel tight binding band structure calculation [@whangbo] for comparison. Note that the energy scale of the calculation was multiplied by a factor of 2.2 in order to roughly match the dispersion of the experiment.](fig2_Gweon.eps)
Fig. 3 compares the $E_F$ crossing line shapes measured on the samples grown by the two different methods, with panel (a) showing the new data in the vertical geometry for the temperature gradient flux sample, and panel (b) showing data from Ref. \[\] taken at the same photon energy for the electrolyte reduction sample. In each panel, the data are presented with the spectra for the various k-values overplotted to better show the approach and $E_F$ crossing of peak C. As explained in the previous paragraph, a small difference of the band B dispersion arises from the slightly different k-paths. The general features of the two sets of spectra are nearly identical, except that, as mentioned already, in (a) the strength of band C relative to that of the non-$E_F$ crossing band B is greater than in (b). Therefore the intrinsic line shape features of band C, which we have shown [@denlinger; @gweon2; @gweon3; @allen; @gweon4] to be well described by the LL line shape theory, are now even more clearly visible. These include the spinon edge and the holon peak, which disperse with different velocities, the diminution of intensity as $E_F$ is approached, and the back dispersing edge after the peak has crossed $E_F$. One is now forced to conclude that the large disagreement of the overall band dispersions and $E_F$ crossing line shapes found previously [@gweon1] for the ARPES data reported by Xue et al.[@xue; @smith] and those reported by ourselves and others [@grioni; @denlinger; @gweon1; @gweon2; @gweon3; @allen; @gweon4] do not stem from the sample growth method.
![Identical nature of LL ARPES lineshapes obtained for samples of Li$_{0.9}$Mo$_6$O$_{17}$ prepared by (a) the temperature gradient flux growth and (b) electrolyte reduction methods. The data in (b) is from Ref. \[\]. In both panels, the momentum increment is 2.6% of $\Gamma$–Y.](fig3_Gweon.eps)
Before concluding, we note that samples prepared by us \[JH, RJ and DM\] in the same way as for those used in the ARPES reported here, have also been used for new measurements of the temperature dependences of the resistivity, specific heat, magnetic susceptibility and optical properties [@musfeldt]. These results have re-confirmed that no gap opening is associated with the low $T$ resistivity rise and have been interpreted as showing the probable importance of localization effects for the properties below $T_X$. Although the lower energy limit of the new optical study is 10 meV, larger than the minimum energy of 1 meV of a previous optical study [@degiorgi-purple-optics], it is nonetheless smaller than the energy resolutions used in any ARPES studies on the material to date ($\geq 15$ meV). Further, the new optical study found that the spectral weight [*increases*]{} below $T_X$ in the low energy sector ($< 100$ meV) for which previous ARPES studies [@xue; @smith_reply; @smith] on temperature gradient flux samples found a large gap opening (2$\Delta \approx$ 80 meV).
To summarize, we have shown that the ARPES spectra of Li$_{0.9}$Mo$_6$O$_{17}$ samples prepared by temperature gradient flux growth display LL behavior the same as seen for samples prepared by the electrolyte reduction method, thus augmenting further the strong case for LL ARPES lineshapes already established by our past ARPES work on this material.
This work was supported by the U.S. NSF grant DMR-99-71611 and the U.S. DoE contract DE-FG02-90ER45416 at U. Mich. The ORNL is managed by UT-Battelle, LLC, for the U.S. DoE under contract DE-AC05-00OR22725. Work at UT was supported by the NSF Grant DMR 00-72998. The SRC is supported by the NSF Grant DMR-0084402.
$^{\dagger}$ Current address: MS 2-200, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, 94720; Electronic address: [email protected].
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---
author:
- |
Arturo J. Nic May$^*$ and Eric J. Avila Vales$^*$\
[$^*$ Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte,]{}\
[Tablaje 13615, Mérida, Yucatán C.P. 97119, México]{}\
[E-mail addresses: arturo\_javier\[email protected], [email protected]]{}
title: Dynamics of tritrophic interaction with volatile compounds in plants
---
**Abstract**
In this paper we will consider a mathematical model that describes, the tritrophic interaction between plants, herbivores and their natural enemies, where volatiles organic compounds (VOCs) released by plants play an important role. We show positivity and boundedness of the system solutions, existence of positive equilibrium and its local stability, we analyse global stability of positive equilibrium via the geometrical approach of Li and Muldowney. We pay attention to parameters in order to discuss different types of bifurcations. Finally, we present some numerical simulations to justify our analytical results.\
\
[ ***Keywords—*** Tritrophic model, Global stability, Bifurcation. ]{}\
\
[ ***Classification—*** 92D40, 34D23, 34C23. ]{}
Introduction
============
In agronomy, tritrophic interactions between crop, herbivores and their natural enemies are one of the drivers of the crop yield. Understanding and manipulating these interactions in order to produce food more sustainably is the basic principle of biological control of pest \[1\]. The plants emit a blend of different Volatile Organic Compounds (VOCs), Some applications of plant VOCs in agriculture are: isoprenoids emitted by leaves can exert a protective effect against abiotic stresses by quenching ROS or by strengthening the cell membranes, some VOCs are able to inhibit germination and growth of plant pathogens in vitro, herbivore repellency and attraction of herbivores parasitoids on infested plants are probably the most known capacity of VOCs \[2\]. For example, when spider mites damage lima beans and apple plants, they attract predatory mites by generating VOCs \[3\]. Corn and cotton plants also propagate volatiles to call hymenopterous parasitoids which demolish larvae of several Lepidoptera species \[4\].\
\
The use of the products chemicals in agriculture has caused serious problems with food safety and environmental pollution. Thus the agriculture is called to provide new solutions to increase yields while preserving natural resources and the environment \[2\]. For this, various models \[5,6,7\] have addressed on indirect defense mechanism of plant population (Vocs). Unlike the models proposed, we consider the attraction constant, due to VOCs.\
\
In this paper, we consider the model proposed in \[1\], given by three ordinary differential equations describing the tritrophic interaction between crop, pest and the pest natural enemy, in which the release of Volatile Organic Compounds (VOCs) by crop to attract the pest natural enemy is explicitly taken into account. Our purpose is to perform a more detailed mathematical analysis of the model proposed that includes an analysis of different types of bifurcations.\
\
The rest of the paper is organized as follows: The model is introduced in Section 2. Positivity and boundedness of solutions of system are given in Section 3. Dynamical behavior of the system are investigated in Section 4. Bifurcation phenomenon, is established in Section 5. Numerical examples are presented in Section 6. A brief discussion is presented in Section 7.
Model
=====
The model of tritrophic interaction among plants, herbivores and carnivores are described by following three Ordinary Differential Equations: $$\begin{aligned}
\frac{\text{d}x}{\text{d}t}&=& rx\left(1-\frac{x}{K}\right)-a\frac{xy}{h+x} \nonumber\\
\frac{\text{d}y}{\text{d}t}&=& y\left(ae\frac{x}{h+x}-m-p\frac{z}{l+y}\right) \nonumber\\
\frac{\text{d}z}{\text{d}t}&=& x\left( b+c\frac{y}{k+y}\right)+z\left(pq\frac{y}{l+y}-n\right). \end{aligned}$$ Where all the parameters are positive except $b \geq 0$ and $c \geq 0$, and biological significance are given below:
- $x$ is the crop population size.
- $y$ is the aphid population size.
- $z$ is the aphid-natural enemy population size.
- $r$ is the crop growth rate.
- $K$ is the crop carrying capacity.
- $a$ is the maximal harvesting rate of crop by aphids.
- $e$ is the crop to aphids conversion (yield).
- $m$ is the aphids’ natural mortality rate.
- is the $p$ maximal uptake rate of aphid by aphid-natural enemy.
- $h$, $k$ and $l$ are the half saturation constants.
- $b$ is the attraction constant due to VOCs.
- $c$ is the enhanced attraction rate of aphid-natural enemy by VOCs released by crops under aphid attack.
- $q$ is the aphids to aphid-natural enemy conversion (yield).
- $n$ is the aphid-natural enemy mortality rate.
Positivity and boundedness of solutions
=======================================
In this section, we shall first show positivity and boundedness of solutions of system (1). These are very important so far as the validity of the model is related. We first study the positivity.
All solutions $(x(t), y(t), z(t))$ of system (1) with initial value\
$(x_0 , y_0 , z_0 ) \in \mathbb{R}^3_+$, remains positive for all $t > 0$.
The positivity of $x(t)$ and $y(t)$ can be verified by the equations $$\begin{aligned}
x(t)&=&x_0 \exp\left(\int_0^t \left[ r-\frac{rx(s)}{K}-a\frac{y(s)}{h+x(s)}\right]\text{d}s\right),\nonumber\\
y(t)&=&y_0 \exp\left(\int_0^t \left[ ae\frac{x(s)}{h+x(s)}-m-p\frac{z(s)}{l+y(s)}\right]\text{d}s\right).\nonumber\end{aligned}$$ Also if $x(0) = x_0 > 0$ and $y(0) = y_0 > 0$, then $x(t) > 0$ and $y(t)>0$ for all $t > 0$. The positivity of $z(t)$ can be easily deduced from the third equation of system (1). We observe that $$\frac{\text{d}z}{\text{d}t}\geq z \left(pq\frac{y}{l+y}-n\right). \nonumber$$ Then. $$z(t)\geq z_0 \exp\left(\int_0^t \left[ pq\frac{y}{l+y}-n\right]\text{d}s\right). \nonumber$$ if $z(0) = z_0 > 0$, then $z(t)>0$ for all $t>0$.
All the solutions of system (1) will lie in the region $\Omega=\{(x,y,z) | x \leq K_1, \ ex+y+\frac{1}{q}z \leq \left( er+\frac{b+c}{q}+1 \right)\frac{K_1}{\delta} \}$, where $\delta=\min \{\frac{1}{e}, \ m, \ n \}$ and $K_1=\max \{x_0, K \}$.
Let $(x(t), y(t), z(t))$ be any solution of system (1) with positive initial conditions $(x_0, y_0, z_0)$ . Since, $\displaystyle \frac{dx}{dt} \leq rx(1 - \frac{x}{K})$, by a standard comparison theorem we have, $\displaystyle \lim_{t\to \infty} \sup x(t)\leq K_1$.\
Let $N(t)=ex+y+\frac{1}{q}z$, Then $$\begin{aligned}
\dot{N}&=&e\left(rx\left(1-\frac{x}{K}\right)-a\frac{xy}{h+x} \right)+y\left(ae\frac{x}{h+x}-m-p\frac{z}{l+y}\right)\nonumber \\
&&+\frac{1}{q}\left(x\left( b+c\frac{y}{k+y}\right)+z\left(pq\frac{y}{l+y}-n\right) \right)\nonumber \\
&=&e\left(rx\left(1-\frac{x}{K}\right)\right)-my+\frac{1}{q}\left(x\left( b+c\frac{y}{k+y}\right)-nz \right)\nonumber \\
&\leq& \left( er+\frac{b}{q}+\frac{c}{q} \right) x -my-\frac{n}{q}z\nonumber\\
&=&\left( er+\frac{b+c}{q}+1 \right) x-x -my-\frac{n}{q}z\nonumber\\
&\leq & \left( er+\frac{b+c}{q}+1 \right)K_1-\delta N \nonumber.\end{aligned}$$ By using the Comparison Theorem we have $0 \leq N(t) \leq \left( er+\frac{b+c}{q}+1 \right)\frac{K_1}{\delta}$ for t sufficiently large, so all solutions of $(1)$ are ultimately bounded and enter the region $\Omega $.
Dynamical behavior
==================
Equilibria
----------
Here we discuss existence condition of interior equilibrium point of system (1). The system has one trivial equilibrium point (the ecosystem collapse) $E_0 = (0,0, 0)$, the aphid-free point $E_1 = (x_1, 0, z_1)$. Where,
$x_1=K$, $z_1= \displaystyle \frac{b}{n}K$
It follows that the point $E_1$ always exists. And coexistence $E^* = (x^*, y^*, z^*)$, where,\
$$y^*=\frac{1}{a}\left[r\left(1-\frac{x}{K}\right)(h+x) \right].$$ Which is nonnegative only for $0 \leq x \leq K$, $$z^*=\frac{l+y}{p}\left[ae\frac{x}{h+x}-m \right].$$ This function is nonnegative if $aex \geq m(h + x)$.\
\
Then $y^*$ and $z^*$ are nonnegative if and only if $ae>m$ and $\frac{mh}{ae-m}\leq x^* \leq K$. With $x^*$ being determined by the roots of the equation. $$\begin{aligned}
H(x)&=&x\left( b+c\frac{y}{k+y}\right)+z\left(pq\frac{y}{l+y}-n\right)\nonumber\\
&=&x\left( b+c\frac{\frac{1}{a}\left[r\left(1-\frac{x}{K}\right)(h+x)) \right]}{k+\frac{1}{a}\left[r\left(1-\frac{x}{K}\right)(h+x)) \right] }\right)+\left( \frac{l+\frac{1}{a}\left[r\left(1-\frac{x}{K}\right)(h+x)) \right]}{p}\right)\nonumber \\
&&\times \left[ae\frac{x}{h+x}-m \right]\left(pq\frac{\frac{1}{a}\left[r\left(1-\frac{x}{K}\right)(h+x)) \right]}{l+\frac{1}{a}\left[r\left(1-\frac{x}{K}\right)(h+x)) \right]}-n\right). \nonumber\end{aligned}$$ note that $H\left(\frac{mh}{ae-m}\right)>0$ and $$\begin{aligned}
H(K)&=&bK-l \frac{n}{p}\left(ae\frac{K}{h+K}-m \right).\end{aligned}$$ If
$ae>m$ and $\displaystyle \frac{aeK}{h+K}\geq m+b\frac{Kp}{ln}.$
Then $H(K)\leq 0$ and by its continuity, the function f must have a zero $x^*$ in the interval $[\frac{mh}{ae-m}, K]$.
Local stability
---------------
We now study the local stability of $E_0$, $E_1$ and $E^*$ of Model (1).
*$E_0=(0,0,0)$ is unstable.*
The Jacobian matrix of the model, we get as follows:
$$\begin{aligned}
J:= \left( \begin{matrix}
r-2\frac{rx}{K}-\frac{ay}{h+x}+\frac{axy}{(h+x)^2} & -\frac{ax}{h+x} &0 \\
\frac{aehy}{(h+x)^2} & \frac{aex}{h+x}-\frac{lpz}{(l+y)^2}-m&-\frac{py}{l+y} \\
b+c\frac{y}{k+y}& \frac{ckx}{(k+y)^2}+\frac{lpqz}{(l+y)^2} & \frac{pqy}{l+y}-n \end{matrix} \right).\end{aligned}$$
$$\begin{aligned}
J_{E_0}:= \left( \begin{matrix}
r & 0 &0 \\
0 & -m&0 \\
b& 0 & -n
\end{matrix} \right).\end{aligned}$$
The characteristic equation at $E_0$ is $$(\lambda-r)(\lambda+m)(\lambda+n)=0$$ Since one of the roots of the above equation is positive, $E_0$ is unstable.
If $\frac{aeK}{h+K}<\frac{pbK}{nl}+m$, then $E_1=(K,0,\displaystyle \frac{b}{n}K)$ is locally asymptotically stable. If $\frac{aeK}{h+K}>\frac{pbK}{nl}+m$, then $E_1$ is unstable.
$$\begin{aligned}
J_{E_1}:= \left( \begin{matrix}
-r & -\frac{aK}{h+K} &0 \\
0 & \frac{aeK}{h+K}-\frac{pbK}{nl}-m& 0 \\
b& \frac{cK}{k}+\frac{pqbK}{nl} & -n
\end{matrix} \right).\end{aligned}$$
The characteristic equation at $E_0$ is $$(\lambda+r)(\lambda-\frac{aeK}{h+K}+\frac{pbK}{nl}+m)(\lambda+n)=0.$$ If $\frac{aeK}{h+K}<\frac{pbK}{nl}+m$ then all the roots of the above equation are negative and hence $E_1$ is locally asymptotically stable. If $\frac{aeK}{h+K}>\frac{pbK}{nl}+m$, since one of the roots of the above equation is positive, then $E_1$ is unstable.\
Suppose that $r+\frac{ax^*y^*}{(h+x^*)^2}<2\frac{rx^*}{K}+\frac{ay^*}{h+x^*}$, $\frac{aex^*}{h+x^*}<\frac{lpz^*}{(l+y^*)^2}+m$ and $\frac{pqy^*}{l+y^*}<n$ and $-A_{11}A_{22}A_{33}-A_{12}A_{23}A_{31}+A_{12}A_{21}A_{33}+A_{11}A_{23}A_{32}>0$, then $E^*$ is locally asymptotically stable.\
\
Where, $$\begin{aligned}
A_{11}&=&r-2\frac{rx^*}{K}-\frac{ay^*}{h+x^*}+\frac{ax^*y^*}{(h+x)^2}<0 \nonumber \\
A_{12}&=&-\frac{ax^*}{h+x^*}<0 \nonumber \\
A_{21}&=&\frac{aehy}{(h+x^*)^2}>0 \nonumber\\
A_{22}&=&\frac{aex^*}{h+x^*}-\frac{lpz^*}{(l+y^*)^2}-m<0 \nonumber\\
A_{23}&=&-\frac{py^*}{l+y^*}<0 \nonumber\\
A_{31}&=&b+c\frac{y^*}{k+y^*}>0 \nonumber\\
A_{32}&=&\frac{ckx^*}{(k+y^*)^2}+\frac{lpqz^*}{(l+y^*)^2}>0 \nonumber\\
A_{33}&=&\frac{pqy^*}{l+y^*}-n<0. \nonumber\end{aligned}$$
The Jacobian matrix of the model, we get as follows:
$$\begin{aligned}
J_{E^*}:= \left( \begin{matrix}
A_{11} & A_{12} &0 \\
A_{21} & A_{22} & A_{23} \\
A_{31}& A_{32} & A_{33} \end{matrix} \right).\end{aligned}$$
The characteristic equation at $E^*$ is $$\lambda^3 +a_1\lambda^2 +a_2 \lambda+a_3=0.$$ Where $$\begin{aligned}
a_1 &=&-A_{11}-A_{22}-A_{33}\nonumber\\
a_2&=&A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}A_{21}-A_{23}A_{32} \nonumber\\
a_3&=&-A_{11}A_{22}A_{33}-A_{12}A_{23}A_{31}+A_{12}A_{21}A_{33}+A_{11}A_{23}A_{32}.
\nonumber\end{aligned}$$ Clearly $a_i > 0$ for $i = 1, 2, 3$ by the assumption of the theorem. Again $$\begin{aligned}
a_1 a_2-a_3&=&\left( -A_{11}-A_{22}-A_{33}\right)\left(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}A_{21}-A_{23}A_{32} \right)\nonumber\\
&+&A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}-A_{12}A_{21}A_{33}-A_{11}A_{23}A_{31}\nonumber \\
&=&-A_{11}A_{22}A_{33}+A_{11}A_{23}A_{32} + -A_{11}\left(A_{11}A_{22}+A_{11}A_{33}+-A_{12}A_{21} \right)\nonumber\\
&&-A_{22}\left(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}A_{21}-A_{23}A_{32} \right)\nonumber\\
&&+A_{33}A_{12}A_{21} -A_{33}\left(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{23}A_{32} \right)\nonumber\\
&+&A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}-A_{12}A_{21}A_{33}-A_{11}A_{23}A_{32}\nonumber \\
&=&-A_{11}\left(A_{11}A_{22}+A_{11}A_{33}+-A_{12}A_{21} \right)\nonumber\\
&&-A_{22}\left(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}A_{21}-A_{23}A_{32} \right)\nonumber\\
&&-A_{33}\left(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{23}A_{32} \right)+A_{12}A_{23}A_{31}>0.\nonumber\end{aligned}$$ Applying the Routh–Hurwitz criterion, we see that all roots of $\lambda^3 +a_1\lambda^2 +a_2 \lambda+a_3=0$ have negative real parts therefore $E^*$ is stable.
The theorem 3 is valid if $a_i > 0$ for $i = 1, 2, 3$ and $a_1a_2 - a_3 > 0$.
Global stability
----------------
We now study the global stability of endemic equilibria of model (1). We used a high-dimensional Bendixson criterion of Li and Muldowney \[8\].
Suppose $\frac{aeK}{h+K}>\frac{pbK}{nl}+m$ then system (1) is uniformly persistent.
Suppose $x_1$ is a point in the positive octant and $o(x_1)$ is the orbit through $x_1$ and $\omega $ is the omega limit set of the orbit through $x_1$. Note that $\omega(x_1)$ is bounded (Lemma 2) .We claim that $E_0 \notin \omega(x_1)$. If $E_0 \in \omega(x_1)$ then by Butler- McGehee lemma \[9\], there exists a point P in $\omega(x_1)\cap W^s(E_0)$ (which denotes stable manifold $of E_0$). Since $o(P)$ lies in $\omega(x_1)$ and $W^s(E_0)$ is the $y-z$ plane hence unbounded orbit lies in $\omega(x_1)$ a contradiction. Next, we show that $E_1 \notin \omega(x_1)$. Since $\frac{aeK}{h+K}>\frac{pbK}{nl}+m$, $E_1$ is a saddle point. $W^s(E_1)$ is the $x-z$ plane and hence orbits in the plane emanate from either $E_0$ or an unbounded orbit lies in $\omega(x_1)$, once more a contradiction. There does not exist any equilibria in the two dimensional plane. Thus, $\omega(x_1)$ does not intersect any of the coordinate planes and hence system (1) is persistent. Since (1) is bounded, by main theorem in Butler et al. \[10\], this implies that the system is uniformly persistent.
We will make use of the following theorem.
Suppose that the system $\dot{x}=f(x)$, with $f:D\subset\mathbb{R}^n\to\mathbb{R}^n$, satisfies the following:
- $D$ is a simply connected open set,
- there is a compact absorbing set $K\subset D$,
- $x^*$ is the only equilibrium in $D$.
Then the equilibrium $x^*$ is globally stable in $D$ if there exists a Lozinskiĭ measure $\mu_1$ such that $$\limsup_{t\to\infty}\sup_{x_0\in K}\frac1t\int_0^t\mu_1\big(B(x(s,x_0))\big)\,\textup{d}s<0, \nonumber$$ Where, $$B=Q_fQ^{-1}+Q\frac{\partial f}{\partial x}^{[2]}Q^{-1} \nonumber$$ And $Q\mapsto Q(x)$ is an ${n\choose2}\times{n\choose2}$ matrix-valued function.
In our case, system (1) can be written as $\dot{x}=f(x)$ with $f:D\subset\mathbb{R}^3\to\mathbb{R}^3$ and $D$ being the interior of the feasible region $\Omega$. The existence of a compact absorbing set $K\subset D$ is equivalent to proving that (1) is uniformly persistent (Theorem 4). Hence, **(H1)** and **(H2)** hold for system (1), and by assuming the uniqueness of the endemic equilibrium in $D$, we can prove its global stability with the aid of Theorem 5.
If
- There exist positive numbers $\alpha$ and $\beta$ such that $$\max \{ N_{11} +\frac{\alpha}{\beta} N_{12},\frac{\beta}{\alpha} N_{21} + N_{22}+\frac{\beta}{\zeta} N_{23},\frac{\zeta}{\alpha}N_{31} +\frac{\zeta}{\beta}N_{32}+N_{33}\}<0.$$
- $\displaystyle \frac{aeK}{h+K}>\frac{pbK}{nl}+m.$
Then $E^*$ is globally stable in $\mathbb{R}^3$.
suppose that $x^*$ is the only equilibrium point in the interior of $\Omega $. By lemma 2 all solution of (1) is bounded, exists a time $T$ such that $x(t)<K_1$, $y(t)\leq M$, and $z(t)\leq qM$ (where $M=\left( er+\frac{b+c}{q}+1 \right)\frac{K_1}{\delta} )$, for $t > T$ and assumption (H2) implies that system (1) is uniformly persistent (Theorem 4) and hence there exists a time $T$ such that $x(t), y(t), z(t) > \eta(0 < \eta )$ for $t > T$.\
\
Starting with the Jacobian matrix $J$ of (1). The Jacobian matrix of the model, we get as follows:
$$\begin{aligned}
J_{E^*}:= \left( \begin{matrix}
a_{11} & a_{12} &0 \\
a_{21} & a_{22} & a_{23} \\
a_{31}& a_{32} & a_{33} \end{matrix} \right).\end{aligned}$$
Where, $$\begin{aligned}
a_{11}&=&r-2\frac{rx}{K}-\frac{ahy}{(h+x)^2} \nonumber \\
a_{12}&=&-\frac{ax}{h+x} \nonumber \\
a_{21}&=&\frac{aehy}{(h+x)^2} \nonumber\\
a_{22}&=&\frac{aex}{h+x}-\frac{lpz}{(l+y)^2}-m \nonumber\\
a_{23}&=&-\frac{py}{l+y} \nonumber\\
a_{31}&=&b+c\frac{y}{k+y} \nonumber\\
a_{32}&=&\frac{ckx}{(k+y)^2}+\frac{lpqz}{(l+y)^2} \nonumber\\
a_{33}&=&\frac{pqy}{l+y}-n. \nonumber\end{aligned}$$ The second additive compound matrix of $J$ is given as follows: $$\begin{aligned}
M:= \left( \begin{matrix}
M_{11} & M_{12} &0 \\
M_{21} & M_{22} & M_{23} \\
M_{31}& M_{32} & M_{33} \end{matrix} \right).\end{aligned}$$ Where, $$\begin{aligned}
M_{11}&=&r-2\frac{rx}{K}-\frac{ahy}{(h+x)^2}+\frac{aex}{h+x}-\frac{lpz}{(l+y)^2}-m \nonumber \\
M_{12}&=&-\frac{py}{l+y} \nonumber \\
M_{21}&=&\frac{ckx}{(k+y)^2}+\frac{lpqz}{(l+y)^2} \nonumber\\
M_{22}&=&r-2\frac{rx}{K}-\frac{ahy}{(h+x)^2}+\frac{pqy}{l+y}-n \nonumber\\
M_{23}&=&-\frac{ax}{h+x} \nonumber\\
M_{31}&=&-b-c\frac{y}{k+y} \nonumber\\
M_{32}&=& \frac{aehy}{(h+x)^2} \nonumber\\
M_{33}&=&\frac{aex}{h+x}-\frac{lpz}{(l+y)^2}-m+\frac{pqy}{l+y}-n. \nonumber \end{aligned}$$ Note that, $$\begin{aligned}
M_{11}&\leq&r-2\frac{r\eta}{K}-\frac{ah\eta}{(h+K_1)^2}+\frac{aeK_1}{h+\eta}-\frac{lp\eta}{(l+M)^2}-m=N_{11} \nonumber \\
M_{12}&\leq&-\frac{p\eta}{l+\eta}=N_{12} \nonumber \\
M_{21}&\leq&\frac{ck K_1}{(k+\eta)^2}+\frac{lpq^2 M}{(l+\eta)^2}=N_{21} \nonumber\\
M_{22}&\leq&r-2\frac{r \eta }{K}-\frac{ah\eta}{(h+K_1)^2}+\frac{pqM}{l+M}-n=N_{22} \nonumber\\
M_{23}&\leq&-\frac{a\eta}{h+\eta}=N_{23} \nonumber\\
M_{31}&\leq&-b-c\frac{\eta}{k+\eta}=N_{31} \nonumber\\
M_{32}&\leq& \frac{aeh K_1}{(h+\eta)^2}=N_{32} \nonumber\\
M_{33}&\leq&\frac{ae K_1}{h+ K_1}-\frac{lp\eta}{(l+M)^2}-m+\frac{pq M}{l+M}-n=N_{33}. \nonumber \end{aligned}$$\
We consider the following norm on $\mathbb{R}^3$. $$\|z \|=\max \{\alpha |z_1| , \beta |z_2|,\zeta |z_3| \} \ \text{where $\alpha$,$\beta$, $\zeta>0$}.$$ The Lozinskiï measure $\bar{\mu}$ can be evaluate as, $$\bar{\mu}(Z)=\inf \{\bar{k}: D_{+} \|z \|\leq \bar{k} \|z \|,\ \text{for all solutions of $z'=Bz$} \}$$ Where $D_+$ is the right-hand derivative. The basic idea of the proof is to the obtain the estimate of the right-hand derivative $D_{+}\|z \|$ of the norm (11), we need to discuss three case.
- Case 1: $\alpha |z_1|\geq \beta |z_2|,\zeta|z_3|.$
Then $\|z\|=\alpha |z_1|$.\
Thus, we have,\
$$\begin{aligned}
D_{+} \|z \|&=&\alpha \frac{z_1}{|z_1|}z'_1\nonumber\\
&\leq&\alpha M_{11} z_1+\alpha M_{12}z_2 \nonumber\\
&\leq&\left( M_{11} +\frac{\alpha}{\beta} M_{12}\right) \|z\| \nonumber\\
&\leq&\left( N_{11} +\frac{\alpha}{\beta} N_{12}\right) \|z\|. \nonumber\end{aligned}$$
- Case 2: $\beta |z_2|\geq \alpha|z_1|, \zeta|z_3|.$
Then $\|z\|=\beta |z_2|$.\
Thus, we have,\
$$\begin{aligned}
D_{+} \|z \|&=&\beta \frac{z_2}{|z_2|}z'_2\nonumber\\
&\leq& \beta M_{21} z_1+\beta M_{22}z_2+\beta M_{23}z_3 \nonumber\\
&\leq&\left( \frac{\beta}{\alpha} M_{21} + M_{22}+\frac{\beta}{\zeta} M_{23}\right) \|z\| \nonumber\\
&\leq&\left( \frac{\beta}{\alpha} N_{21} + N_{22}+\frac{\beta}{\zeta} N_{23}\right) \|z\|. \nonumber\end{aligned}$$
- Case 3: $\zeta |z_3|\geq \alpha |z_1|,\beta |z_2|$.
Then $\|z\|=\zeta |z_3|$.\
Thus, we have,\
$$\begin{aligned}
D_{+} \|z \|&=&\zeta \frac{z_3}{|z_3|}z'_3\nonumber\\
&\leq& \zeta M_{31} z_1+\zeta M_{32}z_2+\zeta M_{33} z_3 \nonumber\\
&\leq& \left( \frac{\zeta}{\alpha}M_{31} +\frac{\zeta}{\beta}M_{32}+M_{33}\right) \|z\| \nonumber\\
&\leq& \left( \frac{\zeta}{\alpha} N_{31}+\frac{\zeta}{\beta}N_{32}+N_{33}\right) \|z\|. \nonumber\end{aligned}$$ Therefore $$D_{+} \|z \|\leq L \|z \|.$$ Where: $$L=\max \{ N_{11} +\frac{\alpha}{\beta} N_{12},\frac{\beta}{\alpha} N_{21} + N_{22}+\frac{\beta}{\zeta} N_{23},\frac{\zeta}{\alpha}N_{31} +\frac{\zeta}{\beta}N_{32}+N_{33}\}<0.$$ So $$\limsup_{t \to \infty} \sup_{x_0\in \omega} \frac{1}{t}\int^t_0 \bar{\mu}(M) ds\leq \limsup_{t \to \infty} \sup_{x_0\in \omega} \frac{1}{t}\int^t_0 L ds=L<0. \nonumber$$ By LI & Muldowney\[8\] and theorem 5, the positive equilibrium point $E^*$ is globally stable in $\mathbb{R}^3_+$.
Bifurcation
===========
In this section we discuss various types of bifurcation of system (1) around different steady states.
If $\frac{2aeh}{(h+x_1)^2}v^{[1]}+\frac{2pz_1}{l^2}-\frac{2p}{l}v^{[3]}\neq 0$, where $v^{[1]}=-\frac{aK}{r(h+K)}$ and $v^{[3]}=\frac{-\frac{baK}{h+K}+\frac{rcK}{k}+\frac{rpqbK}{nl}}{rn}$. Then the system (1) possesses a transcritical bifurcation at the equilibrium point $E_1$ as the parameter $m$ crosses the critical value $m^*=\frac{aeK}{h+K}-\frac{pbK}{nl}$.
Let $X=(x,y,z)$ and $$\begin{aligned}
f(X,m)=\left( \begin{matrix}
rx\left(1-\frac{x}{K}\right)-a\frac{xy}{h+x}\\
y\left(ae\frac{x}{h+x}-m-p\frac{z}{l+y}\right)\\
x\left( b+c\frac{y}{k+y}\right)+z\left(pq\frac{y}{l+y}-n \right)\end{matrix}\right).\nonumber \end{aligned}$$ $$\begin{aligned}
f_m(X,m)=\left( \begin{matrix}
0\\
-y\\
0\end{matrix}\right).\nonumber \end{aligned}$$
$$\begin{aligned}
Df(X,m)= \left( \begin{matrix}
r-2\frac{rx}{K}-\frac{ay}{h+x}+\frac{axy}{(h+x)^2} & -\frac{ax}{h+x} &0 \\
\frac{aehy}{(h+x)^2} & \frac{aex}{h+x}-\frac{lpz}{(l+y)^2}-m&-\frac{py}{l+y} \\
b+c\frac{y}{k+y}& \frac{ckx}{(k+y)^2}+\frac{lpqz}{(l+y)^2} & \frac{pqy}{l+y}-n \end{matrix} \right). \nonumber\end{aligned}$$
$$\begin{aligned}
Df_m(X,m)= \left( \begin{matrix}
0 & 0 &0 \\
0 & -1 &0 \\
0& 0 & 0 \nonumber \end{matrix} \right).\end{aligned}$$
Then $$\begin{aligned}
f_m(E_1,m)=\left( \begin{matrix}
0\\
0\\
0\end{matrix}\right).\nonumber \end{aligned}$$
$$\begin{aligned}
A=Df(E_1,m^*):= \left( \begin{matrix}
-r & -\frac{aK}{h+K} &0 \\
0 & 0& 0 \\
b& \frac{cK}{k}+\frac{pqbK}{nl} & -n
\end{matrix} \right).\end{aligned}$$
$A$ has a simple eigenvalue $\lambda=0$ with eigenvector $v=(v^{[1]},1,v^{[3]})^T$, where $v^{[1]}=-\frac{aK}{r(h+K)}$ and $v^{[3]}=\frac{-\frac{baK}{h+K}+\frac{rcK}{k}+\frac{rpqbK}{nl}}{rn}$. Also, $A^T$ has an eigenvector $w=(0,1,0)^T$ that correspondent to the eigenvalue $ \lambda =
0$.\
Also: $$w^T[f_m(E_1,m^*)]=0.$$ $$\begin{aligned}
w^T[Df_m(X,m)v]=(0,1,0)\left[ \left( \begin{matrix}
0 & 0 &0 \\
0 & -1 &0 \\
0& 0 & 0 \end{matrix} \right) \left( \begin{matrix}
v^{[1]} \\
1 \\
v^{[3]} \end{matrix} \right) \right]=-1\neq 0. \nonumber\end{aligned}$$
$$\begin{aligned}
w^T[D^2f(E_1,m^*)(v,v)]&=&(0,1,0)\nonumber\\
&&\times \left( \begin{matrix}
-\frac{2r}{K} v^{[1]}v^{[1]}-\frac{2 a h}{(h+x_1)^2}v^{[1]} \\
\frac{2aeh}{(h+x_1)^2}v^{[1]}+\frac{2pz_1}{l^2}-\frac{2p}{l}v^{[3]}\\
\frac{2 c}{k}v^{[1]}+\frac{2pq}{l}v^{[3]}-\frac{2cx_1}{k^2}-\frac{2pqz_1}{l^2} \end{matrix} \right) \nonumber\\
&=&\frac{2aeh}{(h+x_1)^2}v^{[1]}+\frac{2pz_1}{l^2}-\frac{2p}{l}v^{[3]}\neq 0. \nonumber\end{aligned}$$
By Sotomayor theorem \[11\], the system (1) experiences a transcritical bifurcation at the equilibrium point $E_1$ as the parameter $m$ varies through the bifurcation value $m=m^*$.
If $\frac{2aeh}{(h+x_1)^2}v^{[1]}+\frac{2pz_1}{l^2}-\frac{2p}{l}v^{[3]}\neq 0$, where $v^{[1]}=-\frac{aK}{r(h+K)}$ and $v^{[3]}=\frac{-\frac{b^*aK}{h+K}+\frac{rcK}{k}+\frac{rpqb^*K}{nl}}{rn}$. Then the system (1) possesses a transcritical bifurcation at the equilibrium point $E_1$ as the parameter $b$ crosses the critical value $b^*=\left(\frac{aeK}{h+K}-m \right)\frac{nl}{pK }$.
Let $X=(x,y,z)$ and $$\begin{aligned}
f(X,b)=\left( \begin{matrix}
rx\left(1-\frac{x}{K}\right)-a\frac{xy}{h+x}\\
y\left(ae\frac{x}{h+x}-m-p\frac{z}{l+y}\right)\\
x\left( b+c\frac{y}{k+y}\right)+z\left(pq\frac{y}{l+y}-n \right)\end{matrix}\right).\nonumber \end{aligned}$$ $$\begin{aligned}
f_b(X,b)=\left( \begin{matrix}
0\\
0\\
x\end{matrix}\right).\nonumber \end{aligned}$$
$$\begin{aligned}
Df(X,b)= \left( \begin{matrix}
r-2\frac{rx}{K}-\frac{ay}{h+x}+\frac{axy}{(h+x)^2} & -\frac{ax}{h+x} &0 \\
\frac{aehy}{(h+x)^2} & \frac{aex}{h+x}-\frac{lpz}{(l+y)^2}-m&-\frac{py}{l+y} \\
b+c\frac{y}{k+y}& \frac{ckx}{(k+y)^2}+\frac{lpqz}{(l+y)^2} & \frac{pqy}{l+y}-n \end{matrix} \right). \nonumber\end{aligned}$$
$$\begin{aligned}
Df_b(E_1,b^*)= \left( \begin{matrix}
0 & 0 &0 \\
0 & -\frac{pk}{nl} &0 \\
1& \frac{pqk}{nl} & 0 \nonumber \end{matrix} \right).\end{aligned}$$
Then $$\begin{aligned}
f_b(E_1,b^*)=\left( \begin{matrix}
0\\
0\\
x_1
\end{matrix}\right).\nonumber \end{aligned}$$
$$\begin{aligned}
A=Df(E_1,b^*):= \left( \begin{matrix}
-r & -\frac{aK}{h+K} &0 \\
0 & 0& 0 \\
b^*& \frac{cK}{k}+\frac{pqb^*K}{nl} & -n
\end{matrix} \right).\end{aligned}$$
$A$ has a simple eigenvalue $\lambda=0$ with eigenvector $v=(v^{[1]},1,v^{[3]})^T$, where $v^{[1]}=-\frac{aK}{r(h+K)}$ and $v^{[3]}=\frac{-\frac{b^*aK}{h+K}+\frac{rcK}{k}+\frac{rpqb^*K}{nl}}{rn}$. Also, $A^T$ has an eigenvector $w=(0,1,0)^T$ that correspondent to the eigenvalue $ \lambda =
0$.\
Also: $$w^T[f_b(E_1,b^*)]=0.$$ $$\begin{aligned}
w^T[Df_b(X,b)v]=(0,1,0)\left[ \left( \begin{matrix}
0 & 0 &0 \\
0 & -\frac{pk}{nl} &0 \\
1& -\frac{pqk}{nl} & 0 \end{matrix} \right) \left( \begin{matrix}
v^{[1]} \\
1 \\
v^{[3]} \end{matrix} \right) \right]=-\frac{pk}{nl}\neq 0. \nonumber\end{aligned}$$
$$\begin{aligned}
w^T[D^2f(E_1,b^*)(v,v)]&=&(0,1,0)\nonumber\\
&&\times \left( \begin{matrix}
-\frac{2r}{K} v^{[1]}v^{[1]}-\frac{2 a h}{(h+x_1)^2}v^{[1]} \\
\frac{2aeh}{(h+x_1)^2}v^{[1]}+\frac{2pz_1}{l^2}-\frac{2p}{l}v^{[3]}\\
\frac{2 c}{k}v^{[1]}+\frac{2pq}{l}v^{[3]}-\frac{2cx_1}{k^2}-\frac{2pqz_1}{l^2} \end{matrix} \right) \nonumber\\
&=&\frac{2aeh}{(h+x_1)^2}v^{[1]}+\frac{2pz_1}{l^2}-\frac{2p}{l}v^{[3]}\neq 0. \nonumber\end{aligned}$$
By Sotomayor theorem \[11\], the system (1) experiences a transcritical bifurcation at the equilibrium point $E_1$ as the parameter $b$ varies through the bifurcation value $b=b^*$.
If $b=\tilde{b}=\frac{-A_{11}A_{22}A_{33}+A_{12}A_{21}A_{33}+A_{11}A_{23}A_{32}}{A_{12}A_{23}}-\frac{cy^*}{k+y^*}$ and\
[$w^{[1]}\left(\left(-\frac{2r}{K}+\frac{2hay^*}{(h+x^*)^3}\right) v^{[1]}v^{[1]}-\frac{2 a h}{(h+x^*)^2}v^{[1]}v^{[2]}\right)+w^{[2]}\left(-\frac{2haey^*}{(h+x^*)^2}v^{[1]}v^{[1]}+\frac{2aeh}{(h+x^*)^2}v^{[1]}v^{[2]}+\frac{2lpz^*}{(l+y^*)^3}v^{[2]}v^{[2]}-\frac{2pl}{(l+y^*)^2}v^{[2]}\right)+\frac{2 ck}{(k+y^*)^2}v^{[1]}v^{[2]}+\frac{2lpq}{(l+y^*)^2}v^{[2]}-\left(\frac{2ckx^*}{(k+y)^3}+\frac{2lpqz^*}{(l+y^3)^3}\right)v^{[2]}v^{[2]}\neq 0$]{}
Where $v^{[1]}=-\frac{A_{12}}{A_{11}}$, $v^{[2]}=\frac{-A_{23}A_{11}}{A_{11}A_{22}-A_{12}A_{21}}$, $w^{[2]}=-\frac{A_{11}A_{32}-A_{31}A_{12}}{A_{11}A_{22}-A_{12}A_{21}}$ and $w^{[1]}=-\frac{A_{21}w^{[2]}}{A_{11}}-\frac{A_{31}}{A_{11}}$.\
\
Then system (1) possesses a saddle-node bifurcation at the equilibrium point $E^*=(x(\tilde{b}),y(\tilde{b}),z(\tilde{b}))$, as the parameter $b$ crosses the critical value $\tilde{b}$.
Let $X=(x,y,z)$ and $$\begin{aligned}
f(X,b)=\left( \begin{matrix}
rx\left(1-\frac{x}{K}\right)-a\frac{xy}{h+x}\\
y\left(ae\frac{x}{h+x}-m-p\frac{z}{l+y}\right)\\
x\left( b+c\frac{y}{k+y}\right)+z\left(pq\frac{y}{l+y}-n \right)\end{matrix}\right).\nonumber \end{aligned}$$ $$\begin{aligned}
f_b(X,b)=\left( \begin{matrix}
0\\
0\\
x\end{matrix}\right).\nonumber \end{aligned}$$
$$\begin{aligned}
Df(X,b)= \left( \begin{matrix}
r-2\frac{rx}{K}-\frac{ay}{h+x}+\frac{axy}{(h+x)^2} & -\frac{ax}{h+x} &0 \\
\frac{aehy}{(h+x)^2} & \frac{aex}{h+x}-\frac{lpz}{(l+y)^2}-m&-\frac{py}{l+y} \\
b+c\frac{y}{k+y}& \frac{ckx}{(k+y)^2}+\frac{lpqz}{(l+y)^2} & \frac{pqy}{l+y}-n \end{matrix} \right). \nonumber\end{aligned}$$
Then $$\begin{aligned}
f_b(E^*,\tilde{b})=\left( \begin{matrix}
0\\
0\\
x^*
\end{matrix}\right).\nonumber \end{aligned}$$
$$\begin{aligned}
A=Df(E^*,\tilde{b}):= \left( \begin{matrix}
A_{11} & A_{22} &0 \\
A_{21} & A_{22}& A_{23} \\
A_{31}& A_{32} & A_{33}
\end{matrix} \right). \nonumber\end{aligned}$$
Also if $b=\tilde{b}$, then $a_3=0$ and the characteristic equation at $E^*$ is $$\lambda^3 +a_1\lambda^2 +a_2 \lambda=\lambda(\lambda^2 +a_1\lambda +a_2)=0.$$ Then $A$ has a simple eigenvalue $\lambda=0$ with eigenvector $v=(v^{[1]},v^{[2]},1)^T$, where $v^{[1]}=-\frac{A_{12}}{A_{11}}$ and $v^{[2]}=\frac{-A_{23}A_{11}}{A_{11}A_{22}-A_{12}A_{21}}$. Also, $A^T$ has an eigenvector $w=(w^{[1]},w^{[2]},1)^T$, where $w^{[2]}=-\frac{A_{11}A_{32}-A_{31}A_{12}}{A_{11}A_{22}-A_{12}A_{21}}$ and $w^{[1]}=-\frac{A_{21}w^{[2]}}{A_{11}}-\frac{A_{31}}{A_{11}}$, that correspondent to the eigenvalue $ \lambda =0$.\
Also: $$w^T[f_b(E^*,b^*)]=x^*\neq 0.$$ $$\begin{aligned}
w^T[D^2f(E_1,b^*)(v,v)]&=&(w^{[1]},w^{[2]},1)\nonumber\\
&&\times \left( \begin{matrix}
\left(-\frac{2r}{K}+\frac{2hay^*}{(h+x^*)^3}\right) v^{[1]}v^{[1]}-\frac{2 a h}{(h+x^*)^2}v^{[1]}v^{[2]} \\
-\frac{2haey^*}{(h+x^*)^2}v^{[1]}v^{[1]}+\frac{2aeh}{(h+x^*)^2}v^{[1]}v^{[2]}+\frac{2lpz^*}{(l+y^*)^3}v^{[2]}v^{[2]}-\frac{2pl}{(l+y^*)^2}v^{[2]}\\
\frac{2 ck}{(k+y^*)^2}v^{[1]}v^{[2]}+\frac{2lpq}{(l+y^*)^2}v^{[2]}-\left(\frac{2ckx^*}{(k+y)^3}+\frac{2lpqz^*}{(l+y^3)^3}\right)v^{[2]}v^{[2]} \end{matrix} \right) \nonumber\\
&=&w^{[1]}\left(\left(-\frac{2r}{K}+\frac{2hay^*}{(h+x^*)^3}\right) v^{[1]}v^{[1]}-\frac{2 a h}{(h+x^*)^2}v^{[1]}v^{[2]}\right)\nonumber \\
&&+w^{[2]}\left(-\frac{2haey^*}{(h+x^*)^2}v^{[1]}v^{[1]}+\frac{2aeh}{(h+x^*)^2}v^{[1]}v^{[2]}\right. \nonumber\\
&&\left.+\frac{2lpz^*}{(l+y^*)^3}v^{[2]}v^{[2]}-\frac{2pl}{(l+y^*)^2}v^{[2]}\right)+\frac{2 ck}{(k+y^*)^2}v^{[1]}v^{[2]} \nonumber\\
&&+\frac{2lpq}{(l+y^*)^2}v^{[2]}-\left(\frac{2ckx^*}{(k+y)^3}+\frac{2lpqz^*}{(l+y^3)^3}\right)v^{[2]}v^{[2]}\neq 0\nonumber\end{aligned}$$ By Sotomayor theorem \[11\], the system (1) experiences a saddle-node bifurcation at the equilibrium point $E^*$ as the parameter $b$ varies through the bifurcation value $b=\tilde{b}$.
We now investigate Hopf bifurcation around $E^*$. We consider $b$ as a bifurcation parameter and define $$g(b)=a_1(b)a_2(b)-a_3(b)$$ Note that if $g(b)=0$, then $b=\bar{b}=-\frac{a_1 a_2+A_{11}A_{22}A_{33}-A_{12}A_{21}A_{33}-A_{11}A_{23}A_{32}}{A_{12}A_{23}}-\frac{cy^*}{k+y^*}$. Now, we will show that the Hopf bifurcation occurs for the system (1) at $ b = \bar{b} $.
If there exists $b =\bar{b}$. Then the positive equilibrium point $E^*=(x^*(b), y^* (b), z^*(b))$ is locally stable if $b>\bar{b}$ but it is unstable for $b<\bar{b}$ and a Hopf bifurcation of periodic solution occurs at $b=\bar{b}$.
We assume that $E^*$ is locally asymptotically stable, let $$g(b) = a_1(b)a_2(b) - a_3(b).$$ Then $a_1(\bar{b})>0$, $g(\bar{b})=0$ and $g'(\bar{b})=A_{12}A_{23}>0$ by a similar argument to the proof of Theorem 4 in \[12\] the proof is completed.
Numerical simulations
=====================
In this section, we will make some numerical simulations to verify the results obtained in section 4 and give examples to illustrate theorems in section 5. In system (1), we set:
$r=$0.1, $K=$1, $h=$0.5, $a=$0.1, $e=$0.4, $m=0.01$, $p=$0.01, $l=$0.5, $c=$0.44, $k=$0.5, $q=$0.5 and $n=$0.3.
In system (1), we set $b=$0.26, then $\frac{pbK}{nl} +m =$0.0273 and $\frac{aeK}{h+K}=$0.0267. By theorem 2, $E_1 = (K, 0, \frac{bK}{n}) \approx$(1,0,0.8667) is locally asymptotically stable, see Figure 1.
In system (1), we set $b =$0.24, then $\frac{pbK}{nl} +m =$0.026 and $\frac{aeK}{h+K} =$0.0267. Then $a_1 =$0.3934, $a_2 =$0.0286, $a_3 =$ 1.16$\times 10^{-5}$ and $a_1a_2-a_3 =$0.0112. By theorem 3, $E^*\approx$(0.9707,0.0431,0.8908) is locally asymptotically stable, see Figure 2.
In system (1), we set $K=$1, $b=$0.23, $c=$0.44, $m=$0.01 and $e=$0.4. We have that $ H (x) $ has a only root in the interval $\left(\frac{mh}{ae-m},1 \right)$ (see Figure 3), then $E^*\approx$(0.2664,0.5622,0.4147) is the only equilibrium point in the interior of $ \Omega $. Besides, we choose $\eta=$0.2, $\alpha=$4 and $\beta=\zeta=1$, then $\frac{aeK}{h+K} =$0.0267, $\frac{pbK}{nl} +m=0.0253$, $N_{11}=$0.1027, $N_{12}=$-0.0286, $N_{21}=$1.0561, $N_{22}=$-0.2395, $N_{23}=$-0.0286, $N_{31}=$-0.3557, $N_{32}=$0.0408, $N_{33}=$-0.2787 and $L=\{\text{-0.0116,-0.0040,-0.3265}\}$. By theorem 6, $E^*$ is globally asymptotically stable, see Figure 4.
In system (1), we set $b =$0.26. If we increase the value of the parameter $m$ and keeping all other parameters value fixed, we observe that transcritical bifurcation arises when $m^* =$ 0.00933, see Figure 5.
In system (1). If we increase the value of the parameter $b$ and keeping all other parameters value fixed, we observe that transcritical bifurcation arises when $b^*=$ 0.25, see Figure 6.
In system (1) we observe that if $b=0.24$ then $E^*_1\approx$(0.9707,0.0431,0.8908) is locally asymptotically stable and $E^*_2\approx$(0.8852,0.1591,1.0256) is unstable. Also if we increase the value of the parameter $b$ and keeping all other parameters value fixed, we observe that saddle-node bifurcation occurs at $b =\tilde{b} \approx$0.23574214, see Figure 7.
In system (1). If we increase the value of the parameter $b$ and keeping all other parameters value fixed, we observe that Hopf bifurcation arises when $b =$0.1906989, see Figure 8.
Discusssion
===========
In this paper we have considered a mathematical model to describe the tritrophic interaction between crop, pest and the pest natural enemy, in which the release of Volatile Organic Compounds (VOCs) by crop is explicitly taken into account. We obtained three equilibrium points:
- The ecosystem collapse is at point $E_0 = (0, 0, 0)$.
- The aphid-free is at point $E_1 =\left(K,0,\frac{b}{n}K \right)$.
- The coexistence is at point $E^*$.
We have investigated the topics of existence and non-existence of various equilibria and their stabilities. More precisely, we have proved the following:
- $E_0 = (0, 0, 0)$ is unstable.
- If $\frac{aeK}{h+K}<\frac{pbK}{nl}+m$, then $E_1 = (K, 0,\frac{b}{n}K)$ is locally asymptotically stable. If $\frac{aeK}{h+K}>\frac{pbK}{nl}+m$, then $E_1$ is unstable.
- $E^*$ it is locally asymptotically stable if $r+\frac{ax^*y^*}{(h+x^*)^2}<2\frac{rx^*}{K}+\frac{ay^*}{h+x^*}$, $\frac{aex^*}{h+x^*}<\frac{lpz^*}{(l+y^*)^2+m}$ and $\frac{pqy^*}{l+y^*}<n$ and $-A_{11}A_{22}A_{33}-A_{12}A_{23}A_{31} + A_{12}A_{21}A_{33} +
A_{11}A_{23}A_{32} > 0$ or $a_i > 0$ for $i =$ 1, 2, 3 and $a_1a_2 - a_3 > 0$.
We also show the globally stability of the positive equilibrium by high-dimensional Bendixson criterion. We used the Sotomayor’s theorem to ensure the existence of saddle-node bifurcation and transcritical bifurcation (this type of bifurcation transforms a herbivore free equilibrium point from stable situation to a unstable). In this paper, we have chosen the parameters $m$ and $b$ arbitrarily for obtaining this type bifurcation. From Hopf bifurcation analysis we observed that $b$ (the attraction constant due to VOCs.) decreasing destabilizes the system.\
\
Thus, $ b $ is an important parameter for our model, because the aphid-free point ($E_1$) is locally asymptotically stable for $b$ sufficiently large. We also found three critical values for b ($ b^* $, $ \tilde{b} $ and $\bar {b}$) and we got that
- If $b>b^*$, then $E_1$ is locally asymptotically stable and If $b<b^*$, then $E_1$ is unstable.
- If $b=b^*$, then a transcritical bifurcation occurs.
- If $\tilde{b}<b<b^*$, then there are 2 positive equilibrium points $ E_1 ^ * $ (locally asymptotically stable) and $ E_2 ^ * $ (unstable).
- If $b=\tilde{b}$, then a saddle-node bifurcation occurs.
- $\bar{b}<b<\tilde{b}$, then there is only one positive equilibrium point $E^*$ that is globally asymptotically stable.
- $b=\bar{b}$, then a Hopf bifurcation occurs.
- $b<\bar{b}$, then there is only one positive equilibrium point $E^*$ that is unstable.
Therefore, VOCs possess a beneficial effect on the environment since their release may be able to stabilize the model dynamics. This could reduce the use of synthetic pesticides.\
\
**Acknowledgments**This work was supported by Sistema Nacional de Investigadores (15284) and Conacyt-Becas.
References {#references .unnumbered}
==========
- Buonomo B., Giannino F., Saussure S. and Venturino E. (2019). Effects of limited volatiles release by plants in tritrophic interactions. Mathematical Biosciences and Engineering, 16(5):3331-3344.
- Brilli F., Loreto F. and Baccelli I. (2019). Exploiting Plant Volatile Organic Compounds (VOCs) in Agriculture to Improve Sustainable Defense Strategies and Productivity of Crops. Frontiers In Plant Science, 10:264.
- Takabayashi J. and Dicke M. (1996). Plant—carnivore mutualism through herbivore-induced carnivore attractants. Trends In Plant Science, 1(4), 109-113.
- Tollsten L., Mller P. (1996): Volatile organic compounds emitted from beech leaves. Phytochemistry 43(4),759-762.
- Mukherjee D (2018) Dynamics of defensive volatile of plant modeling tritrophic interactions. International Journal of Nonlinear Science 25(2):76-86.
- Mondal R., Kesh D. and Mukherjee D.(2019). Role of Induced Volatile Emission Modelling Tritrophic Interaction. Differential Equations And Dynamical Systems.
- Mondal R., Kesh D. and Mukherjee D.(2018). Influence of induced plant volatile and refuge in tritrophic model. Energy Ecology and Environment 3:171–184
- Li M. and Muldowney J. (1996). A geometric approach to global-stability problems. SIAM Journal on Mathematical Analysis 27, 1070-1083.
- Freedman H. I. and Waltman P. (1984). Persistence in models of three interacting predator-prey populations. Mathematical Biosciences, 68 : 213-231.
- Butler G., Freedman H. and Waltman P. (1986). Uniformly persistent systems. Proceedings of the American Mathematical Society 96: 425-430.
- Perko L. (2013). Differential equations and dynamical systems. Springer Science & Business Media, vol. 7.
- Mukherjee D. (2016). The effect of refuge and immigration in a predator–prey system in the presence of a competitor for the prey. Nonlinear Analysis: Real World Applications 31:277–287.
|
---
abstract: 'Spatial indexing of astronomical databases generally uses quadrature methods, which partition the sky into cells to create an index (usually a B-tree) written as a database column. We report the results of a study to compare the performance of two common indexing methods, HTM and HEALPix, on Solaris and Windows database servers installed with PostgreSQL, and a Windows Server installed with MS SQL Server. The indexing was applied to the 2MASS All-Sky Catalog and to the Hubble Source Catalog, which approximate the diversity of catalogs common in astronomy. The study used a dedicated software package in ANSI-C for creating database indices and constructing queries, which will be released in winter 2017. On each server, the study compared indexing performance by submitting 1 million queries at each index level with random sky positions and random cone search radius, which was computed on a logarithmic scale between 1 arcsec and 1 degree, and measuring the time to complete the query and write the output. These simulated queries, intended to model realistic use patterns, were run in a uniform way on many combinations of indexing method and indexing depth. The query times in all simulations are strongly I/O-bound and are linear with number of records returned for large numbers of sources. There are, however, considerable differences between simulations, which reveal that hardware I/O throughput is a more important factor in managing the performance of a DBMS than the choice of indexing scheme. The choice of index itself is relatively unimportant: for comparable index levels, the performance is consistent within the scatter of the timings. At small index levels (large cells; e.g. level 4; cell size 3.7 deg), there is large scatter in the timings because of wide variations in the number of sources found in the cells. At larger index levels, performance improves and scatter decreases, but the improvement at level 8 (14 arcmin) and higher is masked to some extent in the timing scatter caused by the range of query sizes. At very high levels (20; 0.0004 arsec), the granularity of the cells becomes so high that a large number of extraneous empty cells begin to degrade performance. Thus, for the use patterns studied here, the database performance is not critically dependent on the exact choices of index or level.'
author:
- 'G. Bruce Berriman,$^1$ John C. Good,$^1$ Bernie Shiao,$^2$ and Tom Donaldson $^2$'
bibliography:
- 'O4-6.bib'
title: A Study of the Efficiency of Spatial Indexing Methods Applied to Large Astronomical Databases
---
Introduction
============
Spatial indexing methods used in astronomy are usually based on quadrature methods. These methods partition the sky into cells, use the cell numbers to create an index, usually a binary tree (B-tree), and write the index as a database column. What determines the performance of a database index: the choice of index? Its depth? The choice of hardware?
We have therefore compared the performance of two indexing schemes, the Hierarchical Triangular Mesh (HTM, @2000ASPC..216..141K) and the Hierarchical Equal Area iso-Latitude Pixelation of the sphere (HEALPix, @2005ApJ...622..759G), on Solaris, Windows and Red Hat Linux platforms. The indexing was applied to the 2MASS All-Sky Point Source Catalog (470 million records), and the non-merged Hubble Source Catalog (384 million records), installed in PostgreSQL and SQL Server databases. Table 1 summarizes the platforms and databases used. The study was aimed at comparing the performance of indexing schemes, and therefore, neither caching of records in memory nor clustering (ordering) of data within the database was permitted.
[llll]{} Center & Database & OS/Compiler & Server\
STScI & PostgreSQL 9.5 & Win 2012 server OS & 2 processors @3.46 GHz\
& & Cygwin 2.5.2 DLL & 6 cores.\
STScI & SQL Server 2012 &Win 2012 server OS & 2 processors @3.46 GHz\
& & Cygwin 2.5.2 DLL & 6 cores\
IPAC &PostgreSQL 9.3.5 & Solaris 10 & 2 processors @ 2.27 GHz.\
& & & 4 cores\
IPAC &PostgreSQL 9.4 & Red Hat Linux 4.2.1 & 8 processors @ 2.66 GHz\
\
Methodology
===========
The methodology was simple. The 2MASS and HSC catalogs were stripped of all fields except source designation, right ascension and declination. Added to each record were a spatial index value for a specified HTM/HEALPix level between 4 and 20, and the x, y, z spherical 3-vector values. The table was ingested into the database, and a B-tree index was created for the index column. For 1 million queries with random sky positions and cone search radii, selected on a log scale between 1 arcsec and 1 degree, the locations and radii were used to create a list of spatials bins that intersect each region, and the sources in these bins were filtered with the 3-vector values to derive the final source count within the input cone. The computations were performed with a dedicated software package written in ANSI-C and including all support libraries. Release of the code, attached with a BSD 3-clause license, is planned for winter 2017.
What Do The Experiments Show?
=============================
It will be easiest to state the conclusions first, and then refer to the figures. Where comparable, the results are consistent with those of @O07-3_adassxxvi.
- The total query time is dominated by I/O and is linear with the number of records returned for large numbers of sources. For each platform, there is a start-up time that is nearly constant for all experiments. See Figure \[Fig 1\].
- Depth of tesselation and hardware configuration have greater impact on database performance than the choice of tesselation scheme. See Figure \[Fig 1\].
- The optimum indexing level corresponds roughly to when the average search radius is of the order of the cell size. Performance does not improve as the indexing level becomes deeper. See Figures \[Fig 2\] and \[Fig 3\]. At shallow index depths, performance is slowed down in dense regions because large numbers of sources have to be filtered post-query. The numbers of such extraneous sources decline as the index becomes deeper, until near level 20, the performance slows down as the query must search through an ever larger number of empty cells.
- The optimum depth of indexing level depends only weakly on the spatial distribution of queries. See Figure \[Fig 4\].
Funding for the NASA Astronomical Virtual Observatories (NAVO)NAVO is provided by NASA through the Astrophysics Data Curation and Archival Research (ADCAR) program. We thank Mr. Ricardo Ebert and Mr. Scott Terek for making the Solaris server available.
|
LPENSL-TH-12-04\
YITP-SB-12-12
.1in
[**N. Grosjean**]{}, [**J. M. Maillet**]{}, [**G. Niccoli**]{}
**Abstract**
Introduction\[INTR\]
====================
The computation of general matrix elements and correlation functions of local operators is one of the fundamental problems of quantum field theory and statistical mechanics. These objects contain indeed the key dynamical and measurable quantities of the corresponding physical systems, see e.g. [@VanHo54; @VanHo54a; @Kub57; @KubTH85]. In the integrable (low dimensional) situation [@Heise28; @Bet31; @Hul38; @Orb58; @Wal59; @LieSM61; @LieM66], thanks to the existence of powerful algebraic structures related to the Yang-Baxter algebra (see e.g. [Yang67,FadST79,FadT79,Tha81,Bax82L,GauL83,BogIK93L,FadLH96,JimM95L]{} and references therein), significant progress towards their exact determination has been obtained in the last thirty years. Such results concern in particular models solvable by means of the algebraic Bethe ansatz like the $XXZ$ Heisenberg spin chain [@Heise28; @Bet31; @Hul38; @Orb58; @Wal59]. They were first obtained at the free fermion point, namely for the Ising model and the Heisenberg chain (for anisotropy $\Delta$ equal to zero) [Mcc68,McCTW77,SatMJ78,McCPW81]{}. Going beyond such a free fermion case involves a deep use of the Yang-Baxter algebra. After historical attempts for the finite chain in the framework of the Bethe ansatz (see [@BogIK93L] and references therein) but leading in fact to implicit representations in terms of dual fields, explicit representations for form factors and correlation functions were first obtained directly in the infinite chain (in the massive regime) [@JimMMN92; @JimM95L]. The underlying quantum algebra structure was instrumental there together with a few assumptions on the representation of the Hamiltonian and of the local spin operators within the representation theory of this quantum affine algebra. Similar results for the disordered regime were then derived using the assumption that correlation functions should satisfy $q$-deformed KZ equations in close analogy with the massive regime [@JimM96]. The derivation of these results (both for massive and massless regimes and in the presence of a magnetic field) was later obtained in the framework of the algebraic Bethe ansatz, starting from finite size systems, thanks to the resolution of the quantum inverse scattering problem [@KitMT99; @KitMT00; @MaiT00]. Further investigations led to the extension of these results to the non-zero temperature case [@GohKS04; @GohKS05] and also to the case of non-trivial (integrable) boundary conditions [KitKMNST07,KitKMNST08]{}. The computation of physical (two-point) correlation functions required sophisticated summation techniques of the elementary blocks of correlation functions [KitMST02a,KitMST05a,KitMST05b,KitKMST07]{} that ultimately led to the exact computation of their asymptotic behavior [KitKMST09a,KitKMST09b,KozMS11a,KozMS11b]{}. It is also worth mentioning that controlled numerical summation techniques combined with these exact results on form factors led to the determination of dynamical structure factors in very good agreement with actual neutron scattering experiments on magnetic crystals, see e.g. [@CauM05; @CauHM05; @PerSCHMWA06; @PerSCHMWA07]. Another important line of research revealed powerful hidden Grassmann structures that could also be used in the analysis of the conformal limit of these models [BooJMST07,BooJMST09,BooJMS09a,JimMS09,BooJMS09b,JimMS11a,JimMS11b]{}. Finally, it was recently shown how to obtain the asymptotic behavior of the correlation functions for critical systems through their expansion in terms of form factors in the finite volume [@KitKMST11a; @KitKMST11b; @KitKMST12a].
For more sophisticated systems, like lattice integrable models related to higher rank algebras or for integrable relativistic quantum field theories, the present state of the art is slightly less satisfactory despite considerable efforts.
On the one hand, the bootstrap approach has provided great insights into the structure of the exact scattering matrices [Zam77,ZamZ78,ZamZ79,KarTTW77,KarT77,BerKKW78,Kar79]{} and form factors of such integrable massive quantum field theories [KarW78,BerKW79,Smi84,Smi86,Smi92b]{}. Perturbed conformal field theory[^1] (CFT) [Zam87,Zam88,Zam89,Zam86,Car88,Smi89,ResS90,CarM90,Smi90,AlZam91,Mus92,GM96]{} has also been used in this context together with a new understanding of the integrable structure of CFT [@BazLZ96; @BazLZ97; @BazLZ99]. Several important attempts have also been made to use Sklyanin’s separation of variable method (SOV) [@Skl85; @Skl92; @Skl95; @KuzS98; @Skl99; @BabBS96; @BabBS97] in these more complicated algebraic situations and in particular in the case of infinite dimensional representations associated to the quantum fields [@Skl89; @Skl89a; @Smi98; @Luk01; @Bab04; @BytT06; @Tes08; @BytT09]. One should also mention that the new fermionic structures mentioned above in the case of the $XXZ$ lattice model have been used recently to investigate the structure of matrix elements of the sine-Gordon model in the infinite volume limit [JimMS11a,JimMS11b]{}.
On the other hand, although these advances lead to invaluable informations on these theories, a direct computation of the form factors and correlation functions starting from the description of their local fields and using the given Hamiltonian dynamics is still missing. The main reason is that although these approaches succeeded to describe more and more efficiently both the space of states of such models and the space of their form factors, the connection between the set of operators used to reach this goal and the local operators of the theory one is interested in has not been found yet. Consequently, the matrix elements and correlation functions of these local fields can at best be identified through indirect arguments. In other words, contrary to the above mentioned finite dimensional cases, the solution of the quantum inverse scattering problem is in general not known in the field theory situation; moreover, it appears very often in these more complicated models (either on the lattice or in the continuum) that the usual algebraic Bethe ansatz is no longer applicable due to the lack of a proper reference state, and that the SOV framework [@Skl85; @Skl92; @Skl95] has to be considered instead. Although this beautiful method is quite general and powerful to describe the spectrum of these models, the problem of reconstructing the local operators of the corresponding theories in terms of the separate variables is in general still open (see however [@Bab04]).
To explain in more detail the main issues of this problem, let us consider in particular the sine-Gordon model which will be the main example analyzed in this article. As we just recall, integrable massive field theories in infinite volume can be solved *on-shell*[^2] through the determination of their exact S-matrices [Zam77,ZamZ78,ZamZ79,KarTTW77,KarT77,BerKKW78,Kar79]{} which completely characterize the particle dynamics. In this particle formulation of the theory a direct access to the local fields is missing and any information about them needs to be extracted from the particle dynamics. The monodromy properties and the singularity structure provide a set of functional equations [@KarW78; @BerKW79; @Smi84; @Smi86; @Smi92b] for the form factors of the local fields on asymptotic particle states. These form factor equations are uniquely fixed by the knowledge of the exact S-matrix and the space of their solutions is expected to coincide with that of the local operators of the theory; many results are known on the form factors of local fields, see for example [@Smi89; @ResS90; @CarM90; @Smi90], [FriMS90,FMS93-2,KouM93,AhnDM93,MusS94,Kub94,DelfM95,DelfSC96,BabFKZ99,BabFK02,BabFK02-1]{} and references therein for some literature related to sine-Gordon model. However, it is worth pointing out that the form factor equations only contain information on symmetry data of the fields (like charges and spin) and after fixing them to some values, we are still left with an infinite dimensional space of form factor solutions which should correspond to the infinite dimensional space of local fields sharing these data. There is a large literature dedicated to the longstanding problem of the identification of the local fields in the scattering formulation of quantum field theories. Different methods have been introduced to address it; one important line of research is related to the description of massive integrable quantum field theories as (super-renormalizable) perturbations of conformal field theories by relevant local fields [Zam87,Zam88,Zam89,Zam86,Car88,Smi89,ResS90,CarM90,Smi90,AlZam91,Mus92,GM96]{}. This characterization has led to the expectation that the perturbations do not change the structure of the local fields in this way leading to the attempt to classify the local field content of massive theories by that of the corresponding ultraviolet conformal field theories. The latter issue was first addressed in [@CarM90] in the simplest massive free theory, namely the Ising model. There a conjecture was introduced defining a correspondence between mild asymptotic behavior at high energy of form factors and chiral local fields. Such a conjecture was justified showing the isomorphism of the space of chiral local fields in the massive and conformal models. The extension of the chiral isomorphism to interacting massive integrable theories was done in [@Kub95] for several massive deformations of minimal conformal models, in [@Smi96] for the sine-Gordon model and in [@JMT03] for all its reductions to unitary minimal models[^3]. The problem to extend the classification also to non-chiral local fields was analyzed in a series of works [@DN05-1; @DN05-2; @DN06; @DN08]; in particular, for the massive Lee-Yang model, the first proof that the operator space determined by the particle dynamics coincides with that prescribed by conformal symmetry at criticality was given in [@DN08]. While these are indubitably important results on the global structure of the operator space in massive theories it is worth pointing out that they do not lead to the full identification of particular local fields[^4]. In [@BabBS96] a criterion has been introduced based on the quasi-classical characterization of the local fields; it has been fully described in the special cases of the restricted sine-Gordon model at the reflectionless points for chiral fields and verified on the basis of counting arguments [@BabBS97].
This makes clear that, in the S-matrix formulation of massive quantum integrable theories in infinite volume, the main open problem remains the absence of a direct reconstruction of the local fields. One of our motivation for the present work is to define an exact setup to solve this problem for one of the most paradigmatic integrable quantum field theory, namely the sine-Gordon model. As a first step we will consider the discretized version of the sine-Gordon field theory on a finite lattice. Moreover we will simplify its dynamics by considering the case for which the representation space of the exponents of the field and conjugated momentum is an arbitrary finite dimensional cyclic representation of the quantum algebra, namely the case where the parameter q = $e^{i\text{h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}}}$ is a root of unity. This case is interesting in its own right [@NicT10; @Nic10; @Nic11] and we also believe that the treatment of the full continuum theory could then be reached by taking the needed limits; at least, we expect to be able to identify the main ingredients and structures necessary to reach this goal and learn enough to extend it to other models on the lattice or in the continuum.
It is worth to describe schematically the microscopic approach that we intend to follow to solve integrable quantum field theories by the complete characterization of their spectrum and of their dynamics.
The first goal is the solution of the spectral problem, for the lattice and the continuum theories:\
i) Solution of the spectral problem for the integrable lattice regularization by the construction of the eigenstates and eigenvalues of the transfer matrix using the SOV method.\
ii) Reformulation of the spectrum in terms of nonlinear integral equations (of thermodynamic Bethe ansatz type) and definition of finite volume quantum field theories in the continuum limit.\
iii) Derivation of the S matrix and particle description of the spectrum in the infinite volume (IR) limit.\
iv) Derivation of the renormalization group fixed point conformal spectrum in the UV limit.
The second goal is the solution of the dynamics along the following steps:\
i) Determinant formulae for the scalar product of states in particular involving transfer matrix eigenstates.\
ii) Reconstruction of the local operators in terms of the quantum separate variables.\
iii) Computation of matrix elements of local operators in the eigenstates basis of the transfer matrix.\
iv) Thermodynamic behavior of the above quantities and computation of the physical correlation functions.
In this paper, we develop partly this program for the lattice regularization of the sine-Gordon model while the completion of it taking into account the required limits to the continuum theory in finite and infinite volume case will be addressed in future publications. Let us remark that the possibility to apply the SOV method for the discretized version of the sine-Gordon model in finite dimensional cyclic representations was demonstrated recently in [@NicT10], hence opening the question of the computations of matrix elements of local fields in a completely controlled (finite dimensional) setting. The first result of the present paper is to show that the scalar products of states can be computed in this case as finite dimensional determinants involving in particular, for eigenstates of the Hamiltonian, the corresponding eigenvalues of the Baxter Q-operator; orthogonality of different eigenstates can be proven directly from these expressions. Further, we will show that the lattice discretization of the local fields of the sine-Gordon model can be reconstructed explicitly in terms of the separate variables. These two ingredients finally lead to the determination of the matrix elements of the exponential of the local fields of the model between arbitrary eigenstates of the Hamiltonian.
This article is organized as follows. In Section 2 we define the sine-Gordon model on a finite lattice in the cyclic representations and recall the main ingredients of the SOV method in that context. In Section 3 we show how to compute the scalar products of states in the SOV representations. The next section is devoted to the reconstruction of the local fields in terms of the separate variables. In Section 5 we use these results to compute the form factors of the local fields in terms of finite size determinants. In the last section we comment on these results and compare them to the existing literature.
The sine-Gordon model
=====================
We use this section to recall the main results derived in [@NicT10; @Nic10] on the spectrum description of the lattice sine-Gordon model.
Definitions
-----------
### Classical model
The classical sine-Gordon model can be characterized by the following Hamiltonian density:$$\text{H}_{SG}\equiv \left( \partial _{x}\phi \right) ^{2}+\Pi ^{2}+8\pi \mu
\cos 2\beta \phi$$where the field $\phi (x,t)$ is defined for $(x,t)\in \lbrack 0,R]\times \
\mathbb{R}$ with periodic boundary conditions $\phi (x+R,t)=\phi (x,t)$. The dynamics of the model in the Hamiltonian formalism is defined in terms of variables $\phi (x,t),$ $\Pi (x,t)$ with the following Poisson brackets:$$\{\Pi (x,t),\phi (y,t)\}=2\pi \delta (x-y).$$The classical integrability of the sine-Gordon model is assured thanks to the representation of the equation of motion by a zerocurvature condition:$$\lbrack \partial _{t}-V(x,t;\lambda ),\partial _{x}-U(x,t;\lambda )]=0,$$where, by using the Pauli matrices, we have defined:$$\begin{aligned}
U& =\text{k}_{1}\sigma _{1}\cos \beta \phi +\text{k}_{2}\sigma _{2}\sin
\beta \phi -\text{k}_{3}\sigma _{3}\Pi , \\
V& =-\text{k}_{2}\sigma _{1}\cos \beta \phi -\text{k}_{1}\sigma _{2}\sin
\beta \phi -\text{k}_{3}\sigma _{3}\partial _{x}\phi , \\
\text{k}_{1}& =i\beta \left( \pi \mu \right) ^{1/2}(\lambda -\lambda ^{-1}),\text{ \ \ \ k}_{2}=i\beta \left( \pi \mu \right) ^{1/2}(\lambda +\lambda
^{-1}),\text{ k}_{3}\equiv i\frac{\beta }{2}.\end{aligned}$$
### Quantum lattice regularization
In order to regularize the ultraviolet divergences that arise in the quantization of the model a lattice discretization is introduced. The field variables are discretized according to the standard recipe: $$\phi _{n}\equiv \phi (n\Delta ),\text{ \ \ \ }\Pi _{n}\equiv \Delta \Pi
(n\Delta ),$$where $\Delta =R/N$ is the lattice spacing. Then, the canonical quantization is defined by imposing that $\phi _{n}$ and $\Pi _{n}$ are self-adjoint operators satisfying the commutation relations:$$\lbrack \phi _{n},\Pi _{m}]=2\pi i\delta _{n,m}. \label{canonical-CR}$$The quantum lattice regularization of the sine-Gordon model[^5] that we use here goes back to [FadST79,IzeK82,TarTF83]{} and is related to formulations which have more recently been studied in [@Fad94; @FadV94]. Here, the basic operators are the exponentials of the fields and of the conjugate momenta: $$\mathsf{v}_{n}\equiv e^{-i\beta \phi _{n}},\text{ \ \ }\mathsf{u}_{n}\equiv
e^{i\beta \Pi _{n}/2}.$$Then, the commutation relations (\[canonical-CR\]) imply that the couples of unitary operators ($\mathsf{u}_{n}$,$\mathsf{v}_{n}$) generate $\mathsf{N}
$ independent Weyl algebras $\mathcal{W}_{n}$:$$\mathsf{u}_{n}\mathsf{v}_{m}=q^{\delta _{nm}}\mathsf{v}_{m}\mathsf{u}_{n}\,,\text{ \ \ }q\equiv e^{-i\pi \beta ^{2}}, \label{Weyl}$$for any $n\in \{1,...,\mathsf{N}\}$. In terms of these basic operators the quantum lattice sine-Gordon model can be characterized by the following Lax matrix[^6]: $$\mathsf{L}_{n}(\lambda )=\kappa _{n}\left(
\begin{array}{cc}
\mathsf{u}_{n}(q^{-1/2}\mathsf{v}_{n}\kappa _{n}+q^{1/2}\mathsf{v}_{n}^{-1}\kappa _{n}^{-1}) & (\lambda _{n}\mathsf{v}_{n}-(\mathsf{v}_{n}\lambda _{n})^{-1})/i \\
(\lambda _{n}/\mathsf{v}_{n}-\mathsf{v}_{n}/\lambda _{n})/i & \mathsf{u}_{n}^{-1}(q^{1/2}\mathsf{v}_{n}\kappa _{n}^{-1}+q^{-1/2}\mathsf{v}_{n}^{-1}\kappa _{n})\end{array}\right) \label{Lax}$$where $\lambda _{n}\equiv \lambda /\xi _{n}$ for any $n\in \{1,...,\mathsf{N}\}$ with $\xi _{n}$ and $\kappa _{n}$ parameters of the model.
The monodromy matrix $\mathsf{M}(\lambda )$ is defined by:$$\mathsf{M}(\lambda )\equiv \left(
\begin{array}{cc}
\mathsf{A}(\lambda ) & \mathsf{B}(\lambda ) \\
\mathsf{C}(\lambda ) & \mathsf{D}(\lambda )\end{array}\right) \equiv \mathsf{L}_{\mathsf{N}}(\lambda )\cdots \mathsf{L}_{1}(\lambda ),$$and satisfies the quadratic commutation relations:$$R(\lambda /\mu )\,(\mathsf{M}(\lambda )\otimes 1)\,(1\otimes \mathsf{M}(\mu
))\,=\,(1\otimes \mathsf{M}(\mu ))\,(\mathsf{M}(\lambda )\otimes 1)R(\lambda
/\mu )\,, \label{YBA}$$with the 6-vertex $R$-matrix: $$R(\lambda )=\left(
\begin{array}{cccc}
q\lambda -q^{-1}\lambda ^{-1} & & & \\[-1mm]
& \lambda -\lambda ^{-1} & q-q^{-1} & \\[-1mm]
& q-q^{-1} & \lambda -\lambda ^{-1} & \\[-1mm]
& & & q\lambda -q^{-1}\lambda ^{-1}\end{array}\right) \,. \label{Rsg}$$The elements of $\mathsf{M}(\lambda )$ are the generators of the so-called Yang-Baxter algebra and the commutation relations (\[YBA\]) imply the mutual commutativity of the elements of the following one-parameter family of operators:$$\mathsf{T}(\lambda )\equiv \text{tr}_{\mathbb{C}^{2}}\mathsf{M}(\lambda ),
\label{Tdef}$$the so-called transfer matrix. In the case of a lattice with $\mathsf{N}$ even quantum sites, we can also introduce the operator:$$\Theta \equiv \prod_{n=1}^{\mathsf{N}}\mathsf{v}_{n}^{(-1)^{n}},
\label{topological-charge}$$which plays the role of a *grading operator* in the Yang-Baxter algebra:
**Proposition 6 of [@NicT10]*** For the even chain, the charge $\Theta $ commutes with the transfer matrix and satisfies the following commutation relations with the generators of the Yang-Baxter algebra:$$\begin{aligned}
\Theta \mathsf{C}(\lambda ) &=&q\mathsf{C}(\lambda )\Theta \text{, \ \ \ }[\mathsf{A}(\lambda ),\Theta ]=0, \\
\mathsf{B}(\lambda )\Theta &=&q\Theta \mathsf{B}(\lambda ),\text{ \ \ }[\mathsf{D}(\lambda ),\Theta ]=0.\end{aligned}$$Moreover, the $\Theta $-charge allows to express the asymptotics of the leading generators $\mathsf{A}(\lambda )$ and $\mathsf{D}(\lambda )$:$$\lim_{\log \lambda \rightarrow \mp \infty }\lambda ^{\pm \mathsf{N}}\mathsf{A}(\lambda )=\left( \prod_{a=1}^{\mathsf{N}}\frac{\kappa _{a}\xi _{a}^{\pm 1}}{i}\right) \Theta ^{\mp 1},\text{ \ \ \ \ \ }\lim_{\log \lambda \rightarrow
\mp \infty }\lambda ^{\pm \mathsf{N}}\mathsf{D}(\lambda )=\left(
\prod_{a=1}^{\mathsf{N}}\frac{\kappa _{a}\xi _{a}^{\pm 1}}{i}\right) \Theta
^{\pm 1}$$and hence the asymptotic behavior of the transfer matrix:$$\lim_{\log \lambda \rightarrow \mp \infty }\lambda ^{\pm \mathsf{N}}\mathsf{T}(\lambda )=\left( \prod_{a=1}^{\mathsf{N}}\frac{\kappa _{a}\xi _{a}^{\pm 1}}{i}\right) \left( \Theta +\Theta ^{-1}\right) . \label{asymptotics-T}$$*
Cyclic representations\[Def-cyclic-rep\]
----------------------------------------
In the present article, we will consider representations where both $\mathsf{v}_{m}$ and $\mathsf{u}_{n}\,$ have discrete spectra; in particular, we will restrict our attention to the case in which $q$ is a root of unity:$$\beta ^{2}\,=\,\frac{p^{\prime }}{p}\,,\qquad p,p^{\prime }\in \mathbb{Z}^{>0}\,, \label{beta}$$with $p$ odd and $p^{\prime }$ even so that $q^{p}=1$. The condition ([beta]{}) implies that the powers $p$ of the generators $\mathsf{u}_{n}$ and $\mathsf{v}_{n}$ are central elements of each Weyl algebra $\mathcal{W}_{n}$. In this case, we can associate a $p$-dimensional linear space R$_{n}$ to any site $n$ of the chain and we can define on it a cyclic representation of $\mathcal{W}_{n}$ as follows:$$\mathsf{v}_{n}|k_{n}\rangle =v_{n}q^{k_{n}}|k_{n}\rangle ,\text{ \ }\mathsf{u}_{n}|k_{n}\rangle =u_{n}|k_{n}-1\rangle ,\text{\ \ \ \ }\forall k_{n}\in
\{0,...,p-1\}, \label{v-eigenbasis}$$with the cyclicity condition, $$|k_{n}+p\rangle =|k_{n}\rangle .$$The vectors $|k_{n}\rangle $ define a $\mathsf{v}_{n}$-eigenbasis of the local space R$_{n}$ and the parameters $v_{n}$ and $u_{n}$ characterize the central elements of the $\mathcal{W}_{n}$-representation:$$\mathsf{v}_{n}^{p}=v_{n}^{p},\text{ \ }\mathsf{u}_{n}^{p}=u_{n}^{p}.$$Here, we require the unitarity of the generators $\mathsf{u}_{n}$ and $\mathsf{v}_{n}$ and the orthonormality of the elements of the $\mathsf{v}_{n} $-eigenbasis w.r.t. the scalar product introduced in R$_{n}$.
Let L$_{n}$ be the linear space dual of R$_{n}$ and let $\langle k_{n}|$ be the elements of the dual basis defined by: $$\langle k_{n}|k_{n}^{\prime }\rangle =(|k_{n}\rangle ,|k_{n}^{\prime
}\rangle )\equiv \delta _{k_{n},k_{n}^{\prime }}\text{ \ \ }\forall
k_{n},k_{n}^{\prime }\in \{0,...,p-1\}.$$From the unitarity of the generators $\mathsf{u}_{n}$ and $\mathsf{v}_{n}$, the covectors $\langle k_{n}|$ define a $\mathsf{v}_{n}$-eigenbasis in the dual space L$_{n}$ and the following left representation of Weyl algebra $\mathcal{W}_{n}$ :$$\langle k_{n}|\mathsf{v}_{n}=v_{n}q^{k_{n}}\langle k_{n}|,\text{ \ }\langle
k_{n}|\mathsf{u}_{n}=u_{n}\langle k_{n}+1|,\text{\ \ \ \ }\forall k_{n}\in
\{0,...,p-1\},$$with cyclicity condition: $$\langle k_{n}|=\langle k_{n}+p|.$$
In the *left* and *right* linear spaces:$$\mathcal{L}_{\mathsf{N}}\equiv \otimes _{n=1}^{\mathsf{N}}\text{L}_{n},\text{
\ \ \ \ }\mathcal{R}_{\mathsf{N}}\equiv \otimes _{n=1}^{\mathsf{N}}\text{R}_{n},$$the representations of the Weyl algebras $\mathcal{W}_{n}$ induce cyclic left and right representations of dimension $p^{\mathsf{N}}$ of the monodromy matrix elements, i.e. cyclic representations of the Yang-Baxter algebra. Such representations are characterized by the $4\mathsf{N}$ parameters $\kappa =(\kappa _{1},\dots ,\kappa _{\mathsf{N}})$, $\xi =(\xi
_{1},\dots ,\xi _{\mathsf{N}})$, $u=(u_{1},\dots ,u_{\mathsf{N}})$ and $v=(v_{1},\dots ,v_{\mathsf{N}})$. The unitarity of the Weyl algebra generators $\mathsf{u}_{n}$ and $\mathsf{v}_{n}$ implies that the parameters $v$ and $u$ are pure phases and the following Hermitian conjugation properties of the generators of Yang-Baxter algebra hold:
**Lemma 1 of [@Nic10]** *If the parameters of the representation* $\kappa _{n}^{2}$* and* $\xi _{n}^{2}$* are real for any* $n=1,\dots ,\mathsf{N}$* and satisfy the constrains*$$\varepsilon \equiv -(\kappa _{n}\xi _{n})/\left( \kappa _{n}^{\ast }\xi
_{n}^{\ast }\right) \,\,\,\text{ \textit{is uniform along the chain,}}
\label{cond-T-Normality}$$
*then it holds:* $$\mathsf{M}(\lambda )^{\dagger }\equiv \left(
\begin{array}{cc}
\mathsf{A}^{\dagger }(\lambda ) & \mathsf{B}^{\dagger }(\lambda ) \\
\mathsf{C}^{\dagger }(\lambda ) & \mathsf{D}^{\dagger }(\lambda )\end{array}\right) =\left(
\begin{array}{cc}
\mathsf{D}(\lambda ^{\ast }) & \mathsf{C}(\varepsilon \lambda ^{\ast }) \\
\mathsf{B}(\varepsilon \lambda ^{\ast }) & \mathsf{A}(\lambda ^{\ast })\end{array}\right) , \label{Hermit-Monodromy}$$
*which, in particular, implies the self-adjointness of the transfer matrix* $\mathsf{T}(\lambda )$* for real* $\lambda $*.*
Let us define the average value $\mathcal{O}$ of the elements of the monodromy matrix $\mathsf{M}(\lambda )$ as $$\mathcal{O}(\Lambda )\,=\,\prod_{k=1}^{p}\mathsf{O}(q^{k}\lambda )\,,\qquad
\Lambda \,=\,\lambda ^{p}, \label{avdef}$$where $\mathsf{O}$ can be $\mathsf{A}$, $\mathsf{B}$, $\mathsf{C}$ or $\mathsf{D}$ and we have to remark that the commutativity of each family of operators $\mathsf{A}(\lambda )$, $\mathsf{B}(\lambda )$, $\mathsf{C}(\lambda )$ and $\mathsf{D}(\lambda )$ implies that the corresponding average values are functions of $\Lambda $, so that $\mathcal{B}(\Lambda )$, $\mathcal{C}(\Lambda )$ are Laurent polynomials of degree $[\mathsf{N}]$ while $\mathcal{A}(\Lambda )$, $\mathcal{D}(\Lambda )$ are Laurent polynomials of degree $\mathsf{\bar{N}}$ in $\Lambda $, where we are using the notations: $$\text{\ }[\mathsf{N}]\equiv \mathsf{N}-\mathtt{e}_{\mathsf{N}},\text{ \ \ }\bar{\mathsf{N}}\equiv \mathsf{N}+\mathtt{e}_{\mathsf{N}}-1\text{, \ \ \ \ }\mathtt{e}_{\mathsf{N}}\equiv \left\{
\begin{array}{l}
1\text{ for }\mathsf{N}\text{ even,} \\
0\text{ for }\mathsf{N}\text{ odd.}\end{array}\right.$$
\[central\]
- The average values $\mathcal{A}(\Lambda )$, $\mathcal{B}(\Lambda )$, $\mathcal{C}(\Lambda )$, $\mathcal{D}(\Lambda )$ of the monodromy matrix elements are central elements. Furthermore, they satisfy the following relations: $$(\mathcal{A}(\Lambda ))^{\ast }\equiv \mathcal{D}(\Lambda ^{\ast }),\ \ \ \
\ (\mathcal{B}(\Lambda ))^{\ast }\equiv \epsilon \mathcal{C}(\Lambda ^{\ast
}), \label{H-cj-A-D}$$under complex conjugation in the case of self-adjoint representations.
- Let$$\mathcal{M}(\Lambda )\,\equiv \,\left(
\begin{array}{cc}
\mathcal{A}(\Lambda ) & \mathcal{B}(\Lambda ) \\
\mathcal{C}(\Lambda ) & \mathcal{D}(\Lambda )\end{array}\right)$$be the 2$\times $2 matrix whose elements are the average values of the elements of the monodromy matrix $\mathsf{M}(\lambda )$, then we have, $$\mathcal{M}(\Lambda )\,=\,\mathcal{L}_{\mathsf{N}}(\Lambda )\,\mathcal{L}_{\mathsf{N}-1}(\Lambda )\,\dots \,\mathcal{L}_{1}(\Lambda )\,,$$where $$\mathcal{L}_{n}(\Lambda )\equiv \left(
\begin{array}{cc}
q^{p/2}u_{n}^{p}(\kappa _{2n}^{p}v_{n}^{p}+v_{n}^{-p}) & \kappa
_{n}^{p}(\Lambda v_{n}^{p}/\xi _{n}^{p}-\xi _{n}^{p}/\Lambda v_{n}^{p})/i^{p}
\\
\kappa _{n}^{p}(\Lambda /v_{n}^{p}\xi _{n}^{p}-\xi _{n}^{p}v_{n}^{p}/\Lambda
)/i^{p} & q^{p/2} u_{n}^{-p}(\kappa _{2n}^{p}v_{n}^{-p}+v_{n}^{p})\end{array}\right) , \label{Average-L}$$is the 2$\times $2 matrix whose elements are the average values of the elements of the Lax matrix $\mathsf{L}_{n}(\lambda )$.
A similar statement was first proven in [@Ta]. In the present paper, we will be mainly restricted to the case:$$u_{n}=1,\,\,\,v_{n}=1\text{ \ \ for \ }n=1,\dots ,\mathsf{N}.
\label{Special-rep}$$In these representations it holds:
\[zeros-B\] The power 2p of the zeros of $(\lambda )$ are real numbers with possible complex conjugate couples.
The previous proposition and the equality $$\mathcal{C}(\Lambda )=\mathcal{B}(\Lambda ),$$which holds for the sine-Gordon representations with $\mathsf{u}_{n}^{p}=\mathsf{v}_{n}^{p}=1$, as proven in [@NicT10], imply that$$\left( \mathcal{B}(\Lambda )\right) ^{\ast }=\epsilon\mathcal{B}(\Lambda
^{\ast }),$$and so the statement of the lemma.
SOV-representations of the Yang-Baxter algebra {#SOV-Left}
----------------------------------------------
According to Sklyanin’s method [@Skl85; @Skl92; @Skl95], a separation of variables (SOV) representation for the spectral problem of the transfer matrix $\mathsf{T}(\lambda )$ is defined as a representation where the commutative family of operators $\mathsf{B}(\lambda )$ is diagonal. In [NicT10]{}, the following theorem has been shown:
**Theorem 1 of [@NicT10]** *For almost all the values of the parameters* $\kappa $* and* $\xi $* of the representation, there exists a SOV representation for the sine-Gordon model, i.e.* $\mathsf{B}(\lambda )$* is diagonalizable and with simple spectrum.*
The proof of this has been given by a recursive construction of the left cyclic SOV-representations for the sine-Gordon model. Let us recall here the left SOV-representations of the generators of the Yang-Baxter algebra. The Proposition \[central\] fixes the average values of $\mathsf{B}(\lambda )$:$$\mathcal{B}(\Lambda )=\left( \prod_{n=1}^{\mathsf{N}}\frac{\kappa _{n}}{i}\right) ^{p}Z_{\mathsf{N}}^{\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}(\Lambda /Z_{a}-Z_{a}/\Lambda )$$in terms of the parameters of the representations. Note that the simplicity of the spectrum of $\mathsf{B}(\lambda )$ is equivalent to the requirement $Z_{a}\neq Z_{b}$ for any $a\neq b\in \{1,\dots ,\mathsf{N}-\mathtt{e}_{\mathsf{N}}\}$. Moreover, the reality condition of the polynomial $\mathcal{B}(\Lambda )$, proven in Lemma \[zeros-B\], implies that we can chose a $\mathsf{N}$-tupla $\{\eta _{1}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\}$ of $p$-roots of $\{Z_{1},...,Z_{\mathsf{N}}\}$ which satisfy the requirements:$$\left( \eta _{a}^{(0)}\right) ^{2}\in \mathcal{R}\text{ or }\left( \eta
_{a}^{(0)}\right) ^{2}\notin \mathcal{R}\rightarrow \text{ }\exists b\in
\{1,...,\mathsf{N}\}\backslash a:\left( \eta _{a}^{(0)}\right) ^{2}=\left(
\left( \eta _{b}^{(0)}\right) ^{2}\right) ^{\ast }\text{.}$$Let $\langle \eta _{_{1}}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|\in \mathcal{L}_{\mathsf{N}}$ be the generic element of a basis of left eigenstates of $\mathsf{B}(\lambda )$:$$\langle \mathbf{k}|\mathsf{B}(\lambda )=\mathsf{b}_{\mathbf{k}}(\lambda
)\langle \mathbf{k}|\text{ \ with \ }\mathsf{b}_{\mathbf{k}}(\lambda )\equiv
\left( \prod_{n=1}^{\mathsf{N}}\frac{\kappa _{n}}{i}\right) \eta _{\mathsf{N}}^{(k_{\mathsf{N}})\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}(\lambda /\eta _{a}^{(k_{a})}-\eta _{a}^{(k_{a})}/\lambda ),
\label{SOV-B-L}$$ where we have used the notation$$\langle \mathbf{k}|\equiv \langle \eta _{_{1}}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|,\text{ \ \ }\,\eta _{a}^{(k_{a})}\equiv q^{k_{a}}\eta
_{a}^{(0)}\text{ }\forall a\in \{1,\dots ,\mathsf{N}\}\,,\text{ \ \ }\mathbf{k}\equiv (k_{1},\dots ,k_{\mathsf{N}})\in \mathbb{Z}_{p}^{\mathsf{N}}\,.$$Then the left action of the other Yang-Baxter generators reads:$$\begin{aligned}
\langle \mathbf{k}|\mathsf{A}(\lambda )& =\mathtt{e}_{\mathsf{N}}\frac{\mathsf{b}_{\mathbf{k}}(\lambda )}{\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}}(\frac{\lambda }{\eta _{\mathbf{k,}\mathsf{A}}}\langle \mathbf{k}|\mathsf{T}_{\mathsf{N}}^{-}-\frac{\eta _{\mathbf{k,}\mathsf{A}}}{\lambda }\langle
\mathbf{k}|\mathsf{T}_{\mathsf{N}}^{+})+\sum_{a=1}^{[\mathsf{N}]}\prod_{b\neq a}\frac{(\frac{\lambda }{\eta _{b}^{(k_{b})}}-\frac{\eta
_{b}^{(k_{b})}}{\lambda })}{(\frac{\eta _{a}^{(k_{a})}}{\eta _{b}^{(k_{b})}}-\frac{\eta _{b}^{(k_{b})}}{\eta _{a}^{(k_{a})}})}a(\eta
_{a}^{(k_{a})})\langle \mathbf{k}|\mathsf{T}_{a}^{-}, \label{SOV-A-L} \\
& \notag \\
\langle \mathbf{k}|\mathsf{D}(\lambda )& =\mathtt{e}_{\mathsf{N}}\frac{\mathsf{b}_{\mathbf{k}}(\lambda )}{\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}}(\frac{\lambda }{\eta _{\mathbf{k,}\mathsf{D}}}\langle \mathbf{k}|\mathsf{T}_{\mathsf{N}}^{+}-\frac{\eta _{\mathbf{k,}\mathsf{D}}}{\lambda }\langle
\mathbf{k}|\mathsf{T}_{\mathsf{N}}^{-})+\sum_{a=1}^{[\mathsf{N}]}\prod_{b\neq a}\frac{(\frac{\lambda }{\eta _{b}^{(k_{b})}}-\frac{\eta
_{b}^{(k_{b})}}{\lambda })}{(\frac{\eta _{a}^{(k_{a})}}{\eta _{b}^{(k_{b})}}-\frac{\eta _{b}^{(k_{b})}}{\eta _{a}^{(k_{a})}})}d(\eta
_{a}^{(k_{a})})\langle \mathbf{k}|\mathsf{T}_{a}^{+}, \label{SOV-D-L}\end{aligned}$$where $\eta _{\mathbf{k,}\mathsf{A}}$ and $\eta _{\mathbf{k,}\mathsf{D}}$ are defined by:$$\eta _{\mathbf{k,}\mathsf{A}}\equiv \eta _{\mathbf{k,}\mathsf{D}}\equiv
\frac{\prod_{n=1}^{\mathsf{N}}\xi _{n}}{\prod_{n=1}^{\mathsf{N-1}}\eta
_{n}^{(k_{n})}}, \label{eta_A}$$and the shift operators $\mathsf{T}_{n}^{\pm }$ are defined by: $$\langle \eta _{_{1}}^{(k_{1})},...,\eta _{n}^{(k_{n})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|\mathsf{T}_{n}^{\pm }\equiv \langle \eta
_{_{1}}^{(k_{1})},...,\eta _{n}^{(k_{n}\pm 1)},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|.$$The operator family $\mathsf{C}(\lambda )$ is uniquely[^7] defined by the quantum determinant relation:$$\mathrm{det_{q}}\mathsf{M}(\lambda )\,\equiv \,\mathsf{A}(\lambda )\mathsf{D}(q^{-1}\lambda )-\mathsf{B}(\lambda )\mathsf{C}(q^{-1}\lambda ),
\label{qdetdef}$$where $\mathrm{det_{q}}\mathsf{M}(\lambda )$ is a central element[^8] of the Yang-Baxter algebra (\[YBA\]) which reads:$$\mathrm{det_{q}}\mathsf{M}(\lambda )\equiv \prod_{n=1}^{\mathsf{N}}\kappa
_{n}^{2}(\lambda /\mu _{n,+}-\mu _{n,+}/\lambda )(\lambda /\mu _{n,-}-\mu
_{n,-}/\lambda ), \label{q-det-f}$$where $\mu _{n,\pm }\equiv i\kappa _{n}^{\pm 1}q^{1/2}\xi _{n}$. In the representations which satisfy (\[Special-rep\]) the coefficients of the SOV-representations can be fixed by introducing the following Laurent polynomials:$$a(\lambda )=(-i)^{\mathsf{N}}\prod_{n=1}^{\mathsf{N}}\frac{\kappa _{n}}{\lambda _{n}}(1+iq^{-1/2}\lambda _{n}\kappa _{n})(1+iq^{-1/2}\lambda
_{n}/\kappa _{n}),\text{ \ \ }d(\lambda )=q^{\mathsf{N}}a(-\lambda q),
\label{a&d-def}$$Note that these coefficients are related to the quantum determinant by:$$\mathrm{det_{q}}\mathsf{M}(\lambda )=a(\lambda )d(\lambda /q).
\label{q-det-ad}$$Note that for the choice of the parameters $\{\kappa _{n}\}\in i\mathbb{R}^{\mathsf{N}}$ and $\{\xi _{n}\}\in \mathbb{R}^{\mathsf{N}}$ the numbers <span style="font-variant:small-caps;">k</span>$\equiv \prod_{n=1}^{\mathsf{N}}\kappa _{n}/i$ and $\mu _{n,\pm }^{p}$ are real.
SOV-characterization of the $\mathsf{T}$-spectrum
-------------------------------------------------
Let us denote by $\Sigma _{\mathsf{T}}$ the set of the eigenvalue functions $t(\lambda )$ of the transfer matrix $\mathsf{T}(\lambda )$. From the definitions (\[Lax\]) and (\[Tdef\]), $\Sigma _{\mathsf{T}}$ is a subset of $\mathbb{R}[\lambda ^{2},\lambda ^{-2}]_{\bar{\mathsf{N}}/2}$ where $\mathbb{R}[x,x^{-1}]_{\mathsf{M}}$ denotes the linear space over the field $\mathbb{R}$ of the *real* Laurent polynomials of degree $\mathsf{M}$ in the variable $x$: $$(f(x))^{\ast }=f(x^{\ast })\,\,\,\forall x\in \mathbb{C} \text{ \ \ with \ }f(x)\in \mathbb{R}[x,x^{-1}]_{\mathsf{M}}.$$
Note that in the case of $\mathsf{N}$ even, the $\Theta $-charge naturally induces the grading $\Sigma _{\mathsf{T}}=\bigcup_{k=0}^{(p-1)/2}\Sigma _{\mathsf{T}}^{k}$, where: $$\Sigma _{\mathsf{T}}^{k}\equiv \left\{ t(\lambda )\in \Sigma _{\mathsf{T}}:\lim_{\log \lambda \rightarrow \mp \infty }\lambda ^{\pm \mathsf{N}}t(\lambda )=\left( \prod_{a=1}^{\mathsf{N}}\frac{\kappa _{a}\xi _{a}^{\pm 1}}{i}\right) (q^{k}+q^{-k})\right\} . \label{asymptotics-t}$$This simply follows from the asymptotics of $\mathsf{T}(\lambda )$ and from its commutativity with $\Theta $.
In the SOV-representations the spectral problem for $\mathsf{T}(\lambda )$ is reduced to the following discrete system of Baxter-like equations in the wave-function $\Psi _{t}(\mathbf{k})\equiv \langle \mathbf{k}|\,t\,\rangle $ of a $\mathsf{T}$-eigenstate $|\,t\,\rangle $: $$t(\eta _{r}^{(k_{r})})\Psi _{t}(\mathbf{k})=a(\eta _{r}^{(k_{r})})\Psi _{t}(\mathsf{T}_{r}^{-}(\mathbf{k}))+d(\eta _{r}^{(k_{r})})\Psi _{t}(\mathsf{T}_{r}^{+}(\mathbf{k}))\,\qquad \text{ }\forall r\in \{1,...,[\mathsf{N}]\}.
\label{SOVBax1}$$Here, we have denoted by $\mathsf{T}_{r}^{\pm }(\mathbf{k})\equiv
(k_{1},\dots ,k_{r}\pm 1,\dots ,k_{\mathsf{N}})$ and $a(\eta _{r}^{(k_{r})})$ and $d(\eta _{r}^{(k_{r})})$ the coefficients of the SOV-representation as defined in (\[a&d-def\]). In the case of $\mathsf{N}$ even, from the asymptotics of $\mathsf{T}(\lambda )$ given in (\[asymptotics-T\]), we have to add to the system (\[SOVBax1\]) the following equations in the variable $\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}$: $$\Psi _{t,\pm k}(\mathsf{T}_{\mathsf{N}}^{+}(\mathbf{k}))\,=\,q^{\mp k}\Psi
_{t,\pm k}(\mathbf{k}), \label{SOVBax2}$$for the wave function $\Psi _{t,\pm k}(\mathbf{k})\equiv \langle \mathbf{k}|\,t_{\pm k}\rangle $, where $|\,t_{\pm k}\rangle $ is a simultaneous eigenstate of $\mathsf{T}(\lambda )$ and $\Theta $ corresponding to $t(\lambda )\in \Sigma _{\mathsf{T}}^{k}$ and $\Theta $-eigenvalue $q^{\pm k}$ with $k\in \{0,...,(p-1)/2\}$.
In the paper [@Nic10], a complete characterization of the $\mathsf{T}$-spectrum (eigenvalues and eigenstates) has been given in terms of a certain class of polynomial solutions of a given functional equation. Let us recall here these results; to this aim let us introduce the one-parameter family $D(\lambda )$ of $p\times p$ matrices: $$D(\lambda )\equiv
\begin{pmatrix}
t(\lambda ) & -d(\lambda ) & 0 & \cdots & 0 & -a(\lambda ) \\
-a(q\lambda ) & t(q\lambda ) & -d(q\lambda ) & 0 & \cdots & 0 \\
0 & {\quad }\ddots & & & & \vdots \\
\vdots & & \cdots & & & \vdots \\
\vdots & & & \cdots & & \vdots \\
\vdots & & & & \ddots {\qquad } & 0 \\
0 & \ldots & 0 & -a(q^{p-2}\lambda ) & t(q^{p-2}\lambda ) &
-d(q^{p-2}\lambda ) \\
-d(q^{p-1}\lambda ) & 0 & \ldots & 0 & -a(q^{p-1}\lambda ) &
t(q^{p-1}\lambda )\end{pmatrix}
\label{D-matrix}$$where for now $t(\lambda )$ is just a real even Laurent polynomial of degree $\mathsf{\bar{N}}$ in $\lambda $. Then the determinant of the matrix $D(\lambda )$ is an even Laurent polynomial of maximal degree $\mathsf{\bar{N}}$ in $\Lambda \equiv \lambda ^{p}$ and we have the following complete characterization of the transfer matrix spectrum:
\[SOV-T-Q\]The set $\Sigma _{\mathsf{T}}$ coincides with the set of all the $t(\lambda )\in \mathbb{R}[\lambda ^{2},\lambda ^{-2}]_{\mathsf{\bar{N}}/2}$ solutions of the functional equation: $$\det_{p}\text{$D$}(\Lambda )=0,\text{ \ \ }\forall \Lambda \in \mathbb{C}.
\label{I-Functional-eq}$$Moreover, for $\mathsf{N}$ odd the spectrum of $\mathsf{T}(\lambda )$ is simple and the eigenstate $|\,t\rangle $ corresponding to $t(\lambda )\in
\Sigma _{\mathsf{T}}$ is characterized by:$$\Psi _{t}(\mathbf{k})\equiv \langle \,\eta _{1}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}\,|\,t\,\rangle =\prod_{r=1}^{\mathsf{N}}Q_{t}(\eta
_{r}^{(k_{r})}), \label{Qeigenstate-odd}$$while for $\mathsf{N}$ even the simultaneous spectrum of $\mathsf{T}(\lambda
)$ and $\Theta $ is simple and the eigenstate $|\,t_{\pm k}\rangle $ corresponding to $t(\lambda )\in \Sigma _{\mathsf{T}}^{k}$ and $\Theta $-eigenvalue $q^{\pm k}$ with $k\in \{0,...,(p-1)/2\}$ is characterized by:$$\Psi _{t,\pm k}(\mathbf{k})\equiv \langle \eta _{1}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}\,|\,t_{\pm k}\rangle =(\eta _{\mathsf{N}}^{(k_{\mathsf{N}})})^{\mp k}\prod_{r=1}^{\mathsf{N}-1}Q_{t}(\eta
_{r}^{(k_{r})}). \label{Qeigenstate-even}$$Here,$$\begin{aligned}
Q_{t}(\lambda )& =\lambda ^{a_{t}}\prod_{h=1}^{(p-1)\mathsf{N}-(b_{t}+a_{t})}(\lambda _{h}-\lambda ),\,\,\,\,\,\,\,\,0\leq a_{t}\leq
p-1,\,\,0\leq b_{t}\leq (p-1)\mathsf{N}, \label{Q_t-definition} \\
a_{t}& =0,\,\,\,\,\,\,\,\,\,b_{t}=0\,\,\mathsf{mod}\,p,\text{ \ \ \ for }\mathsf{N}\text{\textsf{\ }odd} \label{Q-odd} \\
\text{ }a_{t}& =\pm k\,\,\mathsf{mod}\,p,\,\,\,\,\,\,\,\,\,b_{t}=\pm k\,\,\mathsf{mod}\,p,\text{ \ \ \ for }\mathsf{N}\text{\textsf{\ }even
and\thinspace\ }t(\lambda )\in \Sigma _{\mathsf{T}}^{k}\text{, \ }
\label{Q-even}\end{aligned}$$is the unique (up to quasi-constants) real polynomial solution of the Baxter functional equation: $$t(\lambda )Q_{t}(\lambda )=a(\lambda )Q_{t}(\lambda q^{-1})+d(\lambda
)Q_{t}(\lambda q)\ \ \ \ \forall \lambda \in \mathbb{C}, \label{EQ-Baxter-R}$$corresponding to the given $t(\lambda )$, which has been constructed in terms of the cofactors of the matrix $D$$(\Lambda )$, in Theorems 2 and 3 of [@Nic10].
Decomposition of the identity in the SOV-basis
==============================================
The main goal of this section is to obtain the decomposition of the identity in the SOV basis. This is an important step towards the decomposition of any correlation function in terms of matrix elements of local operators in this basis. In the previous sections we recalled how to define a left SOV basis together with the left action of the Yang-Baxter operators on it. To achieve the identity decomposition we need to define a corresponding right SOV basis.
Traditionally, the right states are constructed from the left ones by means of hermitian conjugation. In the case of hermitian $\mathsf{B}$ operators, this procedure would lead to right states that are still eigenstates of $\mathsf{B}$ by means of its right action. However, from , we know that the hermitian conjugate of the $\mathsf{B}$ operator is the operator $\mathsf{C}$ not equal to $\mathsf{B}$. Hence such a procedure (hermitian conjugation of the left SOV basis) would lead to a right SOV basis with respect to the $\mathsf{C}$ operator, namely it would be an eigenstate basis for $\mathsf{C}$. Although such a route could eventually lead to an interesting decomposition of the identity, it would result in a very complicated structure of the scalar product of states (in particular, left and right states would have no simple orthogonality properties); moreover, we would not be able to obtain, at least by obvious means, the coefficients in such an identity decomposition in terms of simple determinants. As a consequence, following this path, the structure of the matrix elements of the local operators would hardly be given in terms of determinants as these are usually obtained as rather simple and localized deformations of the corresponding scalar product formula.
Therefore, our strategy will be here quite different : we will construct the right SOV basis such that it will be an eigenstate basis for the right action of the $\mathsf{B}$ operators. Hence it will not be obtained as the hermitian conjugate of the left SOV basis; however, due to the simplicity of the $\mathsf{B}$-spectrum, left states will admit natural orthogonality properties with respect to right states. Hence, this procedure will lead to a decomposition of the identity as a single sum over the SOV basis, with coefficients that are not necessarily positive numbers but that are computable by means of rather simple and universal determinant formula. To show this, we will have to compute the action of left $\mathsf{B}$-eigenstates (covectors) on right $\mathsf{B}$-eigenstates (vectors); up to an overall constant these actions are completely fixed by the left and right SOV-representations of the Yang-Baxter algebras when the gauges in the SOV-representations are chosen.
Let us first define the right SOV-representation with respect to the $\mathsf{B}$ operator by the following actions: $$\begin{aligned}
\mathsf{B}(\lambda )|\mathbf{k}\rangle & =|\mathbf{k}\rangle \mathsf{b}_{\mathbf{k}}(\lambda ), \label{SOV-B-R} \\
& \notag \\
\mathsf{A}(\lambda )|\mathbf{k}\rangle & =\mathtt{e}_{\mathsf{N}}(\mathsf{T}_{\mathsf{N}}^{+}|\mathbf{k}\rangle \frac{\lambda }{\eta _{\mathbf{k,}\mathsf{A}}}-\mathsf{T}_{\mathsf{N}}^{-}|\mathbf{k}\rangle \frac{\eta _{\mathbf{k,}\mathsf{A}}}{\lambda })\frac{\mathsf{b}_{\mathbf{k}}(\lambda )}{\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}}+\sum_{a=1}^{[\mathsf{N}]}\mathsf{T}_{a}^{+}|\mathbf{k}\rangle \prod_{b\neq a}\frac{(\frac{\lambda }{\eta
_{b}^{(k_{b})}}-\frac{\eta _{b}^{(k_{b})}}{\lambda })}{(\frac{\eta
_{a}^{(k_{a})}}{\eta _{b}^{(k_{b})}}-\frac{\eta _{b}^{(k_{b})}}{\eta
_{a}^{(k_{a})}})}\bar{a}(\eta _{a}^{(k_{a})}), \label{SOV-A-R} \\
& \notag \\
\mathsf{D}(\lambda )|\mathbf{k}\rangle & =\mathtt{e}_{\mathsf{N}}(\mathsf{T}_{\mathsf{N}}^{-}|\mathbf{k}\rangle \frac{\lambda }{\eta _{\mathbf{k,}\mathsf{D}}}-\mathsf{T}_{\mathsf{N}}^{+}|\mathbf{k}\rangle \frac{\eta _{\mathbf{k,}\mathsf{D}}}{\lambda })\frac{\mathsf{b}_{\mathbf{k}}(\lambda )}{\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}}+\sum_{a=1}^{[\mathsf{N}]}\mathsf{T}_{a}^{-}|\mathbf{k}\rangle \prod_{b\neq a}\frac{(\frac{\lambda }{\eta
_{b}^{(k_{b})}}-\frac{\eta _{b}^{(k_{b})}}{\lambda })}{(\frac{\eta
_{a}^{(k_{a})}}{\eta _{b}^{(k_{b})}}-\frac{\eta _{b}^{(k_{b})}}{\eta
_{a}^{(k_{a})}})}\bar{d}(\eta _{a}^{(k_{a})}), \label{SOV-D-R}\end{aligned}$$on the generic right $\mathsf{B}$-eigenstate $|\mathbf{k}\rangle \equiv
|\eta _{1}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}\rangle \in
\mathcal{R}_{\mathsf{N}}$. Here, the coefficients $\bar{a}(\eta _{a})$ and $\bar{d}(\eta _{a})$ of the representation are fixed up to the gauge by the condition:$$\mathrm{det_{q}}\mathsf{M}(\lambda )=\bar{d}(\lambda )\bar{a}(\lambda /q);$$while $\mathsf{C}(\lambda )$ is uniquely defined by the quantum determinant relation $(\ref{q-det-f})$.
Action of left $\mathsf{B}$-eigenstates on right $\mathsf{B}$-eigenstates
-------------------------------------------------------------------------
It is worth remarking that both for the right and left SOV-representations we are explicitly asking that the coefficients of the representations of $\mathsf{A}(\lambda )$ and $\mathsf{D}(\lambda )$ depend only on the zeros of $\mathsf{B}(\lambda )$ on which the corresponding shift operator acts; i.e. they are separated w.r.t. the zeros of $\mathsf{B}(\lambda )$. Naturally such requirement is compatible with the Yang-Baxter algebra and the quantum determinant relation but it is not implied by them. The main point that we are going to prove here is that this separation requirement completely fixes the actions of the generic covector in the left SOV-basis on the generic vector in the right SOV-basis. It may be helpful to explain this last statement in terms of the properties of the matrices which define the change of basis to the SOV-basis. Let us define the following isomorphism: $$\varkappa :\left( h_{1},...,h_{\mathsf{N}}\right) \in \{1,...,p\}^{\mathsf{N}}\rightarrow j=\varkappa \left( h_{1},...,h_{\mathsf{N}}\right) \equiv
h_{1}+\sum_{a=2}^{\mathsf{N}}p^{(a-1)}(h_{a}-1)\in \{1,...,p^{\mathsf{N}}\},
\label{corrisp}$$then we can write:$$\langle \text{\textbf{y}}_{j}|=\langle \text{\textbf{x}}_{j}|U^{(L)}=\sum_{i=1}^{p^{\mathsf{N}}}U_{j,i}^{(L)}\langle \text{\textbf{x}}_{i}|\text{
\ \ and\ \ \ }|\text{\textbf{y}}_{j}\rangle =U^{(R)}|\text{\textbf{x}}_{j}\rangle =\sum_{i=1}^{p^{\mathsf{N}}}U_{i,j}^{(R)}|\text{\textbf{x}}_{i}\rangle ,$$where we have used the notations:$$\langle \text{\textbf{y}}_{j}|\equiv \langle \eta _{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|\text{ \ and \ }|\text{\textbf{y}}_{j}\rangle
\equiv |\eta _{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle
\text{, for }j=\varkappa \left( h_{1},...,h_{\mathsf{N}}\right) \text{,}$$to represent, respectively, the states of the left and right SOV-basis and:$$\langle \text{\textbf{x}}_{j}|\equiv \otimes _{n=1}^{\mathsf{N}}\langle
h_{n}|\text{ \ \ \ \ and \ \ \ }|\text{\textbf{x}}_{j}\rangle \equiv \otimes
_{n=1}^{\mathsf{N}}|h_{n}\rangle \text{, for }j=\varkappa \left(
h_{1},...,h_{\mathsf{N}}\right) \text{,}$$to represent, respectively, the states of the left and right original $\mathsf{v}_{n}$-orthonormal basis. Here, $U^{(L)}$ and $U^{(R)}$ are the $p^{\mathsf{N}}\times p^{\mathsf{N}}$ matrices for which it holds:$$U^{(L)}\mathsf{B}(\lambda )=\Delta _{\mathsf{B}}(\lambda )U^{(L)},\text{ \ \
}\mathsf{B}(\lambda )U^{(R)}=U^{(R)}\Delta _{\mathsf{B}}(\lambda ),$$where $\Delta _{\mathsf{B}}(\lambda )$ is a diagonal $p^{\mathsf{N}}\times
p^{\mathsf{N}}$ matrix. The diagonalizability and simplicity of the $\mathsf{B}$-spectrum imply the invertibility of the matrices $U^{(L)}$ and $U^{(R)}$ and the fact that all the diagonal entries of $\Delta _{\mathsf{B}}(\lambda )$ are Laurent polynomials in $\lambda $ with different zeros. Then the following proposition holds:
Right and left SOV-basis are right and left $\mathsf{B}$-eigenbasis such that the $p^{\mathsf{N}}\times p^{\mathsf{N}}$ matrix:$$M\equiv U^{(L)}U^{(R)}$$is diagonal and characterized by:$$M_{jj}=\langle \text{\textbf{y}}_{j}|\text{\textbf{y}}_{j}\rangle =\langle
\eta _{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|\eta
_{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle =\frac{C_{\mathsf{N}}\prod_{a=1}^{[\mathsf{N}]}\prod_{k_{a}=1}^{h_{a}}a(\eta
_{a}^{(k_{a})})/\bar{a}(\eta _{a}^{(k_{a}-1)})}{\prod_{1\leq b<a\leq \lbrack
\mathsf{N}]}(\eta _{a}^{(h_{a})}/\eta _{b}^{(h_{b})}-\eta
_{b}^{(h_{b})}/\eta _{a}^{(h_{a})})}, \label{M_jj}$$where we have denoted:$$j=\varkappa \left( h_{1},...,h_{\mathsf{N}}\right)$$and $C_{\mathsf{N}}$ is a constant characteristic of the chosen representations.
The fact that the matrix $M$ is diagonal is a trivial consequence of the orthogonality of left and right eigenstates corresponding to different eigenvalue of $\mathsf{B}(\lambda )$ and of the simplicity of the $\mathsf{B}$-spectrum. Indeed, defined $i=\varkappa (k_{1},...,k_{\mathsf{N}})$ and $j=\varkappa (h_{1},...,h_{\mathsf{N}})$, the following identities hold:$$\begin{aligned}
\mathsf{b}_{\mathbf{k}}(\lambda )M_{ij} &=&\langle \eta
_{1}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|\mathsf{B}(\lambda
)|\eta _{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle ,
\label{Ortho-0} \\
\mathsf{b}_{\mathbf{h}}(\lambda )M_{ij} &=&\langle \eta
_{1}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|\mathsf{B}(\lambda
)|\eta _{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle ,
\label{Ortho-1}\end{aligned}$$where the identity $\left( \ref{Ortho-0}\right) $ follows acting with $\mathsf{B}(\lambda )$ on the left while the identity $\left( \ref{Ortho-1}\right) $ follows acting with $\mathsf{B}(\lambda )$ on the right. Then in the case $(k_{1},...,k_{\mathsf{N}})\neq (h_{1},...,h_{\mathsf{N}})$ the identities $\left( \ref{Ortho-0}\right) $ and $\left( \ref{Ortho-1}\right) $ imply that $M_{ij}=0$ for any $i\neq j\in \{1,...,p^{\mathsf{N}}\}.$
Now, let us compute the matrix element $\theta _{a}\equiv \langle \eta
_{_{1}}^{(0)},...,\eta _{a}^{(1)},...,\eta _{\mathsf{N}}^{(0)}|\mathsf{A}(\eta _{a}^{(1)})|\eta _{_{1}}^{(0)},...,\eta _{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle $, where $a\in \{1,...,[\mathsf{N}]\}$. Then using the left action of the operator $\mathsf{A}(\bar{\eta}_{a})$ we get:$$\theta _{a}=a(\eta _{a}^{(1)})\langle \eta _{_{1}}^{(0)},...,\eta
_{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},...,\eta
_{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle$$while using the right action of the operator $\mathsf{A}(\bar{\eta}_{a})$ and the orthogonality of right and left $\mathsf{B}$-eigenstates corresponding to different eigenvalues we get:$$\theta _{a}=\prod_{b\neq a,b=1}^{[\mathsf{N}]}\frac{(\eta _{a}^{(1)}/\eta
_{b}^{(0)}-\eta _{b}^{(0)}/\eta _{a}^{(1)})}{(\eta _{a}^{(0)}/\eta
_{b}^{(0)}-\eta _{b}^{(0)}/\eta _{a}^{(0)})}\bar{a}(\eta _{a}^{(0)})\langle
\eta _{_{1}}^{(0)},...,\eta _{a}^{(1)},...,\eta _{\mathsf{N}}^{(0)}|\eta
_{_{1}}^{(0)},...,\eta _{a}^{(1)},...,\eta _{\mathsf{N}}^{(0)}\rangle$$and so:$$\frac{\langle \eta _{_{1}}^{(0)},...,\eta _{a}^{(1)},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},...,\eta _{a}^{(1)},...,\eta _{\mathsf{N}}^{(0)}\rangle }{\langle \eta _{_{1}}^{(0)},...,\eta _{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},...,\eta _{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle }=\frac{a(\eta _{a}^{(1)})}{\bar{a}(\eta _{a}^{(0)})}\prod_{b\neq a,b=1}^{[\mathsf{N}]}\frac{(\eta _{a}^{(0)}/\eta
_{b}^{(0)}-\eta _{b}^{(0)}/\eta _{a}^{(0)})}{(\eta _{a}^{(1)}/\eta
_{b}^{(0)}-\eta _{b}^{(1)}/\eta _{a}^{(0)})}. \label{F1}$$Then by applying (\[F1\]) $h_{a}$ times, we get:$$\frac{\langle \eta _{_{1}}^{(0)},...,\eta _{a}^{(h_{a})},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},...,\eta _{a}^{(h_{a})},...,\eta _{\mathsf{N}}^{(0)}\rangle }{\langle \eta _{_{1}}^{(0)},...,\eta _{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},...,\eta _{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle }=\prod_{k_{a}=1}^{h_{a}}\frac{a(\eta _{a}^{(k_{a})})}{\bar{a}(\eta _{a}^{(k_{a}-1)})}\prod_{b\neq a,b=1}^{[\mathsf{N}]}\frac{(\eta
_{a}^{(0)}/\eta _{b}^{(0)}-\eta _{b}^{(0)}/\eta _{a}^{(0)})}{(\eta
_{a}^{(h_{a})}/\eta _{b}^{(0)}-\eta _{b}^{(h_{a})}/\eta _{a}^{(0)})}.
\label{F2}$$Now, let us consider explicitly the case of even $\mathsf{N}$. In this case we still have to define the recurrence for $a=\mathsf{N}$. We compute the matrix element:$$\theta _{\mathsf{N}}(\lambda )\equiv \langle \eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(1)}|\mathsf{A}(\lambda )|\eta
_{1}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(0)}\rangle ,$$acting with $\mathsf{A}(\lambda )$ on the right we get:$$\theta _{\mathsf{N}}(\lambda )\equiv \frac{\lambda \mathsf{b}_{\mathbf{k}}(\lambda )}{\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}\eta _{\mathbf{k,}\mathsf{A}}}\langle \eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(1)}|\eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(1)}\rangle ,$$while acting on the left we get:$$\theta _{\mathsf{N}}(\lambda )\equiv \frac{\lambda \mathsf{b}_{\mathbf{k}}(\lambda )}{\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}\eta _{\mathbf{k,}\mathsf{A}}}\langle \eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(1)}|\eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(1)}\rangle ,$$which simply implies:$$\langle \eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(1)}|\eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(1)}\rangle =\langle \eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},...,\eta _{\mathsf{N}-1}^{(0)},\eta _{\mathsf{N}}^{(0)}\rangle .$$The previous formula implies, for both $\mathsf{N}$ even and odd:$$\frac{\langle \eta _{_{1}}^{(h_{a})},...,\eta _{a}^{(h_{a})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|\eta _{_{1}}^{(h_{a})},...,\eta
_{a}^{(h_{a})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle }{\langle
\eta _{_{1}}^{(0)},...,\eta _{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta
_{_{1}}^{(0)},...,\eta _{a}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle }=\prod_{a=1}^{[\mathsf{N}]}\prod_{k_{a}=1}^{h_{a}}\frac{a(\eta
_{a}^{(k_{a})})}{\bar{a}(\eta _{a}^{(k_{a}-1)})}\prod_{1\leq b<a\leq \lbrack
\mathsf{N}]}\frac{(\frac{\eta _{a}^{(0)}}{\eta _{b}^{(0)}}-\frac{\eta
_{b}^{(0)}}{\eta _{a}^{(0)}})}{(\frac{\eta _{a}^{(k_{a})}}{\eta
_{b}^{(k_{b})}}-\frac{\eta _{b}^{(k_{b})}}{\eta _{a}^{(k_{a})}})}.
\label{F3}$$To prove it let us observe that (\[F2\]) coincides with (\[F3\]) for $h_{2}=...=h_{\mathsf{N}}=0$. Now in the case $h_{3}=...=h_{\mathsf{N}}=0$ with $h_{1}\neq 0$ and $h_{2}\neq 0$, we get (\[F3\]) by the following factorization:$$\begin{aligned}
\frac{\langle \eta _{_{1}}^{(h_{1})},\eta _{2}^{(h_{2})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|\eta _{_{1}}^{(h_{1})},\eta _{2}^{(h_{2})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle }{\langle \eta _{_{1}}^{(0)},\eta
_{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},\eta
_{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle }& =\frac{\langle \eta
_{_{1}}^{(h_{1})},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|\eta _{_{1}}^{(h_{1})},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle }{\langle \eta _{_{1}}^{(0)},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle } \notag \\
& \times \frac{\langle \eta _{_{1}}^{(h_{1})},\eta _{2}^{(h_{2})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|\eta _{_{1}}^{(h_{1})},\eta
_{2}^{(h_{2})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle }{\langle
\eta _{_{1}}^{(h_{1})},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta
_{_{1}}^{(h_{1})},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle }
\notag \\
& =\prod_{a=1}^{2}\prod_{k_{a}=1}^{h_{a}}\frac{a(\eta _{a}^{(k_{a})})}{\bar{a}(\eta _{a}^{(k_{a}-1)})}\prod_{b=2}^{[\mathsf{N}]}\frac{(\eta
_{1}^{(0)}/\eta _{b}^{(0)}-\eta _{b}^{(0)}/\eta _{1}^{(0)})}{(\eta
_{1}^{(h_{1})}/\eta _{b}^{(0)}-\eta _{b}^{(0)}/\eta _{1}^{(h_{1})})} \notag
\\
& \times \frac{(\eta _{2}^{(0)}/\eta _{1}^{(h_{1})}-\eta _{1}^{(h_{1})}/\eta
_{2}^{(0)})}{(\eta _{2}^{(h_{2})}/\eta _{1}^{(h_{1})}-\eta
_{1}^{(h_{1})}/\eta _{2}^{(h_{2})})}\prod_{b=3}^{[\mathsf{N}]}\frac{(\eta
_{2}^{(0)}/\eta _{b}^{(0)}-\eta _{b}^{(0)}/\eta _{2}^{(0)})}{(\eta
_{2}^{(h_{2})}/\eta _{b}^{(0)}-\eta _{b}^{(0)}/\eta _{2}^{(h_{2})})},\end{aligned}$$and so on for the generic case. Finally, from (\[F3\]) the statement of the proposition follows being by definition:$$\frac{M_{ji}}{\langle \text{\textbf{y}}_{p^{\mathsf{N}}}|\text{\textbf{y}}_{p^{\mathsf{N}}}\rangle }\equiv \delta _{i,j}\frac{\langle \eta
_{_{1}}^{(h_{1})},\eta _{2}^{(h_{2})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|\eta _{_{1}}^{(h_{1})},\eta _{2}^{(h_{2})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle }{\langle \eta _{_{1}}^{(0)},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}|\eta _{_{1}}^{(0)},\eta _{2}^{(0)},...,\eta _{\mathsf{N}}^{(0)}\rangle }.$$
**Remark 2.** The diagonal matrix $M$ is mainly fixed by the requirement that the left and right representations have a separate form in the zeros of $\mathsf{B}(\lambda )$. Indeed, this fixes completely the denominator in all the entries of $M$. Moreover, the constant $C_{\mathsf{N}} $ is a function of the representation only via the central elements $(Z_{1},...,Z_{\mathsf{N}})$. In the following, we will fix:$$C_{\mathsf{N}}\equiv (\eta _{\mathsf{N}}^{(0)}p^{1/2})^{\mathtt{e}_{\mathsf{N}}},$$this choice just amounts to an overall renormalization of the states. Finally, the products of $a(\lambda )/\bar{a}(\lambda q^{-1})$ in the numerator of each matrix element is fixed from the choice of the gauge done in the SOV-representations. Note that we are always free to take the following gauge:$$\bar{a}(\lambda q^{-1})\equiv a(\lambda ), \label{L-R-gauge}$$for which the numerator in (\[M\_jj\]) is $C_{\mathsf{N}}$.
SOV-decomposition of the identity
---------------------------------
The previous results allow to write the following spectral decomposition of the identity $\mathbb{I}$:$$\mathbb{I}\equiv \sum_{i=1}^{p^{\mathsf{N}}}\mu _{i}|\text{\textbf{y}}_{i}\rangle \langle \text{\textbf{y}}_{i}|,$$in terms of the left and right SOV-basis. Here,$$\mu _{i}\equiv \frac{1}{\langle \text{\textbf{y}}_{i}|\text{\textbf{y}}_{i}\rangle },$$is the required analogue of Sklyanin’s *measure*[^9], which is discrete for the cyclic representations of the sine-Gordon model. Explicitly, the SOV-decomposition of the identity reads: $$\mathbb{I}\equiv \sum_{h_{1},...,h_{\mathsf{N}}=1}^{p}\prod_{1\leq b<a\leq
\lbrack \mathsf{N}]}((\eta _{a}^{(h_{a})})^{2}-(\eta _{b}^{(h_{b})})^{2})\frac{|\eta _{_{1}}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle \langle \eta _{_{1}}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|}{C_{\mathsf{N}}\prod_{b=1}^{[\mathsf{N}]}\omega _{b}(\eta
_{b}^{(h_{b})})},$$where the separate functions $\omega _{b}(\eta _{b}^{(h_{b})})$ reproduce the numerator in (\[M\_jj\]):$$\omega _{b}(\eta _{b}^{(h_{b})})\equiv \left( \eta _{b}^{(h_{b})}\right) ^{[\mathsf{N}]-1}\prod_{k_{b}=1}^{h_{b}}a(\eta _{b}^{(k_{b})})/\bar{a}(\eta
_{b}^{(k_{b}-1)})$$and they are gauge dependent parameters.
**Remark 3.** Sklyanin’s measure[^10] has been first introduced by Sklyanin in his article on quantum Toda chain [Skl85]{}. There, it has been derived as a consequence of the self-adjointness of the transfer matrix w.r.t. the scalar product. In particular, the Hermitian properties of the operator zeros and their conjugate shift operators have been fixed to assure the self-adjointness of the transfer matrix. In the similar but more involved sinh-Gordon model [@BytT06], the problem related to the uniqueness of the definition of this measure has been analyzed. There, it has been proven that the measure is in fact uniquely determined once the positive self-adjointness of the generators $\mathsf{A}(\lambda )$ and $\mathsf{D}(\lambda )$ is required. In the compact case of the sine-Gordon model the method used here insures that the analogue of Sklyanin’s measure is uniquely determined up to an overall constant and the choice of gauge, as discussed in the previous remark. Let us mention that the Sklyanin’s measure has already been derived in [GIPST07]{} for cyclic representations of the related $\tau _{2}$-model[^11] [@BS; @BBP; @B04]. There the measure has been obtained by a different approach, i.e. by a recursive construction which needs to go through the recursion in the construction of left and right SOV-basis. It is interesting to remark that in our purely algebraic derivation we skip these model dependent features so that the approach we used is suitable for general compact SOV-representations of the 6-vertex Yang-Baxter algebra.
SOV-representation of left and right $\mathsf{T}$-eigenstates \[SOV-T-eigenstates\]
-----------------------------------------------------------------------------------
Up to an overall normalization, the SOV-decomposition of the identity and the SOV-characterization of the transfer matrix spectrum lead to the following representations of the right eigenstate of the transfer matrix $\mathsf{T}(\lambda )$:$$\begin{aligned}
|t_{\pm k}\rangle & =\sum_{i=1}^{p^{\mathsf{N}}}\mu _{i}\langle \text{\textbf{y}}_{i}|t_{\pm k}\rangle |\text{\textbf{y}}_{i}\rangle \notag \\
& =\sum_{h_{1},...,h_{\mathsf{N}}=1}^{p}\left( \frac{q^{\mp kh_{\mathsf{N}}}}{p^{1/2}}\right) ^{\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}Q_{t}(\eta _{a}^{(h_{a})})\prod_{1\leq b<a\leq \lbrack \mathsf{N}]}((\eta
_{a}^{(h_{a})})^{2}-(\eta _{b}^{(h_{b})})^{2})\frac{|\eta
_{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle }{\prod_{b=1}^{[\mathsf{N}]}\omega _{b}(\eta _{b}^{(h_{b})})},
\label{eigenT-r}\end{aligned}$$corresponding to the eigenvalue $t(\lambda )\in \Sigma _{\mathsf{T}}^{k}$, where the index $k\in \{0,...,(p-1)/2\}$ appears only for $\mathsf{N}$ even and indicates that $|t_{\pm k}\rangle $ is also a $\Theta $-eigenstate with eigenvalue $q^{\pm k}$. Here $Q_{t}(\lambda )$ is the solution of the Baxter equation (\[EQ-Baxter-L\]) defined in Proposition \[SOV-T-Q\]. In a similar way we can prove that, up to an overall normalization, the left $\mathsf{T}$-eigenstate has the following SOV-representation:$$\begin{aligned}
\langle t_{\pm k}|& =\sum_{i=1}^{p^{\mathsf{N}}}\mu _{i}\langle t_{\pm k}|\text{\textbf{y}}_{i}\rangle \langle \text{\textbf{y}}_{i}| \notag \\
& =\sum_{h_{1},...,h_{\mathsf{N}}=1}^{p}\left( \frac{q^{\pm kh_{\mathsf{N}}}}{p^{1/2}}\right) ^{\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}\bar{Q}_{t}(\eta _{a}^{(h_{a})})\prod_{1\leq b<a\leq \lbrack \mathsf{N}]}(\eta
_{a}^{(h_{a})}-\eta _{b}^{(h_{b})})\frac{\langle \eta
_{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|}{\prod_{b=1}^{[\mathsf{N}]}\omega _{b}(\eta _{b}^{(h_{b})})}, \label{eigenT-l}\end{aligned}$$where $\bar{Q}_{t}(\lambda )$ is the unique (up to quasi-constants) polynomial solution of the Baxter functional equation:$$t(\lambda )\bar{Q}_{t}(\lambda )=\bar{d}(\lambda )\bar{Q}_{t}(\lambda
q^{-1})+\bar{a}(\lambda )\bar{Q}_{t}(\lambda q). \label{EQ-Baxter-L}$$**Remark 4.** In the gauge fixed by (\[L-R-gauge\]), this Baxter equation reads:$$t(\lambda )\bar{Q}_{t}(-\lambda )=q^{\mathsf{N}}a(\lambda )\bar{Q}_{t}(-\lambda q^{-1})+q^{-\mathsf{N}}d(\lambda )\bar{Q}_{t}(-\lambda q),$$and so, up to quasi-constants, we have:$$\bar{Q}_{t}(\lambda )=\lambda ^{\chi _{\mathsf{N}}}Q_{t}(-\lambda
),\,\,\,\,\,\,\,\,\chi _{\mathsf{N}}=\mathsf{N}\,\,\mathsf{mod}\,p,\text{ }0\leq \chi _{\mathsf{N}}\leq p-1.$$
Decomposition of the identity in the $\mathsf{T}$-eigenbasis
============================================================
Here we use the results of the previous section to compute the action of covectors on vectors which in the left and right SOV-basis have a *separate form* similar to that of the transfer matrix eigenstates.
Action of left separate states on right separate states
-------------------------------------------------------
Let us start giving the following definition of a left $\langle \alpha _{k}|
$ and a right $|\beta _{k}\rangle $ separate states; they are respectively a covector and a vector which have the following *separate form* in terms of the SOV-decomposition of the identity:$$\begin{aligned}
\langle \alpha _{k}|& =\sum_{h_{1},...,h_{\mathsf{N}}=1}^{p}\left( \frac{q^{kh_{\mathsf{N}}}}{p^{1/2}}\right) ^{\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}\alpha _{a}(\eta _{a}^{(h_{a})})\prod_{1\leq b<a\leq \lbrack
\mathsf{N}]}(\left( \eta _{a}^{(h_{a})}\right) ^{2}-\left( \eta
_{b}^{(h_{b})}\right) ^{2})\frac{\langle \eta _{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}|}{\prod_{b=1}^{[\mathsf{N}]}\omega _{b}(\eta
_{b}^{(h_{b})})}, \label{Fact-left-SOV} \\
|\beta _{k}\rangle & =\sum_{h_{1},...,h_{\mathsf{N}}=1}^{p}\left( \frac{q^{-kh_{\mathsf{N}}}}{p^{1/2}}\right) ^{\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}\beta _{a}(\eta _{a}^{(h_{a})})\prod_{1\leq
b<a\leq \lbrack \mathsf{N}]}(\left( \eta _{a}^{(h_{a})}\right) ^{2}-\left(
\eta _{b}^{(h_{b})}\right) ^{2})\frac{|\eta _{1}^{(h_{1})},...,\eta _{\mathsf{N}}^{(h_{\mathsf{N}})}\rangle }{\prod_{b=1}^{[\mathsf{N}]}\omega
_{b}(\eta _{b}^{(h_{b})})}, \label{Fact-right-SOV}\end{aligned}$$where the index $k$ appears only for $\mathsf{N}$ even. It is easy to see that such states generate the whole space of states of the model (in particular the $\mathsf{T}$-eigenbasis is just of this type). The interest toward these kind of states is due to the following:
Let $\langle \alpha _{k}|$ and $|\beta _{k'}\rangle $ two separate states of the form $(\ref{Fact-left-SOV})$ and $(\ref{Fact-right-SOV})$, respectively, then it holds:$$\langle \alpha _{k}|\beta _{k'}\rangle =\delta _{k,k'}^{\mathtt{e}_{\mathsf{N}}}\det_{[\mathsf{N}]}||\mathcal{M}_{a,b}^{\left( \alpha ,\beta \right) }||\text{ \ \ with \ }\mathcal{M}_{a,b}^{\left( \alpha ,\beta \right) }\equiv
\left( \eta _{a}^{(0)}\right) ^{2(b-1)}\sum_{h=1}^{p}\frac{\alpha _{a}(\eta
_{a}^{(h)})\beta _{a}(\eta _{a}^{(h)})}{\omega _{a}(\eta _{a}^{(h)})}q^{2(b-1)h}.$$
From the SOV-decomposition, we have:$$\langle \alpha _{k}|\beta _{k'}\rangle =\left( \sum_{h_{\mathsf{N}}=1}^{p}\frac{q^{(k-k')h_{\mathsf{N}}}}{p}\right) ^{\mathtt{e}_{\mathsf{N}}}\sum_{h_{1},...,h_{[\mathsf{N}]}=1}^{p}V(\left( \eta _{1}^{(h_{1})}\right)
^{2},...,\left( \eta _{\lbrack \mathsf{N}]}^{(h_{[\mathsf{N}]})}\right)
^{2})\prod_{a=1}^{[\mathsf{N}]}\frac{\alpha _{a}(\eta _{a}^{(h_{a})})\beta
_{a}(\eta _{a}^{(h_{a})})}{\omega _{a}(\eta _{a}^{(h_{a})})},$$where $V(x_{1},...,x_{\mathsf{N}})\equiv \prod_{1\leq b<a\leq \lbrack
\mathsf{N}]}(x_{a}-x_{b})$ is the Vandermonde determinant. From this formula by using the multilinearity of the determinant w.r.t. the rows we prove the proposition.
In Section \[Def-cyclic-rep\], we have associated to the linear space $\mathcal{R}_{\mathsf{N}}$ the structure of an Hilbert space by introducing a scalar product. Then it is clear that the above determinant formula represents also the formula for the scalar product of the two vectors $\left( \langle \alpha _{k}|\right) ^{\dagger }$ and $|\beta _{h}\rangle $ in $\mathcal{R}_{\mathsf{N}}$. It is worth pointing out that the vector $\left(
\langle \alpha _{k}|\right) ^{\dagger }\in \mathcal{R}_{\mathsf{N}}$ is a separate state in the right $\mathsf{C}$-eigenbasis, as it simply follows from the hermitian conjugation properties of the Yang-Baxter generators reported in Section \[Def-cyclic-rep\]. Then these results can be considered as the SOV analogue of the scalar product formulae [Salv89,KitMT99]{} computed in the framework of the algebraic Bethe ansatz. However, we want to stress that the determinant formulae obtained here are not restricted to the case in which one of the two states is an eigenstate of the transfer matrix, on the contrary to what happens for the scalar product formulae in the framework of the algebraic Bethe ansatz. Finally, we can prove directly from these formula the following:
Transfer matrix eigenstates corresponding to different eigenvalues are orthogonal states.
Let us denote with $|t_{k}\rangle $ and $|t_{h}^{\prime }\rangle $ two eigenstates of $\mathsf{T}(\lambda )$ with eigenvalues $t_{k}(\lambda )$ and $t_{h}^{\prime }(\lambda )$ for $\mathsf{N}$ odd and with $\Theta $ eigenvalues $q^{k}$ and $q^{h}$ for $\mathsf{N}$ even. To prove the corollary, we have to prove that:$$\det_{\lbrack \mathsf{N}]}||\Phi _{a,b}^{\left( t,t^{\prime }\right) }||=0\text{ \ \ with \ }\Phi _{a,b}^{\left( t,t^{\prime }\right) }\equiv \left(
\eta _{a}^{(0)}\right) ^{2(b-1)}\sum_{c=1}^{p}\frac{Q_{t^{\prime }}(\eta
_{a}^{(c)})\bar{Q}_{t}(\eta _{a}^{(c)})}{\omega _{a}(\eta _{a}^{(c)})}q^{2(b-1)c}, \label{orth-cond}$$with $h=k$ for $\mathsf{N}$ even. To prove (\[L-R-gauge\]), it is enough to show the existence of a non-zero vector V$^{\left( t,t^{\prime }\right) }$ such that:$$\sum_{b=1}^{[\mathsf{N}]}\Phi _{a,b}^{\left( t,t^{\prime }\right) }\text{V}_{b}^{\left( t,t^{\prime }\right) }=0\text{ \ \ \ \ }\forall a\in \{1,...,[\mathsf{N}]\}. \label{zero-eigenvector}$$For simplicity, we construct this vector in the gauge (\[L-R-gauge\]) where it results:$$\omega _{a}(\eta _{a}^{(h)})=\left( \eta _{a}^{(h)}\right) ^{[\mathsf{N}]-1}.$$Let us recall that the eigenvalues of the transfer matrix are even Laurent polynomials of degree $\bar{\mathsf{N}}$ of the form:$$\begin{aligned}
t_{h}(\lambda ) &=&\mathtt{e}_{\mathsf{N}}\left( \prod_{a=1}^{\mathsf{N}}\frac{\kappa _{a}\xi _{a}^{\pm 1}}{i}\right) (q^{h}+q^{-h})(\lambda ^{\mathsf{N}}+\lambda ^{-\mathsf{N}})+\sum_{b=1}^{[\mathsf{N}]}c_{b}\lambda
^{-[\mathsf{N}]-1+2b}, \\
t_{h}^{\prime }(\lambda ) &=&\mathtt{e}_{\mathsf{N}}\left( \prod_{a=1}^{\mathsf{N}}\frac{\kappa _{a}\xi _{a}^{\pm 1}}{i}\right)
(q^{h}+q^{-h})(\lambda ^{\mathsf{N}}+\lambda ^{-\mathsf{N}})+\sum_{b=1}^{[\mathsf{N}]}c_{b}^{\prime }\lambda ^{-[\mathsf{N}]-1+2b},\end{aligned}$$so if we define:$$\text{V}_{b}^{\left( t,t^{\prime }\right) }\equiv c_{b}^{\prime }-c_{b}\text{\ \ \ }\forall b\in \{1,...,[\mathsf{N}]\},$$it results:$$\sum_{b=1}^{[\mathsf{N}]}\Phi _{a,b}^{\left( t,t^{\prime }\right) }\text{V}_{b}^{\left( t,t^{\prime }\right) }=\sum_{c=1}^{p}Q_{t^{\prime }}(\eta
_{a}^{(c)})\bar{Q}_{t}(\eta _{a}^{(c)})(t_{h}^{\prime }(\eta
_{a}^{(c)})-t_{h}(\eta _{a}^{(c)})). \label{zero-eigenvector-1}$$We can use now the Baxter equations (\[EQ-Baxter-R\]) and ([EQ-Baxter-L]{}), with the chosen gauge (\[L-R-gauge\]), to rewrite:$$\begin{aligned}
Q_{t^{\prime }}(\eta _{a}^{(h_{a})})\bar{Q}_{t}(\eta
_{a}^{(h_{a})})(t^{\prime }(\eta _{a}^{(h_{a})})-t(\eta _{a}^{(h_{a})}))&
=a(\eta _{a}^{(h_{a})})Q_{t^{\prime }}(\eta _{a}^{(h_{a}-1)})\bar{Q}_{t}(\eta _{a}^{(h_{a})})\notag \\
&+d(\eta _{a}^{(h_{a})})Q_{t^{\prime }}(\eta
_{a}^{(h_{a}+1)})\bar{Q}_{t}(\eta _{a}^{(h_{a})}) \notag \\
& -d(\eta _{a}^{(h_{a}-1)})Q_{t^{\prime }}(\eta _{a}^{(h_{a})})\bar{Q}_{t}(\eta _{a}^{(h_{a}-1)})\notag \\
&-a(\eta _{a}^{(h_{a}+1)})Q_{t^{\prime }}(\eta
_{a}^{(h_{a})})\bar{Q}_{t}(\eta _{a}^{(h_{a}+1)}),\end{aligned}$$and by substituting it in (\[zero-eigenvector-1\]) we get ([zero-eigenvector]{}).
Decomposition of the identity in the -eigenbasis
------------------------------------------------
Let us remark that the diagonalizability and simplicity of the transfer matrix spectrum implies the following decomposition of the identity in the left and right $\mathsf{T}$-eigenbasis:$$\mathbb{I=}\sum_{k=0}^{\mathtt{e}_{\mathsf{N}}\left( p-1\right)
/2}\sum_{t_{k}(\lambda )\in \Sigma _{\mathsf{T}}^{k}}\frac{|t_{k}\rangle
\langle t_{k}|}{\langle t_{k}|t_{k}\rangle }, \label{Id-decomp}$$where$$\langle t_{k}|t_{k}\rangle =\det_{[\mathsf{N}]}||\Phi _{a,b}^{\left(
t_{k},t_{k}\right) }||\text{ \ with }\Phi _{a,b}^{\left( t_{k},t_{k}\right)
}\equiv (\eta _{a}^{(0)})^{2(b-1)}\sum_{c=1}^{p}\frac{Q_{t}(\eta _{a}^{(c)})\bar{Q}_{t}(\eta _{a}^{(c)})}{\omega _{a}(\eta _{a}^{(c)})}q^{2(b-1)c},$$is the action of the covector $\langle t_{k}|$ on the vector $|t_{k}\rangle $ as defined in Section \[SOV-T-eigenstates\]. It is worth to note that in the representations which defines a normal transfer matrix $\mathsf{T}(\lambda )$, the generic covector $\equiv \left(
|t_{k}\rangle \right) ^{\dagger }$, dual to the right $\mathsf{T}$-eigenstate $|t_{k}\rangle $, is itself a left eigenstate of $\mathsf{T}\left( \lambda \right) $ which taking into account the simplicity of the $\mathsf{T}$-spectrum has to satisfy the following identity $\equiv \alpha _{t_{k}}\langle t_{k}|$, where $\langle t_{k}|$ is the left $\mathsf{T}$-eigenstate defined in (\[eigenT-l\]). Of course, in these representations the following identities hold: $$\frac{|t_{k}\rangle \langle t_{k}|}{\langle t_{k}|t_{k}\rangle }=\frac{|t_{k}\rangle \underline{\langle t_{k}|}}{\left\Vert |t_{k}\rangle
\right\Vert ^{2}}$$where $\left\Vert |t_{k}\rangle \right\Vert $ is the positive norm of the eigenvector $|t_{k}\rangle $ in the Hilbert space $\mathcal{R}_{\mathsf{N}}$. The above discussion implies the relevance of computing explicitly the norm of the transfer matrix eigenstates (\[eigenT-r\]) as it allows to fix the relative normalization $\alpha _{t_{k}}$ thanks to the identity $\alpha_{t_{k}}=\left\Vert |t_{k}\rangle \right\Vert ^{2}/\langle t_{k}|t_{k}\rangle $ and then it allows to take these left and right states as the one being the exact dual of the other. This interesting issue is currently under analysis.
SOV-representation of local operators
=====================================
The determination of the scalar product formulae, presented in the previous section, is the first main step to compute matrix elements of local operators. The second one is to get the reconstruction of local operators in terms of the generators of the Yang-Baxter algebra, i.e. to invert the map which from the local operators in the Lax matrices leads to the monodromy matrix elements. Indeed, the solution of such an inverse problem allows to compute the action of local operators on the eigenstates of the transfer matrix. Together with the scalar product formulae it leads to the determination of the matrix elements of local operators.
The first reconstruction of local operators has been achieved in [KitMT99]{}, in the case of the XXZ spin 1/2 chain. In [@MaiT00], it has been extended to fundamental lattice models, i.e. those with isomorphicauxiliary and local quantum space, for which the monodromy matrix becomes the permutation operator at a special value of the spectral parameter. The reconstruction also applies to non-fundamental lattice models, as it was shown in [@MaiT00] for the higher spin XXX chains by using the fusion procedure [@KulRS81]. In the case of the sine-Gordon model this type of reconstruction is still missing and the only known results are those given by T. Oota based on the use of quantum projectors [@Oota03]. However, it is worth recalling that Oota’s results only lead to the reconstruction of some local operators of the lattice sine-Gordon model. In this section, we will show how to obtain all the local operators of the sine-Gordon model for the cyclic representations which occur at rational values of the coupling constant $\beta ^{2}$.
Oota’s reconstruction of a class of local operators
---------------------------------------------------
Here we recall some of the results of Oota [@Oota03] which lead to the reconstruction of a certain class of local operators in the sine-Gordon model.
The Lax operator $\mathsf{L}_{n}(\lambda )$ has the following factorization in terms of quantum projectors:$$\begin{aligned}
\mathsf{L}_{n}(\mu _{n,+})& =P_{n,+}Q_{n,+}\equiv \kappa _{n}\left(
\begin{array}{c}
\mathsf{u}_{n}^{1/2}\left( \mathsf{v}_{n}\kappa _{n}+\mathsf{v}_{n}^{-1}\kappa _{n}^{-1}\right) \\
\mathsf{u}_{n}^{-1/2}\left( \mathsf{v}_{n}\kappa _{n}^{-1}+\mathsf{v}_{n}^{-1}\kappa _{n}\right)\end{array}\right) \left(
\begin{array}{cc}
\mathsf{u}_{n}^{1/2} & \mathsf{u}_{n}^{-1/2}\end{array}\right) , \label{L-f+} \\
\mathsf{L}_{n}(\mu _{n,-})& =P_{n,-}Q_{n,-}\equiv \kappa _{n}\left(
\begin{array}{c}
\mathsf{u}_{n}^{1/2} \\
\mathsf{u}_{n}^{-1/2}\end{array}\right) \left(
\begin{array}{cc}
\left( \mathsf{v}_{n}\kappa _{n}+\mathsf{v}_{n}^{-1}\kappa _{n}^{-1}\right)
\mathsf{u}_{n}^{1/2} & \left( \mathsf{v}_{n}\kappa _{n}^{-1}+\mathsf{v}_{n}^{-1}\kappa _{n}\right) \mathsf{u}_{n}^{-1/2}\end{array}\right) , \label{L-f-}\end{aligned}$$when computed in the zeros $\mu _{n,\pm }$ of the quantum determinant; such factorization properties are at the basis of the following Oota’s reconstruction.
The local operators $\mathsf{u}_{n}$ and $\alpha _{0,n}\equiv \left( (q^{-1}\mathsf{v}_{n}^{2}+\kappa _{n}^{2})/\left( q^{-1}\mathsf{v}_{n}^{2}\kappa
_{n}^{2}+1\right) \right) \mathsf{u}_{n}^{-1}$ admit the reconstructions:$$\begin{aligned}
\mathsf{u}_{n} &=&\mathsf{U}_{n}\mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu
_{n,+})\mathsf{U}_{n}^{-1}=\mathsf{U}_{n}\mathsf{D}^{-1}(\mu _{n,+})\mathsf{C}(\mu _{n,+})\mathsf{U}_{n}^{-1}, \label{IPS-1} \\
&& \notag \\
\alpha _{0,n} &=&\mathsf{U}_{n}\mathsf{A}^{-1}(\mu _{n,-})\mathsf{B}(\mu
_{n,-})\mathsf{U}_{n}^{-1}=\mathsf{U}_{n}\mathsf{C}^{-1}(\mu _{n,-})\mathsf{D}(\mu _{n,-})\mathsf{U}_{n}^{-1}, \label{IPS-2}\end{aligned}$$where the shift operator $\mathsf{U}_{n}$ brings the quantum sites from $1$ to $n-1$ to the right end of the chain:$$\mathsf{U}_{n}\mathsf{M}_{1,...,\mathsf{N}}(\lambda )\mathsf{U}_{n}^{-1}\equiv \mathsf{M}_{n,...,\mathsf{N},1,...,n-1}(\lambda )\equiv
\mathsf{L}_{n-1}(\lambda )\cdots \mathsf{L}_{1}(\lambda )\mathsf{L}_{\mathsf{N}}(\lambda )\cdots \mathsf{L}_{n}(\lambda ). \label{Def-Un}$$
It is then clear that the formulae (\[IPS-1\])-(\[IPS-2\]) allow to reconstruct all the powers $\mathsf{u}_{n}^{k}=\mathsf{U}_{n}\left( \mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu _{n,+})\right) ^{k}\mathsf{U}_{n}^{-1}$ of the local operators $\mathsf{u}_{n}$ but they do not give a direct reconstruction of the local operators $\mathsf{v}_{n}$; indeed, only rational functions like $(q^{-1}\mathsf{v}_{n}^{2}+\kappa _{n}^{2})/\left(
q^{-1}\mathsf{v}_{n}^{2}\kappa _{n}^{2}+1\right) $ are obtained.
Let us make some comments on the shift operators $\mathsf{U}_{n}$. The definition (\[Def-Un\]) characterizes the shift operator $_{n}$ up to a constant and implies, by the cyclicity invariance of the trace, their commutativity with the transfer matrix $\mathsf{T}(\lambda )$. Moreover, in the case of even chain, the shift operators $\mathsf{U}_{n}$ clearly commute also with the $\Theta $-charge. Then, in the cyclic representations of the sine-Gordon model under consideration, the simplicity of the transfer matrix spectrum implies:$$\mathsf{U}_{n}|t_{k}\rangle =\varphi _{n}^{(t_{k})}|t_{k}\rangle ,$$where $|t_{k}\rangle $ is the generic eigenstate of $\mathsf{T}(\lambda )$ for odd chain and of ($\mathsf{T}(\lambda )$,$\Theta $) for the even chain. In particular, this implies that the shift operators only produce a prefactor in the form factors of local operators which is one if left and right eigenstates are dual of each other[^12]. It is worth remarking that in[^13] [TarTF83]{}, for the special case of highest weight representations of the even lattice sine-Gordon model, it has been shown that:$$\mathsf{U}_{n}|t\rangle \propto \prod_{a=1}^{n-1}t(\mu _{a})|t\rangle
\propto \prod_{a=1}^{n-1}\frac{Q_{t}(\mu _{a}/q)}{Q_{t}(\mu _{a})}|t\rangle ,
\label{Un-eigenvalues}$$where $\mu _{a}$ are zeros of the quantum determinant in these representations. This result is interesting as it shows that the shift operators $_{n}$ for non-fundamental lattice models, like the sine-Gordon model, are characterized by the same type of eigenvalues they have in fundamental lattice models, like the XXZ spin 1/2 chain. However, the proof of (\[Un-eigenvalues\]) presented in [@TarTF83] is representation dependent, as it is based on the algebraic Bethe ansatz representation of the transfer matrix eigenstates. Then, an independent proof is required for cyclic representations of the sine-Gordon model and it will be given directly in a future publication [@GroMN12] in the more general cyclic representations of the 6-vertex Yang-Baxter algebra associated to the $\tau_2$-model.
Inverse problem solution for all local operators
------------------------------------------------
Here, we show how to complete the reconstruction of local operators by solving the inverse problem for the local operators $\mathsf{v}_{n}$ and their powers. The main ingredient used will be the cyclicity of the representations of the sine-Gordon model here analyzed.
Let us define the local operators:$$\beta _{k,n}\equiv \mathsf{u}_{n}^{k}\alpha _{0,n}\mathsf{u}_{n}^{1-k}=\frac{q^{2k-1}\mathsf{v}_{n}^{2}+\kappa _{n}^{2}}{q^{2k-1}\mathsf{v}_{n}^{2}\kappa
_{n}^{2}+1},$$then the following proposition holds:
\[IPS\]In the cyclic representations of the sine-Gordon model, the local operators $\mathsf{v}_{n}^{2k}$ admit the following reconstruction:$$\mathsf{v}_{n}^{2k}=\frac{\left( -1\right) ^{k}(v_{n}^{2p}\kappa _{n}^{2p}+1)}{p\kappa _{n}^{2k}(\kappa _{n}^{2}-\kappa _{n}^{-2})}\sum_{a=0}^{p-1}q^{-k(2a-1)}\beta _{a,n},\text{ \ \ \ for }k\in
\{1,...,p-1\}, \label{IPS-3}$$where:$$\beta _{k,n}=\mathsf{U}_{n}\left[ \left( \mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu _{n,+})\right) ^{k}\mathsf{A}^{-1}(\mu _{n,-})\mathsf{B}(\mu
_{n,-})\left( \mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu _{n,+})\right) ^{1-k}\right] \mathsf{U}_{n}^{-1}. \label{IPS-4}$$
In our cyclic representations the local operators $\mathsf{u}_{n}$ and $\mathsf{v}_{n}$ satisfy the property that $\mathsf{u}_{n}^{p}$ and $\mathsf{v}_{n}^{p}$ are central, i.e. $\mathsf{u}_{n}^{p}$ and $\mathsf{v}_{n}^{p}$ are just numbers $u_{n}^{p}$ and $v_{n}^{p}$ which characterize our representations. Then the following identity holds:$$\prod_{j=0}^{p-1}(q^{2j-1}\mathsf{v}_{n}^{2}\kappa
_{n}^{2}+1)=1+v_{n}^{2p}\kappa _{n}^{2p},$$and so:$$\frac{v_{n}^{2p}\kappa _{n}^{2p}+1}{q^{2k-1}\mathsf{v}_{n}^{2}\kappa
_{n}^{2}+1}=\sum_{a=0}^{p-1}(-1)^{a}q^{a(2k-1)}\mathsf{v}_{n}^{2a}\kappa
_{n}^{2a}.$$The previous formula allows to rewrite the rational function $\beta _{k,n}$ as a finite sum in powers of $\mathsf{v}_{n}^{2}$:$$\beta _{k,n}=\frac{v_{n}^{2p}\kappa _{n}^{2(p-1)}+\kappa _{n}^{2}+(\kappa
_{n}^{2}-\kappa _{n}^{-2})\sum_{a=1}^{p-1}(-1)^{a}q^{a(2k-1)}\mathsf{v}_{n}^{2a}\kappa _{n}^{2a}}{v_{n}^{2p}\kappa _{n}^{2p}+1},$$then by taking the discrete Fourier transformation, we get the reconstructions (\[IPS-3\]), plus the sum rules:$$\sum_{a=0}^{p-1}\beta _{a,n}=\frac{p v_{n}^{2p}\kappa _{n}^{2(p-1)}+\kappa
_{n}^{2}}{v_{n}^{2p}\kappa _{n}^{2p}+1}.$$Finally, the representation (\[IPS-4\]) for the $\beta _{a,n}$ are trivially derived by (\[IPS-1\])-(\[IPS-2\]).
Note that thanks to the identities $\mathsf{v}_{n}^{k}=\mathsf{v}_{n}^{2h}/v_{n}^{p}$ for $k=2h-p$ odd integer smaller than $p$, the formulae (\[IPS-3\]) indeed lead to the reconstruction of all the powers $\mathsf{v}_{n}^{k}$ for $k\in \{1,...,p-1\}$. Then the previous proposition together with Oota’s reconstructions leads to the announced complete reconstruction of local operators for cyclic representations of the sine-Gordon model.
SOV-representations of all local operators
------------------------------------------
In order to compute the action of the local operators $\mathsf{v}_{n}^{k}$ and $\mathsf{u}_{n}^{k}$ on eigenstates of the transfer matrix and eventually obtain their form factors, it is necessary to first derive their SOV-representations[^14]. Here, we show how these SOV-representations can be obtained from the previously given solutions of the inverse problem. In order to simplify the presentation, we introduce here explicitly the operators[^15] $\eta _{1},...,\eta _{\mathsf{N}},\eta _{\mathsf{A}}$ and $\eta _{\mathsf{D}}$ defined by the following actions on the generic element $\langle \mathbf{k}|\equiv \langle \eta
_{1}^{\left( k_{1}\right) },...,\eta _{\mathsf{N}}^{\left( k_{\mathsf{N}}\right) }|$ of the left $\mathsf{B}$-eigenbasis:$$\langle \mathbf{k}|\eta _{a}=\eta _{a}^{\left( k_{a}\right) }\langle \mathbf{k}|,\text{ \ }\forall a\in \{1,...,\mathsf{N}\}\text{, \ \ }\langle \mathbf{k}|\eta _{\mathsf{A}}=\eta _{\mathbf{k,}\mathsf{A}}\langle \mathbf{k}|\text{
\ and \ \ }\langle \mathbf{k}|\eta _{\mathsf{D}}=\eta _{\mathbf{k,}\mathsf{D}}\langle \mathbf{k}|,$$where the corresponding eigenvalues $\eta _{a}^{\left( k_{a}\right) }$, $\eta _{\mathbf{k,}\mathsf{A}}$ and $\eta _{\mathbf{k,}\mathsf{D}}$ are the complex numbers defined in Section \[SOV-Left\]. The following lemma is important as it solves the combinatorial problem related to the computations of the SOV-representations of monomials in the Yang-Baxter generators:
The operator $\left( \mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )\right)
^{k}$ has the following left SOV-representation:$$\begin{aligned}
\left( \mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )\right) ^{k}& =\text{\textsc{k}}^{-k}\sum_{\substack{ \bar{\alpha}\equiv \{\alpha _{1},...,\alpha
_{\mathsf{N}}\}\in \mathbb{N}_{0}^{\mathsf{N}}: \\ \sum_{h=1}^{\mathsf{N}}\alpha _{h}=k}}\left[
\begin{array}{c}
k \\
\bar{\alpha}\end{array}\right] \prod_{j=1}^{\mathsf{N}}\left( \prod_{h=0}^{\alpha _{j}-1}\frac{a(\eta _{j}q^{-h})}{(\lambda q^{h}/\eta _{j}-\eta _{j}/\lambda q^{h})}\right.
\notag \\
& \times \left. \prod_{i\neq j,i=1}^{\mathsf{N}}\prod_{h=\alpha _{i}-\alpha
_{j}+1}^{\alpha _{i}}\frac{1}{(\eta _{j}q^{h}/\eta _{i}-\eta _{i}/\eta
_{j}q^{h})}\right) \prod_{j=1}^{\mathsf{N}}\mathsf{T}_{j}^{-\alpha _{j}},
\label{A/B^k}\end{aligned}$$acting on the state $\langle \eta _{1},...,\eta _{\mathsf{N}}|$, where:$$\left[
\begin{array}{c}
k \\
\bar{\alpha}\end{array}\right] \equiv \frac{\lbrack k]!}{\prod_{j=1}^{\mathsf{N}}\left[ \alpha _{j}\right] !},\text{ }[k]!\equiv \lbrack k][k-1]\cdots \lbrack 1],\text{ }[a]\equiv \frac{q^{a}-q^{-a}}{q-q^{-1}}.$$
Here we use the commutation relations:$$\mathsf{T}_{a}^{\pm }\eta _{b}=q^{\pm \delta _{ab}}\eta _{b}\mathsf{T}_{a}^{\pm },$$and the following left SOV-representation:$$\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )=\text{\textsc{k}}^{-1}\sum_{a=1}^{\mathsf{N}}\frac{a(\eta _{a})}{({\lambda }/{\eta _{a}}-{\eta _{a}}/{\lambda })}\prod_{b\neq a}\frac{1}{({\eta _{a}}/{\eta _{b}}-{\eta _{b}}/{\eta _{a}})}\mathsf{T}_{a}^{-1},$$then the lemma holds for $k=1$ and we prove it by induction for $k>1$. Let us take $\mathsf{N}$ integers $\alpha _{i}$: $$\sum_{a=1}^{\mathsf{N}}\alpha _{i}=k,$$from which we define the set of integers $I=\{i\in \{1,...,\mathsf{N}\}:\alpha _{i}\neq 0\}$ and $C_{{\bar{\alpha}}}^{(k)}$ as the coefficient of $\prod \mathsf{T}_{i}^{-\alpha _{i}}$ in the expansion of the $k$-th power of $\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )$. By writing $(\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda ))^{k}=(\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda ))^{k-1}\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )$ and by using the induction hypothesis for the power $k-1$ of $\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )$, we have:$$\begin{aligned}
C_{{\bar{\alpha}}}^{(k)}& =\text{\textsc{k}}^{-k}\sum_{a\in I}{\left[
\begin{array}{c}
{k-1} \\
{\bar{\alpha}-\bar{\delta}_{a}}\end{array}\right] } \notag \\
& \prod_{j=1}^{N}\prod_{h=0}^{\alpha _{j}-\delta _{a,j}-1}\left( \frac{a(\eta _{j}q^{-h})}{({\lambda q^{h}}/{\eta _{j}}-{\eta _{j}}/{\lambda q^{h}})}\times \prod_{i\neq j,i=1}^{N}\frac{1}{{q^{\alpha _{i}-\delta _{a,i}-h}\eta
_{j}}/{\eta _{i}}-{\eta _{i}}/{q^{\alpha _{i}-\delta _{a,i}-h}\eta _{j}}}\right) \notag \\
& \times \frac{a(\eta _{a}q^{-\alpha _{a}+1})}{({\lambda q^{\alpha _{a}-1}}/{\eta _{a}}-{\eta _{a}}/{\lambda q^{\alpha _{a}-1}})}\prod_{i\in I\backslash
\{a\}}\frac{1}{{q^{\alpha _{a}-\alpha _{i}-1}\eta _{i}}/{\eta _{a}}-{\eta
_{a}}/{q^{\alpha _{a}-\alpha _{i}-1}\eta _{i}}},\end{aligned}$$with $\bar{\delta}_{a}\equiv (\delta _{1,a},\dots ,\delta _{\mathsf{N},a})$. The first term in the r.h.s. is the coefficient of $\prod \mathsf{T}_{i}^{-\alpha _{i}+\delta _{a,i}}$ in $(\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda ))^{k-1}$ and the second is the coefficient of $\mathsf{T}_{a}^{-1}$ in $\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )$ once the commutations between $\prod \mathsf{T}_{i}^{-\alpha _{i}+\delta _{a,i}}$ and the $\eta
_{i}$ have been performed. This can be rewritten as:$$\begin{aligned}
C_{{\bar{\alpha}}}^{(k)}=\text{\textsc{k}}^{-k}\frac{[k-1]!}{\prod [\alpha
_{i}]!}& \left( \prod_{j=1}^{N}\prod_{h=0}^{\alpha _{j}-1}(\prod_{i\neq
j,i=1}^{N}\frac{1}{{q^{\alpha _{i}-h}\eta _{j}}/{\eta _{i}}-{\eta _{i}}/{q^{\alpha _{i}-h}\eta _{j}}})\frac{a(q^{-h}\eta _{j})}{({\lambda q^{h}}/{\eta _{j}}-{\eta _{j}}/{\lambda q^{h}})}\right) \notag \\
& \times \sum_{a\in I}([\alpha _{a}]\prod_{i\in I\backslash \{a\}}\frac{{q^{\alpha _{a}}\eta _{i}}/{\eta _{a}}-{\eta _{a}}/{q^{\alpha _{a}}\eta _{i}}}{{q^{\alpha _{a}-\alpha _{i}}\eta _{i}}/{\eta _{a}}-{\eta _{a}}/{q^{\alpha
_{a}-\alpha _{i}}\eta _{i}}}),\end{aligned}$$which leads to our result when we use the relation:$$\sum_{a=1}^{n}[\alpha _{a}]\prod_{i\neq a}\frac{{q^{\alpha _{a}}\eta _{i}}/{\eta _{a}}-{\eta _{a}}/{q^{\alpha _{a}}\eta _{i}}}{{q^{\alpha _{a}-\alpha
_{i}}\eta _{i}}/{\eta _{a}}-{\eta _{a}}/{q^{\alpha _{a}-\alpha _{i}}\eta _{i}}}=\left[ \sum_{a=1}^{n}\alpha _{a}\right] .$$The above formula holds for any $n$, for any set of numbers $\eta _{i}$ and for any non-negative integers $\alpha _{i}$, all of them being in generic position. This is shown by studying the contour integral and the residues of the function: $$g(z)=\frac{1}{z}\prod_{i=1}^n \frac{z-\eta _{i}^{2}}{z-q^{-2\alpha _{i}}\eta _{i}^{2}}.$$
**Remark 5.** Let us point out that the power $p$ of $\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )$ is a central element of the Yang-Baxter algebra and it reads:$$(\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda ))^{p}=\mathcal{B}(\Lambda
)^{-1}\mathcal{A}(\Lambda ), \label{A/B^p-a}$$as it simply follows from the commutations relations:$$\mathsf{B}^{-1}(q\lambda )\mathsf{A}(q\lambda )=\mathsf{A}(\lambda )\mathsf{B}^{-1}(\lambda ).$$Then, it is important to verify that the same result follows from the previous lemma for $k=p$. In order to prove it, it is enough to use the following properties of the quantum binomials:$$\text{ }{\left[
\begin{array}{c}
p \\
{\bar{\alpha}}\end{array}\right] }=\left\{
\begin{array}{l}
1\text{ \ \ if }\exists i\in \{1,...,\mathsf{N}\}:\alpha _{i}=p\delta _{a,i}\text{\ }\forall a\in \{1,...,\mathsf{N}\}, \\
0\text{ \ \ otherwise,}\end{array}\right.$$from which the formula (\[A/B\^k\]) for $k=p$ reads: $$(\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda ))^{p}=\text{\textsc{k}}^{-p}\sum_{a=1}^{p}\frac{\prod_{k=1}^{p}a(q^{k}\eta _{a})}{({\Lambda }/{Z_{a}}-{Z_{a}}/{\Lambda })}\prod_{b\neq a}\frac{1}{({Z_{b}}/{Z_{a}}-{Z_{a}}/{Z_{b}})}, \label{A/B^p-b}$$ and to observe that the r.h.s. of (\[A/B\^p-b\]) indeed coincides with $\mathcal{B}(\Lambda )^{-1}\mathcal{A}(\Lambda )$.
Note that the above lemma gives directly the SOV-representations of the local operators $\mathsf{u}_{n}^{k}$ and $\alpha _{0,n}^{-k}$ with $k\in
\{1,...,p-1\}$ when we fix the parameter $\lambda $ to $\mu _{n,\varepsilon
} $ with $\varepsilon =+$ and $\varepsilon =-$, respectively. Moreover, it allows to derive also the SOV-representations of the local operators $\mathsf{v}_{n}^{k}$ as it follows:
The local operators $\mathsf{v}_{n}^{2k}$ with $k\in \{1,...,p-1\}$ have the following left SOV-representation:$$\begin{aligned}
\mathsf{U}_{n}^{-1}\mathsf{v}_{n}^{2k}\mathsf{U}_{n}& =\text{\textsc{v}}_{n}^{(2k)}+\sum_{a=1}^{\mathsf{N}}\sum_{\substack{ \bar{\alpha}\equiv
\{\alpha _{1},...,\alpha _{\mathsf{N}}\}\in \mathbb{\mathsf{N}}_{0}^{\mathsf{N}}: \\ \sum_{h=1}^{\mathsf{N}}\alpha _{h}=p-1}}\left[
\begin{array}{c}
p-1 \\
\bar{\alpha}\end{array}\right] \prod_{j=1}^{\mathsf{N}}\left( \prod_{h=0}^{\alpha _{j}-1}\frac{a(\eta _{j}q^{-h})}{(\mu _{n,-}q^{h}/\eta _{j}-\eta _{j}/\mu _{n,-}q^{h})}\right. \notag \\
& \times \left. \prod_{i\neq j,i=1}^{\mathsf{N}}\prod_{h=\alpha _{i}-(\alpha
_{j}+\delta _{j,a})+1}^{\alpha _{i}}\frac{1}{(\eta _{j}q^{h}/\eta _{i}-\eta
_{i}/\eta _{j}q^{h})}\right) \text{\textsc{v}}_{n,(a,\bar{\alpha})}^{(2k)}\prod_{j=1}^{\mathsf{N}}\text{T}_{j}^{-(\alpha _{j}+\delta _{j,a})},\end{aligned}$$where:$$\begin{aligned}
\text{\textsc{v}}_{n}^{(2k)}& \equiv \frac{\left( -1\right) ^{k}(\kappa
_{n}^{2p}+1)}{p\kappa _{n}^{2k}(\kappa _{n}^{2}-\kappa _{n}^{-2})}\sum_{r=1}^{p}\frac{q^{-k(2r-1)}(q^{r}-q^{-r})}{q^{r}\kappa
_{n}^{2}-q^{-r}\kappa _{n}^{-2}}, \\
& \notag \\
\text{\textsc{v}}_{n,(a,\bar{\alpha})}^{(2k)}& \equiv \frac{\left( -1\right)
^{k}(\kappa _{n}^{2p}+1)}{p\kappa _{n}^{2k}(\kappa _{n}^{2}-\kappa _{n}^{-2})}\sum_{r=1}^{p}\frac{q^{-k(2r-1)}(\kappa _{n}^{2}-\kappa _{n}^{-2})a(\eta
_{a}q^{-\alpha _{a}})}{\left( q^{r}\kappa _{n}^{2}-q^{-r}\kappa
_{n}^{-2}\right) (\mu _{n,+}q^{\alpha _{a}+r}/\eta _{a}-\eta _{a}/\mu
_{n,+}q^{\alpha _{a}+r})} \notag \\
& \times \prod_{j=1}^{\mathsf{N}}\prod_{h=0}^{r-1}\frac{(\mu _{n,+}q^{\alpha
_{j}+h}/\eta _{j}-\eta _{j}/\mu _{n,+}q^{\alpha _{j}+h})}{(\mu
_{n,+}q^{h}/\eta _{j}-\eta _{j}/\mu _{n,+}q^{h})}.\end{aligned}$$
This is a consequence of the previous lemma and of the identities:$$\mathsf{U}_{n}^{-1}\beta _{k,n}\mathsf{U}_{n}=\frac{\kappa _{n}^{2}-\kappa
_{n}^{-2}}{q^{k}\kappa _{n}^{2}-q^{-k}\kappa _{n}^{-2}}\gamma _{k,n}+\frac{q^{k}-q^{-k}}{q^{k}\kappa _{n}^{2}-q^{-k}\kappa _{n}^{-2}},\text{ \ for }k\in \{1,...,p-1\},$$where:$$\gamma _{k,n}=\mathsf{B}^{-1}(\mu _{n,+}^{p})\prod_{j=k}^{p-1}\mathsf{B}(\mu
_{n,+}q^{j})\mathsf{A}^{-1}(\mu _{n,-})\mathsf{B}(\mu _{n,-})\mathsf{B}^{-1}(\mu _{n,+}q^{k})\mathsf{A}(\mu _{n,+}q^{k})\prod_{j=0}^{k-1}\mathsf{B}(\mu _{n,+}q^{j}),$$Now by using the relations:$$\mathsf{A}^{-1}(\mu _{n,-})\mathsf{B}(\mu _{n,-})=\left( \mathsf{B}^{-1}(\mu
_{n,-})\mathsf{A}(\mu _{n,-})\right) ^{p-1},$$we get the following representations: $$\begin{aligned}
\gamma _{k,n}& =\sum_{a=1}^{\mathsf{N}}\sum_{\substack{ \bar{\alpha}\equiv
\{\alpha _{1},...,\alpha _{\mathsf{N}}\}\in \mathbb{\mathsf{N}}_{0}^{\mathsf{N}}: \\ \sum_{h=1}^{\mathsf{N}}\alpha _{h}=p-1}}\left[
\begin{array}{c}
p-1 \\
\bar{\alpha}\end{array}\right] \prod_{j=1}^{\mathsf{N}}\left( \prod_{h=0}^{\alpha _{j}-1}\frac{a(\eta _{j}q^{-h})}{(\mu _{n,-}q^{h}/\eta _{j}-\eta _{j}/\mu _{n,-}q^{h})}\right. \notag \\
& \times \left. \prod_{i\neq j,i=1}^{\mathsf{N}}\prod_{h=\alpha _{i}-(\alpha
_{j}+\delta _{j,a})+1}^{\alpha _{i}}\frac{1}{(\eta _{j}q^{h}/\eta _{i}-\eta
_{i}/\eta _{j}q^{h})}\right) \frac{a(\eta _{a}q^{-\alpha _{a}})}{(\mu
_{n,+}q^{\alpha _{a}+r}/\eta _{a}-\eta _{a}/\mu _{n,+}q^{\alpha _{a}+r})}
\notag \\
& \times \prod_{j=1}^{\mathsf{N}}\prod_{h=0}^{r-1}\frac{(\mu _{n,+}q^{\alpha
_{j}+h}/\eta _{j}-\eta _{j}/\mu _{n,+}q^{\alpha _{j}+h})}{(\mu
_{n,+}q^{h}/\eta _{j}-\eta _{j}/\mu _{n,+}q^{h})}\prod_{j=1}^{\mathsf{N}}\text{T}_{j}^{-(\alpha _{j}+\delta _{j,a})},\end{aligned}$$from which our result follows.
Form factors of local operators\[FF-loc-Op\]
============================================
In the following we will compute matrix elements (form factors) of the form[^16]: $$\langle t|O_{n}|t^{\prime }\rangle \label{General-ME}$$which by definition are the action of a covector $\langle t|\in \mathcal{L}_{\mathsf{N}}$, a left $\mathsf{T}$-eigenstate defined in (\[eigenT-l\]), on the vector obtained by the action of the local operator $O_{n}$ on the right $\mathsf{T}$-eigenstate $|t^{\prime }\rangle \in \mathcal{R}_{\mathsf{N}}$ defined in (\[eigenT-r\]). Of course, these form factors depend on the normalization of the states $\langle t|$ and $|t^{\prime }\rangle $ and then it is worth pointing out that nevertheless we can use them to expand m-point functions like:$$\frac{\langle t|O_{n_{1}}\cdots O_{n_{\text{m}}}|t\rangle }{\langle
t|t\rangle }. \label{General-n-point-F}$$Indeed, by definition these m-point functions are normalization independent and by using m-1 times the decomposition of the identity $\left( \ref{Id-decomp}\right) $, we get the expansions:$$\frac{\langle t|O_{n_{1}}\cdots O_{n_{\text{m}}}|t\rangle }{\langle
t|t\rangle }=\sum_{t^{\left( 1\right) }(\lambda ),...,t^{\left( \text{m}-1\right) }(\lambda )\in \Sigma _{\mathsf{T}}}\frac{\langle
t|O_{n_{1}}|t^{(1)}\rangle \langle t^{(\text{m}-1)}|O_{n_{\text{m}}}|t\rangle \prod_{a=2}^{\text{m}-1}\langle t^{\left( a-1\right)
}|O_{n_{a}}|t^{\left( a\right) }\rangle }{\langle t|t\rangle \prod_{a=1}^{\text{m}-1}\langle t^{\left( a\right) }|t^{\left( a\right) }\rangle },
\label{FF-expansion}$$where in the r.h.s there are exactly the form factors $\left( \ref{General-ME}\right) $ that we are going to compute in this paper.
Form factors of $\mathsf{u}_{n}$
--------------------------------
In this section we use the SOV-representations of local operators to give some examples of completely resummed form factors.
\[FF-Prop1\]Let $\langle t_{k}|$ and $|t_{k^{\prime }}^{\prime }\rangle $ be two eigenstates of the transfer matrix $\mathsf{T}(\lambda )$, then it holds:$$\langle t_{k}|\mathsf{u}_{n}|t_{k^{\prime }}^{\prime }\rangle =\frac{\varphi
_{n}^{(t_{k})}}{\varphi _{n}^{(t_{k^{\prime }}^{\prime })}}\left( \delta
_{k,k^{\prime }+1}\right) ^{\mathtt{e}_{\mathsf{N}}}\det_{[\mathsf{N}]}(||\mathcal{U}_{a,b}^{\left( t,t^{\prime }\right) }(\mu _{n,+})||), \label{u}$$where $\varphi _{n}^{(t_{k})}$ and $\varphi _{n}^{(t_{k^{\prime }}^{\prime
})}$ are the eigenvalues of the shift operator $\mathsf{U}_{n}$ and $||\mathcal{U}_{a,b}^{\left( t,t^{\prime }\right) }(\lambda )||$ is the $[\mathsf{N}]\times \lbrack \mathsf{N}]$ matrix:$$\begin{aligned}
\mathcal{U}_{a,b}^{\left( t,t^{\prime }\right) }(\lambda )& \equiv \Phi
_{a,b+1/2}^{\left( t,t^{\prime }\right) }\text{ \ for \ }b\in \{1,...,[\mathsf{N}]-1\}, \\
\mathcal{U}_{a,[\mathsf{N}]}^{\left( t,t^{\prime }\right) }(\lambda )&
\equiv \frac{\left( \eta _{a}^{(0)}\right) ^{[\mathsf{N}]-1}}{\text{\textsc{k}}\eta _{\mathsf{N}}^{(0)}{}^{\mathtt{e}_{\mathsf{N}}}}\sum_{h=1}^{p}\frac{q^{([\mathsf{N}]-1)h}Q_{t^{\prime }}(\eta _{a}^{(h)})}{\omega _{a}(\eta
_{a}^{(h)})}\left[ \frac{\bar{Q}_{t}(\eta _{a}^{(h+1)})}{(\lambda /\eta
_{a}^{(h+1)}-\eta _{a}^{(h+1)}/\lambda )}\bar{a}(\eta _{a}^{(h)})\right.
\notag \\
& +\left. \mathtt{e}_{\mathsf{N}}\bar{Q}_{t}(\eta _{a}^{(h)})\left( \frac{\lambda }{\prod_{j=1}^{\mathsf{N}}\xi _{j}}\left( \eta _{a}^{(h)}\right) ^{[\mathsf{N}]}q^{k^{\prime }}+\frac{\prod_{j=1}^{\mathsf{N}}\xi _{j}}{\lambda }\left( \eta _{a}^{(h)}\right) ^{-[\mathsf{N}]}q^{-k^{\prime }}\right) \right]
,\end{aligned}$$where <span style="font-variant:small-caps;">k</span>$\equiv \prod_{n=1}^{\mathsf{N}}\kappa _{n}/i$.
The right SOV-representation of the operator $\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )$ reads:$$\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )|\mathbf{k}\rangle=\frac{\mathtt{e}_{\mathsf{N}}}{\eta _{\mathsf{N}}}\left( \frac{\lambda }{\eta _{\mathsf{A}}}\mathsf{T}_{\mathsf{N}}^{+}+\frac{\eta _{\mathsf{A}}}{\lambda }\mathsf{T}_{\mathsf{N}}^{-}\right)|\mathbf{k}\rangle +\sum_{a=1}^{[\mathsf{N}]}\mathsf{T}_{a}^{+}|\mathbf{k}\rangle\frac{\bar{a}(\eta
_{a})}{\text{\textsc{k}}\eta _{\mathsf{N}}^{\mathtt{e}_{\mathsf{N}}}(\lambda
/\eta _{a}q-\eta _{a}q/\lambda )}\prod_{b\neq a}\frac{1}{(\eta _{a}/\eta
_{b}-\eta _{b}/\eta _{a})}, \label{ExB-1A}$$Let us denote with $[\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )]$ the second term on the r.h.s. of (\[ExB-1A\]). Then, we observe that from the SOV-decomposition of the $\mathsf{T}$-eigenstates, we have:$$\begin{aligned}
\langle t_{k}|[\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )]|t_{k^{\prime
}}^{\prime }\rangle & =\left( \frac{\sum_{h_{\mathsf{N}}=1}^{p}q^{(k-1-k^{\prime })h_{\mathsf{N}}}}{p\eta^{(0)}_{\mathsf{N}}}\right) ^{\mathtt{e}_{\mathsf{N}}}\sum_{a=1}^{[\mathsf{N}]}\sum_{h_{1},...,h_{[\mathsf{N}]}=1}^{p}V(\left( \eta _{1}^{(h_{1})}\right) ^{2},...,\left( \eta _{\lbrack
\mathsf{N}]}^{(h_{[\mathsf{N}]})}\right) ^{2}) \notag \\
& \times \prod_{b\neq a,b=1}^{[\mathsf{N}]}\frac{\eta
_{b}^{(h_{b})}Q_{t^{\prime }}(\eta _{b}^{(h_{b})})\bar{Q}_{t}(\eta
_{b}^{(h_{b})})}{\omega _{b}(\eta _{b}^{(h_{b})})((\eta
_{a}^{(h_{a})})^{2}-(\eta _{b}^{(h_{b})})^{2})} \notag \\
& \times \frac{\bar{Q}_{t}(\eta _{a}^{(h_{a}+1)})Q_{t^{\prime }}(\eta
_{a}^{(h_{a})})}{\omega _{a}(\eta _{a}^{(h_{a})})\text{\textsc{k}}}\frac{\left( \eta _{a}^{(h_{a})}\right) ^{[\mathsf{N}]-1}\bar{a}(\eta
_{a}^{(h_{a})})}{(\lambda /\eta _{a}^{(h_{a}+1)}-\eta
_{a}^{(h_{a}+1)}/\lambda )},\end{aligned}$$and so:$$\begin{aligned}
\langle t_{k}|[\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )]|t_{k^{\prime
}}^{\prime }\rangle & =\left( \frac{\delta _{k,k^{\prime }+1}}{\eta _{\mathsf{N}}^{(0)}}\right) ^{\mathtt{e}_{\mathsf{N}}}\sum_{a=1}^{[\mathsf{N}]}\sum_{\substack{ h_{1},...,h_{\mathsf{N}}=1 \\ {\small \overbrace{h_{a}\text{ is missing.}}}}}^{p}\underset{\ \ \ {\small \overbrace{(\text{We have
removed the row }a\text{.}})}}{\hat{V}_{a}(\left( \eta _{1}^{(h_{1})}\right)
^{2},...,\left( \eta _{\lbrack \mathsf{N}]}^{(h_{[\mathsf{N}]})}\right) ^{2})} \notag \\
& \times \prod_{b\neq a,b=1}^{[\mathsf{N}]}\frac{\eta
_{b}^{(h_{b})}Q_{t^{\prime }}(\eta _{b}^{(h_{b})})\bar{Q}_{t}(\eta
_{b}^{(h_{b})})}{\omega _{b}(\eta _{b}^{(h_{b})})} \notag \\
& \times (-1)^{[\mathsf{N}]+a}\sum_{h_{a}=1}^{p}\frac{\bar{Q}_{t}(\eta
_{a}^{(h_{a}+1)})Q_{t^{\prime }}(\eta _{a}^{(h_{a})})\left( \eta
_{a}^{(h_{a})}\right) ^{[\mathsf{N}]-1}\bar{a}(\eta _{a}^{(h_{a})})}{\omega
_{a}(\eta _{a}^{(h_{a})})\text{\textsc{k}}(\lambda /\eta
_{a}^{(h_{a}+1)}-\eta _{a}^{(h_{a}+1)}/\lambda )},\end{aligned}$$bringing the sum over ($h_{1},...,\widehat{h_{a}},...,h_{[\mathsf{N}]}$) inside the Vandermonde determinant $\hat{V}_{a}$, we have that the above expression is just the expansion of the determinant:$$\langle t_{k}|[\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )]|t_{k^{\prime
}}^{\prime }\rangle =\left( \delta _{k,k^{\prime }+1}\right) ^{\mathtt{e}_{\mathsf{N}}}\det_{[\mathsf{N}]}(||\left[ \mathcal{U}_{a,b}^{\left(
t,t^{\prime }\right) }(\lambda )\right] ||), \label{FF-Odd-part}$$where $\left[ \mathcal{U}_{a,b}^{\left( t,t^{\prime }\right) }(\lambda )\right] $ coincides with $\Phi _{a,b+1/2}^{\left( t,t^{\prime }\right) }$ for $b\in \{1,...,[\mathsf{N}]-1\}$, while:$$\left[ \mathcal{U}_{a,[\mathsf{N}]}^{\left( t,t^{\prime }\right) }(\lambda )\right] \equiv \frac{\left( \eta _{a}^{(0)}\right) ^{[\mathsf{N}]-1}}{\text{\textsc{k}}\bar{\eta}_{\mathsf{N}}^{\mathtt{e}_{\mathsf{N}}}}\sum_{h=1}^{p}\frac{q^{([\mathsf{N}]-1)h}Q_{t^{\prime }}(\eta _{a}^{(h)})\bar{Q}_{t}(\eta
_{a}^{(h+1)})}{\omega _{a}(\eta _{a}^{(h)})(\lambda /\eta _{a}^{(h+1)}-\eta
_{a}^{(h+1)}/\lambda )}\bar{a}(\eta _{a}^{(h)}).$$Now let us compute the matrix elements:$$\begin{aligned}
\langle t_{k}|\eta _{\mathsf{N}}^{-1}\eta _{\mathsf{A}}^{\mp }\mathsf{T}_{\mathsf{N}}^{\pm }|t_{k^{\prime }}^{\prime }\rangle & =\left( \frac{\sum_{h_{\mathsf{N}}=1}^{p}q^{(k-1-k^{\prime })h_{\mathsf{N}}}}{p\eta _{\mathsf{N}}^{(0)}\prod_{j=1}^{\mathsf{N}}\xi _{j}^{\pm 1}}\right) ^{\mathtt{e}_{\mathsf{N}}}\sum_{h_{1},...,h_{[\mathsf{N}]}=1}^{p}V(\left( \eta
_{1}^{(h_{1})}\right) ^{2},...,\left( \eta _{\lbrack \mathsf{N}]}^{(h_{[\mathsf{N}]})}\right) ^{2}) \notag \\
& \times \prod_{b=1}^{[\mathsf{N}]}\frac{\left( \eta _{b}^{(h_{b}+k^{\prime
})}\right) ^{\pm 1}Q_{t^{\prime }}(\eta _{b}^{(h_{b})})\bar{Q}_{t}(\eta
_{b}^{(h_{b})})}{\omega _{b}(\eta _{b}^{(h_{b})})},\end{aligned}$$and so:$$\langle t_{k}|\eta _{\mathsf{N}}^{-1}\eta _{\mathsf{A}}^{\mp }\mathsf{T}_{\mathsf{N}}^{\pm }|t_{k^{\prime }}^{\prime }\rangle =\left( \frac{q^{\pm
k^{\prime }}\delta _{k,k^{\prime }+1}}{\eta _{\mathsf{N}}^{(0)}\prod_{j=1}^{\mathsf{N}}\xi _{j}^{\pm 1}}\right) ^{\mathtt{e}_{\mathsf{N}}}\det_{[\mathsf{N}]}(||\Phi _{a,b\pm 1/2}^{\left( t,t^{\prime }\right) }||).
\label{FF-Even-part}$$By using the fact that $[\mathsf{N}]-1$ columns are common in the matrix of formula (\[FF-Odd-part\]) and in those of (\[FF-Even-part\]), we get our result. Let us remark that the above result holds for any value of $\lambda $.
**Remark 6.** It is worth pointing out that the form factors of $\mathsf{u}_{n}$ are written in terms of a determinant of a matrix whose elements coincide with those of the scalar product, except for the last line which is modified by the presence of the local operator. It is then interesting to recall that a similar statement holds for the form factors of the local operators in the XXZ spin 1/2 chain.
Suitable operator basis for form factor computations
----------------------------------------------------
In this section we introduce an operator basis which can be conveniently used to describe all local operators. The interest toward this basis is due to the fact that the form factors of its elements are simple being represented by a determinant formula.
### Basis of elementary operators
Let us introduce the following operators:$$\mathcal{O}_{a,k}\equiv \frac{\mathsf{B}(\eta _{a}^{(p+k-1)})\mathsf{B}(\eta
_{a}^{(p+k-2)})\cdots \mathsf{B}(\eta _{a}^{(k+1)})\mathsf{A}(\eta
_{a}^{(k)})}{p\eta _{\mathsf{N}}^{\mathtt{e}_{\mathsf{N}}(p-1)}\text{\textsc{k}}^{(p-1)}\prod_{b\neq a,b=1}^{[\mathsf{N}]}(Z_{a}/Z_{b}-Z_{b}/Z_{a})}\text{
\ with }k\in \{0,...,p-1\},$$where the $\eta _{a}^{(k)}$ are fixed in Section \[SOV-Left\].
The operators $\mathcal{O}_{a,k}$ satisfy the following properties:$$\mathcal{O}_{a,k}\mathcal{O}_{a,h}\text{ is non-zero if and only if }h=k-1,
\label{Prod-O-zeros}$$and$$\mathcal{O}_{a,k}\mathcal{O}_{a,k-1}\cdots \mathcal{O}_{a,k+1-p}\mathcal{O}_{a,k-p}=\frac{\mathcal{A}(Z_{a})}{\prod_{b\neq a,b=1}^{[\mathsf{N}]}(Z_{a}/Z_{b}-Z_{b}/Z_{a})}\mathcal{O}_{a,k}. \label{O-mean-value}$$Moreover the following commutation relations hold:$$\eta _{\mathsf{A}}\mathcal{O}_{a,k}=q\mathcal{O}_{a,k}\eta _{\mathsf{A}},\text{ \ }[\eta _{\mathsf{N}},\mathcal{O}_{a,k}]=[\Theta ,\mathcal{O}_{a,k}]=0,$$and $$\mathcal{O}_{a,k}\mathcal{O}_{b,h}=\frac{(\eta _{a}^{(k-h+1)}/\eta
_{b}^{(0)}-\eta _{b}^{(0)}/\eta _{a}^{(k-h+1)})}{(\eta _{a}^{(k-h-1)}/\eta
_{b}^{(0)}-\eta _{b}^{(0)}/\eta _{a}^{(k-h-1)})}\text{ }\mathcal{O}_{b,h}\mathcal{O}_{a,k} \label{Com-O}$$for $a\neq b\in \{1,...,[\mathsf{N}]\}$.
The first property follows from $\mathcal{B}(Z_{a})=0$, where $\mathcal{B}(\Lambda )$ is the average value of $\mathsf{B}(\lambda )$. By the definition of $\mathcal{O}_{a,k}$ it is clear that:$$\langle \eta _{1}^{(k_{1})},...,\eta _{a}^{(h)},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|\mathcal{O}_{a,k}=\frac{a(\eta _{a}^{(k)})\delta _{h,k}}{\prod_{b\neq a,b=1}^{\mathsf{N}}(\eta _{a}^{(k)}/\eta _{b}^{(k_{b})}-\eta
_{b}^{(k_{b})}/\eta _{a}^{(k)})}\langle \eta _{1}^{(k_{1})},...,\eta
_{a}^{(k-1)},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|, \label{O-Action}$$so that the second property simply follows. To prove the last property we have to use the Yang-Baxter commutation relation:$$(\lambda /\mu -\mu /\lambda )\mathsf{A}(\lambda )\mathsf{B}(\mu )=(\lambda
/q\mu -\mu q/\lambda )\mathsf{B}(\mu )\mathsf{A}(\lambda )+(q-q^{-1})\mathsf{B}(\lambda )\mathsf{A}(\mu ) \label{AB-Yang-Baxter}$$we have before to move the $\mathsf{A}(\eta _{a}^{(k)})$ to the right through all the $\mathsf{B}(\eta _{a}^{(j)})$, remarking that only the first term of the r.h.s of (\[AB-Yang-Baxter\]) survives, and after to move the $\mathsf{A}(\eta _{a}^{(h)})$ to the left.
Let us introduce now the following monomials which we will call *elementary operators:*$$\mathcal{E}_{(k,k_{0})^{\mathtt{e}_{\mathsf{N}}},(a_{1},k_{1}),...,(a_{r},k_{r})}^{(\alpha _{1},...,\alpha _{r})}\equiv
\eta _{\mathsf{N}}^{-\mathtt{e}_{\mathsf{N}}k}\left( \frac{\Theta }{\eta _{\mathsf{A}}}\right) ^{\mathtt{e}_{\mathsf{N}}k_{0}}\mathcal{O}_{a_{1},k_{1}}^{(\alpha _{1})}\cdots \mathcal{O}_{a_{r},k_{r}}^{(\alpha
_{r})}, \label{O-basis}$$where $\sum_{h=1}^{r}\alpha _{h}\leq p,$ $k,k_{i}\in \{0,...,p-1\},\ \
a_{i}<a_{j}\in \{1,...,[\mathsf{N}]\}$ for $i<j\in \{1,...,[\mathsf{N}]\}$ and:$$\mathcal{O}_{a,k}^{(\alpha )}\equiv \mathcal{O}_{a,k}\mathcal{O}_{a,k-1}\cdots \mathcal{O}_{a,k+1-\alpha },\text{ with }\alpha \in
\{1,...,p\}.$$Then the following lemma holds:
For any $n\in \{1,...,\mathsf{N}\}$, the set of the elementary operators dressed by the shift operator $\mathsf{U}_{n}$:$$\mathsf{U}_{n}\mathcal{E}_{(k,k_{0})^{\mathtt{e}_{\mathsf{N}}},(a_{1},k_{1}),...,(a_{r},k_{r})}^{(\alpha _{1},...,\alpha _{r})}\mathsf{U}_{n}^{-1},$$is a basis in the space of the local operators at the quantum site $n$.
The space of the local operators in site $n$ is generated by $\mathsf{u}_{n}^{k}$ and $\mathsf{v}_{n}^{k}$ for $k\in \{1,...,p-1\}$. Proceeding as done in Proposition \[IPS\] we have in particular the possibility to show that an alternative local basis is defined by the operators: $$\begin{aligned}
\mathsf{u}_{n}^{k}& =\mathsf{U}_{n}\left( \mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu _{n,+})\right) ^{k}\mathsf{U}_{n}^{-1}, \label{U-B-basis1} \\
\tilde{\beta}_{k,n}& =\mathsf{U}_{n}\left( \mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu _{n,+})\right) ^{k}\mathsf{B}^{-1}(\mu _{n,-})\mathsf{A}(\mu
_{n,-})\left( \mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu _{n,+})\right)
^{p-1-k}\mathsf{U}_{n}^{-1} \label{U-B-basis2}\end{aligned}$$for $k\in \{1,...,p-1\}$. Then to prove the lemma we just have to show that the above operators are linear combinations of those defined in ([O-basis]{}). Note that for $\lambda ^{p}\neq Z_{a}$ with $a\in \{1,...,[\mathsf{N}]\}$, the operator $\mathsf{B}^{-1}(\lambda )$ is invertible and by the centrality of the average values we can write: $$\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )=\frac{\mathsf{B}(\lambda
q^{p-1})\mathsf{B}(\lambda q^{p-2})\cdots \mathsf{B}(\lambda q)\mathsf{A}(\lambda )}{\mathcal{B}(\Lambda )}.$$Now the operator $\mathsf{B}(\lambda q^{p-1})\mathsf{B}(\lambda
q^{p-2})\cdots \mathsf{B}(\lambda q)\mathsf{A}(\lambda )$ is an even Laurent polynomial of degree:$$(p-1)[\mathsf{N}]+\mathsf{N}-1+\mathtt{e}_{\mathsf{N}}=\left\{
\begin{array}{l}
p\mathsf{N}-1\text{ \ \ \ \ \ \ \ \ \ for }\mathsf{N}\text{ odd} \\
p(\mathsf{N}-1)+1\text{ for }\mathsf{N}\text{ even}\end{array}\right.$$in $\lambda $. So for $\mathsf{N}$ odd to completely characterize it we have to fix its value in $p\mathsf{N}$ distinguished points and we are free to chose these points coinciding with the zeros of the operator $\mathsf{B}(\lambda )$. For $\mathsf{N}$ even, we have to add to the $\mathsf{B}$-zeros the values at the infinity, so that by using the corresponding interpolation formula for $\mathsf{B}(\lambda q^{p-1})\mathsf{B}(\lambda q^{p-2})\cdots
\mathsf{B}(\lambda q)\mathsf{A}(\lambda )$, we derive:$$\mathsf{B}^{-1}(\lambda )\mathsf{A}(\lambda )=\frac{\mathtt{e}_{\mathsf{N}}}{\eta _{\mathsf{N}}}\left( \frac{\lambda \Theta }{\eta _{\mathsf{A}}}+\frac{\eta _{\mathsf{A}}}{\lambda \Theta }\right) +\frac{1}{\eta _{\mathsf{N}}^{\mathtt{e}_{\mathsf{N}}}}\sum_{a=1}^{[\mathsf{N}]}\sum_{k=0}^{p-1}\frac{\mathcal{O}_{a,k}}{(\lambda /\eta _{a}^{(k)}-\eta _{a}^{(k)}/\lambda )}.$$From the previous formula and the representation (\[U-B-basis1\])-([U-B-basis2]{}), we have that the local operators $\mathsf{u}_{n}^{k}$ and $\tilde{\beta}_{k,n}$ are linear combinations of the monomials $\mathsf{U}_{n}\eta _{\mathsf{N}}^{-\mathtt{e}_{\mathsf{N}}h}\left( \frac{\Theta }{\eta
_{\mathsf{A}}}\right) ^{\mathtt{e}_{\mathsf{N}}h_{0}}\mathcal{O}_{a_{1},h_{1}}\cdots \mathcal{O}_{a_{s},h_{s}}\mathsf{U}_{n}^{-1}$ for $s\leq p$, $a_{i}\in \{1,...,[\mathsf{N}]\}$ and $h,h_{i}\in \{0,...,p-1\}$. For any monomial $\mathcal{O}_{a_{1},h_{1}}\cdots \mathcal{O}_{a_{s},h_{s}}$ we can use the commutation rules (\[Com-O\]) to rewrite it in a way that operators with the same index $a$ are adjacent, we can order them in a way that $a_{i}<a_{j}$ for $i<j\in \{1,...,[\mathsf{N}]\}$ and we can apply the rule (\[Prod-O-zeros\]) to say if the monomial is zero or not. Finally, by using the property (\[O-mean-value\]), we have: $$\mathcal{O}_{a,k}^{(p+\alpha )}=\frac{\mathcal{A}(Z_{a})}{\prod_{b\neq
a,b=1}^{\mathsf{N}}(Z_{a}/Z_{b}-Z_{b}/Z_{a})}\mathcal{O}_{a,k}^{(\alpha )},$$and so it is clear that all the non-zero monomials $\mathcal{O}_{a_{1},h_{1}}\cdots \mathcal{O}_{a_{s},h_{s}}$ can be written in the form (\[O-basis\]).
### Form factors of elementary operators
As anticipated the interest in the above definition of elementary operators is the simplicity of their form factors:
Let $\langle t_{k}|$ and $|t_{k^{\prime }}^{\prime }\rangle $ be two eigenstates of the transfer matrix $\mathsf{T}(\lambda )$, then it holds:$$\langle t_{k}|\mathcal{E}_{(h,h_{0})^{\mathtt{e}_{\mathsf{N}}},(a_{1},h_{1}),...,(a_{r},h_{r})}^{(\alpha _{1},...,\alpha
_{r})}|t_{k^{\prime }}^{\prime }\rangle =\frac{\delta _{k,k^{\prime }+h}^{\mathtt{e}_{\mathsf{N}}}q^{\mathtt{e}_{\mathsf{N}}h_{0}k^{\prime }}}{\eta _{\mathsf{N}}^{(0)h\mathtt{e}_{\mathsf{N}}}\prod_{j=1}^{\mathsf{N}}\xi _{j}^{\mathtt{e}_{\mathsf{N}}h_{0}}}\,\mathsf{f}_{(\mathtt{e}_{\mathsf{N}}h_{0},\{\alpha \},\{a\})}\det_{[\mathsf{N}]+rp-g}(||\text{\textsc{O}}_{a,b}^{(\mathtt{e}_{\mathsf{N}}h_{0},\{\alpha \},\{a\})}||),$$where $||$<span style="font-variant:small-caps;">O</span>$_{a,b}^{(\mathtt{e}_{\mathsf{N}}h_{0},\{\alpha
\},\{a\})}||$ is the $\left( [\mathsf{N}]+rp-g\right) \times \left( \lbrack
\mathsf{N}]+rp-g\right) $ matrix of elements:$$\begin{aligned}
\text{\textsc{O}}_{a,\sum_{h=1}^{m-1}(p-\alpha _{h}+1)+j_{m}}^{(\mathtt{e}_{\mathsf{N}}h_{0},\{\alpha \},\{a\})}& \equiv \left( \eta
_{a_{m}}^{(h_{m}+j_{m})}\right) ^{4(a-1)}\text{ \ for }j_{m}\in
\{0,...,p-\alpha _{m}\},\text{ \ }m\in \{1,...,r\},\text{ } \\
\text{\textsc{O}}_{a,\sum_{h=1}^{r}(p-\alpha _{h}+1)+i}^{(\mathtt{e}_{\mathsf{N}}h_{0},\{\alpha \},\{a\})}& \equiv \Phi _{b_{i},a+(\mathtt{e}_{\mathsf{N}}h_{0}+g)/2}^{\left( t,t^{\prime }\right) }\text{ \ \ \ \ \ \ \ \
\ \ for }i\in \{1,...,[\mathsf{N}]-r\},\text{ \ \ \ }g\equiv
\sum_{h=1}^{r}\alpha _{h},\end{aligned}$$for any $a\in \{1,...,[\mathsf{N}]+rp-g\}$. Here, we have defined $\{b_{1},...,b_{[\mathsf{N}]-r}\}\equiv \{1,...,\mathsf{[\mathsf{N}]}\}\backslash \{a_{1},...,a_{r}\}$ with elements ordered by $b_{i}<b_{j}$ for $i<j$ and$$\begin{aligned}
\mathsf{f}_{(\mathtt{e}_{\mathsf{N}}h_{0},\{\alpha \},\{a\})}& \equiv \frac{\prod_{i=1}^{r}Q_{t^{\prime }}(\eta _{a_{i}}^{(h_{i}-\alpha _{i})})\bar{Q}_{t}(\eta _{a_{i}}^{(h_{i})})\frac{\left( \eta _{a_{i}}^{(h_{i})}\right) ^{\mathtt{e}_{\mathsf{N}}h_{0}+\alpha _{i}\left( [\mathsf{N}]-r\right) }}{\omega _{a_{i}}(\eta _{a_{i}}^{(h_{i})})}\prod_{h=0}^{\alpha _{i}-1}a(\eta
_{a_{i}}^{(h_{i}-h)})}{\prod_{i=1}^{r}\prod_{h=0}^{\alpha
_{i}-1}\prod_{j=1}^{i-1}(\frac{\eta _{a_{i}}^{(h_{i}+\alpha _{i}-h)}}{\eta
_{a_{j}}^{(h_{j})}}-\frac{\eta _{a_{j}}^{(h_{j})}}{\eta
_{a_{i}}^{(h_{i}+\alpha _{i}-h)}})\prod_{j=i+1}^{r}(\frac{\eta
_{a_{i}}^{(h_{i})}}{\eta _{a_{j}}^{(h_{j}+h)}}-\frac{\eta
_{a_{j}}^{(h_{j}+h)}}{\eta _{a_{i}}^{(h_{i})}})} \notag \\
& \times \frac{(-1)^{\sum_{i=1}^{r}(a_{i}-i)}\prod_{i=1}^{r}q^{-\left( [\mathsf{N}]-r\right) \alpha _{i}(\alpha _{i}-1)/2}V((\eta
_{a_{1}}^{(h_{1})})^{2},...,(\eta _{a_{r}}^{(h_{r})})^{2})}{\prod_{i=1}^{r}\prod_{j=1}^{[\mathsf{N}]-r}(Z_{a_{i}}^{2}-Z_{b_{j}}^{2})V((\eta _{a_{1}}^{(h_{1})})^{2},...,(\eta _{a_{1}}^{(h_{1}+p-\alpha
_{1})})^{2},...,(\eta _{a_{r}}^{(h_{r})})^{2}...,(\eta
_{a_{r}}^{(h_{r}+p-\alpha _{r})})^{2}))},\end{aligned}$$where $V(x_{1},...,x_{\mathsf{N}})\equiv \prod_{1\leq b<a\leq \mathsf{N}}(x_{a}-x_{b})$ is the Vandermonde determinant.
The operator $\eta _{\mathsf{N}}^{-\mathtt{e}_{\mathsf{N}}h}\left( \frac{\Theta }{\eta _{\mathsf{A}}}\right) ^{\mathtt{e}_{\mathsf{N}}h_{0}}$ act in the following way on the state $\langle t_{k}|$:$$\begin{aligned}
\langle t_{k}|\eta _{\mathsf{N}}^{-\mathtt{e}_{\mathsf{N}}h}\left( \frac{\Theta }{\eta _{\mathsf{A}}}\right) ^{\mathtt{e}_{\mathsf{N}}h_{0}}& =
\notag \\
& =\left( \frac{q^{h_{0}(k-h)}}{(\eta _{\mathsf{N}}^{(0)})^{2}\prod_{j=1}^{\mathsf{N}}\xi _{j}^{h_{0}}}\right) ^{\mathtt{e}_{\mathsf{N}}}\sum_{k_{1},...,k_{\mathsf{N}}=1}^{p}\left( \frac{q^{(k-h)h_{\mathsf{N}}}}{p^{1/2}}\right) ^{\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}(\eta
_{a}^{(k_{a})})^{h_{0}\mathtt{e}_{\mathsf{N}}}\bar{Q}_{t}(\eta
_{a}^{(k_{a})}) \notag \\
& \times \prod_{1\leq b<a\leq \lbrack \mathsf{N}]}((\eta
_{a}^{(k_{a})})^{2}-(\eta _{b}^{(k_{b})})^{2})\frac{\langle \eta
_{1}^{(k_{1})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|}{\prod_{b=1}^{[\mathsf{N}]}\omega _{b}(\eta _{b}^{(k_{b})})}.\end{aligned}$$From the formula (\[O-Action\]), it follows:$$\langle \eta _{1}^{(k_{1})},...,,...,\eta _{a_{i}}^{(f)},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|\mathcal{O}_{a_{i},h_{i}}^{(\alpha _{i})}=\frac{\prod_{h=0}^{\alpha _{i}-1}a(\eta _{a_{i}}^{(h_{i}-h)})\delta
_{f,h_{i}}\langle \eta _{1}^{(k_{1})},...,,...,\eta _{a_{i}}^{(h_{i}-\alpha
_{i})},...,\eta _{\mathsf{N}}^{(k_{\mathsf{N}})}|}{\prod_{b\neq a_{i},b=1}^{[\mathsf{N}]}\prod_{h=0}^{\alpha _{i}-1}(\eta _{a_{i}}^{(h_{i}-h)}/\eta
_{b}^{(k_{b})}-\eta _{b}^{(k_{b})}/\eta _{a_{i}}^{(h_{i}-h)})}.
\label{O-a-Action}$$ So we can compute also the action of $\mathcal{O}_{a_{1},h_{1}}^{(\alpha
_{1})}\cdots \mathcal{O}_{a_{r},h_{r}}^{(\alpha _{r})}$ just taking into account the order of the operators in the monomial which leads by the scalar product formula to:$$\begin{aligned}
\langle t_{k}|\mathcal{E}_{(h,h_{0})^{\mathtt{e}_{\mathsf{N}}},(a_{1},h_{1}),...,(a_{r},h_{r})}^{(\alpha _{1},...,\alpha
_{r})}|t_{k^{\prime }}^{\prime }\rangle & =\left( \frac{q^{h_{0}(k-h)}}{(\eta _{\mathsf{N}}^{(0)})^{h}\prod_{j=1}^{\mathsf{N}}\xi _{j}^{h_{0}}}\right) ^{\mathtt{e}_{\mathsf{N}}}\sum_{k_{1},...,k_{\mathsf{N}}=1}^{p}\left( \frac{q^{\left[ (k-h)-k^{\prime }\right] k_{\mathsf{N}}}}{p}\right) ^{\mathtt{e}_{\mathsf{N}}}\prod_{a=1}^{[\mathsf{N}]}(\eta
_{a}^{(k_{a})})^{h_{0}\mathtt{e}_{\mathsf{N}}} \notag \\
& \times \prod_{i=1}^{r}\frac{\prod_{h=0}^{\alpha _{i}-1}a(\eta
_{a_{i}}^{(h_{i}-h)})\delta _{k_{a_{i}},h_{i}}}{\prod_{j=1}^{[\mathsf{N}]-r}\prod_{h=0}^{\alpha _{i}-1}(\eta _{a_{i}}^{(h_{i}-h)}/\eta
_{b_{j}}^{(k_{b_{j}})}-\eta _{b_{j}}^{(k_{b_{j}})}/\eta _{a_{i}}^{(h_{i}-h)})} \notag \\
& \times \prod_{i=1}^{r}\prod_{h=0}^{\alpha _{i}-1}\frac{\prod_{j=i+1}^{r}(\eta _{a_{i}}^{(h_{i}-h)}/\eta _{a_{j}}^{(h_{j})}-\eta
_{a_{j}}^{(h_{j})}/\eta _{a_{i}}^{(h_{i}-h)})^{-1}}{\prod_{j=1}^{i-1}(\eta
_{a_{i}}^{(h_{i}-h)}/\eta _{a_{j}}^{(h_{j}-\alpha _{j})}-\eta
_{a_{j}}^{(h_{j}-\alpha _{j})}/\eta _{a_{i}}^{(h_{i}-h)})} \notag \\
& \times \prod_{j=1}^{[\mathsf{N}]-r}\frac{Q_{t^{\prime }}(\eta
_{b_{j}}^{(k_{b_{j}})})\bar{Q}_{t}(\eta _{b_{j}}^{(k_{b_{j}})})}{\omega
_{b_{j}}(\eta _{b_{j}}^{(k_{b_{j}})})}\prod_{i=1}^{r}\frac{Q_{t^{\prime
}}(\eta _{a_{i}}^{(h_{i}-\alpha _{i})})\bar{Q}_{t}(\eta _{a_{i}}^{(h_{i})})}{\omega _{a_{i}}(\eta _{a_{i}}^{(h_{i})})} \notag \\
& \times V((\eta
_{1}^{(h_{1})})^{2},...,(\eta _{\lbrack \mathsf{N}]}^{(h_{[\mathsf{N}]})})^{2}).\end{aligned}$$Let us remark that the sum $\sum_{k_{1},...,k_{\mathsf{N}}=1}^{p}$ reduces to $\delta _{k,k^{\prime }+h}^{\mathtt{e}_{\mathsf{N}}}$ times the sum $\sum_{k_{b_{1}},...,k_{b_{\mathsf{[N]}-r}}=1}^{p}$ for the presence of the $\prod_{i=1}^{r}\delta _{k_{a_{i}},h_{i}}$. Now we multiply each term of the sum by: $$\begin{aligned}
1& =\prod_{\epsilon =\pm 1}\prod_{i=1}^{r}\prod_{j=1}^{[\mathsf{N}]-r}\prod_{h=-p+\alpha _{i}}^{-1}((\eta {}_{a_{i}}^{(h_{i}-h)})^{2}-(\eta
_{b_{j}}^{(k_{b_{j}})})^{2})^{\epsilon } \notag \\
& \times \left( \frac{V((\eta _{a_{1}}^{(h_{1})})^{2},...,(\eta
_{a_{1}}^{(h_{1}+p-\alpha _{1})})^{2},...,(\eta
_{a_{r}}^{(h_{r})})^{2}...,(\eta _{a_{r}}^{(h_{r}+p-\alpha _{r})})^{2}))}{V((\eta _{a_{1}}^{(h_{1})})^{2},...,(\eta _{a_{r}}^{(h_{r})})^{2})}\right)
^{\epsilon }\end{aligned}$$here the power $+1$ leads to the construction of the Vandermonde determinant:$$V(\underset{p-\alpha _{1}+1\text{ columns}}{\underbrace{(\eta
_{a_{1}}^{(h_{1})})^{2},...,(\eta _{a_{1}}^{(h_{1}+p-\alpha _{1})})^{2}}},...,\underset{p-\alpha _{r}+1\text{ columns}}{\underbrace{(\eta
_{a_{r}}^{(h_{r})})^{2},...,(\eta _{a_{r}}^{(h_{r}+p-\alpha _{r})})^{2}}},\underset{[\mathsf{N}]-r\text{ columns}}{\underbrace{(\eta
_{b_{1}}^{(k_{b_{1}})})^{2},...,(\eta _{b_{[\mathsf{N}]-r}}^{(k_{b_{[\mathsf{N}]-r}})})^{2}}}),$$and the sum $\sum_{k_{b_{1}},...,k_{b_{\mathsf{[N]}-r}}=1}^{p}$ becomes sum over columns and after some algebra we get our formula.
It is interesting to point out that the last $[\mathsf{N}]-r$ columns of the matrix $||\text{\textsc{O}}_{a,b}^{(\mathtt{e}_{\mathsf{N}}h_0,\{\alpha
\},\{a\})}||$ are just those of the scalar product.
**Remark 7.** For the similarity of the model and representations considered, it is natural to cite the series of works [GIPST07,GIPS06,GIPST08,GIPS09]{}. There, in the framework of cyclic SOV-representations, first results on the matrix elements of local operators appear. However, it is worth saying that these quantities are there computed only for the restriction[^17] of the $\tau _{2}$-model to the generalized Ising model. In particular, the matrix elements of $\mathsf{u}_{1}$ are computed and the results are not presented in a determinant form.
Conclusion and outlook
======================
Results
-------
In this article we have considered the lattice sine-Gordon model in cyclic representations and we have solved in this case two fundamental problems for the computation of matrix elements of local operators:
- Scalar products: determinant of $\mathsf{N}\times\mathsf{N}$ matrices whose matrix elements are sums over the spectrum of each quantum separate variable of the product of the coefficients of states, this being for all the left/right separate states in the SOV-basis.
- Inverse problem solution: reconstruction of all local operators in terms of standard Sklyanin’s quantum separate variables.
Further, we have shown how these results lead to the computation of matrix elements of all local operators. At first, standard[^18] Sklyanin’s quantum separate variables are suitable for solving the transfer matrix spectral problem. Indeed, the transfer matrix spectrum (eigenvalues & eigenstates) admits a simple and complete characterization in terms of Baxter-equation solutions in this SOV-basis. Then the inverse problem solution allows to write the action of any local operator on transfer matrix eigenstates as finite sums of separate states in the SOV-basis. Hence, the matrix elements of any local operator are written as finite sums of determinants of the resulting scalar product formulae.
We have explicitly developed this program characterizing the matrix elements of the local operators $\mathsf{u}_{n}$ and $\alpha _{n}$ by one determinant formulae in terms of matrices obtained by modifying a single row in the scalar product matrices. Moreover, we have constructed an operator basis whose matrix elements are in turn written by one determinant formulae. The matrices involved have rows which coincide with those of the scalar product matrix or with those of the Vandermonde matrix computed in the spectrum of the separate variables.
Comparison with previous SOV-results
------------------------------------
In the literature of quantum integrable models there exist several results on matrix elements of local operators which can be traced back to applications of separation of variable methods. In this section, we try to recall the most relevant ones as they allow for an explicit comparison with our results. It leads to a universal picture emerging in the characterization of matrix elements by SOV-methods.
### On the reconstruction of local operators
One important motivation for our work was to introduce a well defined setup which allows to solve the longstanding problem of the identification of local operators in the continuum sine-Gordon model thanks to the reconstructions achieved on the lattice. Then, it should also allow for the identification of form factor solutions of the continuous theory by implementing well defined limits from our lattice formulae.
Even if methodologically different, it is worth recalling the semi-classical reconstruction presented by Babelon, Bernard and Smirnov in [@BabBS96] for chiral local operators of the restricted sine-Gordon model (in the infinite volume) at the reflectionless points, $\beta ^{2}=1/(1+\nu )$ with $\nu \in \mathbb{Z}^{\geq 0}$. The classical sine-Gordon model admits a SOV description: each $n$-soliton solution $\varphi (x,t)$ of the equation of motion can be represented in terms of $n$-separate variables A$_{j}$, which in the BBS choice [@BabBS96] lead to the representations:$$e^{i\varphi }=\prod_{j=1}^{n}\frac{\text{A}_{j}}{\text{B}_{j}},
\label{SemiCl-Recontruction}$$where the B$_{j}$ are integrals of motion. The formula ([SemiCl-Recontruction]{}) represents classically a SOV reconstruction of the local fields when restricted to the $n$-soliton sector. In [@BabBS96], this reconstruction has been extended to the quantum model in each $n$-solitons sector by quantizing the separate variables[^19] A$_{j}$ and the conjugate momenta as operators which generate $n $ independent Weyl algebras with parameter $\tilde{q}=e^{i\pi \frac{\beta
^{2}}{1-\beta ^{2}}}=e^{i\frac{\pi }{v}}$. This extension and the consequent identifications of primary fields and their chiral descendants in the perturbed minimal models M$_{1,1+\nu }$ are justified by the following indirect but strong arguments: a) The $n$-multiple integrals of the form (36) in [@BabBS96] which represent the $n$-solitons to $n$-solitons form factors[^20] of chiral left operators at the reflectionless points are reproduced from the semi-classical limit. b) The counting of these form factor solutions allows the reconstruction of the chiral characters of M$_{1,1+\nu }$ [@BabBS97]. Further support to (\[SemiCl-Recontruction\]) was given by Smirnov’s work on semi-classical form factors[^21] of the continuous KdV model in finite volume [@Smi98]; there the form factors of [@BabBS96] were reproduced by taking the infinite volume limit of the KdV semi-classical ones.
Let us remark that in our lattice regularization of the sine-Gordon model, choosing as quantum separate variables standard[^22] Sklyanin’s ones, the reconstruction of the exponential fields is not of the simple form given in ([SemiCl-Recontruction]{}). Then, the following question is relevant: is it possible to find a SOV representation of the quantum lattice sine-Gordon model where the exponential fields are simply written as a product of generators of the SOV representations?
A natural idea can be to implement a change of basis in the quantum separate variables from Sklyanin’s ones to a new set; an interesting example of this approach was used by Babelon in [@Bab04] which has provided a simple reconstruction of the lattice quantum Toda local operators in terms of a set of quantum separate variables defined by a change of variables from the Sklyanin’s ones. However, it is worth pointing out that, for a new SOV representation to be really useful for the computation of matrix elements, it should not only give a simple reconstruction of the local operators but also keep the solution of the transfer matrix spectral problem and the scalar product formulae as simple as for original Sklyanin’s variables. Let us comment that a reconstruction like (\[SemiCl-Recontruction\]) can be formally derived at the quantum level implementing the special limit $$\kappa_{n}/i\rightarrow +\infty \label{L1}$$on the following reconstruction formula of the lattice sine-Gordon model:$$\frac{(q^{-1}\mathsf{v}_{n}^{2}+\kappa _{n}^{2})}{\left( q^{-1}\mathsf{v}_{n}^{2}\kappa _{n}^{2}+1\right) }=\mathsf{U}_{n}\mathsf{A}^{-1}(\mu _{n,-})\mathsf{B}(\mu _{n,-})\mathsf{B}^{-1}(\mu _{n,+})\mathsf{A}(\mu _{n,+})\mathsf{U}_{n}^{-1}\ . \label{Oota^2}$$The result for an even chain reads $$\mathsf{v}_{n}^{-2h}=\frac{\Theta ^{2h}}{\prod_{a\neq n,a=1}^{\mathsf{N}}\xi _{a}^{2h}}\mathsf{U}_{n}\prod_{a=1}^{\mathsf{N}-1}\eta _{a}^{2h}\mathsf{U}_{n}^{-1},\text{ \ }h\in \{1,..,p-1\} \label{Formula+}$$in terms of Sklyanin’s separate variables. It is possible to argue that the previous limit can be consistently interpreted as a chiral deformation of the lattice sine-Gordon model to chiral KdV models [@NovMPZ84; @FadV94]. In a future publication, we will analyze the cyclic representations[^23] of these chiral KdV models showing our statement on the reconstruction formulae and computing the matrix elements. It is worth pointing out that the form factors of the r.h.s. of ([Formula+]{}) are trivial to compute in our SOV framework and are written by one determinant formulae which differ w.r.t. the scalar product only for $h\in \{1,..,p-1\}$ rows.
### On the matrix elements of local operators
In the case of the quantum integrable Toda chain [@Skl85], Smirnov [Smi98]{} has derived in the framework of Sklyanin’s SOV determinant formulae for the matrix elements of a conjectured basis of local operators which look very similar to our formulae. The main difference is due to the different nature of the spectrum of the quantum separate variables in the two models. In fact, in the case of the lattice Toda model, Sklyanin’s measure is continuous (continuous SOV-spectrum) while it is discrete in the case of the cyclic lattice sine-Gordon model. The elements of the matrices whose determinants give the form factor formulae are then expressed as convolutions, over the spectrum of the separate variables, of Baxter equations solutions plus contributions coming from the local operators. In the case of Smirnov’s formulae they are true integrals, the SOV-spectrum being continuous, while in our formulae they are discrete convolutions , the SOV-spectrum being discrete. Let us comment that the need to conjecture[^24] the form of a basis of local operators in [@Smi98] is due to the lack of a direct reconstruction of local operators in terms of Sklyanin’s separate variables.
In the case of the infinite volume quantum sine-Gordon field theory, the form factors of local operators [@Smi92b] have also a form similar to the one predicted by SOV. This similarity can be made explicit considering the $n$-soliton form factors for the restricted sine-Gordon theory at the reflectionless points in formula (31) of [@BabBS96]. Then, for the local fields interpreted as primary operators in [@BabBS96], the corresponding form factors can be easily rewritten as determinants of $n\times n$ matrices whose elements are integral convolutions of $n$-soliton wave functions (the $\psi $-functions (32)) plus contributions coming from the local operators.
The rough picture that seems to emerge is that by performing the IR limit on our lattice form factors the lattice wave functions factorized in terms of Q-operator eigenvalues have to converge to the infinite volume $n$-soliton wave functions. To which extent this picture can be confirmed and clarified by a detailed analysis of the thermodynamic limit starting from our lattice sine-Gordon model results is of course an interesting question to which we would like to answer in the future.
Outlook
-------
It is worth mentioning that we didn’t succeed yet to express the matrix elements of discretized exponential of the sine-Gordon field in terms of one simple determinant formula. Hence our next natural project is the simplification of the present representation; this is also important in view of the attempt to extend our results from the lattice to the continuous finite and infinite volume limits. The main goal here is to derive the known form factors of the IR limit, starting from our lattice form factors, in this way solving the longstanding problem of the identifications of local fields in the S-matrix characterization of the infinite volume sine-Gordon model.
Beyond the sine-Gordon model we want to point out the potential generality of the method we have introduced here to compute matrix elements of local operators for quantum integrable models. The main ingredients used to develop it are the reconstruction of local operators in Sklyanin’s SOV representations and the scalar product formulae for the transfer matrix eigenstates (and general separate states). The emerging picture is the possibility to apply this method to a whole class of integrable quantum model which were not tractable with other methods. This is in particular the case for lattice integrable quantum models to which the algebraic Bethe ansatz does not apply. The first remarkable case is given by the $\tau_2$-model in general representations which are of special interest for their connection to the chiral Potts model. This will be the next model that we will analyze by our technique due to the similarity of its cyclic representations with those of the sine-Gordon model.
There are also many other examples which are interesting and for which, on the one hand, the reconstruction of the local operators can be deduced from [@MaiT00] and, on the other hand, the description of the spectrum can be given by Sklyanin’s quantum separation of variables. For all these models the possibility to apply our method for the computation of matrix elements is very concrete and moreover the results are expected to have a completely similar form to the ones shown in the present article.
**Acknowledgments**
We would like to thank N. Kitanine, K. K. Kozlowski, B. M. McCoy, E. Sklyanin, V. Terras and J. Teschner for their interest in this work. J. M. M. is supported by CNRS. N. G. and J. M. M. are supported by ANR grant ANR-10-BLAN-0120-04-DIADEMS. G. N. is supported by National Science Foundation grants PHY-0969739. G. N. gratefully acknowledge the YITP Institute of Stony Brook for the opportunity to develop his research programs and the privilege to have stimulating discussions on the present paper and on related subjects with B. M. McCoy. Moreover, G. N. would like to thank the Theoretical Physics Group of the Laboratory of Physics at the ENS Lyon and the Mathematical Physics Group at the IMB of the University of Dijon for their hospitality.
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[^1]: See [@Vi70; @BPZ84; @Gin89; @Ca88; @DFMS97] and references therein for some literature on conformal field theories.
[^2]: See for a review [@Mus92] and references therein.
[^3]: An important role in these studies has been played by the fermionic representations of the characters, as derived for different classes of rational conformal field theories in [KKMcM93-1,KKMcM93-2,DKKMcM93,FNO92,NRT93,KMcM95,BMc98]{}.
[^4]: Apart for some of them, like the components of the stress energy tensor, which can be characterized by physical prescription [@DN05-1] and [DN06]{}.
[^5]: The integrability of the time evolution in the lattice sine-Gordon model is known [@IzeK82; @BazBR96; @FadV92; @BobKP93] and the most convenient way to formulate it uses the Baxter Q-operators; these operators have been constructed for the closely related chiral Potts model in [@Ba08], see [@NicT10] for a review and rederivation of these points in the lattice sine-Gordon model.
[^6]: Here, we make our analysis for the lattice sine-Gordon model leaving the $\kappa _{n}$ and $\xi _{n}$ as free parameters. However, the sine-Gordon model in the continuum limit is reproduced taking suitable limits on these parameters by relating them to the mass $\mu $ and the radius $R$ of the compactified space direction of the model.
[^7]: Note that the operator $\mathsf{B}(\lambda )$ is invertible except for $\lambda $ which coincides with one of its zeros, so in general $\mathsf{C}(\lambda )$ is defined by (\[qdetdef\]) just inverting $\mathsf{B}(\lambda
)$. This is enough to fix in an unique way $\mathsf{C}(\lambda )$, as it is a Laurent polynomial of degree \[$\mathsf{N}$\] in $\lambda $.
[^8]: The centrality of the quantum determinant in the Yang-Baxter algebra was first discovered in [@IzeK81], see also [@IzeK09] for an historical note.
[^9]: Note that here it cannot be called a measure as the $\mathsf{B}(\lambda )$ operator is not self-adjoint, meaning that in general the $\mu_i$ are not positive real numbers. However the above procedure indeed defines a proper decomposition of the identity operator with simple and computable coefficients.
[^10]: See also [@Smi98] for further discussions on the measure.
[^11]: The SOV analysis for these representations has been first developed in [GIPS06]{}.
[^12]: Translational invariance for the limit of the homogeneous chain.
[^13]: This result is there attributed to V. Korepin, private communications.
[^14]: To simply the notations we chose to present the results in this subsection only for the case $\mathsf{N}$ odd and for the representations with $\mathsf{v}_{n}^{p}=\mathsf{u}_{n}^{p}=1.$
[^15]: To simplify the exposition, we have decided to keep as simple as possible the notations for these operators, we hope that nevertheless the difference with the corresponding eigenvalues is clear.
[^16]: To simplify the notations in this introduction to Section \[FF-loc-Op\] we are omitting the index $k\in \{0,...,(p-1)/2\}$ in the transfer matrix eigenstates which are required in the case of even chain.
[^17]: It can be compared to the restriction to the case $q^{2}=1$ for the even sine-Gordon chain.
[^18]: Coinciding with the operator-zeros of one of the Yang-Baxter algebra generators, like $B(\lambda )$ or $C(\lambda )$.
[^19]: A fundamental point is the introduction of appropriate Hermitian conjugation properties and the characterization of the spectrum of the Weyl algebra generators.
[^20]: Note that these are solutions of the form factor equations and so they surely represent local fields in the S-matrix formulation of the restricted sine-Gordon model.
[^21]: On the basis of this last Smirnov’s work, Lukyanov has introduced his conjecture for the finite temperature expectations values of exponential fields in finite volume for the shG-model [@Luk01].
[^22]: The same is true if we take the products of the operator zeros of $(\lambda )$ but also of $(\lambda )$ and $(\lambda )$, i.e. for all possibilities to construct the SOV representations by the simplest Sklyanin’s method.
[^23]: Here, we are referring to compact representations of chiral KdV models where the generators of the local Weyl algebras are unitary operators. The spectrum of the non-compact versions was instead analyzed by SOV and Q-operator method in [@BytT09].
[^24]: The consistency of this conjecture is there verified by a counting argument based on the existence of an appropriate set of null conditions for the integral convolutions.
|
---
abstract: 'In this paper, an alternating direction implicit (ADI) difference scheme for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term is presented. The unique solvability of the difference solution is discussed, and the unconditional stability and convergence order of the numerical scheme are analysed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithm.'
address:
- 'Research Center for Computational Science, Northwestern Polytechnical University, Xi’an 710129, China'
- 'College of Science, Henan University of Technology, Zhengzhou 450001, China'
author:
- Jiahui Hu
- Jungang Wang
- Zhanbin Yuan
- Zongze Yang
- Yufeng Nie
bibliography:
- 'Reference.bib'
title: 'Numerical algorithm for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term'
---
Two-dimensional time-fractional wave equation of distributed-order , ADI scheme , Nonlinear source term , Stability , Convergence 35R11,65M06,65M12
Introduction {#sec:introduciton}
============
The idea of distributed-order differential equation was first introduced by Caputo in his work for modeling the stress-strain behavior of an anelastic medium in 1960s [@Caputo]. Being different from the differential equations with the single-order fractional derivative and the ones with sums of fractional derivatives, i.e., multi-term fractional differential equations (FDEs), the distributed-order differential equations are derived by integrating the order of differentiation over a certain range. It can be regarded as a generalization of the aforementioned two classes of FDEs. A typical application of this kind of FDEs is in the retarding sub-diffusion process, where a plume of particles spreads at a logarithmic rate, which leads to ultraslow diffusion (see [@Sinai1983][@ChechkinKlafterSokolov2003][@Kochubei2008]). Another example is the fractional Langevin equation of distributed-order, which was proposed to model the kinetics of retarding sub-diffusion whose scaling exponent decreases in time, and then was applied to simulate the strongly anomalous ultraslow diffusion with the mean square displacement growing as a power of logarithm of time [@EabLim2011]. The distributed-order FDEs were also found playing important role in other various research fields, such as control and signal processing [@JiaoChenPodlubny2012], modelling dielectric induction and diffusion [@Caputo2001], identification of systems [@Hartley1999], and so on.
Till now, there have been many important progresses for the research on analytical solutions of distributed-order FDEs. For the kinetic description of anomalous diffusion and relaxation phenomena, A. V. Chechkin et al. presented the diffusion-like equation with time fractional derivative of distributed-order in [@Chechkin2003], where the positivity of the solutions of the proposed equation was proved and the relation to the continuous-time random walk theory was established. T. M. Atanackovic et al. analysed a Cauchy problem for a time distributed-order diffusion-wave equation by means of the theory of an abstract Volterra equation [@atanackovic2009time]. In [@Gorenflo2013], for the one-dimensional distributed-order diffusion-wave equation, R. Gorenflo et al. gave the interpretation of the fundamental solution of the Cauchy problem as a probability density function of the space variable $x$ evolving in time $t$ in the transform domain by employing the technique of the Fourier and Laplace transforms. Using the Laplace transform method, Z. Li et al investigated the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed-order time-fractional diffusion equations [@LiLuchkoYamamoto2014].
In most instances, the analytical solutions of distributed-order differential equations are not easy to available, thus it stimulates researchers to develop numerical algorithms for approximate solutions. To our knowledge, the research on numerically solving the distributed-order differential equations are still in its infancy. The literatures [@DiethelmFord2009][@Podlubny2013][@Katsikadelis2014] concerned on developing numerical methods for solving distributed-order ordinary differential equations. In terms of the distributed-order partial differential equations, most of the work are about the one-dimensional time distributed-order differential equations, and the integrating range of the order of time derivative is the interval $[0,1]$, which is named as time distributed-order diffusion equation. N. J. Ford et al. developed an implicit finite difference method for the solution of the diffusion equation with distributed order in time [@Ford2015]. By using the Gr[ü]{}nwald-Letnikov formula, Gao et al. proposed two difference schemes to solve the one-dimensional distributed-order differential equations, and the extrapolation method was applied to improve the approximate accuracy [@GaoSun2016111]. In [@GaoSunSun2015], the authors handled the same distributed-order differential equations by employing a weighted and shifted Gr[ü]{}nwald-Letnikov formula to derive several second-order convergent difference schemes. When the order of the time derivative is distributed over the interval $[1,2]$, it is called the time distributed-order wave equation. The study of the numerical solution of this kind of equation is rather more limited. Ye et al. derived and analysed a compact difference scheme for a distributed-order time-fractional wave equation in [@YeLiuAnh2015].
When considering the high-dimensional models, Gao et al. investigated ADI schemes for two-dimensional distributed-order diffusion equations [@GaoSun2015][@GaoSun2016], and they also developed two ADI difference schemes for solving the two-dimensional time distributed-order wave equations [@GaoSun2016a]. Due to the widespread use of the nonlinear models [@RidaEl-SayedArafa2010][@WazwazGorguis2004], M. L. Morgado et al. developed an implicit difference scheme for one-dimensional time distributed-order diffusion equation with a nonlinear source term [@MorgadoRebelo2015]. For further discussion on the numerical approaches for solving the high-dimensional distributed-order partial differential equations, this paper is devoted to develop effective numerical algorithm for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term $$\begin{aligned}
\label{eq:equation}
&\int_1^2p(\beta){}_0^CD_t^\beta u(x,y,t)d\beta=\frac{\partial^2u(x,y,t)}{\partial x^2}+\frac{\partial^2u(x,y,t)}{\partial x^2}
+f\big(x,y,t,u(x,y,t)\big), \nonumber \\
&(x,y)\in \Omega,\quad t\in(0,T], \\ \label{eq:boundaryy}
&u(x,y,t)=\phi(x,y,t),\quad (x,y)\in\partial\Omega,\quad 0\leq t<T,\\ \label{eq:initiall}
&u(x,y,0)=\psi_1(x,y),\quad u_t(x,y,0)=\psi_2(x,y),\quad (x,y)\in \Omega,\end{aligned}$$ where $\Omega=(0,L_1)\times(0,L_2)$, and $\partial \Omega$ is the boundary of $\Omega$. The fractional derivative ${}_0^CD_t^\beta v(t)$ in is given in the Caputo sense $${}_0^CD_t^\beta v(t)=
\left\{
\begin{aligned}
&\frac{\partial v(t)}{\partial t}-\frac{\partial v(0)}{\partial t},\ \beta=1,\\
&\frac{1}{\Gamma{(2-\beta)}}\int_0^t(t-\xi)^{1-\beta}\frac{\partial^2 v(\xi)}{\partial \xi^2}d\xi,\ 1<\beta<2,\\
&\frac{\partial^2v(t)}{\partial t^2},\ \beta=2,
\end{aligned}\right.$$ and the function $p(\beta)$ is served as weight for the order of differentiation such that $p(\beta)>0$ and $\int_1^2p(\beta)d\beta=c_0>0$. We assume that $p(\beta)$, $\phi(x,y,t)$, $\psi_1(x,y)$, $\psi_2(x,y)$ and $f(x,y,t,u)$ are continuous, and the nonlinear source term $f$ satisfies a Lipschitz condition of the form $$\label{eq:lip}
|f(x,y,t,u_1)-f(x,y,t,u_2)|\leq L_f|u_1-u_2|,$$ where $L_{f}$ is a positive constant.
The main procedure of developing numerical scheme for solving problem $-$ is as follows. Firstly a suitable numerical quadrature formula is adopted to discrete the integral in , and a multi-term time fractional wave equation is left whereafter. Then we develop an ADI finite difference scheme which is uniquely solvable for the multi-term time fractional wave equation. By using the discrete energy method, we prove the derived numerical scheme is unconditionally stable and convergent.
The rest of this paper is organized in the following way. In Section 2, the ADI finite difference scheme is constructed and described detailedly. In Section 3, we give analysis on solvability, stability and convergence for the derived difference scheme. Numerical results are illustrated in Section 4 to confirm the effectiveness and accuracy of our method, and some conclusions are drawn in the last section.
The derivation of the ADI scheme {#sec:scheme}
================================
This section focuses on deriving the ADI scheme for the problems $-$.
Let ${M_1}$, ${M_2}$ and $N$ be positive integers, and $h_1=L_1/M_1$, $h_2=L_2/M_2$ and $\tau=T/N$ be the uniform sizes of spatial grid and time step, respectively. Then a spatial and temporal partition can be defined as $x_i=ih_1$ for $i=0,1,\cdots,M_1$, $y_j=jh_2$ for $j=0,1,\cdots,M_2$ and $t_n=n\tau$ for $n=0,1,\cdots,N$. Denote $\bar{\Omega}_h=\{(x_i,y_j)\mid 0\leq i\leq M_1,0\leq j\leq M_2\}$ and $\Omega_\tau=\{t_n\mid t_n=n\tau,0\leq n\leq N\}$, then the domain $\bar{\Omega}\times [0,T]$ is covered by $\bar{\Omega}_h\times\Omega_\tau$. Let $u=\{u_{ij}^n\mid 0\leq i\leq M_1,0\leq j\leq M_2,0\leq n\leq N \}$ be a grid function on $\bar{\Omega}_h\times\Omega_\tau$. We introduce the following notations: $$u_{ij}^{n-\frac{1}{2}}=\frac{1}{2}(u_{ij}^n+u_{ij}^{n-1}),\qquad
\delta_tu_{ij}^{n-\frac{1}{2}}=\frac{1}{\tau}(u_{ij}^n-u_{ij}^{n-1}),$$ $$\delta_xu_{i-\frac{1}{2},j}^n=\frac{1}{h_1}(u_{ij}^n-u_{i-1,j}^n),\qquad
\delta_x^2u_{ij}^n=\frac{1}{h_1}(\delta_xu_{i+\frac{1}{2},j}^n-\delta_xu_{i-\frac{1}{2},j}^n),$$ $$\delta_yu_{i,j-\frac{1}{2}}^n=\frac{1}{h_2}(u_{ij}^n-u_{i,j-1}^n),\qquad
\delta_y^2u_{ij}^n=\frac{1}{h_2}(\delta_xu_{i,j+\frac{1}{2}}^n-\delta_xu_{i,j-\frac{1}{2}}^n),$$ and $$\Delta_hu_{ij}=\delta_x^2u_{ij}+\delta_y^2u_{ij}.$$
Consider Eq. at the point $(x_i,y_j,t_n)$, and we write it as $$\label{eq:point}
\begin{aligned}
&\int_1^2p(\beta){}_0^CD_t^{\beta}u(x_i,y_j,t_n)d\beta\\
=&\frac{\partial^2u(x_i,y_j,t_n)}{\partial x^2}
+\frac{\partial^2u(x_i,y_j,t_n)}{\partial y^2}+f\big(x_i,y_j,t_n,u(x_i,y_j,t_n)\big).
\end{aligned}$$ Take an average of Eq. on time level $t=t_n$ and $t=t_{n-1}$, then we have $$\label{eq:average}
\begin{aligned}
&\frac{1}{2}\bigg(\int_1^2p(\beta){}_0^CD_t^\beta u(x_i,y_j,t_n)d\beta
+\int_1^2p(\beta){}_0^CD_t^\beta u(x_i,y_j,t_{n-1})d\beta\bigg)\\
=& \frac{1}{2}\bigg[\frac{\partial^2u(x_i,y_j,t_n)}{\partial x^2}
+\frac{\partial^2u(x_i,y_j,t_{n-1})}{\partial x^2}\bigg]
+\frac{1}{2}\bigg[\frac{\partial^2u(x_i,y_j,t_n)}{\partial y^2}
+\frac{\partial^2u(x_i,y_j,t_{n-1})}{\partial y^2}\bigg]\\
&+\frac{1}{2}\bigg[f\big(x_i,y_j,t_n,u(x_i,y_j,t_n)\big)+f\big(x_i,y_j,t_{n-1},u(x_i,y_j,t_{n-1})\big)\bigg].
\end{aligned}$$ Denote by $U_{ij}^n=u(x_i,y_j,t_n)$ the grid functions on $\bar{\Omega}_h\times\Omega_\tau$ with $0\leq i\leq M_1$, $0\leq j\leq M_2$, $0\leq n\leq N$. Eq. can be expressed as $$\label{eq:gridU}
\begin{aligned}
\int_1^2p(\beta){}_0^CD_t^\beta U_{ij}^{n-\frac{1}{2}}d\beta= & \frac{\partial^2}
{\partial x^2}U_{ij}^{n-\frac{1}{2}}+\frac{\partial^2}{\partial y^2}U_{ij}^{n-\frac{1}{2}}\\
&+\frac{1}{2}\bigg[f\big(x_i,y_j,t_n,U_{ij}^n\big)+f\big(x_i,y_j,t_{n-1},U_{ij}^{n-1}\big)\bigg]
\end{aligned}$$
Firstly we discretize the integral term in . Suppose $p(\beta)\in C^2[1,2]$, ${}_0^CD_t^\beta u(x_i,y_j,t)|_{t=t_{n-1}}$ and ${}_0^CD_t^\beta u(x_i,y_j,t)|_{t=t_n}\in C^2[1,2]$. Let $K$ be a positive integer, and $\Delta\beta=1/K$ be the uniform step size. Take $\beta_l=1+\frac{2l-1}{2}\Delta\beta$, $1\leq l \leq K$, then the mid-point quadrature rule is used for approximating the integral in $$\label{eq:multii}
\begin{aligned}
&\Delta\beta\sum_{l=1}^{K}p(\beta_l){}_0^{C}D_t^{\beta_l}U_{ij}^{n-\frac{1}{2}}+R_1
=\frac{\partial^2}{\partial x^2}U_{ij}^{n-\frac{1}{2}}+\frac{\partial^2}{\partial y^2}U_{ij}^{n-\frac{1}{2}}\\
&+\frac{1}{2}\bigg[f\big(x_i,y_j,t_n,U(x_i,y_j,t_n)\big)+f\big(x_i,y_j,t_{n-1},U(x_i,y_j,t_{n-1})\big)\bigg],
\end{aligned}$$ where $R_1=\mathcal{O}(\Delta\beta^2)$.
Next, we solve the multi-term time fractional wave equation with the initial and boundary conditions and . Suppose $u(x,y,t)\in C_{x,y,t}^{4,4,3}(\bar{\Omega}\times [0,T])$. According to Theorem 8.2.5 in [@Sun2009], the Caputo derivative ${}_0^CD_t^{\beta_l}U_{ij}^{n-\frac{1}{2}}$, $1 < \beta_l< 2$ have the fully discrete difference scheme $$\label{eq:fractional}
\begin{aligned}
&{}_0^CD_t^{\beta_l}U_{ij}^{n-\frac{1}{2}}\\
=&\frac{\tau^{1-\beta_l}}{\Gamma(3-\beta_l)}\bigg[a_0^{(\beta_l)}\delta_tU_{ij}^{n-\frac{1}{2}}
-\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-
a_{n-k}^{(\beta_l)}\big)\delta_tU_{ij}^{k-\frac{1}{2}}-a_{n-1}^{(\beta_l)}\psi_2(x_i,y_j)\bigg]+R_2^l,
\end{aligned}$$ where $$a_k^{(\beta_l)}=(k+1)^{2-\beta_l}-k^{2-\beta_l},\quad k=0,1,2,\cdots,$$ and $$\label{eq:lefterm}
\begin{aligned}
\mid R_2^l\mid\leq & \frac{1}{\Gamma(3-\beta_l)}\bigg[\frac{2-\beta_l}{12}+\frac{2^{3-\beta_l}}{3-\beta_l}
-(1+2^{1-\beta_l})+\frac{1}{12}\bigg]\cdot\\
& \max_{0\leq t\leq t_n}\mid\frac{\partial^3u(x_i,y_j,t)}{\partial t^3}\mid\tau^{3-\beta_l},\quad
l=1,2,\cdots,K.
\end{aligned}$$ In the meantime, using the second order finite difference $$\frac{\partial^2g(x_i)}{\partial x^2}=\frac{g(x_{i+1})-2g(x_i)+g(x_{i-1})}{(\Delta x)^2}
-\frac{(\Delta x)^2}{12}\frac{\partial^4g(\xi_i)}{\partial x^4},\quad \xi_i \in (x_{i-1},x_{i+1})$$ to approximate the second order derivatives in , it is obtained $$\label{eq:leftnon}
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{\tau^{1-\beta_l}}{\Gamma(3-\beta_l)}
\bigg[a_0^{(\beta_l)}\delta_tU_{ij}^{n-\frac{1}{2}}-\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-
a_{n-k}^{(\beta_l)}\big)\delta_tU_{ij}^{k-\frac{1}{2}}\\
&-a_{n-1}^{(\beta_l)}\psi_2(x_i,y_j)\bigg]+\sum_{l=1}^K\Delta\beta p(\beta_l)R_2^l+R_1\\
=& \delta_x^2U_{ij}^{n-\frac{1}{2}}+\delta_y^2U_{ij}^{n-\frac{1}{2}}
+\frac{1}{2}\Big(f\big(x_i,y_j,t_{n-1},U_{ij}^{n-1}\big)+f\big(x_i,y_j,t_{n},U_{ij}^{n}\big)\Big)+R_3,
\end{aligned}$$ where $R_3=\mathcal{O}(h_1^2+h_2^2)$. Subsequently, the nonlinear source term is dealt with in the following manner to avoid a system of nonlinear equations when computing: $$\label{eq:nondis}
f(x_i,y_j,t_n,U_{ij}^n)=f(x_i,y_j,t_{n-1},U_{ij}^{n-1})+\mathcal{O}(\tau).$$ Substituting in , we are left with $$\label{eq:midsch}
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{\tau^{1-\beta_l}}{\Gamma(3-\beta_l)}
\bigg[a_0^{(\beta_l)}\delta_tU_{ij}^{n-\frac{1}{2}}-\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-
a_{n-k}^{(\beta_l)}\big)\delta_tU_{ij}^{k-\frac{1}{2}}-a_{n-1}^{(\beta_l)}\psi_2(x_i,y_j)\bigg]\\
&=\delta_x^2U_{ij}^{n-\frac{1}{2}}+\delta_y^2U_{ij}^{n-\frac{1}{2}}
+f\big(x_i,y_j,t_{n-1},U_{ij}^{n-1}\big)+R_{ij}^{n-\frac{1}{2}}+\widetilde{R}_{ij}^{n-\frac{1}{2}},
\end{aligned}$$ where $$R_{ij}^{n-\frac{1}{2}}=-\sum_{l=1}^K\Delta\beta p(\beta_l)R_2^l+\mathcal{O}(h_1^2+h_2^2)
+\mathcal{O}(\Delta\beta^2)$$ and $$\widetilde{R}_{ij}^{n-\frac{1}{2}}=\mathcal{O}(\tau).$$ From , we can deduce that there exists a positive constant $C_1$ such that $$\bigg|-\sum_{l=1}^K\Delta\beta p(\beta_l)R_2^l \bigg| \leq C_1\tau^{1+\frac{1}{2}\Delta\beta}
\sum_{l=1}^K\Delta\beta p(\beta_l).$$ Since $$\sum_{l=1}^K\Delta\beta p(\beta_l)\sim \int_1^2p(\beta)d\beta=c_0,$$ we get $$\sum_{l=1}^K\Delta\beta p(\beta_l)\leq C_2,$$ where $C_2$ is a positive constant. Thus there exists a positive constant $C_3$ such that $$\Big|R_{ij}^{n-\frac{1}{2}}\Big|\leq C_3\left(\tau^{1+\frac{1}{2}\Delta\beta}
+h_1^2+h_2^2+\Delta\beta^2\right).$$ Besides, it is obvious that $$\Big|\widetilde{R}_{ij}^{n-\frac{1}{2}}\Big| \leq C_4\tau,$$ where $C_4$ is a positive constant.
Denote $$\mu=\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\tau^{\beta_l}\Gamma(3-\beta_l)}.$$ Since $$\nonumber
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\tau^{\beta_l}\Gamma(3-\beta_l)}\\
\sim & \int_1^2p(\beta)\frac{1}{\tau^\beta\Gamma(3-\beta)}d\beta\\
=&\frac{p(\beta^\ast)}{\Gamma(3-\beta^\ast)}\int_1^2\frac{1}{\tau^\beta}d\beta\\
=&\frac{p(\beta^\ast)}{\Gamma(3-\beta^\ast)}\frac{1-\tau}{\tau^2\mid \ln\tau \mid},
\end{aligned}$$ it can be concluded that $$\mu=\frac{1}{\mathcal{O}(\tau^2| \ln\tau|)}.$$ In addition, $|\ln\tau| \leq C\tau^{-\varepsilon}$ for any positive and small $\varepsilon$ when $\tau$ is sufficiently small, thus the term $\mathcal{O}(\tau^2|\ln\tau |)$ is almost the same as $\mathcal{O}(\tau^2)$ when $\tau$ is sufficiently small. Adding the high order term $$\frac{\tau}{4\mu}\delta_x^2\delta_y^2\frac{U_{ij}^n-U_{ij}^{n-1}}{\tau}$$ on both sides of , we derive $$\label{eq:U}
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{\tau^{1-\beta_l}}{\Gamma(3-\beta_l)}
\bigg[a_0^{(\beta_l)}\delta_tU_{ij}^{n-\frac{1}{2}}-\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-
a_{n-k}^{(\beta_l)}\big)\delta_tU_{ij}^{k-\frac{1}{2}}-a_{n-1}^{(\beta_l)}\psi_2(x_i,y_j)\bigg]\\
&+\frac{\tau}{4\mu}\delta_x^2\delta_y^2\frac{U_{ij}^n-U_{ij}^{n-1}}{\tau}\\
=&\delta_x^2U_{ij}^{n-\frac{1}{2}}+\delta_y^2U_{ij}^{n-\frac{1}{2}}
+f\big(x_i,y_j,t_{n-1},U_{ij}^{n-1}\big)+R_{ij}^{n-\frac{1}{2}}+\widetilde{R}_{ij}^{n-\frac{1}{2}}+\widehat{R}_{ij}^{n-\frac{1}{2}},
\end{aligned}$$ where $$\widehat{R}_{ij}^{n-\frac{1}{2}}=\frac{\tau}{4\mu}\delta_x^2\delta_y^2\frac{U_{ij}^n-U_{ij}^{n-1}}{\tau},$$ and it is clear that $$\Big|\widehat{R}_{ij}^{n-\frac{1}{2}}\Big|\leq C_5\tau^3|\ln\tau|.$$ Also, for the initial and boundary value conditions, we have $$\begin{aligned}
&U_{ij}^0=\psi_1(x_i,y_j),\ (x_i,y_j)\in \Omega ,
\end{aligned}$$ $$\label{eq:Ubou}
\begin{aligned}
&U_{ij}^n=\phi(x_i,y_j,t_n),\ (x_i,y_j)\in \partial\Omega,\ 0\leq n\leq N.
\end{aligned}$$
Let $u_{ij}^n$ be the numerical approximation to $u(x_i,y_j,t_n)$. Neglecting the small term $R_{ij}^{n-\frac{1}{2}}$, $\widetilde{R}_{ij}^{n-\frac{1}{2}}$ and $\widehat{R}_{ij}^{n-\frac{1}{2}}$ in , and using $u_{ij}^n$ instead of $U_{ij}^n$ in $-$, we construct the difference scheme for $-$ as follows: $$\begin{aligned}
\label{eq:scheme}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{\tau^{1-\beta_l}}{\Gamma(3-\beta_l)}
\bigg[a_0^{(\beta_l)}\delta_tu_{ij}^{n-\frac{1}{2}}-\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-
a_{n-k}^{(\beta_l)}\big)\delta_tu_{ij}^{k-\frac{1}{2}}\nonumber\\
&-a_{n-1}^{(\beta_l)}(\psi_2)_{ij}\bigg]\nonumber
+\frac{\tau}{4\mu}\delta_x^2\delta_y^2\frac{u_{ij}^n-u_{ij}^{n-1}}{\tau}\nonumber\\
=&\delta_x^2u_{ij}^{n-\frac{1}{2}}+\delta_y^2u_{ij}^{n-\frac{1}{2}}
+f\big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\big),\nonumber\\
&1\leq i\leq M_1-1,\ 1\leq j\leq M_2-1,\ 1\leq n\leq N,\\ \label{eq:schini}
&u_{ij}^0=(\psi_1)_{ij},\ 1 \leq i\leq M_1-1,\ 1\leq j\leq M_2-1,\\ \label{eq:schboun}
&u_{ij}^n=\phi_{ij}^n, \ (i,j)\in \gamma=\big\{(i,j)\ |\ (x_i,y_j)\in\partial\Omega\big\},\ 0 \leq n\leq N,\end{aligned}$$ where $$(\psi_1)_{ij}=\psi_1(x_i,y_j),\ (\psi_2)_{ij}=\psi_2(x_i,y_j),\ 1 \leq i\leq M_1-1,\ 1\leq j\leq M_2-1,$$ and $$\phi_{ij}^n=\phi(x_i,y_j,t_n),\quad (i,j)\in\gamma,\quad 0 \leq n\leq N.$$
Notice $a_0^{(\beta_l)}=1$, then Eq. can be rewritten as: $$\label{eq:add}
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\tau^{\beta_l}\Gamma{(3-\beta_l)}}u_{ij}^n
-\frac{1}{2}\delta_x^2u_{ij}^n-\frac{1}{2}\delta_y^2u_{ij}^n
+\frac{1}{4\mu}\delta_x^2\delta_y^2u_{ij}^n\\
=&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\tau^{\beta_l}\Gamma{(3-\beta_l)}}
\bigg[u_{ij}^{n-1}+\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-a_{n-k}^{(\beta_l)}\big)\big(u_{ij}^k-u_{ij}^{k-1}\big)\\
&+\tau a_{n-1}^{(\beta_l)}(\psi_2)_{ij}\bigg]+\frac{1}{2}\delta_x^2u_{ij}^{n-1}+\frac{1}{2}\delta_y^2u_{ij}^{n-1}
+\frac{1}{4\mu}\delta_x^2\delta_y^2u_{ij}^{n-1}+f\big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\big),
\end{aligned}$$ or $$\begin{aligned}
&\left(\sqrt{\mu}I-\frac{1}{2\sqrt{\mu}}\delta_x^2\right)\bigg(\sqrt{\mu}I-\frac{1}{2\sqrt{\mu}}\delta_y^2\bigg)u_{ij}^n\\
=&\left(\sqrt{\mu}I+\frac{1}{2\sqrt{\mu}}\delta_x^2\right)\bigg(\sqrt{\mu}I+\frac{1}{2\sqrt{\mu}}\delta_y^2\bigg)u_{ij}^{n-1}
+\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\tau^{\beta_l}\Gamma(3-\beta_l)}\cdot\\
&\bigg[\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-a_{n-k}^{(\beta_l)}\big)\big(u_{ij}^k-u_{ij}^{k-1}\big)+\tau a_{n-1}^{(\beta_l)}(\psi_2)_{ij}\bigg]
+f\big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\big),
\end{aligned}$$ where $I$ denotes the identity operator.
Let $$u_{ij}^{\ast}=\left(\sqrt{\mu}I-\frac{1}{2\sqrt{\mu}}\delta_y^2\right)u_{ij}^n.$$ Together with and the ADI difference scheme is derived, and the procedure can be executed as follows:
On each time level $t=t_n$ $(1\leq n\leq N)$, firstly, for all fixed $y=y_j$ $(1\leq j\leq M_2-1)$, solving a set of $M_1-1$ equations at the mesh points $x_i$ $(1\leq i\leq M_1-1)$ to get the intermediate solution $u_{ij}^{\ast}$: $$\label{eq:x}
\left\{
\begin{aligned}
&\left(\sqrt{\mu}I-\frac{1}{2\sqrt{\mu}}\delta_x^2\right)u_{ij}^\ast
=\left(\sqrt{\mu}I+\frac{1}{2\sqrt{\mu}}\delta_x^2\right)\bigg(\sqrt{\mu}I+\frac{1}{2\sqrt{\mu}}\delta_y^2\bigg)u_{ij}^{n-1}\\
&+\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\tau^{\beta_l}\Gamma(3-\beta_l)}
\bigg[\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-a_{n-k}^{(\beta_l)}\big)\big(u_{ij}^k-u_{ij}^{k-1}\big)+\tau a_{n-1}^{(\beta_l)}(\psi_2)_{ij}\bigg]\\
&+f\big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\big),\quad 1\leq i\leq M_1-1,\\
&u_{0j}^\ast=\bigg(\sqrt{\mu}I-\frac{1}{2\sqrt{\mu}}\delta_y^2\bigg)u_{0j}^n,\quad
u_{M_1j}^\ast=\bigg(\sqrt{\mu}I-\frac{1}{2\sqrt{\mu}}\delta_y^2\bigg)u_{M_1j}^n;
\end{aligned}\right.$$ afterwards, for all fixed $x=x_i$ $(1\leq i\leq M_1-1)$, by computing a set of $M_2-1$ equations at the mesh points $y_j$ $(1\leq j\leq M_2-1)$, the solution $u_{ij}^n$ can be obtained: $$\label{eq:y}
\left\{
\begin{aligned}
&\bigg(\sqrt{\mu}I-\frac{1}{2\sqrt{\mu}}\delta_y^2\bigg)u_{ij}^n=u_{ij}^\ast, \quad 1\leq j\leq M_2-1,\\
&u_{i0}^n=\phi(x_i,y_0,t_n),\quad u_{iM_2}^n=\phi(x_i,y_{M_2},t_n).
\end{aligned}\right.$$
Analysis of the ADI difference scheme {#sec:analysis}
=====================================
Solvability
-----------
It is clear that the ADI scheme $-$ is a linear tridiagonal system in unknowns, and the coefficient matrices are strictly diagonally dominant. Thus the scheme $-$ has a unique solution. This result can be written as following.
The ADI difference scheme $-$ is uniquely solvable.
Stability
---------
In this subsection we prove the unconditional stability and the convergence of the difference scheme $-$. We start with some auxiliary definitions and useful results.
Denote the space of grid functions on $\bar{\Omega}_h$ $$\mathcal{V}_h=\{v\mid v=\{v_{ij}\mid (x_i,y_j)\in \bar{\Omega}_h\}\ and\ v_{ij}=0\ if\ (x_i,y_j)\in
\partial\Omega_h\}.$$ For any grid function $v\in \mathcal{V}_h$, the following discrete norms and Sobolev seminorm are introduced: $$\|v\|=\sqrt{h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}|v_{ij}|^2},\quad
\|\delta_x\delta_yv\|=\sqrt{h_1h_2\sum_{i=1}^{M_1}\sum_{j=1}^{M_2}|\delta_x\delta_yv_{i-\frac{1}{2},j-\frac{1}{2}}|^2},$$ $$\|\delta_xv\|=\sqrt{h_1h_2\sum_{i=1}^{M_1}\sum_{j=1}^{M_2-1}|\delta_xv_{i-\frac{1}{2},j}|^2},\quad
\|\delta_yv\|=\sqrt{h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2}|\delta_yv_{i,j-\frac{1}{2}}|^2},$$ $$\|\Delta_hv\|=\sqrt{h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}|\Delta_hv_{ij}|^2},\quad
|v|_1=\sqrt{\|\delta_xv\|^2+\|\delta_yv\|^2}.$$
\[lem:L2\] [@SamarskiiAndreev1976] For any grid function $v\in \mathcal{V}_h$, $\|v\|\leq \frac{1}{2\sqrt{3}}|v|_1$.
\[lem:L22\] [@Sun2009] For any grid function $v\in \mathcal{V}_h$, $|v|_1\leq \frac{1}{2\sqrt{3}}\|\Delta_hv\|$.
\[lem:126\] [@Sun2009] For any $G=\{G_1,G_2,G_3,\ldots\}$ and $q$, we have $$\begin{aligned}
&\sum_{n=1}^{m}\left[b_0G_n-\sum_{k=1}^{n-1}(b_{n-k-1}-b_{n-k})G_k-b_{n-1}q\right]G_n\\
\geq & \frac{t_m^{1-\alpha}}{2}\tau\sum_{n=1}^mG_n^2-\frac{t_m^{2-\alpha}}{2(2-\alpha)}q^2,\qquad m=1,2,3,\cdots,
\end{aligned}$$ where $$b_l=\frac{\tau^{2-\alpha}}{2-\alpha}[(l+1)^{2-\alpha}-l^{2-\alpha}],\qquad l=0,1,2,\cdots.$$
The discrete Gronwall’s inequality is also introduced below since it is necessary to prove the stability and convergence of the proposed method.
\[lem:gronwall\] [@QuarteroniValli2008] Assume that $k_n$ and $p_n$ are nonnegative sequences, and the sequence $\Phi_n$ satisfies $$\Phi_0\leq g_0,\qquad \Phi_n\leq g_0+\sum_{l=0}^{n-1}p_l+\sum_{l=0}^{n-1}k_l\Phi_l,\qquad n\geq 1,$$ where $g_0\geq 0$. Then the sequence $\Phi_n$ satisfies $$\Phi_n\leq\left(g_0+\sum_{l=0}^{l-1}p_l\right)\exp\left(\sum_{l=0}^{n-1}k_l\right),\qquad n\geq 1.$$
Since the ADI difference scheme $-$ is equivalent to $-$ if the intermediate variable $u^\ast$ is eliminated, we analyze the stability and convergence by employing the difference scheme $-$.
Assume that $\widetilde u_{ij}^n$ is the approximate solution of $u_{ij}^n$, which is the exact solution of the scheme $-$. Denote $\varepsilon_{ij}^n=u_{ij}^n-\widetilde u_{ij}^n,\ 0 \leq i\leq M_1,\ 0 \leq j\leq M_2, \ 0 \leq n\leq N$, then we have the perturbation error equations $$\begin{aligned}
\label{eq:pertt}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{\tau^{1-\beta_l}}{\Gamma(3-\beta_l)}
\bigg[a_0^{(\beta_l)}\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}-\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-
a_{n-k}^{(\beta_l)}\big)\delta_t\varepsilon_{ij}^{k-\frac{1}{2}}-a_{n-1}^{(\beta_l)}(\psi_2^\ast)_{ij}\bigg]\nonumber\\
&+\frac{\tau}{4\mu}\delta_x^2\delta_y^2\frac{\varepsilon_{ij}^n-\varepsilon_{ij}^{n-1}}{\tau}\nonumber\\
&=\delta_x^2\varepsilon_{ij}^{n-\frac{1}{2}}+\delta_y^2\varepsilon_{ij}^{n-\frac{1}{2}}
+f\big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\big)-f\big(x_i,y_j,t_{n-1},\widetilde u_{ij}^{n-1}\big),\nonumber\\
&1\leq i\leq M_1-1,\ 1\leq j\leq M_2-1,\ 1\leq n\leq N,\\
&\varepsilon_{ij}^0=(\psi_1)_{ij}-(\widetilde\psi_1)_{ij},\ 1 \leq i\leq M_1-1,\ 1\leq j\leq M_2-1,\nonumber\\
&\varepsilon_{ij}^n=0, \ (i,j)\in \gamma,\ 0 \leq n\leq N \nonumber,\end{aligned}$$ where $$(\psi_2^\ast)_{ij}=(\psi_2)_{ij}-(\widetilde\psi_2)_{ij}.$$
\[them:stability\] Assume that the condition is satisfied, then the difference scheme $-$ is unconditionally stable.
Let $$b_k^{(\beta_l)}=\frac{\tau^{2-{\beta_l}}}{2-{\beta_l}}a_k^{(\beta_l)},\quad 1\leq l\leq K,$$ then Eq. is equivalent to $$\label{eq:perterror}
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\Gamma(2-\beta_l)\tau}
\bigg[b_0^{(\beta_l)}\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}-\sum_{k=1}^{n-1}\big(b_{n-k-1}^{\beta_l}-
b_{n-k}^{(\beta_l)}\big)\delta_t\varepsilon_{ij}^{k-\frac{1}{2}}-b_{n-1}^{(\beta_l)}(\psi_2^\ast)_{ij}\bigg]\\
&+\frac{\tau}{4\mu}\delta_x^2\delta_y^2\frac{\varepsilon_{ij}^n-\varepsilon_{ij}^{n-1}}{\tau}\\
&=\delta_x^2\varepsilon_{ij}^{n-\frac{1}{2}}+\delta_y^2\varepsilon_{ij}^{n-\frac{1}{2}}
+f\big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\big)-f\big(x_i,y_j,t_{n-1},\widetilde u_{ij}^{n-1}\big),\\
&1\leq i\leq M_1-1,\ 1\leq j\leq M_2-1,\ 1\leq n\leq N.
\end{aligned}$$
Multiplying by $h_1h_2\tau\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}$, summing up for $i$ from $1$ to $M_1-1$, for $j$ from $1$ to $M_2-1$ and for $n$ from $1$ to $m$, we analyze each term in the derived equation. Firstly, by employing Lemma \[lem:126\], we have $$\label{eq:left1}
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\Gamma(2-\beta_l)}h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}
\bigg\{\sum_{n=1}^m\Big[b_0^{(\beta_l)}\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}-\\
&\sum_{k=1}^{n-1}\big(b_{n-k-1}^{(\beta_l)}-b_{n-k}^{(\beta_l)}\big)\delta_t\varepsilon_{ij}^{k-\frac{1}{2}}
-b_{n-1}^{(\beta_l)}(\psi_2^\ast)_{ij}\Big]
\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\bigg\}\\
\geq & \Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\Gamma(2-\beta_l)}\Big[\frac{1}{2}t_m^{1-\beta_l}\tau
\sum_{n=1}^m\big\|\delta_t\varepsilon^{n-\frac{1}{2}}\big\|^2\\
&-\frac{t_m^{2-\beta_l}}{2(2-\beta_l)}h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}(\psi_2^\ast)_{ij}^2\Big]\\
=&\frac{1}{2}\tau K_m\sum_{n=1}^m\big\|\delta_t\varepsilon^{n-\frac{1}{2}}\big\|^2
-\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{t_m^{2-\beta_l}}{2\Gamma(3-\beta_l)}\big\|\psi_2^\ast\big\|^2,
\end{aligned}$$ where $$K_m=\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{t_m^{1-\beta_l}}{\Gamma(2-\beta_l)}>0.$$ Whereafter using the discrete Green formula, we get $$\begin{aligned}
&h_1h_2\tau\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\sum_{n=1}^m\frac{\tau}{4\mu}\delta_x^2\delta_y^2
\frac{\varepsilon_{ij}^{n}-\varepsilon_{ij}^{n-1}}{\tau}\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\\
=&\frac{1}{4\mu}\sum_{n=1}^mh_1h_2\sum_{i=1}^{M_1}\sum_{j=1}^{M_2}\bigg(\delta_x\delta_y
\Big(\varepsilon_{i-\frac{1}{2},j-\frac{1}{2}}^n-\varepsilon_{i-\frac{1}{2},j-\frac{1}{2}}^{n-1}\Big)\bigg)
\bigg(\delta_x\delta_y\Big(\varepsilon_{i-\frac{1}{2},j-\frac{1}{2}}^n-\varepsilon_{i-\frac{1}{2},j-\frac{1}{2}}^{n-1}\Big)\bigg)\\
=&\frac{1}{4\mu}\sum_{n=1}^m\big\|\delta_x\delta_y(\varepsilon^n-\varepsilon^{n-1})\big\|^2\geq0,
\end{aligned}$$ and $$\label{eq:deltax}
\begin{aligned}
&\tau\sum_{n=1}^m\bigg[h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\Big(\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\Big)
\Big(\delta_x^2\varepsilon_{ij}^{n-\frac{1}{2}}\Big)\bigg]\\
=&-\tau\sum_{n=1}^m\bigg[h_1h_2\sum_{j=1}^{M_2-1}\sum_{i=1}^{M_1}\Big(\delta_x\varepsilon_{i-\frac{1}{2},j}^{n-\frac{1}{2}}\Big)
\Big(\delta_t\delta_x\varepsilon_{i-\frac{1}{2},j}^{n-\frac{1}{2}}\Big)\bigg]\\
=&-\tau\sum_{n=1}^m\bigg[h_1h_2\sum_{j=1}^{M_2-1}\sum_{i=1}^{M_1}\bigg(\frac{\delta_x\varepsilon_{i-\frac{1}{2},j}^n
+\delta_x\varepsilon_{i-\frac{1}{2},j}^{n-1}}{2}\bigg)\bigg(\frac{\delta_x\varepsilon_{i-\frac{1}{2},j}^n
-\delta_x\varepsilon_{i-\frac{1}{2},j}^{n-1}}{\tau}\bigg)\bigg]\\
=&-\frac{1}{2}\Big[\big\|\delta_x\varepsilon^m\big\|^2-\big\|\delta_x\varepsilon^0\big\|^2\Big].
\end{aligned}$$ Analogous to , it is also obtained $$\begin{aligned}
&\tau\sum_{n=1}^m\bigg[h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\Big(\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\Big)
\Big(\delta_y^2\varepsilon_{ij}^{n-\frac{1}{2}}\Big)\bigg]\\
=&-\frac{1}{2}\Big[\big\|\delta_y\varepsilon^m\big\|^2-\big\|\delta_y\varepsilon^0\big\|^2\Big].
\end{aligned}$$ On the basis of , there holds that $$\label{eq:right3}
\begin{aligned}
&h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\bigg[\tau\sum_{n=1}^m\Big(\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\Big)
\Big|f\Big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\Big)-f\Big(x_i,y_j,t_{n-1},\widetilde{u}_{ij}^{n-1}\Big)\Big|\bigg]\\
\leq & h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\bigg[\tau\sum_{n=1}^m\Big(\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\Big)
L_f\Big|u_{ij}^{n-1}-\widetilde{u}_{ij}^{n-1}\Big|\bigg]\\
\leq & L_fh_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\bigg[\tau\sum_{n=1}^m\Big(\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\Big)
\Big|\varepsilon_{ij}^{n-1}\Big|\bigg]\\
\leq & L_fh_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\tau\sum_{n=1}^m
\bigg[\frac{K_m}{2L_f}\Big(\delta_t\varepsilon_{ij}^{n-\frac{1}{2}}\Big)^2
+\frac{L_f}{2K_m}\Big(\varepsilon_{ij}^{n-1}\Big)^2\bigg]\\
=&\frac{\tau K_m}{2}\sum_{n=1}^m\big\|\delta_t\varepsilon^{n-\frac{1}{2}}\big\|^2
+\frac{\tau L_f^2}{2K_m}\sum_{n=1}^m\big\|\varepsilon^{n-1}\big\|^2.
\end{aligned}$$
From Equations $-$, the inequality below is derived $$\label{eq:inequ}
\begin{aligned}
&\big\|\delta_x\varepsilon^m\big\|^2+\big\|\delta_y\varepsilon^m\big\|^2\leq \big\|\delta_x\varepsilon^0\big\|^2+\big\|\delta_y\varepsilon^0\big\|^2\\
+& \Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{t_m^{2-\beta_l}}{\Gamma(3-\beta_l)}\big\|\psi_2^\ast\big\|^2
+\frac{\tau L_f^2}{K_m}\sum_{n=1}^m\big\|\varepsilon^{n-1}\big\|^2.
\end{aligned}$$ According to Lemma \[lem:L2\] and Lemma \[lem:L22\], we deduce from that $$\begin{aligned}
\|\varepsilon^n\|^2\leq & \frac{1}{144}\|\Delta_h\varepsilon^0\|^2+\frac{1}{12}\Delta\beta\sum_{l=1}^Kp(\beta_l)
\frac{T^{2-\beta_l}}{\Gamma(3-\beta_l)}\|\psi_2^\ast\|^2\\
+& \frac{\tau L_f^2}{12\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{T^{1-\beta_l}}{\Gamma(2-\beta_l)}}\sum_{k=1}^n\|\varepsilon^{k-1}\|^2,
\qquad 1\leq n\leq N.
\end{aligned}$$ Finally, taking Lemma \[lem:gronwall\], it follows that $$\begin{aligned}
\|\varepsilon^n\|^2\leq & \bigg(\frac{1}{144}\|\Delta_h\varepsilon^0\|^2+\frac{1}{12}\Delta\beta\sum_{l=1}^Kp(\beta_l)
\frac{T^{2-\beta_l}}{\Gamma(3-\beta_l)}\|\psi_2^\ast\|^2\bigg)\cdot\\
&\exp{\Bigg(\frac{L_f^2}{12\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{T^{-\beta_l}}{\Gamma(2-\beta_l)}}}\Bigg).
\end{aligned}$$ This completes the proof.
In the following we consider the convergence of the difference approximation. Noticing that $U_{ij}^n$ is the exact solution of the system $-$ and $u_{ij}^n$ is the numerical solution of the difference scheme $-$, we denote the error $$e_{ij}^n=U_{ij}^n-u_{ij}^n,\quad 0\leq i\leq M_1,\quad 0\leq j\leq M_2,\quad 0\leq n\leq N.$$ Subscribing $-$ from $-$, we get the error equations $$\begin{aligned}
\label{eq:error}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{\tau^{1-\beta_l}}{\Gamma(3-\beta_l)}\Big[a_0^{(\beta_l)}\delta_te_{ij}^{n-\frac{1}{2}}
-\sum_{k=1}^{n-1}\big(a_{n-k-1}^{(\beta_l)}-a_{n-k}^{(\beta_l)}\big)\delta_te_{ij}^{n-\frac{1}{2}}\Big]\nonumber\\
&+\frac{\tau}{4\mu}\delta_x^2\delta_y^2\frac{e_{ij}^n-e_{ij}^{n-1}}{\tau}\nonumber\\
= & \delta_x^2e_{ij}^{n-\frac{1}{2}}+\delta_y^2e_{ij}^{n-\frac{1}{2}}+f\big(x_i,y_j,t_{n-1},U_{ij}^{n-1}\big)
-f\big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\big)\nonumber\\
&+R_{ij}^{n-\frac{1}{2}}+\widehat{R}_{ij}^{n-\frac{1}{2}}+\widetilde{R}_{ij}^{n-\frac{1}{2}},
\ 1\leq i\leq M_1-1,\ 1\leq j\leq M_2-1,\ 1\leq n\leq N,\\
&e_{ij}^0=0,\quad 1\leq i\leq M_1-1,\quad 1\leq j\leq M_2-1,\nonumber\\
&e_{ij}^n=0,\quad (i,j)\in \gamma,\quad 0\leq n\leq N.\nonumber\end{aligned}$$
Suppose that the continuous problem $-$ has solution $u(x,y,t)\in C_{x,y,t}^{4,4,3}(\bar{\Omega}\times [0,T])$. Then there is a positive constant $C$ such that $$\nonumber
\big\|e^n\big\|\leq C (\tau+h_1^2+h_2^2+\Delta\beta^2).$$
The proof of convergence is similar to that of Theorem \[them:stability\]. Multiplying by $h_1h_2\tau\delta_t e_{ij}^{n-\frac{1}{2}}$, summing up for $i$ from $1$ to $M_1-1$, for $j$ from $1$ to $M_2-1$ and for $n$ from $1$ to $m$, we estimate each term in the resulted equation.
By using analogous strategies as $-$, we get $-$ correspondingly. $$\label{eq:convleft}
\begin{aligned}
&\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{\tau^{2-\beta_l}}{\Gamma(3-\beta_l)}h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}
\bigg\{\sum_{n=1}^m\bigg[a_0^{(\beta_l)}\delta_te_{ij}^{n-\frac{1}{2}}-\\
&\sum_{k=1}^{n-1}\Big(a_{n-k-1}^{(\beta_l)}-a_{n-k}^{(\beta_l)}\Big)\delta_te_{ij}^{k-\frac{1}{2}}\bigg]
\delta_te_{ij}^{n-\frac{1}{2}}\bigg\}\\
\geq &\frac{1}{2}\tau K_m\sum_{n=1}^m\Big\|\delta_te^{n-\frac{1}{2}}\Big\|^2,
\end{aligned}$$ $$\begin{aligned}
&h_1h_2\tau\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\sum_{n=1}^m\frac{\tau}{4\mu}\delta_x^2\delta_y^2
\frac{e_{ij}^{n}-e_{ij}^{n-1}}{\tau}\delta_te_{ij}^{n-\frac{1}{2}}\\
=&\frac{1}{4\mu}\sum_{n=1}^m\big\|\delta_x\delta_y(e^n-e^{n-1})\big\|^2\geq0,
\end{aligned}$$ $$\begin{aligned}
\tau\sum_{n=1}^m\Big[h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\Big(\delta_te_{ij}^{n-\frac{1}{2}}\Big)
\Big(\delta_x^2e_{ij}^{n-\frac{1}{2}}\Big)\Big]
=-\frac{1}{2}\big\|\delta_xe^m\big\|^2,
\end{aligned}$$ $$\begin{aligned}
\tau\sum_{n=1}^m\Big[h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\Big(\delta_te_{ij}^{n-\frac{1}{2}}\Big)
\Big(\delta_y^2e_{ij}^{n-\frac{1}{2}}\Big)\Big]
=-\frac{1}{2}\big\|\delta_ye^m\big\|^2,
\end{aligned}$$ and $$\label{eq:convright1}
\begin{aligned}
&h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\bigg[\tau\sum_{n=1}^m\Big(\delta_te_{ij}^{n-\frac{1}{2}}\Big)
\Big|f\Big(x_i,y_j,t_{n-1},U_{ij}^{n-1}\Big)-f\Big(x_i,y_j,t_{n-1},u_{ij}^{n-1}\Big)\Big|\bigg]\\
\leq & L_fh_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\bigg[\tau\sum_{n=1}^m\Big(\delta_te_{ij}^{n-\frac{1}{2}}\Big)
\Big|e_{ij}^{n-1}\Big|\bigg]\\
\leq & L_fh_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\tau\sum_{n=1}^m
\bigg[\frac{K_m}{4L_f}\Big(\delta_te_{ij}^{n-\frac{1}{2}}\Big)^2
+\frac{L_f}{K_m}\Big(e_{ij}^{n-1}\Big)^2\bigg]\\
=&\frac{\tau K_m}{4}\sum_{n=1}^m\big\|\delta_te^{n-\frac{1}{2}}\big\|^2
+\frac{\tau L_f^2}{K_m}\sum_{n=1}^m\big\|e^{n-1}\big\|^2.
\end{aligned}$$ As for the remainder, it is deduced that $$\label{eq:convright}
\begin{aligned}
&h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\sum_{n=1}^m\tau\big(\delta_te_{ij}^{n-\frac{1}{2}}\big)
\big(R_{ij}^{n-\frac{1}{2}}+\widetilde{R}_{ij}^{n-\frac{1}{2}}+\widehat{R}_{ij}^{n-\frac{1}{2}}\big)\\
\leq & h_1h_2\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\sum_{n=1}^m\tau\Big(\frac{K_m}{4}\big(\delta_te_{ij}^{n-\frac{1}{2}}\big)^2
+\frac{1}{K_m}\big(R_{ij}^{n-\frac{1}{2}}+\widetilde{R}_{ij}^{n-\frac{1}{2}}+\widehat{R}_{ij}^{n-\frac{1}{2}}\big)^2\Big)\\
\leq & \frac{\tau K_m}{4}\sum_{n=1}^m\big\|\delta_te^{n-\frac{1}{2}}\big\|^2+\frac{\tau h_1h_2}{K_m}\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\sum_{n=1}^m
\Big[C_3\big(\tau^{1+\frac{\Delta\beta}{2}}+h_1^2+h_2^2+\Delta\beta^2\big)\\
& +C_4\tau^3\big|\ln\tau\big|+C_5\tau\Big]^2\\
\leq & \frac{\tau K_m}{4}\sum_{n=1}^m\big\|\delta_te^{n-\frac{1}{2}}\big\|^2+\frac{\tau h_1h_2}{K_m}\sum_{i=1}^{M_1-1}\sum_{j=1}^{M_2-1}\sum_{n=1}^m
\Big[\big(C_3+C_4+C_5\big)\big(\tau+h_1^2+h_2^2+\Delta\beta^2\big)\Big]^2\\
\leq & \frac{\tau K_m}{4}\sum_{n=1}^m\big\|\delta_te^{n-\frac{1}{2}}\big\|^2
+\frac{TL_1L_2}{K_m}\Big[\big(C_3+C_4+C_5\big)\big(\tau+h_1^2+h_2^2+\Delta\beta^2\big)\Big]^2.
\end{aligned}$$ From $-$ it follows that $$\begin{aligned}
\frac{1}{2}\big(\big\|\delta_xe^m\big\|^2+\big\|\delta_ye^m\big\|^2\big)
\leq & \frac{TL_1L_2}{K_m}\Big[(C_3+C_4+C_5\big)\big(\tau+h_1^2+h_2^2+\Delta\beta^2\big)\Big]^2\\
& +\frac{\tau L_f^2}{K_m}\sum_{n=1}^m\big\|e^{n-1}\big\|^2,
\end{aligned}$$ i.e., $$\begin{aligned}
\frac{1}{2}\big|e^m\big|_1^2\leq & \frac{TL_1L_2}{\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\Gamma(2-\beta_l)}T^{1-\beta_l}}
\Big[(C_3+C_4+C_5\big)\big(\tau+h_1^2+h_2^2+\Delta\beta^2\big)\Big]^2\\
& +\frac{\tau L_f^2}{K_m}\sum_{n=1}^m\big\|e^{n-1}\big\|^2.
\end{aligned}$$ According to Lemma \[lem:L2\], we obtain $$\begin{aligned}
\big\|e^n\big\|^2\leq & \frac{TL_1L_2}{6\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\Gamma(2-\beta_l)}T^{1-\beta_l}}
\Big[(C_3+C_4+C_5\big)\big(\tau+h_1^2+h_2^2+\Delta\beta^2\big)\Big]^2\\
& +\frac{\tau L_f^2}{6K_n}\sum_{k=1}^n\big\|e^{k-1}\big\|^2,\quad 0\leq n\leq N.
\end{aligned}$$ Therefore, $$\begin{aligned}
\big\|e^n\big\|^2\leq & \frac{TL_1L_2}{6\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\Gamma(2-\beta_l)}T^{1-\beta_l}}
\Big[(C_3+C_4+C_5\big)\big(\tau+h_1^2+h_2^2+\Delta\beta^2\big)\Big]^2\cdot\\
& \exp{\bigg(\frac{L_f^2}{6\Delta\beta\sum_{l=1}^Kp(\beta_l)\frac{1}{\Gamma(2-\beta_l)}T^{-\beta_l}}}\bigg),
\end{aligned}$$ where Lemma \[lem:gronwall\] is applied. This completes the proof.
Numerical results {#sec:examples}
=================
In this section, a numerical example is tested to demonstrate the effectiveness of the proposed scheme, and verify the theoretical results including convergence orders and numerical stability. The discrete $L^2$ and $L^\infty$ norms are both taken to measure the numerical errors. Denote $$\big\|e^N\big\|_{L^2}:=\Bigg(\sum_{j=1}^{M_2-1}\sum_{i=1}^{M_1-1}\big|U_{ij}^N-u_{ij}^N\big|^2h_1h_2\Bigg)^{\frac{1}{2}},$$ and $$\big\|e^N\big\|_{L^\infty}:=\max_{1\leq j\leq M_2-1, 1\leq i\leq M_1-1}\big|U_{ij}^N-u_{ij}^N\big|.$$
\[exam:a\] $$\begin{aligned}
&\int_1^2\Gamma(4-\beta){}_0^CD_t^{\beta}u(x,y,t)d\beta= \frac{\partial^2u(x,y,t)}{\partial x^2}
+\frac{\partial^2u(x,y,t)}{\partial y^2}\\
&+\sin x\sin y\bigg[2(t^3+2t+4)+\frac{6t^2-6t}{\ln t}\bigg]
-(t^3+2t+4)^2\sin^2 x\sin^2 y+u^2, \\
&0<t<1/2,\quad (x,y)\in \Omega=(0,\pi)\times (0,\pi),\\
&u(x,y,t)=0,\quad (x,y)\in \partial\Omega,\quad 0<t<1/2,\\
&u(x,y,0)=4\sin x\sin y,\quad u_t(x,y,0)=2\sin x\sin y, \quad (x,y)\in\Omega,
\end{aligned}$$ whose analytical solution is known and is given by $$u(x,y,t)=(t^3+2t+4)\sin x\sin y.$$
In Figure \[fig:error\] we illustrate the relative error, which verifies the convergence of the algorithm we proposed.
In Figure \[fig:solution\] we present a comparison of the exact and numerical solutions. It can be seen that the numerical solution is in good agreement with the exact solution.
![Relative error at $T=0.5$, obtained by algorithm $-$ with mesh $h_1=h_2=\frac{\pi}{64}$, $\Delta\beta=\frac{1}{64}$, and $\tau=\frac{1}{4096}$. []{data-label="fig:error"}](mesherror.jpg "fig:"){width="1.2\linewidth"}\
![Exact solution (a) and approximate solution (b) obtained by algorithm $-$ at $T=0.5$ with mesh $h_1=h_2=\frac{\pi}{64}$, $\Delta\beta=\frac{1}{64}$, and $\tau=\frac{1}{4096}$.[]{data-label="fig:solution"}](meshexactu.jpg "fig:"){width="1\linewidth"}\
(a)
![Exact solution (a) and approximate solution (b) obtained by algorithm $-$ at $T=0.5$ with mesh $h_1=h_2=\frac{\pi}{64}$, $\Delta\beta=\frac{1}{64}$, and $\tau=\frac{1}{4096}$.[]{data-label="fig:solution"}](meshP1.jpg "fig:"){width="1\linewidth"}\
(b)
In Table \[table:a\], the numerical accuracy of difference scheme $-$ in time is recorded. Let the step sizes $h_1$, $h_2$, and $\Delta\beta$ be fixed and small enough such that the dominated error arise from the approximation of the time derivatives. Varying the step sizes in time, the numerical errors in discrete both $L^\infty$ and $L^2$ norms and the associated convergence orders are shown in this table respectively, which can be found in agreement with the theoretical analysis.
In Table \[table:b\], we take the fixed and small enough step sizes in space, and adopt an optimal step size ratio in time and distributed order. As $\Delta\beta$ and $\tau$ vary, we compute the errors and convergence orders listed in the table, which indicates that the convergence order in time and distributed order are about one and two, respectively.
Table \[table:c\] displays the computational results with an optimal step size ratio in time, space and distributed order. We can conclude from this table that the convergence orders with respect to time, space and distributed order are approximately one, two and two, respectively, which is in good agreement with our theoretical results analyzed in Section \[sec:analysis\].
$\tau$ $\big\|e^N\big\|_{L^\infty} $ Order $\big\|e^N\big\|_{L^2}$ Order
-------- ------------------------------- -------- ------------------------- --------
1/10 0.0839 - 0.1225 -
1/20 0.0439 0.9344 0.0634 0.9502
1/40 0.0227 0.9515 0.0326 0.9596
1/80 0.0117 0.9526 0.0167 0.9650
1/160 0.0059 0.9877 0.0085 0.9743
: Errors and convergence orders for Example \[exam:a\] in temporal direction with $h_1=h_2=\frac{\pi}{500}$ and $\Delta\beta=\frac{1}{160}$.[]{data-label="table:a"}
$\tau$ $\Delta\beta $ $\big\|e^N\big\|_{L^\infty}$ Order $\big\|e^N\big\|_{L^2}$ Order
-------- ---------------- ------------------------------ -------- ------------------------- --------
1/100 1/10 0.0093 - 0.0133 -
1/400 1/20 0.0024 1.9542 0.0034 1.9678
1/1600 1/40 6.0481e-04 1.9885 8.6411e-04 1.9762
1/6400 1/80 1.4751e-04 2.0357 2.1076e-04 2.0365
: Errors and convergence orders for Example \[exam:a\] with an optimal step size ratio for $\tau$ and $\Delta\beta$, and $h_1=h_2=\frac{\pi}{500}$.[]{data-label="table:b"}
$\tau$ $h_1=h_2$ $\Delta\beta$ $\big\|e^N\big\|_{L^\infty}$ Order $\big\|e^N\big\|_{L^2}$ Order
--------- ----------- --------------- ------------------------------ -------- ------------------------- --------
1/64 $\pi/2$ 1/8 0.4602 - 0.7230 -
1/256 $\pi/4$ 1/16 0.1195 1.9453 0.1689 2.0978
1/1024 $\pi/8$ 1/32 0.0301 1.9892 0.0426 1.9872
1/4096 $\pi/16$ 1/64 0.0075 2.0048 0.0107 1.9932
1/16384 $\pi/32$ 1/128 0.0019 1.9809 0.0027 1.9866
1/65536 $\pi/64$ 1/256 4.7098e-04 2.0123 6.6801e-04 2.0150
: Errors and convergence orders for Example \[exam:a\] with an optimal step size ratio for $\tau$, $h_1$, $h_2$, and $\Delta\beta$.[]{data-label="table:c"}
Conclusion {#sec:conclusion}
==========
In this paper, we construct efficient numerical scheme for solving two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term, and provide the theoretical analysis on stability and convergence by the discrete energy method. Numerical results are provided by figures and tables, which show the algorithm proposed in this work is effective and feasible. In the future work, the promotion of computational efficiency will be considered so that the more complicated problems can be handled.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by National Natural Science Foundations of China (No.11471262). The authors would like to express their gratitude to the referees for their very helpful comments and suggestions on the manuscript.
|
[ Nobuchika Okada$^{~a}$ and Satomi Okada ]{}
[ *$^a$Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL35487, USA* ]{}
Motivated by the recent CMB $B$-mode observation announced by the BICEP2 collaboration, we study simple inflationary models in the Randall-Sundrum brane-world cosmology. Brane-world cosmological effects alter the inflationary predictions of the spectral index ($n_s$) and the tensor-to-scalar ratio ($r$) from those obtained in the standard cosmology. In particular, the tensor-to-scalar ratio is enhanced in the presence of the 5th dimension, and simple inflationary models which predict small $r$ values in the standard cosmology can yield $r$ values being compatible with the BICEP2 result, $r=0.2^{+0.07}_{-0.05}$. Confirmation of the BICEP2 result and more precise measurements of $n_s$ and $r$ in the near future allow us to constrain the 5-dimensional Planck mass ($M_5$) of the brane-world scenario. We also discuss the post inflationary scenario, namely, reheating of the universe through inflaton decay to the Standard Model particles. When we require the renormalizability of the inflationary models, the inflaton only couples with the Higgs doublet among the Standard Model fields, through which reheating process occurs. If $M_5$ is as low as $10^6$ GeV, the mass of inflaton becomes of ${\cal O}$(100 GeV), providing the inflationary predictions being consistent with the BICEP2 and Planck results. Such a light inflaton has some impacts on phenomenology at the electroweak scale through the coupling with the Higgs doublet.
Introduction
============
Inflationary universe is the standard paradigm in the modern cosmology [@inflation1; @inflation2; @chaotic_inflation; @inflation4] which provides not only solutions to various problems in the standard big-bang cosmology, such as the flatness and horizon problems, but also the primordial density fluctuations as seeds of the large scale structure observed in the present universe. Various inflation models have been proposed with typical inflationary predictions for the primordial perturbations. Recent cosmological observations by, in particular, the Wilkinson Microwave Anisotropy Probe (WMAP) [@WMAP9] and the Planck satellite [@Planck2013] experiments have measured the cosmological parameters precisely and provided constraints on the inflationary predictions for the spectral index ($n_s$), the tensor-to-scalar ratio ($r$), the running of the spectral index ($\alpha=d n_s/d \ln k$), and non-Gaussianity of the primordial perturbations. Future cosmological observations are expected to be more precise towards discriminating inflationary models.
Recently the Background Imaging of Cosmic Extragalactic Polarization (BICEP2) collaboration has reported the observation of CMB $B$-mode polarization [@BICEP2], which is interpreted as the primordial gravity waves with $r=0.20^{+0.07}_{-0.05}$ (68% confidence level) generated by inflation. However, future observations of the $B$-mode polarization with different frequencies are crucial to verify the BICEP2 result, since uncertainty of dust polarization could dominate the excess observed by the BICEP2 experiment [@dust_uncertainty]. If confirmed, the BICEP2 result is the direct evidence of the cosmological inflation in the early universe and provide us with great progress in understanding inflationary universe. The observed value of the tensor-to-scalar ratio favors the chaotic inflation model [@chaotic_inflation]. In the light of the BICEP2 result of the large tensor-to-scalar ratio of ${\cal O}(0.1)$, various inflation models and their predictions have been reexamined/updated. In this paper, we study simple inflationary models in the context of the brane-world cosmology (see, for example, Ref. [@OSS2014] for an update of the inflationary predictions of simple models in the standard cosmology).
The brane-world cosmology is based on a 5-dimensional model first proposed by Randall and Sundrum (RS) [@RS2], the so-called RS II model, where the standard model particles are confined on a “3-brane” at a boundary embedded in 5-dimensional anti-de Sitter (AdS) space-time. Although the 5th dimensional coordinate tends to infinity, the physical volume of the extra-dimension is finite because of the AdS space-time geometry. The 4-dimensional massless graviton is localizing around the brane on which the Standard Model particles reside, while the massive Kaluza-Klein gravitons are delocalized toward infinity, and as a result, the 4-dimensional Einstein-Hilbert action is reproduced at low energies. Cosmology in the context of the RS II setup has been intensively studied [@braneworld] since the finding of a cosmological solution in the RS II setup [@RS2solution]. Interestingly, the Friedmann equation in the RS cosmology leads to a non-standard expansion law of our 4-dimensional universe at high energies, while reproducing the standard cosmological law at low energies. This non-standard evolution of the early universe causes modifications of a variety of phenomena in particle cosmology, such as the dark matter relic abundance [@DM_BC] (see also [@DM_GB] for the modification of dark matter physics in a more general brane-world cosmology, the Gauss-Bonnet brane-world cosmology [@GB]), baryogensis via leptogenesis [@LG_BC], and gravitino productions in the early universe [@gravitino_BC].
The modified Friedmann equation also affects inflationary scenario. A chaotic inflation with a quadratic inflaton potential has been examined in [@inflation_BC] and it has been shown that the inflationary predictions are modified from those in the 4-dimensional standard cosmology. Remarkably, the power spectrum of tensor fluctuation is found to be enhanced in the presence of the 5-dimensional bulk [@PT_BC]. Taking this brane-cosmological effect into account, the textbook chaotic inflation models with the quadratic and quartic potentials have been analyzed in [@inflation_models_BC].
In the next section, we first update the inflationary predictions of the textbook inflationary models for various values of the 5-dimensional Planck mass ($M_5$) in the light of the BICEP2 result. Next we analyze the Higgs potential and the Coleman-Weinberg potential modes with various values of the inflaton vacuum expectation value (VEV) and $M_5$. We will show that the brane-world cosmological effect dramatically alters the inflationary predictions from those in the standard cosmology. In particular, the tensor-to-scalar ratio which is predicted to be small in the standard cosmology can be enhanced so as to be compatible with the BICEP2 result when $M_5$ is much smaller than the 4-dimensional Planck mass. For such small $M_5$ values, we will find a relation between the inflaton mass and the 5-dimensional Planck mass to be $m \simeq 10^{-4} M_5$, while the inflationary predictions are consistent with the cosmological observations. Thus, when $M_5$ is as low as $10^6$ GeV, a realistic inflationary scenario can be obtained even for the inflaton mass of ${\cal O}$(100 GeV).
We will also discuss reheating after inflation through the inflaton decay to the Standard Model particles. At the renormalizable level, the inflaton only couples to the Higgs doublet among the Standard Model fields, so that this coupling plays a crucial role in reheating process. When an inflaton mass is much higher than the electroweak scale as usual in the standard inflationary models, the inflaton decays to a pair of Higgs doublets to thermalize the universe. When an inflaton is lighter than the Standard Model Higgs boson, the inflaton mainly decays to a pair of bottom quarks through a mass mixing with the Higgs boson. Such a light inflaton has an impact on Higgs boson phenomenology, since the Higgs boson can decay to a pair of inflatons. Precision measurement of the Higgs boson properties at future collider experiments, such as the International $e^+ e^-$ Linear Collider (ILC), may reveal the existence of an extra scalar field produced through the Higgs boson decay. In the context of the brane-world cosmology, this scalar field can play the role of inflaton, providing the inflationary predictions being consistent with the current cosmological observations. Another possibility we will discuss with the light inflaton is a unified picture of the inflaton and the dark matter particle. Introducing a $Z_2$ parity to ensure the stability of the inflaton, the light inflaton can play the role of the so-called Higgs portal dark matter. Since the inflaton cannot decay in this scenario, we consider preheating to transmit the inflaton energy density into the radiation by explosive production of the Higgs doublets via parametric resonance during the oscillation of the inflaton around its potential minimum. Then, the universe is thermalized by the decay products of the Higgs doublet.
Simple inflationary models in the brane-world cosmology
=======================================================
In the RS II brane-world cosmology, the Friedmann equation for a spatially flat universe is found to be [@RS2solution] $$H^2 = \frac{\rho}{3 M_P} \left(1+\frac{\rho}{\rho_0} \right) + \frac{C}{a^4},
\label{BraneFriedmannEq}$$ where $M_P=2.435 \times 10^{18}$ GeV is the reduced Planck mass, $$\begin{aligned}
\rho_0 = 12 \frac{M_5^6}{M_P^2},
\label{rho_0}\end{aligned}$$ with $M_5$ being the 5-dimensional Planck mass, a constant $C$ is referred to as the “dark radiation,” and we have omitted the 4-dimensional cosmological constant. Note that the Friedmann equation in the standard cosmology is reproduced for $\rho/\rho_0 \ll 1$ and $\rho/(3 M_P^2) \gg C/a^4$. There are phenomenological, model-independent constraints for these new parameters from Big Bang Nucleosynthesis (BBN), which provides successful explanations for synthesizing light nuclei in the early universe. In order not to ruin the success, the expansion law of the universe must obey the standard cosmological one at the BBN era with a temperature of the universe $T_{\rm BBN} \simeq 1$ MeV. Since the constraint on the dark radiation is very severe [@dark_radiation], we simply set $C=0$ in the following analysis. We estimate a lower bound on $\rho_0$ by $\rho_0^{1/4} > T_{\rm BBN}$ and find $M_5 > 8.8$ TeV.[^1] The energy density of the universe is high enough to satisfy $\rho/\rho_0 \gtrsim1$, the expansion law becomes non-standard, and this brane-world cosmological effect alters results obtained in the context of the standard cosmology.
Let us now consider inflationary scenario in the brane-world cosmology with the modified Friedmann equation. In slow-roll inflation, the Hubble parameter is approximately given by (from now on, we use the Planck unit, $M_P=1$) $$\begin{aligned}
H^2 = \frac{V}{3} \left( 1+ \frac{V}{\rho_0}\right),
\label{H_BC}\end{aligned}$$ where $V(\phi)$ is a potential of the inflaton $\phi$. Since the inflaton is confined on the brane, the power spectrum of scalar perturbation obeys the same formula as in the standard cosmology, except for the modification of the Hubble parameter [@inflation_BC], $$\begin{aligned}
{\cal P}_{\cal S} =\frac{9}{4 \pi^2} \frac{H^6}{(V')^2}, \end{aligned}$$ where the prime denotes the derivative with respect to the inflaton field $\phi$. The Planck satellite experiment [@Planck2013] constrains the power spectrum as $ {\cal P}_{\cal S}(k_0) = 2.215 \times 10^{-9}$ for the pivot scale chosen at $k_0=0.05$ Mpc$^{-1}$. The spectral index is given by $$\begin{aligned}
n_s -1 = \frac{d \ln {\cal P}_{\cal S}}{d\ln k} =-6 \epsilon + 2 \eta \end{aligned}$$ with the slow-roll parameters defined as $$\begin{aligned}
\epsilon = \frac{V^\prime}{6 H^2} \left( \ln H^2 \right)^\prime, \; \; \eta =\frac{V^{\prime \prime}}{3 H^2} .\end{aligned}$$ The running of the spectral index, $\alpha=dn_s/d\ln k$, is given by $$\begin{aligned}
\alpha=\frac{dn_s}{d\ln k} =\frac{V^\prime}{3 H^2} \left( 6 \epsilon^\prime - 2 \eta^\prime \right). \end{aligned}$$ On the other hand, in the presence of the extra dimension where graviton resides, the power spectrum of tensor perturbation is modified to be [@PT_BC] $$\begin{aligned}
{\cal P}_{\cal T} =8 \left( \frac{H}{2 \pi} \right)^2 F(x_0)^2,
\label{PT}
\label{P_T}\end{aligned}$$ where $x_0 = 2 \sqrt{3 H^2/\rho_0}$, and $$\begin{aligned}
F(x)= \left( \sqrt{1+x^2} - x^2 \ln\left[ \frac{1}{x}+\sqrt{1+\frac{1}{x^2}} \right] \right)^{-1/2}. \end{aligned}$$ For $x_0 \ll 1$ (or $V/\rho_0 \ll1$), $F(x_0) \simeq 1$, and Eq. (\[P\_T\]) reduces to the formula in the standard cosmology. For $x_0 \gg 1$ (or $V/\rho_0 \gg1$), $F(x_0) \simeq \sqrt{3 x_0/2} \simeq \sqrt{3 V/\rho_0} \gg 1$. The tensor-to-scalar ratio is defined as $r = {\cal P}_{\cal T}/ {\cal P}_{\cal S}$.
The e-folding number is given by $$\begin{aligned}
N_0 = \int_{\phi_e}^{\phi_0} d\phi \; \frac{3 H^2}{V^\prime} = \int_{\phi_e}^{\phi_0} d\phi \; \frac{V}{V^\prime} \left(1+\frac{V}{\rho_0} \right), \end{aligned}$$ where $\phi_0$ is the inflaton VEV at horizon exit of the scale corresponding to $k_0$, and $\phi_e$ is the inflaton VEV at the end of inflation, which is defined by ${\rm max}[\epsilon(\phi_e), | \eta(\phi_e)| ]=1$. In the standard cosmology, we usually consider $N_0=50-60$ in order to solve the horizon problem. Since the expansion rate in the brane-world cosmology is larger than the standard cosmology case, we may expect a larger value of the e-folding number. For the model-independent lower bound, $\rho_0^{1/4} > 1$ MeV, the upper bound $N_0 <75$ was found in [@N_BC]. In what follows, we consider $N_0=50$, $60$, and $70$, as reference values.
Textbook inflationary models
----------------------------
We first analyze the textbook chaotic inflation model with a quadratic potential [@chaotic_inflation], $$\begin{aligned}
V =\frac{1}{2} m^2 \phi^2. \end{aligned}$$ In the standard cosmology, simple calculations lead to the following inflationary predictions: $$\begin{aligned}
n_s=1-\frac{4}{2 N_0+1}, \; \;
r=\frac{16}{2 N_0+1}, \; \;
\alpha= - \frac{8}{(2 N_0+1)^2}. \end{aligned}$$ The inflaton mass is determined so as to satisfy the power spectrum measured by the Planck satellite experiment, ${\cal P}_{\cal S}(k_0)=2.215 \times 10^{-9}$: $$\begin{aligned}
m [{\rm GeV}]= 1.46 \times 10^{13} \left( \frac{60.5}{N_0+1/2}\right). \end{aligned}$$
In the brane-world cosmology, these inflationary predictions in the standard cosmology are altered due to the modified Friedmann equation. In the limit, $V/\rho_0 \gg 1$, the Hubble parameter is simplified as $H^2 \simeq V^2/(3 \rho_0)$, and we can easily find $$\begin{aligned}
n_s=1-\frac{5}{2 N_0+1}, \; \;
r=\frac{24}{2 N_0+1}, \; \;
\alpha= - \frac{10}{(2 N _0+1)^2}.
\label{phi2_BClimit}\end{aligned}$$ The initial ($\phi_0$) and the final ($\phi_e$) inflaton VEVs are found to be $$\begin{aligned}
\phi_0^4 =96 \frac{M_5^6}{m^2} (2 N_0+1), \; \; \phi_e^4 =96 \frac{M_5^6}{m^2} .
\label{phi2_int}\end{aligned}$$ For a common $N_0$ value, the spectral index is reduced, while $r$ and $|\alpha |$ are found to be larger than those predicted in the standard cosmology. In the brane-world cosmology, once $\rho_0$ is fixed, equivalently, $M_5$ is fixed through Eq. (\[rho\_0\]), the inflaton mass is determined by the constraint ${\cal P}_{\cal S}(k_0)=2.215 \times 10^{-9}$. For the limit $V/\rho_0 \gg 1$, we find [@inflation_BC] $$\begin{aligned}
\frac{m}{M_5} \simeq 1.26 \times 10^{-4} \left( \frac{60.5}{N_0+1/2}\right)^{5/6} .
\label{m/M5}\end{aligned}$$ Note that the model-independent BBN bound on $M_5 > 8.8$ TeV leads to $m > 1.1$ GeV for $N_0=60$. Therefore, unlike inflationary scenario in the standard cosmology, the inflaton can be very light, providing the inflationary predictions being compatible with the current observation (see Fig. \[fig:phi2\]). This analysis is valid for $V(\phi_0)/\rho_0 \gg 1$, in other words, $M_5 \ll 0.01$ by using Eqs. (\[rho\_0\]), (\[phi2\_int\]), and (\[m/M5\]).
![ The inflationary predictions for the quadratic potential model: $n_s$ vs. $r$ (left panel) and $n_s$ vs. $\alpha$ (right panel) for various $M_5$ values with $N_0=50$, $60$ and $70$ (from left to right), along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2) [@BICEP2]. The black triangles are the predictions of the textbook quadratic potential model in the standard cosmology, which are reproduced for $M_5 \gtrsim 1$. As $M_5$ is lowered, the inflationary predictions approach the values in Eq. (\[phi2\_BClimit\]), denoted by the black squares. In each line, the turning point appears for $V(\phi_0)/\rho_0 \simeq 1$. []{data-label="fig:phi2"}](Phi21.eps "fig:"){width="45.00000%"} ![ The inflationary predictions for the quadratic potential model: $n_s$ vs. $r$ (left panel) and $n_s$ vs. $\alpha$ (right panel) for various $M_5$ values with $N_0=50$, $60$ and $70$ (from left to right), along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2) [@BICEP2]. The black triangles are the predictions of the textbook quadratic potential model in the standard cosmology, which are reproduced for $M_5 \gtrsim 1$. As $M_5$ is lowered, the inflationary predictions approach the values in Eq. (\[phi2\_BClimit\]), denoted by the black squares. In each line, the turning point appears for $V(\phi_0)/\rho_0 \simeq 1$. []{data-label="fig:phi2"}](Phi22.eps "fig:"){width="45.00000%"}
![ Relations between $n_s$ and $M_5$ (left panel) and between $n_s$ and $m$ (right-panel), for $N_0=50$, $60$ and $70$ from left to right. The black triangles denote the predictions in the standard cosmology. []{data-label="fig:phi2_mass"}](Phi2M5.eps "fig:"){width="45.00000%"} ![ Relations between $n_s$ and $M_5$ (left panel) and between $n_s$ and $m$ (right-panel), for $N_0=50$, $60$ and $70$ from left to right. The black triangles denote the predictions in the standard cosmology. []{data-label="fig:phi2_mass"}](Phi2m.eps "fig:"){width="45.00000%"}
$M_5 ({\rm GeV})$ $m ({\rm GeV})$ $V(\phi_0)^{1/4}$ (GeV) $\phi_0$ $\phi_e$ $n_s$ $r$ $-\alpha\ (10^{-4})$
------------------------ ------------------------- ------------------------- ---------- ----------- ---------- --------- ----------------------
$ \infty $ $ 1.46 \times 10^{13} $ $1.97 \times 10^{16}$ $15.6$ $1.41 $ $0.967 $ $0.132$ $ 5.46 $
$ 1.02 \times 10^{17}$ $ 1.38 \times 10^{13} $ $1.91 \times 10^{16}$ $15.3$ $1.41 $ $0.965 $ $0.160$ $ 5.50 $
$ 7.72 \times 10^{16}$ $ 1.20 \times 10^{13} $ $1.75 \times 10^{16}$ $14.8$ $1.41 $ $0.962 $ $0.189$ $ 5.58 $
$ 5.87 \times 10^{16}$ $ 9.41 \times 10^{13} $ $1.49 \times 10^{16}$ $13.7$ $1.41 $ $0.958 $ $0.212$ $ 6.47 $
$ 3.79 \times 10^{16}$ $ 5.69 \times 10^{13} $ $1.04 \times 10^{16}$ $11.1$ $1.41 $ $0.956 $ $0.219$ $ 7.28 $
$ 2.29 \times 10^{16}$ $ 3.14 \times 10^{12} $ $6.47 \times 10^{15}$ $7.77$ $1.40 $ $0.957 $ $0.210$ $ 7.24 $
$ 2.19 \times 10^{15}$ $ 2.77 \times 10^{11} $ $6.30 \times 10^{14}$ $0.834$ $0.250 $ $0.959 $ $0.198$ $ 6.84 $
$ 1.23 \times 10^{15}$ $ 1.55 \times 10^{11} $ $3.54 \times 10^{14}$ $0.469$ $0.141 $ $0.959 $ $0.198$ $ 6.83 $
$ 5.00 \times 10^{14}$ $ 6.30 \times 10^{10} $ $1.44 \times 10^{14}$ $0.190$ $0.0573 $ $0.959 $ $0.198$ $ 6.83 $
: The values of parameters for the potential $V=(1/2) m^2 \phi^2$ for $N_0=60$, in the Planck unit ($M_P=1$) unless otherwise stated. []{data-label="table:phi2"}
We calculate the inflationary predictions for various values of $M_5$ with fixed e-folding numbers, and show the results in Fig. \[fig:phi2\]. In the left panel, the inflationary predictions for $N_0=50$, $60$, $70$ from left to right are shown, along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2) [@BICEP2]. The black triangles represent the predictions of the quadratic potential model in the standard cosmology, while the black squares are the predictions in the limit of $V/\rho_0 \gg 1. $ For various values of $M_5$, the inflationary predictions stay inside of the contour of the BICEP2 result at 95% confidence level. The results for the running of the spectral index ($n_s$ vs. $\alpha$) is shown in the right panel, for $N_0=50$, $60$, $70$ from left to right. We also show our results for the 5-dimensional Planck mass ($n_s$ vs. $M_5$) and the inflaton mass ($n_s$ vs. $m$) in Fig. \[fig:phi2\_mass\]. In each solid line, the turning point appears for $V(\phi_0)/\rho_0 \simeq 1$. For $N_0=60$, numerical values for selected $M_5$ values are listed in Table \[table:phi2\], for readers convenience.
Next we analyze the textbook quartic potential model, $$\begin{aligned}
V =\frac{\lambda}{4!} \phi^4. \end{aligned}$$ In the standard cosmology, we find the following inflationary predictions: $$\begin{aligned}
n_s=1-\frac{6}{2 N_0+3}, \; \;
r=\frac{32}{2 N_0+3}, \; \;
\alpha= - \frac{12}{(2 N_0+3)^2}. \end{aligned}$$ The quartic coupling ($\lambda$) is determined by the power spectrum measured by the Planck satellite experiment, ${\cal P}_{\cal S}(k_0)=2.215 \times 10^{-9}$ at the pivot scale $k_0=0.05$ Mpc$^{-1}$, as $$\begin{aligned}
\lambda = 8.46 \times 10^{-13} \left( \frac{123}{2 N_0+3}\right)^3. \end{aligned}$$
When the limit $V/\rho_0 \gg 1$ is satisfied during the inflation, we find in the brane-world cosmology $$\begin{aligned}
n_s=1-\frac{9}{3 N_0+2}, \; \;
r=\frac{48}{3 N_0+ 2}, \; \;
\alpha= - \frac{27}{(3 N _0+2)^2}.
\label{phi4_BClimit}\end{aligned}$$ The predicted values are very close to those in the standard cosmology for $N_0 \gg 1$. Using ${\cal P}_{\cal S}(k_0)=2.215 \times 10^{-9}$, we find $$\begin{aligned}
\lambda = 3.26 \times 10^{-14} \left( \frac{182}{3 N_0+ 2}\right)^{3} , \end{aligned}$$ which is independent of $M_5$.
We calculate the inflationary predictions for various values of $M_5$ with fixed e-folding numbers. Our results are shown in Fig. \[fig:phi4\]. In the left panel, the inflationary predictions for $N_0=50$, $60$, $70$ from left to right are shown, along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2). The results for the running of the spectral index ($n_s$ vs. $\alpha$) is shown in the right panel, for $N_0=$50, 60, 70 from left to right. The black points represent the predictions in the standard cosmology. In Fig. \[fig:phi4\], the inflationary predictions are moving anti-clockwise along the contours as $M_5$ is lowered. The turning point on each contour appears for $V(\phi_0)/\rho_0 \simeq 1$, and as $M_5$ is further lowered, the inflationary predictions go back closer to those in the standard cosmology. The brane-world cosmological effect cannot improve the fit of the BICEP2 result. We also show corresponding results for the 5-dimensional Planck mass ($n_s$ vs. $M_5$) and the quartic coupling ($n_s$ vs. $\lambda$) in Fig. \[fig:phi4\_mass\]. For $N_0=70$, numerical values for selected $M_5$ values are listed in Table \[table:phi4\].
![ The inflationary predictions for the quartic potential model: $n_s$ vs. $r$ (left panel) and $n_s$ vs. $\alpha$ (right panel) with various $M_5$ values for $N_0=50$, $60$ and $70$ (from left to right), along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2). The black points are the predictions of the textbook quartic potential model in the standard cosmology, which are reproduced for $M_5 \gtrsim 1$. As $M_5$ is lowered, the inflationary predictions approach the values in Eq. (\[phi4\_BClimit\]) which are in fact very close to those in the standard cosmology. In each line, the turning point appears for $V(\phi_0)/\rho_0 \simeq 1$. []{data-label="fig:phi4"}](Phi41.eps "fig:"){width="45.00000%"} ![ The inflationary predictions for the quartic potential model: $n_s$ vs. $r$ (left panel) and $n_s$ vs. $\alpha$ (right panel) with various $M_5$ values for $N_0=50$, $60$ and $70$ (from left to right), along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2). The black points are the predictions of the textbook quartic potential model in the standard cosmology, which are reproduced for $M_5 \gtrsim 1$. As $M_5$ is lowered, the inflationary predictions approach the values in Eq. (\[phi4\_BClimit\]) which are in fact very close to those in the standard cosmology. In each line, the turning point appears for $V(\phi_0)/\rho_0 \simeq 1$. []{data-label="fig:phi4"}](Phi42.eps "fig:"){width="45.00000%"}
![ Relations between $n_s$ and $M_5$ (left panel) and between $n_s$ and $\lambda$ (right-panel), for $N_0=50$, $60$ and $70$ from left to right. The black points denote the values obtained in the standard cosmology. []{data-label="fig:phi4_mass"}](Phi4M5.eps "fig:"){width="45.00000%"} ![ Relations between $n_s$ and $M_5$ (left panel) and between $n_s$ and $\lambda$ (right-panel), for $N_0=50$, $60$ and $70$ from left to right. The black points denote the values obtained in the standard cosmology. []{data-label="fig:phi4_mass"}](Phi4lam.eps "fig:"){width="45.00000%"}
$M_5 ({\rm GeV})$ $ \lambda $ $V(\phi_0)^{1/4}$ (GeV) $\phi_0$ $\phi_e$ $n_s$ $r$ $-\alpha\ (10^{-4})$
------------------------ -------------------------- ------------------------- ---------- ---------- ---------- ---------- ----------------------
$ \infty $ $ 5.38 \times 10^{-13} $ $2.25 \times 10^{16}$ $23.9$ $3.46 $ $0.958 $ $0.224$ $ 5.87 $
$1.09 \times 10^{17}$ $4.75 \times 10^{-13}$ $2.16 \times 10^{16}$ $23.7$ $3.46$ $0.955$ $0.272 $ $5.56$
$9.07 \times 10^{16} $ $4.06 \times 10^{-13} $ $2.05 \times 10^{16}$ $23.3$ $3.46$ $0.952$ $0.299$ $5.57$
$5.31\times 10^{16} $ $1.62\times 10^{-13}$ $1.45 \times 10^{16} $ $20.8$ $3.46$ $0.944$ $0.328$ $7.79$
$3.39 \times 10^{16}$ $7.00 \times 10^{-14}$ $9.69\times 10^{15}$ $17.1$ $3.46$ $0.947$ $0.295$ $8.11$
$2.39\times 10^{16} $ $4.24\times 10^{-14}$ $6.88 \times 10^{15}$ $13.8$ $3.43$ $0.951$ $0.267$ $7.42$
$1.59\times 10^{16}$ $2.93 \times 10^{-14}$ $4.60 \times 10^{15} $ $10.1$ $3.24$ $0.954$ $0.246$ $6.71$
$2.20 \times 10^{15}$ $2.08\times 10^{-14}$ $6.39 \times 10^{14}$ $1.53$ $0.698$ $0.957$ $0.227$ $6.02$
: The values of parameters for the potential $V=\lambda \phi^4/4!$ for the e-folding number $N_0=70$, in the Planck unit ($M_P=1$) unless otherwise stated. []{data-label="table:phi4"}
Higgs potential model
---------------------
Next we consider an inflationary scenario based on the Higgs potential of the form [@Higgs_potential_model] $$\begin{aligned}
V= \frac{\lambda}{8} \left( \phi^2 -v^2 \right)^2, \end{aligned}$$ where $\lambda$ is a real, positive coupling constant, $v$ is a VEV of the inflaton $\phi$. For simplicity, we assume the inflaton is a real scalar in this paper, but this model is easily extended to the Higgs model in which the inflaton field breaks a gauge symmetry by its VEV. See, for example, Refs. [@hp_corrections; @BL_inflation] for recent discussion, where quantum corrections of the Higgs potential are also taken into account.
For analysis of this inflationary scenario, we can consider two cases for the initial inflaton VEVs: (i) $\phi_0 < v$ and (ii) $\phi_0 > v$. In the case (i), the inflationary prediction for the tensor-to-scalar ratio is found to be small for $v < 1$, since the potential energy during inflation never exceeds $\lambda v^4/8$. This Higgs potential model reduces to the textbook quadratic potential model in the limit $v \gg 1$. To see this, we rewrite the potential in terms of $\phi=\chi +v$ with an inflaton $\chi$ around the potential minimum at $\phi=v$, $$\begin{aligned}
V= \frac{\lambda}{8} \left( 4 v^2 \chi^2 +4 v \chi^3 +\chi^4 \right). \end{aligned}$$ It is clear that if a condition, $\chi_0/v \ll 1$, for an initial value of inflaton ($\chi_0$) is satisfied, the inflaton potential is dominated by the quadratic term. Recall that $\chi_0 > 1$ in the textbook quadratic potential model and hence $v \gg 1$ is necessary to satisfy the condition. In the case (ii), it is clear that the model reduces to the textbook quartic potential model for $ v \ll 1$. For the limit $v \gg 1$, we apply the same discussion in the case (i), so that the model reduces to the textbook quadratic potential model. Therefore, the inflationary predictions of the model in the case (ii) interpolate the inflationary predictions of the textbook quadratic and quartic potential models by varying the inflation VEV from $v=0$ to $v \gg 1$.
We now consider the brane-world cosmological effects on the Higgs potential model. As we have observed in the previous subsection, the inflationary predictions of the textbook models are dramatically altered in the brane-world cosmology, and the quadratic potential model nicely fits the BICEP2 result, while the fitting by the quartic potential model is worse than the fitting in the standard cosmology. Thus, in the following, we concentrate on the case (i) with $\phi_0 < v$. Even in the brane-world cosmology, the above discussion for (ii) is applicable, namely, the Higgs potential model reduces to the textbook models for the very small or large VEV limit and the inflationary predictions for (ii) interpolate those in the two limiting cases. Thus, once we have obtained the inflationary predictions for the case (i), we can imaginary interpolate them to the results for the quartic potential model presented in the previous subsection.
![ The inflationary predictions for the Higgs potential model: $n_s$ vs. $r$ (left panel) and $n_s$ vs. $\alpha$ (right panel) for various $M_5$ values with fixed $v=10$, $20$, $30$ and $200$ from left to right, along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2). Here we have fixed the number of e-foldings $N_0=60$. The dashed line denotes the inflationary predictions for various values of $v$ from $10$ to $200$ in the standard cosmology. As $v$ is raised, the predicted values of $n_s$ and $r$ approach those of the quadratic potential model along the dashed line (the position marked by the the black triangle). For $M_5 \gtrsim 1$, the brane-world cosmological effects are negligible, and the predicted values of $n_s$ and $r$ lie on the dashed line. As $M_5$ is lowered, the inflationary predictions are deviating from the values on the dashed line, and all solid lines are converging to the point marked by the black squares, which are the predictions of the quadratic potential model for $M_5 \ll 1$. []{data-label="fig:hp"}](HP1.eps "fig:"){width="45.00000%"} ![ The inflationary predictions for the Higgs potential model: $n_s$ vs. $r$ (left panel) and $n_s$ vs. $\alpha$ (right panel) for various $M_5$ values with fixed $v=10$, $20$, $30$ and $200$ from left to right, along with the contours (at the confidence levels of 68% and 95%) given by the BICEP2 collaboration (Planck+WP+highL+BICEP2). Here we have fixed the number of e-foldings $N_0=60$. The dashed line denotes the inflationary predictions for various values of $v$ from $10$ to $200$ in the standard cosmology. As $v$ is raised, the predicted values of $n_s$ and $r$ approach those of the quadratic potential model along the dashed line (the position marked by the the black triangle). For $M_5 \gtrsim 1$, the brane-world cosmological effects are negligible, and the predicted values of $n_s$ and $r$ lie on the dashed line. As $M_5$ is lowered, the inflationary predictions are deviating from the values on the dashed line, and all solid lines are converging to the point marked by the black squares, which are the predictions of the quadratic potential model for $M_5 \ll 1$. []{data-label="fig:hp"}](HP2.eps "fig:"){width="45.00000%"}
![ Relations between $n_s$ and $M_5$ (left panel) and between $n_s$ and $\lambda$ (right panel), corresponding to Fig. \[fig:hp\]. []{data-label="fig:hp_mass"}](HPM5.eps "fig:"){width="45.00000%"} ![ Relations between $n_s$ and $M_5$ (left panel) and between $n_s$ and $\lambda$ (right panel), corresponding to Fig. \[fig:hp\]. []{data-label="fig:hp_mass"}](HPlam.eps "fig:"){width="45.00000%"}
We calculate the inflationary predictions for various values of $v$ and $M_5$, and the results are shown in Fig. \[fig:hp\] for $N_0=60$. The dashed line denotes the results in the standard cosmology for various values of $v$. The black triangles represent the inflationary predictions in the quadratic potential model, and we have confirmed that the inflationary predictions are approaching the triangles along the dashed line, as $v$ is raised. The solid lines show the results for various values of $M_5$ with $v=10$, $20$, $30$ and $200$ from left to right. For $M_5 \gtrsim 1$, the brane-world effect is negligible and the inflationary predictions lie on the dashed line. As $M_5$ is lowered, the inflationary predictions are deviating from those in the standard cosmology and they approach the values obtained by the quadratic potential model in the brane-world cosmology (shown as the black squares). We can see that the brane-world cosmological effect enhances the tensor-to-scalar ratio. Fig. \[fig:hp\_mass\] shows corresponding results in ($n_s, M_5$)-plane (left panel) and ($n_s, \lambda$)-plane (right panel). In the right panel, the dashed line denotes the results in the standard cosmology. Numerical values for selected M5 values are listed in Table \[table:hp\] for $N_0 = 60$ and $v=20$.
$M_5 ({\rm GeV})$ $ \lambda $ $V(\phi_0)^{1/4}$ (GeV) $\phi_0$ $\phi_e$ $n_s$ $r$ $-\alpha\ (10^{-4})$
------------------------ -------------------------- ------------------------- ---------- ---------- --------- ---------- ----------------------
$ \infty $ $ 1.11 \times 10^{-13} $ $1.56 \times 10^{16}$ $7.05$ $18.6$ $0.964$ $0.0519$ $ 4.54 $
$6.99 \times 10^{16}$ $8.47 \times 10^{-14}$ $1.44 \times 10^{16}$ $7.59$ $18.6$ $0.964$ $0.0806$ $5.17$
$5.08 \times 10^{16} $ $4.97 \times 10^{-14} $ $1.23 \times 10^{16}$ $8.68$ $18.6$ $0.963$ $0.111$ $5.95$
$3.70 \times 10^{16}$ $2.33 \times 10^{-14}$ $9.70\times 10^{15}$ $10.3$ $18.6$ $0.962$ $0.138$ $6.46$
$2.01\times 10^{16} $ $4.78\times 10^{-15}$ $5.60 \times 10^{15}$ $13.5$ $18.7$ $0.961$ $0.169$ $6.70$
$1.85\times 10^{14}$ $2.29\times 10^{-19}$ $5.30 \times 10^{13} $ $19.9$ $20.0$ $0.959$ $0.198$ $6.83$
: The values of parameters for the Higgs potential for $N_0=60$ and $v=20$, in the Planck unit ($M_P=1$) unless otherwise stated. []{data-label="table:hp"}
It is worth mentioning that in the brane-world cosmology, we can realize an inflationary scenario even for $v \ll 1$. This is a sharp contrast with the standard cosmological case. In the standard cosmology, the slow-roll parameters are given by $$\begin{aligned}
\epsilon(\phi_0) = \frac{1}{2} \left(\frac{V'(\phi_0)}{V(\phi_0)} \right)^2=8 \frac{\phi_0^2}{(\phi_0^2-v^2)^2}, \; \;
\eta(\phi_0) = \frac{V''(\phi_0)}{V(\phi_0)} =4 \frac{3 \phi_0^2-v^2}{(\phi_0^2-v^2)^2}. \end{aligned}$$ Hence, for $\phi_0 < v$, $v \gg 1$ is necessary to satisfy the slow-roll conditions, $\epsilon(\phi_0) \ll 1$ and $|\eta(\phi_0)| \ll 1$, simultaneously. On the other hand, in the limit of $V/\rho_0 \gg 1$ in the brane-world cosmology, we have $$\begin{aligned}
\epsilon(\phi_0) = 12 \frac{M_5^6}{V(\phi_0)} \left(\frac{V'(\phi_0)}{V(\phi_0)} \right)^2, \; \;
\eta(\phi_0) = 12 \frac{M_5^6}{V(\phi_0)} \left(\frac{V''(\phi_0)}{V(\phi_0)} \right). \end{aligned}$$ The slow-roll conditions can be satisfied even for $v \ll 1$ if $M_5 \ll v$.
When the Higgs potential model reduces to the quadratic potential model and $V/\rho_0 \ll 1$, we find from Eq. (\[m/M5\]) $$\begin{aligned}
\lambda \simeq 10^{-8} \left( \frac{M_5}{v}\right)^2. \end{aligned}$$ The condition $\chi_0/v \ll 1$ leads to $M_5 \ll 10^{-3} v$ (see Eqs. (\[phi2\_int\]) and (\[m/M5\])). For $M_5 \sim 10^6$ GeV, the inflaton has mass of order of the electroweak scale. In this case, we may fix the other model parameters as $v =10^{10}$ GeV and $\lambda =10^{-16}$ so as to satisfy the above theoretical consistency conditions. Because of the brane-world cosmological effect, this inflationary scenario can be consistent with the BICEP2 and Planck results with mass parameters far below the 4-dimensional Planck mass. In the next section, we will discuss the reheating process after inflation and implications of such a light inflaton to phenomenology at the electroweak scale.
Coleman-Weinberg potential
--------------------------
Finally, we discuss an inflationary scenario based on a potential with a radiative symmetry breaking [@Shafi_Vilenkin] via the Coleman-Weinberg mechanism [@Coleman_Weinberg]. We express the Coleman-Weinberg potential of the form, $$\begin{aligned}
V= \lambda \phi^4 \left[ \ln \left( \frac{\phi}{v}\right)-\frac{1}{4}\right]+\frac{\lambda v^4}{4}, \end{aligned}$$ where $\lambda$ is a coupling constant, and $ v$ is the inflaton VEV. This potential has a minimum at $\phi=v$ with a vanishing cosmological constant. The inflationary predictions of the Coleman-Weinberg potential model in the brane-world cosmology have recently been analyzed in [@CW_BC]. However, the modification of the power spectrum of tensor perturbation in Eq. (\[PT\]), namely, the function $F$ was not taken into account in the analysis. In the following we will correct the results in [@CW_BC] by taking the function $F$ into account. We will see an enhancement of the tensor-to-scalar ratio due to the function $F$.
Analysis for the Coleman-Weinberg potential model is analogous to the one of the Higgs potential model presented in the previous subsection. As same in the Higgs potential model, we can consider two cases, (i) $\phi_0 < v$ and (ii) $\phi_0 >v$, for the initial VEV of the inflaton. In the same reason as in our discussion about the Higgs potential model, we only consider the case (i) also for the Coleman-Weinberg potential model. Since the inflationary predictions in the case (ii) interpolate the predictions of the quartic potential model to those of the quadratic potential model as $v$ is raised (with a fixed $M_5$), one can easily imagine the results in the case (ii).
We show in Fig. \[fig:CW\] the inflationary predictions of the Colman-Weinberg potential model for various values of $v$ and $M_5$ with $N_0=60$. The dashed line denotes the results in the standard cosmology for various values of $v$. The black triangles denote the results in the quadratic potential model, and we have confirmed that the dashed line approaches the triangles as $v$ is increasing. The solid lines show the results for various values of $M_5$ with $v=10$, $20$, $30$ and $200$ from left to right. For $M_5 \gtrsim 1$, the brane-world effect is negligible and the inflationary predictions lie on the dashed line. As $M_5$ is lowered, the inflationary predictions are deviating from those in the standard cosmology to approach the values obtained by the quadratic potential model in the brane-world cosmology (shown as the black squares). Like in the Higgs potential model, for $M_5 \ll v$ the inflationary predictions of the Coleman-Weinberg potential model approach those of the quadratic potential model in the brane-world cosmology. Fig. \[fig:CW\_mass\] shows corresponding results in ($n_s, M_5$)-plane (left panel) and ($n_s, \lambda$)-plane (right panel). In the right panel, the dashed line denotes the results in the standard cosmology. For $N_0 = 60$ and $v=20$, numerical values for selected $M_5$ values are listed in Table \[table:CW\].
![ Same as Fig. \[fig:hp\] but for the Coleman-Weinberg potential model. []{data-label="fig:CW"}](CW1.eps "fig:"){width="45.00000%"} ![ Same as Fig. \[fig:hp\] but for the Coleman-Weinberg potential model. []{data-label="fig:CW"}](CW2.eps "fig:"){width="45.00000%"}
![ Same as Fig. \[fig:hp\_mass\] but for the Coleman-Weinberg potential model. []{data-label="fig:CW_mass"}](CWM5.eps "fig:"){width="45.00000%"} ![ Same as Fig. \[fig:hp\_mass\] but for the Coleman-Weinberg potential model. []{data-label="fig:CW_mass"}](CWlam.eps "fig:"){width="45.00000%"}
$M_5 ({\rm GeV})$ $ \lambda $ $V(\phi_0)^{1/4}$ (GeV) $\phi_0$ $\phi_e$ $n_s$ $r$ $-\alpha\ (10^{-4})$
------------------------ -------------------------- ------------------------- ---------- ---------- --------- ---------- ----------------------
$ \infty $ $ 2.82 \times 10^{-14} $ $1.36 \times 10^{16}$ $8.46$ $18.7$ $0.963$ $0.0295$ $ 4.90 $
$5.38 \times 10^{16}$ $1.76 \times 10^{-14}$ $1.19 \times 10^{16}$ $9.37$ $18.7$ $0.964$ $0.0617$ $5.44$
$3.76 \times 10^{16} $ $8.19\times 10^{-15} $ $9.48 \times 10^{15}$ $10.8$ $18.7$ $0.964$ $0.0956$ $5.99$
$2.49 \times 10^{16}$ $2.74 \times 10^{-15}$ $6.72\times 10^{15}$ $12.8$ $18.7$ $0.963$ $0.131$ $6.34$
$1.26\times 10^{16} $ $4.51\times 10^{-16}$ $3.53 \times 10^{15}$ $15.8$ $18.9$ $0.962$ $0.166$ $6.55$
$1.31\times 10^{14}$ $2.87\times 10^{-20}$ $3.75 \times 10^{13} $ $19.9$ $20.0$ $0.959$ $0.198$ $6.83$
: The values of parameters for the Coleman-Weinberg potential for $N_0=60$ and $v=20$, in the Planck unit ($M_P=1$) unless otherwise stated. []{data-label="table:CW"}
Post inflationary scenario, a light inflaton and phenomenology at the electroweak scale
=======================================================================================
For completion of our scenario, we discuss, in this section, post inflationary scenario, namely, reheating process after inflation in the brane-world cosmology. In our discussion about reheating, we require the renormalizability of inflationary models, and thus the inflaton only couples with the Higgs doublet among the Standard Model fields. In this section, we will consider two example models, the Higgs potential model and the quadratic potential model.
For the Higgs potential model, we introduce the following scalar potential involving the Higgs doublet and inflaton, $$\begin{aligned}
V =\frac{1}{8} \lambda (\phi^\dagger \phi -v^2)^2 + \frac{1}{2} \lambda_h \left( \Phi^\dagger \Phi - \frac{v_{\rm EW}^2}{2}\right)^2
+ \frac{1}{2} \lambda_{\phi h} (\phi^\dagger \phi -v^2) \left( \Phi^\dagger \Phi- \frac{v^2}{2}\right), \end{aligned}$$ where $\phi$ is the inflaton field, $\Phi$ is the Standard Model Higgs doublet, $v_{\rm EW}=246$ GeV is the Higgs VEV in the Standard Model, and $\lambda$s are quartic coupling constants being real and positive.[^2] This potential has a vacuum at $\langle \phi \rangle =v$ and $\langle \Phi_0 \rangle =v_{\rm EW}/\sqrt{2}$, where $\Phi_0$ is an electric charge neutral component in the Higgs doublet. We rewrite the potential around the vacuum with $\phi=\chi +v$ and $\Phi_0=(h+v_{\rm EW})/\sqrt{2}$, $$\begin{aligned}
V&=&\frac{1}{2} m_\phi^2 \chi^2 +\frac{1}{2} m_h^2 h^2 + m_{\phi h}^2 \chi h
+ \frac{m_\phi^2}{2 v} \chi^3 + \frac{m_{\phi h}^2}{2v} \chi^2 h +\frac{m_{\phi h}^2}{2v_{\rm EW}} \chi h^2+\frac{m_h^2}{2 v_{\rm EW}}h^3
\nonumber\\
&+& \frac{1}{8} \left( \frac{m_\phi}{v} \right)^2 \chi^4 + \frac{1}{8} \left(\frac{m_h}{v_{\rm EW}}\right)^2 h^4
+ \frac{1}{4} \left( \frac{m_{\phi h}^2}{v v_{\rm EW}} \right) \chi^2 h^2,
\label{potential2}\end{aligned}$$ where we have introduced three mass parameters given by $m_\phi=\sqrt{\lambda}v$, $m_h=\sqrt{\lambda_h} v_{\rm EW}$ and $m_{\phi h}=\sqrt{\lambda_{\phi h} v_{\rm EW} v}$.
The inflaton mixes with the Standard Model Higgs boson through the mass term $m_{\phi h}^2 \chi h$. Here we consider the case in which the mixing is so small that $h$ is almost identical to the Standard Model Higgs boson. In this case, the mass matrix of $\phi$ and $h$ is diagonalized by the mass eigenstates $\Phi_1$ and $\Phi_2$ defined as $$\begin{aligned}
h \simeq \Phi_1 + \delta \; \Phi_2, \; \; \chi \simeq - \delta\; \Phi_1 + \Phi_2,
\label{mass_eigenstate}\end{aligned}$$ with the mass eigenvalues, $m_1 \simeq m_h$ and $m_2 \simeq m_\phi$, and a small mixing parameter $|\delta| \ll 1$ defined as $$\begin{aligned}
\delta \simeq \frac{m_{\phi h}^2}{m_\phi^2-m_h^2}. \end{aligned}$$
Now we consider reheating process through the inflaton decay. When the inflaton is much heavier than the Higgs boson ($m_1 \ll m_2$), the inflaton decays to a pair of the Higgs doublets through the interaction $$\begin{aligned}
{\cal L}_{\rm int} \supset -\frac{1}{2} \lambda_{\phi h} v \phi (\Phi^\dagger \Phi) \simeq
-\frac{1}{2} \delta \frac{m_\phi^2}{v_{\rm EW}} \phi (\Phi^\dagger \Phi) .\end{aligned}$$ For the process $\phi \to \Phi^\dagger \Phi$, we have the decay width $$\begin{aligned}
\Gamma(\phi \to \Phi^\dagger \Phi) = \frac{1}{8 \pi} \left( \delta \frac{m_\phi}{v_{\rm EW}}\right)^2 m_\phi. \end{aligned}$$ In evaluating reheating temperature, we compare this decay width with the Hubble parameter $$\begin{aligned}
H=\sqrt{\frac{1}{3} \rho \left( 1+ \frac{\rho}{\rho_0} \right)} \simeq \frac{\rho}{6 M_5^3}, \end{aligned}$$ where $\rho=(\pi^2/30) g_* T_{\rm RH}^4$ with a total number of degrees of freedom ($g_*\simeq 100$). For example, if we take $m_\phi=1$ TeV and $\delta =0.01$, we find $T_{\rm RH} \simeq 4.9 \times 10^{4}$ GeV, by using Eq. (\[m/M5\]).
Since in the brane-world cosmology the inflaton mass can be as low as the electroweak scale, let us consider a radical case with $m_\phi < m_h/2$, so that the Standard Model-like Higgs boson can decay to a pair of inflatons. Rewriting the scalar potential in Eq. (\[potential2\]) by the mass eigenstate of Eq. (\[mass\_eigenstate\]), we find an interaction term, $$\begin{aligned}
{\cal L}_{\rm int} \supset -\frac{1}{2}\left( \frac{2 m_\phi^2+m_h^2}{v_{\rm EW}} \right)\delta^2 \Phi_1 \Phi_2^2, \end{aligned}$$ where $\Phi_1$ is the Standard Model-like Higgs boson, and $\Phi_2$ is the inflaton. Thus we have a partial decay width of the Higgs boson to a pair of inflatons as $$\begin{aligned}
\Gamma(\Phi_1 \to \Phi_2 \Phi_2) \simeq \frac{\delta^4}{32 \pi m_h} \left( \frac{2 m_\phi^2+m_h^2}{v_{\rm EW}} \right)^2
\sqrt{1-\frac{4 m_\phi^2}{m_h^2}}. \end{aligned}$$ When we take as reference values $m_\phi=50$ GeV and $\delta =0.105$ for a Higgs boson mass $m_h=125$ GeV, we find $\Gamma(\Phi_1 \to \Phi_2 \Phi_2) \simeq 0.01 \Gamma_h$, where $\Gamma_h = 4.07$ MeV is the total Higgs boson decay width [@Gamma_h]. Once a Higgs boson is produced in collider experiments, it decays to a pair of inflatons with 1% branching fraction. Then, each produced inflaton mainly decays to $b \bar{b}$ through the mixing with the Standard Model Higgs boson (see the next paragraph for discussion about this decay process). Note that this process, $\Phi_1 \to \Phi_2 \Phi_2 \to 2 b 2{\bar b}$, is similar to the Higgs boson pair production, followed by the decay of each Higgs boson to $b \bar{b}$. Precision measurement of the Higgs boson properties is one of major tasks of future collider experiments such as the ILC. Here we refer ILC studies [@ILC_Higgs] on the measurement of the Higgs boson self-coupling through a Higgs pair production process associated with $Z$ boson, $e^+e^- \to Z^* \to Z h^* \to Z h h $, followed by $h \to b {\bar b}$. The production cross section of this process is $\sim 0.1$ fb for a collider energy $0.5-1$ TeV. On the other hand, in our case, a pair of inflatons can be produced via $e^+e^- \to Z^* \to Z \Phi_1$, followed by the Standard Model-like Higgs boson decay to a pair of inflatons, $\Phi_1 \to \Phi_2 \Phi_2$. Since the Higgs boson production associated with $Z$ boson has a cross section$\sim 10$ fb, the inflaton pair production cross section from the 1% branching fraction of the Higgs boson decay is comparable to the Higgs pair production cross section with the same final states. Therefore, precision measurements of the Higgs self-coupling at the ILC can reveal the inflaton in the brane-world cosmology, although it is difficult to distinguish the inflaton from a light singlet scalar added to the Standard Model.
Now we go back to discussion about reheating. When the inflaton is lighter than the Higgs boson, its main decay mode is to $b {\bar b}$ through the mixing with the Higgs boson. The decay width is calculated to be $$\begin{aligned}
\Gamma (\Phi_2 \to b {\bar b} )\simeq \frac{3}{8 \pi} \delta^2 \left( \frac{m_b}{v_{\rm EW}}\right)^2 m_\phi
\left(1-\frac{4 m_b^2}{m_\phi^2} \right)^{3/2}, \end{aligned}$$ where $m_b=4.2$ GeV is the bottom quark mass. Using the condition, $\Gamma (\Phi_2 \to b {\bar b} )=H(T_{\rm RH})$, in the limit $\rho/\rho_0 \gg 1$, we find $T_{\rm RH} \simeq 676$ GeV for $m_\phi=50$ GeV, $\delta=0.105$ and $M_5 =4.0 \times 10^5$ GeV (see Eq. (\[m/M5\])). In this case, the “transition temperature” ($T_t$), at which the evolution of universe transits from the brane-world cosmology to the standard cosmology, is calculated as $T_t \simeq 0.1$ GeV by $\rho(T_t)/\rho_0=1$. The transition temperature is so low that the brane-world cosmological effects dramatically alter the results obtained in the standard particle cosmology [@DM_BC; @LG_BC; @gravitino_BC].
The next example is the quadratic potential model. When we introduce general renormalizable terms between the inflaton and Higgs doublet such as $\phi \Phi^\dagger \Phi$, our discussion is analogous to that for the first example. Here, let us examine a novel scenario, namely, a unified picture of inflaton and the dark matter particle. For $M_5 \sim 10^6$ GeV, the inflaton has mass at the electroweak scale, providing the inflationary predictions being consistent with the current observations. When we introduce a $Z_2$ parity and assign odd-parity for the inflaton while even-parity for all of the Standard Model particles, the inflaton with the coupling to the Higgs doublet plays the role of the so-called Higgs portal dark matter [@Higgs_portal_DM]. A unified picture of the inflaton and the Higgs portal dark matter has been realized in an inflationary model with non-minimal gravitational coupling [@WIMP_inflaton1; @WIMP_inflaton2]. Now we consider the unified picture in the brane-world scenario.
Because of the $Z_2$ invariance and the requirement of the renormalizability, a unique interaction term of the inflaton with the Higgs doublet is $$\begin{aligned}
{\cal L}_{\rm int} = g^2 \phi^2 |\Phi|^2 ,
\label{I-H int}\end{aligned}$$ where $g$ is a coupling constant.[^3] Through this interaction term, a pair of the inflatons annihilate into the Standard Model particles via the Higgs boson exchange in the $s-$channel and, if kinematically allowed, into a pair of the Higgs bosons. For the inflaton/dark matter mass $\sim$TeV (which means $M_5 \sim 10^7$ GeV) for example, $g^2 \sim 0.1$ reproduces the observed relic abundance of the dark matter particle (see, for example, [@KMNO] for analysis of the dark matter relic abundance and the direct dark matter detection experiments, as well as the production of the Higgs portal dark matter particle at the Large Hadron Collider).[^4]
After inflation, the inflaton oscillates around the potential minimum. During this oscillation phase, the inflaton energy density is transmitted to relativistic particles to thermalize the universe. Since the inflaton cannot decay due to the $Z_2$-parity, preheating [@preheating] through the interaction in Eq. (\[I-H int\]) plays the crucial role. We expect an explosive Higgs doublet production through the parametric resonance during the oscillation phase and the decay products of Higgs doublet get the universe thermalized [@WIMP_inflaton2]. Reheating temperature in this case is estimated by $$\begin{aligned}
\Gamma_h = H(T_{\rm RH}) \simeq \frac{\rho(T_{\rm RH})}{6 M_5^3}, \end{aligned}$$ where $\Gamma_h=4.07$ MeV is the total Higgs boson decay width. We find $T_{\rm RH} \simeq 2.9 \times 10^4$ GeV, which is high enough to get the inflaton in thermal equilibrium, and the standard discussion about thermal relic dark matter particle follows.[^5]
Conclusions
===========
Motivated by the recent observation of the CMS $B$-mode polarization by the BICEP2 collaboration, we have studied simple inflationary models based on the quadratic, quartic, Higgs and Coleman-Weinberg potentials in the context of the brane-world cosmology. For the 5-dimensional Planck mass $M_5 < M_P$, the brane-world cosmological effect alters inflationary predictions from those in the standard cosmology. We have found that all simple models except the quartic potential model can fit the results by the BICEP2 and Planck satellite experiments with an enhancement of the tensor-to-scalar ratio in the presence of the 5-dimensional bulk. Keeping the inflationary predictions to be consistent with the observations, the inflaton can become lighter as the 5-dimensional Planck mass is lowered. Requiring the renormalizability of the inflationary models, the inflaton only couples with the Higgs doublet among the Standard Model fields, and the coupling plays the crucial role in reheating process after inflation. When an inflaton has mass at the electroweak scale, this light inflaton has some impacts on phenomenology at the electroweak scale. We have discussed two scenarios. In the first scenario, the Higgs boson decay can have a 1% branching fraction to a pair of inflatons, which can be tested in the future collider experiments. The second scenario offers a unified picture of the inflaton and the dark matter particle, where the light inflaton plays the role of the Higgs portal dark matter.
Finally we comment on a tension between the BICEP2 result of $r \simeq 0.2$ and a constraint by the Planck measurement on $r < 0.11$. As discussed by the BICEP2 collaboration [@BICEP2], a way to reconcile the two results is to introduce the running spectral index $\alpha=d n_s/d \ln k \simeq -0.022$. The contours given by the BICEP2 collaboration (Planck+WP+highL+BICEP2) shown in Figures in this paper are obtained by setting $\alpha \simeq -0.022$. On the other hand, as we have shown in Figures, the simple inflationary models predict $-\alpha={\cal O}(10^{-3})$, which is too small to make the BICEP2 result compatible with the constraint quoted by the Planck measurement. Thus, rigorously speaking, inflationary predictions with $r \sim 0.2$ in the simple modes we have examined are not the best fit. We may regard the contours given by the BICEP2 collaboration as a reference, and the tensor-to-scalar ratio $r \sim 0.1$ may be more reasonable to discuss the fitting of the BICEP2 and Planck results. In this regard, the experimental results favor the Higgs potential and the Coleman-Weinberg potential models with a mild brane-world cosmological effect, $V/\rho_0 \lesssim1$ (see Figs. \[fig:hp\] & \[fig:CW\] and Tables \[table:hp\] & \[table:CW\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Andy Okada for his encouragement.
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[^1]: A more sever constraint is obtained from the precision measurements of the gravitational law in sub-millimeter range. Through the vanishing cosmological constant condition, we find $\rho_0^{1/4} > 1.3$ TeV, equivalently, $M_5 > 1.1 \times 10^8$ GeV as discussed in the original paper by Randall and Sundrum [@RS2]. However we note that this constraint is model-dependent in general and can be moderated when, for example, a scalar field is introduced in the 5-dimensional bulk (see, for example, [@maeda_wands]). Hence, in this paper we impose the model-independent BBN bound, $M_5 > 8.8$ TeV.
[^2]: Here we have restricted the scalar potential to simplify our discussion. If the inflaton field is a complex scalar in some gauge extension of the model, this scalar potential is the most general, renormalizable one with a vanishing cosmological constant. However, the inflaton is a real scalar in our simple setup, and one can add, in general, more terms such as $\phi (\Phi^\dagger \Phi)$ and $\phi^3$ to the scalar potential.
[^3]: This term induces an inflaton quartic coupling $\sim g^4/(4 \pi)^2$ at the quantum-level. We practically adjust an inflaton quartic coupling at the tree-level to cancel the induced coupling to make an effective quartic coupling negligibly small. Thus, the inflation is governed by the quadratic potential.
[^4]: Precisely speaking, this result in the standard cosmology should be altered in the brane-world cosmology. However, for $M_5 \sim 10^7$ GeV, the transition temperature is close to the decoupling temperature of the dark matter particle, and the brane-world cosmological effect is not significant [@DM_BC].
[^5]: Our discussion about preheating and subsequent thermalization of the universe here is naive and in fact, these processes are very complicated. For detailed analysis, see, for example, Ref. [@PreheatDetail].
|
---
abstract: 'Temporal cavity solitons (CS) are optical pulses that can persist in passive resonators, and they play a key role in the generation of coherent microresonator frequency combs. In resonators made of amorphous materials, such as fused silica, they can exhibit a spectral red-shift due to stimulated Raman scattering. Here we show that this Raman-induced self-frequency-shift imposes a fundamental limit on the duration and bandwidth of temporal CSs. Specifically, we theoretically predict that stimulated Raman scattering introduces a previously unidentified Hopf bifurcation that leads to destabilization of CSs at large pump-cavity detunings, limiting the range of detunings over which they can exist. We have confirmed our theoretical predictions by performing extensive experiments in several different synchronously-driven fiber ring resonators, obtaining results in excellent agreement with numerical simulations. Our results could have significant implications for the future design of Kerr frequency comb systems based on amorphous microresonators.'
author:
- Yadong Wang
- Miles Anderson
- Stéphane Coen
- 'Stuart G. Murdoch'
- Miro Erkintalo
title: Stimulated Raman Scattering Imposes Fundamental Limits to the Duration and Bandwidth of Temporal Cavity Solitons
---
Temporal cavity solitons (CSs) are pulses of light that can circulate indefinitely in passive driven nonlinear resonators. They were first observed and studied in macroscopic fiber ring cavities, and proposed as ideal candidates for bits in all-optical buffers [@Wab1; @leo1; @Jae1; @Jae2]. More recently, studies have demonstrated that temporal CSs can also manifest themselves in monolithic microresonators [@herr2; @Xue1; @Yi1; @webb5], where they play a central role in the generation of stable, low noise, wide bandwidth optical frequency combs [@Del1; @Kip; @Del2]. Because such microresonator frequency combs are very attractive for applications ranging from spectroscopy to telecommunications [@gohle1; @jones1; @hill1; @pfeifle1], there is a growing interest to better understand the dynamics and characteristics of temporal CSs in realistic systems.
=-1 Temporal CSs are able to persist without changes in their shape or energy thanks to a delicate double-balance [@Akhm]. The material Kerr nonlinearity compensates for the solitons’ dispersive spreading, while the energy they lose due to intrinsic resonator loss and coupling is replenished through nonlinear interactions with the continuous wave (cw) field driving the cavity. In addition to these fundamental interactions, the precise characteristics of temporal CSs (e.g. duration, temporal profile, center wavelength) can be further influenced by a variety of “higher-order” effects, such as perturbations induced by higher-order dispersion [@Jang; @Wang] or avoided mode crossings [@Herr1]. In resonators made of amorphous materials, such as silica glass, an effect of particular significance is stimulated Raman scattering (SRS), which causes temporal CSs to be spectrally red-shifted relative to the driving wavelength [@Milian; @kob; @webb5; @Yang]. Signatures of such CS self-frequency-shift were first observed experimentally in the context of frequency comb generation in silicon nitride microresonators [@kob], and subsequently in experiments using silica wedge resonators [@Yi1] and silica microspheres [@webb5]. Indirect time domain signatures have also been reported in macroscopic fiber ring resonators [@Ande].
In addition to shifting the CS center frequency, SRS can also impact on the range of CS existence. Indeed, in a pioneering theoretical work [@Milian], Milián et al. observed that, in the presence of SRS, temporal CSs may not exist over the entire range of parameters where they are expected to do so in the absence of SRS. Yet, the precise fashion in which SRS affects the existence and stability of CSs – particularly in the context of experimentally realistic systems – has not to date been investigated. Notably, to the best of our knowledge, no experimental studies have been reported that would demonstrate the impact of SRS on CS existence.
=-1 In this work, we theoretically and experimentally demonstrate that SRS can significantly restrict the range of CS existence. In particular, we show that, due to SRS, CSs undergo a previously unidentified Hopf bifurcation at large cavity detunings, leading to instability dynamics that limit the range of parameters over which CSs can exist. We show that this new instability sets a fundamental limit for the minimum duration that a CS can possess for given resonator parameters, thereby setting an upper limit for the frequency comb bandwidth that can be achieved in microresonators made of amorphous materials. We have confirmed our theoretical predictions by performing extensive experimental investigations in several different fiber ring resonators; our experimental findings are in very good agreement with numerical simulations.
We begin by presenting a general theoretical analysis of the impact of SRS on the existence and stability of temporal CSs. To this end, we analyze the generalized mean-field Lugiato-Lefever equation (LLE) [@lugiato9; @Coen1; @Chembo] that includes the delayed Raman nonlinearity [@Milian]. This model is well known to allow for the examination of temporal CS dynamics [@Coen1; @kob; @webb5]. In dimensional form, the equation reads [@webb5]: $$\small
\begin{split}
t_R\frac{\partial E(t,\tau)}{\partial t}=&\left[ -\alpha-i\delta_\mathrm{0}-\frac{iL\beta_2}{2}\frac{\partial^2}{\partial \tau^2}\right]E+\sqrt{\theta}E_{\text{in}}\\&+i\gamma L\left[(1-f_R)|E|^2+f_Rh_R(\tau)*|E|^2\right]E.
\end{split}
\label{LEE1}$$ Here, $t$ is a *slow* time variable that describes the evolution of the intracavity field envelope $E(t,\tau)$ over consecutive round trips, while $\tau$ is a *fast* time variable defined in a co-moving reference frame that describes the envelope’s temporal profile over a single round trip. $t_\mathrm{R}$ is the cavity round trip time, $\alpha$ corresponds to half the total power loss per round trip, $\delta_\mathrm{0}$ is the phase detuning of the driving field $E_\mathrm{in}$ from the closest cavity resonance, $L$ is the cavity round trip length, $\beta_2$ and $\gamma$ are the usual group velocity dispersion and Kerr nonlinearity coefficients, respectively, and $\theta$ is the coupling power transmission coefficient. Finally, $h_\mathrm{R}(\tau)$ is a time-domain response function that characterizes the Raman nonlinearity of the resonator, with $f_\mathrm{R}$ the corresponding Raman fraction. For silica glass, $f_\mathrm{R} \approx 0.18$ and the form of the response function is well known [@Stolen; @Hollenbeck]. In the calculations that follow, we will assume a silica glass resonator for simplicity, yet emphasize that our general findings are likely to be applicable to arbitrary resonators where CSs exhibit self-frequency-shift.
![(a) Peak amplitude of the intracavity field, $|E|_\mathrm{max}$, as a function of cavity detuning $\Delta$ for $X=130$. Black curves represent cw solutions, while red and blue curves show CS solutions with and without SRS, respectively. The dashed curves correspond to unstable solutions. (b, c) Temporal (b) and spectral (c) CS profiles for $\Delta = 62$. As in (a), red and blue curves correspond to CS solutions with and without SRS, respectively. The corresponding solutions are marked with crosses in (a). The green and red shaded areas in (a) indicate regions of mono- and bistability of the cw solutions, respectively.[]{data-label="fig:Fig1"}](Fig1.eps){width="\linewidth"}
To gain general insights, we cast Eq. into normalized form via the following transformation of variables [@leo1; @step1]: $\alpha t/t_R\rightarrow t$, $\tau\sqrt{2\alpha/(|\beta_2|L)}\rightarrow\tau$ and $E\sqrt{\gamma L/\alpha}\rightarrow E$. The dimensionless equation reads: $$\small
\begin{split}
\frac{\partial E(t,\tau)}{\partial t}=&\left[ -1-i\Delta+i\frac{\partial^2}{\partial \tau^2}\right]E+S\\&+i\Big[(1-f_\mathrm{R})|E|^2+f_\mathrm{R}\left[\Gamma(\tau,\tau_\mathrm{s})*|E|^2\right]\Big]E,
\end{split}
\label{LEE2}$$ =-1 where we have assumed anomalous dispersion ($\beta_2<0$). The normalized detuning and driving variables are defined as $\Delta=\delta_\mathrm{0}/\alpha$, $S=E_\mathrm{in}\sqrt{\gamma L\theta/\alpha^3}$, respectively, while the normalized Raman response function $\Gamma(\tau,\tau_\mathrm{s})=\tau_\mathrm{s}h_\mathrm{R}(\tau\tau_\mathrm{s})$ where $\tau_\mathrm{s}=\sqrt{|\beta_2|L/(2\alpha)}$ is the fast time normalization time scale.
In the absence of SRS ($f_\mathrm{R} = 0$), the dynamics and solutions of Eq. depend on two parameters only: the cavity detuning $\Delta$ and driving strength $S$ [@step1]. However, the delayed nature of the Raman response leads to an additional dependence on the fast time normalization time scale $\tau_\mathrm{s}$. This can be readily understood by noting that the impact of SRS depends on the physical durations (and bandwidths) of the intracavity field features. Indeed, to first order, the convolution term can be approximated as: $$\Gamma(\tau,\tau_\mathrm{s})*|E|^2\approx|E|^2-\frac{T_\mathrm{R}}{f_\mathrm{R}\tau_\mathrm{s}}\frac{\partial|E|^2}{\partial \tau},
\label{aprox}$$ where $T_\mathrm{R}$ is the Raman time-scale, which is related to the slope of the Raman gain spectrum at zero frequency [@miro]. This approximation shows that, to first order, the strength of the Raman term scales as $\tau_\mathrm{s}^{-1}$.
=-1 To illustrate how SRS affects the stability and existence of CSs, we show in Fig. \[fig:Fig1\](a) the steady-state CS solutions of Eq. for a constant driving power $X =|S|^2= 130$ in the presence ($f_\mathrm{R} \approx 0.18$, red curve) and absence ($f_\mathrm{R} = 0$, blue curve) of SRS. Also shown are temporal \[Fig. \[fig:Fig1\](b)\] and spectral \[Fig. \[fig:Fig1\](c)\] profiles for a CS at a typical detuning $\Delta = 62$. The solutions were obtained by finding the time-localized steady-state solutions of Eq. using a continuation scheme based on the Newton-Raphson method [@Coen1]; a normalization time scale $\tau_\mathrm{s} = 1.9$ $\mathrm{ps}$ (corresponding to one of the experiments that will follow) was used for calculations when including SRS. \[Note that, in the presence of SRS, the CSs exhibit a time-domain temporal drift, which is accounted for by trivially adjusting the frame of reference of Eq. .\] To facilitate our discussion, we also show in Fig. \[fig:Fig1\](a) the steady-state cw solutions (black curves) of Eq. . The cw solutions exhibit a pronounced bistability, with the middle branch being unconditionally unstable (dashed black curve).
=-1 As can be seen, for small detunings SRS does not significantly perturb the CS solutions: in both cases (with and without SRS), the solutions exist for detunings above the up-switching point $\Delta_{\uparrow}$, exhibit well known unstable behaviours for small detunings (dashed blue and red curves) [@leo2; @Ande2; @Myu], and become stable through an inverse Hopf bifurcation ($\Delta_\mathrm{H1}$) as the detuning increases. However, as $\Delta$ increases further, we see a distinct deviation between the solutions obtained in the presence and absence of SRS. This can be understood by noting that the duration of a CS scales inversely with $\sqrt{\Delta}$ [@Coen1]: for large detunings the CSs are temporally narrower, and hence spectrally broader, resulting in stronger overlap with the Raman gain spectrum.
=-1 In the presence of SRS, CSs exhibit lower peak powers, longer durations, and their center frequency is down-shifted \[c.f. Fig. \[fig:Fig1\](b) and (c)\]. But in addition to perturbing their characteristics, it is evident from Fig. \[fig:Fig1\](a) that SRS also impacts on their range of stability and existence. In particular, we find that, in the presence of SRS, CSs can undergo a second Hopf bifurcation at large detunings \[denoted as $\Delta_{\mathrm{H2}}$ in Fig. \[fig:Fig1\](a)\], leading to unstable dynamics. Dynamical (split-step) simulations of Eq. reveal that these previously unidentified, large-$\Delta$ *unstable* CSs, exhibit behaviors qualitatively similar to those observed for unstable CSs below the first Hopf bifurcation point $\Delta_\mathrm{H1}$ [@leo2; @Ande2]: for detunings very slightly above $\Delta_\mathrm{H2}$, the CSs exhibit oscillatory behaviour, but as $\Delta$ increases further, they experience an abrupt collapse to the cw state \[see Supplementary Material\]. The results in Fig. \[fig:Fig1\](a) also show that, in the presence of SRS, the CS solutions cease to exist altogether at a detuning significantly smaller \[$\Delta_\mathrm{max}$ in Fig. \[fig:Fig1\](a)\] than the theoretical limit of $\pi^2X/8$ observed in the absence of SRS [@bara]. We emphasize, however, that $\Delta_\mathrm{max}$ only represents a theoretical upper limit of existence of CS solutions: in practice CSs cannot be sustained for detunings far above $\Delta_{\mathrm{H2}}$ because of the nature of the instability dynamics (see Supplementary Material). At this point we also note that, due to vast differences in the analyzed parameter regions, the instabilities uncovered in Fig. \[fig:Fig1\](a) do not appear straightforwardly related to those described by Milián et al. [@Milian].
![(a) Cavity detunings $\Delta_{\mathrm{H2}}$ and $\Delta_{\mathrm{max}}$, where the CS solution loses its stability and ceases to exist, respectively, as a function of driving power $X$ and for a variety of $\tau_\mathrm{s}$. The squares correspond to simulated points, while the solid curves are guide to eye. As can be seen, each $\tau_\mathrm{s}$ is associated with a maximum $\Delta_{\mathrm{H2}}$ beyond which CSs cannot remain stable regardless of the driving power. Black curves indicate limits of CS existence in the absence of SRS. (b) Maximum detuning $\Delta_{\mathrm{H2}}$ as a function of the normalization time scale $\tau_\mathrm{s}$. Dashed red curve shows a linear fit. At the bottom of (b), we highlight *four* different normalization time scales corresponding to real resonators. $\tau_\mathrm{S0}$ is from [@Yi1] while the others are realized in this work.[]{data-label="fig:Fig2"}](Fig2.eps){width="\linewidth"}
The precise detunings $\Delta_{\mathrm{H2}}$ and $\Delta_{\mathrm{max}}$ (where the CS respectively loses its stability and ceases to exist) depend on the normalization time scale $\tau_\mathrm{s}$ and the driving power $X$. To gain more insight, we have evaluated the CS branches as a function of $\Delta$ \[as in Fig. \[fig:Fig1\](a)\] for a wide range of $\tau_\mathrm{s}$ and $X$, and extracted $\Delta_{\mathrm{H2}}$ and $\Delta_{\mathrm{max}}$ for each set of parameters. Figure \[fig:Fig2\](a) summarizes our findings. Here we show $\Delta_{\mathrm{H2}}$ (blue curves) and $\Delta_{\mathrm{max}}$ (red curves) as a function of $X$ for five different values of $\tau_\mathrm{s}$. The parameter boundaries, between which CSs can exist in the absence of SRS [@leo2], are also displayed (black curves).
For small driving powers $X$ (and/or large normalization time scales $\tau_\mathrm{s}$), we observe no secondary Hopf bifurcation and the upper limit of CS existence follows closely the expected value of $\pi^2X/8$. However, for larger driving powers (and/or shorter normalization time scales $\tau_\mathrm{s}$), we see clearly that the CS solution loses its stability at large $\Delta$, and that the upper limit of their existence is significantly reduced from the Raman-free values. Surprisingly, while that upper limit $\Delta_\mathrm{max}$ continues to increase with driving power $X$, the upper limit of CS stability (determined by the second Hopf bifurcation point $\Delta_{\mathrm{H2}}$) can be seen to saturate to a constant value that depends exclusively on the normalization time scale $\tau_\mathrm{s}$.
=-1 The results in Fig. \[fig:Fig2\](a) suggest that, because of SRS, there exist a maximum detuning $\Delta_{\mathrm{H2}}$ above which CSs can no longer remain stable in a given resonator, regardless of the driving power. Furthermore, because of the CS instability dynamics, this detuning also approximates well the upper limit of practical CS existence. As alluded to in Fig. \[fig:Fig2\](a), $\Delta_{\mathrm{H2}}$ depends on the resonator characteristics solely through the normalization time scale $\tau_\mathrm{s}$, and in Fig. \[fig:Fig2\](b) we plot $\Delta_{\mathrm{H2}}$ as a function of $\tau_\mathrm{s}$ as extracted from our calculations. As can be seen, $\Delta_{\mathrm{H2}}$ increases linearly for large $\tau_\mathrm{s}$. For smaller $\tau_\mathrm{s}<0.5\ \text{ps}$, the dependence becomes nonlinear; we speculate this is because the soliton bandwidth becomes comparable with the Raman gain bandwidth.
The observation that SRS imposes a practical upper limit for CS detunings also implies a lower limit for their temporal durations. Indeed, it is well known that, in physical units, the duration of a temporal CS is approximately given by $\tau_0=\tau_\mathrm{s}/\sqrt{\Delta}$ [@Coen1; @Wab1; @herr2]. Thus, in the presence of SRS, the minimum duration that a (stable) CS can possess is $\tau_\mathrm{min}\approx \tau_\mathrm{s}/\sqrt{\Delta_\mathrm{H2}}$. For large $\tau_\mathrm{s} > 1~\mathrm{ps}$ we can further approximate \[c. f. Fig. \[fig:Fig2\](b)\]: $$\tau_\mathrm{min} = \frac{\tau_\mathrm{s}}{\sqrt{a\tau_\mathrm{s} + b}},
\label{est}$$ where $a = 27.3\ \mathrm{ps^{-1}}$ and $b = 20.4$ are extracted from the linear fit shown as the dashed red line in Fig. \[fig:Fig2\](b). *This simple linear approximation can be used to estimate the minimum CS duration achievable in silica resonators.* (For other amorphous resonators associated with different Raman responses, the coefficients $a$ and $b$ will likely be different.) To illustrate that this width is consistent with previous experimental findings, we note as an example that the typical duration of CSs observed in a silica wedge resonator is $\tau_0=250\ \mathrm{fs}$ [@Yi1]. For the parameters of the resonator, $\tau_\mathrm{S0} \approx 1\ \mathrm{ps}$, yielding a maximum detuning $\Delta_\mathrm{H2} = 47$ and a minimum achievable CS duration $139\ \mathrm{fs}$.
To confirm our theoretical analysis, we have performed experiments using three macroscopic fiber ring resonators associated with different normalization time scales $\tau_\mathrm{s}$. A general schematic of the experimental configurations is depicted in Fig. \[fig:Fig3\]. The cavities are made up of single-mode optical fiber (SMF) laid in a ring configuration and closed on themselves by a 95/5 coupler. Each cavity also incorporates a 99/1 tap coupler through which the intracavity dynamics can be monitored in real time.
Because the cavities contain no other elements, they display very high finesse $\mathcal{F}$. This allows us to reach very high values of normalized driving power $X \propto \mathcal{F}^3$, as required for the study of SRS-induced limits of CS existence \[see Fig. 2(a)\]. To study the effect of the normalization time scale $\tau_s$, our three different cavities have different round trip lengths of $L = 13\ \mathrm{m}, 25\ \mathrm{m}\ \text{and}\ 50\ \mathrm{m}$, corresponding to normalization time scales $\tau_{\text{S1}}=1.9$ $\mathrm{ps}$, $\tau_{\text{S2}}=2.6$ $\mathrm{ps}$ and $\tau_{\text{S3}}=3.7$ $\mathrm{ps}$, and finesses $\mathcal{F}_\text{1}=77$, $\mathcal{F}_\text{2}=77$ and $\mathcal{F}_\text{3}=69$, respectively.
![Experimental setup. FG, function generator; CW, cw laser; IM, intensity modulator; EDFA, Erbium-doped fiber amplifier; BPF, bandpass filter; PC, polarization controller; OSC, oscilloscope.[]{data-label="fig:Fig3"}](Fig3.eps){width="\linewidth"}
=-1 We coherently drive our cavities with flattop nanosecond pulses whose repetition rate is synchronized to the respective cavity round trip time [@Coen2; @Copie; @Ande]. These pump pulses have a duration of $1.2\ \mathrm{ns}$, and they are generated by modulating the output of a narrow linewidth cw laser with a $12$ $\mathrm{GHz}$ intensity modulator. Note that the duration of the pump pulses is sufficiently short to avoid detrimental effects induced by stimulated Brillouin scattering [@Agrawal]. Before the pulses are injected into the cavity, they are amplified using an Erbium-doped fiber amplifier (EDFA), and spectrally filtered to remove amplified spontaneous emission. Together with the high cavity finesse, this pulse pumping method allows us to achieve very high normalized driving powers up to $X\approx 200$.
To experimentally explore the limits of CS existence, we linearly tune the cw laser frequency so as to continuously scan the cavity detuning across individual resonances. By simultaneously measuring the cavity output (extracted by the $1\%$ tap coupler) with a fast $12.5\ \mathrm{GHz}$ photodetector, we are able to monitor the intracavity dynamics in real time. This allows us to observe the creation and annihilation of CSs as the detuning is scanned [@Luo], and from the acquired data, we can extract their limits of existence.
![(a) False colour plot showing the intracavity dynamics as the detuning is linearly scanned from 0 to $110$ (top $x$-axis) during about $7000$ round trips (bottom $x$-axis) with a constant driving power $X = 130$. The cavity is 13 m long and has a normalization time constant $\tau_\mathrm{S1} = 1.9\ \mathrm{ps}$. Dashed black and white vertical lines indicate the zero detuning $\Delta_\mathrm{0}$ and the limit detuning $\Delta_\mathrm{lim}$ at which the CSs cease to exist, respectively. (b) Limit detuning $\Delta_\mathrm{lim}$ at which CSs cease to exist as a function of driving power $X$ and for three different normalization time scales $\tau_\mathrm{s}$ as indicated. The circle markers show values extracted from scanning experiments similar to that in (a), while the dashed curves correspond to results from numerical simulations of the LLE. The solid black line shows the theoretical CS existence limit in the absence of SRS, i.e., $\Delta_\mathrm{max} = \pi^2 X/8$.[]{data-label="fig:Fig4"}](Fig4.eps){width="\linewidth"}
Figure \[fig:Fig4\](a) shows an example of the measured intracavity dynamics as the cavity detuning $\Delta$ is linearly scanned from 0 to 110 at a constant driving power $X=130$ ($\sim11.3\ \mathrm{W}$ peak power). As is well known, for low detunings the field corresponds to an extended modulation instability pattern which is visible in Fig. \[fig:Fig4\](a) as a solid bright band [@Marc]. Out of this chaotic signal, CSs emerge as the detuning increases above $\Delta\approx 30$. The CSs can be seen to exhibit curved trajectories as the detuning increases, which is a known effect of SRS [@Ande]. When the detuning reaches $\Delta_\mathrm{lim} \approx 101$, we can see that all the CSs disappear almost simultaneously, indicating the limit detuning beyond which the solitons can no longer be sustained. The observed value is significantly smaller than the theoretical limit $\pi^2X/8 \approx 160$ expected without SRS. Accordingly, the experiment in Fig. \[fig:Fig4\](a) already provides support to our hypothesis: SRS reduces the range of CS existence.
To more comprehensively test our theoretical predictions, we have repeated the above experiment for a wide range of driving powers $X$ and using all three of our resonators to sample different normalization time scales $\tau_\mathrm{s}$. For each experiment, we perform a detuning scan \[as in Fig. \[fig:Fig4\](a)\], and extract the limit detuning beyond which CSs no longer exist. Our experimental findings are summarized in Fig. \[fig:Fig4\](b). Here the circle markers correspond to experimental data acquired for the different cavities, while the dashed curves correspond to results extracted from realistic numerical simulations of the LLE (the simulations use experimental parameters with no free-running variables). There are several important conclusions to be drawn from Fig. \[fig:Fig4\](b). First, we observe that our numerical simulations are in excellent agreement with experimental findings. Second, in agreement with our theoretical predictions \[see Fig. \[fig:Fig2\](a)\] the limit detuning $\Delta_\mathrm{lim}$ initially increases with $X$, but eventually saturates to a constant value. This saturation occurs in all of our resonators, but results in different saturated limit detunings due to the different normalization time scales $\tau_\mathrm{s}$. Overall, these measurements confirm of our main hypotheses: SRS limits the range of CS existence, and gives rise to a maximum detuning beyond which CSs cannot exist in a given resonator.
The experimental results summarized in Fig. \[fig:Fig4\](b) are in excellent qualitative agreement with the theoretical findings presented in Fig. \[fig:Fig2\]. However, more careful analysis shows that the agreement is not quantitative, and that our experiments consistently show the upper limit of CSs existence to be greater than the theoretically predicted second Hopf bifurcation point $\Delta_\mathrm{H2}$. For example, the scan of the $13\ \mathrm{m}$ cavity ($\tau_{\mathrm{S1}}=1.9\ \mathrm{ps}$) shown in Fig. \[fig:Fig4\](a) demonstrates a limit detuning $\Delta_\mathrm{lim}=101$, which is higher than the theoretically predicted value $\Delta_\mathrm{H2} = 76$. This discrepancy arises predominantly because the detuning is continuously increased in our experiments. Indeed, the CSs persist briefly even after passing the second Hopf bifurcation point $\Delta_\mathrm{H2}$, and so a continuously increasing detuning naturally leads to overestimation of their existence range. We have confirmed this hypothesis by performing additional experiments where the detuning is initially scanned and then stopped at different points close to the CS existence limit. Results are summarised in supplementary material: they clearly demonstrate that the second Hopf bifurcation point $\Delta_\mathrm{H2}$ represents not only the upper limit of CS stability, but also the practical upper limit of CS existence.
=-1 To summarize, we have investigated the dynamics of temporal CSs in the presence of stimulated Raman scattering. We have theoretically shown that, due to SRS, temporal CSs can lose their stability through a previously unidentified Hopf bifurcation that occurs for large detunings. Furthermore, we have shown that this instability gives rise to a maximum detuning, which depends solely on the parameters of the resonator, above which CSs cannot exist. Because the duration of temporal CSs scales inversely with $\sqrt{\Delta}$, our theoretical analysis reveals that SRS imposes a fundamental limit on CS durations that can be achieved in resonators made of amorphous materials. We have confirmed our theoretical analyses by performing comprehensive experiments in three different fiber ring resonators. In addition to unveiling a new type of CS instability, our results could significantly impact on the design of frequency comb generators based on microresonators where SRS plays a role, such as silica microspheres [@webb5], wedge resonators [@Yi], fiber-based Fabry-Perot resonators [@Obrzud], or silicon nitride microresonators [@kob].
We acknowledge support from the Marsden Fund of the Royal Society of New Zealand. M. Erkintalo further acknowledges support from the Rutherford Discovery Fellowships of the Royal Society of New Zealand. Y. Wang acknowledges the scholarship from the Dodd-Walls Centre for Photonic and Quantum Technologies.
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**Supplementary Information – Stimulated Raman Scattering Imposes Fundamental Limits to the Duration and Bandwidth of Temporal Cavity Solitons\
**
[+1.1cm]{}[+1.1cm]{} This article contains supplementary information to the manuscript entitled “*Stimulated Raman Scattering Imposes Fundamental Limits to the Duration and Bandwidth of Temporal Cavity Solitons*”. Specifically, we report on numerical simulations of the mean-field Lugiato-Lefever equation that unveil the dynamical behaviours of unstable cavity solitons in the regime of large pump-cavity detuning. Furthermore, we present additional experimental results that showcase how the second Hopf bifurcation point $\Delta_\mathrm{H2}$ indeed represents a practical upper limit for CS existence.\
Oscillation Behaviors {#oscillation-behaviors .unnumbered}
=====================
We begin by reporting results from split-step simulations of the normalized Lugiato-Lefever equation (LLE, Eq. of our main manuscript) that illustrate the CS instability dynamics above the second Hopf bifurcation point $\Delta_\mathrm{H2}$. To this end, we first show, in Fig. \[rings\](a), the normalized CS peak amplitude $|E|_\mathrm{max}$ as a function of the cavity detuning $\Delta$ for a constant driving power $X = 130$ and a normalization time scale $\tau_{\mathrm{S1}}=1.9\ \text{ps}$, obtained using a Newton-Raphson continuation algorithm. (Note that the same data is shown in Fig. \[fig:Fig1\](a) of our main manuscript.) Here, to more clearly distinguish the different regimes, we plot the stable CS solutions as a *solid black* curve, whilst unstable solutions are highlighted with *dashed red* curves. As can be seen, the CSs become unstable at large detunings through a second Hopf bifurcation: the blue cross in Fig. \[rings\](a) indicates the detuning $\Delta_\mathrm{H2}$ at which the Hopf bifurcation (approximately) takes place.
To study how the CS instabilities manifest themselves in the vicinity of the new Hopf bifurcation point $\Delta_\mathrm{H2}$, we have performed split-step simulations of the LLE at several different detunings. In Figs. \[rings\](b)–(d), we show three illustrative examples of the different temporal dynamics observed in our simulations. The detunings $\Delta_\mathrm{b}$, $\Delta_\mathrm{c}$, and $\Delta_\mathrm{d}$ used in these simulations are highlighted in Fig. \[rings\](a), and their respective values are quoted in the figure caption. We note that the CS evolutions are plotted in the reference frame of the soliton, which deviates from the natural reference frame of the LLE due to the well known temporal drift arising from the interplay of Raman-induced red-shift and group velocity dispersion [@Milian2].
=-1 Results in Fig. \[rings\](b) are obtained with $\Delta_\mathrm{b} = 59.4$, and they show the typical evolution of a stable CS over $65$ photon lifetimes (corresponding to about 800 round trips in one of our experiments). Despite the presence of SRS, the temporal profile of the CS remains constant. In contrast, at a detuning very close to the second Hopf bifurcation point $\Delta_\mathrm{c}=73.5$, the CS starts to oscillate \[see Fig. \[rings\](c)\], exhibiting a zigzag-type trajectory in the time domain. These oscillations seem to persist indefinitely; however, as the detuning increases further away from $\Delta_\mathrm{H2}$, we find that, following a short period of transient oscillations, the CS instability manifests itself as an abrupt collapse to the cw state \[see Fig. \[rings\](d)\]. In general, we find that the time it takes for the collapse to develop shortens as the detuning further increases beyond $\Delta_\mathrm{H2}$.This collapse behavior represents the predominant CS instability dynamics above the second Hopf bifurcation point: persistent oscillations \[like those shown in Fig. \[rings\](c)\] only manifest themselves in the *immediate* vicinity of $\Delta_\mathrm{H2}$, while collapses occur at all other higher detunings. For example, using parameters typical to our experiments (normalization time scale $\tau_{\mathrm{S1}}=1.9\ \text{ps}$ and driving power $X=130$), we find that the CSs start to collapse already when the detuning increases above $\Delta = 74$ (the Hopf bifurcation point $\Delta_\mathrm{H2} \approx 73.5$). This observation clearly indicates that the second Hopf bifurcation point $\Delta_\mathrm{H2}$ can be considered not only the upper of limit of CS stability, but also the practical upper limit of CS existence.
Scan-and-stop Experiment {#scan-and-stop-experiment .unnumbered}
========================
=-1 As highlighted in our main manuscript, the experimentally measured limit detunings $\Delta_\mathrm{lim}$ at which CSs cease to exist are systematically larger than the predicted values of the second Hopf bifurcation point $\Delta_\mathrm{H2}$. We argued that this discrepancy arises from the fact that the detuning is continuously scanned in our experiments. Here we provide supportive evidence for our explanation by reporting on additional experiments where we stop the detuning scan in the vicinity of the CS collapse point. By stopping the scan at various cavity detunings while assessing whether CSs can continue to persist or not, we are able to refine the experimentally deduced limit detuning.
=-1 Out experimental scan-and-stop method is similar to that in ref. [@Ande3]. An electrical ramp signal linearly sweeps the wavelength of the cw laser up to a set value, after which point the wavelength is then held constant. By monitoring the intracavity field after the detuning sweep has been stopped, we can readily gauge whether the CSs persist or not; repeating the experiment at several different stop points allows us to refine the limit detuning above which CSs no longer exist.
Figure \[scanstop\] shows a typical example of experimentally measured intracavity dynamics during a scan-and-stop sequence. The cavity used in this experiment is $13$ m long corresponding to a normalization time scale $\tau_{\mathrm{S1}}=1.9$ $\mathrm{ps}$, and the normalized driving power is set to $X = 130$. Using a detuning scan rate similar to that in Fig. \[fig:Fig4\](a) of our main manuscript, we sweep the detuning from $\Delta_0\approx 0$ to $\Delta_\mathrm{stop} = 90$, and then stop the scan and let the cavity evolve freely. As seen in Fig. \[scanstop\], all of the CSs decay within about $1000$ round trips after the detuning scan is stopped. This measurement thus reveals that, for these experimental parameters, the true limit detuning beyond which CSs cannot exist is smaller than $\Delta_\mathrm{stop} = 90$. We note that this value is already smaller (and closer to $\Delta_\mathrm{H2} = 76$) than the value obtained with continuous scanning, namely $\Delta_\mathrm{lim} = 101$. Note that, because of the limited resolution of our photodiode ($\sim60$ ps impulse response), we are unable to fully resolve the CS instability dynamics. Nevertheless, as shown in the inset of Fig. \[scanstop\], some preliminary traces of oscillatory behaviours can be observed before the collapse.
To systematically refine the limit value of CS existence, we repeat the scan-and-stop measurement above at several different stop detunings. Furthermore, we have performed these measurements for three different pump powers. The results are summarized in Fig. \[Noname\]. Here, the solid black curves show CS existence boundaries in the absence of SRS. The dashed blue curve and circle markers show numerical and experimental data, respectively, of the limit detuning at which the CSs collapse with continuous scanning (same data is shown in Fig. \[fig:Fig4\](a) of our manuscript), and the dashed red curve shows $\Delta_{\mathrm{H2}}$ extracted from Newton-Raphson calculations (same data is shown in Fig. \[fig:Fig2\](a) of our manuscript). New experimental data from our scan-and-stop measurements are shown as black diamonds and red crosses: the former correspond to detuning stop points where the CSs persist stably, while the latter correspond to stop points where the CSs decay (as in Fig. \[scanstop\]). This data supports our hypothesis regarding the influence of continuous scanning of the cavity detuning. Indeed, we see that experiments (and simulations) with continuously increasing detuning overestimate the range of CS existence: if the detuning scan is stopped just below the limit detuning found with continuous scanning, the CSs are found to be unstable and eventually collapse. Our experiments further show that the CSs cease to collapse only when the detuning is stopped very slightly above the theoretically predicted second Hopf bifurcation point $\Delta_{\mathrm{H2}}$; below that point the CSs are always found to be stable.
=-1 Overall our experimental data strongly confirms our main hypothesis: the second Hopf bifurcation point $\Delta_{\mathrm{H2}}$ – induced by stimulated Raman scattering – represents a practical upper limit of CS existence.
[10]{} C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, “Solitons and frequency combs in silica microring resonators: Interplay of the Raman and higher-order dispersion effects,” Phys. Rev. A **92**, 033851 (2015).
M. Anderson, Y. Wang, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Super cavity solitons and the coexistence of multiple nonlinear states in a tristable passive Kerr resonator,” Phys. Rev. X **7**, 031031 (2017).
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—————————————————————————————- \#1[\#1]{} §[[**S**]{}]{} ¶[[****]{}]{} [**H**]{}\_[qm]{}
NEIP-95-003
March 1995\
\
Ph. Jacquod [^1] and J.-P. Amiet[^2]\
Institut de Physique\
Université de Neuchâtel\
1, Rue A.L. Breguet\
CH - 2000 Neuchâtel\
[**Abstract** ]{}
We study the statistical properties of the spectrum of a quantum dynamical system whose classical counterpart has a mixed phase space structure consisting of two regular regions separated by a chaotical one. We make use of a simple symmetry of the system to separate the eigenstates of the time-evolution operator into two classes in agreement with the Percival classification scheme [@Per]. We then use a method firstly developed by Bohigas et. al. [@BoUlTo] to evaluate the fractional measure of states belonging to the regular class, and finally present the level spacings statistics for each class. The level spacings distribution of states belonging to the irregular part of the spectra as well as that of the complete set of levels corroborate the Berry-Robnik surmise [@BeRo]. We further present a statistical study of the regular levels. The presence of intermediate states - states which belong to neither class as long as $\hbar$ is finite, phase spatially mixed among the set of regular ones, together with the small fractional measure of regular states strongly affects the corresponding level spacings statistics, resulting in a non negligible deviation from the expected Poisson distribution. We see however the remarkable agreement of the irregular level spacings statistics as a direct confirmation of the Berry-Robnik surmised.
Introduction
============
For more than a decade, the study of quantum mechanical systems whose classical counterpart exhibits chaos has attracted much interest. One motivation for this study is the paradoxical fact that while the correspondence principle, as we understand it, should imply a quantum manifestation of classical chaos, the Schrödinger equation is linear. As a consequence, the time-evolution operator is unitary, and this suppresses any exponential divergence in the time evolution of quantum states. As a spectacular manifestation of this fact, time-reversal invariant models show no lost of memory : reversing the time at a certain moment T brings us back to the initial situation after another time interval T, while this would require infinite precision in a classical chaotic system. Thus a basic manifestation of classical chaos seems to have no place in quantum mechanics.
On the other hand, the destruction of an integral of motion, of a quantum number, has striking effects on the statistical properties of quantum spectras. It is today taken as granted that in a classically integrable system, the levels are uncorrelated, and so have a poissonian level spacings distribution [@BeTa] (a remarkable exception being the one-dimensional harmonic oscillator) and that in classically fully chaotic models, the level spacings distribution has a dramatically different shape : it obeys predictions of random matrix theory, i.e. it exhibits level repulsion [@Pech]. The situation in mixed systems, where regular and chaotic regions coexist in the classical phase space, is more intricated. In an old paper Percival [@Per] classified the eigenfunctions of the Schrödinger equation into two classes belonging to either the regular regions, where the invariant tori are not destroyed, or the chaotic one. This classification was based mostly on the correspondence principle and has been numerically confirmed a few years ago by Bohigas et. al. [@BoUlTo]. While the eigenfunctions that are mostly confined on classically regular regions - we will call them the regular eigenfunctions - tend to concentrate on invariant tori, the irregular ones tend to spread uniformly over the chaotic region as $\hbar \rightarrow 0$, as has been rigorously demonstrated by Shnirelman [@Shni]. This picture is assumed to reflect reality in the semiclassical limit $\hbar = 0$. Following this classification, Berry and Robnik postulated that the part of the spectrum that corresponds to regular eigenfunctions, has a poissonian level spacings distribution in opposition to the one corresponding to the irregular eigenstates which exhibits level repulsion [@BeRo]. This surmise led them to an expression for the level spacings distribution for mixed systems that has been observed convincingly only recently [@ProRo1] for the case of the kicked rotator on a torus. As pointed out by Prosen & Robnik and Li & Robnik [@ProRo] [@RoBao], reasons for this difficulty of observation could be that we are not deep enough in the semi-classical regime. As long as $\hbar$ is finite, a certain number of wave-functions belong neither to the regular nor to the irregular set of eigenfunctions. We may think of states making use of the Heisenberg uncertainty to overlap the frontier between the regular and irregular regions of the classical phase space, or states located on the regular region which, due to the finiteness of the Planck constant, do not yet belong to the set of regular states. Consequently, the Berry-Robnik regime should be observable only in the far semiclassical limit. We will come back to this point later.
In this paper we present a spin model allowing a precise study of a mixed regime. The reasons for this are first that, in a special regime, an approximate simple symmetry of the phase space structure, namely $S_{z}
\rightarrow -S_{z}$, allows the separation of regular states from the irregular ones, and secondly that the frontier between the regular and the chaotic zones is rather sharp, thus minimizing the number of intermediate eigenstates. This enables us to compute the level spacings statistics independently for the regular and irregular states. We interpret the fact that these statistics obey quite well the poissonian distribution and the GOE respectively as a direct confirmation of the Berry-Robnik surmise.
We study the quantum system defined by the following Hamiltonian : $$\begin{aligned}
\H := \frac{m}{2} ((1-z^2) \S_{z}^{2} - z^2 \S_{x}^{2}) + \kappa
\S_{z} \Delta_{T}\end{aligned}$$ and the corresponding unitary time evolution operator : $$\begin{aligned}
\U:= \exp(-\frac{i}{\hbar}\kappa \S_{z}) \exp(-\frac{i}{\hbar} \frac{m}{2}
((1-z^2) \S_{z}^{2} - z^2 \S_{x}^{2}) T)\end{aligned}$$ where $\vec{\S} = (\S_{x},\S_{y},\S_{z}) = \hbar (\s_{x},\s_{y},\s_{z}) =
\hbar \vec{\s}$ are spin operators satisfying the usual commutation rules ($\epsilon^{ijk}$ is the total antisymmetric tensor of 3$^{rd}$ order): $$\begin{aligned}
\left[ \S_{i},\S_{j} \right] = i \hbar \epsilon^{ijk} \S_{k}\end{aligned}$$ $0 \leq \kappa\leq 2 \pi $, $ \Delta_{T}:=\sum_{n=-\infty}^{+\infty} \delta(t-n T)$, \[$m$\] = energy$^{-1}$ time$^{-2}$ and $0\leq z \leq 1$ . Models of this kind have been extensively studied [@haa]. They represent a spin which evolves under the influence of a classically integrable Hamiltonian $\H^{0} = \frac{m}{2}
((1-z^2) \S_{z}^{2} - z^2 \S_{x}^{2})$ during a time $T$ after which the spin undergoes a rotation of angle $\kappa$ around the z-axis. The regime we consider is defined by $\kappa = 1.1$, $T=\frac{19}{m
S}$ and z$^{2} = \frac{1}{2}$. Classically there are two regular zones around the north and south poles surrounding a chaotic region which is fairly well symmetric under S$_{z}$ reflection (Fig.1). In the semiclassical limit which corresponds to $\hbar s = S = $ constant, $\hbar = s^{-1} \rightarrow 0$, states which are located on the chaotic region tend to cover it homogeneously according to Shnirelman’s theorem [@Shni]. Since this region is symmetric under S$_{z}$ reflection, the expectation value $<\Psi_{chaos}|\s_{z}|\Psi_{chaos}>$ of such a state tends to disappear as we approach the semi-classical limit. For small but finite $\hbar$, the distribution of $<\Psi_{k}|\s_{z}|\Psi_{k}>$, where $|\Psi_{k}>$ is an eigenstate of the operator $\U$ defined in (2), will then present a sharp peak around zero corresponding to the irregular states surrounded by two smaller bumps corresponding to regular states (Fig.2). This allows us to separate easily the regular states from the irregular ones, the validity of this selection being confirmed by a numerical semiclassical argument presented in section 3 as well as an extensive study of the Husimi distributions of the selected states [@AmJa].
The paper is organized as follows : Section 2 is devoted to a short presentation of the classical model. In section 3 we derive some useful semiclassical quantities such as the density of states and the expression for the action. This will allow us to estimate the number of regular states, and give a check of our selection criterion. In section 4 we present the quantum mechanical model as well as our numerical results for a spin magnitude $s$=500. All of them were obtained using direct diagonalization techniques. Conclusions and further remarks are given in section 5.
Classical model
===============
The unperturbed classical Hamiltonian $$\begin{aligned}
H_{cl}^{0} := \frac{m}{2} ((1-z^2) S_{z}^{2} - z^2 S_{x}^{2})\end{aligned}$$ has two degrees of freedom and is an integral of motion. The trajectories are confined to the intersections of the sphere $|\vec{S}| = S$ with the cones of constant energy $E = \frac{m}{2} ((1-z^2) S_{z}^{2} - z^2 S_{x}^{2})$. The perturbation : $$\begin{aligned}
H_{cl}^{1} := \kappa S_{z} \Delta_{T}\end{aligned}$$ corresponds to a rotation of angle $\kappa$ around the z-axis performed at time intervals $T$. Its addition leads to the destruction of the energy surfaces, and allows more and more trajectories to wander chaotically on the sphere of constant spin magnitude as $\kappa$ and $T$ grow. Expanding $H_{cl}^{0}$ up to the first order in $\delta
S_{z}:=S-S_{z}$ near the poles $S_{z} = \pm S$,we get a one-dimensional harmonic oscillator of period $T=2 \sqrt{2} \frac{\pi}{m S}$. In particular we have $\dot{\delta S_{z}}=O(\delta S_{z}^{2})$ : in this approximation $\delta
S_{z}$ is an integral of motion and is furthermore conserved by the perturbation $H_{cl}^{1}$ too. It is thus conceivable that the invariant torii near the poles will offer more resistance to the perturbation than those located away from them. We use this property to find a regime in which there are two regular islands around the poles approximately related by the operation $S_{z} \rightarrow -S_{z}$ and separated by a chaotic region. This we achieved by setting $\kappa = 1.1, T=\frac{19}{m
S}, z^{2}=0.5$ (Fig.1). The regular islands occupy in a good approximation the region $ 0.22 S^{2} \leq E \leq E_{max} = 0.25 S^{2} $.
Semiclassical approach
======================
We compute the Green function for a trajectory of positive energy and the density of states for the unperturbed case $T=\frac{19}{m S}, z^{2}=0.5$. We follow the lines drawn in [@Rei]. We first write the unperturbed Hamiltonian in canonical variables (S$_{z},\phi$) for the chosen regime : $$\begin{aligned}
H_{0} = \frac{m}{4} (S_{z}^{2} (1+\cos^{2}(\phi)) - \vec{S}^{2} \cos^{2}(\phi))\end{aligned}$$ The action integral for a trajectory of energy $E$ starting at $\phi_{0}$ and ending at $\phi$ reads ($\vec{S} = \hbar \vec{s}$, $e=\frac{E}{\hbar^{2} m}$): $$\begin{aligned}
{\cal S}_{\beta}(\phi_{0},\phi,e) =
\hbar \left[\int_{\phi_{0}}^{\phi^{*}} \sqrt{\frac{4 e+s^{2}
\cos^{2}(\phi')}{1+\cos^{2}(\phi')}} d\phi'+
n_{\beta} \int_{0}^{2 \pi} \sqrt{\frac{4 e+s^{2}
\cos^{2}(\phi')}{1+\cos^{2}(\phi')}} d\phi'\right]\end{aligned}$$ where we have set $\phi^{*} = \phi_{0} + (\phi-\phi_{0}) $ mod $2 \pi$, and $n_{\beta}$ is the number of complete revolutions accomplished between $\phi_{0}$ and $\phi$ ($\phi= 2 \pi n_{\beta} + \phi*$). The sum runs over classical orbits $\beta$ of constant energy. This leads us to the expression for the corresponding Green function : $$\begin{aligned}
{\cal G}(\phi_{0},\phi,e) & = & -\frac{i}{\hbar} \sum_{\beta} \sqrt{|{\det \cal
D}_{1,\beta}(\phi_{0},\phi)|} \exp(\frac{i}{\hbar}
{\cal S}_{\beta}(\phi_{0},\phi,e)-\frac{i \pi}{2} l_{\beta}) \nonumber \\
& =: & \sum_{n} \frac{\psi_{n}^{*}(\phi) \psi_{n}(\phi_{0})}{(E-E_{n}+i
\epsilon)}\end{aligned}$$ with $$\begin{aligned}
& & \hspace{-2cm} \det {\cal D}_{1,\beta} (\phi_{0},\phi) =
\frac{1}{m^{2}}\left({\partial^{2}{{\cal S}_{\beta}} \over \partial \phi
\partial \phi_{0}} {\partial^{2}{{\cal S}_{\beta}} \over \partial e^{2}} -
{\partial^{2}{{\cal S}_{\beta}} \over \partial \phi \partial
e}{\partial^{2}{{\cal S}_{\beta}} \over \partial \phi_{0} \partial e} \right) =
\nonumber \\
& & \hspace{-2cm} \frac{-4}{m^{2} \sqrt{(4 e+s^{2} \cos^{2}(\phi_{0}))
(1+\cos^{2}(\phi_{0})) (4 e+s^{2} \cos^{2}(\phi)) (1+\cos^{2}(\phi))}}\end{aligned}$$ this result being obtained by partial differentiations of (7). Since this latter value never changes sign, the Maslov index $l_{\beta}$ vanishes and so the divergence of the Green function leads to the following semiclassical quantization condition : $$\begin{aligned}
\int_{0}^{2 \pi} \sqrt{\frac{4 e+s^{2} \cos^{2}(\phi')}{1+\cos^{2}(\phi')}}
d\phi' = 2 \pi M\end{aligned}$$ for any integer $0 \leq M \leq s$. We have then for the averaged density of states (N is the number of states) : $$\begin{aligned}
\overline{\rho}(e) & = & -\frac{1}{N \pi} Im\left[ \int_{0}^{2 \pi} d\phi
{\cal G} (\phi,\phi,e) \right] \nonumber\\
{} & = & \frac{1}{\pi \hbar} \int_{0}^{2 \pi} d\phi \sqrt{|{\det \cal
D}_{1,\beta}(\phi,\phi)|}\nonumber \\
{} & = & \frac{2}{\pi \hbar m} \int_{0}^{2 \pi} \frac{d\phi}{\sqrt{
(4 e+s^{2} \cos^{2}(\phi)) (1+\cos^{2}(\phi)))}}\end{aligned}$$ The last equation states in particular that the averaged density of states is proportional to the classical orbit period. Fig. 3 shows the agreement of this semiclassical result with the numerically obtained density of states for the unperturbed quantum model at $s$=1000. Using (11) we estimate the number of states occupying the regular region of figure 1 : $$\begin{aligned}
N_{reg} \approx \int_{0.22 s^{2}}^{0.25 s^{2}} \overline{\rho}(e) de\approx
\frac{1}{24} (2 s+1)\end{aligned}$$ The number of states occupying this region in absence of perturbation is $\frac{1}{24}$ times the total number of states. This gives us a first approximation for the number of regular states we must select. A better approximation in presence of perturbation is given using a method developed by Bohigas et. al. [@BoUlTo]. We must evaluate the number N$_{reg}$ of trajectories that satisfy the condition : $$\begin{aligned}
\sum_{i=0}^{P} \left[ \int_{\phi_{i}^{+}}^{\phi_{i}^{-}} s_{z}(\phi') d\phi'
+ s_{z}(\phi_{i}^{+}) \kappa \right] = 2 \pi M\end{aligned}$$ for some integers $M$ and $P$, while it nearly closes on itself after the $P^{th}$ kick, i.e. : $ s_{z}(\phi_{M}^{+}) \approx s_{z}(\phi_{0}^{+})$. $\phi_{i}^{-}$ is the angle between the x and the y component of the spin just before the i$^{th}$ kick while $\phi_{i}^{+}$ refers to the same angle right after this kick.
This condition means that the action integral must still be an integer multiple of 2 $\pi$, and that simultaneously, the orbit must be closed. This condition has meaning only on regular regions were the invariant torii are not destroyed, so that the integrals make sense. We transform this condition and compute the number of trajectories satisfying : $$\begin{aligned}
\frac{\sum_{i=0}^{P} \left[ \int_{\phi_{i}^{+}}^{\phi_{i+1}^{-}} s_{z}(\phi')
d\phi'
+ s_{z}(\phi_{i}^{+}) \kappa \right]}{\kappa P + \sum_{i=1}^{P} \left(
\phi_{i}^{-} - \phi_{i-1}^{+} \right)} \approx M\end{aligned}$$ for integers $M$ and $P$, and $P$ sufficiently large. With this we replace two conditions by only one numerically more tractable condition. Since our task is to evaluate the number of regular semiclassical levels, and not to determine them precisely, we believe that condition (14) is sufficient. The number of regular states we numerically estimated with (14) is 50 $\pm$ 4 for $s$=500, i.e. slightly larger than that estimated with (12). In the next section, we will consider this estimated number of regular states as a check of the validity of our selection criterion.
Quantum Model
=============
In this section we study the statistical properties of the spectrum of the quantum Hamiltonian (1) for integer spin magnitude. Since the perturbation term is time-dependent, the energy is no longer a good quantum number, and we are led to define quasi-energies and quasi-energy eigenstates. The Schrödinger equation leads to the following time evolution from right after a kick to right after the next one: $$\begin{aligned}
\Psi (T^{+}) = \U \Psi(0^{+}) =
\exp(-\frac{i}{\hbar} \kappa \S_{z}) \exp(-\frac{i}{\hbar} \H^{0} T)
\Psi(0^{+})\end{aligned}$$ Quasi-energies $\lambda$ and quasi-energy eigenstates $\Psi_{\lambda}$ are then defined by : $$\begin{aligned}
\U \Psi_{\lambda} = \exp(-i \lambda) \Psi_{\lambda}\end{aligned}$$ Since $\U$ is unitary, the $\lambda$’s are real and defined modulo 2 $\pi$. We introduce two parities : $$\begin{aligned}
\P | \mu > = |- \mu> \\
\T |\mu> = (-1)^{s-\mu} |\mu>\end{aligned}$$ We can express the time-reversal operator [**T**]{} in term of these two parity operators : $$\begin{aligned}
\P \circ \T | \mu> = {\bf T} | \mu > = (-1)^{s-\mu} |- \mu>\end{aligned}$$ In the integer spin case the eigenstates $|\Psi>$ of $\U^{0} :=
\exp(-\frac{i}{\hbar} \H^{0} T) $ satisfy the conditions : $$\begin{aligned}
\P |\Psi> = \pm |\Psi> \\
\T |\Psi> = \pm |\Psi>\end{aligned}$$ So $ \H^{0} $ and $ \U^{0} $ are in particular time-reversible. The perturbation breaks the $\P$-parity but leaves the $\T$-parity unbroken . We will concentrate on the study of even states, i.e. those states satisfying : $$\begin{aligned}
\T |\Psi> = |\Psi>\end{aligned}$$ However partial results obtained for the odd set of states corroborate the results presented here. The key point is now to find a clear quantum manifestation of the approximate symmetry $S_{z} \rightarrow -S_{z}$ of the classical phase space structure. A practical solution is given by Shnirelman’s theorem which states that in the semiclassical limit, the quantum states that are confined on the classically chaotic region of the phase space tend to cover it uniformly. To get an insight in this statement we use the following resolution of unity [@Pelo] $${\bf 1} = \frac{2 s+1}{\pi} \int d\theta d\phi \sin \theta
|\theta,\phi><\theta,\phi|$$ where we introduced coherent states of the spin $SU(2)$ group : $$|\theta,\phi> := \sum_{\mu=-s}^{s}\sqrt{\left(^{\hspace{0.15cm}2 s}_{s-\mu}
\right)} \sin(\frac{\theta}{2})^{s-\mu} \cos(\frac{\theta}{2})^{s+\mu} e^{i
(s-\mu) \phi} |\mu >$$ These are states that are centered on the point ($\theta,\phi$) of the sphere and which minimalize the quantum uncertainty. $\theta $ is defined by $S_{z} =
S \cos(\theta)$. Using (22), the symmetry of the chaotical region and Shnirelman’s theorem [@Shni] : $$<\Psi_{chaos}|\theta,\phi> \longrightarrow \cases{
0 & on the regular region\cr
const & on the chaotic region
}$$ it is then easy to show that $$<\Psi_{chaos}|\s_{z}|\Psi_{chaos}> \rightarrow 0$$ in the semiclassical limit. This translates into Fig.2 where we plotted an histogram of the expectation value of $\s_{z}$ taken over quasi-energy eigenfunctions for $s$=500, $\kappa=1.1$, z$^{2}$=0.5, and $m$=1. The central peak clearly reflects our reasoning, while the two smaller bumps surrounding it are mainly due to the regular states that are confined to the classical stability islands. The gap in-between is a consequence of the uniform distribution of irregular states. It is remarkable that this gap overlaps the classical frontier between regular and chaotic region.
We used this property to part the irregular states from the regular ones and then study separately the statistical properties of the spectrums of each class of states. We believe this criterion is justified since the fluctuations $$\begin{aligned}
\Delta \s_{z} = \sqrt{<\s_{z}^{2}>-<\s_{z}>^{2}}\end{aligned}$$ of regular states is much smaller than the “Shnirelman gap” appearing in the histogram of Fig. 2 between the huge central peak and the smaller bumps. As a consequence only very few regular levels will be selected with the set of irregular ones, while maybe more irregular will be counted with the regular ones. Moreover, the fact that the number of selected regular states is in complete agreement with the numerical semiclassical evaluation given by (14) confirms the relevance of this selection criterion.
We now turn our attention to the study of the spectral properties of the time-evolution operator (16). Due to the $\T$-symmetry (18), $\U$ belongs to the circular orthogonal ensemble, and not to the circular unitary ensemble as would be expected from the fact that the perturbation breaks the time-reversal symmetry. This situation is similar to the one encountered by Berry & Robnik in certain Aharonov-Bohm billiards [@RoBe], or by Delande & Gay in the Hydrogen atom in a magnetic field [@DelGay] where the system violates the time-reversal symmetry, but possesses an invariance under a combination of the time-reversal and another symmetry, in our case the $\P$-symmetry. We thus expect a linear repulsion for the part of the spectrum belonging to the irregular states.
The results of our study for a spin magnitude $s$=500 are plotted in Fig. 4 to 9. Fig. 4 shows a plot of the level spacings statistics for 4233 irregular level spacings computed by diagonalizing ten different evolution matrices for $T=\frac{19}{m S} $ and $ 1.05 \leq \kappa \leq 1.15$. The solid line is the predicted Wigner distribution. The agreement is excellent. In Fig. 5 we plotted the corresponding cumulative level spacings distribution defined in term of the level spacings distribution $P(\underline{s})$ by $$\begin{aligned}
W(\underline{s}) = \int_{0}^{\underline{s}}dt P(t)\end{aligned}$$ Also shown are the Poisson and the Wigner distributions. As shown in inset, small deviations from the Wigner distribution appear only around spacings $\underline{s} \approx$2, but have no significance to our opinion.
Fig.6 and 7 show the level spacings and the cumulative level spacings distribution for 572 regular levels taken from twenty different evolution matrices for $T=\frac{19}{m S} $ and $ 1.05 \leq \kappa \leq 1.15$. The difficulty here is the relatively small number of regular states ($\approx$ 25). Accordingly, only few intermediate or irregular states can have relatively big effects on the statistics. In a convenient basis, $\U$ can be represented by the following matrix : $$\begin{aligned}
\U := \left( \begin{array}{c c} R & K \\ K & C \end{array} \right)\end{aligned}$$ $R = R_{i} \delta_{i,j}$ is a diagonal N$_{reg} \times$ N$_{reg}$ matrix which corresponds to the N$_{reg}$ regular states, $C = C_{i} \delta_{i,j}$ is a diagonal N$_{chaos} \times$ N$_{chaos}$ matrix which corresponds to the N$_{chaos}$ irregular states, and $K = {\rm O}(\hbar)$ couples the two subspaces as long as $\hbar$ is finite. In our picture $K$ disappears in the semiclassical limit, and the $R_{i}$ and $C_{i}$ satisfy a poissonian statistics and a GOE statistics respectively. It would be of course a hopeless task to try to determine K for finite $\hbar$. The important point is to recognize that as long as $\hbar$ is finite but small enough, $K$ couples only few regular states with irregular ones, this fact resulting in a deviation from the Berry-Robnik surmise. This deviation is then naturally much more important for the regular part of the spectrum, since it contains much fewer levels than the irregular part. We believe that this is the reason for the deviation of the statistics of the set of levels we have selected as regular from the poissonian predicted behaviour. We must recall that our whole reasoning is based on the assumption of two classically homogeneous stability islands. In such a case, semiclassical wave-function would mimic classical orbits and would therefore fit together as concentric circles. The presence of hyperbolic fixed points or cantori may change this picture, possibly turning regular states into intermediate ones as long as $\hbar$ is finite. The semiclassical wave-function overlap and thus interact at certain regions, and this, in the Pechukas picture [@Pech], modify very sensibly the equations of motions governing the evolution of the quasi-energies $\lambda$ as $\kappa$ or $T$ is modified, resulting in the appearance of level repulsion. So some intermediate states are phase spatially mixed among the set of states we have selected as regular and consequently modify the corresponding statistics. Their effect is furthermore enhanced by the small ratio of regular levels. A current investigation of the Husimi densities of the selected regular states corroborates this reasoning [@AmJa]. Finally we show in Fig.8 and 9 level spacings and cumulative level spacings statistics for the complete set of levels. We compare our results with the Berry-Robnik prediction for a fractional measure of regular states as approximated by (12). The agreement is amazing, and corroborates our picture. The $\chi^{2}$-test for both graphs - $\chi^{2}$=25, i.e. half the number of boxes for Fig.8, and $\chi^{2}$=1480, i.e. 3.3 times less than the number of levels for Fig.9 - gives full statistical significance to these last graphs. We see them as a good evidence for the validity of the Berry-Robnik surmise in our model.
Conclusion
==========
We studied the statistical properties of a quantum spin model whose classical counterpart exhibits a mixed phase space configuration. Due to a simple approximate symmetry, whose effect on the quantum system is drastically enhanced by Shnirelman’s theorem, we were able to separate the irregular from the regular levels, thereby confirming implicitely the validity of the Percival classification. We then performed a separated statistical study of these levels. The results confirm the Berry-Robnik surmise : while the irregular set of quasienergies exhibits a clear wigner-like shape, the regular part of the spectrum has a clearly different shape, though its spacings distribution does not follow strictly a poissonian law. This deviation is interpreted as the presence of both irregular and intermediate states among the selected regular ones, their effect being enhanced by the relatively small number of the latter. Nevertheless, due to the small number of regular states we believe that the irregular statistics is much more significant, and see our results as a good confirmation of the validity of the Berry-Robnik surmise in our model.\
\
\
**[Acknowledgements]{}\
\
One of us (P.J.) gratefully acknowledges fruitfull discussions with J. Bellissard, C. Rouvinez and D. Shepelyansky , as well as the hospitality of the theoretical physics division of the “Laboratoire de Physique Quantique, Université Paul Sabatier” in Toulouse extended to him during his visit when part of this work has been done. We are grateful to T. Prosen for having drawn our attention on reference [@ProRo1]. Work supported by the Swiss National Science Foundation.**
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Fox R. F. and Elston T. C., Phys. Rev. E [**50**]{}, 2553, (1994) This is a weak convergence. As $\hbar \rightarrow 0$, the Husimi density of such a state tends to a constant over the classically irregular region in the sense of a distribution. See the addendum of A.I. Shnirelman in : KAM Theory and Semiclassical Approximations to Eigenfunctions, V.F Lazutkin, Springer (1993). Baowen Li and Marko Robnik, Preprint CAMTP/94-10 “Separating the regular and irregular energy levels and their statistics in Hamiltonian system with mixed classical dynamics ” and Preprint CAMTP/94-11 “Supplement to the paper : Separating the regular and irregular energy levels and their statistics in Hamiltonian system with mixed classical dynamics ” , to be published in J. Phys. A : Math. Gen. Perelomov A., Generalized Coherent States and Their Applications, Springer (1986). Amiet J.-P. & Jacquod Ph., in preparation.
Figure Captions {#figure-captions .unnumbered}
===============
: Orthogonal projection of the classical phase space on the ($S_{x}$,$S_{y}$) plane for the case T=$\frac{19}{m S}$, $\kappa=1.1$ and z$^{2}$=0.5.\
\
: Histogram of the expectation value of $S_{z}$ taken over the eigenstates of the unitary time evolution operator defined in (2).\
\
: Density of states for the unperturbed Hamiltonian according to (11) (solid line) as compared to numerically obtained datas for the case $s$=1000 (squares).\
\
: Level spacings distribution for 4233 irregular level spacings obtained through direct diagonalization of ten evolution matrices in the parameter range $T=\frac{19}{m S}$ and $1.05 \leq \kappa \leq 1.15$.\
\
: Cumulative level spacings distribution for the same case as fig.4. In inset : regions of small deviation relatively to the Wigner-distribution.\
\
: Level spacings distribution for a set of 472 regular level spacings obtained through direct diagonalization of twenty evolution matrices in the parameter range $T=\frac{19}{m S}$ and $1.095 \leq \kappa \leq 1.105$. The solid line is the predicted Poisson distribution.\
\
: Cumulative level spacings distribution for the same levels as Fig.6 compared to the Poisson distribution.\
\
: Level spacings distribution for a set of 5000 regular and irregular level spacings obtained through direct diagonalization of ten evolution matrices in the parameter range $T=\frac{19}{m S}$ and $1.095 \leq \kappa \leq 1.105$. The solid line is the predicted Berry-Robnik distribution with fractional measure of regular states $\rho_{1} = 0.08$. $\chi^{2}$=25 is half the number of boxes.\
\
: Cumulative level spacings distribution for the same levels as Fig.8 compared to the poissonian and the Berry-Robnik predicted distribution. In inset : Same curve compared to the Wigner distribution. $\chi^{2}$=1480 is 3.3 times less than the number of levels.
[^1]: e-mail [email protected]
[^2]: e-mail [email protected]
|
---
abstract: 'We exhibit a bridge between the theory of *cellular categories*, used in algebraic topology and homological algebra, and the model-theoretic notion of *stable independence*. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin-Eklof-Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický.'
address:
- 'Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic'
- 'Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno, Czech Republic'
- |
Department of Mathematics\
Harvard University\
Cambridge, Massachusetts, USA
author:
- Michael Lieberman
- Jiří Rosický
- Sebastien Vasey
bibliography:
- 'more-indep.bib'
date: |
\
AMS 2010 Subject Classification: Primary: 18C35. Secondary: 03C45, 03C48, 03C52, 03C55, 16B50, 55U35.
title: Cellular categories and stable independence
---
Introduction
============
Cellular categories were introduced in [@mr] as cocomplete categories equipped with a class of morphisms (called cellular) containing all isomorphisms and closed under pushouts and transfinite compositions. These categories are abundant in homotopy theory because any Quillen model category carries two cellular structures given by cofibrations and trivial cofibrations respectively. These cellular categories are, in addition, retract-closed (in the category of morphisms). A retract-closed cellular category is cofibrantly generated if its is generated by a set of morphisms using pushouts, transfinite compositions and retracts. In locally presentable categories, this implies that cellular morphisms form a left part of a weak factorization system. In [@mr], retract-closed cofibrantly generated cellular locally presentable categories were called *combinatorial*. The main result of [@mr] is that combinatorial categories are closed under 2-limits, in particular under pseudopullbacks. A consequence is that combinatorial categories are left-induced in a sense that, given a colimit preserving functor $F:{\mathcal {K}}\to{\mathcal {L}}$ from a locally presentable category ${\mathcal {K}}$ to a combinatorial category ${\mathcal {L}}$ then preimages of cellular morphisms form a combinatorial structure on ${\mathcal {K}}$. This was later used, e.g., in [@induced-model-jtop]. The proof is quite delicate and depends on Lurie’s concept of a good colimit (see [@fat-small-obj]).
The main result of the present paper is that, in the special case when cellular morphisms are coherent and $\aleph_0$-continuous, a retract-closed cellular category is combinatorial if and only if it carries a stable independence notion (Theorem \[cofib-gen\]). The latter concept stems from model theory and a purely category-theoretic definition was given in [@indep-categ-advances]. Roughly, a stable independence notion in a given category ${\mathcal {K}}$ is a class of commutative squares (called *independent*) that satisfies certain properties. In particular, the category whose objects are morphisms of ${\mathcal {K}}$ and whose morphisms are independent squares should be accessible. In our situation, independent squares coincide with *cellular* squares; that is, squares of cellular morphisms such that the unique morphism from the pushout is cellular. These squares are also used in [@henry-induced]. Since a pre-image of an accessible category is accessible, this yields a simple proof that coherent and $\aleph_0$-continuous combinatorial categories are left-induced (see Corollary \[lazy\]). While coherence is quite common, especially for trivial cofibrations, $\aleph_0$-continuity is more limiting. Nevertheless, our theorem covers many situations. In particular, we will show (Theorem \[ext-thm\]) that the abstract elementary classes of “roots of Ext” studied in [@bet] (for example the AEC of flat modules with flat monomorphisms) have a stable independence notion. Note, too, that since pure monomorphisms in a locally finitely presentable category are coherent and $\aleph_0$-continuous, the result of [@lprv-purecofgen-v3], the proof of which relies on [@mr], actually falls within the framework of this paper.
Concerning terminology, we will refer freely to [@adamek-rosicky], [@mr] and [@indep-categ-advances] (concerning accessible categories, cellular categories, and stable independence respectively). A more detailed version of the present paper, with more background, can be found at <https://arxiv.org/abs/1904.05691v2>.
Acknowledgments
---------------
We thank Jan Trlifaj for helpful conversations about roots of Ext and to Simon Henry for sharing [@henry-induced] with us. We also thank John Baldwin, Marcos Mazari-Armida, Misha Gavrilovich, and a referee for useful feedback.
Cellular categories
===================
Recall that a cocomplete category ${\mathcal {K}}$ is called *cellular* if it is equipped with a class ${\mathcal {M}}$ of morphisms containing all isomorphisms and closed under pushouts and transfinite compositions (see [@mr]).
\[subcat\] A composition of two morphisms is a special case of a transfinite composition. Thus a cellular category $({\mathcal {K}},{\mathcal {M}})$ induces a subcategory ${\mathcal {K}}_{\mathcal {M}}$ of the category ${\mathcal {K}}$ whose objects are those in ${\mathcal {K}}$ and whose morphisms are precisely those of ${\mathcal {M}}$. Since ${\mathcal {M}}$ contains all isomorphisms, the subcategory ${\mathcal {K}}_{\mathcal {M}}$ is *isomorphism-closed*. Still, ${\mathcal {K}}_{\mathcal {M}}$ need not have pushouts.
In order to explain this, recall that ${\mathcal {M}}$ is closed under pushouts whenever, given a pushout square $$\xymatrix@=3pc{
C \ar[r]^h & P \\
A \ar [u]^g \ar[r]_f & B \ar[u]
}$$ in ${\mathcal {K}}$ with $f\in{\mathcal {M}}$, then $h\in{\mathcal {M}}$. But this does not mean that, if also $g\in{\mathcal {M}}$, that this square is a pushout square in ${\mathcal {K}}_{\mathcal {M}}$. The latter means that given another commutative square in ${\mathcal {K}}_{\mathcal {M}}$, as below, with $u,v\in {\mathcal {M}}$, $$\xymatrix@=3pc{
& & D \\
B \ar[r]\ar@/^/[rru]^{u} & P \ar[ru]_t {\save*!/dl-1pc/dl:(1,-1)@^{|-}\restore}& \\
A \ar [u]^g \ar[r]_f & C \ar[u]\ar@/_/[ruu]_{v} &
}$$ then the induced morphism $t$ is in ${\mathcal {M}}$.
Similarly, although ${\mathcal {M}}$ is closed under transfinite compositions, these composition does not to be colimits in ${\mathcal {K}}_{\mathcal {M}}$. In the latter case, ${\mathcal {K}}_{\mathcal {M}}$ would be closed under colimits of smooth chains, which implies closure under all directed colimits (see [@adamek-rosicky 1.7]).
\[cellular-square\] Let $({\mathcal {K}},{\mathcal {M}})$ be a cellular category. A commutative square $$\xymatrix@=3pc{
C \ar[r]^u & D \\
A \ar [u]^g \ar[r]_f & B \ar[u]_v
}$$ is called *cellular* if the induced morphism $t:P\to D$ from the pushout (see above) belongs to ${\mathcal {M}}$ .
Cellular squares could also be called ${\mathcal {M}}$-effective. In the special case in which ${\mathcal {M}}$ is the class of regular monomorphisms, this corresponds precisely to the effective squares considered in [@indep-categ-advances], and originating in [@effective-unions].
\[coherent-def\] A cellular category $({\mathcal {K}},{\mathcal {M}})$ will be called
1. *coherent* if whenever $f$ and $g$ are composable morphisms, $gf \in {\mathcal{M}}$ and $g \in {\mathcal{M}}$, then $f \in {\mathcal{M}}$,
2. *left cancellable* if $gf \in {\mathcal{M}}$ implies $f \in {\mathcal{M}}$,
3. *$\lambda$-continuous* if ${\mathcal {K}}_{{\mathcal{M}}}$ is closed under $\lambda$-directed colimits in ${\mathcal {K}}$,
4. *$\lambda$-accessible* it is $\lambda$-continuous and both ${\mathcal {K}}$ and ${\mathcal {K}}_{{\mathcal{M}}}$ are $\lambda$-accessible.
5. *accessible* if it is $\lambda$-accessible for some $\lambda$.
\[coherent-prop\]
1. Since a cellular category is cocomplete, an accessible cellular category has ${\mathcal {K}}$ locally presentable.
2. \[coherent-prop-2\] It is easy to see that $({\mathcal {K}},{\mathcal {M}})$ is $\lambda$-continuous provided that ${\mathcal {M}}$ is closed under $\lambda$-directed colimits in ${\mathcal {K}}^2$. In fact, given a $\lambda$-directed diagram $D:I\to {\mathcal {K}}_{\mathcal {M}}$ and its colimit $\delta_i:Di\to K$ in ${\mathcal {K}}$, then $\delta_i={\operatorname{colim}}_{i\leq j\in I}D_{i,j}$, where the $D_{i,j}:Di\to Dj$ are the appropriate diagram maps. Similarly, given a cocone $\gamma_i:Di\to L$ in ${\mathcal {K}}_{\mathcal {M}}$ then the induced morphism $g:K\to L$ is precisely ${\operatorname{colim}}_i\gamma_i$.
\[independence\]
1. In [@indep-categ-advances], we defined an *independence relation* (or *independence notion*) in a category ${\mathcal {K}}$ as a class ${\unionstick}$ of commutative square (called *${\unionstick}$-independent*, or just *independent*, squares) such that, for any commutative diagram
$$\xymatrix@=3pc{
& & E \\
B \ar[r]\ar@/^/[rru] & D \ar[ru] & \\
A \ar [u] \ar[r] & C \ar[u]\ar@/_/[ruu] &
}$$
the square spanning $A, B, C,$ and $D$ is independent if and only if the square spanning $A, B, C,$ and $E$ is independent. A subcategory of ${\mathcal {K}}^2$ (the category of morphisms in ${\mathcal {K}}$, whose objects are morphisms and morphisms are commutative squares) whose objects are morphisms and morphisms are independent squares was denoted as ${\mathcal {K}}_{{\operatorname{NF}}}$. Here, we will denote it as ${\mathcal {K}}_\downarrow$.
2. In [@indep-categ-advances], as independence relation ${\unionstick}$ was defined to be *stable* if it is symmetric, transitive, accessible, has existence, and has uniqueness. In case ${\unionstick}$ satisfies all of the above conditions except accessibility, we say that it is *weakly stable*.
3. Accessibility of ${\unionstick}$ means that the category ${\mathcal {K}}_\downarrow$ is accessible, which implies, in particular, that it is closed in ${\mathcal {K}}^2$ under $\lambda$-directed colimits for some $\lambda$ (see [@indep-categ-advances 3.26]). If ${\unionstick}$ satisfies the latter closure condition, we say that it is $\lambda$-*continuous*.
4. \[independence-3\] Accessibility of ${\unionstick}$ also implies that ${\mathcal {K}}$ is accessible (see [@indep-categ-advances 3.27]).
\[weak-stable-thm\] If $({\mathcal {K}},{\mathcal{M}})$ is a cellular category, then cellular squares form a weakly stable independence relation in ${\mathcal {K}}_{{\mathcal{M}}}$.
We first check that cellular squares form an independence notion. Assume that $(A, B, C, D)$ is a commutative square[^1] in ${\mathcal {K}}_{{\mathcal{M}}}$ and we are given a morphism $D \to E$ in ${\mathcal{M}}$. If $(A, B, C, D)$ is cellular, then closure of ${\mathcal{M}}$ under composition yields that $(A, B, C, E)$ is cellular. Conversely, if $(A, B, C, E)$ is cellular, then the map $P \to E$ from the pushout is in ${\mathcal{M}}$ by assumption, and also $D \to E$ is in ${\mathcal{M}}$, so by coherence also the map $P \to D$ is in ${\mathcal{M}}$. Thus $(A, B, C, D)$ is cellular.
This concludes the proof that cellular squares form an independence notion. Of course, the relation is also symmetric. Existence follows from closure under pushouts (and the fact that the identity map is an isomorphism, hence in ${\mathcal{M}}$). In order to prove the uniqueness property, consider cellular squares $(A, B, C, D^1)$ and $(A,B,C,D^2)$ with the same span $B \leftarrow A\to C$. Form the pushout $$\xymatrix@=3pc{
B \ar[r]^{} & P \\
A \ar [u]^{} \ar [r]_{} &
C \ar[u]_{}
}$$ and take the induced morphisms $P\to D^1$ and $P\to D^2$. They are in ${\mathcal{M}}$ by cellularity. Then the pushout
$$\xymatrix@=3pc{
D^1 \ar[r]^{} & D \\
P \ar [u]^{} \ar [r]_{} &
D^2 \ar[u]_{}
}$$
amalgamates the starting diagram.
To prove transitivity, consider:
$$\xymatrix@=3pc{
B \ar[r]^{} & D \ar[r]^{} & F \\
A \ar [u]^{f} \ar [r]^{g} & C \ar[u]_{} \ar[r]^{g'} & E \ar[u]_{}
}$$ where both squares are cellular. We have to show that the outer rectangle is cellular. Thus we have to show that the induced morphism $p:P\to F$ from the pushout $$\xymatrix@=3pc{
B \ar[r]^{} & P {\save*!/dl-1pc/dl:(1,-1)@^{|-}\restore}\\
A \ar [u]^{f} \ar [r]^{g'g} &
E \ar[u]_{}
}$$ is in ${\mathcal{M}}$. This pushout is a composition of pushouts $$\xymatrix@=3pc{
B \ar[r]^{u} & Q \ar[r]^{u'}{\save*!/dl-1pc/dl:(1,-1)@^{|-}\restore}& P {\save*!/dl-1pc/dl:(1,-1)@^{|-}\restore}\\
A \ar [u]^{f} \ar [r]^{g} & C \ar[u]_{v} \ar[r]^{g'} & E \ar[u]_{v'}
}$$ Recalling the left square of the starting diagram, we have an induced morphism $q:Q\to D$. Consider the pushout $$\xymatrix@=3pc{
D \ar[r]^{} & P' \\
Q \ar [u]^{q} \ar [r]^{u'} &
P \ar[u]_{\bar{q}}
}$$ Since the left square of the starting diagram is cellular, $q$ is in ${\mathcal{M}}$ and thus $\bar{q}$ is in ${\mathcal{M}}$. Composing this pushout with the right pushout square in the diagram above it, we obtain the pushout $$\xymatrix@=3pc{
D \ar[r]^{} & P' \\
C \ar [u]^{} \ar [r]_{} &
E \ar[u]_{}
}$$ The right square in the starting diagram is cellular, so the induced morphism $p':P'\to F$ is in ${\mathcal{M}}$. Thus $p=p'\bar{q}$ is in ${\mathcal{M}}$.
In the proof, we have not used the full strength of the assumption that ${\mathcal {M}}$ is closed under transfinite compositions: here finite compositions suffice. Coherence is used only once, in the proof that cellular squares form an independence notion (specifically, in the proof that the top right corner can be made “smaller”). Instead of coherence, we could also have assumed the dual property, cocoherence: indeed, we know in the proof that the maps $C \to D$ and $C \to P$ are in ${\mathcal{M}}$, so cocoherence would give us immediately that $P \to D$ is in ${\mathcal{M}}$. Note, however, that if ${\mathcal{M}}$ is a class of monomorphisms, cocoherence is too strong an assumption: if a section $i: A \to B$ is in ${\mathcal{M}}$, cocoherence would imply that the corresponding retract $r: B \to A$ is in ${\mathcal{M}}$, and so $r$ would have to be an isomorphism.
For a cellular category $({\mathcal {K}},{\mathcal{M}})$, we write ${\mathcal {K}}_{{\mathcal{M}},{\downarrow}}$ for $\left({\mathcal {K}}_{{\mathcal{M}}}\right)_{{\downarrow}}$.
In a cellular category, cellular squares form a class of morphisms in ${\mathcal {K}}^2$. Following Theorem \[weak-stable-thm\] this class is closed under composition, by transitivity of the associated weakly stable independence notion. Using [@indep-categ-advances 3.18], it is isomorphism-closed. Using [@indep-categ-advances 3.20, 3.21], cellular squares are left-cancellable.
\[dir-colim-lem\] If $({\mathcal {K}},{\mathcal{M}})$ is a $\lambda$-continuous cellular category, then the independence relation given by cellular squares is $\lambda$-continuous.
Let $({\mathcal {K}},{\mathcal{M}})$ be $\lambda$-continuous. Let $D:I\to {\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ be a $\lambda$-directed diagram where $Di$ is $f_i:A_i\to B_i$. Let $f:A\to B$ be a colimit of $D$ in $\left({\mathcal {K}}_{{\mathcal{M}}}\right)^2$. For each $i\in I$, the pushout of the colimit coprojection $A_i\to A$ along $f_i$, i.e. $$\xymatrix@=3pc{
A \ar[r]^{g} & P \\
A_i \ar [u]^{} \ar [r]_{f_i} &
B_i \ar[u]_{}
}$$ is a $\lambda$-directed colimit of pushouts $$\xymatrix@=3pc{
A_{i'} \ar[r]^{g_{i'}} & P_{i'} \\
A_i \ar [u]^{} \ar [r]_{f_i} &
B_i \ar[u]_{}
}$$ Thus the induced morphism $p:P\to B$ is a $\lambda$-directed colimit of induced morphisms $p_{i'}:P_{i'}\to B_{i'}$. Since ${\mathcal{M}}$ is $\lambda$-continuous, it follows that $p \in {\mathcal{M}}$. This shows that all the maps of the cocone $(f_i \to g)_{i \in I}$ are independent squares. Similarly, one can check that this is a colimit cocone in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$. Thus ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ is closed under $\lambda$-directed colimits in $\left({\mathcal {K}}_{{\mathcal{M}}}\right)^2$.
Combinatorial categories
========================
A cellular category $({\mathcal {K}},{\mathcal {M}})$ is said to be *retract-closed* if ${\mathcal {M}}$ is closed under retracts in the category ${\mathcal {K}}^2$. A retract-closed cellular category is called *combinatorial* if it is *cofibrantly generated*, i.e., if ${\mathcal {M}}$ is the closure of a set ${\mathcal {X}}$ of morphisms under pushouts, transfinite compositions and retracts. In particular, ${\mathcal {M}}={\operatorname{cof}}({\mathcal {X}})$, where $${\operatorname{cof}}({\mathcal {X}})={\operatorname{Rt}}({\operatorname{Tc}}({\operatorname{Po}}({\mathcal {X}})))={\operatorname{Rt}}({\operatorname{cell}}({\mathcal {X}}))$$ where ${\operatorname{Po}}$ denotes the closure under pushouts, ${\operatorname{Tc}}$ under transfinite compositions and ${\operatorname{Rt}}$ under retracts (see [@mr]).
For $\lambda$ a regular cardinal, we write ${\mathcal {K}}_\lambda$ for the full subcategory of ${\mathcal {K}}$ consisting of $\lambda$-presentable objects. We similarly denote by ${\mathcal {K}}_\lambda^2$ the full subcategory of ${\mathcal {K}}^2$ consisting of morphisms with $\lambda$-presentable domains and codomains. We will also write, for example, ${\mathcal{M}}_\lambda := {\mathcal{M}}\cap {\mathcal {K}}_\lambda^2$.
The next result, the main theorem of this paper, characterizes when cellular squares form a stable independence notion in terms of cofibrant generation of the corresponding class of morphisms.
To go from stable independence to cofibrant generation, we require a technical result from [@internal-improved-v2 §9] concerning the existence of *filtrations*. Recall that the *presentability rank* of an object $A$ is the least regular cardinal $\lambda$ such that $A$ is $\lambda$-presentable. We say that $A$ is *filtrable* if it can be written as the directed colimit of a chain of objects with lower presentability rank than $A$. We say that $A$ is *almost filtrable* if it is a retract of such a chain. The chain is *smooth* if directed colimits are taken at every limit ordinal. By [@internal-improved-v2 9.12], in any accessible category with directed colimits, there exists a regular cardinal $\lambda$ such that any object with presentability rank at least $\lambda$ is almost filtrable (and, moreover, the chain in the filtration can be chosen to be smooth). We say that a category satisfying the latter condition is *almost well $\lambda$-filtrable*.
\[cofib-gen\] Let $({\mathcal {K}},{\mathcal {M}})$ be an accessible cellular category which is retract-closed, coherent and $\aleph_0$-continuous. The following are equivalent:
1. \[cofib-gen-1\] ${\mathcal {K}}_{{\mathcal{M}}}$ has a stable independence notion.
2. \[cofib-gen-2\] Cellular squares form a stable independence notion in ${\mathcal {K}}_{{\mathcal{M}}}$.
3. \[cofib-gen-3\] $({\mathcal {K}},{\mathcal{M}})$ is combinatorial.
(\[cofib-gen-1\]) implies (\[cofib-gen-2\]): If ${\mathcal {K}}_{{\mathcal{M}}}$ has a stable independence notion, then canonicity (Theorem \[canon-thm\] – note that ${\mathcal {K}}_{{\mathcal{M}}}$ has directed colimits, since ${\mathcal{M}}$ is $\aleph_0$-continuous) together with Theorem \[weak-stable-thm\] ensures that it is given by cellular squares. Note that if we know that all morphisms in ${\mathcal{M}}$ are monos, then we do not need Theorem \[canon-thm\] and can use [@indep-categ-advances 9.1] instead.
\[cofib-gen-2\]) implies (\[cofib-gen-3\]): Assume that ${\mathcal {K}}_{{\mathcal{M}}}$ has a stable independence ${\unionstick}$ given by cellular squares. Thus ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ is accessible and has directed colimits (by Lemma \[dir-colim-lem\]). By Remark \[independence\](\[independence-3\]), ${\mathcal {K}}_{{\mathcal{M}}}$ is accessible, so ${\mathcal{M}}$ is accessible. Using the preceding discussion, pick a regular uncountable cardinal $\lambda$ such both ${\mathcal {K}}$ and ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ are $\lambda$-accessible and almost well $\lambda$-filtrable. Let ${\mathcal{M}}_\lambda$ be the collection of morphisms in ${\mathcal{M}}$ whose domains and codomains are $\lambda$-presentable (in ${\mathcal {K}}$). We will show that for each infinite cardinal $\mu$, ${\mathcal{M}}_{\mu^+} \subseteq {\operatorname{cof}}({\mathcal{M}}_\lambda)$. We proceed by induction on $\mu$. When $\mu < \lambda$, this is trivial, so assume that $\mu \ge \lambda$. Note that, playing with pushouts, it is straightforward to check that the $\mu^+$-presentable objects in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ are exactly the morphisms of ${\mathcal{M}}_{\mu^+}$.
Every morphism $h$ in ${\mathcal{M}}_{\mu^+}$ must be a retract of a filtrable object in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$. Now, retracts in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ are retracts in ${\mathcal {K}}^2$, so since we are looking at ${\operatorname{cof}}({\mathcal{M}}_\lambda)$ it suffices to show that any morphism $h$ in ${\mathcal{M}}_{\mu^+}$ which *is* filtrable in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ is in ${\operatorname{cof}}({\mathcal{M}}_\lambda)$. So take such a morphism. Write $h = h_0:K_0\to L$. We will show that $h_0\in{\operatorname{cof}}({\mathcal{M}}_\lambda)$. Express $h_0$ as a colimit of a smooth chain of morphisms $t_{0i}\in{\operatorname{cof}}({\mathcal{M}}_\lambda)$, $i<{\text{cf} (\mu)}$, between $(<\mu^+)$-presentable objects in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$. $$\xymatrix@=3pc{
K_0 \ar@{}\ar[r]^{h_0} & L \\
K_{0i} \ar [u]^{k_{0i}} \ar [r]_{t_{0i}} &
L_{0i} \ar[u]_{l_{0i}}
}$$
Form a pushout $$\xymatrix@=3pc{
K_0 \ar@{}\ar[r]^{h_{01}} & K_1 \\
K_{00} \ar [u]^{k_{00}} \ar [r]_{t_{00}} &
L_{00} \ar[u]_{\bar{k}_{00}}
}$$ and take the induced morphism $h_1:K_1\to L$. Since the starting square is cellular, $h_1$ is in ${\mathcal{M}}$. Note also that $K_1$ is $\mu^+$-presentable. We have a commutative square $$\xymatrix@=3pc{
K_1 \ar@{}\ar[r]^{h_1} & L \\
K_{01} \ar [u]^{h_{01}k_{01}} \ar [r]_{t_{01}} &
L_{01} \ar[u]_{l_{01}}
}$$ because $h_1 h_{01} k_{01} = h_0 k_{01} = l_{01} t_{01}$. We can express $h_1$ as a colimit of a smooth chain of morphisms $t_{1i}\in{\operatorname{cof}}({\mathcal{M}}_\lambda)$, $1\leq i<{\text{cf} (\mu)}$, between $<\mu^+$-presentable objects in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ which are above $t_{01}$
$$\xymatrix@C=3pc@R=3pc{
K_1 \ar [r]^{h_1} & L \\
K_{1i} \ar[r]^{t_{1i}} \ar [u]^{k_{1i}} & L_{1i} \ar [u]_{l_{1i}}\\
K_{01} \ar [r]_{t_{01}} \ar [u]^{} & L_{01} \ar [u]_{}
}$$
Form a pushout $$\xymatrix@=3pc{
K_1 \ar@{}\ar[r]^{h_{12}} & K_2 \\
K_{11} \ar [u]^{k_{11}} \ar [r]_{t_{11}} &
L_{11} \ar[u]_{\bar{k}_{11}}
}$$ and take the induced morphisms $h_2:K_2\to L$. Again, by cellularity, $h_2$ is in ${\mathcal{M}}$. In $$K_0 \xrightarrow{\ h_{01}\ } K_1\xrightarrow{\ h_{12}\ } K_2\xrightarrow{\ h_2} L$$ we put $h_{02}=h_{12}h_{01}$ and continue transfinitely. This means that for $i<{\text{cf} (\mu)}$ we express $h_i$ as a colimit of a smooth chain of morphisms $t_{ij}\in{\operatorname{cof}}({\mathcal{M}}_\lambda)$, $i\leq j<{\text{cf} (\mu)}$, between $(<\mu^+)$-presentable objects in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ which are above $t_{0i}$
$$\xymatrix@C=3pc@R=3pc{
K_i \ar [r]^{h_i} & L \\
K_{i+1,j} \ar[r]^{t_{i+1,j}} \ar [u]^{k_{i+1,j}} & L_{i+1,j} \ar [u]_{l_{i+1,j}}\\
K_{0i} \ar [r]_{t_{0i}} \ar [u]^{} & L_{0i} \ar [u]_{}
}$$
Form a pushout $$\xymatrix@=3pc{
K_i \ar@{}\ar[r]^{h_{i,i+1}} & K_{i+1} \\
K_{i+1,i+1} \ar [u]^{k_{i+1,i+1}} \ar [r]_{t_{i+1,i+1}} &
L_{i,i+1} \ar[u]_{\bar{k}_{i+1,i+1}}
}$$ and take the induced morphisms $h_{i+1}:K_{i+1}\to L$. By cellularity, $h_{i+1}$ is in ${\mathcal{M}}$. We put $h_{k,i+1}=h_{i,i+1}h_{ik}$. At limit steps we take colimits. Then by construction $L=K_{{\text{cf} (\mu)}}$ and $h_0$ is the transfinite composition of $(h_{ij})_{i<j<{\text{cf} (\mu)}}$. We have just observed that each $h_{ij}$ is in ${\operatorname{cof}}({\mathcal{M}}_\lambda)$, so $h_0$ also is.
(\[cofib-gen-3\]) implies (\[cofib-gen-1\]): Assume that ${\mathcal{M}}$ is accessible and cofibrantly generated in ${\mathcal {K}}$. Let ${\mathcal{X}}$ be a subset of ${\mathcal{M}}$ so that ${\mathcal{M}}= {\operatorname{cof}}({\mathcal{X}})$. Let $\lambda$ be a big-enough uncountable regular cardinal such that ${\mathcal {K}}$ and ${\mathcal {K}}_{{\mathcal{M}}}$ are $\lambda$-accessible, and all the morphisms in ${\mathcal{X}}$ have $\lambda$-presentable domain and codomain. Note that, by coherence, for any regular $\mu \ge \lambda$, an object which is $\mu$-presentable in ${\mathcal {K}}$ is $\mu$-presentable in ${\mathcal {K}}_{{\mathcal{M}}}$. We claim that ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ is $\lambda$-accessible. First, ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ is closed under directed colimits in ${\mathcal {K}}_{{\mathcal{M}}}$ by Lemma \[dir-colim-lem\]. Now let ${\mathcal{M}}_\lambda$ be the class of morphisms in ${\mathcal{M}}$ with $\lambda$-presentable domain and codomain and let ${\mathcal{M}}^\ast$ be the class of morphisms in ${\mathcal{M}}$ that are $\lambda$-directed colimit (in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$) of morphisms in ${\mathcal{M}}_\lambda$. It suffices to see that ${\mathcal{M}}^\ast = {\mathcal{M}}$.
First, any pushout of a morphism in ${\mathcal{M}}_\lambda$ is in ${\mathcal{M}}^\ast$. Consider such a pushout $$\xymatrix@=3pc{
K \ar@{}\ar[r]^h{} & L \\
K_0 \ar [u]^{k_0} \ar [r]_{h_0} &
L_0 \ar[u]_{l_0}
}$$ where $K_0$ and $L_0$ are $\lambda$-presentable. Then $K$ is a $\lambda$-directed colimits of $\lambda$-presentable objects $K_i$ above $K_0$ in ${\mathcal {K}}_{{\mathcal{M}}}$. Consider pushouts $$\xymatrix@C=3pc@R=3pc{
K \ar [r]^{h} & L \\
K_i \ar[r]^{h_i} \ar [u]^{} & L_i \ar [u]_{}\\
K_0 \ar [r]_{h_0} \ar [u]^{} & L_0 \ar [u]_{}
}$$ It is easy to check that the $L_i$’s are also $\lambda$-presentable and that $h={\operatorname{colim}}h_i$ in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$. Thus $h \in {\mathcal{M}}^\ast$.
Second, ${\mathcal{M}}^\ast$ is closed under compositions of morphisms from ${\operatorname{Po}}_\lambda$ where ${\operatorname{Po}}_\lambda$ consists of pushouts of morphisms from ${\mathcal{M}}_\lambda$. Let $f:K\to L$ and $g:L\to M$ belong to ${\operatorname{Po}}_\lambda$. As above, $f$ is a $\lambda$-directed colimit (in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$), $(k_i,l_i):f_i\to f$ of $f_i\in{\mathcal{M}}_\lambda$, $f_i : K_i \to L_i$. Moreover, $g$ is a pushout of $g_0:L_0\to M_0$ having $L_0$ and $M_0$ both $\lambda$-presentable. Without loss of generality, we can assume that $L_0\to L$ factors through the $L_i$. We then take pushouts as above $$\xymatrix@C=3pc@R=3pc{
L \ar [r]^{g} & M \\
L_i \ar[r]^{g_i} \ar [u]^{} & M_i \ar [u]_{}\\
L_0 \ar [r]_{g_0} \ar [u]^{} & M_0 \ar [u]_{}
}$$ This shows that $gf$ is a $\lambda$-directed colimit of the $g_if_i$’s in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$.
Third, ${\mathcal{M}}^\ast$ is closed under transfinite compositions of morphisms from ${\operatorname{Po}}_\lambda$. Let $(f_{ij})_{i,j\leq \alpha}$ be such a transfinite composition. At limit steps, $f_{0i}$ is the following directed colimit in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$: $$\xymatrix@=3pc{
K_0 \ar@{}\ar[r]^{f_{0i}} & K_i \\
K_0 \ar [u]^{{\operatorname{id}}} \ar [r]_{f_{0j}} &
K_j \ar[u]_{f_{ji}}
}$$
This shows that $f_{0,i}$ is in ${\mathcal{M}}^\ast$ (we used [@indep-categ-advances], 3.12).
We have shown that any transfinite composition of pushouts from ${\mathcal{M}}_\lambda$ is in ${\mathcal{M}}^\ast$. That is, ${\operatorname{cell}}({\mathcal{M}}_\lambda) = {\operatorname{Tc}}({\operatorname{Po}}({\mathcal{M}}_\lambda)) \subseteq {\mathcal{M}}^\ast$. Since ${\mathcal{M}}$ is closed under pushouts, retracts, and transfinite compositions, ${\operatorname{cof}}({\mathcal{X}}) \cap {\mathcal {K}}_\lambda^2 \subseteq {\mathcal{M}}_\lambda$. By [@fat-small-obj], B1, it follows that ${\mathcal{M}}= {\operatorname{cof}}({\mathcal{X}}) = {\operatorname{cell}}({\mathcal{M}}_\lambda)$. We deduce that ${\mathcal{M}}= {\mathcal{M}}^\ast$, as desired.
1. On any locally presentable category ${\mathcal {K}}$, there are two trivial cellular structures – the *discrete* $({\mathcal {K}},{\operatorname{Iso}})$ and the *indiscrete* $({\mathcal {K}},{\mathcal {K}}^2)$. They are both combinatorial (see [@mr]), coherent and $\aleph_0$-continuous. The first one is not accessible because ${\mathcal {K}}_{{\operatorname{Iso}}}$ is not accessible (as long as ${\mathcal {K}}$ is not small, in any case). The second is accessible and yields a stable independence relation where every commutative square is independent.
2. On every locally presentable category ${\mathcal {K}}$, there is a cellular structure where ${\mathcal {M}}$ consists of regular monomorphisms. This cellular category is accessible, retract-closed and coherent. If ${\mathcal {K}}$ is locally finitely presentable, it is $\aleph_0$-continuous. Concrete examples include graphs with induced subgraph embeddings, groups, Banach spaces, boolean algebras, Hilbert spaces, and any Grothendieck topos. The last two are combinatorial, hence have a stable independence notion. See [@indep-categ-advances] for more details.
3. On every locally finitely presentable category ${\mathcal {K}}$, there is a cellular structure where ${\mathcal {M}}$ consists of pure monomorphisms. This cellular category is accesible, retract-closed, coherent and $\aleph_0$-continuous. When this cellular structure is combinatorial is discussed in [@purity-algebra] and [@lprv-purecofgen-v3]. For example, the latter shows that $({\mathcal {K}}, {\mathcal {M}})$ is combinatorial for any additive category ${\mathcal {K}}$.
Often, it is natural to look not at all objects, but just those objects $A$ so that $0 \to A$ is in ${\mathcal{M}}$ (where $0$ is an initial object):
\[fib-def\] Let $({\mathcal {K}},{\mathcal {M}})$ be a cellular category. An object $A$ is called *cellular* if $0\to A$ is cellular. Let ${\mathcal {C}}$ denote the full subcategory of ${\mathcal {K}}$ consisting of cellular objects.
\[fib-rmk\] Let ${\mathcal {M}}_0$ be the class of cellular morphisms with a cellular domain (then the codomain is cellular too). Then $({\mathcal {C}},{\mathcal {M}}_0)$ satisfies all properties of a cellular category up to cocompleteness of ${\mathcal {C}}$. Thus it induces a subcategory ${\mathcal {C}}_{{\mathcal {M}}_0}$ of ${\mathcal {C}}$ consisting of cellular objects and cellular morphisms.
If ${\mathcal {M}}$ is coherent, then every cellular morphism $A\to B$ with $B\in{\mathcal {C}}$ has $A\in{\mathcal {C}}$.
We have the following version of Theorem \[cofib-gen\] for cofibrant objects. Its advantage is that we do not need to assume that $({\mathcal {K}},{\mathcal {M}})$ itself is accessible: it suffices to have ${\mathcal {K}}$ accessible.
\[cofib-gen-cof\] Let ${\mathcal {K}}$ be a retract-closed, coherent and $\aleph_0$-continuous cellular category such that ${\mathcal {K}}$ is accessible. The following are equivalent:
1. ${\mathcal{C}}_{{\mathcal{M}}_0}$ has a stable independence notion.
2. ${\mathcal{M}}_0$-effective squares form a stable independence notion in ${\mathcal{C}}_{{\mathcal{M}}_0}$.
3. ${\mathcal{M}}_0$ is cofibrantly generated in ${\mathcal{C}}$.
Similar to the proof of Theorem \[cofib-gen\]. Following [@fat-small-obj] 5.2, (3) implies that ${\mathcal {C}}_{{\mathcal {M}}_0}$ is accessible.
In many cases, the cellular squares will be pullback squares:
Let $({\mathcal {K}},{\mathcal{M}})$ be a cellular category where every cellular morphism is a monomorphism. If:
1. A pullback of two morphisms in ${\mathcal{M}}$ is again in ${\mathcal{M}}$.
2. Every epimorphism in ${\mathcal{M}}$ is an isomorphism.
Then every cellular square is a pullback square.
Conversely, it is natural to ask whether every pullback square is cellular. When ${\mathcal{M}}$ is the class of regular monomorphisms, categories with this property are said to have *effective unions*, a condition isolated by Barr [@effective-unions]. The connections of this special case with stable independence were investigated in [@indep-categ-advances §5], where it was shown that having effective unions implies that effective squares form a stable independence notion. We show that the definition can be naturally parameterized by ${\mathcal{M}}$ (this was done already for pure morphisms in [@purity-algebra 2.2]), and the corresponding results generalized.
\[effective-unions-def\] We say that a cellular category $({\mathcal {K}},{\mathcal {M}})$ has *effective unions* if
1. The pullback of any two morphisms in ${\mathcal{M}}$ with common codomain exists and the projections are again in ${\mathcal{M}}$.
2. Any pullback square with morphisms in ${\mathcal{M}}$ is cellular.
\[effective-unions-thm\] Let $({\mathcal {K}},{\mathcal {M}})$ be a cellular category which is coherent, has effective unions, and with ${\mathcal {K}}$ accessible. Then $({\mathcal {K}}, {\mathcal{M}})$ is accessible if and only if cellular squares form a stable independence notion in ${\mathcal {K}}_{{\mathcal{M}}}$.
If there is a stable independence notion in ${\mathcal {K}}_{{\mathcal{M}}}$, then by Remark \[independence\](\[independence-3\]), $({\mathcal {K}}, {\mathcal{M}})$ is accessible. Let us prove the converse. Pick a regular cardinal $\lambda$ such that $({\mathcal {K}}, {\mathcal{M}})$ is $\lambda$-accessible. By Theorem \[weak-stable-thm\], cellular squares form a weakly stable independence notion and by Lemma \[dir-colim-lem\] this independence notion is $\lambda$-continuous. It remains to see that ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ is accessible. Consider an object $C \to D$ of ${\mathcal {K}}_{{\mathcal{M}},{\downarrow}}$. Since ${\mathcal{M}}$ is $\lambda$-accessible, $D$ can be written as a $\lambda$-directed colimit ${\langle D_i : i \in I \rangle}$ of $\lambda$-presentable objects. Let $C_i$ be the pullback of $C$ and $D_i$ over $D$. Then the resulting maps $C_i \to D_i$ form a $\lambda$-directed system. Since $\lambda$-directed colimits commute with finite limits (see [@adamek-rosicky 1.59], the pullback functor is accessible so must preserve arbitrarily large presentability ranks. Thus there is a bound on the presentability rank of $C_i$ that depends only on $\lambda$. This shows that ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ is accessible.
Note that, as opposed to Theorem \[cofib-gen\], we did *not* need to assume that $({\mathcal {K}}, {\mathcal{M}})$ was $\aleph_0$-continuous (nor that $({\mathcal {K}}, {\mathcal{M}})$ was retract-closed). However, a category may fail to have effective unions even if the effective squares form a stable independence notion (this is the case, for example, in locally finite graphs with regular monos, see [@indep-categ-advances 5.7]).
As a corollary, we obtain a quick proof that having effective unions implies cofibrant generation. This had been done “by hand” before for several special classes of morphisms [@beke-sheafifiable 1.12], [@purity-algebra 2.4].
If $({\mathcal {K}},{\mathcal {M}})$ is an accessible cellular category which is coherent, $\aleph_0$-continuous, and has effective unions, then it is combinatorial.
By Theorem \[effective-unions-thm\] cellular squares form a stable independence notion, so by Theorem \[cofib-gen\] (noting that retract-closedness is not used for this direction) $({\mathcal {K}}, {\mathcal{M}})$ is cofibrantly generated.
\[left-induced\] Let $F:{\mathcal {K}}\to{\mathcal {L}}$ be a colimit-preserving functor from a locally presentable categopry ${\mathcal {K}}$ to a combinatorial category ${\mathcal {L}}$. We get a cellular structure on ${\mathcal {K}}$ where $f$ is cellular if and only if $Ff$ is cellular. This cellular structure is called *left-induced* (see [@mr 3.8]). It was shown in [@mr], using a great deal of heavy machinery, that such left-induced cellular structures are combinatorial. With the aid of Theorem \[cofib-gen\], we obtain a special case of this result without any effort.
\[lazy\] Let $F:{\mathcal {K}}\to{\mathcal {L}}$ be a colimit preserving functor from a locally presentable category to a combinatorial category. If ${\mathcal {L}}$ is coherent and $\aleph_0$-continuous, then ${\mathcal {K}}$ is combinatorial.
Preimages of cellular squares are cellular and the left-induced cellular category ${\mathcal {K}}$ is clearly retract-closed, coherent and $\aleph_0$-continuous. We have a pseudopullback $$\xymatrix@=3pc{
{\mathcal {K}}^2 \ar[r]^{F^2} &{\mathcal {L}}^2 \\
{\mathcal {K}}_\downarrow \ar [u] \ar[r]_{F_\downarrow} & {\mathcal {L}}_\downarrow \ar[u]
}$$ Since a pseudopullback of accessible categories is accessible (see [@adamek-rosicky Ex. 2n]), ${\unionstick}$ in ${\mathcal {K}}$ is accessible. The now result follows from \[cofib-gen\].
Abstract elementary classes of roots of Ext
===========================================
Abstract elementary classes (or AECs) are a framework for abstract model theory introduced by Shelah [@sh88]. We will use the category-theoretic characterization of Beke-Rosický [@beke-rosicky]: they are accessible categories with directed colimits and with all morphisms monomorphisms which embed “nicely” into finitely accessible categories.
\[aec\] Let $({\mathcal {K}},{\mathcal {M}})$ be an accessible cellular category which is coherent and $\aleph_0$-continuous. Assume that ${\mathcal {K}}$ is finitely accessible and all morphisms in ${\mathcal{M}}$ are monomorphisms.
1. ${\mathcal {K}}_{{\mathcal {M}}}$ is an abstract elementary class.
2. If $({\mathcal {K}}, {\mathcal{M}})$ is combinatorial, then ${\mathcal {C}}_{{\mathcal {M}}_0}$ (see Definition \[fib-def\], Remark \[fib-rmk\]) is an abstract elementary class.
It is easy to verify that ${\mathcal {K}}_{\mathcal {M}}$ satisfies the conditions in [@beke-rosicky 5.7]. When $({\mathcal {K}}, {\mathcal{M}})$ is combinatorial, one can use [@fat-small-obj 5.2] to see that ${\mathcal {C}}_{{\mathcal {M}}_0}$ is an AEC as well.
In what follows, we will apply our main theorem to the AECs studied in [@bet]. For a fixed (associative and unital) ring $R$, let ${R\text{-}\operatorname{\bf {Mod}}}$ denote the category of (left) $R$-modules with homomorphisms. It is a locally finitely presentable category.
Given a class ${\mathcal {B}}$ of $R$-modules, we define its *Ext-orthogonality class*, ${{}^{\perp_\infty}{\mathcal {B}}}$, as follows:
$${{}^{\perp_\infty}{\mathcal {B}}}=\{A\,:\,\mbox{Ext}^i(A,N)=0\mbox{ for all } 1 \le i < \omega \text{ and all } N\in {\mathcal {B}}\}$$
Roughly speaking, ${{}^{\perp_\infty}{\mathcal {B}}}$ is the collection of $R$-modules that do not admit nontrivial extensions by modules in ${\mathcal {B}}$. For example, when ${\mathcal {B}}$ is the class of all pure injective modules, then ${{}^{\perp_\infty}{\mathcal {B}}}$ is exactly the class of flat modules (see [@enochs-jenda 5.3.22, 7.1.4]).
From now on, we assume that ${\mathcal {B}}$ is a class of pure injective modules. Let ${\mathcal {K}}:= {R\text{-}\operatorname{\bf {Mod}}}$, and let ${\mathcal{C}}$ be the full subcategory of ${R\text{-}\operatorname{\bf {Mod}}}$ with objects from ${{}^{\perp_\infty}{\mathcal {B}}}$. Let ${\mathcal{M}}$ be the class of monomorphisms (in ${R\text{-}\operatorname{\bf {Mod}}}$) whose cokernel is in ${{}^{\perp_\infty}{\mathcal {B}}}$. That is, a monomorphism $A \xrightarrow{f} B$ is in ${\mathcal{M}}$ if and only if $B / f[A]$ is in ${{}^{\perp_\infty}{\mathcal {B}}}$. Let ${\mathcal{M}}_0$ be the class of elements in ${\mathcal{M}}$ with domain and codomain in ${\mathcal{C}}$. Note that this coincides with the notation from Definition \[fib-def\], Remark \[fib-rmk\].
The category ${\mathcal{C}}_{{\mathcal{M}}_0}$ is studied from the point of view of model theory by Baldwin-Eklof-Trlifaj [@bet], where they prove it is an AEC. They ask (see [@bet 4.1(1)]) what one can say about tameness and stability in ${\mathcal{C}}_{{\mathcal{M}}_0}$ (see, for example, [@baldwinbook09] for the relevant definitions). We now show, using our main theorem and known facts, that ${\mathcal{C}}_{{\mathcal{M}}_0}$ has a stable independence notion, hence (by [@indep-categ-advances 8.16]) it will *always* be stable and tame.
\[ext-thm\] $({\mathcal {K}}, {\mathcal{M}})$ is a coherent, $\aleph_0$-continuous, and retract-closed cellular category. Moreover, ${\mathcal{C}}_{{\mathcal{M}}_0}$ is cofibrantly generated in ${\mathcal{C}}$. In particular, ${\mathcal{C}}_{{\mathcal{M}}_0}$ is an AEC with a stable independence notion.
The “in particular” part of the statement follows from Theorem \[cofib-gen-cof\] and Lemma \[aec\]. For the first sentence, following [@flat-covers-factorizations 4.2], $({\mathcal {K}}, {\mathcal{M}})$ is a retract-closed cellular category. The coherence was observed in [@bet 1.14] and $\aleph_0$-continuity in [@bet 1.6]. In fact, the latter follows from \[coherent-prop\](\[coherent-prop-2\]) because ${\mathcal {K}}_{{\mathcal{M}}}$ is closed under directed colimits in ${R\text{-}\operatorname{\bf {Mod}}}$ (as outlined in, for example, [@bet §1]). It remains to see that ${\mathcal{C}}_{{\mathcal{M}}_0}$ is cofibrantly generated in ${\mathcal{C}}$.
By [@flat-cover Proposition 2], [@covers-ext Theorem 8], ${\mathcal{C}}_{{\mathcal{M}}_0}$ has refinements. This means there exists a regular cardinal $\theta$ so that any object of ${\mathcal{C}}$ can be written as the union of an increasing smooth chain ${\langle A_i : i < \alpha \rangle}$ of submodules, with $A_0$ the zero module and for all $i < \alpha$, $A_{i + 1} / A_i$ in ${\mathcal{C}}$ and $\theta$-presentable.
By the proof of [@flat-covers-factorizations 4.5], ${\mathcal{M}}$ is cofibrantly generated by a set of maps $f$ so that $0 \to A \xrightarrow{f} F \to B \to 0$ is a short exact sequence, $F$ is a free module, and $B$ is a $\theta$-presentable object of ${\mathcal {L}}$. Since $F$ is free, $F \in {\mathcal{C}}$ as well, hence $A \in {\mathcal{C}}$. Thus $f \in {\mathcal{M}}_0$. Thus ${\mathcal{M}}$ is cofibrantly generated in ${\mathcal {K}}$ by a subset of ${\mathcal{M}}_0$, showing in particular that ${\mathcal{M}}_0$ is cofibrantly generated in ${\mathcal{C}}$.
Canonicity of stable independence
=================================
We prove here canonicity of stable independence without the hypothesis, present in [@indep-categ-advances 9.1], that all morphisms are monomorphisms. This does not depend on the rest of the paper. Our proof is a category-theoretic version of the argument in [@bgkv-apal] which shows somewhat more transparently what is going on there. The key notion is that of an independent sequence:
Let ${\mathcal {K}}$ be a category and let ${\unionstick}$ be an independence notion on ${\mathcal {K}}$. Let $f : M_0 \to M$ be a morphism in ${\mathcal {K}}$. An *${\unionstick}$-independent sequence for $f$* consists of a nonzero ordinal $\alpha$ and morphisms $(f_i)_{i \le \alpha}$ and $(g_{i, j})_{i \le j \le \alpha}$ such that for $i \le j \le k \le \alpha$:
- $f = f_0$ and $N_0=M$.
- $f_i : M \to N_i$ for $0<i$.
- $g_{i, j} : N_i \to N_j$.
- $g_{j,k} g_{i, j} = g_{i, k}$, $g_{i, i} = {\operatorname{id}}_{N_i}$.
- When $i < j$, the following square commutes and, when $j < \alpha$, is ${\unionstick}$-independent:
$$\xymatrix{
M \ar[r]^{f_j} & N_j \\
M_0 \ar[u]_{f_0} \ar[r]_{g_{0, i} f_0} & N_i \ar[u]_{g_{i, j}} \\
}$$
We call $\alpha$ the *length* of the sequence. For a regular cardinal $\lambda$, we say the independent sequence is *$\lambda$-smooth* if whenever ${\text{cf} (i)} \ge \lambda$, $N_i$ is the colimit of the system $(g_{j, k})_{j \le k < i}$. We say it is smooth if it is $\aleph_0$-smooth.
For example, an independent sequence of length one for $f: M_0 \to M$ consists of $f_0 = f$, $f_1 : M \to N_1$, $g_{0, 1} : M = N_0 \to N_1$ such that $f_1 f_0 = g_{0, 1} f_0$. Since there are no independence requirements, it is essentially just the morphism $f_0$ (the additional data is only relevant when $\alpha$ is limit; we could have taken $N_1 = N_0 = M, f_1 = {\operatorname{id}}_{M}$). More interestingly, an independent sequence of length two consists essentially (because $N_0 = M$ and $g_{0,0} f_0 = f_0$) of an independent square:
$$\xymatrix{
M \ar[r]^{f_1} & N_1 \\
M_0 \ar[u]_{f_0} \ar[r]_{f_0} & M \ar[u]_{g_{0, 1}} \\
}$$
Thus it consists of two “independent copies” of $M$.
An independent sequence of length three will look like:
$$\xymatrix{
& N_2 & & \\
& & N_1 \ar[ul]_{g_{1, 2}} & \\
M \ar[uur]_{f_2} & M \ar[ur]_{f_1} & & M \ar[ul]_{g_{0, 1}} \\
& & M_0 \ar[ull]_{f_0} \ar[ul]_{f_0} \ar[ur]_{f_0} &
}$$
where the inner diamond $(M_0, M, M, N_1)$ and the outer diamond $(M_0, M, N_1, N_2)$ is independent (in fact, if ${\unionstick}$ is monotonic, all commutative subsquares of the diagram will be independent). Essentially, the leftmost “copy” of $M$ is independent of the two rightmost copies (in fact it is independent of $N_1$).
Existence allows us to build independent sequences. Recall that a category ${\mathcal {K}}$ has *chain bounds* if any chain has a compatible cocone.
\[exist-indep\] If ${\mathcal {K}}$ has $\lambda$-directed colimits, chain bounds, and ${\unionstick}$ is a monotonic independence notion with existence, then for any morphism $f: M_0 \to M$ and any ordinal $\alpha$, there exists a $\lambda$-smooth independent sequence for $f$ of length $\alpha$. More generally, any independent sequence of length $\alpha_0 < \alpha$ extends to one of length $\alpha$ (in the natural sense).
By repeated use of existence.
The following local character lemma will be handy:
\[loc-character\] Let ${\mathcal {K}}$ be a category, ${\unionstick}$ an independence relation such that ${\mathcal {K}}_{{\downarrow}}$ is a $\lambda$-accessible category. Let $(M_i \to N_i)_{i < \lambda^+}$ be a system of $\lambda^+$-presentable objects in ${\mathcal {K}}^2$ with colimit $M \to N$. Then there exists $i < \lambda^+$ such that the square
$$\xymatrix{
N_i \ar[r] & N \\
M_i \ar[u] \ar[r] & M \ar[u] \\
}$$
is independent.
Write $I$ for $\lambda^+$ with the usual ordering. By taking colimits at ordinals of cofinality $\lambda$ and adding them to the system, we can assume without loss of generality that the system is $\lambda$-smooth: for any $i \in I$ of cofinality $\lambda$, $M_i$ is the colimit of $(M_{i_0})_{i_0 < i}$.
Let $(M_j' \to N_j')_{j \in J}$ be a $\lambda^+$-directed system of $\lambda^+$-presentable objects whose colimit in ${\mathcal {K}}_{{\downarrow}}$ is $M \to N$; we know that ${\mathcal {K}}_{{\downarrow}}$ is $\lambda^+$-accessible. We build $(i_\alpha, j_\alpha)_{\alpha < \lambda}$ such that for all $\alpha < \lambda$:
1. $i_\alpha \in I$, $j_\alpha \in J$.
2. $i_\alpha < i_{\alpha + 1}$.
3. The map from $M_{i_\alpha} \to N_{i_\alpha}$ to $M \to N$ factors through $M_{j_\alpha}' \to N_{j_\alpha}'$.
4. The map from $M_{j_\alpha}' \to N_{j_\alpha}'$ to $M \to N$ factors through $M_{i_{\alpha + 1}} \to N_{i_{\alpha + 1}}$.
This is possible since $I$ and $J$ are $\lambda^+$-directed and $M_i \to N_i$, $M_j' \to N_j'$ are $\lambda^+$-presentable. Now, let $i := \sup_{\alpha < \lambda} i_\alpha$. The colimit in ${\mathcal {K}}^2$ of $(M_{i_\alpha} \to N_{i_\alpha})_{\alpha < \lambda}$ and $(M_{j_\alpha}' \to N_{j_\alpha}')_{\alpha < \lambda}$ coincide and by $\lambda$-smoothness must be $M_i \to N_i$. By assumption, for all $\alpha < \lambda$, the square
$$\xymatrix{
N_{j_\alpha}' \ar[r] & N \\
M_{j_\alpha}' \ar[u] \ar[r] & M \ar[u] \\
}$$
is independent. Since ${\mathcal {K}}_{{\downarrow}}$ has $\lambda$-directed colimits, this means that the square
$$\xymatrix{
N_i \ar[r] & N \\
M_i \ar[u] \ar[r] & M \ar[u] \\
}$$
is also independent.
A much simpler result than the canonicity theorem is:
\[one-dir-lem\] Assume ${\mathcal {K}}$ is a category, ${\unionstick}^1$, ${\unionstick}^2$ are independence notions such that ${\unionstick}^1 \subseteq {\unionstick}^2$, ${\unionstick}^1$ has existence, and ${\unionstick}^2$ has uniqueness. Then ${\unionstick}^1 = {\unionstick}^2$.
Given a square $M_0, M_1, M_2, M_3$ that is ${\unionstick}^2$-independent, use existence for ${\unionstick}^1$ to ${\unionstick}^1$-amalgamate the span $M_0 \to M_1$, $M_0 \to M_2$, giving maps $M_1 \to M_3'$, $M_2 \to M_3'$. Now by uniqueness for ${\unionstick}^2$, the amalgam involving $M_3$ and the one involving $M_3'$ must be equivalent, hence $M_0, M_1, M_2, M_3$ is also ${\unionstick}^1$-independent.
We can now prove the canonicity theorem. The idea is to use a generalization of the fact that, in a vector space, if $I$ is linearly independent and $a$ is a vector, there exists a finite subset $I_0 \subseteq I$ such that $(I - I_0) \cup \{a\}$ is independent. Thus we can remove a small subset of $I$ and get something independent.
\[canon-key-lem\] Assume ${\mathcal {K}}$ has chain bounds, and ${\unionstick}^1$, ${\unionstick}^2$ are independence notions with existence such that:
1. ${\unionstick}^1$ is right monotonic.
2. ${\unionstick}^2$ is transitive, left monotonic, and right accessible.
Then any span has an amalgam that is both ${\unionstick}^1$-independent and ${\unionstick}^2$-independent. In particular, if ${\unionstick}^1$ has uniqueness then ${\unionstick}^1 \subseteq {\unionstick}^2$.
Consider a span $M_0 \xrightarrow[f_0]{} M$, $M_0 \xrightarrow[f_0']{} M'$. Fix a regular cardinal $\lambda$ such that ${\mathcal {K}}_{{\downarrow}^2}$ (the arrow category induced by ${\unionstick}^2$) is $\lambda$-accessible and $M_0, M, M', f_0, f_0'$ are $\lambda$-presentable in all relevant categories.
Using Lemma \[exist-indep\], build a $({\unionstick}^2)^d$-independent sequence for $f_0$, $(f_i : M \to N_i)_{i \le \lambda^+}$, $(g_{i, j}: N_i \to N_j)_{i \le j \le \lambda^+}$, where $N_{\lambda^+}$ is the colimit of $(N_i)_{i < \lambda^+}$. Observe that $$f_{\lambda^+}f_0=g_{0,\lambda^+}f_0.$$ Along the way, we ensure that $N_i$ is $\lambda^+$-presentable for $i < \lambda^+$. Now ${\unionstick}^1$-amalgamate the span $M_0 \to N_{\lambda^+}$, $M_0 \to M'$, giving an ${\unionstick}^1$-independent square:
$$\xymatrix{
M' \ar[r]^{h'} & N_{\lambda^+}' \\
M_0 \ar[u]^{f_0'} \ar[r]_{g_{0,\lambda^+}f_0} & N_{\lambda^+} \ar[u]_{h} \\
}$$
with $N_{\lambda^+}'$ a $\lambda^{++}$-presentable object. Reworking the proof of [@rosicky-sat-jsl Lemma 1]—which requires directed colimits—to use the chain bounds available to us here, we can write $N_{\lambda^+}'$ as a colimit of $\lambda^+$-presentables $(g_{i,j}':N_i'\to N_{j}')_{i\leq j < \lambda^+}$, where:
1. There is an arrow $h_i:N_i \to N_i'$ for each $i < \lambda^+$.
2. The $N_i'$ lie above $M'$, in the sense that $h':M'\to N_{\lambda^+}'$ factors as $$M'\stackrel{u_i}{\longrightarrow}N_i'\stackrel{g_{i,\lambda^+}'}{\longrightarrow}N_{\lambda^+}'$$ and, moreover, that the morphisms $h'f_0'=hg_{0,\lambda^+}f_0:M_0\to N_{\lambda^+}'$ factor identically through $g_{i,\lambda^+}'$, i.e. $$h_if_if_0=u_if_0'.$$ Here we use $\lambda$-presentability of $M_0$, $M'$, and $\lambda^+$-directedness of the chain.
Then $h$ is a colimit of the $h_i$ in ${\mathcal {K}}^2$ and by Lemma \[loc-character\], there exists $i < \lambda^+$ such that the square
$$\xymatrix{
N_i' \ar[r]^{g_{i,\lambda^+}'} & N_{\lambda^+}' \\
N_i \ar[u]^{h_i} \ar[r]_{g_{i,\lambda^+}} & N_{\lambda^+} \ar[u]_{h} \\
}$$
is ${\unionstick}^2$-independent. By definition of an $({\unionstick}^2)^d$-independent sequence, the square
$$\xymatrix{
N_i \ar[r]^{g_{i,\lambda^+}} & N_{\lambda^+} \\
M_0 \ar[u]^{f_if_0} \ar[r]_{f_0} & M \ar[u]_{f_{\lambda^+}} \\
}$$
is ${\unionstick}^2$-independent. By left transitivity, we obtain that the following is ${\unionstick}^2$-independent.
$$\xymatrix{
N_i' \ar[r]^{g_{i,\lambda^+}'} & N_{\lambda^+}' \\
M_0 \ar[u]^{h_if_if_0} \ar[r]_{f_0} & M \ar[u]_{hf_{\lambda^+}} \\
}$$
A chase through the diagrams above reveals that $$g_{i,\lambda^+}'h_if_if_0=h'f_0'=hg_{0,\lambda^+}f_0=hf_{\lambda^+}f_0,$$ meaning that the outer square and the large upper triangle in the following diagram commute:
$$\xymatrix{
N_i' \ar[rr]^{g_{i,\lambda^+}'} & & N_{\lambda^+}' \\
& M'\ar[ur]_{h'}\ar[ul]^{u_i} & \\
M_0 \ar[uu]^{h_if_if_0}\ar[ur]_{f_0'} \ar[rr]_{f_0} & & M \ar[uu]_{hf_{\lambda^+}} \\
}$$
Thus the square $$\xymatrix{
N_i' \ar[r]^{g_{i,\lambda^+}'} & N_{\lambda^+}' \\
M_0 \ar[u]^{u_if'_0} \ar[r]_{f_0} & M \ar[u]_{hf_{\lambda^+}} \\
}$$ is ${\unionstick}^2$-independent.
By left monotonicity for ${\unionstick}^2$, then, the following is also ${\unionstick}^2$-independent:
$$\xymatrix{
M' \ar[r]^{h'} & N_{\lambda^+}' \\
M_0 \ar[u]^{f_0'} \ar[r]_{f_0} & M \ar[u]_{hf_{\lambda^+}} \\
}$$
Note, however, that the morphism from $M$ to $N_{\lambda^+}'$ in the diagram above is not the same as the one in the ${\unionstick}^1$-amalgam of $M_0 \to N_{\lambda^+}$, $M_0 \to M'$. In fact, we have a diagram of the form:
$$\xymatrix{
M' \ar[rr]^{h'} & & N_{\lambda^+}' \\
M_0 \ar[u]^{f_0'} \ar[r]^{f_0} \ar[rd]_{f_0} & M \ar[r]^{g_0,\lambda^+} & N_{\lambda^+} \ar[u]_{h} \\
& M \ar[ru]_{f_{\lambda^+}} & \\
}$$
where the upper rectangle is ${\unionstick}^1$-independent and the outer “square” $(f_0', f_0, h f_{\lambda^+}, h')$ is ${\unionstick}^2$-independent. By right monotonicity for ${\unionstick}^1$, we get that $(f_0', f_0, h f_{\lambda^+}, h')$ is also ${\unionstick}^1$-independent. Thus it is the desired amalgam of $f_0', f_0$.
\[canon-thm\] Assume ${\mathcal {K}}$ has chain bounds, and ${\unionstick}^1$, ${\unionstick}^2$ are independence notions with existence and uniqueness such that:
1. ${\unionstick}^1$ is right monotonic.
2. ${\unionstick}^2$ is transitive and right accessible.
Then ${\unionstick}^1 = {\unionstick}^2$. In particular, ${\mathcal {K}}$ has at most one stable independence notion.
Combine Lemmas \[one-dir-lem\] and \[canon-key-lem\]. Note that right monotonicity for ${\unionstick}^2$ follows from existence, uniqueness, and transitivity [@indep-categ-advances 3.20].
Assume ${\mathcal {K}}$ has chain bounds. If ${\unionstick}$ is a transitive and right accessible independence notion with existence and uniqueness, then ${\unionstick}$ is a stable independence notion. In particular, it is symmetric.
It suffices to see that ${\unionstick}= {\unionstick}^d$. For this, apply Theorem \[canon-thm\] with ${\unionstick}^1 = {\unionstick}$ and ${\unionstick}^2 = {\unionstick}^d$ (again, ${\unionstick}^2$ is right monotonic by [@indep-categ-advances 3.20]).
Instead of chain bounds, it suffices to be able to build the appropriate independent sequences. See [@indep-categ-advances 9.6].
[^1]: We occasionally economize by not explicitly naming the morphisms involved when there is no danger of confusion.
|
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author:
- |
H. B. Benaoum\
Institut für Physik , Theoretische Elementarteilchenphysik,\
Johannes Gutenberg–Universität, 55099 Mainz, Germany\
email : [email protected]
date: 30 March 2000
title: '**On noncommutative and commutative equivalence for BFYM theory : Seiberg–Witten map** '
---
0.0in -0.0in -0.2in 6.4in 9.0in plus 1pt
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\
[**MZ–TH/00–12** ]{} \
\
It is by now clear that our naive picture on space–time, as space with commuting coordinates, has to be modified at the Planck scale. Indeed a unification between quantum mechanics and general relativity requires to go beyond this picture.\
Recently it has been found that noncommutative geometry [@con] arises naturally in string theory [@dou; @sei]. In a certain limit, the low energy effective theories is intimately connected to gauge theories on noncommutative spaces.\
The action for gauge theories on noncommutative space is obtained from the ordinary ( commutative ) Yang–Mills theories by replacing the usual product of the fields by the $\ast$–product [@sei].\
It has also been shown in [@sei], that one can get ordinary or noncommutative Yang–Mills theory from the same two dimensional field theory just by using two different regularizations procedure. Indeed, Yang–Mills theory on noncommutative space results from the use of a point splitting regularization whereas the ordinary one follows as a consequence of the use of Pauli–Villars regularization scheme.\
Consequently, they argued that there should exist a transformation from ordinary to noncommutative Yang–Mills which maps the noncommutative field $\hat{A}$ with gauge transformation $\hat{\lambda}$ to the ordinary gauge field $A$ with gauge transformation $\lambda$. More details about this equivalence between the two descriptions can be found in [@sei] ( see also [@ish] –[@wes] ).\
Perturbative aspects of noncommutative theories on noncommutative ${\mathbb{R}}^d$ have been recently adressed in [@mar]–[@min]. In particular, it has been found that noncommutativity leads to a strange mixing between IR and UV effects [@min] which has no analog in the usual quantum field theory.\
In [@ben], we have introduced the $U(1)$ BF–Yang Mills ( BFYM ) on noncommutative ${\mathbb{R}}^4$ and in this formulation the $U(1)$ Yang–Mills theory on noncommutative ${\mathbb{R}}^4$ is seen as a deformation of the pure BF theory. We recall that BFYM on commutative ${\mathbb{R}}^4$ has been introduced and studied extensively in [@fuc].\
We have also performed a one–loop calculation of the $U(1)$ BFYM on noncommutative ${\mathbb{R}}^4$ and concluded that it is asymtotically free and its UV–behaviour in the computation of the $\beta$–function is like the usual $SU( N )$ commutative BFYM and Yang–Mills theories.\
Here we will go a step further and study the equivalence between the noncommutative and commutative for BFYM. We consider first rank $N$ noncommutative and ordinary BFYM and construct the gauge–equivalent Seiberg–Witten transformation. This means that ordinary and noncommutative BFYM theories can be induced by the same field theory regularized in two different ways.\
In particular for rank 1, this mapping is more simple and for this case, we write the BFYM noncommutative Lagrangian in terms of the ordinary fields.\
\
Noncommutative ${\mathbb{R}}^4$ is described by an algebra generated by the coordinates $x^i~( i = 1, \cdots 4 )$ obeying the commutation relations : $$\begin{aligned}
x^i \ast x^j - x^j \ast x^i & = & i~\theta^{i j} \end{aligned}$$ where $\theta^{i j} = - \theta^{j i}$ is a real and constant antisymmetric matrix. In the dual language, the algebra of functions on ${\mathbb{R}}^4$ vanishing at infinity is equipped with a deformed multiplication known as a Moyal $\ast$–product, $$\begin{aligned}
( f \ast g ) ( x ) & = & e^{\frac{i}{2}~\theta^{i j}
\frac{\partial}{\partial \xi^{i}} \frac{\partial}{\partial \zeta^{j}}}~
f( x + \xi ) g( x + \zeta) |_{\xi = \zeta = 0} \end{aligned}$$ which is noncommutative, associative and obeys the Leibniz rule. It reduces for the first order in $\theta$ to, $$\begin{aligned}
( f \ast g ) ( x ) & = & f g + \frac{i}{2}~\theta^{i j}~\partial_i f
\partial_j g + O\left( \theta^2 \right) \end{aligned}$$ \
We consider the BFYM on noncommutative ${\mathbb{R}}^4$ introduced in [@ben] by replacing the usual product of fields by the $\ast$–product, $$\begin{aligned}
\hat{S} & = & \int d^4 x~Tr \left( \frac{i}{2}~ \epsilon^{i j k l}
\hat{B}_{i j} \ast \hat{F}_{k l} + g^2 \hat{B}_{i j} \ast \hat{B}^{i j}
\right) ( x ).\end{aligned}$$ where $\hat{F}_{i j} = \partial_i \hat{A}_j - \partial_j \hat{A}_i +
\hat{A}_i \ast \hat{A}_j - \hat{A}_j \ast \hat{A}_i$ is the field strength of the antihermitian gauge field $\hat{A}_i$ and $\hat{B}_{i j}$ is an antisymmetric tensor. We take here the gauge fields $\hat{A}_i$ and $\hat{B}_{i j}$ to be arbitrary rank $N$, so that all the fields and gauge parameters are $N \times N$ matrices and $Tr$ is the ordinary trace of $N \times N$ matrices.\
This BFYM on noncommutative ${\mathbb{R}}^4$ is invariant under the gauge transformation, $$\begin{aligned}
\hat{\delta}_{\hat{\lambda}} \hat{A}_i & = & \partial_i \hat{\lambda} +
\hat{\lambda} \ast \hat{A}_i - \hat{A}_i \ast \hat{\lambda} \nonumber \\
\hat{\delta}_{\hat{\lambda}} \hat{B}_{i j} & = & \hat{B}_{i j} \ast
\hat{\lambda} - \hat{\lambda} \ast \hat{B}_{i j} \nonumber \\
\hat{\delta}_{\hat{\lambda}} \hat{F}_{i j} & = & \hat{F}_{i j} \ast
\hat{\lambda} - \hat{\lambda} \ast \hat{F}_{i j} \end{aligned}$$ Using the first order expansion in $\theta$ of the Moyal $\ast$–product ( 3), the above formulas for noncommutative gauge field $\hat{A}_i$ and $\hat{B}_{i j}$ read, $$\begin{aligned}
\hat{\delta}_{\hat{\lambda}} \hat{A}_i & = & \partial_i \hat{\lambda} +
[ \hat{\lambda}, \hat{A}_i ] +
\frac{i}{2}~\theta^{k l}~\{ \partial_k \hat{\lambda},
\partial_l \hat{A}_i \} + O \left( \theta^2 \right) \nonumber \\
\hat{\delta}_{\hat{\lambda}} \hat{B}_{i j} & = & [ \hat{B}_{i j}, \hat{\lambda} ]
+ \frac{i}{2}~\theta^{k l}~\{ \hat{B}_{i j}, \hat{\lambda} \} +
O \left( \theta^2 \right) \end{aligned}$$ where $[~,~]$ and $\{~,~\}$ are the usual commutator and anticommutator for noncommutative fields and gauge parameter $\hat{\lambda}$ respectively.\
Similarly, the BFYM on commutative ${\mathbb{R}}^4$ is, $$\begin{aligned}
S & = & \int d^4 x~Tr \left( \frac{i}{2}~ \epsilon^{i j k l}
B_{i j} F_{k l} + g^2 B_{i j} B^{i j}
\right) ( x ).\end{aligned}$$ Its infinitesimal gauge transformations for the different fields are of the form, $$\begin{aligned}
\delta_{\lambda} A_i & = & \partial_i \lambda + [ \lambda, A_i ] \nonumber \\
\delta_{\lambda} B_{i j} & = & [ B_{i j}, \lambda ] \nonumber \\
\delta_{\lambda} F_{i j} & = & [ F_{i j}, \lambda ]\end{aligned}$$ \
Following Seiberg and Witten [@sei], we consider the BFYM and look for a gauge–equivalent mapping between the noncommutative fields $\hat{A}_i$ and $\hat{B}_{i j}$ with gauge parameter $\hat{\lambda}$ and the ordinary fields $A_i$ and $B_{i j}$ with gauge parameter $\lambda$ such that the following diagrams are commutative :
(400,50) (40,20)(120,20) (30,-30)(30,10) (40,-40)(120,-40) (140,-30)(140,10) (80,-10)(15,-30, 60) (20,-5)\[\][$\hat{\delta}_{\hat{\lambda}}$]{} (150,-5)\[\][$\delta_{\lambda}$]{} (20,20)\[\][$\hat{\delta}_{\lambda} \hat{A}_i$]{} (30,-40)\[\][$\hat{A}_i$]{} (140,-40)\[\][$A_i$]{} (140,20)\[\][$\delta_{\lambda} A_i$]{} (220,20)(300,20) (210,-30)(210,10) (220,-40)(300,-40) (320,-30)(320,10) (270,-10)(15,-30,60) (200,-5)\[\][$\hat{\delta}_{\hat{\lambda}}$]{} (330,-5)\[\][$\delta_{\lambda}$]{} (200,20)\[\][$\hat{\delta}_{\hat{\lambda}} \hat{B}_{i j}$]{} (210,-40)\[\][$\hat{B}_{i j}$]{} (320,-40)\[\][$B_{i j}$]{} (320,20)\[\][$\delta_{\lambda} B_{i j}$]{}
\
\
\
\
This means that the two following relations hold : $$\begin{aligned}
\hat{A}_i ( A ) + \hat{\delta}_{\hat{\lambda}} \hat{A}_i ( A ) & = &
\hat{A}_i ( A + \delta_{\lambda} A ) \nonumber \\
\hat{B}_{i j} ( B ) + \hat{\delta}_{\hat{\lambda}} \hat{B}_{i j} ( B ) & = &
\hat{B}_{i j} ( B + \delta_{\lambda} B ) \end{aligned}$$ By writing $\hat{A} = A + A' ( A ) + O \left( \theta^2 \right),
\hat{\lambda} ( \lambda , A ) = \lambda + \lambda' ( \lambda, A) +
O \left( \theta^2 \right),
\hat{B} = B + B' ( B, A ) + O \left( \theta^2 \right)$ and expanding (9 ) to the first order in $\theta$, we get for gauge field, $$\begin{aligned}
A'_i ( A + \delta A) - A'_i ( A ) - \partial_i \lambda' -
[ \lambda', A_i ] - [ \lambda, A'_i ] & = &
- \frac{i}{2}~\theta^{k l} \{ \partial_k \lambda, \partial_l A_i \} +
O \left( \theta^2 \right) \end{aligned}$$ and for antisymmetric field, $$\begin{aligned}
B'_{i j} ( B + \delta_{\lambda} B ) - B_{i j} ( B ) - [ B_{i j}, \lambda' ] -
[ B'_{i j}, \lambda ] & = &
- \frac{i}{2}~\theta^{k l} \{ \partial_k B_{i j}, \partial_l \lambda \} +
O \left( \theta^2 \right). \end{aligned}$$ By solving (10) and (11), the following solutions for the noncommutative gauge field $\hat{A}$, gauge parameter $\hat{\lambda}$ and antisymmetric field $\hat{B}$ are obtained : $$\begin{aligned}
\hat{A}_i ( A ) & = & A_i + A'_i ( A ) = A_i - \frac{i}{4}~\theta^{k l}~
\{ A_k , \partial_l A_i + F_{l i} \} + O \left( \theta^2 \right) \nonumber \\
\hat{\lambda} ( \lambda, A ) & = & \lambda + \lambda' ( \lambda, A ) =
\lambda + \frac{i}{4}~\theta^{k l} \{ \partial_k \lambda, A_l \} +
O \left( \theta^2 \right) \nonumber \\
\hat{B}_{i j} ( B, A ) & = & B_{i j} + B'_{i j} ( B, A ) \nonumber \\
& = & B_{i j} + \frac{i}{4}~\theta^{k l}~\left(~2~\{ B_{i k}, B_{j l} \}
- \{ A_k, D_l B_{i j} + \partial_l B_{i j} \} \right) +
O \left( \theta^2 \right) .\end{aligned}$$ where $D_l B_{i j} = \partial_l B_{i j} + [ A_l, B_{i j} ]$.\
From the gauge field $\hat{A}_i$, the noncommutative field strength $\hat{F}_{i j}$ is given as : $$\begin{aligned}
\hat{F}_{i j} & = & F_{i j} + \frac{i}{4}~\theta^{k l}~
\left(~2~\{ F_{i k}, F_{j l} \} -
\{ A_k, D_l F_{i j} + \partial_l F_{i j} \} \right) +
O \left( \theta^2 \right) . \end{aligned}$$ This means that knowing the commutative fields $A_i, B_{i j}$ and $F_{i j}$ with gauge parameter $\lambda$, the noncommutative one are expressed by (12) and (13) up to first order in $\theta$.\
Now it is easy to extract the differential equation for noncommutative fields $\hat{A}, \hat{B}$ and $\hat{F}$ with gauge parameter $\hat{\lambda}$ corresponding to two infinitesimally close values of the deformation parameter $\theta$ and $\theta + \delta \theta$.\
They are given as follows : $$\begin{aligned}
\delta \hat{A}_i & = & - \frac{i}{4}~\delta \theta^{k l}~[~
\hat{A}_k \ast ( \partial_l \hat{A}_i + \hat{F}_{l i} ) +
( \partial_l \hat{A}_i + \hat{F}_{l i} ) \ast \hat{A}_k ~] \nonumber \\
\delta \hat{\lambda } & = & \frac{i}{4}~\delta \theta^{k l}~(
\partial_k \hat{\lambda} \ast \hat{A}_l + \hat{A}_l \ast
\partial_k \hat{\lambda} ) \nonumber \\
\delta \hat{B}_{i j} & = & \frac{i}{4}~\delta \theta^{k l}~[~
2 \hat{B}_{i k} \ast \hat{B}_{j l} + 2 \hat{B}_{j l} \ast \hat{B}_{i k} -
\hat{A}_k \ast ( \hat{D}_l \hat{B}_{i j} + \partial_l \hat{B}_{i j} )
\nonumber \\
& & - ( \hat{D}_l \hat{B}_{i j} + \partial_l \hat{B}_{i j} ) \ast \hat{A}_k~]
\nonumber \\
\delta \hat{F}_{i j} & = & \frac{i}{4}~\delta \theta^{k l}~[~
2 \hat{F}_{i k} \ast \hat{F}_{j l} + 2 \hat{F}_{j l} \ast \hat{F}_{i k} -
\hat{A}_k \ast ( \hat{D}_l \hat{F}_{i j} + \partial_l \hat{F}_{i j} )
\nonumber \\
& & - ( \hat{D}_l \hat{F}_{i j} + \partial_l \hat{F}_{i j} ) \ast \hat{A}_k~] \end{aligned}$$ where $\hat{D}_l \hat{B}_{i j} = \partial_l \hat{B}_{i j} +
\hat{A}_l \ast \hat{B}_{i j} - \hat{B}_{i j} \ast \hat{A}_l$.\
The above equations should in principle give the noncommutative fields and gauge parameter as powers series in $\theta$.\
\
For rank 1, i.e. $U( 1)$, the Seiberg–Witten map (12) for different fields reduces to : $$\begin{aligned}
\hat{A}_i & = & A_i - \frac{i}{2} \theta^{k l} A_k ( \partial_l A_i +
F_{l i} ) + O \left( \theta^2 \right) \nonumber \\
\hat{\lambda} & = & \lambda + \frac{i}{2} \theta^{k l} \partial_k \lambda
A_l + O \left( \theta^2 \right) \nonumber \\
\hat{B}_{i j} & = & B_{i j} + i \theta^{k l} ( B_{i k} B_{j l} - A_k
\partial_l B_{i j} ) + O \left( \theta^2 \right) \nonumber \\
\hat{F}_{i j} & = & F_{i j} + i \theta^{k l} ( F_{i k} F_{j l} - A_k
\partial_l F_{i j} ) + O \left( \theta^2 \right) \end{aligned}$$ \
The noncommutative action $\hat{S}$ in terms of the fields $\hat{A}, \hat{B}$ and $\hat{F}$ and the commutative one $S$ in terms of the fields $A, B$ and $F$ for BFYM are related as : $$\begin{aligned}
\hat{S} & = & \int d^4 x~ \hat{ \cal L} (\hat{B}, \hat{F} ) \nonumber \\
& = & \int d^4 x~ \{~\frac{i}{2}~\epsilon^{i j k l}
B_{i j} F_{k l} + g^2 B_{i j}
B^{i j} - \frac{1}{2}~\theta^{n m}~\epsilon^{i j k l} F_{k l} B_{i n}
B_{j m} \nonumber \\
& & + \frac{1}{2}~\theta^{n m}~\epsilon^{i j k l} B_{i j} F_{n m}
F_{k l} - \frac{1}{2}~\theta^{n m}~\epsilon^{i j k l} B_{i j} F_{k n}
F_{l m} + 2 i g^2 \theta^{n m} B^{i j} B_{i n} B_{j m} \nonumber \\
& & - i g^2 \theta^{n m} B^{i j} F_{n m} B_{i j}
+ total~derivative~+ O\left( \theta^2 \right)~ \} \nonumber \\
& = & \int d^4 x~ \frac{1}{1 + i F \theta}~\{~\frac{i}{2}~\epsilon^{i j k l}
\left( \frac{1}{1 + i B \theta}~B \right)_{i j}~\left(
\frac{1}{1 + i F \theta}~F \right)_{k l} \nonumber \\
& & + g^2~\left( \frac{1}{1 + i B \theta}~B \right)_{i j}~
\left( \frac{1}{1 + i B \theta}~B \right)^{i j} +
total~derivative~+ O\left( \theta^2 \right)~ \} \end{aligned}$$ The $U(1)$ BFYM theory on noncommutative ${\mathbb{R}}^4$ with noncommutative fields $\hat{B}, \hat{F}$ written in terms of the ordinary fields as in (16) is equivalent to $U(1)$ BFYM on commutative ${\mathbb{R}}^4$ with antisymmetric field $\frac{1}{1 + i B \theta}~B$ and strength field $\frac{1}{1 + i F \theta}
~F$ where the factor $\frac{1}{1 + i F \theta}$ results from the partial derivation on the fields $B$ and $F$ in (15).\
The solutions, $$\begin{aligned}
\hat{F}~=~\frac{1}{1 + i F \theta}~F~,~~~
\hat{B} = \frac{1}{1 + i B \theta}~B.\end{aligned}$$ follows also from the resolution of the differential equation (14) for constant fields $\hat{B}, \hat{F}$.\
For this case, (16) can be written as : $$\begin{aligned}
\hat{S} & = & \int d^4 x~\hat{ \cal L} ( \hat{B}, \hat{F} ) \nonumber \\
& = & \int d^4 x~\{~{\cal L} ( \frac{1}{1 + i B \theta}~B,
\frac{1}{1 + i F \theta}~F ) + O \left( \theta^2 \right)~\}.\end{aligned}$$ \
We close this paper by saying that the existence of a Seibeg–Witten gauge–equivalent map, relating noncommutative fields to commutative one for BFYM theory, means that noncommutative or ordinary BFYM may be seen as an effective theory which arises from the same field theory regularized with two different schemes.
Acknowledgment {#acknowledgment .unnumbered}
==============
I would like to thank the DAAD for its financial support and Y. Okawa for his comments and remarks.
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|
---
abstract: |
[@mcfadden-richter] and later [@McFadden05] show that the *Axiom of Revealed Stochastic Preference* characterizes rationalizability of choice probabilities through random utility models on finite universal choice spaces. This note proves the result in one short, elementary paragraph and extends it to set valued choice. The latter requires a different axiom than is reported in [@McFadden05].
**Keywords:** Revealed Stochastic Preference; Random Utility.
**JEL Code:** D11.
author:
- 'Jörg Stoye[^1]'
bibliography:
- 'KMS.bib'
date: 'January 16th, 2019'
title: |
Revealed Stochastic Preference:\
A One-Paragraph Proof and Generalization
---
Introduction
============
[@mcfadden-richter] and [@McFadden05] show that, for a finite choice universe and singleton-valued choice, the *Axiom of Revealed Stochastic Preference* (ARSP) is necessary and sufficient for rationalizability of stochastic choice through a random utility model. This note proves the same result in one short and “chatty" paragraph. Indeed, setting up an efficient notation makes the result so immediate that one might hesitate to report it. However, the precise argument appears to have been overlooked.[^2] Furthermore, it immediately applies to set-valued choice. The resulting generalization is novel; in particular, if “decisiveness" of choice were dropped in [@McFadden05], the axiom reported there would be too weak.
The Classic Setting and Result
==============================
Consider a set of choice problems defined as subsets of a finite choice universe.[^3] Suppose a researcher observes the corresponding choice probabilities. The question is whether these probabilities can be rationalized through maximization of a random utility function. The random utility could reflect randomness in an individual’s utility assessment but also unobserved heterogeneity if data are from a repeated cross-section. Also, it will become clear that the underlying notion of nonstochastic utility is very flexible.
It will be useful to formalize everything in terms of vectors. Thus, identify the finite choice universe with components of a vector $X=(x_1,\dots,x_K)$. Any choice problem $C$ is a subvector of $X$. For any observable $C$, the corresponding choice probabilities will be collected in a vector $\pi(C) \in \Delta^{|C|-1}$, the $(|C|-1)$-dimensional unit simplex. Let $\mathcal{C}=(C_1,\dots,C_J)$ collect all observable choice problems in arbitrary (but henceforth fixed) order. We can then represent the totality of observed choice probabilities by the vector $\Pi \equiv (\pi(C_1),\dots,\pi(C_J))$. Note that $\Pi \in [0,1]^I$, where $I \equiv \sum_{j=1}^J{\vert C_j\vert}$, and that the components of $\Pi$ sum to $J$.
A set of particular interest will be the collection of rationalizable *nonstochastic* choice behaviors, i.e. the set of rationalizable choice functions on $\mathcal{C}$. For example, rationalizability could be defined through independence of irrelevant alternatives for individual choice [@Arrow59] or (in a demand context) the Generalized Axiom of Revealed Preference [@Afriat67]; in either case, the nonstochastic rationality concept then is utility maximization. But “rationalizability" is used in a very broad sense – one could easily define more narrow, more lenient, or also nonnested criteria.[^4] Any choice function can be expressed as a vector $R \in \{0,1\}^I$ that is interpreted just like $\Pi$ but whose entries are binary, with exactly one entry of $1$ for each choice problem. Let the set $\mathcal{R}$, which is necessarily finite, collect the vector representations of all rationalizable choice functions.
We call $\Pi$ *stochastically rationalizable* if it can be generated by the “random utility" extension of the notion of rationalizability embodied in $\mathcal{R}$, that is, if the choice probabilities $\Pi$ can be generated by drawing a nonstochastic “choice type" $R$ from some (arbitrary but fixed) distribution on $\mathcal{R}$ and then executing this type’s nonstochastic choices. In short, the set of stochastically rationalizable choice probabilities $\Pi$ is the convex hull of $\mathcal{R}$, a finite polytope that is henceforth denoted $\text{co}(\mathcal{R})$.
[@mcfadden-richter] characterize this notion of rationalizability through the ARSP, an axiom about sequences of subsets of choice problems and corresponding choice probabilities. The present notation allows for a succinct statement. To this purpose, let a *trial* $T \in \{0,1\}^I$ be a vector whose “$1$" entries correspond to (some or all) elements of the same choice problem. Intuitively, any trial characterizes a specific subset of a specific choice problem. (If the same subset of $X$ appears as subset of distinct choice problems, this gives rise to distinct trials.) Let the finite set ${\mathcal{T}}$ collect all possible trials. Then we have:
**Axiom of Revealed Stochastic Preference (ARSP)**
For any finite sequence $(T_1,\dots,T_M) \in {\mathcal{T}}^M$, $M \leq \infty$, one has $$\sum_{m=1}^M{T_m \Pi} \leq \max_{R \in \mathcal{R}}{\sum_{m=1}^M{T_m R}}. $$
In words, the choice probabilities corresponding to any finite sequence of trials cannot add up to more than the maximal analogous sum induced by a nonstochastic choice function in ${\mathcal{R}}$. We then have:
\[T1\] $\Pi$ is stochastically rationalizable if, and only if, ARSP holds.
The sequence $(T_1,\dots,T_M)$ matters only through $\sum_{m=1}^M T_m$. This sum is an $I$-vector with integer components and (because the canonical basis of $\mathbb{R}^I$ is in ${\mathcal{T}}$) any $I$-vector with integer components can be expressed as such a sum. Hence, ARSP can be rewritten as $$\begin{aligned}
&& T\Pi \leq \max_{R \in \mathcal{R}}{TR}~~\text{for all vectors}~T \in \mathbb{R}_+^I~\text{with integer components} \\
&\overset{(1)}{\Leftrightarrow} & T\Pi \leq \max_{R \in \mathcal{R}}{TR}~~\text{for all vectors}~T \in \mathbb{R}_+^I~\text{with rational components} \\
&\overset{(2)}{\Leftrightarrow} & T\Pi \leq \max_{R \in \mathcal{R}}{TR}~~\text{for all vectors}~T \in \mathbb{R}_+^I \\
&\overset{(3)}{\Leftrightarrow} & T\Pi \leq \max_{R \in \mathcal{R}}{TR}~~\text{for all vectors}~T \in \mathbb{R}^I \\
&\overset{(4)}{\Leftrightarrow} & \Pi \in \text{co}({\mathcal{R}}).\end{aligned}$$ Here, $(1)$ holds because inequalities can be multiplied by positive scalars; $(2)$ holds because weak inequalities are preserved under limit taking and all vectors in $\mathbb{R}_+^I$ can be approximated by rational ones. To see $(3)$, let $\bm{1} \equiv (1,\dots,1) \in \mathbb{R}^I$, then $\bm{1}\Pi=J$ but also $\bm{1}R=J$ for any $R \in \mathcal{R}$. Thus, $$T\Pi \leq \max_{R \in \mathcal{R}}{TR} \Leftrightarrow (T+\bm{1} \cdot \Vert T \Vert_\infty)\Pi \leq \max_{R \in \mathcal{R}}{(T+\bm{1} \cdot \Vert T \Vert_\infty)R},$$ and $T+\bm{1} \cdot \Vert T \Vert_\infty \in \mathbb{R}^I_+$ even if $T$ is not. (Recall that $\Vert T \Vert_\infty$ is the largest absolute value taken by any component of $T$.) Finally, $(4)$ is an elementary property of convex hulls.
The theorem is not new but the proof reflects two contributions. First, its almost embarrassing simplicity is partly due to efficient notation, i.e. to recognizing that with a finite universal choice space, all important quantities can be expressed as vectors and choice probabilities then as inner products. That ARSP prevents $\Pi$ from being separable from $\text{co}({\mathcal{R}})$ in any positive direction is then near immediate and has also been anticipated in the literature. The second contribution is to notice that adding-up constraints on $\Pi$ and $R$ relate separation in *any* direction to separation in a positive direction.
The result’s exposition also renders immediate some (previously known) facts about ARSP and $\text{co}({\mathcal{R}})$. For example, while there are infinitely many conceivable sequences $(T_1,...,T_M)$, a finite subset of “essential" sequences suffices to characterize stochastic rationalizability. In particular, the essential sequences correspond to gradients of full-dimensional faces of $\text{co}({\mathcal{R}})$. These gradients are rational because $\text{co}({\mathcal{R}})$ is spanned by binary vectors; therefore, the essential sequences are indeed finite. The gradients are known in some special cases: For binary choice and $K \leq 5$, they correspond to the Block-Marschak triangle inequalities [@Dridi1980], and they can be explicitly computed for the consumer choice problem with finitely many goods and $3$ budgets [@KS display 4.7] or with finitely many budgets and $2$ goods [@HS15]. However, computing them is equivalent to deriving the halfspace representation of the polytope $\text{co}(\mathcal{R})$ from its vertex representation, and this transformation is in general computationally prohibitive.[^5]
Extension to Set Valued Choice
==============================
With a finite choice universe, the above analysis is easily extended to choice correspondences, i.e. set-valued choice functions. Specifically, define a new choice universe $\tilde{X}\equiv 2^X$, the subsets of $X$. Any choice problem $C$ in the original problem can be identified with a new problem $\tilde{C} \equiv 2^C$. Then any data set concerning set valued choice on $(C_1,...,C_J)$ can be interpreted as data set with unique choice on $(\tilde{C}_1,...,\tilde{C}_J)$. A choice probability $\tilde{\Pi}$, a collection $\tilde{{\mathcal{R}}}$ of rationalizable nonstochastic behaviors, and a set $\tilde{{\mathcal{T}}}$ of possible trials can be defined as before. Asserting the ARSP for these new quantities yields:
**Generalized Axiom of Revealed Stochastic Preference (GARSP)**
For any finite sequence $(\tilde{T}_1,\dots,\tilde{T}_M) \in \tilde{{\mathcal{T}}}^M$, $M \leq \infty$, one has $$\sum_{m=1}^M{\tilde{T}_m \tilde{\Pi}} \leq \max_{\tilde{R} \in \tilde{{\mathcal{R}}}}{\sum_{m=1}^M{\tilde{T}_m \tilde{R}}}.$$
Then the previous proof establishes:
\[T2\] $\tilde{\Pi}$ is stochastically rationalizable if, and only if, GARSP holds.
Trivial as it may seem, this extension appears novel and contains two insights.
- Contrary to most of the literature, the empty set is included in the choice universe and all choice problems. Thus, Theorem \[T2\] applies to the stochastic extension of incomplete preference models and the like. If the underlying nonstochastic choice model excludes the empty set, this will be reflected in $\tilde{{\mathcal{R}}}$ and, therefore, in all rationalizable $\tilde{\Pi}$.[^6]
- [@mcfadden-richter] restrict attention to singleton-valued choice “for simplicity" (footnote 2); [@McFadden05] presents an extremely general setup and passes to singleton-valued choice only after defining ARSP. While neither formally characterize set-valued stochastic choice, the reader might infer that, for example, Theorem 3.3 in [@McFadden05] holds with ARSP as reported there even if the choice function is not singleton valued. This is not the case. Both papers define choice probability as the probability that a set *or any of its subsets* are chosen from the corresponding budget. This does not lose information and makes no difference at all for singleton-valued choice. But with set-valued choice, asserting ARSP for this probability restricts trials to query a set *and all its subsets* within a specific choice problem. This is a small subset of the trial sequences that are actually needed, so the restriction to single-valued choice is crucial and the axiom is otherwise too weak.[^7]
Conclusion
==========
This note concisely proves that rationalizability of stochastic choice is characterized by (G)ARSP. Most of the work is done by a vector notation which obviates that the axiom really restricts separating hyperplanes. Beyond its simplicity, the approach (i) further clarifies some known facts about stochastic rationalizability and (ii) immediately generalizes to set-valued choice, where it allows for indecisive (empty valued) choice and clarifies the appropriate axiom.
[^1]: Department of Economics, Cornell University, [email protected]. This note was motivated by joint work with Yuichi Kitamura. Thanks to an anonymous referee for helpful comments. Financial support through NSF Grants SES-1260980 and SES-1824375 is gratefully acknowledged.
[^2]: The proofs in [@mcfadden-richter] and [@border] take several pages. The proof in [@McFadden05] is one dense paragraph but goes through several intermediate characterizations and invokes duality, though some of that is to also establish other characterizations. [@GulPesendorfer06 p.123] call the ARSP and related axioms “complicated restrictions on arbitrary finite collections of decision problems" that are “difficult to interpret." They establish the result by relating ARSP to their own axioms. Within the considerable literature inspired by [@Falmagne], the basic geometry of random utility models is folk knowledge, but the focus is on other questions.
[^3]: This setting encompasses a classic demand setting with finitely many budgets because all revealed preference information is then contained in choice probabilities corresponding to a certain finite partition of the choice universe. See [@McFadden05] and [@KS].
[^4]: See [@DKQS] for a recent random utility model of the latter kind.
[^5]: See [@Ziegler] for more on these representations. The difficulty of this computation is illustrated by the intricate literature on finding facet defining inequalities for the linear order polytope [@Fishburn92].
[^6]: The same applies if $\tilde{{\mathcal{R}}}$ forces choice of singletons, in which case Theorem \[T1\] is recovered.
[^7]: Consider any setting where at least one choice problem is not a singleton, any $\tilde{{\mathcal{R}}}$ that restricts choice to be singleton-valued, and the choice probability $\tilde{\Pi}$ that always selects the entire choice problem. For any trial that queries an entire choice problem and all its subsets, $\tilde{\Pi}$ as well as any $\tilde{R} \in \tilde{{\mathcal{R}}}$ return value $1$. For any other trial, $\tilde{\Pi}$ returns $0$. Thus ARSP as defined in [@McFadden05 section 2.3.3] is fulfilled, but $\tilde{\Pi} \notin \text{co}({\mathcal{R}})$. For that paper’s Theorem 3.3 to apply to set-valued choice, choice probabilities should be defined as explained above and the set inclusions in expressions (1) and (2) should be equalities.
|
AS-TEXONO/00-06\
W.P. Lai$^{a,b}$, K.C. Cheng$^a$, H.B. Li$^{a,c}$, H.Y. Sheng$^{a,d}$, B.A. Zhuang$^{a,d}$, C.Y. Chang$^e$,\
C.P. Chen$^a$, Y.P. Chen$^a$, H.C. Hsu$^a$, J. Li$^d$, C.Y. Liang$^f$, Y. Liu$^d$, Z.S. Liu$^g$,\
C.S. Luo$^a$, F. Shi$^d$, R.F. Su$^h$, P.K. Teng$^a$, P.L. Wang$^d$, H.T. Wong$^{a,}$[^1],\
Z.Y. Zhang$^g$, D.X. Zhao$^{a,d}$, J.W. Zhao$^d$, P.P. Zhao$^d$, Z.Y. Zhou$^i$\
[The TEXONO[^2] Collaboration ]{}
$^a$ Institute of Physics, Academia Sinica, Taipei, Taiwan.\
$^b$ Department of Management Information Systems, Chung Kuo Institute of Technology,\
Taipei, Taiwan.\
$^c$ Department of Physics, National Taiwan University, Taipei, Taiwan.\
$^d$ Institute of High Energy Physics, Beijing, China.\
$^e$ Department of Physics, University of Maryland, College Park, U.S.A.\
$^f$ Center of Material Science, National Tsing Hua University, Hsinchu, Taiwan.\
$^g$ Department of Electronics, Institute of Radiation Protection, Taiyuan, China.\
$^h$ Nuclear Engineering Division, Kuo-sheng Nuclear Power Station,\
Taiwan Power Company, Taiwan.\
$^i$ Department of Nuclear Physics, Institute of Atomic Energy, Beijing, China.\
[**Abstract** ]{}
A 500 kg CsI(Tl) scintillating crystal detector is under construction for the studies of low-energy neutrino physics. The requirements, design, realization and the performance of the associated electronics, trigger, data acquisition and software control systems are described. Possibilities for future extensions are discussed.
[**PACS Codes:**]{} 29.40.Mc, 07.50.Qx, 07.05.M.\
[**Keywords:**]{} Scintillation detectors, Electronics, Data acquisition.\
[*Submitted to Nucl. Instrum. Methods.*]{}
Introduction
============
One of the major directions and experimental challenges in neutrino physics [@bib:nuphys] is to extend the measurement capabilities to the sub-MeV range for the detection of the p-p and $^7$Be solar neutrinos and other topics. The merits of scintillating crystal detector in low-energy low-background experiment were recently discussed [@bib:prospects]. An experiment with a CsI(Tl) detector placed near a reactor core to study neutrino interactions at low energy is being constructed [@bib:expt]. In the first data-taking phase, the detector is based on 200 kg of CsI(Tl) scintillating crystals[^3] read out by 200 photo-multipliers (PMTs)[^4] made of special low-activity glass. The set-up is shown schematically in Figure \[fig:setup\]. The eventual goal will be to operate with 500 kg of CsI(Tl) crystals.
The tasks and requirements of the readout systems are to achieve a low detection threshold, large dynamic range and good energy resolution provided by the CsI(Tl) and PMTs, to provide relative timing among individual channels as well as delayed correlated events for background diagnostics, and to record the pulse shape faithfully for pulse shape discrimination between $\gamma$/e and $\alpha$-particles [@bib:prototype]. The sum of the two PMT signals for each crystal module gives the energy while their difference provides the longitudinal position information. The intrinsic rise and decay times for the emissions from CsI(Tl) are typically at the range of 100 ns and $\rm{2~\mu s}$, respectively. A typical signal, as recorded by a 100 MHz digital oscilloscope, is shown in Figure \[fig:signals\]a.
This article describes the requirements, design, construction and performance of the the electronics, data acquisition (DAQ), and software control systems of this experiment.
Electronics System {#sect::ele}
==================
The schematic diagram of the electronics system is depicted in Figure \[fig:Electronics\]. Every channel of the “raw” ‘(CsI(Tl)+PMT) signal is amplified and shaped by the amplifier/shaper (AS), and subsequently digitized by the flash analog-to-digital convertor (FADC). The trigger system selects relevant events to be read out, while the logic control system provides a coherent timing and synchronization for the different electronics modules. The stability and linearity of the system is monitored (and corrected for, if necessary) by the calibration unit. For optimal adaptations to our applications, as well as for cost-effectiveness reasons, almost the entire electronics system is designed, constructed and tested by the Collaboration.
Two industry standards are adopted: NIM-convention for the AS and the various coincidence and veto logic modules, while the VME-protocols for the electronics-data acquisition interface. The trigger modules, logic control unit and calibration pulser are “double height” VME-6U modules, while the FADCs are “triple height” VME-9U modules. All are single-width modules.
The PMT signals pass through discriminators embedded in the AS module, and the NIM-level output are transferred to the trigger system. Simultaneously, the pre-trigger pedestals and the amplified AS pulses are continuously digitized and recorded on the circular buffers of the FADCs running at a 20 MHz (50 ns) rate. A typical FADC output signal of an event originated from the CsI(Tl) crystal is depicted in Figure \[fig:signals\]b. The shaping effects can be seen when compared to the raw CsI(Tl)+PMT pulse in Figure \[fig:signals\]a. The integrated area of the signal from two PMTs added together gives the energy of the event.
Once a signal due to a valid trigger is issued, the logic control system will generate a pre-defined “digitization gate” which determines the FADC digitalization time duration for the PMT signals. At the end of the gate, the FADC recording stops, and an “Interrupt” request is issued to the DAQ software hosted on a Linux-based PC via the VME-bus and a commercial VME-PCI bus bridge controller. The entire FADC system issues a single DAQ interrupt request (instead of each per module). The DAQ software response accordingly, and the data from the FADC circular buffers are read out and saved at computer disks for further processing. A detailed description of the online/offline DAQ system is given in Section \[sect::software\].
The calibration pulser generates and distributes simulation signals to monitor the functionalities and performances of all channels, helps trouble-shooting for locating noisy or dead channels and provides them with correction factors for subsequent analysis.
The components of the electronics system are discussed in the following sub-sections.
Amplifier and Shaper
--------------------
The schematic diagram of the amplifier/shaper (AS) module is shown in Figure \[fig:MAMP\]. The AS receives the PMT signals via LEMO cables and provides them with amplification, filtering, and pulse shaping. In addition, two discriminators per channels are integrated in the same module. A total of 16 channels are grouped into one module for the AS and the FADC.
The discriminator threshold levels for the PMT signals are provided by a separate manually-set threshold voltage adjustor and distributor module. The high one (“HiThr”) is for trigger purposes while the low one (“LoThr”) allows recording of all signals minimally above electronics noise. The settings are constantly monitored by a commercial VME Charge-to-Digital Convertor (QDC) board[^5]. The discriminator signals, “HDTO” and “LDTO” for those passing HiThr and LoThr, respectively, are sent to the trigger for processing by the trigger system.
As shown in Figure \[fig:signals\]a, the raw signals originate as a summation of a collection of single photo-electrons (pe) statistically distributed over a time duration according to the standard light curve of the CsI(Tl). Therefore, they are several $\mu$s in length whose profile are rough with O(10 ns) spikes due to single pe. In comparison, thermal noise from PMT appear as the single pe spike with this O(10 ns) width.
The first stage[^6] of the AS provides shaping at a time constant of 27 ns. It functions as a charge-sensitive amplifier for short \[O(10 ns and less)\] signals while remains as a current-sensitive amplifier for long ($>1~\mu$s) signals. As a result, the single pe spikes are smoothened relative to the digitization time-bin of 50 ns, such that their amplitude are reduced to below the LoThr level. Consequently, the trigger threshold can be reduced, and that the 20 MHz FADC clock rate are adequate and appropriate. The subsequent stage[^7] is a slow device providing effective shaping of about 250 ns and 1.6 $\mu$s in rise and fall times, respectively. The overall gain is 3.2 V/mA.
Each AS module is also equipped with a 16-bit shift register, which can give a hit pattern of the 16 channels on the module. The hit-pattern can be used to generate trigger decisions and to reduce the reading time of the FADC module. However, in a low count-rate experiment like ours, the DAQ dead time is not a of critical concern, and therefore, this functionality of fast hit-pattern measurement is not implemented for the first operational version of the electronics system.
Flash ADC
---------
The main tasks of the FADC are to digitize the pulse from the main amplifier/shaper, and to issue an Interrupt request to the DAQ system. The schematic diagram of the FADC is displayed in Figure \[fig:FADCmodule\]. There are 16 channels per module, matching the differential output signals from the AS delivered to the FADC with flat twisted cables. The resolution is 8-bit and the gain is adjusted such that the full dynamic range corresponds to an input of 2 V.
After an event is read out and the DAQ system is ready to receive the next event, a “Reset” signal is issued to the electronics hardware. The input signal (which are the AS output) are continuously digitized[^8], and the data are put into a circular buffer memory[^9]. of size 4096 bytes (4K in depth and each bin having 8 bit resolution). The digitization frequency is driven by an externally programmable clock. The hardware limiting rate is 30 MHz, while typically a 20 MHz clock is used.
Digitization is terminated when the external clock stops at the end of a valid event: that is, typically 25 $\mu$s after a trigger is issued at “t=0”. Any subsequent events up to a pre-selected time (typically 2 ms) are also recorded. The circular buffer points at the last byte it writes. The valid data on the circular buffer is then read out in the time-reversed sense, starting with the last byte until some pre-trigger time bins (typically 5 $\mu$s before t=0) which are for extracting the pedestal information.
To enhance the reading speed, the 16 channels on one module are divided into four groups, and each group takes a 32-bit word for data transfer since the circular buffer for each channel uses 8-bit per recording. Based on special features to be described in details in Section \[sect::logic\], only those FADC modules with at least one hit above LoThr within the event duration will be read out.
Calibration Pulser
------------------
The calibration pulser provides the information of dynamical range, linearity, and stability for all channels. It helps trouble-shooting to locate problematic channels, and provides a set of online or offline correction parameters.
As illustrated by the schematic diagram in Figure \[fig:Calpulser\], the pulser module consists of two components: calibration signal generator and distributor. The former generates a simulated signal, which is then fanned out via the latter to provide the input of the AS modules for all channels. A non-linearity of better than $0.3\%$ has been achieved for all channels.
During steady-state data taking, the calibration will be performed several times in a day to the electronics system via fully automatic software.
Logic Control Unit {#sect::logic}
------------------
The schematic diagram of the logic control unit is depicted in Figure \[fig:LogicControl\]. Its main function is to coordinate and synchronize the interactive actions of the various components of the electronics and DAQ systems. The designed timing schematic is shown in Figure \[fig:Timing\].
After receiving a valid trigger signal which defines the “START” timing, control signals are distributed by the logic control unit to the various components. The hardware pre-selectable digitization gate, typically of 25 $\mu$s width, is initiated. At the end of the gate, a “STOP” signal is generated which disables the clocking circuit and the FADC digitization will stop.
For detailed diagnostics of the background events, it is desirable to record the possible occurrence of delayed correlated events, such as the $^{214}$Bi-$^{214}$Po decay sequence $$\rm{
^{214}Bi ~ \rightarrow ~ ^{214}Po ~ + ~ \bar{\nu_e} ~ +
~ e^- ~ + ~ \gamma 's ~ (Q=3.28~MeV ~ ; ~ T_{\frac{1}{2}}=19.8~min) ~,
}$$ $$\rm{
^{214}Po ~ \rightarrow ~ ^{210}Pb~ + ~ \alpha ~
(Q=7.69~MeV ~ ; ~ T_{\frac{1}{2}}=164~ \mu s) ~~~.
}$$ Full digitization of the entire period far exceeds the available memory space in the circular buffer. To realize this requirement, a special “cascade-sequence” function is implemented After the completion of the triggered event, the entire system remains at the read-enable state capable of recording the pulse shape as well as timing of any delayed activities above LoThr for the duration of the “STROB” gate, typically set at 2 ms. Delayed events will initiate another cycle of digitization gate during which FADC digitization resumes, starting from the last location of the previous events. Relative arrival times of these delayed events are measured by additional Time-to-Digital-Convertor (TDCs) with 1 $\mu$s resolution. In this way, complete information up to 7 delayed events can be recorded (for a 20 MHz clock rate at the memory depth of 4K). In addition, the computer clock time is read out for every event, providing complementary timing information between events. The measured timing uncertainty is close to the limiting accuracy from the computer clock resolution of 10 ms. Therefore, the detailed and complete timing sequence of all the events can be reconstructed offline, providing powerful diagnostic tools for background understanding and suppression.
A manual switch provides selection of data taking modes either with the cascade-sequence functionalities or in the single-event mode for calibration purposes.
An Interrupt (“INTE”) signal is issued to the DAQ system at the completion of the STROB gate. The DAQ system starts transferring the data from the FADC memory and other locations to the hard disk storage space in the computer. When data acquisition is successfully completed, a “RESET” signal is distributed to the entire system to turn it back to the read-enable state for the next event.
The INTE signal instructs the DAQ system to read out a valid event. For DAQ efficiency and data suppression reasons, only those FADC modules with valid data are read out. This is achieved when an “EVENT” signal is issued from an AS module via its respective FADC module to the DAQ system. This signifies at least one channel having a data record above LoThr, and data will be read out from all 16 channels in this module. Software zero suppression is performed online and instantaneously to those channels without any signals above LoThr.
Trigger System
--------------
The trigger system is responsible for selecting events to be recorded. The schematic diagram is displayed in Figure \[fig:trigger\]. A logic FAN-IN of the signals from all the cosmic veto scintillators, after a coincidence requirement within individual panels, generates the “$\rm{V_{IN}}$” pulse. The veto action is extended over a hardware select-able “veto period”, typically of 100 $\mu$s, to get rid of background due to delayed cosmic-ray neutron-induced activities. The signal-definition line “$\rm{S_{IN}}$” is a logic FAN-in from selected signals from the CsI(Tl) target.
Four software select-able trigger modes are implemented to serve various data taking purposes: $\rm{ T1 = S_{IN} }$, $\rm{ T2 = V_{IN} }$, $\rm{ T3 = S_{IN} + V_{IN}}$, and $\rm{ T4 = S_{IN} + \bar{V}_{IN} }$. The “physics trigger” is provided by T4, while T3 is used for dedicated data taking for cosmic rays, while T1 and T2 are for diagnostics and debugging purposes. In steady-state data taking, the “trigger menu” consists mainly of T4 with a small sample T3 and calibration pulser events taken once every several hours for monitoring and calibration purposes.
The definition of $\rm{S_{IN}}$ depends on the selection requirements based on the hit-pattern input from the various CsI(Tl)+PMT channels. For low count rate applications such as neutrino physics experiments, the trigger conditions can be kept loose and minimal, and can be fine-tuned to be optimized for specific physics focus.
The “minimal bias” trigger is a simple coincidence between both PMTs corresponding to the same crystal module, whose pulse height are above the HiThr discriminator level. This is realized by having the “left” PMTs connected to one AS module, and the “right” ones to another. A coincidence from the HDTO lines of these modules, followed by a logic FAN-IN circuitry provides the $\rm{S_{IN}}$ line. Detailed selection of signal events from the background (for instance, events with a single hit versus those with multiple hits) can be carried out with subsequent offline analysis.
The trigger system is equipped with other diagnostic tools, like hit-pattern units for the cosmic-rays veto panels, scalers for recording various count rates (like the trigger and veto rates), as well as TDCs for measuring various timing (like the time between cascade sequence, as well as that between the $\rm{S_{IN}}$ and $\rm{V_{IN}}$ lines). The time between the most recent veto pulse and a valid trigger is recorded for diagnostic purposes.
Software System {#sect::software}
===============
The tasks of the software system are to perform data acquisition of the electronics discussed in Section \[sect::ele\], to operate a slow control system for the high voltage supplies and other ambient parameters, as well as to provide event display and monitoring graphic output. These functions are constructed in different programming languages and with different software tools, and work together in one single PC which runs on the Linux operation system. For security reasons, remote access and control to the reactor site can only be done with a telephone line dial-up solution.
The schematic framework of the software system, as well as the data flow, are depicted in Figure \[fig:ASNP\]. The FADC data are recorded on hard disks. Only minimal bias selection criteria are applied online. Event display for data-taking quality checks are provided for the on-site data taking.
At steady-state data-taking, the experiment is expected to be attended manually once per week. Filled-up hard disks will be replaced by new ones and brought back to the home-base laboratory where the data are copied into CDs and Exabyte tapes as permanent storage media. Meanwhile HBOOK[@bib:HBOOK] N-tuples are prepared for the storage of refined data which passes some technical event filter criteria and contains useful reconstructed quantities, such as energy depositions and longitudinal positions, from the pulse shape information. The refinement process of raw data to N-tuples can be iterated as many times as needed based on better understanding on experimental running conditions and detector calibrations. The calibrated N-tuples are then distributed to collaborating laboratories for further physics analysis.
Details of the various components are described below.
Data Acquisition
----------------
The schematic architecture of the data acquisition (DAQ) software system is shown in Figure \[fig:SBS\]. The system provides control to the experimental running parameters, accesses valid data from the VME electronics modules, and saves them on to hard disks.
The host computer is a PC running with the Pentium III-500 processor using RedHat Linux as the Operation System. The PC masters the VME slave modules on both the 6U and 9U VME crates via a commercial Adaptor system[^10] which provides communication between the VME-bus and the PCI-bus.
The DAQ software can be divided into two components in kernel and user space. The program written at the Linux kernel space provides low-level access to the VME-PCI Adaptor, and serves as a device driver which uses the kernel functions and provides the entry-point where user-space programs can communicate with the hardware.
The Tk/TcL-based graphics user interface package is equipped with various operational dialogues, including Start/Pause/Stop buttons and pull down menus for other functionalities for electronic module tests and regular DAQ operation. Various input fields are provided, where running conditions and parameters can be configured, file names and directory paths can be specified, and operational comments can be recorded. A function for pre-scheduling different trigger modes is also available which enables automatic selection of trigger conditions within the same data taking period. All control buttons in the dialogues are responded by their respective call-back functions at the kernel level of Linux.
Slow Control
------------
The slow control system serves to record and monitor the high voltage (HV) power supplies for the CsI(Tl) and veto scintillator PMTs, as well as other ambient and operation parameters.
The high voltage system is based on a dedicated main-frame crate[^11] operating on special HV modules[^12]. Each HV supply is distributed to two PMTs at the detector level. There are 12 HV supplies in each module and 16 modules can be inserted into one crate, so that a crate can handle 384 channels altogether.
The crate is internally equipped with a micro processor, adequate RAM space, and communication ports which can receive and execute external commands or return the voltage and current status of the individual channels. The serial port communication are adopted (the other option is the Ethernet) via a RS-232C cable to the DAQ PC. In principle, one PC can host up to 4 serial ports with each port driving one crate, such that future expansion of the system is straight-forward.
A VT100-based socket program is used to communicate with the PC serial ports using multiple Linux threads. Control and monitoring is achieved both locally on the console PC or remotely via the PPP (Point-to-Point Protocol) dial-up connection from home-base laboratories, as described in Section \[sect::connect\].
Slow control data such as output voltages and currents measured by the main-frame processor are recorded by the socket program on the hard disks of the DAQ PC, at a typical frequency of once per 10 minutes. Subsequent monitoring software packages can detect the voltage or current fluctuation, and warning or fatal error messages will be issued in case of alarms. Appropriate actions will be taken automatically, like voltage ramp-up for a tripped HV channel.
Other ambient conditions like temperature readings from thermostats, as well as operating parameters like the HiThr and LoThr values, are read out by a QDC module$^{\ref{qdc}}$. The cumulative count rates over a data-taking period from the CsI(Tl) target and the veto panels (the $\rm{S_{IN}}$ and $\rm{V_{IN}}$ lines, respectively) are measured by a scaler module[^13]. Both of these are read out by the main DAQ program, at a typical frequency of once every hour.
The slow control functions consume very little CPU time, and therefore does not affect the DAQ speed, such that both operations can both be handled by a single PC without interference on the actual performance.
Display and Monitoring Graphics
-------------------------------
A PAW-based [@bib:PAW] event display and monitoring program enable data quality check on-site from the information provided by the pulse shape, energy spectra, hit map, and calibration pulser analysis. Both real-time and on-disk raw data can be accessed. The graphics panels paged with these specific functional buttons are implemented with PAW executable files, and can be linked together and switched with each other on the HIGZ [@bib:HIGZ] window.
A clone of this program is duplicated on an off-site PC with an extra function which allows automatic data transfer (using a batch ftp shell script) from the on-site DAQ PC. To avoid the traffic caused by the low bandwidth of the networking telephone line, as discussed in Section \[sect::connect\], only a summary data in the format of HBOOK N-tuples and histograms are transferred back to home-base laboratories on a daily basis for offline monitoring and analysis.
\[sect::connect\]Network Connection
-----------------------------------
Owing to security reasons, the on-site computer can only be accessed via telephone line from home-based laboratories, as depicted in Figure \[fig:ASNP\], to decouple the connections from the reactor plant’s internal Ethernet network. The PPP system is adopted for the network connection of the on-site DAQ PC and the off-site PC, which serves the entry point for other off-site computers. Under the PPP client-server mode, the on-site PC is configured as the PPP server, whose modem can automatically pick up a dial-up connection request from the off-site PPP client, which get its IP address from the server.
Both the server and client are equipped with 56k bit-per-second(bps) modems, and the bandwidth is limited by the quality (9.6k bps) of the existing internal telephone circuitry of of the reactor plant. An average networking speed is around 4.8k bps is achieved. This is adequate for monitoring purposes when no large amount of data transfer is involved.
Performance
===========
The electronics and data acquisition systems with a total of 200 readout channels has been in robust operating conditions taking data from the prototype crystal modules at the laboratory. The readout scheme essentially preserves all energy and timing information recorded by the detector, providing powerful diagnostic tools for rare-search type experiments.
Using the input from the precision calibration pulser, the overall linearity of the response of the electronics and data acquisition systems is better than 0.3%. The RMS of the pedestal is equivalent to 0.30 FADC-channel corresponding to 2.3 mV baseline noise at the input level of the FADC. The resolution of the measured pulser-charge as a function of pulse amplitude, in the unit of FADC-channel, is displayed in Figure \[fig:pulserreso\]. The readout capabilities are much better than the detector response such that the overall performance parameters of the experiment are limited by the detector hardware.
The typical signals from the CsI(Tl)+PMT and the FADC outputs are displayed in Figure \[fig:signals\]a and \[fig:signals\]b, respectively. The fall times are different between $\gamma$/e and $\alpha$ events, as shown in Figure \[fig:psd\], providing a basis of particle identification. A typical cosmic-ray muon traversing a prototype detector set-up with 7 crystal modules is presented in Figure \[fig:muon\].
The integrated sum of the signals from both ends of a crystal module gives the energy information of the event. A typical energy spectrum due to a $^{137}$Cs source is depicted in Figure \[fig:cs137\]. A energy threshold of less than 50 keV is achieved. The full-width-half-max (FWHM) resolution at 660 keV is about 10%. This energy corresponds to pulses with around 50 FADC-channels in amplitude, which implies, based on Figure \[fig:pulserreso\], a readout contribution of 1.6% to the resolution effects at the $^{137}$Cs energy.
The system dead time is due to the contributions of two factors: (1) veto dead time, which is the veto rate multiplied by the inactive time per event (typically 100 $\mu$s), and (2) DAQ dead time, which is the time needed to read out the FADC and other modules (that is, the time after the STROB gate till the RESET pulse, as displayed in Figure \[fig:Timing\]).
The DAQ dead time depends on the complexity of the events. To illustrate the range, the typical dead time is 16 ms to read out and zero-suppress the simplest event where there is a single hit without delayed cascade, requiring only one FADC per event to be handled. The corresponding dead time for the read-out of 9 FADCs each having 7 delayed cascade per event is 700-800 ms. Typically, in situations like calibration data taking with radioactive sources which give high count rate but simple hit-pattern, and where the delayed cascade functions are disabled, a DAQ rate of 100 Hz can be sustained.
Summary
=======
An electronics and data acquisition system for low energy neutrino physics experiment, based on CsI(Tl) scintillating crystal as detectors with 100$-$500 readout channels, has been successfully designed, built and commissioned. It is modular by design and flexible in applications, such that the system can be adapted, as a whole or in parts, for other purposes.
There are several possible future upgrades which can further enhance its capabilities. The amplifier/shaper modules are equipped with shift registers such that a hit-pattern can be extracted if necessary. The readout time can then be reduced with this prior knowledge of what channels to read. Similarly, the trigger conditions employed for the current experiment are very loose and with minimal bias. With the exact hit-pattern information extracted, more sophisticated trigger selection criteria can be devised. Both functions are not critical for low count rate, low occupancy experiments, but may be desirable for higher trigger rate applications like in a high energy physics environment.
The current application requires a large dynamic range from about 10 keV (several photo-electrons) to 50 MeV (cosmic-ray muons). Although the FADC utilizes an 8-bit digitization scheme, its capabilities are beyond a $2^8$=256 fold dynamic range, since the entire pulse with a few hundred data-points are measured for every event. Over-scale signal manifest itself as event with a “flat-top”, the duration of which provides a measurement to the energy of the event. Using this feature [@bib:dynrange], the desired range can be covered even with a readout system with an 8-bit resolution in amplitude.
The authors are grateful to the technical staff of our institutes for invaluable support. This work was supported by contracts NSC 88-2112-M-001-007, NSC 89-2112-M-001-028 and NSC 89-2112-M-001-056 from the National Science Council, Taiwan, as well as 19975050 from the National Science Foundation, China.
[99]{} For the recent status on neutrino physics and astrophysics, see, for example, Y. Suzuki and Y. Totsuka, eds., Nucl. Phys. [**B**]{} (Procs. Suppl.) [**77**]{}, 335 (1999). H.T. Wong et al., Astropart. Phys. [**14**]{}, 141 (2000). H.B. Li et al., hep-ex/0001001, Nucl. Instrum. Methods, in press (2000). C.P. Chen et al., in preparation for Nucl. Instrum. Methods (2000). R.Brun et al., HBOOK-Statistical Analysis and Histograming, CERN Program Library Long Write-Ups Y250, CERN (1998). CN/ASD Group, PAW Users Guide, CERN Program Library W121, CERN (1993). CN/ASD Group, HIGZ User’s Guide and HPLOT User’s Guide, CERN Program Library Long Write-Ups Q120 and Y251, CERN (1995). W.P. Lai et al., in preparation for Nucl. Instrum. Methods (2000).
[**(a)**]{}
[**(b)**]{}
[^1]: Corresponding author: Email: [email protected]; Tel:+886-2-2789-9682; FAX:+886-2-2788-9828.
[^2]: Taiwan EXperiment On NeutrinO
[^3]: Manufacturer: Unique Crystals, Beijing
[^4]: Hamamatsu CR110-10
[^5]: \[qdc\] LeCroy ADC 1182
[^6]: Based on Harris IC HA-2525
[^7]: Based on Analog Devices IC AD817
[^8]: Based on Analog Devices IC AD775
[^9]: Elite MT IC UM61256FK
[^10]: SBS Bit-3 Adaptor
[^11]: LeCroy 1458
[^12]: LeCroy 1461
[^13]: CAEN V260
|
---
abstract: 'In this paper, we combine thermal effects with Landau-Zener (LZ) quantum tunneling effects in a dynamical Monte Carlo (DMC) framework to produce satisfactory magnetization curves of single-molecule magnet (SMM) systems. We use the giant spin approximation for SMM spins and consider regular lattices of SMMs with magnetic dipolar interactions (MDI). We calculate spin reversal probabilities from thermal-activated barrier hurdling, direct LZ tunneling, and thermal-assisted LZ tunnelings in the presence of sweeping magnetic fields. We do systematical DMC simulations for Mn$_{12}$ systems with various temperatures and sweeping rates. Our simulations produce clear step structures in low-temperature magnetization curves, and our results show that the thermally activated barrier hurdling becomes dominating at high temperature near 3K and the thermal-assisted tunnelings play important roles at intermediate temperature. These are consistent with corresponding experimental results on good Mn$_{12}$ samples (with less disorders) in the presence of little misalignments between the easy axis and applied magnetic fields, and therefore our magnetization curves are satisfactory. Furthermore, our DMC results show that the MDI, with the thermal effects, have important effects on the LZ tunneling processes, but both the MDI and the LZ tunneling give place to the thermal-activated barrier hurdling effect in determining the magnetization curves when the temperature is near 3K. This DMC approach can be applicable to other SMM systems, and could be used to study other properties of SMM systems.'
author:
- 'Gui-Bin Liu'
- 'Bang-Gui Liu'
title: 'Dynamical Monte Carlo investigation of spin reversals and nonequilibrium magnetization of single-molecule magnets'
---
Introduction {#sec1}
============
Single-molecule magnet (SMM) systems attract more and more attention because they can be used to make devices for spintronic applications [@spintr1; @spintr2], quantum computing [@qc], high-density magnetic information storage [@mem] etc [@ad21; @bb; @ad22]. Usually, a SMM can be treated as a large spin with strong magnetic anisotropy at low temperature. The most famous and typical is Mn$_{12}$-ac (\[Mn$_{12}$O$_{12}$(Ac)$_{16}$(H$_2$O)$_4$\]$\cdot$2HAc$\cdot$4H$_2$O, where HAc=accetic acid), or Mn$_{12}$ for short [@Mn12a]. It usually has spin $S=10$ and large anisotropy energy, producing a high spin reversal barrier [@sessoli]. Many interesting phenomena have been observed, such as various dynamical magnetism. One of the most intriguing phenomena observed in SMM systems is a step-wise structure in low-temperature magnetization curves [@mstep1; @mstep2; @mstep3]. Great efforts have been made to investigate this phenomenon and related effects [@add1; @chio; @Mn12tBuAc; @TB; @Mn4b; @book; @Mn12para1; @Mn12E]. The step-wise structure is attributed to Landau-Zener (LZ) quantum tunneling effect [@Landau; @Zener]. This stimulates intensive study on LZ model and its variants [@lz1; @lz2; @lz3; @lzepl; @lz4; @lz5; @lz6; @lz7; @lz8]. Some authors use numeric diagonalization methods [@numeric1; @numeric2] to study many-level LZ models to understand the step structure in experimental magnetization curves. However, it is difficult to consider thermal effects in these approaches to obtain satisfactory magnetization curves comparable to experimental results.
In this paper, we shall combine the classical thermal effects with the quantum LZ tunneling effects in a dynamical Monte Carlo (DMC) framework [@dmc0; @dmc1; @gbliu] in order to produce satisfactory magnetization curves comparable to experimental results. We consider ideal tetragonal body-centered lattices and use the giant spin approximation for spins of SMMs. We consider magnetic dipolar interactions, but neglect other factors such as defects, disorders, and misalignments between the easy axis and applied magnetic field. We calculate spin reversal probabilities from thermal-activated barrier hurdling, direct LZ tunneling effect, and thermal-assisted LZ tunneling effects in the presence of sweeping magnetic fields, and thereby derive a unified probability expression for any temperature and any sweeping field. Taking the Mn$_{12}$ as example, we do systematical DMC simulations with various temperatures and sweeping rates. The step structure appears in our simulated low-temperature magnetization curves, and our simulated magnetization curves are semi-quantitatively consistent with corresponding experimental results on those good Mn$_{12}$ systems (with less disorders) in the presence of little misalignments between the easy axis and applied fields [@Mn12tBuAc; @TB]. Interplays of the LZ tunneling effect, the thermal effects, and the magnetic dipolar interactions are elucidated. These imply that our simple model and DMC method capture the main features of experimental magnetization curves for little misalignments. More detailed results will be presented in the following.
The rest of this paper is organized as follows. In next section we shall define our spin model and describe approximation strategy. In the third section we shall describe our simulation method, present our unified probability formula for the spin reversal from the three spin reversal mechanisms, and give our simulation parameters. In the fourth section we shall present our simulated magnetization curves and some analysis. In the fifth section we shall show the key roles of the dipolar interactions in determining LZ tunneling probabilities. Finally, we shall give our conclusion in the sixth section.
Spin Model and approximation
============================
Without losing generality, we take typical Mn$_{12}$ system as our sample in the following. Under giant spin approximation, every Mn$_{12}$ SMM is represented by a spin $S$=10. Magnetic dipolar interactions are the only inter-SMM interactions, with hyperfine interactions neglected. Mn$_{12}$ SMMs are arranged to form a body-centered tetragonal lattice with experimental lattice parameters[@Mn12abc]. Using a body-centered tetragonal unit cell that consists of two SMMs, we define our lattice as $L_1\times L_2\times L_3$, where $L_1$, $L_2$, and $L_3$ are three positive integers. A longitudinal magnetic field $B_z(t)=B_0+\nu
t$ is applied along the $c$-easy axis of magnetization, where $\nu$ is the field-sweeping rate and $B_0$ is the starting magnetic field. The total Hamiltonian of this system can be expressed as $$\hat{H}=\sum_i\hat{H}_i^0+\frac12\sum_{i\ne j}\hat{H}_{ij}^{\rm
di}~, \label{eq.Htot}$$ where $\hat{H}_i^0$ is the single-body part for the $i$-th single SMM, and $\hat{H}_{ij}^{\rm di}$ describes the magnetic dipolar interaction between the $i$-th and $j$-th SMM. The factor $1/2$ before the sum sign is due to the double counting in the summation. $\hat{H}_i^0$ is given by $$\begin{aligned}
\hat{H}_i^0 = & -D(\hat{S}_i^z)^2 + E[(\hat{S}_i^x)^2-(\hat{S}_i^y)^2] \nonumber\\
& + B_4^0\hat{O}_4^0+B_4^4\hat{O}_4^4 + g\mu_B B_z \hat{S}_i^z ~,
\label{eq.Hi0}\end{aligned}$$ where $\hat{\mathbf{S}}_i\equiv(\hat{S}_i^x,\hat{S}_i^y,\hat{S}_i^z)$ is the spin vector operator for the $i$-th SMM, $g$ the Landé g-factor (here $g=2$ is used), $\mu_B$ the Bohr magneton, $D$, $E$, $B_4^0$ and $B_4^4$ are all anisotropic parameters, and $\hat{O}_4^0$ and $\hat{O}_4^4$ are both Steven operators[@book] defined by $\hat{O}_4^0=35(\hat{S}_i^z)^4-[30S(S+1)-25](\hat{S}_i^z)^2+3S^2(S+1)^2-6S(S+1)$ and $\hat{O}_4^4=[(\hat{S}_i^+)^4+(\hat{S}_i^-)^4]/2$. $\hat{H}_{ij}^{\rm di}$ is defined by $$\hat{H}_{ij}^{\rm
di}=\frac{\mu_0g^2\mu^2_B}{4\pi r^3_{ij}}[\hat{\mathbf{S}}_i\cdot\hat{\mathbf{S}}_j-\frac{3}{r^2_{ij}}(\hat{\mathbf{S}}_i\cdot
\mathbf{r}_{ij})(\hat{\mathbf{S}}_j\cdot \mathbf{r}_{ij})]~,
\label{eq.Hijdi}$$ where $\mu_0$ is the magnetic permeability of vacuum, and $\mathbf{r}_{ij}$ the vector from $i$ to $j$, with $r_{ij}$=$|
\mathbf{r}_{ij}|$ being the distance between $i$ and $j$.
For the $i$-th SMM, we treat all the effects from the other SMMs by classical-spin approximation. As a result, we derive the partial Hamiltonian $\hat{H}_i$ that acts on the $i$-th SMM: $$\begin{aligned}
\! \hat{H}_i &\!\!=\!\!& \hat{H}_i^0+g\mu_B \mathbf{B}_i^{\rm di}\cdot \hat{\mathbf{S}}_i\nonumber\\
&\!\!=\!\!& -D(\hat{S}_i^z)^2+B_4^0\hat{O}_4^0+\hat{H}_i^{\rm tr}
+ g\mu_B (B_z + B_{iz}^{\rm di})\hat{S}_i^z, \label{eq.Hi}\end{aligned}$$ where the transverse part $\hat{H}_i^{\rm tr}$ is defined as $$\begin{aligned}
\hat{H}_i^{\rm tr} = E[(\hat{S}_i^x)^2\!-\!(\hat{S}_i^y)^2] +
B_4^4\hat{O}_4^4 + g\mu_B(B_{ix}^{\rm
di}\hat{S}_i^x\!+\!B_{iy}^{\rm di}\hat{S}_i^y). \label{tran}\end{aligned}$$ For the $i$-th SMM, the dipolar interaction of the other SMMs is equivalent to $\mathbf{B}_i^{\rm di}\equiv(B_{ix}^{\rm di},
B_{iy}^{\rm di}, B_{iz}^{\rm di})= \sum_{j(\ne
i)}\mathbf{B}_{ji}^{\rm}$, where $\mathbf{B}_{ji}^{\rm}$ is the magnetic dipolar field applied by the $j$-th SMM on the $i$-th SMM. It contributes a magnetic field consisting of longitudinal and transverse parts.
{width="13cm"}
Simulation method and parameters
================================
As we show in Fig. 1, there are three main mechanisms related to the reversal of a SMM spin [@sessoli; @book; @mstep1; @mstep2; @mstep3; @add1; @lz1; @lzepl]: (a) thermal-activated barrier-hurdling, (b) direct LZ tunneling, and (c) thermal-assisted LZ tunneling. The thermal-activated barrier hurdling dominates at high temperature (if the blocking temperature $T_B\sim$3.3K for Mn$_{12}$ [@TB] is treated as high temperature), the direct LZ tunneling at low temperature, and the thermal-assisted LZ tunneling at intermediate temperature. For any temperature, we consider all the three spin reversal mechanisms simultaneously. For the time scale we are interested, we do not need to treat phonon-related interactions directly, but shall use an effective transition-state theory to calculate the thermal-activated spin-reversal rates. We shall use a DMC method to combine the quantum LZ tunneling effects with the classical thermal effects. Various kinetic Monte Carlo (KMC) methods[@witten; @kmc1; @kmc2; @lbg99; @kmc3], essentially similar to this DMC method, have been used to simulate atomic kinetics during epitaxial growth for many years. On the other hand, MC simulation has been used to study Glauber dynamics of kinetic Ising models[@glauber; @ad23; @ad24]. We shall present a detailed description of this DMC simulation method in the following.
Thermal-activated spin reversal probability
-------------------------------------------
We need the thermal-activated energy barrier in order to calculate the thermal-activated spin-reversal rate. When calculating the thermal-activated energy barrier we ignore the small transverse part $\hat{H}_i^{\rm tr}$ and use classical approximation for the spin operators. The large spin $S=10$ of Mn$_{12}$ further supports the approximations. As a result, the energy of the $i$-th SMM can be expressed as $$\bar{E}_i=-D_2(S_i^z)^2-D_4(S_i^z)^4+h_iS_i^z~, \label{eq.Ei}$$ where $S_i^z$ is the classical variable for the spin operator $\hat{S}_i^z$, $h_i=g\mu_B (B_z+B_{iz}^{\rm di})$, $D_2=D+[30S(S+1)-25]B_4^0$, and $D_4=-35B_4^0$. Because $h_i$ is dependent on time $t$, $\bar{E}_i$ changes with $t$.
We define our MC steps by the time points, $t_n=\Delta t\cdot n$, where $n$ takes nonnegative integers in sequence. For the $n$-th MC step, we use $\bar{E}_{i,n}$, $h_{i,n}$, and $S_{i,n}^z$ to replace $\bar{E}_i$, $h_i$, and $S_i^z$. Because each of the spins has two equilibrium orientations along the easy axis, we assume every spin takes either $S$ or $-S$ at each of the times $t_n$. Within the $n$-th MC step ($t$: from $t_n$ to $t_{n+1}$), we use an angle variable $\theta_{i,n}$ to describe the $i$-th spin’s deviation from its original ($t_n$) orientation $S_{i,n}^{\rm
eq}$. Naturally, $\theta_{i,n}=0$ corresponds to the original state and $\theta_{i,n}=\pi$ is the reversed state, and then all the other angle values ($0<\theta_{i,n}<\pi$) are treated as transition states. Expressing $S_{i,n}^z$ as $S_{i,n}^{\rm
eq}\cos\theta_{i,n}$, we usually have a maximum in the curve of $\bar{E}_{i,n}(\cos\theta_{i,n})$ as a function of $\cos\theta_{i,n}$, and the maximum determines the energy barrier for the spin reversal mechanism[@liying1; @liying2; @lbg], as shown in Fig. 1(a). We define $x_{i,n}=\cos\theta_{i,n}$ for convenience. We have $-1\leq x_{i,n} \leq 1$ for actual $\theta_{i,n}$, but $x_{i,n}$ can be extended beyond this region in order to always obtain a formal solution $x_{i,n}^{\rm max}$ for the maximum. $|x_{i,n}^{\rm max}|< 1$ implies that there actually exists an energy barrier, and $|x_{i,n}^{\rm max}|\ge 1$ means that there is no barrier for the corresponding process. Under conditions $D_2>0$ and $D_4>0$, the barrier can be expressed as: $$\Delta \bar{E}_{i,n} = \left\{ \begin{array}{lll}
\bar{E}_{i,n}(x_{i,n}^{\rm max}), && |x_{i,n}^{\rm max}|\leq
1\\
\bar{E}_{i,n}(-1)=|2h_{i,n}S_{i,n}^{\rm eq}|, && x_{i,n}^{\rm max}<-1\\
\bar{E}_{i,n}(1)=0, && x_{i,n}^{\rm max}>1
\end{array} \right.$$ where $x_{i,n}^{\rm max}$ is defined by $$x_{i,n}^{\rm max}=\sqrt[3]{-q_{i,n}/2+\sqrt{d_{i,n}}} +
\sqrt[3]{-q_{i,n}/2-\sqrt{d_{i,n}}}$$ and the three parameters are defined by $d_{i,n}=(q_{i,n}/2)^2+(p/3)^3$, $p=D_2/(2D_4S^2)$, and $q_{i,n}=-h_{i,n}S_{i,n}^{\rm eq}/(4D_4S^4)$. These parameters are dependent on the spin configuration and the magnetic field, and then on the time $t_n$ (or $n$).
The spin reversal rate within the $n$-th MC step (between $t_n$ and $t_{n+1}$) can be expressed as $R_{i,n}=R_0\exp(-\Delta
\bar{E}_{i,n}/k_B T)$ in terms of Arrhenius law [@ArrheniusLaw2], where $k_B$ is the Boltzmann constant and $R_0$ the characteristic attempt frequency. We use $P_n(t')$ to describe the probability that the $i$-th spin is reversed between $0$ and $t'$, where $t'$ satisfies the condition $t'\le\Delta t$. It has the initial condition $P_n(t'=0)=0$ and satisfies the equation $[1-P_n(t')]\cdot R_n(t')dt'=P_n(t'+dt')-P_n(t')$, or $$[1-P_n(t')]R_n(t')=\frac{d}{dt'}P_n(t')$$ where $R_n(t')$, the reversal rate at $t'$, is taken as the rate $R_{i,n}$, independent of $t'$ within the region \[0,$\Delta t$\]. Solving the equation, we obtain the probability $P^{\rm
clas}_{i,n}$ defined as $P_n(t'=\Delta t)$ for a classical thermal-activated reversal of the $i$-th spin within the $n$-th MC step: $$\label{Pclas}
P^{\rm clas}_{i,n}=1-\exp(-\Delta t\cdot R_{i,n}).$$ For $\Delta t\ll 1/R_{i,n}$, Eq. (\[Pclas\]) reduces to $P^{\rm
clas}_{i,n}=\Delta t\cdot R_{i,n}$. The probability expression defined in Eq. (\[Pclas\]) is reasonable because $P^{\rm
clas}_{i,n}$ will not exceed unity even when $\Delta t$ is very large with respect to $1/R_{i,n}$.
LZ-tunneling related spin reversal probabilities
------------------------------------------------
When temperature is lower than $T_B$, LZ tunneling begins to contribute to spin reversal. We begin with the effective quantum single-spin Hamiltonian (\[eq.Hi\]) with (\[tran\]). All the effects of other spins are included in the magnetic dipolar field $\mathbf{B}_{i}^{\rm di}$ (depending on the time $t$) and are depending on the magnetic field and the current spin configuration. For the $n$-th MC step, if the transverse term $\hat{H}_i^{\rm tr}$ is removed, Hamiltonian Eq. (\[eq.Hi\]) is diagonal and has $2S+1$ energy levels, $E_{m}^{i,n}$, where $m$ can take any of $S, S-1,\cdots,-(S-1),-S$. If using the continuous time variable $t$, we can express the energy levels as $E_{m}^{i}(t)$ (with $m$ from $S$ to -$S$) and derive their crossing fields \[at which $E_{m}^{i}(t)=E_{m'}^{i}(t)$\]: $$B_{m,m'}=\frac{(m+m')[D_2+D_4(m^2+m'^2)]}{g\mu_B}.$$ The transverse term $\hat{H}_i^{\rm tr}$ will modify the energy levels $E_{m}^{i,n}$, but the $2S+1$ energy levels of Hamiltonian (\[eq.Hi\]) with (\[tran\]), $\tilde{E}_{m}^{i,n}$, can be still labelled by $m=S, S-1,\cdots,-(S-1),-S$. Actually, the difference between $E_{m}^{i,n}$ and $\tilde{E}_{m}^{i,n}$ is small. Due to the existence of the transverse part $\hat{H}_i^{\rm
tr}$, there will be an avoided level crossing between $\tilde{E}_{m}^{i,n}$ and $\tilde{E}_{m'}^{i,n}$ for the $n$-th MC step when the effective field $B_z+B_{iz}^{\rm di}$ equals $B^{i,n}_{m,m'}$, with $m$ and $m'$ taking values among $S,
S-1,\cdots,-(S-1),-S$. The set of all the $B^{i,n}_{m,m'}$ values are the effective field conditions for the avoided-level-crossings. If $E_{m}^{i,n}$ equals $E_{m'}^{i,n}$, $B^{i,n}_{m,m'}$ is approximately equivalent to the crossing field (equaling $B_{m,m'}$). The allowed $(m,m')$ pairs are shown in Fig. 2. This means that when $B_z$ is swept to a right $B_{m,m'}^{i,n}-B_{iz}^{\rm di}$ value, a quantum tunneling occurs between the $m$ and $m'$ states. The tunneling can be well described using LZ tunneling[@gbliu; @Mn4b; @numeric1; @numeric2]. The nonadiabatic LZ tunneling probability $P^{{\rm
LZ},i,n}_{m,m'}$ is given by[@Landau; @Zener] $$P^{{\rm LZ},i,n}_{m,m'}=1-\exp \Bigg[ -\frac{\pi
(\Delta_{m,m'}^{i,n})^2}{2\hbar g\mu_B|m-m'|\nu} \Bigg] ~,
\label{Plz}$$ where the tunnel splitting $\Delta^{i,n}_{m,m'}$ is the energy gap at the avoided crossing of states $m$ and $m'$. $B^{i,n}_{m,m'}$ and $\Delta^{i,n}_{m,m'}$ can be calculated by diagonalizing Eq. (\[eq.Hi\]). If the dipolar field is neglected, $B^{i,n}_{m,m'}$, $\Delta^{i,n}_{m,m'}$, and $P^{{\rm
LZ},i,n}_{m,m'}$ reduce to $B^0_{m,m'}$, $\Delta^0_{m,m'}$, and $P^0_{m,m'}$, those of corresponding isolated SMMs, respectively.
{width="14cm"}
At the beginning of field sweeping, we let all the spins have $m=S$. If $T\!\ll\!T_B$, thermal activations are frozen, and LZ tunnelings only occur at the avoided crossings $(S,m')$, where $m'$ takes one of $-S,-S+1,\cdots,S-1$. This is the direct tunneling shown in Fig. 1(b), and the LZ tunneling probability is given by $$P^{\rm dLZ}_{i,n}=P_{S,m'}^{{\rm dir},i,n}=P_{S,m'}^{{\rm
LZ},i,n}.\label{Pdlz}$$ It is nonzero only when the condition $E_{S}^{i,n}=E_{m'}^{i,n}$ is satisfied. When the temperature is in the intermediate region $0\ll T<T_B$, the thermal-assisted tunneling plays an important role. This process can be represented by $S\rightsquigarrow m
\rightarrow m'$ as shown in Fig. 1(c), in which $S$ and $m$ states lie on one side of the thermal barrier and $m'$ and $-S$ states on the other side. The first process $S\rightsquigarrow m$ means that a spin is thermally activated from $S$ to $m$ state with the probability $P_{S\rightsquigarrow m}^{\text{act},i,n}$, which is given by $P_{S\rightsquigarrow m}^{\text{act},i,n}=1-\exp(-\Delta
t\cdot R^{\text{act}}_{i,n})$, where $R^{\text{act}}_{i,n}$ is given by $R_0\exp[-(E_{m}^{i,n}-E_{S}^{i,n})/k_B T]$. The second process $m\rightarrow m'$ is the LZ tunneling from $m$ to $m'$, with the probability defined in Eq. (\[Plz\]). Therefore, the reversal probability of thermal-assisted LZ tunneling through $m$ is given by $$P^{\rm
taLZ}_{i,n,m}=P_{S\rightsquigarrow m\rightarrow
m'}^{\text{ass},i,n}=P_{S\rightsquigarrow
m}^{\text{act},i,n}P_{m,m'}^{{\rm LZ},i,n}.\label{Ptalz}$$ It is nonzero only when the condition $E_{m}^{i,n}=E_{m'}^{i,n}$ is satisfied.
It must be pointed out that $m'$ in $P_{S,m'}^{{\rm dir},i,n}$ and $P_{S\rightsquigarrow m\rightarrow m'}^{\text{ass},i,n}$ is determined by $E_{m'}^{i,n}=E_{S}^{i,n}$ and $E_{m'}^{i,n}=E_{m}^{i,n}$, respectively, as is shown in Figs. 1(b) and 1(c). If the energy-level condition is satisfied, the probability is larger than zero; or else the probability is equivalent to zero. Therefore, the subscript $m'$ in $P_{S,m'}^{{\rm dir},i,n}$ and $P_{S\rightsquigarrow m\rightarrow m'}^{\text{ass},i,n}$ can be removed, as we have done in $P^{\rm dLZ}_{i,n}$ and $P^{\rm
taLZ}_{i,n,m}$.
Unified spin reversal probability for MC simulation
---------------------------------------------------
Generally speaking, every one of the three spin reversal mechanisms takes action at any given temperature. Actually the LZ tunneling effect dominates at low temperatures and the thermal effects become more important at higher temperatures. For the $n$-th MC step, the probability for the thermal-activated barrier-hurdling reversal of the $i$-th spin is given by $P^{\rm
clas}_{i,n}$ defined in Eq. (\[Pclas\]) \[see Fig. 1(a)\], that for the direct LZ tunneling effect equals $P^{\rm dLZ}_{i,n}$ defined in Eq. (\[Pdlz\]) \[see Fig. 1(b)\], and that for the thermal-assisted LZ tunneling effects through the $m$ state is given by $P^{\rm taLZ}_{i,n,m}$ defined in Eq. (\[Ptalz\]) \[see Fig. 1(c)\]. Here the partial probabilities from the three mechanisms are considered independent of each other. Therefore, we can derive the total probability $P_{i,n}^{\rm tot}$ for the reversal of the $i$-th spin within the $n$-th MC step: $$P_{i,n}^{\rm tot}=1-(1-P_{i,n}^{\rm clas})(1-P^{\rm dLZ}_{i,n})
\!\!\prod_{m_{\rm top}<m<S}\!\!(1-P^{\rm taLZ}_{i,n,m}),
\label{Ptot}$$ where $m_{\rm top}$, depending on the effective field, is determined by the highest level $E^{i,n}_{m_{\rm top}}$ among the $2S$ energy levels, $E^{i,n}_{m}$ ($-S\le m <S$), as we show in Fig. 1(c).
It must be pointed out that $P_{i,n}^{\rm clas}$ is always larger than zero, but the LZ-tunneling related probabilities, $P^{\rm
dLZ}_{i,n}$ and $P^{\rm taLZ}_{i,n,m}$, are nonzero only at some special values of the effective field. As is shown in Fig. 2, there is at most one LZ-tunneling channel, from either direct or thermal-assisted LZ effect, for a given nonzero value of the effective field. As a result, when the effective field is nonzero, we have at most one nonzero value from either $P^{\rm dLZ}_{i,n}$ or one of $P^{\rm taLZ}_{i,n,m}$ ($m_{\rm top}<m<S$). It is only at the zero value of the effective field that both $P^{\rm
dLZ}_{i,n}$ and $P^{\rm taLZ}_{i,n,m}$ ($0<m<S$) ($m_{\rm top}=0$) can be larger than zero so that we can have the direct LZ tunneling and all the thermal-assisted LZ-tunneling channels simultaneously. In our simulations, the processes that a reversed spin is reversed again are also considered, but the probabilities are tiny.
Simulation parameters
---------------------
We use experimental lattice constants, $a=b=17.1668\,\text{\AA}$ and $c=12.2545\, \text{\AA}$, and experimental anisotropy parameters, $D/k_B=0.66$K, $B_4^0/k_B=-3.2\times10^{-5}$K, and $B_4^4/k_B=6\times10^{-5}$K [@Mn12abc; @Mn12tBuAc; @Mn12para1]. As for the second-order transverse parameter $E$, $E/k_B=1.8\times10^{-3}$K is taken from the average of experimental values [@Mn12E]. We describe the time by using both continuous variable $t$ and discrete superscript/subscript $n$. In some cases, the sweeping field can be used to describe the time because it is defined by $B_z(t)=B_0+\nu t$. There is always a nonnegative integer $n$ for any given $t$ value, and there is a $t$ region, \[$t_n,t_{n+1}$\], for any given nonnegative $n$. We take $\Delta t=0.1$ms and $R_0=10^9$/s, which guarantee the good balance between computational demand and precision.
The dipolar fields $(B_{ix}^{\rm di},B_{iy}^{\rm di},B_{iz}^{\rm
di})$ at each SMM are updated whenever any of the SMM spins is reversed. The $\Delta^{i,n}_{m,m'}$ values are recalculated whenever any LZ tunneling happens. In the simulations, the field $B_z$ is swept from -7 to 7T in the forward process, and the full magnetization hysteresis loop is obtained simply by using the loop symmetry. Every magnetization curve is calculated by averaging over 100 runs to make statistical errors small enough. The main results presented in the following are simulated and calculated with lattices consisting approximately of $900\sim
1200$ body-centered unit cells or $1800\sim 2400$ spins. We have test our results with lattices consisting approximately of 100 $\sim$ 6000 body-centered unit cells, or 200 $\sim$ 12000 spins.
simulated magnetization curves
==============================
Presented in Fig. 3 are simulated magnetization curves (with $M$ normalized to the saturated value $M_S$) against the applied sweeping field $B_z$ for ten different temperatures: 0.1, 0.5, 0.6, 0.8, 1.0, 1.5, 2.0, 2.5, 2.8, and 3.2 K. Here, the lattice dimension is $10\times10\times10$ and the field sweeping rate is 0.02T/s. Each of the curves is calculated by averaging over 100 runs. The curves of 0.1K and 0.5K fall in the same curve, which implies that thermal activation is totally frozen when the temperature is below 0.5K. It can be seen in Fig. 3 that the area enclosed by a magnetization loop decreases with the temperature increasing, becoming nearly zero at 3.2 K (near the blocking temperature 3.3K of Mn$_{12}$). There are clear magnetization steps when the temperature is below 2.0K. They are caused by the LZ quantum tunneling effects. For convenience, we describe a step by using a H-part, a vertex, and a V-part. For an ideal step, the H-part is horizontal and the V-part vertical, but for any actual step in a magnetization curve, the H-part is not horizontal and the V-part not vertical because of the dipolar interaction and thermal effects, and the two parts still meet at the vertex. The vertex is convex toward the up-left direction in the right part of a magnetization loop and toward the down-right direction in the left part. At higher temperatures ($\geq 2.0$K), there is no complete step and there are only some kinks that remind us of some LZ tunnelings. This should be caused mainly by thermal effects.
![Simulated magnetic hysteresis loops ($M/M_S$ vs $B_z$) with sweeping rate $\nu=0.02$T/s for ten temperatures: 0.1, 0.5, 0.6, 0.8, 1.0, 1.5, 2.0, 2.5, 2.8, and 3.2K (from outside to inside). The lattice dimension is $10\times10\times10$. Note that the two curves of 0.1K and 0.5K fall in the same curve. []{data-label="fig:3"}](fig3.eps){width="8.5cm"}
Presented in Fig. 4 are the right parts of the magnetization curves against the applied sweeping field for three temperatures, 0.1, 1.5, and 2.5 K, and with three sweeping rates, 0.002, 0.02, and 0.2 T/s. Here, the lattice dimension is $10\times10\times10$. We label a magnetization step by the magnetic field defined by its V-part near its vertex. For $T$=0.1K, only the direct LZ tunnelings change the magnetization, and the magnetization steps from $B_z$=2 to 6T in Fig. 4 correspond to $B_{S,m'}^0$ with $m'$ being from -6 to 2 in Table I. For $T$=1.5K, there are clear steps in the lower parts of the three magnetization curves, but their V-parts deviate substantially from the corresponding $B_{S,m'}^0$ values and the steps are substantially deformed, which show that thermal-assisted LZ tunnelings play an important role. When temperature rises to 2.5K, there is no step structure and only one kink can be seen in the lower part of the magnetization curve in the cases of 0.2T/s and 0.02T/s. This is because the effects of thermal activation become dominating over the LZ tunneling effects. Different sweeping rates lead to substantial changes in the magnetization curves, and the larger the sweeping rate becomes, the larger the hysteresis loops are.
![(Color online.) The right parts of simulated magnetization curves of different sweeping rates 0.002 (dot), 0.02 (dash), and 0.2 (solid)T/s for three temperatures 0.1K, 1.5K, and 2.5K, as labelled. The lattice dimension is $10\times10\times10$. Each of the visible steps and kinks along a magnetization curve corresponds to one of the magnetic fields at which the direct and thermal assisted LZ tunnelings take place. The thin vertical dotted lines show the positions of $B_{S,m'}^0$ for $m'$=-10,-9,$\cdots$,2.[]{data-label="fig:4"}](fig4.eps){width="8.2cm"}
![(Color online.) The right parts of simulated magnetization curves for three temperatures with five different lattice dimensions: $20\times20\times3$ (dash-dot), $12\times12\times8$ (dash), $10\times10\times10$ (solid), $9\times9\times14$ (dot), and $3\times3\times100$ (short-dash). The temperatures are 0.1K, 1.5K, and 2.5K, as labelled. The sweeping rate is 0.02T/s. For comparison, we also present the results without considering dipolar interaction (thin solid line). []{data-label="fig:5"}](fig5.eps){width="8.3cm"}
Presented in Fig. 5 are the right parts of simulated hysteresis loops with $\nu$=0.02 T/s at three temperatures for five different lattice dimensions: $20\times20\times3$, $12\times12\times8$, $10\times10\times10$, $9\times9\times14$, and $3\times3\times100$. The temperatures are 0.1, 1.5, and 2.5K. For comparison, the simulated results without the dipolar interaction are presented too. For $T$=0.1K, there are clear step structures for all the five lattice shapes. The step height varies with the lattice shape, which can be attributed to the dipolar interaction. If the dipolar interaction is switched off, there are only two steps: one tall step at $B_z=4.00$ T and the other very short step at 3.06 T. They correspond to the two transitions from 10 to -2 and -4, respectively. Other transitions from 10 to -4, -6, -8, and -10 have too small probabilities to be seen. When the dipolar interaction is switched on, the transition from 10 to -3 is allowed and the tall step becomes much shorter, resulting in the rich step structures between 3 and 6 T. The steps are caused by the direct LZ tunnelings. When the temperature changes to 1.5K, the hysteresis loops become substantially smaller because of the enhanced thermal effects. In this case, there are deformed step structures in the lower parts of the magnetization curves and there does not exists any clear step structure in the upper parts. The deformed step structures between 1 and 3 T result from the thermally assisted LZ tunnelings. For $T$=2.5K, there does not exist any step structure at all for all the six cases. The effect of the lattice shape is attributed to the long-range property of the dipolar interaction, and can be clearly seen in the magnetization curves only at the low temperatures in the extreme cases of $20\times20\times3$ and $3\times3\times100$. Actually, there is little visible difference between the magnetization curves of the three lattices: $12\times12\times8$, $10\times10\times10$, and $9\times9\times14$. Visible difference can be found at 0.1K and 1.5K only for the two extreme cases: $20\times20\times3$ and $3\times3\times100$. If we define a ratio $r=L_l/L_t$ of longitudinal size to transverse size for $L_t\times
L_t\times L_l$, we have $r$=1 for $10\times10\times10$, $r$=0.67 for $12\times12\times8$, $r$=1.56 for $9\times9\times14$, $r$=0.15 for $20\times20\times3$, and $r$=33 for $3\times3\times100$. Therefore, there is little clear effect of lattice shape as long as the shape parameter $r$ is neither extremely large nor extremely small.
Now we address the statistical errors. We have calculated standard errors $\sigma_M$ of the reduced magnetization $M/M_s$ as functions of the sweeping field for various temperatures and sweeping rates. Our results show that for a given magnetization curve, the statistical errors are very small ($\sigma_M<0.005$) in the region of $B_z$ defined by $|M/M_s|>0.9$, and reach a maximal value $\sigma^{\rm max}_M$ near the point of $B_z$ defined by $M/M_s=0$. The maximal statistical error $\sigma^{\rm max}_M$ is dependent on the temperature and sweeping rate, varying from 0.015 to 0.025 for our simulation parameters. Such statistical errors appear only in a very small region of $B_z$. For any magnetization curve as a whole, the statistical errors are small enough to be acceptable.
Here we discuss effects of lattice sizes on simulated results. The above simulated results are based on the lattice dimensions: $20\times20\times3$, $12\times12\times8$, $10\times10\times10$, $9\times9\times14$, and $3\times3\times100$. They have $900\sim
1200$ body-centered unit cells, or $1800 \sim 2400$ spins. To test our results, we have done a series of simulations for different parameters using lattice dimension defined by $L_t\times L_t\times
L_l$. In the cases of $T=0.1$K, $\nu=0.2$T/s, and $L_t=L_l=L$ with $L=5\sim 20$, the largest size effects appear between 4 and 5.5 T for the right parts of the magnetization curves. For the steps at 4T, the $L$-caused change in the magnetization decreases quickly with increasing $L$, becoming very small when $L$ is larger than 9. Therefore, our lattice sizes of the results presented above are large enough to be reliable.
The above simulated results show that the area enclosed by a magnetization hysteresis loop decreases with the temperature increasing and increases with the sweeping rate increasing. This is completely consistent with the temperature and sweeping-rate dependence of the thermal reversal probability and LZ tunneling probabilities. Thermal activation effects dominate at high temperature. The LZ tunneling effects manifest themselves through the steps and kinks along the magnetization curves. However, there is a limit for the hysteresis loops at the low temperature end for a given sweeping rate. These limiting magnetization curves are caused by the minimal reversal probability set by the direct LZ quantum tunneling effect because the thermal activation probability becomes tiny at such low temperatures. With usual shape parameter $r$, these results are consistent with experimental magnetization curves of good Mn$_{12}$ crystal samples in the presence of little misalignments between the easy axis and applied fields [@Mn12tBuAc; @TB]. In principle, a transverse magnetic field (due to the misalignment of the applied field and the easy axis) can enhance the energy splitting, and as a result will reduce the magnetization loop and smooth some steps [@Mn12tBuAc; @TB; @Mn12added]. These usual (not extreme) shape parameters should reflect real shape factors in experimental samples. The consistence should be satisfactory, especially considering that our theoretical probabilities are calculated under leading order approximation and our model does not include possible defects and disorders in actual materials.
$m'$ $B_{S,m'}^0$(T) $\Delta B^n_{S,m'}$(T) $P_{S,m'}^0$ $\langle P^{\rm LZ,n}_{S,m'}\rangle$ $\sigma_{S,m'}^{n}$
------ ----------------- ------------------------ -------------- -------------------------------------- ---------------------
-10 0.000000 6.4$\times10^{-15}$ 0.00000 0.00000 0.00000
-9 0.564160 1.6$\times10^{-6~}$ 0 0.00000 0.00000
-8 1.099966 3.5$\times10^{-6~}$ 0.00000 0.00000 0.00000
-7 1.612415 5.1$\times10^{-6~}$ 0 0.00000 0.00000
-6 2.106511 6.7$\times10^{-6~}$ 0.00138 0.00138 0.00001
-5 2.587260 7.9$\times10^{-6~}$ 0 0.00002 0.00002
-4 3.059671 8.6$\times10^{-6~}$ 0.01815 0.01838 0.00320
-3 3.528757 8.6$\times10^{-6~}$ 0 0.22194 0.21086
-2 3.999529 7.8$\times10^{-6~}$ 1.00000 1.00000 0.00000
-1 4.476997 6.3$\times10^{-6~}$ 0 0.53746 0.33455
0 4.966165 3.9$\times10^{-6~}$ 1.00000 1.00000 0.00089
1 5.472035 7.4$\times10^{-7~}$ 0 0.99975 0.01091
2 5.999604 3.6$\times10^{-6~}$ 1.00000 1.00000 0.00000
3 6.553867 8.6$\times10^{-6~}$ 0 0.99988 0.00749
: Calculated results of $B_{S,m'}^0$, $\Delta B^n_{S,m'}$, $P_{S,m'}^0$, $\langle P^{\rm LZ,n}_{S,m'}\rangle$, and $\sigma_{S,m'}^{n}$ for the direct LZ tunneling $(S,m')$ when the field $B_z$ is swept to 3.75 T, where $n$ is determined by the field 3.75 T. $T=0.1$ K, $\nu=0.02$ T/s, and the lattice dimension is $10\times10\times10$. []{data-label="tab:1"}
Key roles of dipolar fields
===========================
To investigate the effects of dipolar interactions, we divide the dipolar fields within the $n$-th MC step, $(B_{ix,n}^{\rm di},
B_{iy,n}^{\rm di}, B_{iz,n}^{\rm di})$, into two parts: transverse dipolar field $B_{ix,n}^{\rm di}$ and $B_{iy,n}^{\rm di}$, and longitudinal dipolar field $B_{iz,n}^{\rm di}$. Transverse dipolar field not only modifies $B^{i,n}_{m,m'}$, but also affects $\Delta^{i,n}_{m,m'}$ and $P^{\rm LZ,i,n}_{m,m'}$. In contrast, longitudinal dipolar field affects neither $\Delta^{i,n}_{m,m'}$ nor $P^{\rm LZ,i,n}_{m,m'}$, but shifts $B^{i,n}_{m,m'}$ by $-B_{iz,n}^{\rm di}$. This means that LZ tunnelings actually occur at the field $B^{i,n}_{m,m'}-B_{iz,n}^{\rm di}$, not $B^{i,n}_{m,m'}$. This shift has two effects. First, it broadens the LZ transition and deforms the steps in magnetization curves. Second, the quick changing of $B_{iz,n}^{\rm di}$ results in that the value $B^{i,n}_{m,m'}$ can be missed by the effective field $B_z+B_{iz,n}^{\rm di}$, and therefore the actual percentage of the reversed spins due to the LZ tunneling effect with respect to the total spins is smaller than the LZ probability $P^{\rm
LZ,i,n}_{m,m'}$ given in Eq. (\[Plz\]). This means that the dipolar interaction hinders both the direct LZ tunneling process and the thermal assisted LZ tunneling processes.
![(Color online.) Distributions of dipolar fields $B_{ix}^{\rm di}$ (dashed line + circle), $B_{iy}^{\rm di}$ (dotted line + cross), and $B_{iz}^{\rm di}$ (solid line + square) for the five lattice dimensions $20\times 20\times 3$ (a), $12\times12\times8$ (b), $10\times10\times10$ (c), $9\times
9\times14$ (e), and $3\times 3\times100$ (e) when the field $B_z$ is swept to 3.75 T. The temperature $T$ is 0.1 K and the sweeping rate $\nu$ equals 0.02 T/s. []{data-label="fig:6"}](fig6.eps){width="8cm"}
Without transverse dipolar field, $B_{m,m'}^{i,n}$ becomes $B_{m,m'}^0$, and $P^{\rm LZ,i,n}_{S,m'}$ equals 0 for odd $m'$ values because transverse dipolar field is the only transverse term of odd order in Hamiltonian Eq. (4). Without longitudinal dipolar field, the V-parts of steps remain vertical and the percentage of the reversed spins due to LZ tunneling is strictly equivalent to the LZ probability $P^{\rm LZ,i,n}_{S,m'}$ at low temperatures. These are shown by the thin solid line for 0.1K in Fig. 5. In Table I we also present the average value ($\Delta
B^n_{S,m'}$=$\langle | B^{n}_{S,m'}-B_{S,m'}^0|\rangle$) of dipolar-field fluctuations with respect to $B_{S,m'}^0$, the dipolar-field-free LZ probability $P^0_{S,m'}$, and the average value $\langle P^{\rm LZ,n}_{S,m'}\rangle$ and the corresponding standard error $\sigma_{S,m'}^{n}$ of $P^{\rm LZ,i,n}_{S,m'}$ for the avoided crossing positions of $S$ and $m'$, where $m'$ varies from -10 to 3 and the averaging $\langle X^n \rangle$ of $X^{i,n}$ is calculated over all the spins and all the runs within the $n$-th MC step. It should be pointed out that the $\Delta
B^n_{S,m'}$ values, although very important to LZ tunnelings, are very small, as shown in Table I. It is transverse dipolar field that make $\Delta B^n_{S,m'}$ nonzero and make $P^{\rm
LZ,i,n}_{S,m'}$ ($m'$=-5, -3, -1, 1, 3) change from 0 to nonzero, even nearly reach 1 in the cases $m'=1$ and 3.
In order to elucidate the magnitude and distribution of the dipolar fields, we address the time-dependent distributions of SMMs that have dipolar fields $(B_{ix}^{\rm di}, B_{iy}^{\rm di},
B_{iz}^{\rm di})$ (here the continuous time variable is implied), or in short the distributions of $B_{ix}^{\rm di}$, $B_{iy}^{\rm
di}$, and $B_{iz}^{\rm di}$, in the following. In Fig. 6 we compare the results from five different lattice dimensions: $20\times 20\times 3$, $12\times12\times8$, $10\times10\times10$, $9\times 9\times14$, and $3\times 3\times100$. Here the time is when the field $B_z$ is swept to 3.75 T, the temperature $T$ is 0.1 K, and the sweeping rate $\nu$ equals 0.02 T/s. For all the five lattices, our results show that the distribution of $B_{ix}^{\rm di}$ always is approximately equivalent to that of $B_{iy}^{\rm di}$ and they are both symmetrical and peaked at zero. The peak is sharper for the extremely slab-like $20\times20\times3$ lattice and extremely rod-like $3\times3\times100$ lattice. The peak of the $B_{iz}^{\rm di}$ distribution is wider than that of both $B_{ix}^{\rm di}$ and $B_{iy}^{\rm di}$. It shifts substantially away from zero when the lattice shape is either extremely slab-like or extremely rod-like. The leftward shift of the $B_{iz}^{\rm di}$ peak can be attributed to dipolar-interaction-induced ferromagnetic orders in rod-like systems [@garanin1; @garanin2], and the similar rightward shift to dipolar-interaction-induced antiferromagnetic orders in slab-like systems. Because dipolar interactions are the only inter-SMM interactions in our model, the differences of distributions between the the five lattices are caused by the dipolar fields, or dipolar interactions in essence.
conclusion
==========
In summary, we have combined the thermal effects with the LZ quantum tunneling effects in a DMC framework by using the giant spin approximation for spins of SMMs and considering magnetic dipolar interactions for comparison with experimental results. We consider ideal lattices of SMMs consistent with experimental ones and assume that there are no defects and axis-misalignments therein. We calculate spin reversal probabilities from thermal-activated barrier hurdling, direct LZ tunneling effect, and thermal-assisted LZ tunneling effects in the presence of sweeping magnetic fields. Taking the parameters of experimental Mn$_{12}$ crystals, we do systematical DMC simulations with various temperatures and sweeping rates. Our results show that the step structures can be clearly seen in the low-temperature magnetization curves, the thermally activated barrier hurdling becomes dominating at high temperature near 3K, and the thermal-assisted tunneling effects play important roles at the intermediate temperature. These are consistent with corresponding experimental results on good Mn$_{12}$ samples (with less disorders) in the presence of little misalignments between the easy axis and applied fields [@Mn12tBuAc; @TB], and therefore our magnetization curves are satisfactory.
Furthermore, our DMC results show that the magnetic dipolar interactions, with the thermal effects, have important effects on the LZ magnetization tunneling effects. Their longitudinal parts can partially break the resonance conditions of the LZ tunnelings and their transverse parts can modify the tunneling probabilities. They can clearly manifest themselves when the SMM crystal is extremely rod-like or slab-like. However, both the magnetic dipolar interactions and the LZ tunneling effects have little effects on the magnetization curves when the temperature is near 3K. This DMC approach can be applicable to other SMM systems, and could be used to study other properties of SMM systems.
This work is supported by Nature Science Foundation of China (Grant Nos. 10874232 and 10774180), by the Chinese Academy of Sciences (Grant No. KJCX2.YW.W09-5), and by Chinese Department of Science and Technology (Grant No. 2005CB623602).
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|
---
abstract: |
It has been suggested that, when faced with large amounts of uncertainty in situations of automated control, type-2 fuzzy logic based controllers will out-perform the simpler type-1 varieties due to the latter lacking the flexibility to adapt accordingly. This paper aims to investigate this problem in detail in order to analyse when a type-2 controller will improve upon type-1 performance. A robotic sailing boat is subjected to several experiments in which the uncertainty and difficulty of the sailing problem is increased in order to observe the effects on measured performance. Improved performance is observed but not in every case. The size of the FOU is shown to be have a large effect on performance with potentially severe performance penalties for incorrectly sized footprints.
*Keywords: Interval Type-2 Fuzzy, Robot Boat control, Fuzzy Control, Uncertainty*
author:
-
- 'Naisan Benatar, Uwe Aickelin and Jonathan M. Garibaldi [*Member, IEEE*]{}'
bibliography:
- 'Journal.bib'
title: 'An Investigation into the Relationship Between Type-2 FOU Size and Environmental Uncertainty in Robotic Control'
---
Introduction
============
A fuzzy logic system maps inputs into a fuzzy set by means of a *fuzzifier*. The fuzzy set output is then processed as part of a Fuzzy Inference System (FIS) where, the set is used as an input to an inference system with its associated rule base. This results in a new output set that can itself be *defuzzified* into a value suitable for use as a standard (e.g. PID) controller output.
The way the processing of the fuzzy sets is handled can be varied based upon application specific requirements or restrictions and this gives rise to three main subcategories of fuzzy control: type-1, interval type-2 and general type-2. In this paper we consider type-1 and interval type-2 based systems and apply both to control problems of increasing difficulty with increasing quantities of uncertainty. We thereby hope to develop a method of determining parameters, such as FOU size that will give performance improvements over type-1 based controllers.
The application discussed in this paper is one of robotic sailing using the FLOATs (Fuzzy Logic Operated AuTonomous Sailboat) platform as described in [@Benatar2011], in which a robotic sail boat actuates sail and rudder positions based on received sensor data with the aim of steering an autonomous boat around a predetermined course. Similar boat based applications have been investigated with a variety of approaches such as PID [@Sauze2005], biologically inspired [@Sauze2010] and fuzzy methods as in [@Stelzer-20081]. This application was selected due to the multiple sources of noise and uncertainty in the environment. This maybe useful for discerning under which conditions type-2 based controllers will outperform type-1.
This paper is organised as follows: Section \[sec:background\] provides background and information into the methods and systems used in the rest of the paper. Section \[sec:SEM\] describes the experiments that will be performed and will be followed by Section \[sec:Results\] where we state our numerical results and Section \[sec:Discussion\] where these results are discussed. Finally in Section \[sec:Conclusions\_Future\] we draw some conclusions along with some ideas for future work.
Background {#sec:background}
==========
Type-1 Fuzzy Logic {#sec:t1}
------------------
Type-1 fuzzy logic, introduced by Zadeh in [@Zadeh1975] uses 2D membership functions that are commonly triangular or Gaussian in shape. The $x$ axis represents the set of possible inputs into the system as obtained from the application. The $y$ axis represents the degree to which the given input is a member of the fuzzy set and may have a value between 0 (no membership to the set) and 1 (complete membership to the set).
One of the main shortcomings of type-1 fuzzy logic is its limited flexibility. Moving a type-1 controller tuned in a specific environment into a different environment can often lead to significant performance degradation unless controller changes are made. It has been stated by Mendel [@NileshNKarnik1999] that type-2 control may be advantageous over type-1 in many areas such as robust control in which the information to be processed is uncertain.
Type-2 Fuzzy Logic {#sec:t2}
------------------
Type-2 fuzzy logic is a development of type-1 and with this development comes greater freedom and flexibility as discussed by Wu [@Wu2011a]. This may increase performance in certain situations as explored by Hagras [@hagras2004], at the potential cost of increased computational load. Some of the increased flexibility of this type of system can be attributed to the fact that the membership functions in type-2 systems are represented in three dimensions instead of type-1’s two. This extra dimension gives rise to what is known as the secondary membership function.
In an interval type-2 system the value of the secondary function is binary, allowing interval type-2 systems to be represented by two individual membership functions in the 2D plane. They are termed the upper and lower membership functions (UMF and LMF respectivly). The area that they enclose is termed the footprint of uncertainty (FOU).
With general type-2 fuzzy logic, unlike in interval systems, the secondary membership function is continuous instead of discrete, for a more comprehensive overview of type-2 fuzzy logic the reader is directed at a paper such as [@Mendel2006].
As a type-2 inference system will return at least 1 type-2 fuzzy set and this cannot be directly defuzzified a process known as *type-reduction* is required . This process reduces this type-2 set into type-1 which can then be defuzzified into a usable output.
While both interval and general type-2 are more complex than type-1, methods to reduce processing requirements are being developed which bring the possibility of type-2 based mobile robotic systems closer. For example Wu [@Wu2006] and Wagner and Hagras [@Wagner2011] both present distinct methods to reduce the computational load significantly so that is is manageable by resource constrained systems such as those used on mobile robots.
Experimental Methodology {#sec:SEM}
========================
Problem description
-------------------
The application to be used as our test case is FLOATS, described in [@Benatar2011]. It is an autonomous sailing boat originally based on the work done by Stelzer in [@Stelzer-2008] which can work in both simulation and the real world environments. Wind, location, waypoint location and direction sensors are used to calculate error and delta error values. These are used as inputs to a fuzzy inference system which produces a rudder change output value.
In this paper we have opted to use simulation as the method of experimentation, with real life experimentation planned for future work. The simulator used, Tracksail has been used by others for development and testing of autonomous sailing robot systems including Sauze and Neal [@Sauze200811]. Tracksail is Java based open source software and communicates with boat controllers by means of a standard network socket.
A running rate of 1Hz was fixed in the controller code for all controller configurations. This value was chosen in order to ensure that the more sophisticated controllers could run a complete cycle as there were initial concerns that for type-2 based controllers especially the overhead would be too high for a faster rate. While this low running rate will lead to overall lower performance we believe consistency between controllers is more important in this work. A comparison of type-1 and type-2 controllers running at different speeds is reserved for future work.
The cumulative RMSE (Root-mean square error) between the current and desired heading (measured in degrees) will be the main metric used for comparison. As is usual with control experiments, lower values represent a better performing controller.
Hypothesis {#sec:hyp}
----------
We hypothesize that there will be A point at which the difficulty of the course is sufficient that the type-2 will significantly improve upon the results of the standard type-1 controller. We expect that this behaviour will occur more obviously in situations with a higher uncertainty score as described in Table \[tab:Configs\] and with the extra turns exaggerating the effect further. The uncertainty score is used only for giving an arbitrary ordering for the configurations and is calculated by summing the direction and speed uncertainty scores together.
It is expected that as the various wind configurations are tested the RMSE will increase in a predictable manner — with configurations A and B showing lower RMSE values than the configurations H and I for example. We do not expect a linear increase especially as several configurations have the same uncertainty score.
We also expect to see that as the FOU size is increased the performance will start at type-1 levels. We expect this to be followed by an increase in performance and a subsequent drop as the FOU increases to cover more of the universe of discourse. This will result in extremely bad performance which, in the worst cases will prevent the course from being completed at all.
Experimental Design {#sec:Methodology}
-------------------
The controllers under test maintained the same membership functions throughout all experiments, with the type-1 membership functions shown in \[fig:T1MF\]. The only change in each run was the horizontal movement of the type-2 controller that alters the size of the FOU. We tested six different values for each FOU size, starting at 0 and increasing to a maximum of 25 in increments of 5. An example FOU size 10 is shown in Figure \[fig:Movement10T2\].
In our application we derive our type-2 footprints of uncertainty by introducing a horizontal movement to the type-1 membership functions, with the amount of movement being used will be varied as a parameter value. Figures \[fig:Movement10T2\] and \[fig:Movement20T2\] show examples of this with the type-1 membership function being moved 10 and 20 respectively to give the shown FOUs.
Two separate mechanisms were used to gradually increase the difficulty of completing the course, with the aim of highlighting the differences in performance that can be achieved with the various fuzzy systems under test. The first mechanism is the way point system of the simulator which will allow us to define the number and size of turn that the controller must steer the boat through in order to complete the course. The second mechanism is the introduction of noise into the environment in the form of variations of the wind. Table \[tab:Windspeed\] outlines the configurations of wind that we will be using in this experiment.
Direction
------------------- --- ----- -----
None 0 180 180
Low 1 160 200
High 2 140 220
Speed
None 0 7 7
Low 1 4 10
High 2 1 13
\[tab:Windspeed\]
: Wind Configurations of Experiments. The Uncertainty score is the sum of the directional and speed uncertainty scores as shown in Table \[tab:Windspeed\].
--------------- ------------- ------------- -------
Configuration Speed Direction
Uncertainty Uncertainty Score
A None None 0
B Low None 1
C None Low 1
D Low Low 2
E High None 2
F None High 2
G High Low 3
H Low High 3
I High High 4
--------------- ------------- ------------- -------
: Wind Configurations of Experiments. The Uncertainty score is the sum of the directional and speed uncertainty scores as shown in Table \[tab:Windspeed\].
\[tab:Configs\]
An automated control rig was used to execute batches of 30 runs for each combination of controller, parameter value (FOU size) and course layout. Each piece of software (controller, simulator and common code) maintains its own logging files that can be analysed to produce RMSE values and other interesting statistics.
Course Design
-------------
The courses will be built up from the simplest of all courses — a straight line with a parallel fixed wind, in which the boat must simply move forward in order to complete the course. The difficulty will then be increased by adding turns of varying angles as shown in Figure \[fig:Course\]. It can be observed that the courses under test will containing either one or two turns. The vertical movement required to complete the course, defined as either 0, 25, 50 or 100 metres, will alter the angle that the boat must turn in order to complete the course. The angle required for the first turn are 5.71$^{\circ}$, 11.42$^{\circ}$ and 21.84$^{\circ}$ for 25, 50 and 100 meters vertical movements respectively while the second turn, being twice as large will be 11.4$^{\circ}$, 22.84$^{\circ}$ and 43.68$^{\circ}$.
![Each coloured line represents a course layout under test. The white circles represent end points and the black circle the start point. The angles required for the first turn are 5.71$^{\circ}$, 11.42$^{\circ}$ and 21.84$^{\circ}$ for 25, 50 and 100 meters vertical movements respectively. Not to scale.[]{data-label="fig:Course"}](images/Course)
Every combination of course layout and wind configurations, as shown in Table \[tab:Configs\], will be tested with each controller configuration. We will start with no noise (configuration A) and move towards the most uncertain environment (Configuration I). Every four seconds a wind change will be triggered by the simulator using a Gaussian random number generator to change the values of the wind speed and direction. This gradual increase of noise will allow structured observations to be made about the effects of noise upon the performance of type-2 controllers with varying FOU sizes.
Results {#sec:Results}
=======
One sided Wilcoxon tests were used to test the statistical significance of the difference between two individual batches of experiments. The result for this test is a P-Value with a small value ($<$0.0005) indicating a statistically significant difference. For clarity course layouts are displayed as the vertical distance hyphenated with the number of turns, for example Single-25 would indicate a course which a single turn and 25m of vertical movement.
The first test for all experiments was a comparison of the type-1 controller metric values with the FOU size 0 type-2 controller values in order to ensure the values of each were statistically similar. This allowed us to ensure all parts of the simulator setup were functioning correctly and gave a good sanity check for each experiment.
\[fig:0\_Point\]\
\
Configuration
--------------------- ------ ------ -------
A 5.93 3.56 -2.37
B 8.35 3.91 -4.44
C 6.34 3.31 -3.03
D 5.90 3.20 -2.70
E 7.41 4.08 -3.33
F 4.83 2.84 -1.99
G 6.32 3.46 -2.86
H 5.10 2.44 -2.66
I 4.72 2.66 -2.06
\[tab:50\_2\_Diff\]
: RMSE Difference between Type-1 and a Type-2 Controller with FOU size of 20 on Single-100 course layout
Configuration
--------------- ------- ------- -------
A 15.29 15.70 0.41
B 15.75 22.43 6.69
C 11.84 16.68 4.83
D 12.33 17.15 4.82
E 12.53 25.68 13.15
F 11.67 15.53 3.86
G 14.53 15.50 0.97
H 13.68 22.22 8.54
I 12.97 16.35 3.38
: RMSE Difference between Type-1 and a Type-2 Controller with FOU size of 20 on Single-100 course layout
\[tab:100\_2\_Point\]
\
We first consider Table \[fig:0\_Point\], which shows the results of a benchmark experiment in which the majority of controllers simply maintain a straight course when the FOU size was 0 (equivalent to type-1). The average RMSE was expected at 0.0 with no statistically significant differences except with the very widest FOU sizes where performance decreases significantly as shown by the RMSE increasing in Figure \[fig:0\_Config\]. We believe that these results are caused by the fact that the controller does not need to execute any turns or course corrections in order to complete the task. This means any performance benefits/penalties a controller may have when turning do not have a chance to become apparent.
The next set of data to be considered is shown in Figures \[fig:2\_Config\] and \[fig:3\_Config\]. These show how the RMSE value (on the $y$ axis) change as the FOU size is increased from 0 to 25 (as shown on the $x$ axis) for each wind configuration (each coloured line). Each course configuration is shown in a separate Figure. In each of the figures we can see some similarities. In general the RMSE increases (signifying decreasing performance) as FOU size exceeds 20. We can also observe that improvements in performance happen before this point, usually at a FOU size of 20 — this is most obvious in Figure \[fig:50\_Config\], but can also be observed in Figure \[fig:25\_Config\] and Figure \[fig:3\_50\_Config\].
Figures \[fig:2\_Gain\] and \[fig:3\_Gain\] plot the difference between the type-1 controller and the best performing type-2 controller which as stated previously commonly occurs when the FOU is 20. The $x$ axis shows the different wind configurations under test while the $y$ axis shows difference in RMSE between the type-1 and type-2 controllers. The data for vertical movements 50m and 100m with a single turn are shown in tables \[tab:50\_2\_Diff\] and \[tab:100\_2\_Point\]. It is hard to observe any obvious patterns in these plots suggesting that the noise level is not directly linked to the overall performance difference of the controller. However in Figures \[fig:25\_Gain\] and \[fig:50\_Gain\] all points have a negative difference, shown by green circles, representing an improvement in performance over the type-1 RMSE value. In all other cases very few points show improvement over the type-1 value, shown by mostly red and black symbols.
Figures \[fig:2\_Course\_plot\] and \[fig:3\_Course\_Plot\] show example course plots of single and double turn courses with all the various wind configurations under test represented by coloured lines and the white circles indicating way points that must be reached to complete the course. We can see a rise in difficulty of the course with both angle and number of required turns increasing, from left to right, which in turn seems to be leading to more runs showing additional turns such as green line in Figure \[fig:3\_100\_Plot\] being a good example.
-------- -------- -------- ---------- --
Wind Type-1 Type-2
Config RMSE RMSE Movement
A 2.72 1.55 25
B 2.82 1.78 25
C 2.60 1.28 25
D 2.81 1.89 25
E 2.58 1.87 25
F 2.16 1.08 25
G 2.17 1.06 25
H 2.67 1.80 25
I 2.24 0.85 25
A 7.00 3.56 50
B 6.76 3.91 50
C 6.51 3.31 50
D 5.99 3.20 50
E 6.98 4.08 50
F 4.86 2.84 50
G 4.85 2.44 50
H 6.49 3.46 50
I 4.82 2.66 50
-------- -------- -------- ---------- --
: RMSEs and P-Value of best performing FOU sizes in comparison with type-1 FOU size for double turn course configurations.
\[tab:2\_Point\_SI\]
---------------------- -------- -------- ---------- ------ --
Wind Type-1 Type-2 FOU
Config RMSE RMSE Movement Size
A 12.94 11.11 50 20
B 12.79 9.84 50 10
E 12.66 9.23 50 15
I 11.07 10.06 50 15
\[tab:3\_Point\_SI\]
---------------------- -------- -------- ---------- ------ --
: RMSEs and P-Value of best performing FOU sizes in comparison with type-1 FOU size for double turn course configurations.
Tables \[tab:2\_Point\_SI\] and \[tab:3\_Point\_SI\] show the P-Value obtained when comparing the type-1 controller with the best performing FOU size for each wind configuration and vertical movement combination for both single and double turn experiments respectively. If there is no FOU size in which better performance is observed then this configuration is omitted. We can observe firstly that there are no points in which the vertical movement is 100. Secondly that double turn experiments have considerably fewer points than the single turn. An explanation for this will be discussed in the next section.
Discussion {#sec:Discussion}
==========
It can be observed from the results obtained and outlined in the previous section that type-2 based controllers can and do out-perform type-1 controllers in several circumstances. However this does not occur in the majority of cases. It is, in fact, more common for the performance to be similar to the type-1 value (statistically so in many but not all of cases).
If we enumerate the number of cases we find only 23 of the total of 324 (comprised of nine wind configurations, six different FOU sizes, three different vertical movement values and two different turn counts) show statistical improvement equating to approximately 8%. This relativity low percentage shows that those researchers who moved from type-1 to type-2 expecting a large increase in performance are more than likely to see at best the same performance but in most cases significantly worse unless considerable design effort is undertaken.
Our results are supported by other works in which type-2 performance is compared with type-1 such as the work by Musikasuwan et al in [@GaribaldiOzen0401] where a type-1 controller outperforms, albeit by a small margin, a type-2 based controller. While this work was more focussed on number of model parameters in each controller the essential result — that type-1 can out perform type-2 under the correct circumstances agrees with the finding of this paper.
![RMSE Values for Each wind configuration for each experiment[]{data-label="fig:ConfigNoise"}](images/ConfigVsNoise)
The ordering of the individual wind configurations in each of Figures \[fig:2\_Config\] and \[fig:3\_Config\] does not match with our expected hypotheses, in that the higher noise levels do not produce significantly higher RMSE values. This can be better seen in Figure \[fig:ConfigNoise\] in which the RMSE for each wind configuration, vertical movement and turn count combination is plotted with the FOU size being held at 20. In the majority of cases wind configuration ’B’ (red crosses) tends to have one of the the highest RMSE over the entire range of FOU sizes. This contrasts with wind configuration ’I’ (orange points) which seem to often appear at the bottom of the graph indicating the best performance. This seems counter to what might be expected, which would be for wind configuration A to have the lowest RMSE and configuration I to have the highest (as common sense would seem to indicate that more noisy environments are more difficult to sail in). Whether this conclusion is a general result or an artefact of the nature of this specific control problem is not yet known but will be the subject of future research.
We also observe the spread of the results for different wind configurations increases with the course difficulty. When the vertical movement is 25 with a single turn, the results are much closer together with a difference between highest and lowers RMSE value of 1.04. This contrasts significantly with the 100-double turn experiment in which the difference is 9.98. These results can be observed when comparing Figures \[fig:2\_25\_Plot\] and \[fig:3\_100\_Plot\]. This is an expected result as with each increase in course difficulty the number of course corrections that must be done by each controller increases, meaning there is greater scope for a controller to demonstrate its improved performance (or lack thereof).
The Figures \[fig:2\_Gain\] and \[fig:3\_Gain\] show that there is no obvious correlation between wind configuration (and therefore environmental noise) and the performance change achieved when moving to a type-2 controller. This could be down to the ordering of the configurations, as defined in Table \[tab:Configs\]. Multiple configurations have been given an equal uncertainty score based on the assumed equal weighting of the two noise sources and this may be a faulty assumption. This also contrasts with the findings made by Sepulveda et al [@SEPULVEDA2007] in which type-1 and type-2 controllers are tested and the type-2 out performing the type-1 in all cases. This occurs both with and without uncertainty and the difference in performance seems to have an increasing correlation. This suggests either the difference is down to application. Alternativly they have simply not tried as many noise configurations as we have done here meaning the differences found here have not been able to present themselves.
The addition of turns to increase the difficulty of the course has a significant effect on the performance of the controllers. It can be observed between Figures \[fig:2\_Config\] and \[fig:3\_Config\] that every value is higher in the double turn situation in comparison with the single turn.
Conclusions & Future Work {#sec:Conclusions_Future}
=========================
We have shown that type-2 based controllers can and do out-perform type-1 controllers. However, care must be taken in the design of the type-2 system, especially with regards to the size of the FOU. Too small an FOU and the Type-2 FLC will not improve over the type-1. Too large and it will perform worse. In our selected application an FOU size of 20 seems to be the optimal value over a range of experimental conditions. Further work will be required to determine the reason for this value.
Overall, this work shows the association between performance change and environmental noise to be considerably more complex than previously assumed. The statement that increasing environmental noise will lead to the type-2 improving in performance compared to type-1 is not supported by the results in this paper.
Future Work
-----------
The next step in this work will be to perform these experiments in a real world environment and to observe to what degree the findings presented here in simulation apply to a real world control problem. Based on the results found in these experiments, generalised type-2 may be an avenue for future work.
Acknowledgements
================
The authors would like to thank the School of Computer Science, University of Nottingham for their support in funding this paper.
|
---
abstract: |
We calculate the new dinamic exponent $\theta $ of the 4-state Potts model, using short-time simulations. Our estimates $\theta _{1}=-0.0471(33)$ and $%
\theta _{2}=$ $-0.0429(11)$ obtained by following the behavior of the magnetization or measuring the evolution of the time correlation function of the magnetization corroborate the conjecture by Okano et. al. In addition, these values agree with previous estimate of the same dynamic exponent for the two-dimensional Ising model with three-spin interactions in one direction, that is known to belong to the same universality class as the 4-state Potts model. The anomalous dimension of initial magnetization $%
x_{0}=z\theta +\beta /\nu $ is calculated by an alternative way that mixes two different initial conditions. We have also estimated the values of the static exponents $\beta $ and $\nu $. They are in complete agreement with the pertinent results of the literature.
PACS: 05.50.+q, 05.10.Ln, 05.70.Fh
author:
- Roberto da Silva
- 'J. R. Drugowich de Felício'
title: 'Critical dynamics of the Potts model: short-time Monte Carlo simulations. '
---
Introduction
============
Several results about critical phenomena have been recently obtained using Monte Carlo simulations in short-time regime [@Zheng]. Such simulations are more convenient than traditional ones done in the equilibrium regime because they circumvent a known problem in computational physics: the critical slowing down phenomena.
Investigating Monte Carlo simulations before attaining equilibrium permits to us obtaining the static critical exponents ($\beta $ and $\nu $) but also leads to the less known dynamical ones. The scaling equation for non-equilibrium regime was obtained by Jansen *et al.* [@Jansen], on the basis of renormalization group theory. For systems without conserved quantities like energy and magnetization (model A in the terminology of Halperin and Hohenberg [@Halperin]), it is written as $$M^{(k)}(t,\tau ,L,m_{0})=b^{\frac{-k\beta }{\nu }}M^{(k)}(b^{-z}t,b^{\frac{1%
}{\nu }}\tau ,b^{-1}L,b^{x_{0}}m_{0}) \label{Jansen_eq}$$where $m_{0}$ is the initial magnetization, $x_{0}=$ $x_{0}(m_{0})$ [PRL do Zheng]{} is the anomalous dimension of the initial magnetization, $%
\beta $ and $\nu $ are the known static exponents and $z$ is the dynamic one ($\tau \sim \xi ^{z}$). The values of $M^{(k)}$ are the $k$th moments of magnetization, defined by $M^{(k)}=\left\langle M^{k}\right\rangle $, where $%
\left\langle \cdot \right\rangle $ indicates an average over several samples randomly initialized but satisfying the condition of having the same magnetization $(m_{0})$ at the beggining. The relevance of this result is allowing to include the dependence on the initial conditions of the dynamic systems in their non-equilibriun relaxation. As an important consequence, they could advance the existence of a new critical exponent $\theta $, which is independent of the known set of static exponents and even of the dynamic exponent $z$.
By considering large systems and choosing $\tau =0$ $(T=T_{c})$ ; $b^{-z}t=1
$ in the equation (\[Jansen\_eq\]), we obtain the dynamic scaling law for the magnetization: $$M(t,m_{0})=t^{\frac{-\beta }{\nu z}}M(1,t^{\frac{x_{0}}{z}}m_{0}).
\label{scaling}$$Expanding $M$ around the zero value of the parameter $u=t^{\frac{x_{0}}{z}%
}m_{0}$, one obtains: $$M(t,m_{0})=m_{0}t^{\theta }+O(u^{2}) \label{ordem}$$which leads to the power law: $$M(t)\sim m_{0}t^{\theta } \label{initial slip}$$whereas $t<t_{0}\sim m_{0}^{-\frac{z}{x_{0}}}$ since in this regime $u\ll 1$. This new universal stage, characterized by the exponent $\theta =\left(
x_{0}-\frac{\beta }{\nu }\right) /z$ , has been exhaustively investigated to confirm theoretical predictions and to enlarge our knowledge on phase transitions and critical phenomena.
Indeed the anomalous behavior of the magnetization at the beginning of relaxation, also called critical initial slip, was originally associated to a positive value of $\theta $. This kind of behavior was confirmed in the kinetic $q=2$ and $3-$state Potts models [@Zheng], as well as in irreversible models like the majority voter one [@TTome] and the probabilistic cellular automaton proposed by Tomé and Drugowich de Felício to describe part of the immunological system [@TD]. However, the exponent $\theta $ can also be negative. This possibility was observed in the Blume-Capel model (analytical [@oerding] and numerically [nossopaperphysrevE]{}) and in the Baxter-Wu model (numerically [Drugoarashiro]{}), an exactly solvable model which shares with the $4-$state Potts model the same set of critical exponents. In addition, a negative but close to zero value for $\theta $ was obtained for the two-dimensional (2-D) Ising model with three-spin interactions in just one direction (IMTSI) [@DrugoSimoes], a model which is also known to behave to the $4-$state Potts model universality class.
However, at the best of our knowledge a numerical determination of the exponent $\theta $ for this special case ($q=4$) of the Potts model continues to be lacking. Thus, we decided to investigate the short-time behavior of that model in order to better understand its dynamics and also to learn about the role of the exponent $x_{0}$ of the nonequilibrium magnetization concerning the well established classification of the models (at least in what concerns the static behavior) in the universality classes.
As we mentioned above the estimates for the exponent $\ \theta $ for two models that belong to the same universality class as the 4-state Potts model have revealed discrepant results. Whereas the two-dimensional Ising model with threee spin interactions in one direction (IMTSI) obeys the conjecture by Okano et al. [@Okano]($\theta $ should be negative and close to zero) [@DrugoSimoes] the Baxter-Wu model [@Drugoarashiro] strongly disagrees from that prediction. Putting in numerical values, whereas the IMTSI model exhibits a small and negative value $(-0.03\pm 0.01)$ for $%
\theta $, when studying the BW model we found $\theta =-0.186\pm 0.002$.
In this paper we study the short-time behavior of the two-dimensional four-state Potts model and present numerical estimates for $\theta $ using two different methods. First, we fix the initial value of the magnetization and follow its time evolution to obtain $\theta (m_{0})$ which in turn can be extrapolated to lead to $\theta $ when $m_{0}\rightarrow 0$. Second, we deal with the time correlation of the magnetization defined by [@TTome]:
$$C(t)=\frac{1}{N^{2}}\left\langle M(t)M(0)\right\rangle$$
where the average is done over samples whose initial magnetizations are randomly choosen but obey $\left\langle M(0)\right\rangle =0$. In reference [@TTome] it was shown that this quantity exhibits at $T=T_{c}$ the power-law behavior: $$C(t)\sim t^{\theta }\text{.} \label{corr}$$This approach has several advantages when compared to the other technique but its application was initially restricted to the models which exhibit up-down symmetry. Recently, Tome [@TTomeII] has shown that this result is more general and can be applied whenever the model presents any group of symmetry operations related to the Markovian dynamics. Equation \[corr\] was shown to be valid for instance in the antiferromagnet ordering model, models with one absorvent state and even in the Baxter-Wu model [Drugoarashiro]{} which exhibits a $Z(2)\otimes Z(2)$ symmetry.
It is well known that the $q\neq 2$ Potts Model does not have up-down symmetry. However, based on the paper of Tomé [@TTomeII] we can use this approach to calculate the exponent $\theta $ of the three- and four-state Potts models.
We checked this possibility working with the three-state Potts model. Our estimate is $\theta =0.072(1)$, in good agreement with the result obtained by Zheng [@Zheng] ($\theta =0.070(2)$) and with the estimate obtained by Brunstein and Tome [@Brunstein] using a cellular automaton which exhibits C3v symmetry, the same as the three-state Potts model. A plot of the correlation function of the magnetization $C(t)$ in this case is presented in the Fig. 1.
In the sequence we estimated the exponent $\theta $ of the 4-state Potts model which agrees with the conjectured by Okano et al. In addition, they agree with previous estimates for $\theta $ obtained by Simões and Drugowich de Felício [@DrugoSimoes].
To confirm our first estimate we simulated the magnetization evolution for four different initial values of $m_{0}$ which after extrapolated ($%
m_{0}\rightarrow 0$) led to a similar result.
It is clear that having the exponent $\theta $ the anomalous dimension $%
x_{0} $ follows. But inspired in previous studies [@nosso; @paper; @Phys.; @Letters] we decided to investigate a direct manner in determining $x_{0}$ using mixed initial conditions. In order to build the adequate function we remember that when in contact with a heat bath at $T=T_{c}$ the magnetization of completely ordered samples ($m_{0}=1$) decays like a power law
$$M(t)\sim t^{^{\frac{-\beta }{\nu z}}}\text{.} \label{magnetization decay}$$
Thus, according to the scaling relations (\[initial slip\]) and ([magnetization decay]{}) it is enough to work with the ratio $$F_{3}(t,L)=\frac{\left\langle M\right\rangle _{m_{0}\rightarrow 0}}{%
\left\langle M\right\rangle _{m_{0}=1}} \label{f2}$$to obtain a power law which decays as $t^{\theta +\beta /\nu z}$ $%
=t^{x_{0}/z}$ . Using $z$ as input [@nosso; @paper; @Phys.; @Letters] we can achieve the value of $x_{0}$. On the other hand, by following the relation (\[magnetization decay\]) we estimate the ratio $\beta /\nu z$ which can be compared with the exact result $\beta /\nu =1/12$ after using the value of the dynamical exponent $z$.
Finally using derivatives of the magnetization at an early time and once more the value of $z$ as input we obtain the critical exponent $\nu $ of the correlation length, whose numerical estimates by usual techniques (phenomenological renormalization group, hamiltonian studies and equlibrium Monte Carlo simulations) are always very different from the pertinent result ($\nu =2/3$).
The paper is organized as follows: in the next section we present a brief review and some details of the simulation. The results are presented in Section 3 and our conclusions are in section 4.
The kinetic $4$-states Potts model
===================================
In time-dependent simulations we are interested in finding power laws for physical quantities even when the system is far from equilibrium. In this regimen the magnetization $M(t)$ must be calculated as an average over several samples because the system does not obey any *a priori* probability distribution. The average can be done in different ways: we can prepare all the samples with the same initial magnetization (sharp preparation) or generate samples which satisfy a less restrict criteria like to have mean value of the magnetization equal to zero.
The $q$ states Potts model ferromagnetic without the presence of an external field is defined by the Hamiltonian [@potts]:
$$H=-J\sum \limits_{\left\langle i,j\right\rangle
}\delta_{\sigma_{i},\sigma_{j}} \label{hamiltoniana}$$
where $J$ denotes the interaction between the nearest neighbors $%
\left\langle i,j\right\rangle $ and $\sigma_{i}$ can assume different values $\sigma _{i}=0,1..,q-1$. If two spins are parallel they contribute with energy $-J$, else the energy is null.
The critical temperature of this model, is known exactly [@potts] , $$\frac{J}{k_{B}T}=\log[1+\sqrt{q}]. \label{parcritico}$$
The magnetization, different of other models as Ising and Blume Capel is not only the sum of variables of spin. A general expression used for the magnetization in the Potts model that considers an average over the sites and over the samples is written as $$\left\langle M(t)\right\rangle =\frac{1}{N_{s}(q-1)N}\sum%
\limits_{j=1}^{N_{s}}\sum\limits_{i=1}^{L^{d}}(q\delta _{\sigma
_{i,j}(t),1}-1) \label{mag}$$where $\sigma _{i,j}$ denotes the spin $i$ of the $j$th. sample at the $t$th. MC sweep. Here $N_{s}$ denotes the number of the samples and $L^{d}$ is the volume of the system. This kind of simulation is performed $N_{B}$ times to obtain our final estimates as a function of $t$. In order to prepare a lattice with a given magnetization we need to choose the states $\sigma
_{i}=0,1,2,3$ in each site with equal probability (1/4). Next we measure the magnetization and change states in sites randomly chosen in order to obtain a null value for the magnetization. Finally we change $\delta $ sites occupied by $\sigma _{i}=0,2$ or $3$ and substitute by $\sigma _{i}=1$. The initial magnetization in this case will be given by: $$m_{0}=\frac{4\delta }{3N} \label{initial_magnetization}$$where $N=L^{d}$.
We have chosen to update the spins according to the heat bath algorithm, which means that the probability that the spin $\sigma _{\overrightarrow{x}}$ localized at the site $\overrightarrow{x}$ can assume another value $\sigma
_{\overrightarrow{x}}^{f}$ is given by: $$P(\sigma _{\overrightarrow{x}}\rightarrow \sigma _{\overrightarrow{x}}^{f})=%
\frac{\exp(-\frac{J}{k_{B}T}\sum\limits_{\overrightarrow{k}}\delta _{\sigma
_{\overrightarrow{x}+\overrightarrow{k}}(t),\sigma _{\overrightarrow{x}%
}^{f}})}{\sum\limits_{\sigma _{\overrightarrow{x}}^{f}=0}^{3}\exp(-\frac{J}{%
k_{B}T}\sum\limits_{\overrightarrow{k}}\delta _{\sigma _{\overrightarrow{x}+%
\overrightarrow{k}}(t),\sigma _{\overrightarrow{x}}^{f}})} \label{prob}$$where $\sum\limits_{\overrightarrow{k}}$ denotes the sum about the nearest neighbors sites of the spin at the site $\overrightarrow{x}$ .
Results
=======
We performed Monte Carlo simulations for $N_{B}=5$ different bins and four different initial magnetizations, $m_{0}=4\delta /3N\approx 0.033$, $0.049$, $0.066$ and $0.082$, for a lattice of size $L=90$. These magnetizations correspond respectively to $\delta =200$, $\delta =300$, $\delta =400$, $%
\delta =500$ according to the expression (\[initial\_magnetization\]). To estimate the errors we ran $5$ different bins, with $35000$ samples each one.
In the figs. 2 and 3 we present the log-log plot for two different initial magnetizations and two different values of $m_{0}=0.033$ and $m_{0}=0.082$ respectively. The fit had excellent goodness $(Q=0.99)$ in the entire range $%
[t_{i},t_{f}]\subset \lbrack 0,200]$. We have chosen the range $[10,100]$ to estimate the value to $\theta $ for all initial magnetizations. We must note that the slope is positive for $m_{0}=$ $0.082$ but changes the signal at some value between $0.066$ and $0.049$. This trend to a negative value can be also certified in the Fig. 4 that shows the plot of $\theta =\theta
(m_{0})$. It is worth to mention that there is a clear linearity in the plot of $\theta $ versus the initial magnetization (see fig. 4) but big fluctuations are observed in the evolution of the magnetization in each case.The value found for $\theta _{ext}$ is $-0.0471(33)$ is in fair agreement with the result [@DrugoSimoes] for the IMTSI model and strongly disagrees with the result obtained for the Baxter-Wu model.
We also ran Monte Carlo simulations to obtain the time-evolution correlation $C(t)$, and the the dynamic exponent $\theta $ according to the relation (\[corr\]). For the same interval $[10,100]$ we have gotten the estimate $%
\theta =$ $-0.0429(11)$ that corroborates the above estimate $-0.0471(33)$ obtained from the evolution of the non-equilibrium magnetization. We used $%
35000$ samples and once more 5 different bins to make this estimate. In figure 5 it is shown the log-log plot of $C(t)\times t$ for the four-state Potts model.
Equilibrium simulations for the four-state Potts model are ever acompannied by strong fluctuations. The reason is the presence in the Hamiltonian of a marginal operator characterized by an anomalous dimension equal to zero. If we hope some signal of that anomalous behavior in short-time simulations done far from equilibrium we should look for that in the only new anomalous dimension: that of the initial magnetization $x_{0}$. In order to confirm the null value for the anomalous dimension we estimate directly $x_{0}$ using the reported function $F_{3}$.
Over here we used $N_{b}=5$ bins for each initial condition and again the $%
N_{s}=35000$ realizations. A plot when the initial magnetization is $%
m_{0}=0.082$ can be seen in the fig. 6. Extrapolating $\ $ to $%
m_{0}\rightarrow 0$ we found $\left( x_{0}/z\right) _{ext}=0.0077(33)$ with the same set initial magnetizations $m_{0}$ that was used to extrapolating $%
m_{0}$. Using as input $z=2.290(3)$, we obtain $x_{0}=0.0176(75)$.
The value of $\beta /\nu $ can be obtained directly of magnetization decay of a initial ordered state, considering the relation: $$\frac{\beta }{\nu }=\left( \frac{\beta }{\nu z}\right) _{c}\cdot z\text{.}
\label{betani}$$The value of the exponent obtained from the decay $(M\sim t^{-\frac{\beta }{%
\nu z}})$, was $\left( \frac{\beta }{\nu z}\right) _{c}=0.0547(1)$ with $%
Q=0.84$.
Using as the input the same value for $z$ value we achieved $\frac{\beta }{%
\nu }=0.12526(28)$ that is in a good agreement with the exact result $\frac{%
\beta }{\nu }=0.125$ [@potts].
Finally to find the exponent $\nu $, we calculated the expected decay of derivative
$$\begin{tabular}{lll}
$D(t)$ & $=$ & $\left. \dfrac{\partial}{\partial\tau}\ln
M(t,\tau)\right\vert _{\tau=0}$ \\
\ & \ & \ \\
\ & $\underset{=}{N}$ & $\dfrac{\ln M(t,\tau+\delta)-\ln M(t,\tau-\delta )}{%
2\delta}.$%
\end{tabular}
\ \label{derivative}$$
According to the reference (\[Jansen\_eq\]), for $m_{0}=1$, we expect of $%
D(t)$ the power law behavior$$D(t)\sim t^{\frac{1}{\nu z}} \label{lei de potencia derivada}$$and so $\nu =$ $\left( \dfrac{1}{\nu z}\right) _{c}^{-1}\dfrac{1}{z}$. In the Fig. 7 we illustrate the power law (\[lei de potencia derivada\]), performing simulations with $N_{s}=8000$ samples with $N_{b}=5$ to estimate errors and $t_{\max }=1000$ MC steps.
Our best estimative is $\dfrac{1}{\nu z}=0.65453(46)$, in the interval $%
[10,1000]$ with $Q=0.99$. The value found to $\nu $ at this interval is $\nu
=0.667(1)$. The value $\left( \dfrac{1}{\nu z}\right) _{c}$ also is used to estimate $\beta $. This minimizes the errors because:$$\beta =\left( \frac{\beta }{\nu z}\right) _{c}\left( \dfrac{1}{\nu z}\right)
_{c}^{-1} \label{beta}$$
The value found to $\beta $ is $0.0836(2).$ These values are in complete agreement with conjectured results $\nu =2/3=0.\overline{6}$ and $\beta
=1/12=0.08\overline{3}$.
Conclusions
===========
We calculated the dynamic exponent $\theta $ of the four-state Potts Model using two different techniques: the time-evolution of the magnetization when the samples are prepared with a small and nonzero value of $m_{0}$ for four different values of initial magnetization and through of the time correlation of the magnetization. The values found for $\theta $ in both cases corroborate the conjecture by Okano et al. and are in complete agreement with the estimate obtained for another model which is in the same universality class as the 4-state Potts model: the 2-D Ising model with three-spin interactions in one direction. We also estimated directly the anomalous dimension of the initial magnetization $x_{0}$ which results very closed to zero. The same does not occur in the Baxter-Wu model that, although exhibiting the same leading exponents as the 4-state Potts model, does not own a marginal operator which prevents us from determine with good precision the critical exponents.
We also estimated the static exponents $\beta $ and $\nu $ using the short-time Monte Carlo simulations and our results are in surprisingly agreement with pertinent results.
As an additional contribution we present a new estimate for the exponent $%
\theta $ of the 3-state Potts Model working with the time evolution of the time correlation of the magnetization $C(t)$. This kind of approach has been previoulsy used by Brunstein and Tomé when studying a four-state cellular automaton with C-3$\nu $ symmetry [@Brunstein].The result agrees with estimates obtained from the evolution of the magnetization [Zheng]{} after the extrapolation ($m_{0}\rightarrow 0$).
**Acknowledgments** {#acknowledgments .unnumbered}
===================
The authors thank to CNPq for financial support.
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|
---
abstract: 'Fusion of very neutron rich nuclei may be important to determine the composition and heating of the crust of accreting neutron stars. We present an exploratory study of the effect of the neutron-gas environment on the structure of nuclei and the consequences for pycnonuclear fusion cross-sections in the neutron drip region. We studied the formation and properties of Oxygen and Calcium isotopes embedded in varying neutron-gas densities. We observe that the formed isotope is the drip-line nucleus for the given effective interaction. Increasing the neutron-gas density leads to the swelling of the nuclear density. We have used these densities to study the effect of this swelling on the fusion cross-sections using the São-Paulo potential. At high neutron-gas densities the cross-section is substantially increased but at lower densities the modification is minimal.'
author:
- 'A.S. Umar'
- 'V.E. Oberacker'
- 'C. J. Horowitz'
- 'P.-G. Reinhard'
- 'J.A. Maruhn'
bibliography:
- 'VU\_bibtex\_master.bib'
title: 'Swelling of nuclei embedded in neutron-gas and consequences for fusion'
---
=10000 =10000 =10000
Introduction
============
Recent advances in radioactive beam technologies have opened up new experimental possibilities to study fusion of neutron rich nuclei [@balantekin2014]. Furthermore, near barrier fusion cross sections are relatively large so experiments are feasible with modest beam intensities. In addition, measurements are possible at the TRIUMF ISAC facility and in the near future at the NSCL ReA3-6 reaccelerated beam facility. Other radioactive ion beam facilities include ATLAS-CARIBU at Argonne National Laboratory, SPIRAL2 at GANIL (France), and RIBF at RIKEN (Japan). Note that the dynamics of the neutron rich skin of these nuclei can enhance the cross-section over that predicted by a simple static barrier penetration model. For example, neutrons may be transferred from the neutron rich beam to the stable target. Fusion of very neutron rich nuclei, near the drip line, raise very interesting nuclear structure and nuclear dynamics questions.
Neutron stars, in binary systems, can accrete material from their companions. This material undergoes a variety of nuclear reactions [@haensel2007]. First at low densities, conventional thermonuclear fusion takes place, see for example [@schatz2001]. Next at higher densities, the rising electron Fermi energy induces a series of electron captures [@gupta2007] to produce increasingly neutron rich nuclei. Finally at high densities, these very neutron rich nuclei can fuse via pycnonuclear reactions. Pycnonuclear fusion is induced by quantum zero point motion [@salpeter1969; @schramm1990]. The energy released, and the densities at which these reactions occur, are important for determining the temperature and composition profile of accreting neutron star crusts. The existence of the inner neutron-star crust, in which very neutron rich nuclei are immersed in a gas of neutrons raises the question, what is the impact of this neutron gas on nuclear fusion rates? This neutron drip region is believed to occur for densities in the approximate range of a few $\times10^{11}$ to $8\times10^{13}$ g/cm$^3$. One of the early studies of the inner crust, consisting of drip-line nuclei combined with the background neutron gas, had been done by Negele and Vautherin [@negele1973]. Therefore understanding fusion reactions of neutron rich isotopes near the drip line are important. Horowitz *et al.* [@horowitz2008] calculate the enhancement in fusion rates from strong ion screening using molecular dynamics simulations, and find that $^{24}$O + $^{24}$O can fuse near $10^{11}$ g/cm$^3$, just before neutron drip. Extensive studies of the astrophysical $S(E)$ factors have been done using densities emanating from microscopic calculations and a barrier penetration model for fusion [@beard2010; @afanasjev2012]. Furthermore, this fusion can take place in the background neutron gas that is present in the inner crust of a neutron star. The possible effect of the neutron gas background was discussed in Ref. [@afanasjev2012] by empirically changing the barrier height and width. Here, we study this effect by considering the presence of the background neutron gas microscopically by directly including the neutron gas and the nucleus in the same framework. We explicitly calculate the self-consistent proton and neutron densities of a single nucleus in equilibrium with the background neutron gas. Furthermore, for astrophysical applications, it seems clear that this adiabatic approach is the one that is relevant for calculating fusion rates in the inner crust. The neutron gas should have plenty of time to adjust to the presence of a nucleus.
The paper is organized as follows. Our computational approach to consider the presence of neutron gas together with the nucleus and the model for calculating the fusion cross-sections is discussed in Sec. \[sec2\]. Computational results for the Oxygen and Calcium systems is described in Sec. \[sec3\]. Finally, these results are discussed and we conclude in Sec. \[sec4\].
Computational Details\[sec2\] {#sec.formalism}
=============================
Computational setup
-------------------
Hartree-Fock (HF) calculations were done in a three-dimensional Cartesian geometry with no symmetry assumptions and using the Skyrme effective nucleon-nucleon interaction [@umar2006c]. The infinite neutron star crust environment is simulated by using a three-dimensional Cartesian box with periodic boundary conditions for both the bound and neutron gas states as well as the solution of the Poisson equation for the Coulomb potential, which is performed using Fast-Fourier Transform (FFT) techniques [@maruhn2014]. The Coulomb solution assumes global neutrality which means that the proton charges are compensated by a homogeneous negative electron gas cloud. In practice, this is achieved by setting the zero-momentum part of the Coulomb field in Fourier space to zero. The code uses the basis-spline collocation method for the lattice discretization of the HF equations using periodic boundary-conditions as described in Refs. [@umar1991a; @umar1991b; @bottcher1989]. The HF equations are solved using the damped gradient iteration method. The Skyrme parametrization used was SLy4 [@chabanat1998a]. In addition to providing a good description of nuclei this interaction has been used to produce an equation of state for neutron stars [@douchin2001].
For the choice of initial states to be used in HF minimization we have tried a number of choices, which all resulted in the same identical solution. One can first generate any isotope of the desired nucleus by solving the HF equations as described above and subsequently combine these states with a large number of free neutron gas states and minimize the entire system again. Alternately, one can simply choose a number of free proton states together with a large number of free neutron states and minimize this system. Both methods result in exactly the same numerical solution with a drip-line isotope corresponding to the nucleus with the given number of protons embedded in a given density of neutron gas states. Initial states, $\boldsymbol{\psi}$, are spinors with a non-zero upper component in case of time-reversal invariance. In case of no time-reversal invariance the number of states are doubled by adding spinors having non-zero lower components as well. They satisfy the periodicity condition $$\boldsymbol{\psi}_{\mathbf{n}}(\mathbf{r}+\mathbf{L})=\boldsymbol{\psi}_{\mathbf{n}}(\mathbf{r})\;,$$ where $\mathbf{n}=(n_x,n_y,n_z)$ and $n_a$ taking on integer values $-N_a,\ldots,+N_a$. Free states satisfying the above periodicity condition are simple plane-wave states with the appropriate normalization $$\boldsymbol{\psi}_{\mathbf{n}}(\mathbf{r})=\frac{1}{\sqrt{L_xL_yL_z}}e^{\imath(k_{n_x}x+k_{n_y}y+k_{n_z}z)}
\boldsymbol{\chi}_n\;,$$ where $k_{n_a}=2\pi n_a/L_a$ and $\boldsymbol{\chi}_n$ is an up or down spinor. The initial neutron and proton densities are perfectly uniform filling the entire numerical box. These initial states comprise the total number of states used in the self-consistent HF problem using the Skyrme interaction. For even number of states time-reversal is valid and the HF single-particle Hamiltonian only depends on particle density, $\rho$, kinetic energy density. $\tau$, and the spin-orbit pseudotensor $\mathbf{J}$ through the single-particle states [@chabanat1998a] $$\mathbf{h}\left( \left\{ \boldsymbol{\phi}_{\mu} \right\} \right) \boldsymbol{\phi}_{\lambda}=\epsilon_{\lambda}\boldsymbol{\phi}_{\lambda}
\;\;\;\;\;\;\;\;\;\lambda=1,...,N\;.
\label{tdhf0}$$ As the HF iterations proceed (preserving orthogonality for the entire system) some states evolve to form a bound nuclear system while the others remain as gas states showing some non-uniformity due to the presence of shell effects.
Fusion cross-sections
---------------------
The São Paulo model of fusion calculates an effective nuclear potential based on the density overlap between colliding nuclei [@gasques2004; @chamon2002]. Sub-barrier fusion cross-sections can then be calculated via tunneling. The model can be easily applied to a very large range of fusion reactions and qualitatively reproduces many experimental cross-sections [@gasques2005; @gasques2007]. Recently this model was used to tabulate astrophysical $S$ factors describing fusion of many carbon, oxygen, neon and magnesium isotopes for use in astrophysical simulations [@beard2010], see also Ref. [@yakovlev2010].
In this section we describe the São Paulo barrier penetration model to calculate fusion cross-sections. This starts with the double folding potential $V_F(R)$ [@gasques2004; @chamon2002], $$V_F(R)=\int d^3r_1d^3r_2 \rho_1(r_1)\rho_2(r_2) V_0 \delta(\mathbf{r}_1- \mathbf{r}_2 - \mathbf{R})\, .
\label{eq:vf}$$ Here $\rho_1$ and $\rho_2$ are the densities of the two nuclei and $V_0=-450$ MeV-fm$^{3}$. From $V_F$ a nonlocal potential $V_N(R,E)$ is constructed, $V_N(R,E)=V_F(R)e^{-4v^2/c^2}$, where $v$ is the local relative velocity [@gasques2004; @chamon2002] between the two nuclei at separation $R$ ($c$ is the speed of light) $$v^2(R,E)=\frac{2}{\mu}\left[E-V_C(R)-V_N(R,E)\right]\;.
\label{eq:v2}$$ Here $\mu$ is the reduced mass and $V_C(R)$ is the Coulomb potential at $R$. In practice, we use FFT techniques to calculate $V_F(R)$ as well as the Coulomb potential $V_C(R)$ (instead of using the point Coulomb formula). The velocity equation (\[eq:v2\]) has to be solved by iteration at each value of $R$ and $E$.
Note that the neutron gas background could behave differently for two nuclei in close proximity then it does for only a single nucleus. However for simplicity, in this first study, we consider only a single nucleus in the background gas at a time in order to get the density profiles shown in Fig. \[fig2\]. We then use these profiles in Eq. \[eq:vf\] and assume they are unmodified by the presence of the second nucleus.
The fusion barrier penetrabilities $T_L(E_{\mathrm{c.m.}})$ are obtained by numerical integration of the two-body Schrödinger equation $$\left[ \frac{-\hbar^2}{2\mu}\frac{d^2}{dR^2}+\frac{L(L+1)\hbar^2}{2\mu R^2}+V(R,E)-E\right]\psi=0\;,
\label{eq:xfus}$$ using the [*incoming wave boundary condition*]{} (IWBC) method [@hagino1999]. The potential $V(R,E)$ is the sum of nuclear and Coulomb potentials. IWBC assumes that once the minimum of the potential is reached fusion will occur. In practice, the Schrödinger equation is integrated from the potential minimum, $R_\mathrm{min}$, where only an incoming wave is assumed, to a large asymptotic distance, where it is matched to incoming and outgoing Coulomb wavefunctions. The barrier penetration factor, $T_L(E_{\mathrm{c.m.}})$ is the ratio of the incoming flux at $R_\mathrm{min}$ to the incoming Coulomb flux at large distance. Here, we implement the IWBC method exactly as it is formulated for the coupled-channel code CCFULL described in Ref. [@hagino1999]. This gives us a consistent way for calculating cross-sections at above and below the barrier via $$\sigma_f(E_{\mathrm{c.m.}}) = \frac{\pi}{k^2} \sum_{L=0}^{\infty} (2L+1) T_L(E_{\mathrm{c.m.}})\;.
\label{eq:sigfus}$$
Results\[sec3\]
===============
All the calculations presented here were done using a three-dimensional cubic Cartesian box with $31$ fm sides and $1.0$ fm lattice spacing. With the basis-spline method this gives highly accurate results for the HF problem [@umar1991a]. We studied two systems with $Z=8$ and $Z=20$ protons inside a neutron gas. The resulting nuclei are always spherical and density profiles can be obtained by taking a cut along a particular axis. Our mesh includes the origin in all directions. We have repeated some of these calculations in a cubic bot with $41$ fm sides and for the same neutron-gas density the results were numerically indistinguishable.
![(Color online) Mean-field potentials for neutrons and protons (solid lines) for the system with $Z=8$ and $520$ neutrons. The dashed lines indicate the energies of the bound single-particle states with degeneracies shown in brackets.[]{data-label="fig1"}](fig1.pdf){width="8.6cm"}
$Z=8$ system
------------
In these calculations we start with $8$ proton states and a number of neutron states ranging from $50-1020$. The Skyrme SLy4 force gives $^{28}$O as the slightly bound drip-line nucleus in free-space. For the above range of neutron states we also find the $^{28}$O to be the bound part of the system. As the number of neutron states is increased an overall negative potential is developed permeating the entire box. Figure \[fig1\] shows the neutron and proton mean-field potentials (solid lines) for $Z=8$ and $520$ neutrons. The dashed lines indicate the energies of the bound single-particle states with degeneracies shown in brackets. We define $N_\mathrm{bound}$ as the highest neutron s.p. state below the continuum threshold. Consequently, the bound and gas densites become $$\begin{aligned}
\rho_\mathrm{bound}
&=&
\sum_{\lambda=1}^{N_\mathrm{bound}}
\left|\phi_\lambda\right|^2
\; ,
\\
\rho_\mathrm{free}
&=&
\sum_{\lambda>N_\mathrm{bound}}
\left|\phi_\lambda\right|^2 \; .\end{aligned}$$
![(Color online) (a) Total density profiles for bound states; (b) density profiles for bound neutrons (solid lines) and protons (dashed lines), for the system with $Z=8$ and $n=20,520$, and $1020$ neutrons.[]{data-label="fig2"}](fig2.pdf){width="8.6cm"}
In this case the asymptotic value of the neutron potential is about $-8$ MeV. As it is the case in free-space increasing neutron number leads to deepening of the proton potential. In Fig. \[fig2\] we show the density profiles for neutrons and protons as well as the total density as a function of the number of neutron-gas states. The top frame shows the total density behavior as the neutron-gas density is increased. The curves labeled $n=20$ correspond to free-space $^{28}$O nucleus. As the external neutron-gas density is increased the bound system swells up in a way similar to a density scaling $\rho(r)\rightarrow\rho(sr)$ with $s<1$ as discussed in Ref. [@umar2007a]. While the peak of the total density decreases from the free-space value of $0.16$ fm$^{-3}$ to as low as $0.068$ fm$^{-3}$ for the $1000$ external neutron-gas state case, the tail region flattens and develops a larger spatial extent, since the total integral remains to be $28$. The density profiles are symmetric about $x=0$ and the numerical box extends to larger values then shown in the figure.
![(Color online) Ion-Ion potentials $V_{SP}(R)=V_F(R)+V_C(R)$ for $^{28}$O isotope as a function of the external neutron-gas density. Also shown is the point-Coulomb interaction. Densities are in units of gm/cm$^3$.[]{data-label="fig3"}](fig3.pdf){width="8.6cm"}
Using these densities in Eq. (\[eq:vf\]) we have calculated the corresponding ion-ion folding potentials as well as the Coulomb potential. In Fig. \[fig3\] these potentials are plotted for a range of external neutron-gas densities. What is observed is that for neutron-gas densities in the range $\rho_{gas}=2-4\times 10^{12}$ gm/cm$^3$ the effect of the gas is not changing the ion-ion potential in comparison to the free-space case in a considerable way. However, for gas densities above $10^{13}$ gm/cm$^3$ a very significant change is observed.
![(Color online) Astrophysical $S$ factor versus center of mass energy for fusion of $^{28}$O isotope as a function of external neutron-gas density. Cross-sections are calculated using the São Paulo barriers and the IWBC method.[]{data-label="fig4"}](fig4.pdf){width="8.6cm"}
The free-space barrier has a peak value of $7.87$ MeV located at $R=10.7$ fm. As the external gas density is increased the corresponding barrier height is reduced to $7.69$, $7.23$, and $6.39$ MeV with the peak location moving outward at $10.8$, $11.7$, and $13.3$ fm. This behavior is very similar to what is observed in free-space as one goes up in the oxygen isotope chain [@umar2012a]. Unfortunately, the dynamical density constrained time-dependent Hartree-Fock (DC-TDHF) methodi [@umar2006b; @umar2008a; @keser2012] used in Ref. [@umar2012a] is not applicable in this situation since it requires a fully dynamical calculation.
These together with the calculation of the energy dependence from Eq. (\[eq:v2\]) allows the calculation of fusion cross-sections as a function of external neutron-gas density, using the IWBC method discussed in the previous section. Figure \[fig4\] demonstrates the effect of neutron-gas seen in the potentials on the astrophysical $S$ factor. The S-factor for different external neutron-gas densities start to deviate from each other as soon as the center-of-mass energy falls below the barrier. Even for the lowest gas density of $2.9\times 10^{12}$ gm/cm$^3$ the difference with the free-space value is about a factor of two at the center-of-mass energy of $2$ MeV. The difference at higher gas densities are about $1-3$ orders of magnitude larger then the free-space values at sub-barrier energies.
![(Color online) The cross-sectional density ($y=0$ plane) profile of the $^{28}$O+500n system. The neutron density is low enough to have shell structures leading to the formation of neutron arms. []{data-label="fig5"}](fig5a.pdf){width="8.6cm"}
In Fig. \[fig5\] we plot the cross-sectional density ($y=0$ plane) profile of the $^{28}$O+500n system. In general we see a higher gas density in the vicinity of the nucleus as can also be seen as the dotted line in Fig. \[fig2\](a). The energy of this low density neutron gas is so low that even very small “shell effects” can lead to nonuniform densities. In some of the cases we studied the neutron density is low enough to have shell structures leading to the formation of neutron arms seen in Fig. \[fig5\]. If we replicate this cubic box in three-dimensions one sees a lattice like structure linked by these neutron arms. These effects are driven mainly by the periodic boundary conditions here. Irregularities in the grid of nuclei will produce more irregular structures of these arms. Since the density in the arms is of the order of free density such that they do not affect the analysis of the bound part.
$Z=20$ system
-------------
We have repeated the same study by starting with $20$ proton states and a number of neutron states ranging from $140-1040$. Using the Skyrme SLy4 we get $^{60}$Ca as the slightly bound drip-line nucleus in free-space. For the above range of neutron states we also find the $^{60}$Ca to be the bound part of the system. As the number of neutron states is increased an overall negative potential is developed permeating the entire box. Figure \[fig6\] shows the neutron and proton mean-field potentials (solid lines) for $Z=20$ and $540$ neutrons. The dashed lines show the mean-field potentials in free-space. In this case the asymptotic value of the neutron potential is about $-6.5$ MeV. At higher neutron densities we do observe the tendency for $^{72}$Ca to be the drip-line nucleus but the tendency is very weak and for practical purposes considering $^{60}$Ca is sufficient for the general purposes of this study.
![(Color online) Mean-field potentials for neutrons and protons (solid lines) for the system with $Z=20$ and $540$ neutrons. The dashed lines show the mean-field potential of $^{60}$Ca in free-space.[]{data-label="fig6"}](fig6.pdf){width="8.6cm"}
In Fig. \[fig7\] we plot the density profiles for neutrons and protons as well as the total density as a function of the number of neutron-gas states. The top frame shows the total density behavior as the neutron-gas density is increased. The curves labeled $n=40$ correspond to free-space $^{60}$Ca nucleus. As the external neutron-gas density is increased the bound system swells up as in the $Z=8$ case. While the peak of the total density decreases from the free-space value of $0.165$ fm$^{-3}$ to as low as $0.048$ fm$^{-3}$ for the $1000$ external neutron-gas state case, the tail region flattens and develops a larger spatial extent, since the total integral remains to be $60$. The density profiles are symmetric about $x=0$ and the numerical box extends to larger values then shown in the figure. Corresponding ion-ion folding potentials calculated by using these densities in Eq. (\[eq:vf\]) are plotted in Fig. \[fig8\] for a range of external neutron-gas densities. Again what we observe is that for neutron-gas densities in the range $\rho_{gas}=2-4\times 10^{12}$ gm/cm$^3$ the effect of the gas in changing the ion-ion potential compared to the free-space case is not significant. However, for gas densities above $10^{13}$ gm/cm$^3$ a very significant change is observed. The free-space barrier has a peak value of $47.4$ MeV located at $R=11.3$ fm. As the external gas density is increased the corresponding barrier height is reduced to $46.3$, $42.9$, and $37.4$ MeV with the peak location moving outward at $11.6$, $12.5$, and $14.3$ fm.
![(Color online) (a) Total density profiles for bound states; (b) density profiles for bound neutrons (solid lines) and protons (dashed lines), for the system with $Z=20$ and $n=40,540$, and $1040$ neutrons.[]{data-label="fig7"}](fig7.pdf){width="8.6cm"}
![(Color online) Ion-Ion potentials $V_{SP}(R)=V_F(R)+V_C(R)$ between two $^{60}$Ca isotopes as a function of the external neutron-gas density. Also shown is the point-Coulomb interaction. Densities are in units of gm/cm$^3$.[]{data-label="fig8"}](fig8.pdf){width="8.6cm"}
Figure \[fig9\] shows the fusion cross-sections calculated using the ion-ion potentials shown in Fig. \[fig8\]. The dramatic rise of the cross-section is obvious as the neutron gas density becomes significantly higher than the minimum neutron drip density.
![(Color online) Fusion cross-section as a function of center-of-mass energy for the fusion of two $^{60}$Ca isotopes calculated for changing external neutron-gas density. Cross-sections are calculated using the São Paulo barriers and the IWBC method.[]{data-label="fig9"}](fig9.pdf){width="8.6cm"}
Summary and Discussion\[sec4\]
==============================
Pycnonuclear reactions are expected to occur between very neutron-rich nuclei and in a dense background of of a neutron gas [@afanasjev2012]. We have made an exploratory study of the effect of the external neutron gas on nuclear fusion in the regime between the start of the neutron drip region and the melting region. For our study we have used an approach that treats the nuclei and the extra neutrons in a unified manner. The computations are done in full 3D with periodic boundary conditions and a special treatment of the Coulomb potential with periodic boundary conditions. For the calculation of fusion cross-sections we have used the São-Paulo model. In our calculations we observe that for lower background densities the cross-sections do not change in a very significant manner. On the other hand as we increase the neutron gas density we observe the swelling of the nuclei that results in the lowering of the ion-ion potential barriers and significant increase in the fusion cross-sections. At densities higher than the neutron-drip regime the melting of the nuclei can be observed.
While our methods give us a good understanding of fusion under these conditions more precise computations, including the full effects of pairing and effective interactions tailored for neutron star crust [@erler2013], may reduce some of the observed shell effects and modify some of the results but the main features observed are expected to remain the same. In addition, movement of nuclei inside the neutron gas as they approach each other may cause ripples and waves in the neutron background that can also influence these results. However, in the adiabatic limit these effects are not expected to be very large.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by DOE grant Nos. DE-FG02-87ER40365 and DE-FG02-96ER40975, and by the German BMBF under contract No. 05P12RFFTG.
|
---
abstract: 'We present $UBRI$ photometry and spectra for 60 quasars found within one square degree centered on the J0053+1234 region, which has been the subject of the Caltech Faint Galaxy Redshift Survey. Candidate quasars were selected by their ultraviolet excess with respect to the stellar locus, and confirmed spectroscopically. The quasars span a wide range in brightness ($17.5<B<21.6$) and redshift ($0.43<z<2.38$). These new quasars comprise a grid of absorption probes that can be used to study large-scale structure as well as the correlation between luminous galaxies, non-luminous halos, and Lyman-$\alpha$ absorbers in the direction of the deep pencil-beam galaxy survey. Spectra of 14 emission line galaxies found using the same technique are also presented.'
author:
- 'Daniel H. McIntosh[^1], Chris D. Impey, and Catherine E. Petry'
title: 'Quasars as Absorption Probes of the J0053+1234 Region[^2]'
---
Introduction
============
Pencil-beam galaxy redshift surveys are enormously useful in defining the evolution and clustering of luminous matter out to redshifts $z \gtrsim 1$. An important complement to such surveys is a network of quasar absorbers within the same volume, which allows the detection of matter in absorption that would be impossible to detect in emission. Leveraging a large investment of resources from the Hubble Space Telescope ([*HST*]{}), the two Hubble Deep Fields [HDF-N and HDF-S @williams96; @williams00] have attracted an international campaign of multi-wavelength observations. The more recent Ultra-Deep Field (UDF) is attracting similar attention. The Caltech Faint Galaxy Redshift Survey (CFGRS) has devoted a concentrated effort to two fields, using the Low Resolution Imaging Spectrograph at the Keck Observatory [@cohen96]. One CFGRS field is the northern Hubble Deep Field, and the other is a previously unstudied field centered at $\alpha = 00^{\rm h} 53^{\rm m} 23^{\rm s}$, $\delta = 12\degr 33\arcmin 58\arcsec$ (J2000). Initial spectroscopy has been published by @cohen99a [@cohen99b], and a clustering analysis based on both fields has also been presented [@hogg00]. These studies concentrate on the distribution and evolution of luminous matter along this sightline, and will eventually reveal the evolutionary properties of galaxies of different Hubble types, delineate large-scale structure in redshift, and help define the cosmic history of star formation. Here we provide a grid of quasar probes, which will allow an examination of the cold, diffuse, and dark components of the universe along the J0053+1234 pencil beam.
Regardless of the depth of an imaging survey, it can reveal only the luminous parts of galaxies, which contain about 10–15% of the baryons in the local universe, which in turn form only about 15% of the matter content of the universe [@turner01]. Studying the cold, diffuse, and dark components of the universe in a cosmological volume centered on J0053+1234 provides an important complement to the study of the luminous matter content. This can be accomplished by absorption line spectroscopy, using distant quasars as probes. For example, galaxy halos can be detected indirectly using the $\lambda\lambda$1548, 1550 and $\lambda\lambda$2796, 2800 doublets over the entire range $0 < z
< 4$ [e.g. @meylan95]. Furthermore, Lyman-$\alpha$ absorbers are as numerous as galaxies and they effectively trace the gravitational potential of the underlying dark matter [@miralda96; @croft98]. Given a sufficiently bright background quasar, absorbers can be detected with an efficiency that does not depend on redshift. In this respect, quasar absorption probes have an advantage over galaxy surveys that are always affected by, and sometimes compromised by, effects such as cosmological $(1+z)^4$ dimming, $k$-corrections, and morphological selection that depends on redshift.
The detection of a network of quasar absorbers in a volume that encompasses a deep pencil-beam survey allows measurements of large-scale structure that can be used to relate luminous baryons in galaxies to baryons that are largely intergalactic. Individual quasar sightlines show that metals such as and have correlation power on scales up to 100 $h^{-1}_{100}$ Mpc [@loh01], and multiple sightlines have been used to trace out three-dimensional structures on even larger scales [@dinshaw96; @williger96]. Locally, a grid of quasar probes have been used to trace out diffuse gas structure in three dimensions and relate it to the galaxy distribution in the direction of the galactic poles [@vandenberk99] and in the direction of the Virgo cluster [@impey99]. There is evidence that Lyman-$\alpha$ absorbers are significantly but weakly clustered on scales of 20-30 $h^{-1}_{100}$ Mpc [@williger00; @liske00]. Deep pencil-beam galaxy redshift surveys have shown that around half of the bright galaxies lie in high contrast structures with line of sight separations of 50-300 $h^{-1}_{100}$ Mpc [@cohen96; @cohen99a]. Thus, the spatial relationship between quasar absorbers and luminous galaxies can be studied as a function of redshift. Gas dynamical simulations indicate that low column density absorbers more closely reflect the underlying dark matter mass distribution than galaxies [@cen98].
We have already published first results on a quasar survey in an area encompassing the HDF-N [@liu99; @vandenberk00], and we are completing a multi-color search for quasars in a region centered on the HDF-S [see also @palunas00]. In this paper, we present the observations that have led to the discovery of 60 quasars in a one square degree field centered on the J0053+1234 region of @cohen99a [@cohen99b]. The galaxy redshift survey covered a region 14.6 arcmin$^{2}$ at the center of the field surveyed for quasars. In all three cases, quasar selection is a prelude to absorption line spectroscopy and a comparison of the spatial distribution of absorbers and galaxies. Although quasars as faint as $B\approx21.5$ have traditionally been too challenging for absorption line work, a new generation of large ground-based telescopes and multi-object spectrographs will enable the follow-up observations required for this sample.
We describe the imaging and multi-color photometry leading to the selection of quasar candidates in §2, and the quasar candidate selection in §3. In §4 we describe the spectroscopy. In §5 we summarize the yield of confirmed quasars, the reliability of the redshifts, and present a smaller number of emission line galaxies discovered in the same survey. We close with brief comments on future work on this region. A second paper (Petry, Impey, & McIntosh, in preparation) will present additional quasar candidates for the J0053+1234 field derived from deeper $U$ imaging of the central half square degree, as well as confirmed quasars in the entire field from two recent observing runs. Additional papers will present photometry, quasar candidate lists, and confirmed quasars for both the Hubble Deep Fields, North and South. Spectra have been obtained for quasars in all three fields that are of sufficient quality to measure strong Lyman-$\alpha$ absorbers, so these future papers will also include a first comparison between quasar absorbers and galaxies in the three pencil-beam surveys.
Photometry
==========
Observations
------------
We used the Kitt Peak National Observatory (KPNO) 0.9-meter Telescope with the CCD Mosaic Wide-Field Imager [@boroson94; @muller98] to acquire $UBRI$ observations of the one square-degree region centered on the J0053+1234 region ($\rm \alpha = 00^{h}53^{m}23.2^{s}, \delta= +12\degr 33\arcmin 58\arcsec$) during 1998 September 30 – October 2 UT, under generally good observing conditions.
The Mosaic imager has eight $2048$$\times$$4096$ pixel anti-reflection coated SITe Loral chips. The combined field of view spans $59\arcmin \times 59\arcmin$, and has a pixel scale of $0\farcs43$ (15 $\micron$ pixels). The readout noise for this camera is $5.66 e^-$, the dark current is negligible ($\sim15 e^-$ hour$^{-1}$), and the average single chip gain is $2.86 e^-$ ADU$^{-1}$. Each CCD has been thinned for detecting $U$-band photons; however, the quantum efficiency (QE) still falls off rapidly blueward of 4000 Å. Thus, the spectral response in the $U$-band is not a perfect match to standard @johnson66 $U$-band. The Mosaic camera employs a large (5.75 inches square) par-focal filter set. Each filter’s transmission and response, calculated from the QE, is plotted in Figure \[response\].
An area of sky centered on the J0053+1234 region was imaged in each passband using a standard dither pattern with five separate pointings a few arcminutes apart to remove inter-chip gaps, CCD defects and cosmic rays. This dither pattern ensured at least $80\%$ of the maximum exposure for all regions of a final combined image of 5 exposures. The total integration times were 420, 110, 30, and 70 minutes for the $U$, $B$, $R$ and $I$ bands, respectively. A variety of short and long exposure times were used for individual $U$ and $I$ frames to avoid saturation of the brightest objects. Unfortunately, we did not experience perfectly photometric conditions during our imaging campaign. Nevertheless, for relative photometric calibration we imaged @landolt92 standard star cluster fields, over a range of airmasses, at least three times during each night. The overall photometric zero point consistency was $\sim 10$%. Quasar selection is governed by relative colors with respect to the stellar locus; therefore, zero point determination did not limit our ability to identify quasar candidates with this photometry. We summarize the observations in Table \[obslog\].
Data Reduction
--------------
To select quasar candidates in the J0053+1234 region for spectroscopic follow-up, we require good photometric uniformity across our deep, wide-field images. Homogeneous photometry also requires procedures to deal with bad pixels, cosmic rays, and gaps between CCD’s in each passband. Achieving such high quality images requires stacking dithered Mosaic frames, which in turn places high demands on the initial data reduction steps [^3]. In particular, the data must be well-flattened and carefully corrected for photometric effects of the variable pixel scale. The data are reduced with a customized reduction pipeline that uses the IRAF[^4] environment and adheres to standard image reduction techniques.
We perform basic reduction of the individual Mosaic frames using the IRAF [*mscred*]{} package. This software allows image processing to be performed on multi-chip exposures as if they were single CCD frames. For each single Mosaic exposure, we trim and debias the eight individual chip images separately. A small correction ($<0.3\%$) is necessary due to cross talk between pairs of adjacent chips sharing the same electronics in the Mosaic detector. This correction subtracts a predetermined fraction of the adjacent chip’s $(i,j)^{\rm th}$ pixel value from pixel $(i,j)$ of the current chip. We then remove an averaged zero frame from each Mosaic image. The thinned Mosaic chips require no dark correction. Roughly $0.4\%$ of the full array of $8192\times 8192$ pixels are bad; these are flagged and included in the mask frames during the final image combination.
An important step towards achieving precise photometry is the determination and removal of the response function of the individual CCD’s – [*i.e.*]{} flat fielding the data. Traditionally, dividing each exposure by a uniformally illuminated blank frame will produce an image that has a uniform and flat appearance. Yet, the Mosaic imagers have pixel scales that decrease roughly quadratically such that an individual pixel in a field corner is 6% smaller, and contains only 92% of the flux, compared to a pixel at field center (see MosManual for details). Therefore, although an individual star anywhere on the image will have the same number of photons within the point-spread-function (PSF), the variable pixel scale causes the photometric zero point to vary by 8% over the field of view. We correct for this photometric effect following the recommendations given in the MosManual. Briefly, we flatten each image with a flat-field frame that has [*not*]{} been corrected for the variable pixel scale. Then, following our astrometric calibration and prior to stacking multiple exposures, we re-grid each frame to a tangent-plane projection with pixels of constant angular scale. We note that during the re-gridding, we [*do not*]{} scale each pixel photometrically by the amount each pixel area has changed. In this manner we account for the variable pixel scale and produce uniform images over the entire field of view.
We construct flat-field frames for each passband using a combination of twilight and night-sky flats. First we make a normalized flat with high signal-to-noise (S/N) by averaging a set of twilight illuminated exposures. This component accounts for both the small scale, high frequency (pixel-to-pixel) variations in response, and the spatial variations over large fractions of an image. We then fit a smooth surface to a night-sky flat produced by median combination of a set of unflattened frames of the J0053+1234 region with all objects masked out, and we multiply this smooth surface to the twilight flat frame to produce a high S/N flat with the spectral response of the night sky. We iterate the flat field construction twice to optimize the night-sky flat object masking. Thus, the resultant “super flat” for each passband is spatially and spectrally flat. We divide each exposure of the J0053+1234 region and the standard star cluster image by its super flat to achieve $\lesssim 1$% global flatness over the eight chip array. Good astrometry is required to register and stack each dithered set of exposures. In addition, accurate celestial coordinates are necessary for the follow-up spectroscopy of identified quasar candidates. All Mosaic images have an initial default world coordinate system (WCS) loaded in their header at the time of observation. The WCS maps the image pixel space onto celestial coordinate axes (RA and Dec). However, effects such as global pointing offsets, instrument rotations and differential atmospheric refraction produce the need for corrections to this astrometric calibration. We use the [*msccmatch*]{} package in IRAF to interactively derive accurate (RMS $\lesssim 0.3\arcsec$) astrometric solutions for each dithered exposure by matching $\sim300$ fairly bright stars (blue and red magnitudes $12\lesssim m \lesssim 16.5$), distributed evenly over the Mosaic field, with a reference frame given by their epoch J2000.0 USNOv2.0 coordinates [@monet96]. We map the eight chip exposures for each Mosaic frame onto a single image by rebinning the pixels to a tangent-plane projection, thus producing pixels of constant angular size (as described above). The resultant image is astrometrically calibrated to the J2000.0 celestial equatorial WCS.
Finally, we combine each set of fully processed and registered exposures into a final high S/N image for each passband. We subtract a constant flux level equal to the mode sky value from each image in a dithered set. Higher order terms are unnecessary due to the better than 1% global flat fielding. Before combining a dithered sequence of exposures into a single image, we account for the different photometric depths of individual frames. These differences are due to time-varying effects during observations. The most common effect is the changing airmass over an hour-long dithered sequence. Variable sky transparency due to occasional thin clouds (cirrus) is a second effect. All exposures of a dithered sequence are scaled to the reference image selected to have the lowest airmass and/or best photometric conditions. We calculate each frame’s multiplicative scale factor by comparing simple aperture flux measurements from the set of $\sim300$ astrometric calibration stars common to each image and its reference. The scaling factors are typically of order a few percent. This procedure scales each set of dithered, same passband frames to the same effective airmass and exposure time (given in Table \[photcal\]). We align and median combine (for removal of cosmic rays) each dithered set of registered, scaled, and sky-subtracted J0053+1234 region Mosaic frames to produce calibrated and cosmetically clean images.
Calibrations and Catalogs
-------------------------
We use the source detection and extraction software SExtractor [Source Extractor; @bertin96] to compile catalogs of instrumental magnitudes (MAG\_BEST) for every source in the final $UBRI$ images. In addition, SExtractor provides accurate positions and a variety of photometric measurements for each detected source. We configure SExtractor to detect objects comprised of a minimum of 5 pixels (DEBLEND\_MINAREA) above a background threshold of $3\sigma_{\rm bkg}$ (DETECT\_THRESH). Overlapping sources are deblended into multiple objects if the contrast between flux peaks associated with each object is $\geq0.05$ (DEBLEND\_MINCONT). These parameters provide our working definition of an imaged source. We confirm that these parameters provide good source detection and deblending by visually inspecting random regions from each image. We remove sources from each catalog with saturated, or otherwise corrupted, pixel values as flagged by SExtractor (FLAGS$\ge4$). The primary source of flagged sources are those with bright magnitudes; i.e. have at least one saturated pixel. Additionally, we exclude sources within 140 pixels ($60\arcsec$) of an image edge; stacking and combining dithered frames makes these regions of the final images lower in S/N.
We determine a turn-over magnitude ($m_{\rm TO}$) where the source counts distribution flattens and begins to fall off. This empirical limit provides a rough estimate of the magnitude that all sources, point-like and extended, become incomplete. For point sources we estimate 99% and 90% completeness limits by randomly distributing artificial stars in each image (100 per $\Delta m=0.25$ bin over the magnitude range $m_{\rm TO}-1\leq m \leq m_{\rm TO}+2$), rerunning SExtractor, and determining the number of these stars that are recovered. We use the IRAF [*artdata*]{} package to create artificial stars with characteristics (PSF size, magnitude zero point, gain, and Poisson noise) matched to actual stars on our images. We note that extended source completeness is more difficult to quantify, due mostly to the wide range of galaxy surface brightnesses. For this reason, the completeness of extended sources turns over more slowly than for point sources. As can be seen in Table \[cats\], the 90% completeness limit for point sources is fainter than $m_{\rm TO}$.
We summarize the magnitude limits and number counts for the sources in the separate $U$, $B$, $R$, and $I$ catalogs in Table \[cats\]. To generate the catalogs from which quasar candidates will be selected, we sequentially match the $U$ catalog with the $B$, $R$, and finally the $I$ source catalogs. We note that in all four cases we use the entire (i.e. not magnitude-limited) catalogs. The 1946 $U$ sources are correlated with the $B$ sources resulting in 1730 $UB$ matches with 216 $U$-only sources (see discussion in §\[candsel\]). Comparing the $UB$ matches with the $R$ catalog results in 1642 $UBR$ matches, leaving 88 $UB$ matches without $R$ magnitudes. Comparing the $UBR$ matches with the $I$ catalog results in 1552 $UBRI$ matches, leaving 90 $UBR$ matches without $I$ magnitudes. Our final two source catalogs include all sources with reliable photometry. They are (1) the $UBRI$ catalog with the 1552 matches in the $UBRI$ bands, and (2) the $UB$ catalog consisting of the 178 sources with no $I$ magnitudes, of which 88 have neither $R$ nor $I$ magnitudes.
We transform science image fluxes into apparent magnitudes calibrated to the @landolt92 system. Even though the observing conditions were not photometric, calibrating the photometry to within $\sim25\%$ absolute provides useful flux estimates for the followup spectroscopic observations. The photometric system is defined by the zero point zp, extinction (or airmass) coefficient $\alpha$, and color coefficient $\beta$ for each passband. These coefficients are determined by solving simple, linear transformation equations that relate instrumental magnitudes ($u,b,r,i$) of standard stars observed each night with their published magnitudes ($U,B,R,I$). We measure the fluxes of standard stars within a $14\farcs4$ circular aperture similar to that used by @landolt92. We use IRAF’s [*photcal*]{} package to find the best-fit solutions to the following transformation equations: $$U = u + {\rm zp}_U + \alpha_U X_U + \beta_U (U-B)$$ $$B = b + {\rm zp}_B + \alpha_B X_B + \beta_B (U-B)$$ $$R = r + {\rm zp}_R + \alpha_R X_R + \beta_R (R-I)$$ $$I = i + {\rm zp}_I + \alpha_I X_I + \beta_I (R-I)$$ We give the calibration coefficients and their errors in Table \[photcal\]. The photometric zero point quantifies the gain and the total sensitivity of the telescope plus detector. The airmass term is a measure of the atmospheric extinction as a function of telescope altitude. The color term shows how well the instrumental system matches the @landolt92 system. We note that large uncertainties in coefficients ([*i.e.*]{} systematic zero point and airmass term errors in excess of 0.1 mag) indicate non-photometric conditions during the nights we observed standards.
From the instrumental magnitude $m_A$ of each source in passband $A$, we first convert to apparent magnitude $A=m_A + {\rm zp}_A + \alpha_AX_A$, using the coefficients given in Table \[photcal\]. Next, we calculate the colors of each source by iteratively solving the following equations: $$(U-B)_i = (U-B)_0 + (\beta_U - \beta_B)\cdot(U-B)_{i-1} ,$$ $$(R-I)_i = (R-I)_0 + (\beta_R - \beta_I)\cdot(R-I)_{i-1} .$$ The initial colors $(U-B)_0$ and $(R-I)_0$ are derived simply from the apparent magnitudes. We iterate these calculations using color coefficients from Table \[photcal\] until the difference between successive iterations is $\delta m \leq 0.001$ mag.
Finally, we correct our magnitudes and colors for Galactic extinction by applying reddening corrections using the dust maps of @schlegel98[^5]. These new maps are based on full-sky $100\micron$ emission from [*COBE*]{}/DIRBE and [*IRAS*]{}/ISSA observations and, thus, directly measure the Galactic dust content. The @schlegel98 data have orders of magnitude higher resolution (at $6.1\arcsec$), and are $\sim2$ times more accurate than the traditional @burstein82 reddening estimates based on neutral hydrogen $21$ cm emission. The J0053+1234 field does not have large extinction; the local reddening $E(B-V)$ ranges from 0.058 to 0.075, corresponding to mean extinction corrections of 0.37, 0.29, 0.18, and 0.13 mag for $UBRI$ passbands, respectively. These corrections have a formal uncertainty of 10%.
The systematic errors of our photometry dominate over random errors, even at the faintest magnitudes. The systematic uncertainties in our photometry are due mainly to the zero point calibration ($\lesssim0.10$ mag) and the airmass correction ($\lesssim0.10$ mag) resulting from the variable transparency during our observations. Yet, the obvious stellar locus of main sequence stars in the color-color plots (see §\[candsel\]) illustrate the utility of relative photometry during non-photometric conditions.
Quasar Candidate Selection {#candsel}
==========================
The practice of using multi-color photometry to search for quasars is well known [e.g. @koo86; @hall96; @liu99]. In this paper we present the full $UBRI$ photometry, but only use the $U$, $B$, and $R$ filters for quasar selection by ultra-violet excess, which is most efficient at $z\lesssim2.5$. In future work the $I$ information can and will be used to help select quasars at higher redshift, where the baseline to longer wavelengths is essential but the efficiency of selection is lower.
We plot our photometry for the 1552 sources from the $UBRI$ catalog in two color-color plots: the ($U-B$) vs. ($B-R$) plane in Figure \[ubr\]; and ($U-B$) vs. ($R-I$) in Figure \[ubri\]. These sources have an effective cut of $B=21.3$ mag, corresponding to the limit of photometry accurate enough to define a tight stellar locus. In Figure \[ub\] we plot the ($U-B$) vs. $B$ color-magnitude diagram for the 1552 sources from the $UBRI$ catalog plus the 178 additional sources from the $UB$ catalog (i.e. only $U$ and $B$ detections). We note that the majority of sources from the $UB$ catalog (plotted in Figure \[ub\] as open circles) are bright ($B<17$ mag) and red ($U-B>0$). These bright $UB$ sources are saturated in $R$ and $I$, and they are both too bright and too red to be quasars. Also, 90 of these $UB$ sources have $R$ magnitudes but no $I$ magnitudes, and so were not used in the ($U-B$) vs. ($B-R$) candidate selection process; however, this means that any quasars in this group of 90 will be found at a lower yield. We find 34 $UB$ catalog sources scattered between $16.0<B<19.5$ that are not part of the $UBRI$ catalog for a variety of reasons: (i) 7 are edge sources in the $R$ or $I$ images[^6]; (ii) 10 are flagged and removed from $R$ or $I$ owing to flux contamination from nearby bright red stars that are not contaminated at bluer passbands; and (iii) 17 are $I$-band saturated and thus, among the 90 $UBR$ detections. We note that it is possible that some of the 88 $UB$ sources that were not detected in $R$ may have been detected in $I$; however, given the completeness limits of the survey, such objects must have an inflected spectrum and are therefore unlikely to be quasars with a power-law spectral energy distribution. Lastly, at faint $B>20$ limits there are blue $(U-B)<0.2$ $UB$ catalog sources that are true $R$ and $I$-band drop outs, many (30) of which meet our quasar candidate selection (see below) and were later targeted for spectroscopy (triangles in Figure \[ub\]; 16 are confirmed quasars shown as solid triangles).
Our primary selection strategy is based on ($U-B$) color, and thus depends on point sources detected jointly in at least $U$ and $B$. However, there is an additional category of sources that are potentially of interest in a quasar selection experiment: $U$-only detections where the level of the $B$ non-detection implies a source blue enough to be a quasar. Most of the 216 $U$-only sources are saturated stars or artifacts, and 53 of the 54 that are plausible point sources are faint, $U > 21$. Thus, the $U$-only faint sources are fainter than the sky brightness at longer wavelengths so they were considered too faint for useful spectroscopy. The omission of this category, which may include unrecognized quasars, affects the overall completeness measures that will be considered in an upcoming paper.
As illustrated in Figures \[ubr\] and \[ubri\], the stellar main-sequence is readily observed in color-color plots. The tight locus begins with cool stars at $(U-B) \sim 1.4$ and ends abruptly at a blue color of $(U-B) \sim -0.2$, corresponding to the hottest main-sequence stars and white dwarfs. Quasars with $z\lesssim2$ have colors typically bluer than the stellar locus; therefore, following @liu99 we select quasar candidates based on $(U-B)$ color, $(B-R)$ color, and $B$ magnitude. The exact selection procedure is designed to maximize the efficiency of quasar selection relative to the stellar locus. We select $UBRI$ quasar candidates in the $UBRI$ catalog from two regions in the ($U-B$) vs. ($B-R$) plane as shown in Figure \[ubr\] by the dashed lines: Region 1 defined by $(U-B)\leq -0.2$ and $(B-R) \leq 0.6$; and Region 2 defined by $(U-B) > -0.2$ and $(B-R) \leq 0.4$. In addition, we select the $UB$ quasar candidates from the $UB$ catalog with $(U-B)\leq -0.1$, as shown in Figure \[ub\] by the dashed line.
We do not attempt to remove extended sources from our photometric catalogs. Many of the sources redward of the stellar locus in either color in Figures \[ubr\] and \[ubri\] are resolved; nevertheless, their removal to produce a tighter stellar locus is outweighed by the desire to keep all possible quasar candidates. When observed with sufficient resolution and S/N, many low redshift quasars appear non-stellar because of the presence of a host galaxy. In principle, any deep efficient search for quasars should apply a criterion that filters out the substantial number of faint galaxies while not rejecting quasars whose images are softened by host light. Yet, most of our quasar candidates are within two magnitudes of the detection limit. At these apparent brightnesses, the large CCD pixel scale and a slightly variable PSF (as a function of image position) cause SExtractor to be unreliable at distinguishing stars and galaxies. Specifically, we find that both our candidates and our confirmed quasars span the full range of the SExtractor star/galaxy classifier (CLASS\_STAR). For these reasons, we choose not to select according to CLASS\_STAR, and as a result, our sample has no explicit selection effect against quasars at low redshift or against quasars with luminous host galaxies.
We performed several empirical checks of the quality of our photometry across the large CCD field of view. The surface density of quasar candidates does not vary significantly across the field. Nor does the distribution of photometric errors, as might occur if there were position-dependent sensitivity or background variations. The variation of random photometric error with $B$ magnitude is shown in the lower panel of Figure \[ub\]. All of this affirms the homogeneity of the photometry and the resultant catalogs.
Spectroscopy
============
Observations
------------
We searched for quasars from the candidate list formed as described in §2.4 during two observing runs in the Fall of 2000. For the first run we used the Hydra multi-object spectrograph on the KPNO 3.5-meter WIYN telescope for 5 nights, 29 Sep – 3 Oct 2000. In two overlapping fields centered on the J0053$+$1234 region, we integrated for 13.5 and 18 hours each, observing 62 and 58 targets, respectively. We combined the Simmons camera using the the blue fibers with the G400 grating to give wavelength coverage of about 3410–6610 Å, with a dispersion of 1.56 Å pix$^{-1}$ and 6.8 Å resolution. We assigned approximately one third of the 100 Hydra fibers to observe the sky for optimum sky subtraction. During the second run at the MMT 6.5-meter on 30 Nov 2000 UT, we observed six targets from the candidate list for 20 min to 1 hour each, depending on the target brightness. For these observations, the Blue Channel spectrograph and the 300 l mm$^{-1}$ grating (blazed at 5800 Å) provided wavelength coverage of about 3200–8800 Å, with a dispersion of 1.94 Å pix$^{-1}$ and 8.8 Å resolution.
Data Reduction
--------------
The WIYN and MMT data are reduced with standard IRAF routines. We perform basic reduction of the CCD data using the [*imred.ccdred*]{} package. We trim, subtract the overscan region (fit with a low order Legendre polynomial), and bias subtract each CCD image. The dark current is low enough so that a correction is unnecessary. We flat field the MMT data by fitting a response function in the dispersion direction to a median-combined dome flat image and dividing it into the science images. Since only one exposure was taken of each MMT target, we create a bad pixel mask from the normalized flat field and use it to replace bad pixels in the science frames with flux values interpolated from nearby pixels.
For the WIYN data we perform the spectral extraction and calibration using the IRAF [*dohydra*]{} package in [*imred.hydra*]{}. We obtained dome flats for each of the two target configurations, which we use to trace the positions of the spectra on the chip for extraction as well as to determine the throughput correction. The apertures are referenced to the fibers so that the extracted spectra are properly identified. The flat field correction was performed by fitting an average spectrum over all the fibers in a configuration with a high order function. For wavelength calibration we calculate a wavelength solution from the HeNeAr lamp exposures taken adjacent in time to each science exposure. The solution is applied to the quasar candidate, sky, and standard star spectra. For each configuration, we visually inspect the set of sky spectra to be sure each sky position was free from contamination from non-sky sources. Then we construct an average of good sky spectra and subtract it from the science and standard star spectra. We observed the standard star BD+28$^\circ$4211 for flux calibration of the science spectra. This star was observed only once during the run and the conditions were not photometric, therefore, the spectrophotometric calibration is only accurate to about 30% on average. More accurate spectrophotometry is not required for quasar identification since the distinguishing features are broad emission lines of large equivalent width.
We perform spectral extraction and calibration for the MMT data using [*twodspec.longslit*]{}. We extract the science and standard star spectra by tracing the flux along the dispersion axis and subtracting a value for the sky. The sky is determined from a linear fit to sky pixels in a region transverse to the dispersion direction along the trace. We observed GD 50 as the flux standard to calibrate the science spectra. This star was only observed once during the run in non-photometric conditions. Thus, the flux calibration serves primarily to calibrate the continuum shape for the science targets, and the flux calibration does not yield spectrophotometry.
Results
=======
Quasars
-------
The number of quasar candidates selected from both the $UBRI$ (Regions 1 and 2) and $UB$ catalogs by the procedure described in §\[candsel\] are listed in Table \[qresults\], along with the number of objects that were observed spectroscopically, and the number of confirmed quasars. >From the $UBRI$ selection, Region 1 was the most effective at finding quasars with an efficiency of 75% and a yield of 43 quasars; Region 2 only contributed one additional quasar among 11 targets observed. From the $UB$ selection, 16 quasars were confirmed from 30 targets observed, for an efficiency of 53%. Overall, a total of 60 quasars were confirmed with an efficiency of 61%. To examine whether a color cut that included the blue end of the stellar locus would yield more quasars, we observed 12 of 15 sources spectroscopically with $(U-B)\leq -0.1$ and $0.4<(B-R)\leq 0.6$, but this selection yielded no additional quasars.
We measured the quasar redshifts and their rms errors by cross-correlating their spectra with the LBQS composite spectrum [@fran91] using IRAF’s [*fxcor*]{}. Figure \[qsospec\] contains spectra of all objects where at least one broad emission line is detected, where we are essentially certain of the quasar identification. However, redshifts are not always as certain, particularly when only one emission line is observed. For spectral coverage in the range 3400–6400 Å, most redshifts yield two or more strong lines. The only exception in the range $0.3 < z < 0.8$, where Mg II is the strongest line, C III\] and H$\beta$ are not covered, and \[OII\] might not be visible if it is weak. There are two cases (Q 005141$+$123050 and Q 005344$+$121847) where this occurs, and in each case the single strong line has only one plausible identification. Several other quasars in Figure \[qsospec\] have their second strong line at the noisy, blue end of the spectrum, but inspection of the data before clipping for display shows that the lines are real in each case. We note that three of the 60 spectra in Figure \[qsospec\] were taken with the MMT which provides spectral coverage out to 8800 Å. In Figure \[qsospec\] we clip these three spectra to 6750 Å and note that the only strong line omitted is in Q005501+125932, which has a secure redshift of $z=1.516$.
The names of the new quasars, their J2000 coordinates, redshifts, and colors are listed in Table \[qsos\]. Their spectra are plotted in R.A. order in Figure \[qsospec\]. The spectra have been smoothed with a Gaussian having a FWHM of 3 pixels, and the pixels at the bluest and reddest ends of the spectra which contain no useful information have been trimmed. Figure \[skymap\] shows a sky plot of the 60 quasars with the boundaries of the galaxy redshift survey of @cohen99b superimposed. To see if any quasar had been discovered previously, we performed a search in NED on the position of each quasar using a circular radius of 10 arcsec. Our survey did not rediscover any previously known quasar. NED lists an X-ray counterpart for Q005321+122740 and a radio counterpart for Q005355+121232, but redshifts for neither.
Emission Line Galaxies
----------------------
Fourteen new narrow emission line galaxies were discovered in this survey. None were found in a search of NED. Their redshifts were measured by fitting a Gaussian profile to the three or four strong lines in each spectrum, computing the redshift for each line using the fitted line center, and averaging the redshifts determined for each spectrum. The ELG names, J2000 coordinates, redshifts, and colors are listed in Table \[elgs\]. The second lowest redshift galaxy in the Table, Q005436+122318 has no colors listed because the nucleus and a nearby region were separately selected in the $B$ band and $U$ band images, respectively. The spectra are plotted in Figure \[elgspec\], in R.A. order from left to right and down the page. The spectra have been smoothed with a Gaussian having a FWHM of 3 pixels, and the pixels at the bluest and reddest ends of the spectra which contain no useful information have been trimmed.
Future Work
===========
We have presented the multi-color photometric selection and low-resolution spectroscopic confirmation of 60 new quasars in the J0053$+$1234 region. These sources provide the initial grid of absorption probes for a cosmological volume centered on this region, which has been the subject of a deep redshift survey. These probes will be used in future work involving deep absorption spectroscopy to measure the relationship between massive halos traced by Mg II and C IV absorption, and those traced by luminous galaxies. Moreover, absorption in the spectra of these quasars will allow us to measure the bias between the baryons in the massive halos of luminous galaxies, and the baryons in the intergalactic medium traced by the Lyman-$\alpha$ forest of HI absorption. We have begun the deep absorption spectra observing campaign. Furthermore, we have selected additional quasar candidates based on deeper $U$-band imaging of the central half square degree of this region. The goal of the deeper imaging is to have a denser grid of absorption probes at small impact parameters from the field center, but spectroscopy of these additional quasars will be very challenging. Upcoming papers will address the overall completeness of our survey towards J0053$+$1234, and present additional quasar candidates and spectra.
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[lcrcccc]{} 30 Sep 1998 & $U$ & 175 & 5 & 1.7 & $1.06-1.37$ & Mostly clear\
& $B$ & 30 & 2 & 2.0 & $1.06-1.07$ &\
& $B$ & 30 & 3 & 1.9 & $1.13-1.16$ &\
& $R$ & 30 & 5 & 1.9 & $1.22-1.33$ &\
& $I$ & 10 & 1 & 1.7 & $1.08$ &\
01 Oct 1998 & $U$ & 150 & 5 & 1.9 & $1.09-1.34$ & Mostly clear\
& $I$ & 60 & 5 & 1.6 & $1.40-1.19$ &\
02 Oct 1998 & $U$ & 125 & 5 & 1.8 & $1.12-1.48$ & Some thin cirrus\
& $B$ & 50 & 5 & 1.6 & $1.06-1.07$ &\
\[obslog\]
[lcccccccc]{} $U$ & 35 & 1.06 & 97 & 30 Sep 1998 & $20.00\pm0.12$ & $-0.51\pm0.09$ & $+0.04\pm0.03$ & 0.051\
$B$ & 10 & 1.06 & 462 & 30 Sep 1998 & $21.70\pm0.07$ & $-0.11\pm0.05$ & $+0.12\pm0.01$ & 0.029\
$R$ & 6 & 1.21 & 449 & 30 Sep 1998 & $22.33\pm0.14$ & $-0.33\pm0.10$ & $+0.06\pm0.03$ & 0.036\
$I$ & 12 & 1.19 & 1328 & 1 Oct 1998 & $21.37\pm0.09$ & $+0.01\pm0.06$ & $+0.01\pm0.02$ & 0.030\
\[photcal\]
[lccccccccc]{} $U$ & 1946 & 20.7 & 1283 & $\sim25$ & 21.5 & 1754 & $\sim12$ & 21.0 & 1.7\
$B$ & 2463 & 20.7 & 1575 & $\sim25$ & 21.7 & 2334 & $\sim11$ & 21.2 & 1.8\
$R$ & 4505 & 19.5 & 2080 & $\sim33$ & 20.5 & 3678 & $\sim14$ & 20.2 & 1.9\
$I$ & 7922 & 19.5 & 3493 & $\sim25$ & 20.5 & 6569 & $\sim11$ & 20.2 & 1.7\
\[cats\]
[lrrrr]{} UBRI – Region 1 & 58 & 57 & 43 & 75%\
UBRI – Region 2 & 15 & 11 & 1 & 9%\
UB & 32 & 30 & 16 & 53%\
Combined & 105 & 98 & 60 & 61%\
\[qresults\]
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[^1]: Present address: University of Massachusetts, Lederle Graduate Research Tower, Amherst, MA 01003
[^2]: Observations reported here were obtained at Kitt Peak National Observatory, National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^3]: NOAO CCD Mosaic Imager User Manual (hereafter MosManual), Version Sept. 15, 2000, G. Jacoby; http://www.noao.edu/kpno/mosaic/manual/index.html.
[^4]: IRAF is distributed by the National Optical Astronomical Observatories, which are operated by AURA, Inc. under contract to the NSF.
[^5]: The authors have kindly made available the dust maps with user friendly software that outputs $E(B-V)$ for an input $(l,b)$ at http://astron.berkeley.edu/davis/dust/index.html.
[^6]: Our $UBRI$ images are concentric to within a few tens of pixels ($<10\arcsec$), thus, a slight shift means some objects will be culled from one catalog given our edge proximity flag, yet not culled from another.
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---
abstract: 'We uncover a remarkable quantum scattering phenomenon in two-dimensional Dirac material systems where the manifestations of both classically integrable and chaotic dynamics emerge simultaneously and are electrically controllable. The distinct relativistic quantum fingerprints associated with different electron spin states are due to a physical mechanism analogous to chiroptical effect in the presence of degeneracy breaking. The phenomenon mimics a chimera state in classical complex dynamical systems but here in a relativistic quantum setting - henceforth the term “Dirac quantum chimera,” associated with which are physical phenomena with potentially significant applications such as enhancement of spin polarization, unusual coexisting quasibound states for distinct spin configurations, and spin selective caustics. Experimental observations of these phenomena are possible through, e.g., optical realizations of ballistic Dirac fermion systems.'
author:
- 'Hong-Ya Xu'
- 'Guang-Lei Wang'
- Liang Huang
- 'Ying-Cheng Lai'
title: 'Chaos in Dirac electron optics: Emergence of a relativistic quantum chimera'
---
The tremendous development of two-dimensional (2D) Dirac materials such as graphene, silicene and germanene [@Novoselovetal:2004; @Novoselovetal:2005; @Netoetal:2009; @Wehling:2014; @Wang2015], in which the low-energy excitations follow the relativistic energy-momentum relation and obey the Dirac equation, has led to the emergence of a new area of research: Dirac electron optics [@Cse2007; @Cheianov2007; @Shytov2008; @Beenakker2009; @Moghaddam2010; @Gu2011; @Will2011; @Rickhaus2013; @Liao2013; @Hei2013; @Asm2013; @Wu2014; @Zhao2015; @Peter2015; @Leeetal:2015; @RPeter2015; @Walls2016; @Caridad2016; @Gutierrez2016; @JLee2016; @Chenetal:2016; @Settnes2016; @Liu2017; @Barnard2017; @Jiang2017; @Ghahari2017; @Zhang2017; @Boggild2017]. Theoretically, it was articulated early [@Cheianov2007] that Klein tunneling and the unique gapless conical dispersion relation can be exploited to turn a simply p-n junction into a highly transparent focusing lens with a gate-controlled [*negative refractive index*]{}, producing a Vaselago lens for the chiral Dirac fermions in graphene. The negative refraction of Dirac fermions obeys the Snell’s law in optics and the angularly-resolved transmittances in analogy with the Fresnel coefficients in optics have been recently confirmed experimentally [@Leeetal:2015; @Chenetal:2016]. Other works include various Klein-tunneling junction based electronic counterparts of optical phenomena such as Fabry-Pérot resonances [@Shytov2008; @Rickhaus2013], cloaking [@Gu2011; @Liao2013], waveguide [@Will2011; @Peter2015], Goos-Hänchen effect [@Beenakker2009], Talbot effect [@Walls2016], beam splitter and collimation [@RPeter2015; @Liu2017; @Barnard2017], and even Dirac fermion microscope [@Boggild2017]. A Dirac material based electrostatic potential junction with a closed interface can be effectively tuned to optical guiding and acts as an unusual optical dielectric cavity whose effective refractive index can be electrically modulated, in which phenomena such as gate controlled caustics [@Cse2007], electron Mie scattering [@Hei2013; @Caridad2016; @Gutierrez2016; @JLee2016] and whispering gallery modes [@Wu2014; @Zhao2015; @Jiang2017; @Ghahari2017] can arise. In addition, unconventional electron optical elements have been demonstrated such as valley resolved waveguides [@Wu2011; @ZMC:2011] and beam splitters [@Settnes2016], electronic birefringent superlens [@Asm2013] and spin (current) lens [@Moghaddam2010; @Zhang2017]. Research on Dirac electron optics offers the possibility to control Dirac electron flows in a similar way as for light.
In this Letter, we address the role of chaos in Dirac electron optics. In nonrelativistic quantum mechanics, the interplay between chaos and quantum optics has been studied in microcavity lasers [@NSCGC:1996; @Nockel1997; @GCNNSFSC:1998; @Vahala:2003] and deformed dielectric microcavities with non-Hermitian physics and wave chaos [@Cao2015]. With the development of Dirac electron optics [@Cse2007; @Cheianov2007; @Shytov2008; @Beenakker2009; @Moghaddam2010; @Gu2011; @Will2011; @Rickhaus2013; @Liao2013; @Hei2013; @Asm2013; @Wu2014; @Zhao2015; @Peter2015; @Leeetal:2015; @RPeter2015; @Walls2016; @Caridad2016; @Gutierrez2016; @JLee2016; @Chenetal:2016; @Settnes2016; @Liu2017; @Barnard2017; @Jiang2017; @Ghahari2017; @Zhang2017; @Boggild2017], the relativistic electronic counterparts of deformed optical dielectric cavities/resonators have become accessible. For massless Dirac fermions in ballistic graphene, the interplay between classical dynamics and electrostatic confinement has been studied [@Bardarson2009; @Schneider2011; @Heinl2013; @Schneider2014] with the finding that integrable dynamics lead to sharp transport resonances due to the emergence of bound states while chaos typically removes the resonances. In these works, the uncharged degree of freedom such as electron spin, which is fundamental to relativistic quantum systems, was ignored.
Our focus is on the interplay between ray-path defined classical dynamics and spin in Dirac electron optical systems. To be concrete, we introduce an electrical gate potential defined junction with a ring geometry, in analogy to a dielectric annular cavity. Classically, this system generates integrable and mixed dynamics with the chaotic fraction of the phase space depending on the ring eccentricity and the effective refractive index configuration, where the index can be electrically tuned to negative values to enable Klein tunneling. Inside the gated region, the electron spin degeneracy is lifted through an exchange field from induced ferromagnetism, leading to a class of spin-resolved, electrically tunable quantum systems of electron optics with massless Dirac fermions (by mimicking the photon polarization resolved photonic cavities made from synthesized chiral metamaterials). We develop an analytic wavefunction matching solution scheme and uncover a striking quantum scattering phenomenon: manifestations of classically integrable and chaotic dynamics [*coexist simultaneously*]{} in the system at the same parameter setting, which mimics a chimera state in classical complex dynamical systems [@KB:2002; @AS:2004; @AS:2006; @TNS:2012; @HMRHOS:2012; @MTFH:2013; @YHLZ:2013; @YHGL:2015]. The basic underlying physics is the well-defined, spin-resolved, gate-controllable refraction index that dominantly controls the ballistic motion of short-wavelength Dirac electrons across the junction interface, in which the ray tracing of reflection and refraction associated with particles belonging to different spin states generates distinct classical dynamics inside the junction/scatterer. Especially, with a proper gate potential, the spin-dependent refractive index profile can be controlled to generate regular ray dynamics for one spin state but generically irregular behavior with chaos for the other. A number of highly unusual physical phenomena arise, such as enhanced spin polarization with chaos, simultaneous quasiscarred and whispering gallery type of resonances, and spin-selective lensing with a starkly near-field separation between the local density of states (DOS) for spin up and down particles.
Low energy excitations in 2D Dirac materials are described by the Dirac-Weyl Hamiltonian $H_0 = v_F\bm{\sigma\cdot p}$, where $v_F$ is the Fermi velocity, $\bm{p}=(p_x, p_y)$ is the momentum measured from a given Dirac point and $\bm{\sigma}=(\sigma_x, \sigma_y)$ are Pauli matrices for sublattice pseudospin. In the presence of a gate potential and an exchange field due to the locally induced ferromagnetism inside the whole gated region, the effective Hamiltonian is $H = v_Fs_0\otimes\bm{\sigma\cdot p} + s_0\otimes\sigma_0\mathcal{V}_{gate}
(\bm{r}) - s_z\otimes\sigma_0\mathcal{M}(\bm{r})$, where the Pauli matrix $s_z$ acts on the real electron spin space, $s_0$ and $\sigma_0$ both are identity matrices, $\mathcal{V}_{gate}(\boldsymbol{r})$ and $\mathcal{M}(\boldsymbol{r})$ are the electrostatic and exchange potential, respectively. Due to the pseudospin-momentum locking (i.e., $\bm{\sigma\cdot p}$), a non-uniform potential couples the two pseudospinor components, but the electron spin components are not coupled with each other. The exchange field breaks the twofold spin degeneracy. Since $[s_z\otimes\sigma_0, H]=0$, the Hamiltonian can be simplified as $H_s = H_0 + \mathcal{V}_{gate}(\bm{r})-s\mathcal{M}(\bm{r})$ with $s=\pm$ denoting the electron spin quantum number. Because of $\mathcal{M}$, the Dirac-type Hamiltonian $H_s$ can give rise to spin dependent physical processes.
For the ring configuration in Fig. \[fig:classical\](a) and assuming the potentials are smooth on the scale of the lattice spacing but sharp in comparison with the conducting carriers’ wavelength, in the polar coordinates $\bm{r}=(r,\theta)$, we have $\mathcal{V}_{gate}(\bm{r})=\hbar v_F\nu_1\Theta(R_1 - r)
\Theta(|\bm{r - \xi}|-R_2) + \hbar v_F\nu_2\Theta(R_2 - |\bm{r - \xi}|)$, and $\mathcal{M}(\bm{r})=\hbar v_F\mu\Theta(R_1 - r)$, where $\Theta$ is the Heaviside step function, $R_2$ is the radius of the small disk gated region of strength $\hbar v_F (\nu_2-\nu_1)$ placed inside a larger disk of radius $R_1$ ($>R_2$) and strength $\hbar v_F \nu_1$, the displacement vector between the disk centers is $\bm{\xi}=(\xi,0)$, and the exchange potential has the strength $\hbar v_F \mu$ over the whole gated region. The two circular boundaries divide the domain into three distinct regions: $I$: $r >R_1$; $II$: $r<R_1$ and $|\bm{r-\xi}|>R_2$; $III$; $|\bm{r-\xi}|<R_2$. For given particle energy $E=\hbar v_F\epsilon$, the momenta in the respective regions are $k_s^{I}=|\epsilon|$, $k_s^{II}=|\epsilon - \nu_1 + s\mu|$, and $k_s^{III} = |\epsilon - \nu_2 + s\mu|$. Within the gated region, the exchange potential splits the Dirac cone into two in the vertical direction in the energy domain while the electrostatic potential simply shifts the cone, leading to a spin-resolved, gate-controllable annular junction for massless Dirac electrons.
![ **Scattering system and classical ray dynamics**. (a) Annular shaped scattering region with eccentricity $\xi = \overline{OO'}$, (b) a cross-sectional view, (c,d) chaotic and integrable ray dynamics on the Poincaré surface of section defined by the Birkhoff coordinates ($\theta$, $\sin\beta$) for spin up and down particles, respectively, where $\theta$ denotes the polar angle of a ray’s intersection point with the cavity boundary and $\beta$ is the angle of incidence with respect to the boundary normal. The quantity $\sin\beta$ is proportional to the angular momentum and the critical lines for total internal reflection are given by $\sin\beta_c=\pm1/n_s$.[]{data-label="fig:classical"}](figure1.pdf){width="1\linewidth"}
In the short wavelength limit, locally the curved junction interface appears straight for the electrons, so the gated regions and the surroundings can be treated as optical media. The unusual feature here is that the refractive indices are spin-dependent: $n_s^{II,III} = (\epsilon +s\mu - \nu_{1,2})/\epsilon$, similar to light entering and through a polarization resolved photonic crystal [@Gansel2009; @ZPLLZ2009]. Given the values of $\epsilon$ and $\mu$, depending on the values of $\nu_{1,2}$, the refractive indices for the two spin states can be quite distinct with opposite signs. The system is thus analogous to a chiral photonic metamaterial based cavity, which represents a novel class of Dirac electron optics systems. The classical behaviors of Dirac-like particles in the short wavelength limit can be assessed using the optical analogy, as done previously for circularly curved $p-n$ junctions [@Cse2007; @Boggild2017], where the classical trajectories are defined via the principle of least time. Because of the spin dependent and piecewise constant nature of the index profile, the resulting stationary ray paths for the Dirac electrons are spin-resolved and consist of straight line segments. At a junction interface, there is ray splitting governed by the spin-resolved Snell’s law. On a Poincaré surface of section, the classical dynamics are described by a spin-resolved map $F_s$ relating the dynamical variables $\theta$ and $\beta$ (Fig. \[fig:classical\]) between two successive collisions with the interface: $(\theta_i,\sin\beta_i)\mapsto(\theta_{i+1}, \sin\beta_{i+1})$. The ray-splitting picture is adequate for uncovering the relativistic quantum fingerprints of distinct classical dynamics.
Spin-resolved ray trajectories inside the junction lead to the simultaneous coexistence of distinct classical dynamics. For example, for the parameter setting $\nu_2=-\nu_1=\epsilon=\mu$, i.e., $n_s^{II}=2+s$ and $n_s^{III}= s$, for spin up particles ($s=+$), the junction is an eccentric annular electron cavity characterized by the refractive indices $n_+^{II}=3$ and $n_+^{III}=1$, as exemplified in Fig. \[fig:classical\](b) for $\xi=0.3$. However, for spin down particles ($s=-$), the junction appears as an off-centered negatively refracted circular cavity with $n_-^{II}=1$ and $n_-^{III}=-1$. Figures \[fig:classical\](c) and \[fig:classical\](d) show the corresponding ray dynamics on the Poincaré surface of section for spin up and down particles, respectively, where the former exhibit chaos while the dynamics associated with the latter are integrable with angular momentum being the second constant of motion.
For a spin unpolarized incident beam, the simultaneous occurrence of integrable and chaotic classical dynamics means the coexistence of distinct quantum manifestations, leading to the emergence of a Dirac quantum chimera. To establish this, we carry out a detailed analysis of the scattering matrices for spin-dependent, relativistic quantum scattering and transport through the junction. Using insights from analyzing optical dielectric cavities [@Hack1997; @Hentschel2002] and nonrelativistic quantum billiard systems [@Doron1992; @Doron1995], we develop an analytic wave function matching scheme at the junction interfaces [ (See Supplemental Material [@SI] which includes Refs. [@Wigner1955; @Smith1960; @Schomerus2015; @zwillinger2014table; @Gutierrez2016; @Jiang2017; @Weietal:2016])]{} to solve the Dirac-Weyl equation to obtain the scattering matrix $S$ as a function of the energy $E$ as well as the spin polarization $s$ for given system parameters $R_2/R_1$, $\xi$, $\nu_{1,2}$ and $\mu$. The Wigner-Smith time delay [@Wigner1955; @Smith1960] is defined from the $S$-matrix as $\tau = -i\hbar\textrm{Tr}\left[S^\dag(\partial S/\partial E)\right]$, which is proportional to the DOS of the cavity. Large positive values of $\tau$ signify resonances associated with the quasibound states [@Rotter2017]. Physically, a sharper resonance corresponds to a longer trapping lifetime and scattering time delay. Previous works on wave or quantum chaotic scattering [@BS:1988; @BS:1989a; @JBS:1990; @MRWHG:1992; @LBOG:1992; @Ketzmerick:1996; @KS:1997; @SKGFKDW:1998; @KSFN:1999; @KS:2000; @HKL:2000; @CGM:2000; @deMLBAF:2002; @CSGFBR:2003; @KS:2003; @GS:2006; @BSS:2010; @KFOA:2011; @YHLG:2011a; @WYLG:2013] established that classical chaos can smooth out (broaden) the sharp resonances and reduce the time delay markedly while integrable dynamics can lead to stable, long-lived bound states (or trapping modes).
![ **A Dirac quantum chimera**. (a) top: Contour map of dimensionless Wigner-Smith time delay (on a logarithmic scale) versus energy $E$ and eccentricity $\xi$ for spin down (left) and up (right) cases, where the bright yellow color indicates larger values. Middle and bottom panels: time delay and total cross section averaged over all directions of the incident waves versus $E$, respectively, for $\xi=0.3$. (b) Dependence of the maximum time delay on $\xi$ (red: spin up; blue: spin-down). (c) Energy averaged spin polarization versus $\xi$.[]{data-label="fig:DQC"}](figure2.pdf){width="1\linewidth"}
We present concrete evidence for Dirac quantum chimera. Figure \[fig:DQC\](a) shows, for $R_2/R_1=0.6$, $\mu=-\nu_1=5$ and $\nu_2 = 45$, the dimensionless time delay (on a logarithmic scale) versus the eccentricity $\xi$ and energy $E$ (in units of $\hbar v_F/R_1$). Figure \[fig:DQC\](b) shows the maximum time delay \[within the given energy range in Fig. \[fig:DQC\](a)\] versus $\xi$ for spin-up (red) and spin-down (blue) particles. There are drastic changes in the time delay as the energy is varied, which are characteristic of well-isolated, narrow resonances and imply the existence of relatively long-lived confined modes. There is a key difference in the resonances associated with the spin up and down states: the former depend on the eccentricity parameter $\xi$ and are greatly suppressed for $\xi>0.2$, while the latter are independent of $\xi$. For example, the middle panel of Fig. \[fig:DQC\](a) shows that, for a severely deformed structure ($\xi = 0.3$), there are sharp resonances with high peak values of the time delay for the spin down state, but none for the spin up state. The suppression of resonances associated with the spin up state is consistent with the behavior of the total cross section $\overline{\sigma}_t$ (averaged over the directions of the incident wave) given in terms of the $S$-matrix elements by $\overline{\sigma}_t = (2k)^{-1}\sum_{m,l=-\infty}^{\infty}
\left|S_{ml}-\delta_{ml}\right|^2$, as shown in the bottom panel of Fig. \[fig:DQC\](a). Because the classical dynamics for massless fermions in the spin up and down states are chaotic and integrable, respectively \[c.f., Figs. \[fig:classical\](c,d)\], there is [*simultaneous*]{} occurrence of two characteristically different quantum scattering behaviors for a spin unpolarized beam: one without and another with *sharp* resonances. This striking contrast signifies a Dirac quantum chimera.
Are there unexpected, counterintuitive physical phenomena associated with a Dirac quantum chimera? Yes, there are! Here we present two and point out their applied values.
![ **Spin polarized scarred and regular whispering-gallery-mode resonances as a result of Dirac quantum chimera**. (a,c) Real space probability densities (on a logarithmic scale) of the representative quasibound states for spin-up and spin-down Dirac electrons, respectively. For the spin-up particles, the spinor wave solution is scarred by an unstable periodic ray trajectory obeying the Snell’s law, as indicated by the red dashed path with highlighted pentagram markers. The spin-down Dirac electrons are associated with a whispering gallery ray path due to the continuous total internal reflections denoted by the blue dotted segments. (b,d) The corresponding phase-space representations with regions below the critical black dash lines satisfying the total internal reflection at the boundary. The distinct quasibound modes are from simultaneous resonances under the same system parameters, leading to a relativistic quantum chimera. Further signatures of the chimera state can be seen in the plot of the total cross section versus the particle energy for different spin states (e) and a net spin distribution with a dramatic spin-resolved separation in the real space confined inside the cavity (f).[]{data-label="fig:scar-wgm"}](figure3.pdf){width="1\linewidth"}
The first is spin polarization enhancement, which has potential applications to Dirac material based spintronics. A general way to define spin polarization is through the spin conductivities $G^{\downarrow(\uparrow)}$ as $P_z = (G^{\downarrow}-G^{\uparrow})/(G^{\downarrow}+G^{\uparrow})$. Imagine a system consisting of a set of sparse, randomly distributed, identical junction-type of annular scatterers, and assume that the scatterer concentration is sufficiently low ($n_c\ll1/R_1^2$) so that multiple scattering events can be neglected. In this case, the spin conductivities can be related to the transport cross section as $G^{\downarrow(\uparrow)}/G_0=k/(n_c\sigma_{tr}^{\downarrow(\uparrow)})$, where $G_0$ is the conductance quantum and $\sigma_{tr}^{\downarrow(\uparrow)}$ can be calculated from the $S$-matrix. For a spin unpolarized incident beam along the $x$-axis with equal spin up and down populations, we calculate the average spin polarization over a reasonable Fermi energy range as a function of the eccentricity $\xi$, as shown in Fig. \[fig:DQC\](c). For $\xi>0.2$ so classical chaos is relatively well developed and a Dirac quantum chimera emerges, there is robust enhancement of spin polarization. From the standpoint of classical dynamics, the scattering angle is much more widely distributed for spin up particles (due to chaos) as compared with the angle distribution for spin down particles with integrable dynamics, leading to a larger effective resistance for spin up particles. From an applied perspective, the enhancement of spin polarization brought about by a Dirac quantum chimera can be exploited for developing spin rheostats or filters, where one of the spin resistances, e.g., $R^{\uparrow}\propto 1/G^{\uparrow}$, can be effectively modulated through tuning the deformation parameter $\xi$ so as to induce classically chaotic motion for one type of polarization but integrable dynamics for another.
![ **Spin-selective caustic lens and skew scattering associated with a Dirac quantum chimera**. (a) Caustic patterns resulting from the scattering of a spin unpolarized planar incident wave traveling along the positive $x$-axis ($\theta'=0$) with relatively short wavelength, i.e., $kR_1=70\gg1$, and (c) from scattering of the wave propagating along the direction that makes an angle $\theta'=\pi/4$ with the $x$ axis. (b,d) The corresponding spatially resolved near field net spin distributions measured by the difference $|\psi_\uparrow|^2-|\psi_\downarrow|^2$, respectively. (e) The resulting far-field behavior characterized by the angular distributions of spin-dependent differential cross sections with symmetric profiles for $\theta'=0$ (left inset) and spin-selective asymmetric one for $\theta'=\pi/4$ (right inset), where both insets are plotted by the eighth root of $\sigma_{diff}^{\uparrow(\downarrow)}$ in order to weaken the drastic contrast variation in magnitude for better visualization. Parameters are $\xi=0.27$, $R_2/R_1=0.6$, $\nu_2=\mu=-\nu_1=70$ and $E=70$.[]{data-label="fig:Lensing"}](figure4.pdf){width="1\linewidth"}
The second phenomenon is resonance and lensing associated with a Dirac quantum chimera. Figures \[fig:scar-wgm\](a-f) show, for $\xi=0.27$ (in units of $R_1$), $R_2/R_1=0.6$, $\nu_2=4\nu_1=-4\mu=24.16$ (in units of $1/R_1$) and $E=6.04$ (in units of $\hbar v_F/R_1$), a resonant (quasibound) state, in which the spatially separated, spin resolved local DOS is confined inside the cavity. The spin up state is concentrated about a particular unstable periodic orbit without the rotational symmetry \[Figs. \[fig:scar-wgm\](a) and \[fig:scar-wgm\](b)\] and exhibits a scarring pattern with a relatively short lifetime characterized by a wider resonance profile, as shown in Fig. \[fig:scar-wgm\](e). Spin down particles are trapped inside the inner disk by a regular long-lived whispering gallery mode associated with the integrable dynamics \[Figs. \[fig:scar-wgm\](c) and \[fig:scar-wgm\](d)\]. The Dirac quantum chimera thus manifests itself as the simultaneous occurrence of a magnetic scarred quasibound state and a whispering gallery mode excited by an incident wave with equal populations of spin up and down particles, as shown in Fig. \[fig:scar-wgm\](f), a color-coded spatial distribution of the difference between the local DOS for spin up and down particles.
In the sufficiently short wavelength regime where the ray picture becomes accurate, a spin resolved lensing behavior arises, due to the simultaneous occurrence of two distinct quantum states associated with the chimera state. The cavity can be regarded as an effective electronic Veselago lens with a robust caustic function for spin down particles but the spin up particles encounter simply a conventional lens of an irregular shape. Particularly, for a spin-unpolarized, planar incident wave, a spin-selective caustic behavior arises, as shown in Figs. \[fig:Lensing\](a-d) through the color-coded near-field patterns. There is a pronounced lensing caustic of the cusp type for the spin down state while a qualitatively distinct lensing pattern occurs for the spin up state. A consistent far-field angular distribution of the differential cross section is shown in Fig. \[fig:Lensing\](e), which gives rise to well-oriented/collimated, spin-dependent far-field scattering with the angle resolved profile shrinked into a small range due to the lensing effect. Despite lack of robust lensing, the spin up particles in general undergo asymmetric scattering, which can lead to spin-polarized transverse transport in addition to longitudinal spin filtering.
To summarize, we uncover a Dirac quantum chimera - a type of relativistic quantum scattering states characterized by the simultaneous coexistence of two distinct types of behaviors as the manifestations of classical chaotic and integrable dynamics, respectively. The physical origin of the chimera state is the optical-like behavior of massless Dirac fermions with both spin and pseudospin degrees of freedom, which together define a spin-resolved Snell’s law governing the chiral particles’ ballistic motion. The phenomenon is predicted analytically based on quantum scattering from a gate-defined annular junction structure. The chimera has striking physical consequences such as spin polarization enhancement, unusual quantum resonances, and spin-selective lensing, which are potentially exploitable for developing 2D Dirac material based electronic and spintronic devices.
This work is supported by ONR under Grant No. N00014-16-1-2828. L.H. is also supported by NNSF of China under Grant No. 11775101.
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---
author:
- 'Andrzej M. Oleś and Peter Horsch'
title: 'Orbital fluctuations in the $R$VO$_3$ perovskites'
---
Orbital degrees of freedom in strongly correlated systems {#sec:orbi}
=========================================================
Orbital degrees of freedom play a key role for many intriguing phenomena in strongly correlated transition metal oxides, such as the colossal magnetoresistance in the manganites or the effective reduction of dimensionality in KCuF$_3$ [@Tok00]. Before addressing complex phenomena in doped Mott insulators, it is necessary to describe first the undoped materials, such as LaMnO$_3$ or LaVO$_3$. These two systems are canonical examples of correlated insulators with coexisting magnetic and orbital order [@Tok00]. In both cases large local Coulomb interaction $U$ suppresses charge fluctuations, leading to low-energy effective Hamiltonians with superexchange interactions which stabilize antiferromagnetic (AF) spin order at low temperature [@Fei99; @Kha01]. However, the AF order is different in both cases: ferromagnetic (FM) planes are coupled by AF interactions in the $A$-type AF phase of LaMnO$_3$, while FM chains along the $c$ cubic axis are coupled by AF interactions in the $ab$ planes in the $C$-type AF ($C$-AF) phase of LaVO$_3$. The superexchange Hamiltonians which describe both systems are just examples for the spin-orbital physics [@Ole05], where orbital (pseudospin) operators contribute explicitly to the structure of the superexchange interactions — their actual form depends on the number of $3d$ electrons (holes) at transition metal ions which determines the value of spin $S$, and on the type of active orbital degrees of freedom, $e_g$ or $t_{2g}$. In simple terms, the magnetic structure is determined by the pattern of occupied and empty orbitals, and the associated rules are known as Goodenough-Kanamori rules (GKR). The central focus of this overview are $t_{2g}$ orbital degenerate systems, where quantum fluctuations of orbitals play a central role for the electronic properties [@Kha01; @Miy05; @Yan07] and modify the predictions of the GKR.
In the last two decades several new concepts were developed in the field of orbital physics [@Tok00]. The best known spin-orbital superexchange Hamiltonian is the Kugel-Khomskii model [@Kug82], which describes the $e_g$ orbital $\{x^2-y^2,3z^2-r^2\}$ degrees of freedom coupled to $S=1/2$ spins at Cu$^{2+}$ ($d^9$) ions in KCuF$_3$. The spins interact by either FM and AF exchange interactions, depending on the type of occupied and empty orbitals on two neighboring ions. It has been found that enhanced quantum fluctuations due to orbital degrees of freedom, which contribute to joint spin-orbital dynamics, may destabilize long-range magnetic order near the quantum critical point of the Kugel-Khomskii model [@Fei97]. The orbital part of the superexchange is thereby intrinsically frustrated even on geometrically non-frustrated lattices, as in the perovskite lattice [@Fei97; @vdB04], which is a second important concept in the field of orbital physics. Finally, although spin and orbital operators commute, there are situations where joint spin-orbital dynamics plays a crucial role, and spin and orbital operators cannot be separated from each other. This situation is called spin-orbital entanglement [@Ole06], and its best example are the entangled SU(4) singlets in the one-dimensional (1D) SU(4) model [@Fri99]. There is no doubt that these recent developments in the orbital physics provide many challenges both for the experimental studies and for the theoretical understanding of the experimental consequences of the spin-orbital superexchange.
Let us consider first the orbital part of the superexchange. Its intrinsic frustration results from the directional nature of orbital pseudospin interactions [@Fei97; @vdB04] — they imply that the pair of orbitals which would minimize the energy depends on the direction of a bond $\langle ij\rangle$ in a cubic (peovskite) lattice. In case of $e_g$ orbitals the superexchange interactions are Ising-like as only one orbital flavor allows for electron hopping $t$ and the electron exchange process does not occur. They favor a pair of orthogonal orbitals on both sites of the considered bond [@vdB99], for instance $|z\rangle\sim (3z^2-r^2)/\sqrt{6}$ and $|x\rangle\sim x^2-y^2$ orbital for a bond along the $c$ axis. When the two above orbital states are represented as components of $\tau=1/2$ pseudospin, this configuration gives the energy of $-\frac{1}{4}J$, where $J$ is the superexchange constant. Unlike in the 1D model [@Dag04], such an optimal orbital configuration cannot be realized simultaneously on all the bonds in a two-dimensional (2D) or three-dimensional (3D) system. Thus, in contrast to spin systems, the tendency towards orbital disordered state (orbital liquid) is [*enhanced*]{} with increasing system dimension [@Fei05; @Kha05].
The essence of orbital frustration is captured by the 2D compass model, originally developed as a model for Mott insulators [@Kug82]. Intersite interactions in the compass model are descibed by products $\tau^{\alpha}_i\tau^{\alpha}_j$ of single pseudospin components, $$\tau^x_i = \frac{1}{2}\sigma^{x}_i , \hskip 1cm
\tau^y_i = \frac{1}{2}\sigma^{y}_i , \hskip 1cm
\tau^z_i = \frac{1}{2}\sigma^{z}_i .
\label{t2g}$$ for a bond $\langle ij\rangle\parallel\gamma$, where $\alpha=x,y,z$, rather than by a pseudospin scalar product ${\vec \tau}_i\cdot{\vec \tau}_j$. For instance, in the 2D case of a single $ab$ plane, the compass model [@Kho03], $$H_{2D} =
J_x\sum_{\langle ij\rangle\parallel a}\tau^x_i\tau^x_j
+J_z\sum_{\langle ij\rangle\parallel b}\tau^z_i\tau^z_j\;,
\label{compa}$$ describes the competition between $\tau^x_i\tau^x_j$ and $\tau^z_i\tau^z_j$ interactions for the bonds along $a$ and $b$ axis, respectively. This competition of pseudospin interactions along different directions results in intersite correlations similar to those in the anisotropic XY model, and generates a quantum critical point at $J_x=J_z$, with high degeneracy of the ground state [@Mil05]. So, despite certain similarities of the compass model to ordinary models used in quantum magnetism, an ordered phase with finite magnetization is absent. It is interesting to note that a similar quantum phase transition exists also in the 1D chain compass model [@Brz07] ($N'=N/2$ is the number of unit cells): $$H_{1D}=\sum_{i=1}^{N'}
\left\{ J_x\tau_{2i-1}^x \tau_{2i}^x +
J_z\tau_{2i}^z \tau_{2i+1}^z \right\}\,.
\label{H1}$$ Recently this 1D compass model was solved exactly in the whole range of $\{J_x,J_z\}$ parameters [@Brz07] by mapping to the exactly solvable (quantum) Ising model in transverse field. It provides a beautiful example of a first order quantum phase transition between two phases with large $\langle\tau_{2i-1}^x \tau_{2i}^x\rangle$ or $\langle\tau_{2i}^z \tau_{2i+1}^z\rangle$ correlations, and discontinuous changes of intersite correlation functions.
In realistic spin-orbital superexchange models transitions between different ordered or disordered orbital states are accompanied by magnetic transitions. This field is very rich, and several problems remain unsolved as simple mean-field (MF) approaches do not suffice in general, even for the systems with perovskite lattices [@Ole05]. In this chapter we shall address the physical properties of the $R$VO$_3$ perovskites ($R$=Lu,Yb,$\cdots$,La), where not only the above intrinsic frustration of the orbital superexchange, but also the structure of the spin-orbital superexchange arising from multiplet splittings due to Hund’s exchange plays a role and determines the observed physical properties at finite temperature. Moreover, we shall see that the coupling of the orbitals to the lattice, i.e., via Jahn-Teller (JT) coupling, GdFeO$_3$-like and orthorhombic distortion, are important control parameters. First we analyze the structure of the spin-orbital superexchange in section 2 and show its consequences for the magnetic and optical properties of strongly correlated transition metal compounds. Here we also address the entanglement of spin and orbital variables which is ignored in the MF decoupling, and we point out that it fails in certain situations.
![Magnon dispersion relation obtained by neutron scattering for the $C$-AF phase of YVO$_3$ at $T=85$ K. The lines are interpolation between the experimental points (squares with error bars) along two high symmetry directions in the Brillouin zone. Image courtesy of Clemens Ulrich.[]{data-label="fig:yvo"}](yvo85K.eps){width="6.8cm"}
The coupling between the orbital and spin variables is capable of generating qualitatively new phenomena which do not occur in the absence of orbital interactions, such as anisotropic magnetic interactions, and novel quantum phenomena at finite temperature, discusseed on the example of LaVO$_3$ in section 3. One of such novel and puzzling phenomena is the magnetic phase transition between two different types of AF order observed in YVO$_3$ — this compound has $G$-type AF order (staggered in all three directions, called below $G$-AF phase) at low temperature $T<T_{N2}$, while the magnetic order changes in the first order magnetic transition at $T_{N2}=77$ K to $C$-AF phase which remains stable up to $T_{N1}\simeq 116$ K. The latter $C$-AF phase has rather exotic magnetic properties, and the magnon spectra show dimerization of the FM interactions along the $c$ axis [@Ulr03], see figure \[fig:yvo\]. In fact, the $G$-AF phase occurs in systems with large GdFeO$_3$-like distortion [@Miy03]. In Ref. [@Kha01] an orbital interaction favoring $C$-type alternating orbital ($C$-AO) order was invoked to explain the $G$-AF phase. We also address this problem in section 3 and present arguments that at higher $T>T_{N2}$ $C$-AF phase reappears is due to its higher entropy [@Kha01].
In section 4 we address the experimental phase diagram of the $R$VO$_3$ perovskites. It is quite different from the (also puzzling) phase diagram of the $R$MnO$_3$ perovskites, where the orbital order (OO) appears first at $T_{\rm OO}$ upon lowering the temperature, and spin order follows [@Goo06] at the Néel temperature, $T_{N1}\ll T_{\rm OO}$. In contrast, in the $R$VO$_3$ perovskites the two transitions appear at similar temperature [@Miy06]. For instance, in LaVO$_3$ they occur even almost simultaneously i.e., $T_{N1}\simeq T_{\rm OO}$. However, they become separated from each other in the $R$VO$_3$ systems with smaller ionic radii of $R$ ions — whereas $T_{N1}$ gets reduced for decreasing $r_R$, $T_{\rm OO}$ exhibits a [*nonmonotonic*]{} dependence on $r_R$ [@Miy03]. A short summary is presented in section 5. We also point out a few unsolved problems of currect interest in the field of orbital physics.
Spin-orbital superexchange and entanglement {#sec:som}
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Spin-orbital models derived for real systems are rather complex and the orbital part of the superexchange is typically much more complicated than the compass model (\[compa\]) of section 1 [@Ole05]. The main feature of these superexchange models is that not only the orbital interactions are directional and frustrated, but also spin correlations may influence orbital interactions and [*vice versa*]{}. This is best seen in the Kugel-Khomskii ($d^9$) model [@Kug82], where the $G$-AF and $A$-AF order compete with each other, and the long-range order is destabilized by quantum fluctuations in the vicinity of the quantum critical point $(J_H,E_z)=(0,0)$ [@Fei98]. Here $J_H$ is the local exchange (see below), and $E_z$ is the splitting of two $e_g$ orbitals. Although this model is a possible realization of disordered spin-orbital liquid, its phase diagram remains unexplored beyond the MF approach and simple valence-bond wave functions of Ref. [@Fei97] — it remains one of the challenges in this field.
In this chapter we consider the superexchange derived for an (idealized) perovskite structure of $R$VO$_3$, with V$^{3+}$ ions occupying the cubic lattice. The kinetic energy is given by: $$\label{hkin}
H_{t}=-t\sum_{\langle ij\rangle{\parallel}\gamma}\;
\sum_{\alpha(\gamma),\sigma}
\left(d^{\dagger}_{i\alpha\sigma}d^{}_{j\alpha\sigma}+
d^{\dagger}_{j\alpha\sigma}d^{}_{i\alpha\sigma}\right),$$ where $d^{\dagger}_{i\alpha\sigma}$ is electron creation operator for an electron with spin $\sigma=\uparrow,\downarrow$ in orbital $\alpha$ at site $i$. The summation runs over the bonds $\langle ij\rangle{\parallel}\gamma$ along three cubic axes, $\gamma=a,b,c$, with the hopping elements $t$ between active $t_{2g}$ orbitals. They originate from two subsequent hopping processes via the intermediate $2p_{\pi}$ oxygen orbital along each V–O–V bond. Its value can in principle be derived from the charge-transfer model [@Zaa93], and one expects $t=t_{pd}^2/\Delta\sim 0.2$ eV [@Kha01]. Only two out of three $t_{2g}$ orbitals, labelled by $\alpha(\gamma)$, are active along each bond $\langle ij\rangle$ and contribute to the kinetic energy (\[hkin\]), while the third orbital lies in the plane perpendicular to the $\gamma$ axis and the hopping via the intermediate oxygen $2p_{\pi}$ oxygen is forbidden by symmetry. This motivates a convenient notation used below, $$\label{abc}
|a\rangle\equiv |yz\rangle, \hskip .7cm
|b\rangle\equiv |xz\rangle, \hskip .7cm
|c\rangle\equiv |xy\rangle,$$ where the orbital inactive along a cubic direction $\gamma$ is labelled by its index as $|\gamma\rangle$.
The superexchange model for the $R$VO$_3$ perovskites arises from virtual charge excitations between V$^{3+}$ ions in the high-spin $S=1$ state. The number of $d$ electrons is 2 at each V$^{3+}$ ion ($d^2$ configuration), and the superexchange is derived from all possible virtual $d_i^2d_j^2\rightleftharpoons d_i^3d_j^1$ excitation processes (for more details see Ref. [@Ole07]). It is parametrized by the superexchange constant $J$ and Hund’s parameter $\eta$, $$\label{sex}
J=\frac{4t^2}{U}\,, \hskip 2cm \eta=\frac{J_H}{U}\,,$$ where $U$ is the intraorbital Coulomb interaction and $J_H$ is Hund’s exchange between $t_{2g}$ electrons. Here we use the usual convention and write the local Coulomb interactions between $3d$ electrons at V$^{3+}$ ions by limiting ourselves to intraorbital and two-orbital interaction elements [@Ole05]: $$\begin{aligned}
\label{Hee}
H_{\rm int}&=&
U\sum_{i\alpha}n_{i\alpha \uparrow}n_{i\alpha\downarrow}
+\Big(U-\frac{5}{2}J_H\Big)\sum_{i,\alpha<\beta}n_{i\alpha}n_{i\beta}
-2J_H\sum_{i,\alpha<\beta}\textbf{S}_{i\alpha}\cdot\textbf{S}_{i\beta}
\nonumber \\
&+& J_H\sum_{i,\alpha<\beta}
\left( d^{\dagger}_{i\alpha\uparrow}d^{\dagger}_{i\alpha\downarrow}
d^{ }_{i\beta\downarrow}d^{ }_{i\beta\uparrow}
+d^{\dagger}_{i\beta\uparrow}d^{\dagger}_{i\beta\downarrow}
d^{ }_{i\alpha\downarrow}d^{ }_{i\alpha\uparrow}\right).\end{aligned}$$ When only one type of orbitals is party occupied (as in the present case of the $R$VO$_3$ perovskites or in KCuF$_3$), the two parameters $\{U,J_H\}$ are sufficient to describe these interactions in Eq. (\[Hee\]): ($i$) the intraorbital Coulomb element $U$ and ($ii$) the interorbital (Hund’s) exchange element $J_H$, where $\{A,B,C\}$ are the Racah parameters [@Gri71]. In such cases the above expression is exact; in other cases when both $e_g$ and $t_{2g}$ electrons contribute to charge excitations (as for instance in the $R$MnO$_3$ perovskites), Eq. (\[Hee\]) is only an approximation — the anisotropy on the interorbital interaction elements has to be then included to reproduce accurately the multiplet spectra of the transition metal ions [@Gri71]. The intraorbital interaction is $U=A+4B+3C$, while $J_H$ depends on orbital type — for $t_{2g}$ electrons one finds [@Ole05; @Gri71] $J_H=3B+C$.
The perturbative treatment of intersite charge excitations $d_i^2d_j^2\rightleftharpoons d_i^3d_j^1$ in the regime of $t\ll U$ leads for the $R$VO$_3$ perovskites (and in each similar case [@Ole05]) to the spin-orbital superexchange model: $$\label{HJ}
{\cal H}_J=\sum_{\langle ij\rangle\parallel\gamma}H^{(\gamma)}(ij)
=J\sum_{\langle ij\rangle\parallel\gamma}\left\{
\left({\vec S}_i\cdot {\vec S}_j+S^2\right)
{\hat J}_{ij}^{(\gamma)} +
{\hat K}_{ij}^{(\gamma)} \right\}.$$ The spin interactions $\propto {\vec S}_i\cdot {\vec S}_j$ obey the SU(2) symmetry. In contrast, the orbital interaction operators ${\hat J}_{ij}^{(\gamma)}$ and ${\hat K}_{ij}^{(\gamma)}$ involve directional (here $t_{2g}$) orbitals on each individual bond $\langle ij\rangle\parallel\gamma$, so they have a lower (cubic) symmetry. The above form of the spin-orbital interactions is general and the spin value $S$ depend on the electronic configuration $d^n$ of the involved transition metal ions (here $n=2$ and $S=1$). For convenience, we introduced also a constant $S^2$ in the spin part, so for the classical Néel order the first term $\propto {\hat J}_{ij}^{(\gamma)}$ vanishes.
In the $R$VO$_3$ perovskites one finds the orbital operators [@Ole07]: $$\begin{aligned}
\label{orbj}
{\hat J}_{ij}^{(\gamma)}&=&
\frac{1}{2}\left\{(1+2\eta r_1)
\left({\vec\tau}_i\cdot {\vec\tau}_j
+\frac{1}{4}n_i^{}n_j^{}\right)\right. \nonumber \\
&-&\! \left.\eta r_3
\left({\vec \tau}_i\times{\vec \tau}_j+\frac{1}{4}n_i^{}n_j^{}\right)
-\frac{1}{2}\eta r_1(n_i+n_j)\right\}^{(\gamma)}, \\
%
\label{orbk}
{\hat K}_{ij}^{(\gamma)}&=&\!
\left\{\eta r_1
\left({\vec\tau}_i\cdot {\vec\tau}_j+\frac{1}{4}n_i^{}n_j^{}\right)
+\eta r_3\left({\vec\tau}_i\times {\vec\tau}_j
+\frac{1}{4}n_i^{}n_j^{}\right)\right. \nonumber \\
&-&\left.
\frac{1}{4}(1+\eta r_1)(n_i+n_j)\right\}^{(\gamma)},\end{aligned}$$ where the scalar product $({\vec\tau}_i\cdot {\vec\tau}_j)^{(\gamma)}$ and the cross-product, $$\label{exo1}
\left({\vec\tau}_i\times {\vec\tau}_j\right)^{(\gamma)}=
\frac{1}{2}\left(\tau_i^+\tau_j^++\tau_i^-\tau_j^-\right)
+\tau_i^z\tau_j^z\,,$$ involve orbital (pseudospin) operators corresponding to two active $t_{2g}$ orbitals along the $\gamma$ axis, with ${\vec\tau}_i=\{\tau_i^+,\tau_i^-\tau_i^z\}$, and $$\label{tauz}
\tau_i^z=\textstyle{\frac{1}{2}}(n_{i,yz}-n_{i,zx})\,.$$ They follow from the structure of local Coulomb interaction (\[Hee\]). The latter term (\[exo1\]) leads to the nonconservation of total pseudospin quantum number. Density operators $n_i^{(\gamma)}$ in Eqs. (\[orbj\]) and (\[orbk\]) stand for the number of $d$ electrons in active orbitals for the considered bond $\langle ij\rangle$, e.g. $n_i^{(c)}=n_{ia}+n_{ib}$. The coefficients, $$\label{rr}
r_1=\frac{1}{1-3\eta}\,, \hskip 1.5cm r_3=\frac{1}{1+2\eta}\,,$$ follow from the energies of $d_i^2d_j^2\rightleftharpoons d_i^3d_j^1$ excitations in the units of $U$: ($i$) $r_1$ represents the high-spin $^4A_2$ excitation of energy $(U-3J_H)$, while the low-spin excitations are given by ($ii$) $r_2=1$ originating from the low-spin $^2T_1$ and $^2E$ excitations of energy $U$, and ($iii$) $r_3$ represents the low-spin $^2T_2$ states of energy $(U+2J_H)$.
Magnetic order observed in Mott insulators is usually understood in terms of the GKR which are based on the MF picture and ignore entangled quantum states. These rules state that the pattern of occupied orbitals determines the spin structure. For example, for $180^{\circ}$ bonds (e.g. Mn–O–Mn bonds in LaMnO$_3$) there are two key rules: ($i$) if two partially occupied $3d$ orbitals point towards each other, the interaction is AF, however, ($ii$) if an occupied orbital on one site has a large overlap with an empty orbital on the other site of a bond $\langle ij\rangle$, the interaction is weak and FM due to finite Hund’s exchange. This means that spin order and orbital order are complementary — ferro-like (uniform) orbital (FO) order supports AF spin order, while AO order supports FM spin order. Indeed, these celebrated rules are well followed in LaMnO$_3$ [@Wei04] and in KCuF$_3$ [@Ole00], where strong JT effect stabilizes the orbital order and suppresses the orbital fluctuations. The AO order is here robust in the FM $ab$ planes, while the orbitals obey the FO order along the $c$ axis, supporting the AF coupling and leading to the $A$-AF phase for both systems. In such cases the GKR directly apply. Therefore, one may disentangle the spin and orbital operators, and it has been shown that this procedure is sufficient to explain both the magnetic [@Fei99] and optical [@Ole05] properties of LaMnO$_3$.
As another prominent example of the Goodenough-Kanamori complementarity we would like to mention the AF phases realized in YVO$_3$ [@Ulr03], which are the subject of intense research in recent years. [*A priori,*]{} the orbital interactions between V$^{3+}$ ions in $d^2$ configuration obey the cubic symmetry, if the $t_{2g}$ orbitals are randomly occupied. However, the symmetry breaking at the structural transition where the symmetry is reduced from cubic to orthorhombic, which persists in the magnetic phases, suggests that the electronic configuration is different. Indeed, the GdFeO$_3$ distortions in the $R$VO$_3$ structure break the symmetry in the orbital space, and both the electronic structure calculations [@And07] and the analysis using the point charge model [@Hor08] indicate that the electronic configuration $(xy)^1(yz,zx)^1$ is induced at every site, i.e., $$\label{nn} n_{ic}=1\,, \hskip 1.5cm n_{ia}+n_{ib}=1\,.$$ The partly filled $\{a,b\}$ orbitals are both active along the $c$ axis, and may lead either to FO or to AO order. Indeed, depending on this orbital pattern, the magnetic correlations are there either AF or FM, explaining the origin of the two observed types of AF order: ($i$) the $C$-AF phase, and ($ii$) the $G$-AF phase. However, the situation is more subtle as both orbitals in the orbital doublet $\{|yz\rangle,|xz\rangle\}\equiv\{|a\rangle,|b\rangle\}$ at each site $i$ are active on the bonds along the $c$ axis. This demonstrates an important difference between the $e_g$ (with one electron or one hole in active $e_g$ orbital at each site [@Dag04]) and a $t_{2g}$ system, such as $R$VO$_3$ perovskite vanadates, where electrons occupying two active $t_{2g}$ orbitals may fluctuate and form an [*orbital singlet*]{} [@Kha01]. The cubic symmetry is thus broken as both orbital flavors are active only along the $c$ axis, and the bonds in the $ab$ planes and along the $c$ axis are nonequivalent. Consequently, superexchange orbital operators (\[orbj\]) and (\[orbk\]) take different forms along these two distinct directions, $$\begin{aligned}
\label{orbjc}
{\hat J}_{ij}^{(c)}&=&
\frac{1}{2}\left\{(1+2\eta r_1)
\left({\vec\tau}_i\cdot {\vec\tau}_j +\frac{1}{4}\right)
-\eta r_3\left({\vec\tau}_i\times{\vec\tau}_j+\frac{1}{4}\right)
-\eta r_1\right\}\,, \\
%
\label{orbkc}
{\hat K}_{ij}^{(c)}&=&
\left\{\eta r_1
\left({\vec\tau}_i\cdot {\vec\tau}_j+\frac{1}{4}\right)
+\eta r_3\left({\vec\tau}_i\times {\vec\tau}_j
+\frac{1}{4}\right)
-\frac{1}{2}(1+\eta r_1)\right\}\,, \\
\label{orbja}
{\hat J}_{ij}^{(a)}&=&
\frac{1}{4}\left\{(1-\eta r_3)(1+n_{ib}n_{jb})
-r_1(n_{ib}-n_{jb})^2\right\}\,, \\
%
\label{orbka}
{\hat K}_{ij}^{(a)}&=&
\frac{1}{2}\eta(\eta r_1+r_3)(1+n_{ib}n_{jb})\,.\end{aligned}$$
The general form of spin-orbital superexchange model (\[HJ\]) suggests that the above symmetry breaking leads indeed to an effective spin model with broken symmetry between magnetic interactions along different cubic axes. By averaging over the orbital operators one finds indeed different effective magnetic exchange interactions, $J_c$ along the $c$ axis and $J_{ab}$ within the $ab$ planes: $$\label{Jij}
J_{c} =\left\langle{\hat J}_{ij}^{(c)}\right\rangle\,, \hskip 2cm
J_{ab}=\left\langle{\hat J}_{ij}^{(a)}\right\rangle\,.$$ The interactions in the $ab$ planes could in principle still take two different values in case of finite lattice strain discussed below, making both $\{a,b\}$ axes inequivalent, but here we want just to point out the symmetry breaking between the $c$ axis and the $ab$ planes, which follows from the density distribution (\[nn\]) and explains the nonequivalence of spin interactions in the $C$-AF phase of the $R$VO$_3$ perovskites [@Kha01].
Apart from the superexchange there are in general also interactions due to the couplings to the lattice that control the orbitals. In the cubic vanadates these interactions are expected to be weak, but nevertheless they influence significantly the spin-orbital fluctuations and decide about the observed properties in the $R$VO$_3$ family. We write the orbital interactions, $\propto\tau^z_i\tau^z_j$, induced by the GdFeO$_3$ distortions and by the JT distortions of the lattice using two parameters, $V_{ab}$ and $V_c$, $$\label{HJT}
{\cal H}_V =
V_{ab}\sum_{\langle ij\rangle\parallel c}\tau^z_i\tau^z_j
-V_c\sum_{\langle ij\rangle\parallel c}\tau^z_i\tau^z_j\,.$$ The orbital interaction along the $c$ axis $V_c$ plays here a crucial role and allows one to switch between the two types of magnetic order, $C$-AF and $G$-AF phase [@Hor03], stabilizing simultaneously either $G$-AO or $C$-AO order.
However, the description in terms of the GKR does not suffice and the ground state of spin-orbital model for the $R$VO$_3$ perovskites, which consists of the superexchange and the effective orbital interactions, $$\label{som} {\cal H}_{S\tau} ={\cal H}_J + {\cal H}_V\,.$$ may also be entangled due to the quantum coupling between spin $S=1$ and orbital $\tau=1/2$ operators along the $c$ axis, see Eq. (\[orbjc\]). In constrast, the orbital fluctuations in the $ab$ planes are quenched due to the occupied $c$ orbitals at each site (\[nn\]), so spins and orbitals disentangle. Possible entanglement between spin $({\vec S}_i\cdot {\vec S}_j)$ and orbital $({\vec\tau}_i\cdot {\vec\tau}_j)$ operators along the bonds $\langle ij\rangle\parallel c$ in the $R$VO$_3$ perovskites, and the applicability of the GKR to these systems, may be investigated by evaluating intersite spin and orbital correlations (to make these two functions comparable, we renormalized the spin correlations by the factor $\frac14$), $$\begin{aligned}
\label{sij}
S_{ij}&=&\frac14\langle{\vec S}_i\cdot{\vec S}_j\rangle\,, \\
\label{tij}
T_{ij}&=&\langle{\vec\tau}_i\cdot{\vec\tau}_j\rangle\,,\end{aligned}$$ and comparing them with each other. A key quantity that measures spin-orbital entanglement is the composite correlation function [@Ole06], $$\label{cij}
C_{ij}=\frac14
\left\{\big\langle({\vec S}_i\cdot{\vec S}_j)
({\vec\tau}_i\cdot{\vec\tau}_j)\big\rangle
-\big\langle{\vec S}_i\cdot{\vec S}_j \big\rangle
\big\langle{\vec\tau}_i\cdot{\vec\tau}_j \big\rangle\right\}\,.$$ When $C_{ij}=0$, the spin and orbital operators are disentangled and their MF decoupling is exact, while if $C_{ij}<0$ — spin and orbital operators are entangled and the MF decoupling not justified.
![ Evolution of intesite correlations and exchange constants along the $c$ axis obtained by exact diagonalizaton of spin-orbital model on a chain of $N=4$ sites with periodic boundary conditions, with ${\hat J}_{ij}^{(c)}$ and ${\hat K}_{ij}^{(c)}$ given by Eqs. (\[orbjc\]) and (\[orbkc\]), for increasing Hund’s exchange $\eta$: (a),(b) intersite spin $S_{ij}$ (\[sij\]) (filled circles), orbital (\[tij\]) (empty circles), and spin-orbital $C_{ij}$ (\[cij\]) ($\times$) correlations; (c),(d) the corresponding spin exchange constants $J_{ij}$ (\[Jij\]). In the shaded areas of (c) and (d) the spin correlations $S_{ij}<0$ do not follow the sign of the exchange constant $J_{ij}<0$, and the classical GKR are violated. Parameters: (a),(c) $V_c=0$, and (b),(d) $V_c=J$. []{data-label="fig:enta"}](allcorr2.eps){width="\textwidth"}
![ Evolution of intesite correlations and exchange constants along the $c$ axis obtained by exact diagonalizaton of spin-orbital model on a chain of $N=4$ sites with periodic boundary conditions, with ${\hat J}_{ij}^{(c)}$ and ${\hat K}_{ij}^{(c)}$ given by Eqs. (\[orbjc\]) and (\[orbkc\]), for increasing Hund’s exchange $\eta$: (a),(b) intersite spin $S_{ij}$ (\[sij\]) (filled circles), orbital (\[tij\]) (empty circles), and spin-orbital $C_{ij}$ (\[cij\]) ($\times$) correlations; (c),(d) the corresponding spin exchange constants $J_{ij}$ (\[Jij\]). In the shaded areas of (c) and (d) the spin correlations $S_{ij}<0$ do not follow the sign of the exchange constant $J_{ij}<0$, and the classical GKR are violated. Parameters: (a),(c) $V_c=0$, and (b),(d) $V_c=J$. []{data-label="fig:enta"}](allex2.eps){width="\textwidth"}
The numerical results for a 1D chain along the $c$ axis described by vanadate spin-orbital model (\[som\]) are shown in Fig. \[fig:enta\]. One finds entangled spin-orbital states with all three $S_{ij}$, $T_{ij}$ and $C_{ij}$ correlations being negative in the spin-singlet ($S=0$) regime of fluctuating $yz$ and $zx$ orbitals, obtained for $\eta< 0.07$ \[Fig. \[fig:enta\](a)\]. Therefore, the complementary behavior of spin (\[sij\]) and orbital (\[tij\]) correlations is absent in this regime of parameters and the GKR are violated. In addition, composite spin-orbital correlations (\[cij\]) are here finite ($C_{ij}<0$), so spin and orbital variables are entangled, and the MF factorization of spin-orbital operators fails. In a similar $d^1$ model for the perovskite titanates (with $S=1/2$) one finds even somewhat stronger spin-orbital entanglement and the regime of $\eta$ with $C_{ij}<0$ is broader (i.e., $\eta< 0.21$) [@Ole06]. At the point $\eta=0$ one recovers then the SU(4) model with $S_{ij}=T_{ij}=C_{ij}=-0.25$, and the ground state is an entangled SU(4) singlet, involving a linear combination of (spin singlet/orbital triplet) and (spin triplet/orbital singlet) states.
To provide further evidence that the GKR do not apply to spin-orbital model (\[som\]) in the regime of small $\eta$, we compare spin exchange constants $J_{ij}$ (\[Jij\]) shown in Fig. \[fig:enta\](c) with spin correlations $S_{ij}$ (\[sij\]), see Fig. \[fig:enta\](a). One finds that exchange interaction is formally FM ($J_{ij}<0$) in the orbital-disordered phase in the regime of $\eta<0.07$, but it is accompanied by AF spin correlations ($S_{ij}<0$). Therefore $J_{ij}S_{ij}>0$ and the ground state energy would be enhanced in an ordered state, when calculated in the MF decoupling of spin-orbital operators [@Ole06]. This at first instance somewhat surprising result is a consequence of [*‘dynamical’*]{} nature of exchange constants $\hat{J}_{ij}^{(c)}$ which exhibit large fluctuations [@Ole06], measured by the second moment, $\delta J=\{\langle (\hat{J}_{ij}^{(\gamma)})^2\rangle-J_{ij}^2\}^{1/2}$. For instance, in $d^2$ model (\[som\]) the orbital bond correlations change dynamically from singlet to triplet, resulting in large $\delta J=\frac{1}{4}\{1-(2T_{ij}+\frac{1}{2})^2\}^{1/2}\simeq 0.247$, i.e., $\delta J>|J_{ij}|$.
Remarkably, finite spin-orbital correlations $C_{ij}<0$ and similar violation of the GKR are found also at finite orbital interaction (\[HJT\]) induced by the lattice, $V_c>0$. Representative results obtained for $V_c=J$ are shown in Figs. \[fig:enta\](b) and \[fig:enta\](d). At small $\eta$ FO order is induced, and in this regime the GKR are followed by the AF/FO phase (similar to the FM/AO phase at large $\eta$ which also follows the GKR). However, for intermediate Hund’s exchange $\eta\sim 0.07$ FO order is destabilized and the entangled AF/AO phase appears, with similar spin, orbital and composite spin-orbital coorrelations as found before at $V_c=0$ and $\eta=0$ \[Figs. \[fig:enta\](a)\]. Also in this case FM exchange ($J_{ij}<0$) coexists with AF spin correlations ($S_{ij}<0$). Thus we conclude that orbital interactions induced by the lattice modify the regime of entangled spin-orbital states in the intermediate AF/AO phase which may be moved to more realistic values of $\eta\sim 0.1$, and cannot eliminate it completely. In addition, the transition between the FO/AF and AO/AF phase is [*continuous*]{} [@Ole06] due to the structure of orbital superexchange which contains terms (\[exo1\]) responsible for non-conservation of orbital quantum numbers.
Experimental evidence of orbital fluctuations in LaVO$_3$/YVO$_3$ {#sec:rvo}
=================================================================
Before discussing the exotic magnetic properties and the phase diagram of the $R$VO$_3$ perovskites, we will consider the influence of magnetism on the optical spectra of LaVO$_3$, starting with a general formulation of the theory. While exchange constants may be extracted from the spin-orbital superexchange model (\[Jij\]), it is frequently not realized that virtual charge excitations that contribute to the superexchange are responsible as well for the optical absorption, thus the superexchange and the optical absorption are intimately related to each other via the optical sum rule [@Bae86]. This is not so surprising as when electrons are almost localized in a Mott insulator, the only kinetic energy which is left and decides about the optical spectral weight is associated with virtual excitations contributing to superexchange. Therefore, in Mott insulators the thermal evolution of optical spectral weight can be deduced from the superexchange [@Aic02]. In a system with orbital degeneracy the optical spectra consist of several multiplet transitions, and the kinetic energy $K_n^{(\gamma)}$ (due to $d-d$ excitations) associated with each of them can be determined from the superexchange (\[HJ\]) using the Hellman-Feynman theorem [@Kha04], $$\label{hefa} K_n^{(\gamma)}=2\left\langle
H_n^{(\gamma)}(ij)\right\rangle\,.$$ Note that $K_n^{(\gamma)}$ is negative and corresponds to the $n$’th multiplet state of the transition metal ion, created by a charge excitation along a bond $\langle ij\rangle\parallel\gamma$. It is obvious that the thermal excitation values $\langle\cdots\rangle$ depend sensitively on the magnetic structure, i.e., whether spin correlations on a bond $\langle ij\rangle$ are FM or AF.
Thus it is natural to decompose the optical sum rule which is usually formulated in terms of the [*total*]{} kinetic energy for polarization $\gamma$, $$\label{opsatot}
K^{(\gamma)}=2J\sum_n\big\langle H_n^{(\gamma)}(ij)\big\rangle,$$ into [*partial optical sum rules*]{} for individual Hubbard subbands [@Kha04], $$\label{opsa}
\frac{a_0\hbar^2}{e^2}\int_0^{\infty}\sigma_n^{(\gamma)}(\omega)
d\omega=-\frac{\pi}{2}K_n^{(\gamma)}=-\pi\left\langle
H_n^{(\gamma)}(ij)\right\rangle\,,$$ where $\sigma_n^{(\gamma)}(\omega)$ is the contribution of band $n$ to the optical conductivity for polarization along the $\gamma$ axis, $a_0$ is the distance between transition metal ions, and the tight-binding model with nearest neighbor hopping is assumed. Equation (\[opsa\]) provides a practical way of calculating the optical spectral weights from spin-orbital superexchange models, such as the one derived for the $R$VO$_3$ perovskites (\[som\]). Note that the total optical intensity (\[opsatot\]) is of less interest here as it has a much weaker temperature dependence and does not allow one for a direct insight into the nature of the electronic structure. In addition, it might be also more difficult to resolve from experiment.
In order to apply the above theory to the $R$VO$_3$ perovskites, we write the superexchange operator $H^{(\gamma)}(ij)$ for a bond $\langle ij\rangle\parallel\gamma$, contributing to operator ${\cal H}_J$ (\[HJ\]), as a superposition of $d_i^2d_j^2\rightleftharpoons d_i^3d_j^1$ charge excitations to different upper Hubbard subbands labelled by $n$ [@Kha04], $$\label{Hn} H^{(\gamma)}(ij)=\sum_n H_n^{(\gamma)}(ij)\,.$$ One finds the superexchange terms $H^{(c)}_n(ij)$ for a bond ${\langle ij\rangle}$ along the $c$ axis [@Kha04], $$\begin{aligned}
\label{H1c} H_1^{(c)}(ij)&=&-\frac{1}{3}Jr_1 (2+\vec
S_i\!\cdot\!\vec S_j)
\left(\frac{1}{4}-\vec \tau_i\cdot\vec \tau_j\right)\,, \\
\label{H2c} H_2^{(c)}(ij)&=&-\frac{1}{12}J(1-\vec S_i\!\cdot\!\vec
S_j) \left(\frac{7}{4}-\tau_i^z\tau_j^z-\tau_i^x\tau_j^x
+5\tau_i^y \tau_j^y\right)\,, \\
\label{H3c} H_3^{(c)}(ij)&=&-\frac{1}{4}Jr_3 (1-\vec
S_i\!\cdot\!\vec S_j)
\left(\frac{1}{4}+\tau_i^z\tau_j^z+\tau_i^x\tau_j^x -\tau_i^y
\tau_j^y\right)\,,\end{aligned}$$ and $H^{(ab)}_n(ij)$ for a bond in the $ab$ plane, $$\begin{aligned}
\label{H1a} H_1^{(ab)}(ij)&=&-\frac{1}{6}Jr_1\left(2+\vec
S_i\!\cdot\!\vec S_j\right)
\left(\frac{1}{4}-\tau_i^z\tau_j^z\right)\,, \\
\label{H2a} H_2^{(ab)}(ij)&=&-\frac{1}{8}J\left(1-\vec
S_i\!\cdot\!\vec S_j\right) \left(\frac{19}{12}\mp
\frac{1}{2}\tau_i^z
\mp \frac{1}{2}\tau_j^z-\frac{1}{3}\tau_i^z\tau_j^z\right)\,, \\
\label{H3a} H_3^{(ab)}(ij)&=&-\frac{1}{8}Jr_3\left(1-\vec
S_i\!\cdot\!\vec S_j\right) \left(\frac{5}{4}\mp \frac{1}{2}\tau_i^z
\mp \frac{1}{2}\tau_j^z+\tau_i^z\tau_j^z\right)\,.\end{aligned}$$ These expressions show that the spin correlations along the $c$ axis and within the $ab$ planes, $$\label{spins}
s_c=\langle\vec{S}_i\cdot\vec{S}_j\rangle_{c}\,, \hskip 1cm
s_{ab}=\langle\vec{S}_i\cdot\vec{S}_j\rangle_{ab}\,,$$ as well as the orbital correlations, play an important role in the intensity distribution in optical spectroscopy. From the form of the above superexchange contributions one sees that high-spin excitations $H_1^{(\gamma)}(ij)$ support the FM coupling while the low-spin ones, $H_2^{(\gamma)}(ij)$ and $H_3^{(\gamma)}(ij)$, contribute with AF couplings.
We have determined the exchange constants in LaVO$_3$ by averaging over the orbital operators, see Eqs. (\[Jij\]). The case of the $ab$ planes is straightforward as only the average densities $\langle n_{ia}\rangle$ and $\langle n_{ib}\rangle$ are needed to determine $J_{ab}$, and at large $\eta$ they follow from the $G$-AO order in these planes. At $\eta=0$ the orbital correlations along the $c$ axis result from orbital fluctuations in the 1D orbital chain. In this limit the orbital correlations are the same as for the AF Heisenberg chain, i.e., $\langle{\vec\tau}_i\cdot{\vec\tau}_j\rangle=-0.4431$ and the ground state is disordered, with $\langle\tau_i^z\rangle=0$. Nevertheless, for this disordered state the result for $J_{ab}$ is similar as for the $G$-AO phase [@Ole07].
![ Exchange constants $J_{ab}$ and $-J_c$ (\[Jij\]) calculated from Eqs. (\[orbja\]) and (\[orbjc\]) in the $C$-AF phase of LaVO$_3$ for increasing $\eta$ (solid lines). Dashed line shows the value of $-J_c$ obtained for classical orbital order (\[Jccaf\]) according to GKR, $\langle\vec\tau_i\cdot\vec\tau_j\rangle=-\frac14$. A representative value of $\eta=0.14$ (for $U=5.0$ and $J_H=0.7$ eV) is marked by dotted line. Parameters of the model (\[som\]): $J=35$ meV, $V_c=V_{ab}=0$. []{data-label="fig:lavo"}](jlavo.eps){width="7.0cm"}
For the disordered (fluctuating) $\{a,b\}$ orbital state at $\eta=0$, the AF exchange interactions in $ab$ planes (see Fig. \[fig:lavo\]) result solely from singly occupied $c$ orbitals (\[nn\]), which are active in these planes and contribute by their double occupancies in excited state with AF superexchange. One expects that the exchange constants along the $c$ axis in the $C$-AF phase could be deduced from Eqs. (\[Jij\]), as spin and orbital order are complementary [@Miy06]. It is quite remarkable that at the same time finite FM interactions $-J_c\simeq 3$ meV are obtained at $\eta=0$ (Fig. \[fig:lavo\]). They follow from the orbital fluctuations which dominate at low values of $\eta$. This mechanism of FM exchange adds to the one known in systems with real orbital order at finite $\eta$ — the latter mechanism gradually takes over when $\eta$ increases and the $G$-AO order develops and reduces the orbital fluctuations. At finite $\eta>0$ we used the linear orbital-wave theory [@vdB99] to determine the intersite orbital correlations $\langle{\vec\tau}_i\cdot{\vec\tau}_j\rangle$ and the order parameter $\langle\tau_i^z\rangle$, for more details see Ref. [@Ole07]. At $\eta=0.14$ representative for LaVO$_3$, the FM interactions are stronger than from AF ones, $|J_c|>J_{ab}$. Indeed, this early prediction of the theory [@Kha01] agrees qualitatively with larger average FM exchange $J_c<0$ in the $C$-AF phase of YVO$_3$ than the AF exchange $J_{ab}>0$ in the $ab$ planes, see below.
We emphasize that the strong FM exchange along the $c$ axis follows from the orbital fluctuations, and the rigid $G$-AO order obtained in the limit of strong orbital interactions $\{V_{ab},V_c\}$ (\[HJT\]) would give a much weaker FM interaction, $$\label{Jccaf} J_c^{G-{\rm AO}}=-\frac{1}{2}\eta r_1J\,,$$ see Fig. \[fig:lavo\]. The FM interaction $J_c^{G-{\rm AO}}$ (\[Jccaf\]) vanishes at $\eta=0$, is triggered by finite Hund’s exchange $\eta$ and increases in lowest order linearly with $\eta$. This behavior follows the conventional mechanism of FM interactions induced by finite Hund’s exchange in the states with AO order, as for instance in KCuF$_3$ [@Fei97] or in LaMnO$_3$ [@Fei99].
![ Kinetic energy $K_n^{(c)}$ (solid lines) for the optical subband $n$ and total $K^{(c)}$ (dashed line) obtained from the spin-orbital model (\[som\]). Filled circles show the effective carrier number $N_{eff}^{(c)}$ (in the energy range $\omega<3$ eV) for LaVO$_3$, presented in Fig. 5 of Ref. [@Miy02]. Dotted line shows $K_1^{(c)}$ obtained from the MF decoupling (\[wc21\]). Parameters: $\eta=0.12$, $V_c=0.9J$, $V_{ab}=0.2J$. []{data-label="fig:osw"}](wlavo.eps){width="7.5cm"}
A crucial test of the present theory which demonstrates that orbital fluctuations are indeed present in LaVO$_3$, concerns the temperature dependence of the low-energy (high-spin) spectral weight in optical absorption along the $c$ axis $-K_1^{(c)}/2J$. According to experiment [@Miy02] it decreases by about 50% between low temperature and $T=300$ K. In contrast, the result obtained by averaging the high-spin superexchange term $H_1^{(c)}(ij)$ (\[H1c\]) for polarization along the $c$ axis assuming robust $G$-AO order is, $$\label{wc21}
%
w_{c1}^{G-{\rm AO}}=\frac{1}{6}r_1\big(s_c+2\big)\,,$$ where the spin correlation function $s_c$ (\[spins\]) is responsible for the entire temperature dependence of the low-energy spectral weight. Equation (\[wc21\]) predicts decrease of $w_{c1}$ of only about 27%, see Fig. \[fig:osw\], and the maximal possible reduction of $K_1^{(c)}$ reached at $s_c=0$ in the limit of $T\to\infty$ is by 33%. This result proves that the scenario with frozen $G$-AO order in LaVO$_3$ is [*excluded by experiment*]{} [@Miy05].
In contrast, when a cluster method which allows to include orbital fluctuations along the $c$ axis is used to determine the optical spectral weight from the high-spin superexchange term (\[H1c\]) [@Kha04], the temperature dependence resulting from the theory follows the experimental data [@Miy02]. This may be considered as a remarkable success of the theory based on the spin-orbital superexchange model derived for the $R$VO$_3$ perovskites.
However, the experimental situation in the cubic vanadates is more complex and full of puzzles. One is connected with the second magnetic transition in YVO$_3$, as we already mentioned in Sec. \[sec:som\]. The magnetic transition at $T_{N2}=77$ K is particularly surprising as the staggered moments are approximately parallel to the $c$ axis in the $G$-AF phase, and rotate above $T_{N2}$ to the $ab$ planes in the $C$-AF phase, with some small alternating $G$-AF component along the $c$ axis [@Ren00]. While the orientation of spins in $C$-AF and $G$-AF phase follow in a straightforward manner from the model, i.e., are consistent with the expected anisotropy due to spin-orbit coupling [@Hor03], the observed magnetization reversal with the weak FM component remains puzzling. Therefore, in spite of the suggested mechanism based on the entropy increase in the $C$-AF phase [@Ole07], the lower magnetic transition in YVO$_3$ remains mysterious. Secondly, the scale of magnetic excitations is considerably reduced for the $C$–AF phase (by a factor close to two) as compared with the exchange constants deduced from magnons measured in the $G$-AF phase [@Ulr03]. In addition, the magnetic order parameter in the $C$-AF phase of LaVO$_3$ is strongly reduced to $\simeq 1.3\mu_B$, which cannot be explained by the quantum fluctuations in the $C$-AF phase (being only 6% for $S=1$ spins [@Rac02]). Finally, the $C$-AF phase of YVO$_3$ is dimerized. Until now, only this last feature found a satisfactory explanation in the theory [@Sir03; @Sir08], see below.
![Spin-wave dispersions $\omega_{\bf k}$ obtained in the LSW theory (\[spinw\]) for the $C$-AF phase of YVO$_3$ (lines), and measured by neutron scattering at $T=85$ K [@Ulr03] (circles). Parameters: $J_{ab}=2.6$ meV, $J_c=-3.1$ meV, $\delta_s=0.35$, and $K_z=0.4$ eV (full lines), $K_z=0$ (dashed lines). The high symmetry points are: $\Gamma=(0,0,0)$, $M=(\pi,\pi,0)$, $R=(\pi,\pi,\pi)$, $Z=(0,0,\pi)$. []{data-label="fig:swcafd"}](swyvo.eps){width="7.5cm"}
We remark that the observed dimerization in the magnon dispersions may be seen as a signature of [*entanglement in excited states*]{} which becomes active at finite temperature. The microscopic reason of the anisotropy in the exchange constants ${\cal J}_{c1}\equiv{\cal J}_c(1+\delta_s)$ and ${\cal J}_{c2}\equiv{\cal J}_c(1-\delta_s)$ is the tendency of the orbital chain to dimerize, activated by thermal fluctuations in the FM spin chain [@Sir08] which support dimerized structure in the orbital sector. As a result one finds alternating stronger $\propto {\cal J}_c(1+\delta_s)$ and weaker $\propto {\cal
J}_c(1-\delta_s)$ FM bonds along the $c$ axis in the dimerized $C$-AF phase (with $\delta_s>0$). The observed spin waves may be explained by the following effective spin Hamiltonian for this phase (assuming again that the spin and orbital operators may be disentangled which is strictly valid only at $T=0$): $$\label{hcafd}
{\cal H}_{s}={\cal J}_{c}\sum_{\langle i,i+1\rangle\parallel c}
\left[1+(-1)^i\delta_s\right]{\vec S}_{i}\cdot{\vec S}_{i+1}
+{\cal J}_{ab}^C\sum_{\langle ij\rangle\parallel ab}
{\vec S}_i\cdot{\vec S}_j
+K_z\sum_i\left(S_i^z\right)^2\,.$$ Following the linear spin-wave theory [@Ole07], the magnon dispersion is given by $$\label{spinw} \omega_{\pm}({\bf k})= 2\left\{\left[2{\cal
J}_{ab}+|{\cal J}_c|+\frac12 K_z \pm {\cal J}_c\eta_{\bf
k}^{1/2}\right]^2 -\big(2{\cal J}_{ab}\gamma_{\bf
k}\big)^2\right\}^{1/2}\,,$$ with $$\begin{aligned}
\label{gamma}
\gamma_{\bf k}&=&\frac12\left(\cos_kx+\cos k_y\right)\;, \\
\label{eta}
\eta_{\bf k}&=&\cos^2k_z+\delta_s^2\sin^2k_z\,.\end{aligned}$$ For the numerical evaluation of figure \[fig:swcafd\] we have used the experimental exchange interactions [@Ulr03]: ${\cal
J}_{ab}=2.6$ meV, ${\cal J}_c=-3.1$ meV, $\delta_s=0.35$. Indeed, large gap is found between two modes halfway in between the $M$ and $R$ points, and between the $Z$ and $\Gamma$ points (not shown). Two modes measured by neutron scattering [@Ulr03] (see also figure 1) and obtained from the present theory in the unfolded Brillouin zone are well reproduced by the dimerized FM exchange couplings in spin Hamiltonian (\[spinw\]). We note that a somewhat different Hamiltonian with more involved interactions was introduced in ref. [@Ulr03], but the essential features seen in the experiment are reproduced already by the present model $H_s$ with a single ion anisotropy term $\propto K_z$.
![ Free energies ${\cal F}_C$ ($C$-AF, solid line) and ${\cal F}_G$ ($G$-AF, dashed line) as obtained for the spin-orbital model (\[som\]) using the experimental values of magnetic exchange constants in both phases [@Ulr03]. The experimental magnetic transition temperatures, $T_{N2}\simeq 77$ K and $T_{N1}\simeq 116$ K, are indicated by arrows. Parameters: $J=40$ meV, $\eta=0.13$, $V_a=0.30J$, $V_c=0.84J$. []{data-label="fig:free"}](phthvar.eps){width="7.5cm"}
As the transition between the two magnetic phases, $G$-AF and $C$-AF phase, occurs in YVO$_3$ at finite temperature, the entropy has to play an important role. As mentioned above, the exchange constants found in the $C$-AF phase of YVO$_3$ (Fig. \[fig:swcafd\]) are considerably lower than the corresponding values in the $G$-AF phase, $J_{ab}=J_c\simeq 5.7$ meV [@Ulr03]. As a result of weaker exchange interactions, the spin entropy of the $C$ phase will grow faster than that of the $G$ phase, and induce the $G\rightarrow C$ transition. However, starting from our model (\[som\]) we do not find this strong reduction of energy scale in the $C$-AF phase. Other mechanism like the fluctuation of $n_{xy}$ occupancy has been invoked to account for this reduction [@Ole07]. Here we will simply adopt the experimental values for the exchange constants in the $C$-AF phase.
Using linear spin-wave and orbital-wave theory, the spin and orbital entropy normalized per one vanadium ion was calculated and compared for both magnetic phases of YVO$_3$ [@Ole07]. Using the experimental parameters [@Ulr03] one finds that: ($i$) the entropy ${\cal S}_C$ for the $C$-AF phase is larger that ${\cal S}_G$ for the $G$-AF phase, and ($ii$) the spin entropy grows significantly faster with temperature than the orbital entropy for each phase. Therefore, we conclude that the spin entropy gives here a leading contribution and is responsible for a fast decrease of the free energy in the $C$-AF phase which is responsible for the observed magnetic transition at $T_{N2}$ [@Ole07], see Fig. \[fig:free\].
Orbital and magnetic transition in the $R$VO$_3$ perovskites {#sec:phd}
============================================================
Spin-orbital-lattice coupling {#subsec:sol}
-----------------------------
Experimental studies have shown that the $C$-AF order is common to the entire family of the $R$VO$_3$ vanadates, where $R$=Lu,$\cdots$,La. In general the structural (orbital) transition occurs first. i.e., $T_{N1}<T_{\rm OO}$, except for LaVO$_3$ with $T_{N1}\simeq T_{\rm OO}$ [@Miy03; @Miy06]. When the ionic radius $r_R$ decreases, the Néel temperature $T_{N1}$ also decreases, while the orbital transition temperature $T_{\rm OO}$ increases first, passes through a maximum close to YVO$_3$, and decreases afterwards when LuVO$_3$ is approached. Knowing that quantum fluctuations and spin-orbital entanglement play so important role in the perovskite vanadates, it is of interest to ask whether the spin-orbital model (\[som\]) is able to describe this variation of $T_{\rm OO}$ and $T_{N1}$ with decreasing radius $r_R$ of $R$ ions in $R$VO$_3$ [@Miy03]. It is clear that the nonmonotonic dependence of $T_{\rm OO}$ on $r_R$ cannot be reproduced just by the superexchange, as a maximum in $T_{\rm OO}$ requires two mechanisms which oppose each other. In fact, the decreasing V–O–V angle ($\Theta$ along the $c$ axis) with decreasing ionic radius $r_R$ along the $R$VO$_3$ perovskites [@Ren03; @Ree06; @Sag06; @Sag07] reduces somewhat both the hopping $t$ and superexchange $J$ (\[sex\]), but we shall ignore this effect here and concentrate ourselves on the leading dependence on orbital correlations which are controlled by lattice distortions.
The model introduced in Ref. [@Hor08] to describe the phase diagram of $R$VO$_3$ includes the spin-orbital-lattice coupling by the terms: ($i$) the superexchange $H_J$ (\[HJ\]) between $V^{3+}$ ions in the $d^2$ configuration with $S=1$ spins [@Kha01], ($ii$) intersite orbital interactions $H_V$ (\[HJT\]) (which originate from the coupling to the lattice and play an important role in the transition between the $C$-AF and $G$-AF phase), ($iii$) the crystal-field splitting $\propto E_z$ between $yz$ and $zx$ orbitals, and ($iv$) orbital-lattice term $\propto gu$ which induces orbital polarization when the lattice strain (distortion) $u$ inreases. The Hamiltonian consists thus of several terms [@Hor08], $$\label{rvo3}
{\cal H}={\cal H}_{J}+{\cal H}_V(\vartheta)+E_z(\vartheta)
\sum_i\!e^{i{\vec R}_i{\vec Q}}\tau_i^z
-gu\sum_i\tau_i^x+\frac12 N K\{u-u_0(\vartheta)\}^2\,.$$ Except for the superexchange ${\cal H}_J$ (\[HJ\]), all the other terms in Eq. (\[rvo3\]) depend on the tilting angle $\vartheta$, which we use to parameterize the $R$VO$_3$ perovskites below. It is related to the V–O–V angle $\Theta=\pi-2\vartheta$, which decreases with increasing ionic radius $r_R$ ($\Theta=180^{\circ}$ corresponds to an ideal perovskite structure). By analyzing the structural data of the $R$VO$_3$ perovskites [@Ren03; @Ree06; @Sag06; @Sag07] one arrives at the following empirical relation between $r_R$ and $\vartheta$: $$\label{rR}
r_R=r_0-\alpha\,\sin^22\vartheta\,,$$ with $r_0=1.5$ Å and $\alpha=0.95$ Å.
The crystal-field splitting of $\{yz,zx\}$ orbitals ($E_z>0$) alternates in the $ab$ planes and is uniform along the $c$ axis, with a modulation vector ${\vec Q}=(\pi,\pi,0)$ in cubic notation — it supports the $C$-AO order, and not the observed (weak) $C$-AO order. The orbital interactions induced by the distortions of the VO$_6$ octahedra and by GdFeO$_3$ distortions of the lattice, $V_{ab}>0$ and $V_c>0$, also favor the $C$-AO order (like $E_z>0$). The orbital interaction $V_c$ counteracts the orbital superexchange $\propto J$ (\[orbkc\]), and has only rather weak dependence on $\vartheta$, so it suffices to choose a constant $V_c=0.26J$ to reproduce an almost simultaneous onset of spin and orbital order in LaVO$_3$, with $T_{\rm OO}\simeq T_{N1}$, as observed [@Miy03]. One finds $T_{N1}^{\rm exp}=147$ K taking $J=200$ K in the present model (\[rvo3\]), which reproduces well the experimental value $T_{N1}^{\rm exp}=143$ K for LaVO$_3$ [@Miy03].
The last two terms in Eq. (\[rvo3\]) descibe the orbital-lattice coupling via the orthorhombic strain $u=(b-a)/a$, where $a$ and $b$ are the lattice parameters of the $Pbnm$ structure, $K$ is the force constant, and $N$ is the number of $V^{3+}$ ions. Unlike $E_z$, the coupling $gu>0$ acts as a transverse field in the pseudospin space and favors that one of the two linear combinations $\frac{1}{\sqrt{2}}(|a\rangle_i\pm|b\rangle_i)$ of active $t_{2g}$ orbitals is occupied at site $i$. By minimizing the energy over $u$, one finds $$\label{geffT}
g_{\rm eff}(\vartheta;T)\equiv gu(\vartheta;T)
=gu_0(\vartheta)+\frac{g^2}{K}\langle\tau^x\rangle_T\,,$$ which shows that the global distortion $u(\vartheta;T)$ consists of ($i$) a pure lattice contribution $u_0(\vartheta)$, and ($ii$) a contribution due the orbital polarization $\propto\langle\tau^x\rangle$ which is determined self-consistently.
Dependence on lattice distortion {#subsec:la}
--------------------------------
In order to investigate the phase diagram of the $R$VO$_3$ perovskites one needs still information on the functional dependence of the parameters $\{E_z,V_{ab},g_{\rm eff}\}$ of the microscopic model (\[rvo3\]) on the tilting angle $\vartheta$. The GdFeO$_3$-like distortion is parametrized by two angles $\{\vartheta,\varphi\}$ describing rotations around the $b$ and $c$ cubic axes, as explained in Ref. [@Pav05]. Here we adopted a representative value of $\varphi=\vartheta/2$, similar as in the perovskite titanates. Therefore, we used only a single rotation angle $\vartheta$ in Eq. (\[rvo3\]), which is related to the ionic size by Eq. (\[rR\]). Functional dependence of the crystal-field splitting $E_z\propto\sin^3\vartheta\cos\vartheta$ on the angle $\vartheta$ may be derived from the point charge model [@Hor08], using the structural data for the $R$VO$_3$ perovskites [@Ren03; @Ree06; @Sag06; @Sag07]. It is expected that the functional dependence of $V_{ab}$ follows the crystal-field term, so we write: $$\begin{aligned}
\label{Ez}
E_z(\vartheta)&=&J\,v_z\,\sin^3\vartheta\cos\vartheta\,, \\
\label{vab}
V_{ab}(\vartheta)&=&J\,v_{ab}\,\sin^3\vartheta\cos\vartheta.\end{aligned}$$ Qualitatively, increasing $E_z$ and $V_{ab}$ with increasing lattice distortion and tilting angle $\vartheta$ do favor the orbital order, so the temperature $T_{\rm OO}$ is expected to increase.
A maximum observed in the dependence of $T_{\rm OO}$ on $r_R$ (or $\vartheta$) may be reproduced within the present model (\[rvo3\]) only when a competing orbital polarization interaction $g_{\rm eff}(\vartheta;T)$ (\[geffT\]) increases faster with $\vartheta$ when the ionic radius $r_R$ is reduced than $\{E_z,V_{ab}\}$. Both $u_0$ and $\langle\tau^x\rangle$ in Eq. (\[geffT\]) are expected to increase with increasing tilting angle $\vartheta$. Below we present the results obtained with a semiempirical relation, $$\label{geff}
g_{\rm eff}(\vartheta)=J\,v_{g}\,\sin^5\vartheta\cos\vartheta\,,$$ as postulated in Ref. [@Hor08]. Altogether, model (\[som\]) depends on three parameters: $\{v_z,v_{ab},v_g\}$ which could be selected [@Hor08] to reproduce the observed dependence of orbital and magnetic transition temperature on the ionic radius $r_R$ in the $R$VO$_3$ perovskites, see below.
Evolution of spin and orbital order in $R$VO$_3$ {#subsec:evo}
-------------------------------------------------
Hamiltonian (\[rvo3\]) poses a many-body problem which includes an interplay between spin, orbital, and lattice degrees of freedom. A standard approach to investigate the onset of spin and orbital order is to use the mean-field (MF) theory with on-site order parameters $\langle S^z\rangle$ (corresponding to $C$-AF phase) and $$\label{taug}
\langle\tau^z\rangle_G\equiv\frac12
\left|\langle\tau^z_i-\tau^z_j\rangle\right|\,,$$ as well as the coupling between them which modifies the MF equations, similar to the situation encountered in the Ashkin-Teller model [@Dit80]. This approach was successfully implemented to determine the orbital and magnetic transition temperature, $T_{\rm OO}$ and $T_{N1}$ in LaMnO$_3$ [@Fei99]. It was also applied to the $R$VO$_3$ perovskites [@Silva] to demonstrate that either spin or orbital order may occur first at decreasing temperature, depending on the amplitude of hopping parameters. However, Such a MF approach uses only on-site order parameters and cannot suffice when orbital fluctuations also contribute, e.g. stabilizing the $C$-AF phase in LaVO$_3$ [@Kha01] — then it becomes essential to determine self-consistently the above on-site order parameters together with orbital singlet correlations (\[tij\]) on the bonds $\langle ij\rangle\parallel c$. The simplest approach which allows us to determine these correlations is a cluster MF theory for a bond coupled to effective spin and orbital symmetry breaking fields which originate from its neighbors in an ordered phase. The respective transition temperatures are obtained when $\langle S^z\rangle>0$ ($\langle S^z\rangle=0$) for $T<T_{N1}$ ($T>T_{N1}$), and $\langle\tau^z\rangle_G>0$ ($\langle\tau^z\rangle_G=0$) for $T<T_{\rm OO}$ ($T>T_{\rm OO}$).
![ The orbital polarization $\langle\tau^x\rangle$ (dashed-dotted lines), $G$-type orbital order parameter $\langle\tau^z\rangle_G$ (\[taug\]) (dashed lines), and spin order parameter $\langle S^z\rangle$ (solid lines) for LaVO$_3$ and SmVO$_3$ (thin and heavy lines). Parameters: $V_{c}=0.26J$, $v_z=17$, $v_{ab}=22$, $v_{g}=740$. []{data-label="fig:ops"}](szhvar.eps){width="7.0cm"}
Making a proper selection of the model parameters $\{v_z,v_{ab},v_g\}$ one is able to reproduce the experimental phase diagram for the onset of $G$-AO and $C$-AF order in the $R$VO$_3$ family in the entire range of $r_R$, see below. We start with presenting an example of the orbital and spin phase transition in LaVO$_3$ and in SmVO$_3$, see Fig. \[fig:ops\]. By selecting $V_{c}=0.26J$ both $G$-AO and $C$-AF order occur simultaneuosly in LaVO$_3$ below $T_{\rm OO}=T_{N1}\simeq 0.73J$. The crystal field splitting $E_z$, orbital interaction $V_{ab}$, and the coupling to the lattice $g_{\rm eff}$ are rather small and do not influence the order in LaVO$_3$. We emphasize that orbital correlations along the $c$ axis are here practically as in the AF Heisenberg chain, $\langle{\vec\tau}_i\cdot{\vec\tau}_j\rangle\simeq -0.44$, and the orbital order is considerably reduced, $\langle\tau^z\rangle_G\simeq
0.32$. The orbital polarization in LaVO$_3$ $\langle\tau^x\rangle\simeq 0.03$ is rather weak at $T_{N1}$, and is further reduced with decreasing $T<T_{\rm OO}$. Note that also spin order parameter is expected to be reduced below $\langle S^z\rangle=1$, but weak quantum fluctuations in the $C$-AF phase [@Rac02] were neglected here. In contrast, in SmVO$_3$ the phase transitions separate: the orbital transition occurs first at $T_{\rm OO}\simeq 0.86J$, and the magnetic one follows at a lower $T_{N1}\simeq 0.65J$. Already in this case the transverse orbital polarization is considerably increased, with $\langle\tau^x\rangle\simeq 0.20$ at $T_{N1}$ (see Fig. \[fig:ops\]), and further increases with decreasing $r_R$ (not shown). Note that the polarization $\langle\tau^x\rangle$ does not change close to $T_{\rm OO}$, and only below $T_{N1}$ gets weakly reduced due to the developing magnetic order, in agreement with experiment [@Sag07]. The $G$-OO parameter is here stronger as the singlet orbital fluctuations are not so pronounced when $T\to 0$, being $\langle\tau^z\rangle_G\simeq 0.37$.
The key features of the present spin-orbital system which drive the observed dependence of $T_{\rm OO}$ and $T_{N1}$ on $r_R$ [@Miy03] is the evolution of intersite orbital correlations are: ($i$) the gradual increase of the orbital interactions $K_{ab}\tau^z_i\tau^z_j$ \[Fig. \[fig:orbi\](a)\], and ($ii$) the reduction of orbital fluctuations on the bonds along the $c$ axis, described by the bond singlet correlations $\langle{\vec\tau}_i\cdot{\vec\tau}_j\rangle$ \[Fig. \[fig:orbi\](b)\]. The parameter $K_{ab}$ in Fig. \[fig:orbi\](a) consists of the superexchange contribution $\propto J$ (\[orbka\]) and orbital interaction $V_{ab}$ (\[HJT\]) induced by the lattice distortion. While the superexchange does not change with decreasing $r_R$, the latter term increases and induces the increase of $T_{\rm OO}$ from LaVO$_3$ to YVO$_3$. This increase is similar to that observed in the $R$MnO$_3$ manganites [@Goo06]. Thereby the bond angle $\Theta$ decreases from $157.4^{\circ}$ in LaVO$_3$ to $144.8^{\circ}$ in YVO$_3$.
While the singlet correlations are drastically suppressed from LaVO$_3$ towards LuVO$_3$, the orbital order parameter $\langle\tau^z\rangle_G$ somewhat increases from LaVO$_3$ to SmVO$_3$ (see also Fig. \[fig:ops\]). At the same time the orbital polarization $\langle\tau^x\rangle$ increases and soon becomes as important as the orbital order parameter, i.e., $\langle\tau^x\rangle\simeq\langle\tau^z\rangle_G$. Further increase of the orbital polarization towards LuVO$_3$ suppresses the $G$-AO parameter, so $\langle\tau^z\rangle_G$ passes through a maximum and decreases for $r_R<1.22$ Å.
![ (a) The width of magnon band $W_{C-{\rm AF}}$ for finite $g_{\rm eff}$ (circles) and without orbital-strain coupling ($g_{\rm eff}=0$, dashed), and orbital interactions in $ab$ planes $K_{ab}$ (squares) in the $C$-AF phase of cubic vanadates (the points correspond to the $R$VO$_3$ compounds of Fig. \[fig:phd\]). (b) Evolution of the orbital order parameter $\langle\tau_i^z\rangle_G$ (filled circles), transverse orbital polarization $\langle\tau_i^x\rangle$ (squares), and orbital intersite correlations $|\langle\vec{\tau}_i\cdot\vec{\tau}_j\rangle|$ (diamonds) along $c$ axis at $T=0$. Parameters: $v_z=17$, $v_{ab}=22$, $v_{g}=740$.[]{data-label="fig:orbi"}](xjall.eps){width="\textwidth"}
![ (a) The width of magnon band $W_{C-{\rm AF}}$ for finite $g_{\rm eff}$ (circles) and without orbital-strain coupling ($g_{\rm eff}=0$, dashed), and orbital interactions in $ab$ planes $K_{ab}$ (squares) in the $C$-AF phase of cubic vanadates (the points correspond to the $R$VO$_3$ compounds of Fig. \[fig:phd\]). (b) Evolution of the orbital order parameter $\langle\tau_i^z\rangle_G$ (filled circles), transverse orbital polarization $\langle\tau_i^x\rangle$ (squares), and orbital intersite correlations $|\langle\vec{\tau}_i\cdot\vec{\tau}_j\rangle|$ (diamonds) along $c$ axis at $T=0$. Parameters: $v_z=17$, $v_{ab}=22$, $v_{g}=740$.[]{data-label="fig:orbi"}](ttc.eps){width="\textwidth"}
It is remarkable that the above changes in orbital correlations induced by the lattice suppress gradually the magnetic interactions in the $C$-AF phase, although the value of $J$ remains unchanged. This is well visible in the total width of the magnon band, $W_{C-{\rm AF}}=4(J_{ab}+|J_c|)$ (at $T=0$) [@Ole07], shown in Fig. \[fig:orbi\](a), being reduced from $\sim 1.84J$ in LaVO$_3$ to $\sim 1.05J$ in YVO$_3$. This large reduction qualitatively agrees with the rather small values of the exchange constants in the $C$-AF phase of YVO$_3$ [@Ulr03], see also Fig. \[fig:swcafd\]. This reduction is caused by the suppression of the singlet orbital correlations $\langle{\vec\tau}_i\cdot{\vec\tau}_j\rangle$ by the increasing coupling to the lattice $g_{\rm eff}(\vartheta)$ when $r_R$ decreases. Note also that this effect would be rather small for $g_{\rm eff}=0$ — this behavior is excluded by experiment.
Following Ref. [@Hor08], we argue that the gradual reduction of the orbital singlet correlations in favor of increasing orbital polarization is responsible for the evolution of the orbital transition temperature $T_{\rm OO}$ in the experimental phase diagram of Fig. \[fig:phd\], which is reproduced by the theory in the entire range of available $r_R$. The transition temperature $T_{\rm OO}$ changes in a nonmonotonic way, similar to the orbital order parameter $\langle\tau^z\rangle_G$ at $T=0$ \[Fig. \[fig:orbi\](b)\]. After analyzing the changes in the orbital correlations, we see that the physical reasons of the decrease of $T_{\rm OO}$ for small (large) $r_R$ are quite different. While the orbital fluctuations dominate and largely suppress the orbital order in LaVO$_3$, the orbital polarization takes over near YVO$_3$ and competes with $G$-AO order.
![ The orbital $T_{\rm OO}$ and magnetic $T_{N1}$ transition temperature for varying $r_R$ in the $R$VO$_3$ perovskites, obtained from model (\[rvo3\]) for: $v_{g}=740$ (solid lines) and $v_g=0$ (dashed lines). Circles show the experimentat data of Ref. [@Miy03]. The inset shows the GdFeO$_3$-type distortion, with the rotation angles $\vartheta$ and $\varphi$. Other parameters as in Fig. \[fig:orbi\]. This figure is reproduced from Ref. [@Hor08]. []{data-label="fig:phd"}](phdh.eps){width="6.8cm"}
![ Experimental distortion (in percent) at $T=0$ ($u_0$, circles) and above $T_{N1}$ ($u_1$, triangles) for the $R$VO$_3$ compounds [@Ren03; @Ree06; @Sag07], compared with the orbital polarization $\langle\tau^x\rangle_{T=0}$ and with $g_{\rm eff}$ (\[geff\]); $g_{\rm eff}$ and $g^2/K$ are in units of $J$. Squares show the upper bound for $g^2/K$ predicted by the theory. Parameters: $v_z=17$, $v_{ab}=22$, $v_{g}=740$. This figure is reproduced from Ref. [@Hor08]. []{data-label="fig:u"}](ur.eps){width="7.5cm"}
While the above fast dependence on the tilting angle $\vartheta$ of VO$_6$ octahedra in the $R$VO$_3$ family was introduced in order to reproduce the experimentally observed dependence of $T_{\rm OO}$ on $r_R$, see Fig. \[fig:phd\], it may be justified [*a posteriori*]{}. It turns out that the dependence of $g_{\rm eff}$ on the ionic radius $r_R$ in Eq. (\[geff\]) follows the actual lattice distortion $u$ in $R$VO$_3$ measured at $T=0$ ($u_0$), or just above $T_{N1}$ ($u_1$) [@Hor08]. Also the orbital polarization $\langle\tau^x\rangle$ is approximately $\propto\sin^5\vartheta\cos\vartheta$, and follows the same fast dependence of $g_{\rm eff}(\vartheta$ for the $R$VO$_3$ perovskites (Fig. \[fig:u\]). This result is somewhat unexpected, as information about the actual lattice distortions has not been used in constructing the miscroscopic model (\[som\]). These results indicate that the bare coupling parameters $\{g,K\}$ are nearly constant and independent of $r_R$, which may be treated as a prediction of the theory to be verified by future experiments.
Summary and outlook {#sec:summa}
===================
Summarizing, spin-orbital superexchange model (\[rvo3\]) augmented by orbital-lattice couplings provides an explanation of the experimental variation of the orbital $T_{\rm OO}$ and magnetic $T_{N1}$ transition temperatures for the whole class of the $R$VO$_3$ perovskites. A more complete theoretical understanding including a description of the second magnetic transition from $C$-AF to $G$-AF phase, which occurs at $T_{N2}$ for small ionic radii $r_R$ [@Maz08], remains to be addressed by future theory, which should include the spin-orbit relativistic coupling [@Hor03].
We conclude by mentioning a few open issues and future directions of reasearch in the field of perovskite vanadates. Rapid progress of the field of orbital physics results mainly from experiment, and is triggered by the synthesis of novel materials. Although experiment is ahead of theory in most cases, there are some exceptions. One of them was a theoretical prediction of the energy and dispersion of orbital excitations [@vdB99; @Ish97; @vdB01]. Only recently orbital excitations (orbitons) could be observed by Raman scattering in the Mott insulators LaTiO$_3$ and YTiO$_3$ [@Ulr06; @Ulr08]. They were also identified in the optical absorption spectra of YVO$_3$ and HoVO$_3$ [@Ben08]. The exchange of two orbitals along the $c$ axis in the intermediate $C$-AF phase was shown to contribute to the optical conductivity $\sigma(\omega)$.
An interesting question which arises in this context is the carrier propagation in a Mott insulator with orbital order. This problem is rather complex as in a spin-orbital polaron, created by doping of a single hole, both spin and orbital excitations contribute to the hole scattering [@Zaa93], which may even become localized by string excitations as in the $t$-$J^z$ model [@Mar91]. Indeed, the coupling to orbitons increases the effective mass of a moving hole in $e_g$ systems [@vdB00]. The orbital part of the superexchange is classical (compass-like) in $t_{2g}$ systems, but nevertheless the hole is not confined as weak quasiparticle dispersion arises from three-site processes [@Dag08; @Woh08].
As in the doped manganites, also in doped $R_{1-x}$(Sr,Ca)$_x$VO$_3$ systems the $G$-AO order gradually disappears [@Fuj05]. The $C$-AF spin order survives, however, in a broad range of doping, in contrast to La$_{1-x}$Sr$_x$MnO$_3$, where FM order replaces the $A$-AF phase already at $x\sim 0.10$, and is accompanied by the $e_g$ orbital liquid [@Ole02] at higher doping. It is quite remarkable that the complementary $G$-AF/$C$-AO order is fragile and disappears in Y$_{1-x}$Ca$_x$VO$_3$ already at $x=0.02$ [@Fuj05]. The doped holes in $C$-AF/$G$-AO phase are localized in polaron-like states [@Fuj08], so the pure electronic model such as the one of Ref. [@Dag08] is too crude to capture both the evolution of the spin-orbital order in doped vanadates and the gradual decrease of the energy scale for spin-orbital fluctuations. Theretical studies at finite hole concentration are still nonexistent in 3D models, but one may expect a transition from a phase with AF order to a phase with FM spin polarization at large Hund’s coupling, as shown both for $e_g$ [@Dag04] and $t_{2g}$ [@Fuj05] systems.
A few representative problems related to the properties of $R$VO$_3$ perovskites discussed above demonstrate that the orbital physics is a very rich field, with intrinsically frustrated interactions and rather exotic ordered or disordered phases, with their behavior dominated by quantum fluctuations. While valuable information about the electronic structure is obtained from density functional theory [@Sol08], the many-body aspects have to be studied simultaneously using models of correlated electrons. The $R$VO$_3$ perovskites remain an interesting field of research, as it turned out that electron-lattice coupling is here not strong enough to suppress (quench) the orbital fluctuations [@Hor08]. Thus the composite quantum fluctuations described by the spin-orbital model (\[rvo3\]) remain active. Nevertheless, there is significant control of the electronic properties due to the electron-lattice coupling. Thus, the lattice distortions may also influence the onset of magnetic order in systems with active orbital degrees of freedom. If they are absent and the lattice is frustrated in addition, a very interesting situation arises, with strong tendency towards truly exotic quantum states [@Kha05]. Examples of this behavior were considered recently for the triangular lattice, both for $e_g$ orbitals in LiNiO$_2$ [@Rei05] and $t_{2g}$ orbitals in NaTiO$_2$ [@Nor08]. None of these models could really be solved, but generic tendency towards dimer correlations with spin singlets on the bonds for particular orbital states has been shown. Yet, the question whether novel types of orbital order, such as e.g. nematic order in spin models [@Sha06], could be found in certain situations remains open.
It is our great pleasure to thank G. Khaliullin and L.F. Feiner for very stimulating collaboration which significantly contributed to our present understanding of the subject. We thank B. Keimer, G.A. Sawatzky, Y. Tokura and particularly C. Ulrich for numerous insightful discussions. A.M. Oleś acknowledges financial support by the Foundation for Polish Science (FNP) and by the Polish Ministry of Science and Education under Project No. N202 068 32/1481.
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|
---
abstract: |
Algorithmic statistics is a part of algorithmic information theory (Kolmogorov complexity theory) that studies the following task: given a finite object $x$ (say, a binary string), find an ‘explanation’ for it, i.e., a simple finite set that contains $x$ and where $x$ is a ‘typical element’. Both notions (‘simple’ and ‘typical’) are defined in terms of Kolmogorov complexity.
It was found that this cannot be achieved for some objects: there are some “non-stochastic” objects that do not have good explanations. In this paper we study the properties of maximally non-stochastic objects; we call them “antistochastic”.
It turns out the antistochastic strings have the following property (Theorem \[th2\]): if an antistochastic string has complexity $k$, then any $k$ bit of information about $x$ are enough to reconstruct $x$ (with logarithmic advice). In particular, if we erase all bits of this antistochastic string except for $k$, the erased bits can be restored from the remaining ones (with logarithmic advice). As a corollary we get the existence of good list-decoding codes with erasures (or other ways of deleting part of the information).
Antistochastic strings can also be used as a source of counterexamples in algorithmic information theory. We show that the symmetry of information property fails for total conditional complexity for antistochastic strings.
author:
- |
Alexey Milovanov\
Moscow State University\
[[email protected]]{}
title: Some properties of antistochastic strings
---
**Keywords:** Kolmogorov complexity, algorithmic statistics, stochastic strings, total conditional complexity, symmetry of information.
Introduction
============
Let us recall the basic notion of algorithmic information theory and algorithmic statistics (see [@shen15; @lv; @suv] for more details).
We consider strings over the binary alphabet $\{0,1\}$. The set of all strings is denoted by $\{0,1\}^*$ and the length of a string $x$ is denoted by $l(x)$. The empty string is denoted by $\Lambda$.
Algorithmic information theory
------------------------------
Let $D$ be a partial computable function mapping pairs of strings to strings. *Conditional Kolmogorov complexity* with respect to $D$ is defined as $$C_D(x {\mskip 1mu|\mskip 1mu}y)=\min\{l(p)\mid D(p,y)=x\}.$$ In this context the function $D$ is called a *description mode* or a *decompressor*. If $D(p,y)=x$ then $p$ is called a *description of $x$ conditional to $y$* or a *program mapping $y$ to $x$*.
A decompressor $D$ is called *universal* if for every other decompressor $D'$ there is a string $c$ such that $D'(p,y)=D(cp,y)$ for all $p,y$. By Solomonoff—Kolmogorov theorem universal decompressors exist. We pick arbitrary universal decompressor $D$ and call $C_D(x {\mskip 1mu|\mskip 1mu}y)$ *the Kolmogorov complexity* of $x$ conditional to $y$, and denote it by $C(x {\mskip 1mu|\mskip 1mu}y)$. Then we define the unconditional Kolmogorov complexity $C(x)$ of $x$ as $C(x {\mskip 1mu|\mskip 1mu}\Lambda)$. (This version of Kolmogorov complexity is called *plain* complexity; there are other versions, e.g., prefix complexity, monotone complexity etc., but for our purposes plain complexity is enough, since all our considerations have logarithmic precision.)
Kolmogorov complexity can be naturally extended to other finite objects (pairs of strings, finite sets of strings, etc.). We fix some computable bijection (“encoding”) between these objects are binary strings and define the complexity of an object as the complexity of the corresponding binary string. It is easy to see that this definition is invariant (change of the encoding changes the complexity only by $O(1)$ additive term).
In particular, we fix some computable bijection between strings and finite subsets of $\{0,1\}^*$; the string that corresponds to a finite $A\subset \{0,1\}^*$ is denoted by $[A]$. Then we understand $C(A)$ as $C([A])$. Similarly, $C(x {\mskip 1mu|\mskip 1mu}A)$ and $C(A {\mskip 1mu|\mskip 1mu}x)$ are understood as $C(x {\mskip 1mu|\mskip 1mu}[A])$ and $C([A] {\mskip 1mu|\mskip 1mu}x)$, etc.
Algorithmic statistics
----------------------
Algorithmic statistics studies explanations of observed data that are good in the algorithmic sense: an explanation should be simple and capture all the algorithmically discoverable regularities in the data. The data is encoded, say, by a binary string $x$. In this paper we consider explanations (statistical hypotheses) of the form “$x$ was drawn at random from a finite set $A$ with uniform distribution”. (As argued in [@vv], the class of general probability distributions reduces to the class of uniform distributions over finite sets.)
Kolmogorov suggested in 1974 [@kolm] to measure the quality of an explanation $A\ni x$ by two parameters, Kolmogorov complexity $C(A)$ of $A$ (the explanation should be simple) and the cardinality $|A|$ of $A$ (the smaller $|A|$ is, the more “exact” the explanation is). Both parameters cannot be very small simultaneously unless the string $x$ has very small Kolmogorov complexity. Indeed, $C(A)+\log_2|A|{\geqslant}C(x)$ with logarithmic precision[^1], since $x$ can be specified by $A$ and its index (ordinal number) in $A$. Kolmogorov called an explanation $A\ni x$ *good* if $C(A)\approx 0$ and $\log_2|A|\approx C(x)$, that is, $\log_2|A|$ is as small as the inequality $C(A)+\log_2|A|{\geqslant}C(x)$ permits given that $C(A)\approx 0$. He called a string *stochastic* if it has such an explanation.
Every string $x$ of length $n$ has two trivial explanations: $A_1=\{x\}$ and $A_2=\{0,1\}^n$. The first explanation is good when the complexity of $x$ is small. The second one is good when the string $x$ is random, that is, its complexity $C(x)$ is close to $n$. Otherwise, when $C(x)$ is far both from $0$ and $n$, neither of them is good.
Informally, non-stochastic strings are those having no good explanation. They were studied in [@gtv; @vv]. To define non-stochasticity rigorously we have to introduce the notion of the *profile* of $x$, which represents the parameters of possible explanations for $x$.
The *profile* of a string $x$ is the set $P_x$ consisting of all pairs $(m, l)$ of natural numbers such that there exists a finite set $A \ni x$ with $C(A) {\leqslant}m$ and $\log_2|A| {\leqslant}l$.
Figure \[f1\] shows how the profile of a string $x$ of length $n$ and complexity $k$ may look like.
![The profile $P_x$ of a string $x$ of length $n$ and complexity $k$.[]{data-label="f1"}](curve1-1.pdf)
The profile of every string $x$ of length $n$ and complexity $k$ has the following three properties.
- First, $P_x$ is upward closed: if $P_x$ contains a pair $(m,l)$, then $P_x$ contains all the pairs $(m',l')$ with $m'{\geqslant}m$ and $l'{\geqslant}l$.
- Second, $P_x$ contains the set $$P_{\mathrm{min}}=\{(m,l)\mid m+l{\geqslant}n \text{ or } m{\geqslant}k\}$$ (the set consisting of all pairs above and to the right of the dashed line on Fig. \[f1\]) and is included into the set $$P_{\mathrm{max}}=\{(m,l)\mid m+l{\geqslant}k\}$$ (the set consisting of all pairs above and to the right of the dotted line on Fig. \[f1\]). In other words, the border line of $P_x$ (Kolmogorov called it the *structure function* of $x$), lies between the dotted line and the dashed line.
Both inclusions are understood with logarithmic precision: the set $P_{\mathrm{min}}$ is included in the $O(\log n)$-neighborhood of the set $P_x$, and $P_x$ is included in the $O(\log n)$-neighborhood of the set $P_{\mathrm{max}}$.
- Finally, $P_x$ has the following property: $$\begin{split}
\text{if a pair $(m,l)$ is in $P_x$,
then}\hspace*{11em}\\
\text{the pair $(m+i+O(\log n),l-i)$ is in $P_x$ for all $i{\leqslant}l$.}
\end{split}$$
If for some strings $x$ and $y$ the inclusion $P_x \subset P_y$ holds, then we can say informally that $y$ is “more stochastic” then $x$. The largest possible profile is close to the set $P_{\mathrm{max}}$. Such a profile is possessed, for instance, by a random string of length $k$ with $n-k$ trailing zeros. As we will see soon, the minimal possible profile is close to $P_{\mathrm{max}}$; this happens for antistochastic strings.
It was shown is [@vv] that every profile that has these three properties is possible for a string of length $n$ and complexity $k$ with logarithmic precision:
\[th1\] Assume that we are given an upward closed set $P$ of pairs of natural numbers which includes $P_{\mathrm{min}}$ and is included into $P_{\mathrm{max}}$ and for all $(m,l)\in P$ and all $i{\leqslant}l$ we have $(m+i,l-i)\in P$. Then there is a string $x$ of length $n$ and complexity $k+O(\log n)$ whose profile is at most $C(P)+O(\log n)$-close to $P$.
In this theorem, we say that two subsets of ${{\Bbb N}}^2$ are *${\varepsilon}$-close* if each of them is contained in the ${\varepsilon}$-neighborhood of the other. This result mentions the complexity of the set $P$ that is not a finite set. Nevertheless, a set $P$ that satisfies the assumption is determined by the function $h(l)=\min\{m\mid (m,l)\in P\}$. This function has only finitely many non-zero values, as $h(k)=h(k+1)=\ldots=0$. Hence $h$ is a finite object, so we define the complexity of $C(P)$ as the complexity of $h$ (a finite object).
For the set $P_{\text{min}}$ the corresponding function $h$ is defined as follows: $h(m)=n-m$ for $m<k$ and $h(k)=h(k+1)=\ldots=0$. Thus the Kolmogorov complexity of this set is $O(\log n)$. Theorem \[th1\] guarantees then that there is a string $x$ of length about $n$ and complexity about $k$ whose profile $P_x$ is close to the set $P_{\text{min}}$. We call such strings *antistochastic*.
The main result of our paper (Theorem \[th2\]) says that an antistochastic string $x$ of length $n$ can be reconstructed with logarithmic advice from every finite set $A$ that contains $x$ and has size $2^{n-k}$ (thus providing $k$ bits of information about $x$). We prove this in Section \[sec:main\].
Then in Section \[sec:erasures\] we show that a known result about list decoding for erasure codes is a simple corollary of the properties of antistochastic strings, as well as some its generalizations.
In Section \[sec:total\] we use antistochastic strings to construct an example where the so-called total conditional complexity is maximally far from standard conditional complexity: a tuple of strings $x_i$ such that conditional complexity $C(x_i {\mskip 1mu|\mskip 1mu}x_j)$ is small while the total conditional complexity of $x_i$ given all other $x_j$ as a condition, is maximal (Theorem \[thltc\]).
Antistochastic strings {#sec:main}
======================
A string $x$ of length $n$ and complexity $k$ is called *${\varepsilon}$-antistochastic* if for all $(m,l)\in P_x$ either $m>k-{\varepsilon}$, or $m+l>n-{\varepsilon}$ (in other words, if $P_x$ is close enough to $P_{\mathrm{min}}$, see Figure \[f2\]).
![The profile of an ${\varepsilon}$-antistochastic string $x$ for a very small ${\varepsilon}$ is close to $P_\mathrm{min}$.[]{data-label="f2"}](curve2.pdf)
By Theorem \[th1\] antistochastic strings exist. More precisely, Theorem \[th1\] has the following corollary:
\[c1\] For all $n$ and all $k{\leqslant}n$ there exists an $O(\log n)$-antistochastic string $x$ of length $n$ and complexity $k+O(\log n)$.
This corollary can be proved more easily than the general statement of Theorem \[th1\], so we reproduce its proof for the sake of completeness.
We first formulate a sufficient condition for antistochasticity.
\[l3\] If the profile of a string $x$ of length $n$ and complexity $k$ does not contain the pair $(k-{\varepsilon},n-k)$, then $x$ is ${\varepsilon}+O(\log n)$-antistochastic.
Notice that the condition of this lemma is a special case of the definition of ${\varepsilon}$-antistochasticity. So Lemma \[l3\] can be considered as an equivalent (with logarithmic precision) definition of ${\varepsilon}$-antistochasticity.
Assume that a pair $(m,l)$ is in the profile of $x$. We will show that either $m>k-{\varepsilon}$ or $m+l> n-{\varepsilon}-O(\log n)$. Assume that $m{\leqslant}k-{\varepsilon}$ and hence $l> n-k$. By the third property of profiles we see that the pair $$(m+(l-(n-k))+O(\log n), n-k)$$ is in its profile as well. Hence we have $$m+l-(n-k)+O(\log n)>k-{\varepsilon}$$ and $$m+l>n-{\varepsilon}-O(\log n).$$
We return now to the proof of Corollary \[c1\]. Consider the family $\mathcal A$ consisting of all finite sets $A$ of complexity less than $k$ and log-cardinality at most $n-k$. The number of such sets is less than $2^k$ (they have descriptions shorter than $k$) and thus the total number of strings in all these sets is less than $2^{k}2^{n-k}=2^n$. Hence there exists a string of length $n$ that does not belong to any of sets from $\mathcal{A}$. Let $x$ be the lexicographically least such string.
Let us show that the complexity of $x$ is $k+O(\log n)$. It is at least $k-O(1)$, as by construction the singleton $\{x\}$ has complexity at least $k$. On the other hand, the complexity of $x$ is at most $\log |\mathcal A|+O(\log n){\leqslant}k+O(\log n)$. Indeed, the family $\mathcal A$ can be found from $k,n$ and $|\mathcal A|$, as we can enumerate $\mathcal A$ until we get $|\mathcal A|$ sets, and the complexity of $|\mathcal{A}|$ is bounded by $\log|\mathcal{A}|+O(1)$, while complexities of $k$ and $n$ are bounded by $O(\log n)$.
By construction $x$ satisfies the condition of Lemma \[l3\] with ${\varepsilon}=O(\log n)$. Hence $x$ is $O(\log n)$-antistochastic.
Before proving our main result, set us recall some tools that are needed for it. For any integer $i$ let $\Omega_i$ denote the number of strings of complexity at most $i$. Knowing $i$ and $\Omega_i$, we can compute a string of Kolmogorov complexity more than $i$, so $C(\Omega_i)=i+O(\log i)$ (in fact, one can show that $C(\Omega_i)=i+O(1)$, but logarithmic precision is enough for us). If $l {\leqslant}m$ then the leading $l$ bits of $\Omega_m$ contain the same information as $\Omega_l$ (see [@vv Theorem VIII.2] and [@suv Problem 367] for the proof):
\[l2\] Assume that $l{\leqslant}m$ and let $(\Omega_m)_{1:l}$ denote the leading $l$ bits of $\Omega_m$. Then both $C((\Omega_m)_{1:l} {\mskip 1mu|\mskip 1mu}\Omega_l)$ and $C(\Omega_l {\mskip 1mu|\mskip 1mu}(\Omega_m)_{1:l})$ are of order $O(\log m)$.
Every antistochastic string $x$ of complexity $k<l(x)-O(\log l(x))$ contains the same information as $\Omega_k$:
\[l1\] There exists a constant $c$ such that the following holds. Let $x$ be an ${\varepsilon}$-antistochastic string of length $n$ and complexity $k<n-{\varepsilon}-c\log n$. Then both $C(\Omega_k {\mskip 1mu|\mskip 1mu}x)$ and $C(x {\mskip 1mu|\mskip 1mu}\Omega_k)$ are less than ${\varepsilon}+ c\log n$.
Actually this lemma is true for all strings whose profile $P_x$ does not contain the pair $(k-{\varepsilon}+O(\log k),{\varepsilon}+O(\log k))$, in which form it was essentially proven in [@gtv]. The lemma goes back to L. Levin (personal communication, see [@vv] for details).
Let us prove first that $C(\Omega_k {\mskip 1mu|\mskip 1mu}x)$ is small. Fix an algorithm that given $k$ enumerates all strings of complexity at most $k$. Let $N$ denote the number of strings that appear after $x$ in the enumeration of all strings of complexity at most $k$ (if $x$ turns out to be the last string in this enumeration, then $N=0$).
Given $x$, $k$ and $N$, we can find $\Omega_k$ just by waiting until $N$ strings appear after $x$. If $N=0$, the statement $C(\Omega_k {\mskip 1mu|\mskip 1mu}x) = O(\log k)$ is obvious, so we assume that $N>0$. Let $l=\lfloor\log N\rfloor$. We claim that $ l{\leqslant}{\varepsilon}+ O(\log n)$ because $x$ is ${\varepsilon}$-antistochastic. Indeed, chop the set of all enumerated strings into portions of size $2^l$. The last portion might be incomplete; however $x$ does not fall in that portion since there are $N{\geqslant}2^l$ elements after $x$. Every complete portion can be described by its ordinal number and $k$. The total number of complete portions is less than $O(2^k/2^l)$. Thus the profile $P_x$ contains the pair $(k-l+O(\log k),l)$. By antistochasticity of $x$, we have $k-l+O(\log k){\geqslant}k-{\varepsilon}$ or $k-l+O(\log k)+l{\geqslant}n-{\varepsilon}$. The first inequality implies that $l{\leqslant}{\varepsilon}+O(\log k)$. The second inequality cannot happen provided the constant $c$ is large enough.
We see that to get $\Omega_k$ from $x$ we need only ${\varepsilon}+O(\log n)$ bits of information since $N$ can be specified by $\log N=l$ bits, and $k$ can be specified by $O(\log k)$ bits.
We have shown that $C(\Omega_k {\mskip 1mu|\mskip 1mu}x)<{\varepsilon}+ O(\log n)$. It remains to use the Kolmogorov–Levin symmetry of information theorem that says that $C(u)-C(u {\mskip 1mu|\mskip 1mu}v)=C(v)-C(v {\mskip 1mu|\mskip 1mu}u)+O(\log C(u,v))$ (see, e.g., [@shen15; @lv; @suv]). Indeed, $$C(x)+C(\Omega_k {\mskip 1mu|\mskip 1mu}x)=C(x {\mskip 1mu|\mskip 1mu}\Omega_k)+C(\Omega_k)+O(\log k).$$ The strings $x$ and $\Omega_k$ have the same complexity with logarithmic precision, so $C(\Omega_k {\mskip 1mu|\mskip 1mu}x)=C(x {\mskip 1mu|\mskip 1mu}\Omega_k)+O(\log n)$.
From this lemma it follows that there are at most $2^{{\varepsilon}+ O(\log n)}$ ${\varepsilon}$-antistochastic strings of complexity $k$ and length $n$. Indeed, we have $C(x {\mskip 1mu|\mskip 1mu}\Omega_k) {\leqslant}{\varepsilon}+ O(\log n)$ for each string $x$ of this type.
Before stating the general result (Theorem \[th2\] below), let us consider its special case as example. Let us prove that every $O(\log n)$-antistochastic string $x$ of length $n$ and complexity $k$ can be restored from its first $k$ bits using $O(\log n)$ advice bits. Indeed, let $A$ consist of all strings of the same length as $x$ and having the same $k$ first bits as $x$. The complexity of $A$ is at most $k+O(\log n)$. On the other hand, the profile of $x$ contains the pair $(C(A), n-k)$. Since $x$ is $O(\log n)$-antistochastic, we have $C(A){\geqslant}k-O(\log n)$. Therefore, $C(A)=k+O(\log n)$. Since $C(A {\mskip 1mu|\mskip 1mu}x)=O(\log n)$, by symmetry of information we have $C(x {\mskip 1mu|\mskip 1mu}A)=O(\log n)$ as well.
The same arguments work for every simple $k$-element subset of indices (instead of first $k$ bits): if $I$ is a $k$-element subset of $\{1,\dots,n\}$ and $C(I)=O(\log n)$, then $x$ can be restored from $x_I$ and some auxiliary logarithmic amount of information. Here $x_I$ denotes the string obtained from $x$ by replacing all the symbols with indices outside $I$ by the blank symbol (a fixed symbol different from $0$ and $1$); note that $x_I$ contains information both about $I$ and the bits of $x$ in $I$-positions.
Surprisingly, the same result is true for *every* $k$-element subset of indices, even if that subset is complex: $C(x {\mskip 1mu|\mskip 1mu}x_I)=O(\log n)$. The following theorem provides an even more general statement.
\[th2\] Let $x$ be an ${\varepsilon}$-antistochastic string of length $n$ and complexity $k$. Assume that a finite set $A$ is given such that $x \in A$ and $|A| {\leqslant}2^{n-k}$. Then $C(x{\mskip 1mu|\mskip 1mu}A) {\leqslant}2{\varepsilon}+ O(\log C(A) +\log n ) $.
Informally, this theorem says that any $k$ bits of information about $x$ that restrict $x$ to some subset of size $2^{n-k}$, are enough to reconstruct $x$. The $O()$-term in the right hand side depends on $C(A)$ that can be very large, but the dependence is logarithmic.
For instance, let $I$ be a $k$-element set of indices and let $A$ be the set of all strings of length $n$ that coincide with $x$ on $I$. Then the complexity of $A$ is $O(n)$ and hence $C(x {\mskip 1mu|\mskip 1mu}A) {\leqslant}2{\varepsilon}+ O(\log n ) $.
We may assume that $k < n - {\varepsilon}- c \log n$ where $c$ is the constant from Lemma \[l1\]. Indeed, otherwise $A$ is so small ($n-k{\leqslant}{\varepsilon}+ c$) that $x$ can be identified by its index in $A$ in ${\varepsilon}+c$ bits. Then by Lemma \[l1\] both $C(\Omega_k {\mskip 1mu|\mskip 1mu}x)$ and $C(x {\mskip 1mu|\mskip 1mu}\Omega_k)$ are less than ${\varepsilon}+ O(\log n)$.
In all the inequalities below we ignore additive terms of order $O(\log C(A)+\log n)$. However, we will not ignore additive terms ${\varepsilon}$ (we do not require ${\varepsilon}$ to be small, though it is the most interesting case).
Let us give a proof sketch first. There are two cases that are considered separately in the proof: $A$ is either “non-stochastic” or “stochastic” — more precisely, appears late or early in the enumeration of all sets of complexity at most $C(A)$. The first case is easy: if $A$ is non-stochastic, then $A$ is informationally close to $\Omega_{C(A)}$ that determines $\Omega_k$ that determines $x$ (up to a small amount of auxiliary information, see the details below).
In the second case $A$ is contained in some simple small family $\mathcal{A}$ of sets; then we consider the set of all $y$ that are covered by many elements of $\mathcal{A}$ as an explanation for $x$, and use the assumption that $x$ is antistochastic to get the bound for the parameters of this explanation. This is main (and less intuitive) part of the argument.
Now let us provide the details for both parts. Run the algorithm that enumerates all finite sets of complexity at most $C(A)$, and consider $\Omega_{C(A)}$ as the number of sets in this enumeration. Let $N$ be the index of $A$ in this enumeration (so $N{\leqslant}\Omega_{C(A)}$). Let $m$ be the number of common leading bits in the binary notations of $N$ and $\Omega_{C(A)}$ and let $l$ be the number of remaining bits. That is, $N=a2^l+b$ and $\Omega_{C(A)}=a2^l+c$ for some integer $a<2^m$ and $b{\leqslant}c<2^l$. For $l>0$ we can estimate $b$ and $c$ better: $ b < 2^{l-1}{\leqslant}c < 2^l$. Note that $l+m$ is equal to the length of the binary notation of $\Omega_{C(A)}$, that is, $C(A)+O(1)$. Now let us distinguish two cases mentioned:
**Case 1**: $m{\geqslant}k$. In this case we use the inequality $C(x {\mskip 1mu|\mskip 1mu}\Omega_k){\leqslant}{\varepsilon}$. (Note that we omit terms of order $O(\log C(A)+\log n)$ here and in the following considerations.) The number $\Omega_k$ can be retrieved from $\Omega_{m}$ since $m{\geqslant}k$ (Lemma \[l2\]), and the latter can be found given $m$ leading bits of $\Omega_{C(A)}$. Finally, $m$ leading bits of $\Omega_{C(A)}$ can be found given $A$, as $m$ leading bits of the index $N$ of the code of $A$ in the enumeration of all strings of complexity at most $C(A)$.
**Case 2**: $m<k$. This case is more elaborated and we need an additional construction.
\[lem:model\] The pair $(m,l+n-k-C(A{\mskip 1mu|\mskip 1mu}x)+{\varepsilon})$ belongs to $P_x$.
As usual, we omit $O(\log C(A)+\log n)$ terms that should be added to both components of the pair this is statement.
We construct a set $B \ni x$ of complexity $m$ and log-size $ l+n-k-C(A{\mskip 1mu|\mskip 1mu}x)+{\varepsilon}$ in two steps.
*First step*. We construct a family $\mathcal A$ of sets that is an explanation for $A$ such that $A\in \mathcal A$ and $C(\mathcal A){\leqslant}m$, $C(\mathcal A {\mskip 1mu|\mskip 1mu}x){\leqslant}{\varepsilon}$ and $|\mathcal A|{\leqslant}2^l$. To this end chop all strings of complexity at most $C(A)$ in chunks of size $2^{l-1}$ (or $1$ if $l=0$) in the order they are enumerated. The last chunk may be incomplete, however, in this case $A$ belongs to the previous (complete) chunk due to the choice of $m$ as the length of common prefix of $\Omega_{C(A)}$ and $N$.
Let $\mathcal A$ be the family of those finite sets that belong to the chunk containing $A$ and have cardinality at most $2^{n-k}$. By construction $|\mathcal A|{\leqslant}2^l$. Since $\mathcal A$ can be found from $a$ (common leading bits in $N$ and $\Omega_{C(A)}$), we have $C(\mathcal A){\leqslant}m$. To prove that $C(\mathcal A {\mskip 1mu|\mskip 1mu}x){\leqslant}{\varepsilon}$ it suffices to show that $C(a {\mskip 1mu|\mskip 1mu}x){\leqslant}{\varepsilon}$. We have $C(\Omega_{k} {\mskip 1mu|\mskip 1mu}x) {\leqslant}{\varepsilon}$ and from $\Omega_k$ we can find $\Omega_m$ and hence the number $a$ as the $m$ leading bits of $\Omega_{C(A)}$ (Lemma \[l2\]).
*Second step*. We claim that $x$ appears in at least $2^{C(A {\mskip 1mu|\mskip 1mu}x) - {\varepsilon}}$ sets from $\mathcal A$. Indeed, assume that $x$ falls in $K$ of them. Given $x$, we need $C(\mathcal{A}{\mskip 1mu|\mskip 1mu}x){\leqslant}{\varepsilon}$ bits to describe $\mathcal{A}$ plus $\log K$ bits to describe $A$ by its ordinal number in the list of elements of $\mathcal{A}$ containing $x$. Therefore, $C(A {\mskip 1mu|\mskip 1mu}x){\leqslant}\log K+{\varepsilon}$.
Let $B$ be the set of all strings that appear in at least $2^{C(A {\mskip 1mu|\mskip 1mu}x)-{\varepsilon}}$ of sets from $\mathcal A$. As shown, $x$ belongs to $B$. As $B$ can be found from $\mathcal A$, we have $C(B){\leqslant}m$. To finish the proof of Lemma \[lem:model\], it remains to estimate the cardinality of $B$. The total number of strings in all sets from $\mathcal A$ is at most $2^{l}\cdot 2^{n - k}$, and each element of $B$ is covered at least $2^{C(A{\mskip 1mu|\mskip 1mu}x)-{\varepsilon}}$ times, so $B$ contains at most $2^{l + n - k-C(A {\mskip 1mu|\mskip 1mu}x)+{\varepsilon}}$ strings.
Since $x$ is ${\varepsilon}$-antistochastic, Lemma \[lem:model\] implies that either $m{\geqslant}k-{\varepsilon}$ or $m+l+n-k-C(A|x)+{\varepsilon}{\geqslant}n-{\varepsilon}$. In the case $m{\geqslant}k-{\varepsilon}$ we can just repeat the arguments from Case 1 and show that $C(x {\mskip 1mu|\mskip 1mu}A){\leqslant}2{\varepsilon}$.
In the case $m+l+n-k-C(A|x)+{\varepsilon}{\geqslant}n-{\varepsilon}$ we recall that $m+l=C(A)$ and by symmetry of information $C(A)-C(A {\mskip 1mu|\mskip 1mu}x)=C(x)-C(x {\mskip 1mu|\mskip 1mu}A)=k-C(x
{\mskip 1mu|\mskip 1mu}A)$. Thus we have $
n-C(x {\mskip 1mu|\mskip 1mu}A)+{\varepsilon}{\geqslant}n-{\varepsilon}.
$
Notice that every string that satisfied the claim of Theorem \[th2\] is $\delta$-antistochastic for $\delta\approx 2{\varepsilon}$. Indeed, if $x$ has length $n$, complexity $k$ and is not $\delta$-antistochastic for some $\delta$, then $x$ belongs to some set $A$ that has $2^{n-k}$ elements and whose complexity is less than $k-\delta+O(\log n)$ (Lemma \[l3\]). Then $C(x {\mskip 1mu|\mskip 1mu}A)$ is large, since $$k=C(x){\leqslant}C(x {\mskip 1mu|\mskip 1mu}A)+C(A)+O(\log n){\leqslant}C(x{\mskip 1mu|\mskip 1mu}A)+k-\delta+O(\log n)$$ and hence $C(x {\mskip 1mu|\mskip 1mu}A){\geqslant}\delta- O(\log n)$ while the claim of Theorem \[th2\] says that $C(x{\mskip 1mu|\mskip 1mu}A) {\leqslant}2{\varepsilon}+O(\log C(A)+\log n)$.
Antistochastic strings and list decoding\
from erasures {#sec:erasures}
=========================================
Theorem \[th2\] implies the existence of good codes. We cannot use antistochastic strings directly as codewords, since there are only few of them. Instead, we consider a weaker property and note that every antistochastic string has this property (so it is non-empty); then we prove that there are many strings with this property and they can be used as codewords.
A string $x$ of length $n$ is called $({\varepsilon},k)$-holographic if for all $k$-element set of indexes $I\subset\{1,\dots,n\}$ we have $C(x {\mskip 1mu|\mskip 1mu}x_I)<{\varepsilon}$.
\[th4\] For all $n$ and all $k{\leqslant}n$ there are at least $2^{k}$ strings of length $n$ that are $(O(\log n),k)$-holographic.
By Corollary \[c1\] and Theorem \[th2\] for all $n$ and $k{\leqslant}n$ there exists an $(O(\log n),k)$-holographic string $x$ of length $n$ and complexity $k$ (with $O(\log n)$ precision). This implies that there are many of them. Indeed, the set of all $(O(\log n),k)$-holographic strings of length $n$ can be identified by $n$ and $k$. More specifically, given $n$ and $k$ we can enumerate all $(O(\log n),k)$-holographic strings and hence $x$ can be identified by $k,n$ and its ordinal number in that enumeration. The complexity of $x$ is at least $k-O(\log n)$, so this ordinal number is at least $k-O(\log n)$, so there are at least $2^{k-O(\log n)}$ holographic strings.
Our claim was a bit stronger: we promised $2^k$ holographic strings, not $2^{k-O(\log n)}$ of them. For this we can take $k'=k+O(\log n)$ and get $2^k$ strings that are $(O(\log n), k')$-holographic. The difference between $k$ and $k'$ can then be moved into the first $O(\log n)$, since the first $k'-k$ erased bits can be provided as an advice of logarithmic size.
Theorem \[th4\] provides a family of codes that are list decodable from erasures. Indeed, consider $2^k$ strings that are $(O(\log n), k)$-holographic, as codewords. This code is list decodable from $n-k$ erasures with list size $2^{O(\log n)}={\mathrm{poly}}(n)$. Indeed, assume that an adversary erases $n-k$ bits of a codeword $x$, so only $x_I$ remains for some set $I$ of $k$ indices. Then $x$ can be reconstructed from $x_I$ by a program of length $O(\log n)$. Applying all programs of that size to $x_I$, we obtain a list of size ${\mathrm{poly}}(n)$ which contains $x$.
Although the existence of list decodable codes with such parameters can be established by the probabilistic method [@guru Theorem 10.9 on p. 258], we find it interesting that a seemingly unrelated notion of antistochasticity provides such codes. In fact, a more general statement where erasures are replaced by any other type of information loss, can be obtained in the same way.
\[escode\] Let $k,n$ be some integers and $k{\leqslant}n$. Let $\mathcal{A}$ be a family of $2^{n-k}$-element subsets of $\{0,1\}^n$ that contains $|\mathcal{A}| = 2^{{\mathrm{poly}}(n)}$ subsets. Then there is a set $S$ of size at least $2^{k - O(\log n)}$ such that every $A \in \mathcal{A}$ contains at most ${\mathrm{poly}}(n)$ strings from $S$.
Theorem \[th4\] is a special case of this theorem: in Theorem \[th4\] the family $\mathcal{A}$ consists of all sets of the form $\{x'\in\{0,1\}^n\mid x'_I=x_I\}$ for different $n$-bit strings $x$ and different sets $I$ of $k$ indexes.
Assume first that Kolmogorov complexity of $\mathcal{A}$ is $O(\log n)$.
We use the same idea as in the proof of Theorem \[th4\]. We may assume without loss of generality that the union of sets in $\mathcal{A}$ contains all strings, by adding some elements to $\mathcal{A}$. It can be done in such a way that $C(\mathcal{A})$ remains $O(\log n)$ and the size of $\mathcal{A}$ is still $2^{{\mathrm{poly}}(n)}$.
Let $x$ be an $O(\log n)$-antistochastic string of length $n$ and complexity $k$. By our assumption the string $x$ belongs to some set in $\mathcal{A}$. The family $\mathcal{A}$ has low complexity and is not very large, hence for every $A \in \mathcal{A}$ we have $C(A) {\leqslant}{\mathrm{poly}}(n)=2^{O(\log n)}$. By Theorem \[th2\] for every $A\in\mathcal{A}$ containing $x$ we have $C(x {\mskip 1mu|\mskip 1mu}A) < D\log n$ for some constant $D$.
Now we define $S$ as the set of all strings $y$ such that $C(y
{\mskip 1mu|\mskip 1mu}A)< D\log n$ for every $A\in \mathcal{A}$ containing $y$. From the definition of $S$ it follows that for every $A\in \mathcal{A}$ there are at most $2^{D \log n}$ strings in $S$ that belong to $A$. So now we need to prove only that $|S| {\geqslant}2^{k - O(\log n)}$.
Since $C(\mathcal{A})=O(\log n)$, we can enumerate $S$ by a program of length $O(\log n)$. The antistochastic string $x$ belongs to $S$; on the other hand, $x$ can be identified by its ordinal number in that enumeration of $S$. So we conclude that the logarithm of this ordinal number (and therefore the log-cardinality of $S$) is at least $k - O(\log n)$.
It remains to get rid of the assumption $C(\mathcal{A})=O(\log n)$. To this end, fix a polynomial $p(n)$ in place of ${\mathrm{poly}}(n)$ in the statement of the theorem. Then for any given $k,n$ with $k{\leqslant}n$ consider the smallest $D=D_{kn}$ such that the statement of the theorem holds for $D\log n$ in place of $O(\log n)$. We have to show that $D_{kn}$ is bounded by a constant. For every $k,n$ the value $D_{kn}$ and a family $\mathcal{A}=\mathcal{A}_{kn}$ witnessing that $D$ cannot be made smaller that $D_{kn}$ can be computed by a brute force from $k,n$. This implies that $C(\mathcal{A}_{kn})=O(\log n)$. Hence $D_{kn}=O(1)$, as $D_{kn}$ is the worst family for $k,n$.
Like Theorem \[th4\], Theorem \[escode\] can also be easily proved by the probabilistic method; see Theorem \[probproof\] in Appendix.
Antistochastic strings and\
total conditional complexity {#sec:total}
============================
The conditional complexity $C(a{\mskip 1mu|\mskip 1mu}b)$ of $a$ given $b$ is defined as a minimal length of a program that maps $b$ to $a$. We may require that the program is total; in this way we get another (bigger) version of conditional complexity that was used, e.g., in [@bauwens].
Total conditional complexity is defined as the shortest length of a total program $p$ mapping $b$ to $a$: $CT(a {\mskip 1mu|\mskip 1mu}b)=\min\{l(p)\mid D(p,b)=a$ and $D(p,y)$ is defined for all $y\}$.
It is easy to show that the total conditional complexity may be much higher than the plain conditional complexity (see, e.g., [@shen12]). Namely, there exist strings $x$ and $y$ of length $n$ such that $CT(x {\mskip 1mu|\mskip 1mu}y) {\geqslant}n $ and $C(x {\mskip 1mu|\mskip 1mu}y) = O(1)$. Antistochastic strings help to extend this result (unfortunately, with slightly worse accuracy):
\[thltc\]
For every $k$ and $n$ there exist strings $x_1 \ldots x_k$ of length $n$ such that:
$C(x_i {\mskip 1mu|\mskip 1mu}x_j) = O(\log k+\log n)$ for every $i$ and $j$.
$CT(x_i {\mskip 1mu|\mskip 1mu}x_1 \ldots x_{i-1}x_{i+1} \ldots x_k) {\geqslant}n - O(\log k+\log n)$ for every $i$.
Let $x$ be an $O(\log (kn))$-antistochastic string of length $kn$ and complexity $n$. We consider $x$ as the concatenation of $k$ strings of length $n$: $$x=x_1 \ldots x_k,\qquad
x_i \in \{0,1\}^n.$$ Let us show that the strings $x_1, \ldots, x_k$ satisfy the requirements of the theorem.
The first statement is a simple corollary of antistochasticity of $x$. Theorem \[th2\] implies that $C(x {\mskip 1mu|\mskip 1mu}x_j) = O(\log (kn))$ for every $j$. As $C(x_i {\mskip 1mu|\mskip 1mu}x) = O(\log (kn)$ for every $i$, we have $C(x_i {\mskip 1mu|\mskip 1mu}x_j)=O(\log (kn))$ for every $i$ and $j$.
To prove the second statement consider a total program $p$ such that $p(x_1\ldots x_{i-1} x_{i+1}\ldots x_k) = x_i$. Our aim is to show that $p$ is long. Change $p$ to a total program $\tilde{p}$ such that $\tilde{p}(x_1\ldots x_{i-1}x_{i+1}\ldots x_k) = x$ and $l(\tilde{p}) {\leqslant}l(p) + O(\log (kn))$. Consider the set $$A := \{\tilde {p}(y)\mid y \in \{0, 1\}^{k(n-1)} \}.$$ Note that $A$ contains antistochastic string $x$ of length $kn$ and complexity $n$ and $\log|A| {\leqslant}k\cdot (n-1)$. By the definition of antistochasticity we get $C(A) {\geqslant}n - O(\log (kn))$. By the construction of $A$ it follows that $$C(A) {\leqslant}l(\tilde{p})+O(\log (kn)) {\leqslant}l(p)+O(\log (kn)).$$ So, we get $l(p) {\geqslant}n-O(\log (kn))$, i.e., $$CT(x_i {\mskip 1mu|\mskip 1mu}x_1 \ldots x_{i-1}x_{i+1}\ldots x_k){\geqslant}n-O(\log(kn)).$$
This example, as well as the example from [@ver], shows that for total conditional complexity the symmetry of information does not hold. Indeed, let $CT(a)=CT(a {\mskip 1mu|\mskip 1mu}\Lambda)=C(a)+O(1)$. Then $$CT(x_1) - CT(x_1
{\mskip 1mu|\mskip 1mu}x)=(n+O(\log kn))-O(\log k)=n+O(\log kn)$$ while $$CT(x) - CT(x {\mskip 1mu|\mskip 1mu}x_1)= (n+O(\log kn))- (n+O(\log kn))=O(\log kn)$$ for strings $x,x_1$ from Theorem \[thltc\].
A big question in time-bounded Kolmogorov complexity is whether the symmetry of information holds for time-bounded Kolmogorov complexity. Partial answers to this question were obtained in [@lm; @lw; @lr]. Total conditional complexity $CT(b
{\mskip 1mu|\mskip 1mu}a)$ is defined as the shortest length of a total program $p$ mapping $b$ to $a$. Being total that program halts on all inputs in time bounded by a total computable function $f_p$ of its input. Thus total conditional complexity may be viewed as a variant of time bounded conditional complexity. Let us stress that the upper bound $f_p$ for time may depend (and does depend) on $p$ in a non-computable way. Thus $CT(b
{\mskip 1mu|\mskip 1mu}a)$ is a rather far approximation to time bounded Kolmogorov complexity.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Alexander Shen and Nikolay Vereshchagin for useful discussions, advice and remarks.
[M]{}
B. Bauwens, [Computability in statistical hypothesis testing, and characterizations of directed influence in time series using Kolmogorov complexity]{}. Ph.D thesis, University of Gent, May 2010.
B. Bauwens, A. Makhlin, N. Vereshchagin, M. Zimand, Short lists with short programs in short time. Proceedings 28-th IEEE Conference on Computational Complexity (CCC), Stanford, CA, pages 98-108, June 2013.
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A. Shen The concept of $(\alpha, \beta)$-stochasticity in the Kolmogorov sense, and its properties. *Soviet Mathematics Doklady*, **271**(1):295–299, 1983
A. Shen, Game Arguments in Computability Theory and Algorithmic Information Theory. *How the World Computes. Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Cambridge, UK, June 18-23, 2012. Proceedings*, LNCS 7318, p. 655–666.
A. Shen, Around Kolmogorov complexity: basic notions and results. *Measures of Complexity. Festschrift for Alexey Chervonenkis*. Editors: V. Vovk, H. Papadoupoulos, A. Gammerman. Springer, 2015. ISBN: 978-3-319-21851-9
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Appendix {#appendix .unnumbered}
========
Here we provide a probabilistic proof of Theorem \[escode\]:
\[probproof\] Let $k,n$ be some integers, $k{\leqslant}n$. Let $\mathcal{A}$ be a finite family of $2^{n-k}$-element subsets of $\{0,1\}^n$. Then there is a set $S$ of size at least $2^k$ such that every $A \in \mathcal{A}$ contains at most $\log|\mathcal{A}|+1$ strings from $S$.
Let us show that a randomly chosen subset of $\{0,1\}^n$ of size approximately $2^k$ has the required property with a positive probability. More precisely, we assume that every $n$-bit string is included in $S$ independently with probability $\frac{1}{2^{n-k-1}}$.
The cardinality of $S$ is the random variable with binomial distrubution. The expectation of $|S|$ is equal to $2^n \cdot \frac{1}{2^{n-k -1}} = 2^{k+1}$. The variance $\sigma^2$ of $|S|$ is equal to $2^n \cdot \frac{1}{2^{n-k -1}} \cdot (1 - \frac{1}{2^{n-k -1}}) {\leqslant}2^{k+1} $. By Chebyshev’s inequality $$P(| {\mskip 1mu|\mskip 1mu}S| - E| {\geqslant}\frac{1}{2} \sigma) {\leqslant}\left(\frac{1}{2}\right)^2 \Rightarrow P(| {\mskip 1mu|\mskip 1mu}S| - 2^{k+1}| {\geqslant}2^k) {\leqslant}\frac{1}{4}.$$ Hence, the event “$|S| < 2^k$” happens with probability at most $\frac{1}{4} < \frac{1}{2}$.
It remains to show that the event “there is a set in $ \mathcal{A}$ containing more than $2^{O(\log n)}$ strings from $S$” happens with probability less than $\frac{1}{2}$. To this end we show that for every $A \in \mathcal{A}$ the event “$A$ contains more than $\log|\mathcal{A}|+1$ elements of $S$” has probability less than $\frac{1}{2} \cdot \frac{1}{|\mathcal{A}|}$.
Fix a $2^{n-k}$-element set $A\in\mathcal A$. For every $i$ the probability that $S$ contains at least $i$ elements from $S$ is at most $$\binom{|A|}{i}\cdot 2^{-(n-k +1)i} {\leqslant}\frac{|A|^{i}}{i!} \cdot 2^{(-n+k-1) \cdot i} = \frac{2^{-i}}{i!}{\leqslant}2^{-i}.$$ This value is less than $\frac{1}{2}\frac{1}{|\mathcal{A}|}$ for $i =\log|\mathcal{A}|+1$.
[^1]: In this paper we consider all the equations and inequalities for Kolmogorov complexities up to additive logarithmic terms ($O(\log n)$ for strings of length at most $n$).
|
---
abstract: 'We find sufficient conditions for commutative non-autonomous systems on certain metric spaces to be topologically stable. In particular, we prove that (i) Every mean equicontinuous, mean expansive system with strong average shadowing property is topologically stable. (ii) Every equicontinuous, recurrently expansive system with almost shadowing property is topologically stable. (iii) Every equicontinuous, expansive system with shadowing property is topologically stable.'
author:
- 'Abdul Gaffar Khan$^{1}$, Pramod Kumar Das$^{2}$ and Tarun Das$^{1}$'
title: 'TOPOLOGICALLY STABLE EQUICONTINUOUS NON-AUTONOMOUS SYSTEMS'
---
Introduction
============
In experiments, it is seldom possible to measure a physical quantity without any error. Therefore, only those properties that are unchanged under small perturbations are physically relevant. In topological dynamics, a meaningful way to perturb a system is through a morphism (continuous map).
A homeomorphism (resp. continuous map) $f$ on a metric space $X$ is said to be topologically stable if for every $\epsilon>0$ there exists $\delta>0$ such that if $h$ is another homeomorphism (resp. continuous map) on $X$ satisfying $d(f(x),h(x))<\delta$ for all $x\in X$ then there is a continuous map $k:X\rightarrow X$ satisfying $f\circ k = k\circ h$ and $d(k(x),x)<\epsilon$ for all $x\in X$.
This particular concept of stability is popularly known as topological stability which was originally introduced [@W] for a diffeomorphism on compact smooth manifold. By looking at the significance of the concept, it is worth to identify those dynamical properties which imply topological stability. One of such result in topological dynamics is “Walters stability theorem" which states that expansive homeomorphisms with shadowing property on compact metric spaces are topologically stable [@WO].
The notion of expansivity expresses the worse case unpredictability of a system. Although such unpredictable behaviour of symbolic flows was recognized earlier, the concept of expansivity for homeomorphisms on general metric spaces was introduced [@UU] in the middle of the twentieth century. The expansive behaviour of continuous maps is popularly known as positive expansivity [@CKE].
For a continuous map $f$ on a metric space $X$ and fixed $x_0\in X$, identifying those $x\in X$ whose orbit follow that of $x_0$ for a long time and hence, understanding the asymptotic behaviour of $f^n(x)$ relative to $f^n(x_0)$ can provide deep insight of the system. Anosov closing lemma [@A] provides us with such information for a differentiable map on compact smooth manifold. Although the terminology “shadowing" was originated from this lemma in differentiable dynamics, the notion of shadowing property has played a central role in topological dynamics. The idea behind the notion of shadowing is to guarantee the existence of an actual orbit with a particular behaviour by giving evidence of the existence of a pseudo orbit with the same behaviour. For general qualitative study on shadowing property, one may refer to [@AHT].
In [@TDT], authors studied notions of expansivity and shadowing property in the context of non-autonomous systems and proved “Walters stability theorem" in this settings. Such systems occur as mathematical models of real life problems affected by two or more distinct external forces in different time span. These systems appear in various branches like informatics, quantum mechanics, biology etcetera. The earliest known example with connection to biology arise while solving the famous “mathematical rabbit problem" which was appeared in the book “Liber Abaci" written by Fibonacci in the year 1202. This problem can be written in the form of second order difference equation [@CD] which is also known as Fibonacci sequence and in this representation every state depends explicitly on the current time and therefore, it can be seen from the context of non-autonomous system. Because of such frequent occurrence of non-autonomous systems in practical problems, it is important to know about stability of such systems.
In this paper, we prove the following results which provide sufficient conditions for non-autonomous systems to be topologically stable.
Let $F$ be a commutative non-autonomous system on a Mandelkern locally compact metric space.\
(i) If $F$ is mean equicontinuous, mean expansive and has strong average shadowing property, then it is topologically stable.\
(ii) If $F$ is equicontinuous, recurrently expansive and has almost shadowing property, then it is topologically stable.\
(iii) If $F$ is equicontinuous, expansive and has shadowing property, then it is topologically stable.
The first part of the above result shows that average shadowing property is not only useful in the investigation of chaos [@BD] but also in the investigation of stability. The study of such useful property of a system was also initiated in the context of flows [@GSX], iterated function systems [@B] and non-autonomous systems [@MRT]. The second part shows the connection of topological stability with an weaker form of shadowing property called almost shadowing property which is useful when one is not interested in the initial behaviour of the system. The final part improves the conclusion of Theorem 4.1 [@TDT] under stronger hypothesis.
Before proving these results we introduce the above mentioned notions and study their general properties.
Definitions and General Properties
==================================
Throughout this paper $\mathbb{Z}$, $\mathbb{N}$ and $\mathbb{N}^+$ denote the set of all integers, the set of all non-negative integers and the set of all positive integers respectively. A pair $(X,d)$ denotes a metric space with metric $d$ and if no confusion arises of the concerned metric $d$, then we simply write $X$ is a metric space. All maps between metric spaces are assumed to be uniformly continuous. We say that the collection $F=\lbrace f_i:X\rightarrow X\rbrace_{i \in \mathbb{N}^{+}}$ is a non-autonomous system (NAS) on $X$ generated by the sequence of maps $\lbrace f_{i}\rbrace_{i\in\mathbb{N}^+}$. $F$ is said to be autonomous system generated by $f$, if $f_{i} = f$ for all $i \in \mathbb{N}^{+}$ and in this case, we simply write $F=\langle f \rangle$. We say that $F$ is periodic with period $m \in \mathbb{N}^{+}$ if $f_{mi+j} = f_{j}$ for all $i,j \in \mathbb{N}^{+}$. $F$ is said to be commutative if $f_{i}\circ f_{j} = f_{j}\circ f_{i}$ for all $i, j \in \mathbb{N}^{+}$. We say that $F$ is surjective if $f_{j}$ is surjective for all $j\in \mathbb{N}^{+}$. The set of all NAS on $X$, the set of all NAS of period $m$ on $X$ and the set of all commutative NAS on $X$ are denoted by $N(X)$, $N_{m}(X)$ and $NC(X)$ respectively.
Let $(X,d)$ and $(Y,p)$ be metric spaces. The product of $F\in N(X)$ and $G\in N(Y)$ is defined as $F\times G = \lbrace f_{i}\times g_{i} : X\times Y \rightarrow X\times Y\rbrace_{i \in \mathbb{N}^{+}}$, where $X\times Y$ is equipped with the metric $q((x_{1}, y_{1}), (x_{2}, y_{2})) =$ max $\lbrace d(x_{1}, x_{2}), p(y_{1}, y_{2})\rbrace$. For $F\in N(X)$ and $n\in\mathbb{N}$, we denote $F_n = f_{n}\circ f_{n-1}\circ . . . f_{1}\circ f_{0}$, where $f_{0}$ denotes the identity map on $X$. For any $j \leq k$, we define $F_{[j, k]} = f_k\circ f_{k-1}\circ . . .\circ f_{j+1}\circ f_j$. For any $k \in \mathbb{N}^{+}$, the $k^{th}$-iterate of $F$ is given by $F^k = \lbrace F_{[(i-1)k+1,ik]}\rbrace_{i \in \mathbb{N}^{+}}$.
$F\in N(X)$ is said to be equicontinuous if the family $\lbrace f_i\rbrace_{i\in\mathbb{N}^{+}}$ is equicontinuous i.e. for every $\epsilon > 0$ there exists $\delta > 0$ such that $d(x,y) < \delta$ implies $d(f_{i}(x), f_{i}(y)) < \epsilon$ for all $i \in \mathbb{N}^{+}$.
We now introduce mean equicontinuous NAS and give examples satisfying the same.
$F\in N(X)$ is said to be mean equicontinuous (ME) if for every $\epsilon > 0$ there exists $\delta > 0$ such that if a pair of sequences $\lbrace x_{i}\rbrace_{i=0}^{\infty}$ and $\lbrace y_{i}\rbrace_{i=0}^{\infty}$ satisfy $\frac{1}{n}\sum_{i=0}^{n-1}d(x_{i}, y_{i})<\delta$, then $\frac{1}{n}\sum_{i=0}^{n-1}d(f_{j}(x_{i}), f_{j}(y_{i})) < \epsilon$ for all $j\in\mathbb{N}^{+}$. A continuous map $f$ is said to be mean continuous (MC) if $F=\langle f\rangle$ is ME. A homeomorphism $f$ is said to be mean equivalence (MEQ), if both $f$ and $f^{-1}$ are MC. \[D2.1\]
We say that $F\in N(X)$ and $G\in N(Y)$ are uniformly conjugate, if there exists a uniform equivalence $h : Y\rightarrow X$ such that $f_{n}\circ h = h\circ g_{n}$ for all $n\in \mathbb{N}$. In addition, if $h$ is MEQ then we say that $F$ and $G$ are mean conjugate. We say that a property of a NAS is a uniform dynamical (resp. mean dynamical) property if it is preserved under uniform (resp. mean) conjugacy. One can easily check that, every mean continuous map is uniformly continuous. Hence every uniform dynamical property is a mean dynamical property.
\(i) An isometry is MEQ and a contraction map is MC.\
(ii) The tent map $f:[0,1]\rightarrow [0,1]$ given by $f(x) = 2 $min$\lbrace x, 1-x\rbrace$ is MC.\
(iii) Let $X =\prod_{i\in \mathbb{Z}} X_{i}$ be equipped with the metric $d(x,y)=\sum_{i=-\infty}^{\infty}\frac{|x_{i} - y_{i}|}{2^{|i|}}$, where $X_{i} = \lbrace 0, 1\rbrace$. Let $f : X\rightarrow X$ be given by $f(x) = y$, where $y_{i} = x_{i+1}$ for all $i\in \mathbb{Z}$. Then, $f$ is MEQ. Further, since for all $x,y\in X$, we have $d(f(x), f(y)) \leq 2d(x, y)$ and $d(f^{-1}(x), f^{-1}(y)) \leq 2d(x, y)$, therefore the system $F=\lbrace f, f^{-1}, f, f^{-1}, \underbrace{f^{-1}, f^{-1}}_\text{$2$-times}, \underbrace{f, f}_\text{$2$-times},
\\
f, f^{-1},\underbrace{f^{-1}, f^{-1}}_\text{$2$-times},\underbrace{f, f}_\text{$2$-times}, \underbrace{f, f, f}_\text{$3$-times},\underbrace{f^{-1}, f^{-1}, f^{-1}}_\text{$3$-times},$ $\underbrace{f^{-1}, f^{-1}, f^{-1}, f^{-1}}_\text{$4$-times},\underbrace{f, f, f, f}_\text{$4$-times}... \rbrace$ is ME.
If $F\in N(X)$ and $G\in N(Y)$, then following statements hold.
1. $F$ and $G$ are ME if and only if $F\times G$ is ME.
2. If $F$ is ME, then $F^{k}$ is ME for all $k\in \mathbb{N}^{+}$.
\[P3.2\]
For the proof of (a), use $p,q \leq max\lbrace p, q\rbrace \leq p + q$ for all $p, q \geq 0$. Proof of (b) follows from the definition.
$F\in N(X)$ is said to be expansive [@TDT] with expansive constant $0 < \mathfrak{c} < 1$ if for each pair of distinct points $x, y\in X$, there exists $n\in \mathbb{N}$ such that $d(F_{n}(x), F_{n}(y)) > \mathfrak{c}$. In literature, if $F=\langle f\rangle$ is expansive, then $f$ is said to be positively expansive. Recall from [@CKE] that, if there exists a continuous injective map $f$ on compact $X$ such that $F=\langle f\rangle$ is expansive, then $X$ is finite. The following examples justify that this is not true for non-autonomous systems.
\(i) Let $X =\lbrace \frac{1}{m}, 1-\frac{1}{m}:m\in \mathbb{N}^{+}\rbrace$ and $f : X\rightarrow X$ be given by $f(0) = 0$, $f(1) = 1$ and $f(x) = x^{+}$ otherwise, where $x^+$ is the immediate right to $x$. Then, one can check that $F = \lbrace f, f^{-1}, f^{-2}, f^{2}, f^{3}, f^{-3}, f^{-4}, f^{4}, . . . \rbrace$ is expansive with expansivity constant $ 0 < \alpha < \frac{1}{6}$.\
(ii) Let $X =\prod_{i\in \mathbb{Z}} X_{i}$ be equipped with the metric $d(x,y)=\sum_{i=-\infty}^{\infty}\frac{|x_{i} - y_{i}|}{2^{|i|}}$, where $X_{i} = \lbrace 0, 1\rbrace$. Let $f : X\rightarrow X$ be given by $f(x) = y$, where $y_{i} = x_{i+1}$ for all $i\in \mathbb{Z}$. One can check that systems $F=\lbrace f, f^{-1}, f, f^{-1}, \underbrace{f^{-1}, f^{-1}}_\text{$2$-times}, \underbrace{f, f}_\text{$2$-times}, f, f^{-1}, \underbrace{f^{-1}, f^{-1}}_\text{$2$-times}, \underbrace{f, f}_\text{$2$-times}, \underbrace{f, f, f}_\text{$3$-times},$ $\underbrace{f^{-1}, f^{-1}, f^{-1}}_\text{$3$-times},$ $\underbrace{f^{-1}, f^{-1}, f^{-1}, f^{-1}}_\text{$4$-times},\underbrace{f, f, f, f}_\text{$4$-times}... \rbrace$ and $G=\lbrace f, f^{-1}, f^{-2}, f^{2}, f^{3}, f^{-3}, f^{-4}, f^{4}
\\
, . . . \rbrace$ are expansive with expansivity constant $ 0 < \alpha < \frac{1}{2}$. \[E2.3\]
This discussion allow us to study the following notions both of which implies expansivity.
\(i) $F\in N(X)$ is said to be recurrently expansive if there exists $\mathfrak{c}\in (0,1)$ such that any distinct pair $x,y\in X$ satisfy $\limsup\limits_{n\rightarrow\infty}d(F_{n}(x), F_{n}(y)) > \mathfrak{c}$.\
(ii) $F\in N(X)$ is said to be mean expansive if there exists $\mathfrak{c}\in (0,1)$ such that any distinct pair $x,y\in X$ satisfy $\limsup\limits_{n\rightarrow \infty} \frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x), F_{i}(y)) > \mathfrak{c}$. \[D2.5\]
$F=\langle f\rangle$ is mean expansive implies it is recurrently expansive implies $f$ is positively expansive injective map. Consequently, there does not exist any autonomous mean expansive or recurrently expansive system on non-discrete compact metric space. \[P3.3\]
Clearly, if $F$ is recurrently expansive with expansive constant $\mathfrak{c}$, then $f$ is positively expansive with expansive constant $\mathfrak{c}$. Now, if $f(x) = f(y)$, then $f^{n}(x) = f^{n}(y)$ for all $n\in \mathbb{N}^{+}$ which implies that $\limsup\limits_{n\rightarrow \infty} d(f^{n}(x), f^{n}(y)) = 0$. Therefore, we must have $x=y$.
Conversely, suppose that $f$ is positively expansive injective map with expansive constant $\mathfrak{c}$. If $x\neq y$ in $X$, then by positive expansivity there exists $n_{1}\in \mathbb{N}$ such that $d(f^{n_{1}}(x), f^{n_{1}}(y))>\mathfrak{c}$. By injectivity and positive expansivity, we can choose $n_{2}\in \mathbb{N}^+$ such that $d(f^{n_{2}+n_{1}+1}(x), f^{n_{2}+n_{1}+1}(y)) > \mathfrak{c}$. Continuing in this way, we can choose a strictly increasing sequence $\lbrace m_{i}\rbrace_{i=1}^{\infty}$ such that $d(f^{m_{i}}(x), f^{m_{i}}(y))>\mathfrak{c}$ for all $i\in \mathbb{N}^{+}$. Therefore, we have $\limsup\limits_{n\rightarrow \infty}d(f^{n}(x), f^{n}(y))>\mathfrak{c}$ and hence, the result.
If $F\in N(X)$ and $G\in N(Y)$, then $F$ and $G$ are recurrently expansive (resp. mean expansive) if and only if $F\times G$ is recurrently expansive (resp. mean expansive). \[P3.7\]
If $F\in N(X)$ is equicontinuous, then $F$ is recurrently expansive if and only if $F^{k}$ is recurrently expansive for all $k\in \mathbb{N}^{+}$. \[P3.8\]
Proof is similar to the proof of Theorem 2.2 [@TDT]
Let $X =\prod_{i\in \mathbb{Z}} X_{i}$ be equipped with the metric $d(x,y) = \sum_{i=-\infty}^{\infty}\frac{|x_{i} - y_{i}|}{2^{|i|}}$, where $X_{i} = \lbrace 0, 1\rbrace$. Let $f : X\rightarrow X$ be given by $f(x) = y$, where $y_{i} = x_{i+1}$ for all $i\in \mathbb{Z}$. Then, $F = \lbrace f, f^{-1}, f^{-2}, f^{2}, f, f^{-1}, f^{-2}, f^{2}, f^{3}, f^{-3}, f^{-4}, f^{4}, f, f^{-1}, f^{-2}, f^{2},$ $f^{3}, f^{-3}, f^{-4}, f^{4}, f^{5}, f^{-5}, f^{-6}, f^{6}, f^{7}, f^{-7}, f^{-8}, f^{8}, . . . \rbrace$ is recurrently expansive with expansive constant $0<\alpha <\frac{1}{2}$ but for each $i\in\mathbb{N}^+$, $F^{2i}$ is not recurrently expansive. Thus, we conclude that Proposition \[P3.8\] is not be true if NAS is not equicontinuous. \[E3.9\]
Let $F\in N(X)$. If for some $k\in\mathbb{N}^+$, $F^{k}$ is mean expansive, then $F$ is mean expansive. \[P3.10\]
Suppose that $F^{k}$ is mean expansive with expansive constant $\mathfrak{c}$. For $x, y\in X$ and $n\in \mathbb{N}^{+}$, we have $\frac{1}{n}\sum_{i=0}^{n-1}d((F^{k})_{i}(x), (F^{k})_{i}(y)) \leq k\frac{1}{nk}\sum_{i=0}^{(n-1)k}d(F_{i}(x), F_{i}(y))\leq k \frac{1}{nk}\sum_{i=0}^{nk-1}d(F_{i}(x), F_{i}(y))$. From this we conclude that $F$ is mean expansive with expansive constant $\frac{\mathfrak{c}}{k}$.
We now give an example showing that the converse of the above result is not true. We further give sufficient condition under which the converse holds.
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ is given by $f(x) = 2x$. Then, $F = \lbrace f, f^{-1}, f^{2}, f^{-2}, f^{3},
\\
f^{-3}, . . .\rbrace$ is mean expansive but $F^{2i}$ is not mean expansive for each $i\in \mathbb{N}^{+}$. \[E3.12\]
Let $F\in N_{m}(X)$ be ME. If $F$ is mean expansive, then for each $k\in\mathbb{N}^+$, $F^{k}$ is mean expansive.
Suppose that $F\in N_{m}(X)$ is mean expansive with expansive constant $\mathfrak{c}$. Fix $k\in \mathbb{N}^{+}$ and choose $\mathfrak{d} > 0$ such that for every pair of sequences $\lbrace x_{i}\rbrace_{i=0}^{\infty}$ and $\lbrace y_{i}\rbrace_{i=0}^{\infty}$, $\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(x_{i}, y_{i}) < m\mathfrak{d}$ implies that $\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{j}(x_{i}), F_{j}(y_{i})) < \frac{\mathfrak{c}}{mk}$ for all $0\leq j\leq (mk - 1)$.
If for $x,y\in X$, $\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d((F^{k})_{i}(x), (F^{k})_{i}(y)) < \mathfrak{d}$, then we must have $\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{mki}(x), F_{mki}(y)) < m\mathfrak{d}$.
By ME, $\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d((F_{mki+j})(x), (F_{mki+j})(y)) < \frac{\mathfrak{c}}{mk}$ for all $0\leq j\leq (mk - 1)$ and hence $$\begin{aligned}
\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x), F_{i}(y)) &\leq \limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{nmk-1}d(F_{i}(x), F_{i}(y)) \\
&\leq \sum_{j=0}^{mk-1}\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{mki+j}(x), F_{mki+j}(y)) \\
&< \mathfrak{c}\end{aligned}$$ By mean expansivity of $F$, we must have $x=y$. Hence, $F^{k}$ is mean expansive with expansive constant $\mathfrak{d}$.
Let $\gamma=\lbrace x_{n}\rbrace_{n\in \mathbb{N}}$ be a sequence of elements of $X$. $\gamma$ is said to be $\delta$-pseudo orbit of $F$, if $d(f_{i+1}(x_{i}), x_{i+1}) < \delta$ for all $i\in \mathbb{N}$. $\gamma$ is said to be $\delta$-average-pseudo orbit of $f$ if there exists $N_{\delta}\in \mathbb{N}^{+}$ such that $\frac{1}{n}\sum_{i=0}^{n-1}d(f_{i+k+1}(x_{i+k}), x_{i+k+1}) < \delta$ for all $n\geq N_{\delta}$ and $k\in\mathbb{N}$. $\gamma$ is said to be $\epsilon$-shadowed by some $z\in X$, if $d(F_{n}(z), x_{n}) < \epsilon$ for all $n\in \mathbb{N}$. $\gamma$ is said to be $\epsilon$-shadowed in average by some $z\in X$, if $\limsup\limits_{n\rightarrow \infty} \frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(z), x_{i})$ $< \epsilon$. $\gamma$ is said to be almost $\epsilon$-shadowed by some $z\in X$ if $d(z, x_{0})< \epsilon$ and $\limsup\limits_{n\rightarrow \infty}d(F_{n}(z), x_{n}) < \epsilon$. $\gamma$ is said to be strongly $\epsilon$-shadowed in average if it is $\epsilon$-shadowed in average by some $z\in X$ such that $d(z, x_{0}) < \epsilon$.
$F\in N(X)$ is said to have shadowing property [@TDT] if for every $\epsilon>0$ there exists $\delta>0$ such that every $\delta$-pseudo orbit of $F$ can be $\epsilon$-shadowed by some point in $X$.
\(i) $F\in N(X)$ is said to have almost shadowing property (ALSP) if for every $\epsilon > 0$ there exists $\delta > 0$ such that every $\delta$-pseudo orbit of $F$ can be almost $\epsilon$-shadowed by some point in $X$.\
(ii) $F\in N(X)$ is said to have strong average shadowing property (SASP) if for every $\epsilon > 0$ there exists $\delta > 0$ such that every $\delta$-average pseudo orbit of $F$ can be strongly $\epsilon$-shadowed in average by some point in $X$. \[D2.7\]
If $F\in N(X)$ is equicontinuous, then $F$ has ALSP if and only if $F^{k}$ has ALSP, for each $k>1$. \[P3.13\]
One can prove similarly as the proof of Theorem 3.3 and Theorem 3.5 in [@TDT].
Let $(X,d)$ and $(Y,p)$ be metric spaces and $F\in N(X)$, $G\in N(Y)$. Then, $F$ and $G$ has SASP (ALSP) if and only if $F\times G$ has SASP (ALSP). \[P3.14\]
Suppose that $F$ and $G$ has SASP. Let $\epsilon > 0$ and $\delta > 0$ be given for $\frac{\epsilon}{2}$ by strong average shadowing property of $F$ and $G$. Let $\lambda = \lbrace (x_{n}, y_{n})\rbrace_{n=0}^{\infty}$ be a $\delta$-average pseudo orbit of $F\times G$. Since $d(f_{n+1}(x_{n}), x_{n+1})\leq q((f_{n+1}\times g_{n+1})(x_{n}, y_{n}), (x_{n+1}, y_{n+1}))$ and $p(g_{n+1}(y_{n}), y_{n+1})\leq q((f_{n+1}\times g_{n+1})(x_{n}, y_{n}), (x_{n+1}, y_{n+1}))$, therefore $\gamma = \lbrace x_{n}\rbrace_{n=0}^{\infty}$ and $\eta = \lbrace y_{n}\rbrace_{n=0}^{\infty}$ are $\delta$-average pseudo orbits of $F$ and $G$ respectively. If $\gamma$ and $\eta$ are strongly $\frac{\epsilon}{2}$-shadowed in average by $x$ and $y$ through $F$ and $G$ respectively, then $\lambda$ is strongly $\epsilon$-shadowed in average by $(x,y)$ through $F\times G$.
Conversely, suppose that $F\times G$ has SASP. Let $\epsilon > 0$ and $\delta > 0$ be given for $\frac{\epsilon}{2}$ by strong average shadowing of $F\times G$. Let $\gamma = \lbrace x_{n}\rbrace_{n=0}^{\infty}$ be a $\delta$-average pseudo orbit of $F$. For some $y\in Y$, let $\eta = \lbrace y_{n} = G_{n}(y) \rbrace_{n=0}^{\infty}$. Clearly, $\lambda = \lbrace (x_{n}, y_{n})\rbrace_{n=0}^{\infty}$ is a $\delta$-average pseudo orbit of $F\times G$ and hence, strongly $\epsilon$-shadowed in average by some point, say $(x, z)$. It is easy to see that, $\gamma$ is strongly $\epsilon$-shadowed in average by $x$ through $F$ and hence, $F$ has SASP. Similarly, one can prove that $G$ has SASP.
Proof of $F$ and $G$ has ALSP if and only if $F\times G$ has ALSP, is left to the reader.
We say that $F$ is transitive if for every pair of non-empty open sets $U$ and $V$, there exists $n\in \mathbb{N}^{+}$ such that $F_{[i, i+(n-1)]}(U)\cap V\neq \phi$ for all $i\in \mathbb{N}^{+}$.
For $x,y\in X$, we write $x\mathsf{R}_{\delta} y$ if there exists $n\in \mathbb{N}^{+}$ such that for each $j\in \mathbb{N}^{+}$ there exists a finite sequence $x = x_{0}^{j},x_{1}^{j}, . . ., x_{n-1}^{j}, x_{n}^{j}=y$ such that $d(f_{j+i}(x_{i}^{j}), x_{i+1}^{j}) < \delta$ for all $0\leq i \leq (n-1)$. We write $x\mathcal{R}_{\delta} y$ if $x \mathsf{R}_{\delta} y$ and $y \mathsf{R}_{\delta} x$ and further, write $x\mathcal{R} y$ if $x\mathcal{R}_{\delta} y$ for all $\delta > 0$. We say that $F$ is chain transitive if $x\mathcal{R} y$ for all $x, y\in X$.
If $F=\langle f\rangle$, then this definition boils down to the following in case of autonomous systems.
$f$ is said to be chain transitive if for every $\delta>0$ and $x,y\in X$, there exists a finite sequence $z_{0} = x, z_{1}, z_{2},..., z_{m+1} = y$ of elements in $X$ such that $d(f(z_{i}), z_{i+1}) < \delta$ for all $0\leq i <m$.
If $F$ is equicontinuous transitive system, then it is chain transitive.
For given $\epsilon > 0$, choose $\delta > 0$ by equicontinuity of $F$. By transitivity, choose $n\in \mathbb{N}^{+}$ such that $F_{[j, j+(n-1)]}(B(x, \delta))\cap B(y, \delta)\neq \phi$ for all $j\in \mathbb{N}^{+}$. Thus for each $j\in \mathbb{N}^{+}$, there exists $z^{j}\in B(x, \delta)$ so that the sequence $x = x_{0}^{j}, x_{1}^{j} = f_{j}(z^{j}), x_{2}^{j} = f_{j+1}\circ f_{j}(z^{j}), . . ., x_{n-1}^{j} = f_{j+(n-2)}\circ . . .\circ f_{j+1}\circ f_{j}(z^{j}), x_{n}^{j} = y$ satisfies $d(f_{j+i}(x_{i}^{j}), x_{i+1}^{j}) < \epsilon$ for all $0\leq i \leq (n-1)$ and hence, $x\mathsf{R}_{\epsilon} y$. Since $j, x, y$ and $\epsilon$ were chosen arbitrarily, we conclude that $F$ is chain transitive.
If $F$ is surjective chain transitive system with shadowing property, then $F$ is transitive.
Let $x, y\in X$ and $\epsilon > 0$. Let $\delta > 0$ be given for $\epsilon$ by the shadowing property of $F$. By chain transitivity of $F$, there exists $n\in\mathbb{N}^+$ such that for each $j\in \mathbb{N}^{+}$ there exists a finite sequence $x = x_{0}^{j},x_{1}^{j}, . . ., x_{n-1}^{j}, x_{n}^{j}=y$ satisfying $d(f_{j+i}(x_{i}^{j}), x_{i+1}^{j}) < \delta$ for all $0\leq i \leq (n-1)$. Extend this sequence to a $\delta$-pseudo orbit $\eta = \lbrace z_{0},. . . , z_{j-3}, z_{j-2}, z_{j-1} = x = x_{0}^{j}, z_{j} = x_{1}^{j}, . . ., z_{j+(n-2)} = x_{n-1}^{j}, z_{j+(n-1)} = x_{n}^{j}= y, z_{j+n} = f_{j+n}(y),z_{j+(n+1)} = f_{j+(n+1)}(z_{j+n}), . . .\rbrace$, where $f_{i}(z_{i-1}) = z_{i}$ for $1\leq i\leq (j-1)$ and $z_{j+(n+k)} = f_{j+(n+k)}(z_{j+(n+k-1)})$ for all $k\geq 2$. By the shadowing property, there exists $w\in X$ such that $d(F_{n}(w), z_{n}) < \epsilon$ for all $n\in \mathbb{N}^{+}$. Therefore, $F_{j-1}(w) \in B(x, \epsilon)$ and $F_{[j, j+(n-1)]}(F_{j-1})(w) \in B(y, \epsilon)$. Hence, $F_{[j, j+(n-1)]}(B(x, \epsilon))\cap B(x, \delta) \neq \phi$. Since $j, x, y$ and $\epsilon$ were chosen arbitrarily, we conclude that $F_{[j, j+(n-1)]}(B(x, \epsilon))\cap B(x, \delta) \neq \phi$ for all $j\in \mathbb{N}^{+}$. Hence the result.
The following two results show relations among shadowing property, almost shadowing property, average shadowing property and strong average shadowing property in case of an autonomous system. Unfortunately, we do not know much in case of a nonautonomous system.
Let $F=\langle f\rangle$ on compact $X$ be chain transitive. Then, $F$ has ALSP if and only if it has shadowing property. \[ET2.6\]
As the converse implication is automatic from the definition, we prove the forward implication. Suppose that $F$ has ALSP but does not have shadowing. For any given $\epsilon > 0$ and for each $n\in\mathbb{N}$, we can choose $\frac{1}{n}$-pseudo orbit $\alpha_{n}$ for $F$ which cannot be $\epsilon$-shadowed. Let $\delta>0$ be given for this $\epsilon$ by ALSP and fix $k\in \mathbb{N}^{+}$ such that $\frac{1}{k}<\delta$. By chain transitivity, choose finite $\frac{1}{m}$-pseudo orbits $\gamma_{m}$ for $f$ such that $\alpha_{m}\gamma_{m}\alpha_{m+1}$ forms a finite $\frac{1}{m}$-pseudo orbits for $f$, for all $m\geq k$. Clearly, $\alpha_{k}\gamma_{k}\alpha_{k+1}\gamma_{k+1}\alpha_{k+2}. . .$ forms a $\delta$-pseudo orbit for $f$ and hence, it can be almost $\epsilon$-shadowed. Therefore, there exists $p\in \mathbb{N}^{+}$ such that for all $j\geq p$, $\alpha_{j}$ is $\epsilon$-shadowed by some point in $X$, a contradiction.
If $f$ is chain transitive continuous surjective map with ALSP on a compact metric space $X$, then $f$ has average shadowing property if and only if $f$ has SASP. \[EC2.7\]
Since the converse implication is automatic, we prove the forward implication. By Theorem \[ET2.6\], $f$ has shadowing property. By Theorem 1 [@KOA], $f$ has specification property and Lemma 12 [@KOA] implies that $f$ has SASP.
Let $X=\mathbb{R}$ be given with the usual metric and choose $m > 1$. Define $g_{m} : X\rightarrow X$ by $g_m(x) = mx$ for all $x\in X$. Consider $F_{m} = \lbrace f_{i}\rbrace_{i\in \mathbb{N}^{+}}$ such that for every $i\in \mathbb{N}^{+}$, exactly one of the triplet $f_{3i+1}, f_{3i+2}, f_{3i+3}$ is $g_{m}$ and the other two are identity maps on $X$. Note that, $F_{m}$ need not be a periodic system. Also, $F_{m}$ is equicontinuous, mean equicontinuous, recurrently expansive and mean expansive. Since $F_{m}^{3} = \langle g_{m}\rangle$, by Proposition \[P3.13\], $F_m$ has ALSP. \[4.13\]
Sufficient Conditions For Topological Stability
===============================================
For a metric space $(X,d)$, define the bounded metric $d_{1}$ by $d_{1}(x,y)=$min$\lbrace d(x,y), 1\rbrace$. Let $(C(X),\eta)$ be the space of all continuous self maps on $X$, where the metric $\eta$ is defined by $\eta(f, g) =$ sup$_{x\in X} d_{1}(f(x), g(x)$). We define a metric $\gamma$ on $N(X)$ as $\gamma(F, G) = $sup$_{i\in \mathbb{N}} \eta( f_i, g_i)$, where $F = \lbrace f_i\rbrace_{i\in \mathbb{N}^{+}}$ and $G = \lbrace g_i\rbrace_{i\in \mathbb{N}^{+}}$.
$F\in NC(X)$ is said to be topologically stable if for every $\epsilon > 0$ there is $ \delta > 0$ such that for any $G\in NC(X)$ satisfying $\gamma(F, G) < \delta$ there exists a continuous map $h: X\rightarrow X$ such that $f_{i}\circ h = h\circ g_{i}$ for all $i\in \mathbb{N}$ and $d(h(x), x) < \epsilon$ for all $x\in X$. \[D4.1\]
Note that if $F\in NC(X)$, then this notion is stronger than the notion of topological stability used in [@TDT].
Let $(X,d)$ and $(Y,p)$ be two metric spaces and $F\in NC(X)$, $H\in NC(Y)$. If $F$ and $G$ are uniformly conjugate, then $F$ is topologically stable if and only if $G$ is topologically stable. In other words, topological stability is a uniform dynamical property. \[T4.2\]
Suppose that $F$ is topologically stable. Let $j:Y\rightarrow X$ be a uniform conjugacy between $F$ and $H$ i.e. $f_{i}\circ j = j\circ h_{i}$ for all $i\in \mathbb{N}$. We want to show that $H$ is topologically stable. For $\epsilon\in (0,1)$, let $\beta\in (0,1)$ be given for $\epsilon$ by the uniform continuity of $j^{-1}$ i.e. $d(x,y)<\beta$ implies $p(j^{-1}(x),j^{-1}(y))<\epsilon$. Let $\alpha\in (0,1)$ be given for $\beta$ by the topological stability of $F$. Further, let $\delta \in (0,1)$ be given for $\alpha$ by the uniform continuity of $j$ i.e. $p(x,y)<\delta$ implies $d(j(x),j(y))<\alpha$. Let $G\in NC(Y)$ be such that $\gamma(H, G) < \delta$. Hence, sup$_{i\in \mathbb{N}}\eta^{Y}(h_{i}, g_{i}) < \delta$ implying $\eta(j^{-1}\circ f_{i}\circ j, g_{i}) < \delta$ for all $i\in \mathbb{N}$. This implies that $d_{1}(j^{-1}\circ f_{i}\circ j, g_{i}) < \delta$ for all $i\in \mathbb{N}$. By uniform continuity of $j$, we get that $d_{1}(f_{i}\circ j(y), j\circ g_{i}\circ j^{-1}(j(y))) < \alpha$ for all $i\in \mathbb{N}$ and all $y\in Y$. Set $G' = \lbrace g'_{i} = j\circ g_{i}\circ j^{-1} \rbrace_{i\in \mathbb{N}^{+}}$. By topological stability of $F$, there exists a continuous map $k : X\rightarrow X$ such that $f_{i}\circ k = k\circ g'_{i}$ for all $i\in \mathbb{N}^{+}$ and $d(k(x), x) < \beta$ for all $x\in X$. If we set $k' = j^{-1}\circ k\circ j$, then $h_{i}\circ k' = h_{i}\circ j^{-1}\circ k\circ j = j^{-1}\circ f_{i}\circ k\circ j = j^{-1}\circ k\circ g'_{i}\circ j = j^{-1}\circ k\circ j\circ g_{i}\circ j^{-1}\circ j = k'\circ g_{i}$ for all $i\in \mathbb{N}$. Also, by uniform continuity of $j^{-1}$, we have $p(k'(y), y) < \epsilon$ for all $y\in Y$. Hence the result.
A proof of the converse implication follows immediately from the fact that $j$ is an uniform equivalence.
We say that a metric space is Mandelkern locally compact [@M] if every bounded subset is contained in a compact set. Observe that, a metric space is Mandelkern locally compact if and only if every closed ball of finite radius is compact. From now onwards, we assume that $X$ is a Mandelkern locally compact metric space. Without loss of generality, we also assume that $0 < \epsilon, \delta, \alpha, \beta, \mathfrak{c}, \mathfrak{c}' < 1$.
Let $F\in NC(X)$ be equicontinuous recurrently expansive with expansive constant $\mathfrak{c}$. If $F$ has ALSP, then $F$ is topologically stable. Moreover, for every $\epsilon\in (0,\frac{\mathfrak{c}}{3})$ there exists $\delta > 0$ such that if $G\in NC(X)$ satisfies $\gamma(F, G) < \delta$, then there exists a unique continuous map $h : X\rightarrow X$ such that $F_{n}\circ h = h\circ G_{n}$ for all $n\in \mathbb{N}$ and $d(h(x), x) < \epsilon$ for all $x\in X$. In addition, if $G$ is expansive with expansive constant $\mathfrak{c'}\geq 3\epsilon$, then the conjugating map $h$ is injective. \[T4.3\]
Let $F$ be recurrently expansive with expansive constant $\mathfrak{c}$ and let $F$ has ALSP. For $\epsilon\in (0,\frac{\mathfrak{c}}{3})$, let $\delta\in (0,\frac{\mathfrak{c}}{3})$ be given by ALSP of $F$. Then, every $\delta$-pseudo orbit of $F$ can be almost $\epsilon$-shadowed by exactly one point. \[L4.4\]
Let $\lbrace x_{n}\rbrace_{n\in\mathbb{N}}$ be a $\delta$-pseudo orbit of $F$ and Suppose that $x,y\in X$ almost $\epsilon$-shadow a $\delta$-pseudo orbit $\lbrace x_n\rbrace_{n\in\mathbb{N}}$ of $F$. Since $d(F_{n}(x), F_{n}(y)) \leq d(F_{n}(x), x_{n}) + d(x_{n}, F_{n}(y))$ for all $n\in \mathbb{N}$, we have $$\begin{aligned}
\limsup\limits_{n\rightarrow \infty}d(F_{n}(x), F_{n}(y)) &\leq \limsup\limits_{n\rightarrow \infty}(d(F_{n}(x), x_{n}) + d(x_{n}, F_{n}(y))) \\
&\leq \limsup\limits_{n\rightarrow \infty}d(F_{n}(x), x_{n}) +\limsup\limits_{n\rightarrow \infty}d(x_{n},F_{n}(y)) \\
&\leq 2\epsilon < \mathfrak{c}\end{aligned}$$ By recurrent expansivity of $F$, we get that $x=y$. Hence the result.
Let $F$ be recurrently expansive with expansive constant $\mathfrak{c}$. For any $x_{0}\in X$ and $\lambda >0$, there exists $N > 0$ such that $d(F_{n}(x_{0}), F_{n}(x)) \leq \mathfrak{c}$ for all $0\leq n\leq N$ implies $d(x_{0}, x) < \lambda$. \[L4.5\]
Choose a sequence $\lbrace x_{N}\rbrace_{N\in \mathbb{N}^{+}}$ in $X$ such that $d(F_{n}(x_{0}), F_{n}(x_{N})) \leq \mathfrak{c}$ for all $0\leq n \leq N$ and $d(x_{0}, x_{N}) \geq \lambda$. Since $B[x_{0}, \mathfrak{c}]$ is compact, we can assume that $x_{N}$ converges to $x$, for some $x\in X$. By continuity of each $F_n$, we have $d(F_{n}(x_{0}), F_{n}(x)) \leq \mathfrak{c}$ for all $n \in \mathbb{N}$ and $d(x_{0}, x) \geq \lambda$, a contradiction to the recurrent expansivity of $F$.
**Proof of Theorem \[T4.3\]** For $\epsilon\in (0,\frac{\mathfrak{c}}{3})$, choose $\beta\in (0,\epsilon)$ by the equicontinuity of $F$. Let $\delta\in (0,\beta)$ be given for $\beta$ by the ALSP of $F$. Let $G\in NC(X)$ be such that $\gamma(F, G) < \delta$ i.e. $\eta(f_{i}(x), g_{i}(x)) < \delta$ for all $i\in \mathbb{N}$ and all $x\in X$. Thus for all $x\in X$, $\lbrace G_{n}(x)\rbrace_{n\in \mathbb{N}}$ forms a $\delta$-pseudo orbit of $F$. By Lemma \[L4.4\], define $h : X\rightarrow X$, where $h(x)$ is a unique almost-$\beta$-tracing point of the $\delta$-pseudo orbit $\lbrace G_{n}(x)\rbrace_{n\in \mathbb{N}}$ i.e. $\limsup\limits_{n\rightarrow \infty} d(F_{n}(h(x)), G_{n}(x))<\beta$ for all $x\in X$, where $d(h(x), x) < \epsilon$. Note that, for all $x\in X$ and all $i\in \mathbb{N}$, we have $$\begin{aligned}
\limsup\limits_{n\rightarrow \infty}d(F_{n}(f_{i}(h(x))), F_{n}(h(g_{i}(x)))) &\leq \limsup\limits_{n\rightarrow \infty}d(f_{i}F_{n}(h(x)), f_{i}G_{n}(x)) \\
&+ \limsup\limits_{n\rightarrow \infty}d(f_{i}G_{n}(x), g_{i}G_{n}(x))\\
&+ \limsup\limits_{n\rightarrow \infty}d( G_{n}(g_{i}(x)), F_{n}h(g_{i}(x)))\\
&< 3\epsilon < \mathfrak{c}\end{aligned}$$ Hence by the recurrent expansivity of $F$, $f_{i}\circ h(x) = h\circ g_{i}(x)$ for all $i\in \mathbb{N}$. Hence, $F_{n}\circ h = h\circ G_{n}$ for all $n\in \mathbb{N}$.
Now we show that $h$ is a continuous map. Let $x_0\in X$ and $\lambda > 0$. By Lemma \[L4.5\], there exists $N > 0$ such that for any $y\in X$, $d(F_{n}h(x_{0}), F_{n}h(y)) \leq \mathfrak{c}$ for all $n\leq N$ implies $d(h(x_{0}), h(y)) < \lambda$. Choose $\alpha > 0$ such that $d(x_{0}, y) < \alpha$ implies $d(G_{n}(x_{0}), G_{n}(y)) < \frac{\mathfrak{c}}{3}$ for all $n \leq N$ and all $y\in X$. Therefore, $d(x_{0}, y) < \alpha$ implies that for all $n \leq N$ and all $y\in X$, $$\begin{aligned}
d(F_{n}h(x_{0}), F_{n}h(y)) = d(hG_{n}(x_{0}), hG_{n}(y)) &\leq d(hG_{n}(x_{0}), G_{n}(x_{0}))
\\
&+ d(G_{n}(x_{0}), G_{n}(y))
\\
&+ d(G_{n}(y), hG_{n}(y))
\\
&< \epsilon + \frac{\mathfrak{c}}{3} + \epsilon < \mathfrak{c} \end{aligned}$$ Thus for all $y\in X$, we get that $d(h(x_{0}), h(y)) < \lambda$, whenever $d(x_{0}, y) < \alpha$ i.e. $h$ is continuous at $x_{0}$. Hence, $h$ is a continuous map.
Assume that there exists another continuous map $h':X \rightarrow X$ such that $F_{n}\circ h' = h'\circ F_{n}$ for all $n\in \mathbb{N}$ and $d(h'(x), x) < \epsilon$ for all $x\in X$. Thus for all $n\in \mathbb{N}$ and all $x\in X$ $$\begin{aligned}
d(F_{n}(h(x)), F_{n}(h'(x)))&\leq d(F_{n}(h(x)), G_{n}(x)) + d(G_{n}(x), F_{n}(h'(x)))\\
&= d(h(G_{n}(x)), G_{n}(x)) + d(G_{n}(x), h'(G_{n}(x))) \\
&< 2\epsilon < \mathfrak{c}\end{aligned}$$
Hence by recurrent expansivity of $F$, we have $h(x) = h'(x)$ for all $x\in X$.
Now assume that $h(x) = h(y)$. Since for each $n\in\mathbb{N}$ $$\begin{aligned}
d(G_{n}(x), G_{n}(y))& \leq d(G_{n}(x), h(G_{n}(x))) + d(hG_{n}(x), hG_{n}(y)) \\
&\hspace*{0.5cm} + d(h(G_{n}(y)), G_{n}(y))\\
&< \epsilon + 0 + \epsilon \\
&= 2\epsilon < \mathfrak{c'}\end{aligned}$$ therefore by the expansivity of $G$ we get that $x = y$.
Let $F\in NC(X)$ be equicontinuous and expansive with expansive constant $\mathfrak{c}$. If $F$ has shadowing property, then $F$ is topologically stable. Moreover, for every $\epsilon\in (0,\frac{\mathfrak{c}}{3})$ there is $\delta > 0$ such that if $G\in NC(X)$ satisfies $\gamma(F, G) < \delta$, then there is a unique continuous map $h : X\rightarrow X$ such that $F_{n}\circ h = h\circ G_{n}$ for all $n\in \mathbb{N}$ and $d(h(x), x) < \epsilon$ for all $x\in X$. In addition, if $G$ is expansive with expansive constant $\mathfrak{c'}\geq 3\epsilon$, then $h$ is injective. \[T4.6\]
Proof is similar to the proof of Theorem \[T4.3\]
Let $F\in NC(X)$ be mean equicontinuous and mean expansive with expansive constant $\mathfrak{c}$. If $F$ has SASP, then $F$ is topologically stable. Moreover, for every $\epsilon\in (0,\frac{\mathfrak{c}}{3})$ there is $\delta > 0$ such that if $G\in NC(X)$ satisfies $\gamma(F, G) < \delta$, then there is a unique continuous map $h : X \rightarrow X$ such that $F_{n}\circ h=h\circ G_{n}$ for all $n\in \mathbb{N}$ and $d(h(x), x)< \epsilon$ for all $x\in X$. In addition, if $G$ is mean expansive with expansive constant $\mathfrak{c'}\geq 3\epsilon$, then $h$ is injective. \[T4.7\]
Let $F$ be mean expansive with expansive constant $\mathfrak{c}$. Let $\delta > 0$ be given for $\epsilon\in (0,\frac{\mathfrak{c}}{3})$ by SASP of $F$. Then every $\delta$-average pseudo orbit of $F$ can be strongly $\epsilon$-shadowed in average uniquely. \[L4.8\]
Let $\lbrace x_{n}\rbrace_{n\in\mathbb{N}}$ be a $\delta$-average pseudo orbit for $F$ and let it be strongly $\epsilon$-shadowed in average by $x,y\in X$. Then $d(F_{n}(x), F_{n}(y)) \leq d(F_{n}(x), x_{n}) + d(x_{n}, F_{n}(y))$ for all $n\in \mathbb{N}$. This implies that $$\begin{aligned}
\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x), F_{i}(y)) &\leq \limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}(d(F_{i}(x), x_{i}) + d(x_{i}, F_{i}(y))) \\
&\leq \limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x), x_{i}) +\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(x_{i},F_{i}(y)) \\
&< 2\epsilon < \mathfrak{c}\end{aligned}$$ By mean expansivity of $F$, we get that $x = y$.
Let $F$ be mean expansive with expansive constant $\mathfrak{c}$. For any $x_{0}\in X$ and $\lambda >0$, there exists $N > 0$ such that if $\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x_{0}), F_{i}(x)) \leq \mathfrak{c}$ for all $n\leq N$, then $d(x_{0}, x) < \lambda$ for all $x\in X$. \[L4.9\]
Choose a sequence $\lbrace x_{N}\rbrace_{N\in \mathbb{N}^{+}}$ in $X$ such that $\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x_{0}), F_{i}(x_{N})) \leq \mathfrak{c}$ for all $n\leq N$ and $d(x_{0}, x_{N}) \geq \lambda$. Since $B[x_{0}, \mathfrak{c}]$ is compact, we can assume that $x_{N}$ converges to $x\in X$. For each $n\in \mathbb{N}$ and $\epsilon > 0$ there exists $N'\geq n$ such that $d(F_{i}(x), F_{i}(x_{N})) < \epsilon$ for all $0\leq i\leq n$ and all $N \geq N'$. For such $N'$, $\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x), F_{i}(x_{N'})) < \epsilon$ and hence, $$\begin{aligned}
\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x_{0}), F_{i}(x)) &\leq \frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x_{0}), F_{i}(x_{N'}))+\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x_{N'}), F_{i}(x)) \leq \mathfrak{c} + \epsilon\end{aligned}$$ Since $n$ and $\epsilon$ was chosen arbitrary, we get that $\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x_{0}), F_{i}(x)) \leq \mathfrak{c}$ and $d(x_{0}, x) \geq \lambda$, a contradiction to the mean expansivity of $F$.
**Proof of Theorem \[T4.7\]** Let $\beta\in (0,\epsilon)$ be given for $\epsilon \in (0,\frac{\mathfrak{c}}{3})$ by mean equicontinuity of $F$. For this $\beta$, choose $\delta\in (0,\beta)$ such that every $\delta$-average pseudo orbit of $F$ is strongly $\beta$-shadowed in average. Let $G\in NC(X)$ be such that $\gamma(F, G) < \delta$ i.e. $\eta(f_{i}(x), g_{i}(x)) < \delta$ for all $x\in X$ and all $i\in \mathbb{N}$. Thus for all $x\in X$, the sequence $\lbrace G_{n}(x)\rbrace_{n\in \mathbb{N}}$ forms a $\delta$-average pseudo orbit for $F$. Define a map $h : X\rightarrow X$, where $h(x)$ is a unique strongly $\beta$-tracing point in average of the $\delta$-average pseudo orbit $\lbrace G_{n}(x)\rbrace_{n\in \mathbb{N}}$ i.e. $\limsup\limits_{n\rightarrow \infty} \frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(h(x)), G_{i}(x)) < \beta$ for all $x\in X$ and $d(h(x), x) < \beta$ for all $x\in X$. Note that, for all $x\in X$ and all $m \in \mathbb{N}$,
$$\begin{aligned}
\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(f_{m}(h(x))), F_{i}(h(g_{m}(x)))) &\leq \limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f_{m}F_{i}(h(x)), f_{m}G_{i}(x)) \\
&+ \limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f_{m}G_{i}(x), g_{m}G_{i}(x)) \\
&+ \limsup\limits_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(G_{i}(g_{m}(x)), F_{i}(h(g_{m}(x)))) \\
&< 3\epsilon < \mathfrak{c}\end{aligned}$$
By mean expansivity of $F$, we get that $f_{m}\circ h=h\circ g_{m}$ for all $m\in \mathbb{N}$ and hence, we have $F_{n}\circ h = h\circ G_{n}$ for all $n\in \mathbb{N}$.
We now show that $h$ is continuous. Let $x_{0}\in X$ and $\lambda > 0$. By Lemma \[L4.9\], there exists $N > 0$ such that for any $y\in X$, $\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(x_{0}), F_{i}(y)) \leq \mathfrak{c}$ for all $n\leq N$ implies $d(x_{0}, y) < \lambda$. Choose $\alpha > 0$ such that $d(x_{0}, y) < \alpha$ implies $d(G_{n}(x_{0}), G_{n}(y)) < \frac{\mathfrak{c}}{3}$ for all $n \leq N$ and all $y\in X$. Therefore, $d(x_{0}, y) < \alpha$ implies that for all $n \leq N$ and all $y\in X$, $$\begin{aligned}
\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}h(x_{0}), F_{i}h(y)) &= \frac{1}{n}\sum_{i=0}^{n-1}d(hG_{i} (x_{0}), hG_{i}(y))\\
&\leq \frac{1}{n}\sum_{i=0}^{n-1}d(hG_{i}(x_{0}), G_{i}(x_{0})) + \frac{1}{n}\sum_{i=0}^{n-1}d(G_{i}(x_{0}), G_{i}(y))\\
&\hspace*{0.5cm}+ \frac{1}{n}\sum_{i=0}^{n-1}d(G_{i}(y), hG_{i}(y))\\
&< \mathfrak{c} \end{aligned}$$ Thus $d(h(x_{0}), h(y)) < \lambda$ whenever $d(x_{0}, y) < \alpha$, for all $y\in X$ i.e. $h$ is continuous at $x_{0}$. So, $h$ is a continuous map.
Assume that there exists another continuous map $h':X \rightarrow X$ satisfying $F_{n}\circ h' = h'\circ G_{n}$ for all $n\in \mathbb{N}$ and $d(h'(x), x) < \epsilon$ for all $x\in X$. Thus for all $n\in \mathbb{N}^{+}$ and all $x\in X$, $$\begin{aligned}
\frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(h(x)), F_{i}(h'(x))) &\leq \frac{1}{n}\sum_{i=0}^{n-1}d(F_{i}(h(x)), G_{i}(x)) +\frac{1}{n}\sum_{i=0}^{n-1}d(G_{i}(x), G_{i}(h'(x)))\\
&= \frac{1}{n}\sum_{i=0}^{n-1}d(h(G_{i}(x)), G_{i}(x)) + \frac{1}{n}\sum_{i=0}^{n-1}d(G_{i}(x), h'(G_{i}(x))) \\
&< 2\epsilon < \mathfrak{c} \end{aligned}$$ Hence by the mean expansivity of $F$, $h(x) = h'(x)$ for all $x\in X$.
Now assume that $h(x) = h(y)$. Since for each $n\in\mathbb{N}$, $$\begin{aligned}
d(G_{n}(x), G_{n}(y))\leq d(G_{n}(x), h(G_{n}(x))) + d(hG_{n}(x), hG_{n}(y))+d(h(G_{n}(y)), G_{n}(y))< \mathfrak{c'}\end{aligned}$$ therefore by the expansivity of $G$ we get that $x = y$.
Every commutative equicontinuous mean expansive NAS with ALSP on a Mandelkern locally compact metric space is topologically stable. \[C4.11\]
[@WO Theorem 4] Every autonomous expansive system with shadowing property on a compact metric space is topologically stable. \[C4.12\]
**Acknowledgements:** The first author is supported by CSIR-Junior Research Fellowship (File No.-09/045(1558)/2018-EMR-I) of Government of India.
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---
abstract: 'We consider the numerical approximation of compressible flow in a pipe network. Appropriate coupling conditions are formulated that allow us to derive a variational characterization of solutions and to prove global balance laws for the conservation of mass and energy on the whole network. This variational principle, which is the basis of our further investigations, is amenable to a conforming Galerkin approximation by mixed finite elements. The resulting semi-discrete problems are well-posed and automatically inherit the global conservation laws for mass and energy from the continuous level. We also consider the subsequent discretization in time by a problem adapted implicit time stepping scheme which leads to conservation of mass and a slight dissipation of energy of the full discretization. The well-posedness of the fully discrete scheme is established and a fixed-point iteration is proposed for the solution of the nonlinear systems arising in every single time step. Some computational results are presented for illustration of our theoretical findings and for demonstration of the robustness and accuracy of the new method.'
address: 'Department of Mathematics, TU Darmstadt, Germany'
author:
- 'H. Egger'
title: |
A robust conservative mixed finite element method\
for compressible flow on pipe networks
---
> [ compressible flow, variational methods, energy estimates, Galerkin approximation, mixed finite elements, implicit time discretization ]{}
> [ 35D30,35R02,37L65,76M10,76N99]{}
Introduction {#sec:intro}
============
This paper addresses the numerical approximation of flow problems governing the propagation of a compressible fluid in a pipeline network. On every single pipe $e$, the conservation of mass and the balance of momentum shall be modeled by $$\begin{aligned}
\dt \rho_e + \dx m_e &= 0,\\
\dt m_e + \dx \Big(\frac{m_e^2}{\rho_e} + p_e\Big) &= a_e \rho_e \dx( \frac{1}{\rho_e^2} \dx m_e) - b_e \frac{|m_e| m_e }{\rho_e}. \end{aligned}$$ Here $\rho_e$ is the density, $m_e=\rho_e u_e$ is the mass flux, and $u_e$ is the velocity of the flow. The parameters $a_e,b_e$ are assumed to be constant and non-negative. As equation of state, relating the density $\rho_e$ to the pressure $p_e$, we utilize $$\begin{aligned}
p_e(\rho) = c_e \rho^\gamma, \qquad \gamma>1, \ c_e > 0,\end{aligned}$$ and we use a corresponding potential energy density of the form [@NovotnyStraskraba04] $$\begin{aligned}
P_e(\rho) = \rho \int_0^\rho p_e(s) / s^2 ds = \frac{c_e}{\gamma-1} \rho^\gamma. \end{aligned}$$ As indicated below, more general situations including the isothermal flow of real gas scan be considered as well. The above equations describe the conservation of mass and energy in isentropic flow of a compressible fluid within the pipe $e$ with possible dissipation of energy through viscous forces and friction at the pipe walls. The particular form of the viscous term will become clear from our analysis. Note that for constant density $\rho = \bar \rho$ we have $$\begin{aligned}
\bar \rho \dx \left(\frac{1}{\bar \rho^2} \dx m\right) = \dxx u.\end{aligned}$$ The viscous term in the above equations thus is in principle of the form as the one usually employed in the compressible Navier-Stokes equations [@Feireisl03; @Lions98; @NovotnyStraskraba04]. In order to correctly describe the conservation of mass and energy at junctions $v$ of several pipes $e \in \E(v)$ of the network, we require the following coupling conditions to hold true at the junctions: $$\begin{aligned}
\sum_{e \in \E(v)} m_e(v) n_e(v) &= 0,\\
\frac{m_e(v)^2}{2 \rho_e(v)^2}+P'_e(\rho_e(v)) &= h_v + \frac{a_e}{\rho_e^2}\dx m_e(v), \qquad \forall e \in \E(v).\end{aligned}$$ Note that the value of $h_v$ is assumed to be independent of $e$. For the inviscid case $a_e=0$, the second equation describes the continuity of the *specific stagnation enthalpy* [@MorinReigstad15; @Reigstad15]; here we additionally take into account the viscous forces. The two coupling conditions imply that the fluxes of mass and energy through a junction $v$ sum up to zero and therefore no mass or energy is generated or annihilated. This will become clear from our analysis below.
There has been intensive discussion about the appropriate coupling conditions for compressible flow in pipe networks. The first condition stated above is equivalent to conservation of mass at the junction and out of doubt; see e.g. [@BrouwerGasserHerty11; @Garavello10; @Osiadacz84]. In contrast to that, various different relations have been considered as a second coupling condition: continuity of the pressure $p$ is used frequently in the modeling and simulation of pipeline networks [@BandaHertyKlar06a; @GugatEtAl12; @Osiadacz84]. As shown in [@ColomboGaravello08; @Reigstad15], this condition may lead to unphysical solutions for junctions of more than two pipes. Also the continuity of the dynamic pressure $m^2/\rho+p(\rho)$ proposed in [@ColomboGaravello06; @ColomboGaravello08; @ColomboHertySachers08] may in general lead to unphysical solutions; see [@MorinReigstad15; @Reigstad15]. The continuity of the stagnation enthalpy on the other hand leads to entropic solutions for all subsonic flow conditions for junctions connecting several pipes of arbitrary cross-sectional area [@Reigstad15]. Our second coupling condition in addition accounts for the viscous forces. In the course of the manuscript, we will provide a derivation of the two coupling conditions based only on the rationale that the mass and energy should be conserved at the junctions.
A vast amount of literature is devoted to numerical methods for compressible flow in a single pipe. Most of the schemes that have been proven to be globally convergent to weak solutions of the compressible Navier-Stokes equations are formulated in Lagrangian coordinates [@ZarnowskiHoff91; @ZhaoHoff94; @ZhaoHoff97]. This makes their generalization to networks practically infeasible. Finite volume methods on the other hand are typically formulated in a Eulerian framework and many partial results on their stability and convergence are available; see [@LeVeque02] and the references given there. The correct handling of coupling conditions in the network context, however, seems not straight forward; see [@BressanEtAl15] for analytical reasons. A globally convergent non-conforming finite element for the compressible Navier-Stokes equations has been proposed and analyzed recently [@GallouetEtAl16; @Karper13; @Karper14]. Due to the somewhat unusual form of the coupling conditions, a direct generalization of this method to pipe networks again seems not feasible.
In this paper, we therefore propose an alternative strategy for the numerical approximation of one-dimensional compressible flow that naturally generalizes to networks and that allows to establish the conservation of mass and energy in a rather direct manner. The two coupling conditions stated above naturally arise in the derivation of a special variational characterization of solutions for the compressible flow problem given below. This variational principle encodes the conservation of mass and energy much more directly as previous formulations and in addition turns out to be amenable to a conforming Galerkin approximation in space. As a particular discretization, we consider in detail a mixed finite element method for which we prove conservation of mass and energy independently of the topology of the network and of the mesh size. In addition, we also investigate the discretization in time by a problem adapted implicit time stepping scheme that allows to establish global balance relations for mass and energy also for the fully discrete scheme. The final method provides exact conservation of mass and a slight dissipation of energy due to numerical dissipation caused by the implicit time stepping strategy. We establish the well-posedness of the fully discrete scheme and consider a problem adapted fixed-point iteration for the numerical solution of the nonlinear systems arising in every time step.
The numerical approximations obtained with our method automatically satisfy uniform energy bounds. This is the first step and main ingredient for the proof of global existence of weak solutions to the compressible Navier-Stokes equations [@Feireisl03; @Lions98; @NovotnyStraskraba04] and also for their systematic numerical approximation [@GallouetEtAl16; @Karper13; @Karper14]. Our method is comparably simple and directly inherits the basic conservation principle for mass and energy from the continuous problem. Moreover, our approach naturally extends to pipeline networks. We strongly belief that with similar arguments as in [@Karper14], it might be possible to obtain a complete convergence analysis also for the method proposed in this paper. This is however left as a topic for future research.
The remainder of the manuscript is organized as follows: In Section \[sec:prelim\], we introduce our notation and provide a complete definition of the compressible flow problem to be considered. Section \[sec:variational\] is then devoted to the derivation of the variational principle, which is the basis for the design of our numerical method. In Section \[sec:galerkin\], we discuss the discretization of the weak formulation in space by a conforming Galerkin approach using mixed finite elements, and we establish conservation of mass and energy for the resulting semi-discretization. Section \[sec:time\] is then devoted to the subsequent discretization in time. In Section \[sec:implementation\], we discuss some details of the implementation of the fully discrete scheme. For illustration of our theoretical results, we also present some preliminary computational results for standard test problems in Section \[sec:numerics\]. We close with a short summary and a discussion of topics for future research.
Notation and problem statement {#sec:prelim}
==============================
Let us briefly summarize our main notation used in the rest of the paper and then provide a complete definition of the flow problem under consideration.
Topology and geometry
---------------------
The topology of the pipe network will be represented by a finite, directed, and connected graph $\G=(\V,\E)$ with vertices $\V=\{v_1,\ldots,v_n\}$ and edges $\E=\{e_1,\ldots,e_m\} \subset \V \times \V$. For any edge $e=(v_1,v_2)$, we denote by $\V(e)=\{v_1,v_2\}$ the set of vertices of the edge $e=(v_1,v_2)$, and we denote by $\E(v)=\{ e=(v,\cdot) \text{ or } e=(\cdot,v)\}$ the set of edges incident on $v$. The set of vertices can be split into inner vertices $\Vi=\{v : |\E(v)| \ge 2\}$ and boundary vertices $\Vb = \V \setminus \Vi$. For any edge $e=(v_1,v_2)$, we further define two numbers $$\begin{aligned}
n_e(v_1)=-1 \qquad \text{and} \qquad n_e(v_2)=1,\end{aligned}$$ marking the in- and outflow vertex and thus defining the orientation of the edge. The matrix $N \in \RR^{n \times m}$ given by $N_{ij} = n_{e_j}(v_i)$ is usually called the *incidence matrix* of the graph [@Berge]. For illustration of the above notions by a particular example, see Figure \[fig:graph\].
(v1) at (0,2) [$v_1$]{}; (v2) at (4,2) [$v_2$]{}; (v3) at (8,4) [$v_3$]{}; (v4) at (8,0) [$v_4$]{}; (v1) – node\[above\] [$e_1$]{} ++(v2); (v2) – node\[above,sloped\] [$e_2$]{} ++(v3); (v2) – node\[above,sloped\] [$e_3$]{} ++(v4);
To each edge $e \in \E$, we further associate a parameter $\ell_e>0$ representing the length of the corresponding pipe. Throughout the presentation, we tacitly identify the interval $[0,\ell_e]$ with the edge $e$ which it corresponds to. The values $\ell_e$ are stored in a vector $\ell=(\ell_e)_{e \in \E}$. The triple $\G=(\V,\E,\ell)$ is called a *geometric graph* [@Mugnolo14] and serves as the basic geometric model for the pipe network for the rest of the manuscript.
Function spaces
---------------
We denote by $$\begin{aligned}
L^2 = L^2(\E) = \{ u : u|_e=u_e \in L^2(e) \quad \forall e \in \E\}\end{aligned}$$ the space of square integrable functions defined over the network. Here and below, we tacitly identify $L^2(e)$ with $L^2(0,l_e)$ and with slight abuse of notation, we write $$\begin{aligned}
(u_e,v_e)_{e}=\int_e u_e v_e dx = \int_{v_1}^{v_2} u_e v_e dx = \int_0^{l_e} u_e v_e dx. \end{aligned}$$ The scalar product of $L^2(\E)$ is then defined by $$\begin{aligned}
(u,v)_\E = \sum\nolimits_{e \in \E} (u_e,v_e)_{e}. \end{aligned}$$ We further denote by $$\begin{aligned}
H^s(\E) = \{ u \in L^2(e): u_e \in H^s(e) \quad \forall e \in \E\} \end{aligned}$$ the broken Sobolev spaces of piecewise smooth functions. The broken derivative of a piecewise smooth function $u \in H^1(\E)$ is denoted by $\dx' u$ and given by $$\begin{aligned}
(\dx' u)|_e = \dx (u|_e) \qquad \text{for all } e \in \E. \end{aligned}$$ This allows us to equivalently define $H^1(\E) = \{v \in L^2(\E) : \dx' v \in L^2(\E)\}$. Note that for any parameter $s>1/2$, functions in $H^s(\E)$ are continuous along every edge $e$, but they may in general be discontinuous across inner vertices $v \in \Vi$. We will further make use of the space $$\begin{aligned}
H_0(\div) &:= \{ m \in H^1(\E) : \sum\nolimits_{e \in \E(v)} m_e(v) n_e(v) = 0\quad \forall v \in \V \}\end{aligned}$$ of *conservative flux functions* which vanish at the boundary of the network. Let us finally note that for all $p,m \in H^1(\E)$ we have the following integration-by-parts formula $$\begin{aligned}
% \label{eq:ibp}
(\dx' p,m)_\E
= -(p, \dx' m)_\E + \sum\nolimits_{v \in \V} \sum\nolimits_{e \in \E(v)} p_e(v) m_e(v) n_e(v).\end{aligned}$$ This identity follows directly from the definition of $(\dx' p,m)_\E =\sum_{e \in \E} (\dx p_e, m_e)_e$, integration-by-parts on every edge, and summation over all elements. The interface terms drop out if, in addition, $p$ is continuous across junctions $v$ and if $m \in H_0(\div)$; this will be utilized below.
Problem description
-------------------
Let $\G=(\V,\E,\ell)$ be a geometric graph as introduced in the previous section. The problem under investigation then consists of the following differential and algebraic equations. We look for functions $\rho$, $m$ of time and space, such that $$\begin{aligned}
\dt \rho_e + \dx m_e &= 0, && \forall e \in \E \label{eq:sys1}\\
\dt m_e + \dx \left(\frac{m_e^2}{\rho_e} + p_e\right) &= a_e \rho_e \dx\left( \frac{1}{\rho_e^2} \dx m_e\right) - b_e \frac{|m_e| m_e }{\rho_e}, && \forall e \in \E. \label{eq:sys2}\end{aligned}$$ Recall that $f_e = f|_e$ denotes the restriction of a function $f$ to the edge $e$. The equations here and below are tacitly assumed to hold for all $t > 0$. We further assume that the pressure and potential energy density are related to the density of the fluid by $$\begin{aligned}
\label{eq:sys3}
p_e(\rho) = c_e \rho^\gamma
\qquad \text{and} \qquad
P_e(\rho) = \frac{c_e}{\gamma-1} \rho^\gamma, \qquad \gamma>1.\end{aligned}$$ More general equations of state can be handled without much difficulty. In our analysis, we will actually only make use of the basic identity [@NovotnyStraskraba04] $$\begin{aligned}
\rho P_e'(\rho) - P_e(\rho) = p_e(\rho).\end{aligned}$$ This allows to apply our basic arguments directly also to the isothermal flow of real gases. The parameters $a_e,b_e,c_e$ are assumed to be non-negative and constant on every pipe.The value of $c_e>0$ may be different on every edge $e$, which allows to deal with pipes of different cross-section. In addition to the conservation laws on the individual pipes, we assume that the following coupling conditions hold at every internal vertex $v \in \Vi$ of the network: $$\begin{aligned}
\sum\nolimits_{e \in \E(v)} m_e(v) n_e(v) &= 0, && \forall v \in \Vi, \label{eq:sys4}\\
\frac{m_e(v)^2}{2 \rho_e(v)^2}+P_e'(\rho_e(v)) &= h_v + \frac{a_e}{\rho_e^2} \dx m_e(v), && \forall v \in \Vi, \ e \in \E(v). \label{eq:sys5}\end{aligned}$$ To complete the system description, additional conditions have to be imposed at the boundary vertices. For ease of presentation, we assume here that the network is closed such that no mass can enter or leave the network via the boundary, i.e., $$\begin{aligned}
m_e(v) n_e(v) &= 0, \qquad \forall v \in \Vb. \label{eq:sys6}\end{aligned}$$ Again, more general boundary conditions could be considered without much difficulty. To uniquely determine the solution, we finally assume access to the initial values $$\begin{aligned}
\rho(0)=\rho_0 \qquad \text{and} \qquad m(0)=m_0 \label{eq:sys7}\end{aligned}$$ for the density and the mass flux. These will only play a minor role in our considerations. To ensure that all equations are well-defined, we have to require that the functions $\rho$, $m$ have certain smoothness properties and that $\rho > 0$ during the evolution.
\[not:smooth\] By [*smooth solution*]{} of –, we understand a pair of functions $$\begin{aligned}
(\rho,m) \in C^1([0,T];L^\infty(\E) \times L^2(\E)) \cap C([0,T];H_+^1(\E) \times H^2(\E)) \end{aligned}$$ satisfying the equations for all $t>0$ and a.e. on $\E$. Here $H^1_+(\E) = \{\rho \in H^1(\E) : \exists \underline \rho>0 : \rho \ge \underline \rho \mbox{ a.e. on } \E \}$ shall denote the space of uniformly positive piecewise smooth functions.
A variational principle {#sec:variational}
=======================
We now present a variational characterization for solutions of – which is directly related to the conservation of mass and energy. This weak formulation will be at the center of our considerations and later on serve as the starting point for the numerical approximation.
A variational characterization
------------------------------
The first equation in the variational principle below directly follows from the continuity equation. To motivate the somewhat unusual form of the second equation, let us consider the change of kinetic energy, given by $$\begin{aligned}
\frac{d}{dt} \Big(\frac{m^2}{2\rho} \Big)
&= \Big(\frac{m}{\rho}\Big) \dt m - \Big(\frac{m^2}{2\rho^2}\Big) \dt \rho
= \Big(\frac{1}{\rho} \dt m - \frac{m}{2\rho^2} \dt \rho \Big) \ m. \end{aligned}$$ The time derivative of the local kinetic energy is thus obtained by a weighted linear combination of the time derivatives arising in and . With this in mind, we arrive at the following variational characterization of solutions to the compressible flow problem.
\[lem:variational\] $ $\
Let $(\rho,m)$ be a smooth solution of – in the sense of Notation \[not:smooth\]. Then $$\begin{aligned}
\left(\dt \rho,q\right)_\E &= - \left(\dx' m,q\right)_\E, \\
% \left(\frac{1}{\gamma-1} \frac{p'(\rho)}{\rho} \dt \rho,q\right)_\E &= - \left(\frac{1}{\gamma-1} \frac{p'(\rho)}{\rho} \dx' m,q\right)_\E, \\
\left(\frac{1}{\rho} \dt m - \frac{m}{2\rho^2} \dt \rho,v\right)_{\!\!\E}
&= \left(\frac{m^2}{2\rho^2} + P'(\rho) - \frac{a}{\rho^2}\dx' m, \dx' v\right)_{\!\!\E}
\! - \! \left(\frac{m}{2\rho^2} \dx' m + b \frac{|m| m}{\rho^2},v\right)_{\!\!\E},\end{aligned}$$ for all $q \in L^2$, all $v \in H_0(\div)$, and all $t>0$. Note that the functions $\rho$ and $m$ here depend on $t$, while the test functions $q$ and $v$ are independent of time.
The first equation follows by multiplying with test functions $q_e \in L^2(e)$, integration over $e$, and summation over all edges. Using , one can see that $$\begin{aligned}
-\frac{m_e}{2\rho_e^2} \dt \rho_e = \frac{m_e}{2\rho_e^2} \dx m_e. \end{aligned}$$ From equation , we then obtain $$\begin{aligned}
\frac{1}{\rho_e} \dt m_e
&= -\frac{1}{\rho_e}\dx \left(\frac{m_e^2}{\rho_e} + p_e(\rho_e)\right) + a_e \dx \left( \frac{1}{\rho_e^2} \dx m_e\right) - b_e \frac{|m_e| m_e}{\rho_e^2}. \end{aligned}$$ The first term on the right hand side of the previous equation can be replaced by $$\begin{aligned}
\frac{1}{\rho_e} \dx \left(\frac{m_e^2}{\rho_e}\right)
&= \dx \left(\frac{m_e^2}{\rho_e^2}\right) + \frac{m_e}{\rho_e^2} \dx m_e. \end{aligned}$$ Using the constitutive equation , we obtain for the second term $$\begin{aligned}
\frac{1}{\rho_e} \dx p_e(\rho_e)
&= \dx \left(\frac{p_e(\rho_e)}{\rho_e}\right) + \frac{p_e(\rho_e)}{\rho_e^2} \dx \rho_e
\\&
= \dx P_e'(\rho_e) - \dx \left(\frac{P_e(\rho_e)}{\rho_e}\right) + \frac{p_e(\rho_e)}{\rho_e^2} \dx \rho_e
= \dx P_e'(\rho_e). \end{aligned}$$ The last identity follows from the relation between pressure and potential energy density. A combination of the four identities stated above directly leads to $$\begin{aligned}
\frac{1}{\rho_e} \dt m_e - \frac{m_e}{2\rho_e^2} \dt \rho_e = -\dx \left(\frac{m_e^2}{2\rho_e^2} + P_e'(\rho_e) - \frac{a_e}{\rho_e^2}\dx m_e\right) - \frac{m_e}{2\rho_e^2} \dx m_e - b_e \frac{|m_e| m_e}{\rho_e^2}. \end{aligned}$$ Next we multiply this equation by a test function $v_e$ on every edge $e$, integrate over $e$, and then sum up over all edges $e$. In compact notation, the resulting identity reads $$\begin{aligned}
\left(\frac{1}{\rho} \dt m - \frac{m}{2\rho^2} \dt \rho,v\right)_{\!\!\E}
% \\&\qquad \qquad
= -\left(\dx' \left( \frac{m^2}{2\rho^2} + P'(\rho) - \frac{a}{\rho^2} \dx' m \right), v\right)_{\!\!\E}
\! - \! \left(\frac{m}{2\rho^2} \dx' m + b \frac{|m| m}{\rho^2},v\right)_{\!\!\E}.\end{aligned}$$ We can now use integration-by-parts for the first term on the right hand side. Using the continuity condition and assuming $v \in H_0(\div)$, all interface terms in the integration-by-parts formula vanish, and we arrive at the second identity of the variational principle.
Conservation of mass and energy
-------------------------------
As an immediate consequence of the variational characterization of smooth solutions, we obtain the following global balance relations.
\[lem:identities\] $ $\
Let $(\rho,m)$ be a smooth solution of – in the sense of Notation \[not:smooth\]. Then $$\begin{aligned}
\frac{d}{dt} \int_\E \rho dx = 0 \end{aligned}$$ and $$\begin{aligned}
\frac{d}{dt} \int_\E \frac{m^2}{2\rho} + P(\rho) dx + \int_\E \frac{a}{\rho^2} |\dx' m|^2 + b \frac{|m|^3}{\rho^2} dx=0, \end{aligned}$$ i.e., the total mass is conserved and the total energy is dissipated at any point in time.
As shown in the previous lemma, any smooth solution satisfies the above variational principle. Testing the first variational equation with $q=1$ leads to $$\begin{aligned}
\frac{d}{dt} \int_\E \rho dx
&= (\dt \rho,1)_\E
= -(\dx' m, 1)_\E
= -\sum_{v \in \V} \sum_{e \in \E(v)} m_e(v) n_e(v) = 0.\end{aligned}$$ Here we used integration-by-parts in the third step, and the coupling and boundary conditions stated in and in the last step. For the second assertion, observe that $$\begin{aligned}
\frac{d}{dt} \int_\E \frac{m^2}{2\rho} + P(\rho) dx
&= \left(\frac{1}{\rho} \dt m - \frac{m}{2\rho^2} \dt \rho,m\right)_{\!\E} + \Big(\dt \rho, P'(\rho)\Big)_{\!\E} = (*). \end{aligned}$$ Testing the variational principle with $q=P'(\rho)$ and $v=m$, we further obtain $$\begin{aligned}
(*) &= \left(\frac{m^2}{2\rho^2} + P'(\rho) - \frac{a}{\rho^2} \dx' m,\dx' m\right)_{\!\!\E}
- \left(\frac{m}{2\rho^2} \dx' m,m\right)_{\!\!\E} - \left(b \frac{|m|m}{\rho^2},m\right)_{\!\!\E} - \Big(\dx' m, P'(\rho)\Big)_{\E}
\\&
= - \left(\frac{a}{\rho^2} \dx' m, \dx' m\right)_{\!\!\E} - \left(b \frac{|m|m}{\rho^2},m\right)_{\!\!\E}.\end{aligned}$$ This shows the energy identity stated in the lemma and completes the proof.
As a direct consequence of the previous lemma, we obtain the following basic identities.
$ $\
Let $(\rho,m)$ be a smooth solution of – in the sense of Notation \[not:smooth\]. Then $$\begin{aligned}
\int_\E \rho(t) dx = \int_\E \rho_0 dx\end{aligned}$$ and $$\begin{aligned}
% E(t) + \int_0^t D(s) ds :=
\int_\E \frac{m(t)^2}{2\rho(t)} + P(\rho(t)) dx + \int_0^t \int_\E a \frac{|\dx' m(s)|^2}{|\rho(s)|^2} + b \frac{|m(s)|^3}{|\rho(s)|^2} dx \; ds
% \\
= \int_\E \frac{m_0^2}{2\rho_0} + P(\rho_0) dx.\end{aligned}$$
This shows that the total mass is conserved and that the energy is decreased only by dissipation due to viscous forces and friction. Note that the sum of total and dissipated energy is conserved as well, which is why we speak of energy conservation also in the presence of viscous forces and friction. The uniform bounds for the energy and the dissipation terms allows us to prove the existence of solutions to finite dimensional approximations later on.
Comments on the variational principle
-------------------------------------
Let us briefly put the results of the previous lemmas into perspective: The proof of Lemma \[lem:identities\] reveals that the variational principle of Lemma \[lem:variational\] is very tightly connected to the global conservation laws for mass and energy. The last step in the proof of Lemma \[lem:variational\] additionally shows that the coupling conditions – are the natural ones for this weak formulation and that they guarantee that no mass or energy is generated or annihilated at the junctions. Note that the convective terms arising on the right hand side of the second variational equation are antisymmetric and therefore disappear when testing with $v=m$. Also the pressure terms arising from the first and second equation in the variational principle annihilate naturally. These inherent symmetry properties are directly related to the particular form of our variational characterization and almost immediately imply the conservation of mass and energy. Finally note that no space derivatives of the density appear in the weak formulation. This substantially simplifies the design and analysis of appropriate numerical approximations in the following sections.
Galerkin approximation in space {#sec:galerkin}
===============================
As a first step towards the numerical solution of problem –, we now consider a conforming Galerkin approximation in space by a mixed finite element method. We only discuss the lowest order approximation in detail. Similar arguments may however also be used for the design and analysis of higher order approximations.
The mesh and polynomial spaces
------------------------------
Let $[0,\ell_e]$ be the interval related to the edge $e$. We denote by $T_h(e) = \{K\}$ a uniform mesh of $e$ with elements $K$ of length $h_K$. The global mesh is defined as $T_h(\E) = \{T_h(e) : e \in \E\}$, and as usual, we denote by $h=\max_{K \in T_h(\E)} h_K$ the global mesh size. Let us define spaces of piecewise polynomials by $$\begin{aligned}
P_k(T_h(\E)) &= \{v \in L^2(\E) : v|_e \in P_k(T_h(e)) \ \forall e \in \E\}. \end{aligned}$$ Here $P_k(T_h(e)) = \{v \in L^2(e) : v|_K \in P_k(K), \ K \in T_h(e)\}$ is the space of piecewise polynomials on the mesh $T_h(e)$ of a single edge $e$, and $P_k(K)$ is the space of polynomials of degree $\le k$ on the subinterval $K$. For approximation of the density $\rho$ and the mass flux $m$, we consider functions in the finite element spaces $$\begin{aligned}
\label{eq:spaces}
V_h = P_{1}(T_h(\E)) \cap H_0(\div)
\quad \text{and} \quad
Q_h = P_{0}(T_h(\E)). \end{aligned}$$ These spaces have good approximation properties and have been used successfully for the numerical approximations of damped wave propagation on networks; see [@EggerKugler16] for details.
The Galerkin semi-discretization
--------------------------------
For the space discretization of the compressible flow problem –, we consider the following finite dimensional approximations.
\[prob:semi\] Find $(\rho_h,m_h) \in C^1(0,T;Q_h \cap V_h)$ with $\rho_h(0) = \rho_{0,h}$ and $m_h(0)=m_{0,h}$, and such that for all $v_h \in V_h$ and $q_h \in Q_h$, and every $t \in [0,T]$, there holds $$\begin{aligned}
\left(\dt \rho_h,q_h\right)_\E
&= - \left(\dx' m_h,q_h\right)_\E, \\
\left(\frac{1}{\rho_h} \dt m_h - \frac{m_h}{2\rho_h^2} \dt \rho_h,v_h\right)_{\!\!\E} \!
&= \! \left(\frac{m_h^2}{2\rho_h^2} +P'(\rho_h) - \frac{a}{\rho_h^2} \dx' m_h, \dx' v_h\right)_{\!\!\E}
\! - \! \left(\frac{m_h}{2\rho_h^2} \dx' m_h + b \frac{|m_h| m_h}{\rho_h^2},v_h\right)_{\!\!\E}\!.\end{aligned}$$ Here $0 < \rho_{0,h} \in Q_h$ and $m_{0,h} \in V_h$ are given approximations for the initial values.
Mass and energy conservation
----------------------------
We will show below that the semi-discrete problem is uniquely solvable. Before doing that, let us present some identities for the conservation of mass and energy, which more or less directly follow from our construction of the variational principle and the fact that we are using a *conforming* Galerkin approximation.
\[lem:identitiesh\] $ $\
Let $(\rho_h,m_h)$ be a solution of Problem \[prob:semi\]. Then $$\begin{aligned}
\frac{d}{dt} \int_\E \rho_h dx = 0 \end{aligned}$$ and $$\begin{aligned}
\frac{d}{dt} \int_\E \frac{m_h^2}{2\rho_h} + P(\rho_h) dx
+ \int_\E \frac{a}{\rho_h^2} |\dx' m_h|^2 + b \frac{|m_h|^3}{\rho_h^2} dx=0, \end{aligned}$$ i.e., the mass and energy identities of Lemma \[lem:identities\] also hold for the semi-discretization.
The assertions follow similar as in the proof of Lemma \[lem:identities\].
Upon integration of the two identities, we obtain the following global conservation laws.
\[lem:identitiesh2\]$ $\
Let $(\rho_h,m_h)$ be a solution of Problem \[prob:semi\]. Then $$\begin{aligned}
\int_\E \rho_h(t) dx = \int_\E \rho_{0,h} dx\end{aligned}$$ and $$\begin{aligned}
\int_\E \frac{m_h(t)^2}{2\rho_h(t)} + P(\rho_h(t)) dx
+ \int_0^t \int_\E a \frac{|\dx' m_h(s)|^2}{|\rho_h(s)|^2} + b \frac{|m_h(s)|^3}{|\rho_h(s)|^2} dx \; ds
% \\
= \int_\E \frac{m_{0,h}^2}{2\rho_{0,h}} + P(\rho_{0,h}) dx.\end{aligned}$$
Let us emphasize that these two identities are a verbatim translation of the conservation laws on the continuous level and they are derived in the very same manner.
Well-posedness of the semi-discrete scheme
------------------------------------------
As a consequence of the uniform bounds for the energy and the dissipation terms resulting from the previous lemma, we are now able to establish the well-posedness of the semi-discretization.
\[lem:wellposedh\] Assume that $\rho_{0,h} \ge \underline\rho>0$. Then there exists a $T>0$ such that Problem \[prob:semi\] has a unique solution $(\rho_h,m_h) \in C^1([0,T);Q_h \times V_h)$ and $\rho_h(t)>0$ for all $0 \le t < T$. If $a \ge \underline a > 0$, then we can choose $T=\infty$, i.e., the solution is global.
Since the spaces $Q_h$ and $V_h$ are finite dimensional and $\rho_h(0)>0$, the local existence of a solution follows readily from the Picard-Lindelöf theorem. Moreover, the solution can be extended uniquely as long as $\rho_h(t)>0$. For any element $K$ of the mesh, we have $$\begin{aligned}
\frac{d}{dt} \int_K \frac{1}{\rho_h} dx
&= -\left( \dt \rho_h, \frac{1}{\rho_h^2}\right)_K
= \left(\dx' m_h, \frac{1}{\rho_h^2}\right)_K \\
&= \left(\frac{1}{\rho_h} \dx' m_h, \frac{1}{\rho_h}\right)_K
\le \left\|\frac{1}{\rho_h} \dx' m_h\right\|_{L^\infty(K)} \int_K \frac{1}{\rho_h} dx. \end{aligned}$$ From the equivalence of norms on finite dimensional spaces, we know that $$\begin{aligned}
\left\|\frac{1}{\rho_h} \dx' m_h\right\|_{L^\infty(K)}
\le \left\|\frac{1}{\rho_h} \dx' m_h\right\|_{L^\infty(\E)}
\le C_h \left\|\frac{1}{\rho_h} \dx' m_h\right\|_{L^2(\E)}.\end{aligned}$$ By the Gronwall lemma, we thus obtain $$\begin{aligned}
\int_K \frac{1}{\rho_h(t)} dx
&\le e^{\int_0^t C_h \left\|\frac{1}{\rho_h(s)} \dx' m_h(s)\right\|_{L^2(\E)} ds} \int_K \frac{1}{\rho_h(0)} dx.\end{aligned}$$ The integral in the exponential term can be further estimated by $$\begin{aligned}
\int_0^t \left\|\frac{1}{\rho_h(s)} \dx' m_h(s)\right\|_{L^2(\E)} ds
\le \sqrt{t} \Big(\int_0^t \left\|\frac{1}{\rho_h(s)} \dx' m_h(s)\right\|_{L^2(\E)}^2 ds \Big)^{1/2}.\end{aligned}$$ If $a \ge \underline a > 0$, the term in parenthesis can be bounded uniformly in $t$ by the estimates for the dissipation term provided by the energy identity of Lemma \[lem:identitiesh2\]. Using that $h_{min} \le |K| \le h_{max}$ for some $h_{max},h_{min}>0$ and all elements $K$, we then obtain $$\begin{aligned}
h_{min} \left\|\frac{1}{\rho_h(t)}\right\|_{L^\infty(\E)}
&\le \max_{K \in T_h(\E)} \int_K \frac{1}{\rho_h(t)} dx
\le e^{\sqrt{t} C} h_{max} \left\|\frac{1}{\rho_{0,h}}\right\|_{L^\infty(\E)}.\end{aligned}$$ This shows that the density stays positive for all time. From the energy identity, one can then further deduce that also $\|\rho_h\|_{L^\infty(\E)}$ and $\|m_h\|_{L^\infty(\E)}$ at most increase exponentially in time, such that the solution can in fact be extended globally.
Time discretization {#sec:time}
===================
As a final step in the discretization procedure, we now consider the time discretization of the Galerkin approximation considered in the previous section. Let $\tau>0$ denote the time-step and set $t^n = n \tau$ for $n \ge 0$. Given a sequence $\{d^n\}_{n \ge 0}$, we denote by $$\begin{aligned}
\dtau d^n := \frac{d^n - d^{n-1}}{\tau} \quad \text{for } n \ge 0,\end{aligned}$$ the backward differences which are taken as approximations for the time derivative terms. Different value $\tau_n>0$ for the individual time steps could in principle be chosen as well.
Fully discrete scheme
---------------------
For the time discretization of the Galerkin approximation stated in Problem \[prob:semi\], we now consider the following implicit time stepping scheme.
\[prob:full\] Let $\rho_{0,h}$ and $m_{0,h}$ be given as in Problem \[prob:semi\], and set $\rho_h^0=\rho_{0,h}$ and $m_h^0=m_{0,h}$. Then for $n \ge 1$ find $(\rho_h^n,m_h^n)\in Q_h\times V_h$, such that $$\begin{aligned}
\left(\dtau \rho_h^n,q_h\right)_\E
&= - \left(\dx' m_h^n,q_h\right)_\E, \\
\left(\frac{1}{\rho_h^{n-1}} \dtau m_h^n - \frac{m_h^n}{2|\rho_h^{n}|^2} \dtau \rho_h^n,v_h\right)_{\!\!\E}
&= \left(\frac{|m_h^n|^2}{2|\rho_h^n|^2} + P'(\rho_h^n) - \frac{a}{|\rho_h^n|^2} \dx' m_h^n, \dx' v_h\right)_{\!\!\E}
\\ & \qquad \qquad \qquad
\! - \! \left(\frac{m_h^n}{2|\rho_h^n|^2} \dx' m_h^n, v_h\right)_{\!\!\E}
\! - \! \left(b \frac{|m_h^n| m_h^n}{|\rho_h^n|^2},v_h\right)_{\!\!\E}\end{aligned}$$ hold for all test functions $q_h \in Q_h$ and $v_h \in V_h$.
The well-posedness of the method will be stated below. Before doing that, let us summarize the basic conservation principles for mass and energy for the full discretization.
Conservation of mass and dissipation of energy
----------------------------------------------
The particular form of the fully discrete variational problem allows us to immediately obtain the following balance relations for mass and energy by appropriate choice of the test functions.
$ $ \[lem:identitieshh\]\
Let $(\rho_h^n,m_h^n)_{n \ge 0}$ be a solution of Problem \[prob:full\]. Then for all $n \ge 0$ we have $$\begin{aligned}
\int_\E \rho_h^n dx = \int_\E \rho_{0,h}\end{aligned}$$ and $$\begin{aligned}
\int_\E \frac{|m_h^n|^2}{2\rho_h^n} + P(\rho_h^n) dx
+ \tau \sum_{k=1}^n \int_\E a\frac{|\dx' m_h^k|^2}{|\rho_h^k|^2} + b \frac{|m_h^k|^3}{|\rho_h^j|^2} dx
% \\
\le \int_\E \frac{m_{0,h}^2}{2\rho_{0,h}} + P(\rho_{0,h}) dx.\end{aligned}$$ The mass is thus conserved and energy is dissipated effectively by viscous forces and friction.
The implicit time integration scheme yields some extra numerical dissipation, which leads to the inequality instead of equality in the energy balance. The uniform bounds for energy and dissipation again plays an important role for the well-posedness of the scheme.
The first identity follows by testing Problem \[prob:full\] with $q_h=1$ and $v_h=0$. Now turn to the second: By elementary manipulations, we can split $$\begin{aligned}
\int_\E \frac{|m_h^n|^2}{2\rho_h^n} dx
&= \int_\E \frac{|m_h^{n-1}|^2}{2\rho_h^{n-1}} dx
+ \int_\E \frac{|m_h^n|^2}{2 \rho_h^n} - \frac{|m_h^n|^2}{2 \rho_h^{n-1}} dx
+ \int_\E \frac{|m_h^n|^2}{2 \rho_h^{n-1}} - \frac{|m_h^{n-1}|^2}{2 \rho_h^{n-1}} dx. \end{aligned}$$ Now recall that for any convex differentiable scalar function $f(y)$, we have $$\begin{aligned}
f(y^n) \le f(y^{n-1}) + f'(y^{n}) (y^n - y^{n-1}). \end{aligned}$$ Applying this to the functions $f(\rho)=\frac{|m_h^n|^2}{2\rho}$ and $f(m)=\frac{m^2}{2 \rho_h^{n-1}}$, which are both convex with respect to their arguments, we obtain $$\begin{aligned}
\int_\E \frac{|m_h^n|^2}{2\rho_h^n} dx
&\le \int_\E \frac{|m_h^{n-1}|^2}{2\rho_h^{n-1}} dx
- \int_\E \frac{|m_h^n|^2}{2 |\rho_h^n|^2} (\rho_h^n - \rho_h^{n-1}) dx
+ \int_\E \frac{m_h^n}{\rho_h^{n-1}} (m_h^n - m_h^{n-1}) dx \\
&= \int_\E \frac{|m_h^{n-1}|^2}{2\rho_h^{n-1}} dx
+ \tau \left(\frac{1}{\rho_h^{n-1}} \dtau m_h^n - \frac{m_h^n}{2 |\rho_h^n|^2} \dtau \rho_h^n, m_h^n\right)_{\!\!\E}. \end{aligned}$$ For the last term we can use the discrete variational principle. Since the potential energy density is again a convex function, one obtains in a similar manner that $$\begin{aligned}
\int_\E P(\rho_h^n) dx
&\le \int_\E P(\rho_h^{n-1}) dx
+ \tau \left(\dtau \rho_h^n, P'(\rho_h^n)\right)_{\E},\end{aligned}$$ and the last term can again be treated by the discrete variational principle. The energy inequality for $n=1$ now follows from the definition of $(\rho_h^n,m_h^n)$ in Problem \[prob:full\] with the same reasoning as in Lemma \[lem:identities\]. The result for $n>1$ is obtained by recursion.
Well-posedness of the fully discrete scheme
-------------------------------------------
Before closing this section, let us establish a basic result concerning the solvability of the nonlinear problems that arise in every single time step of of the fully discrete scheme.
\[lem:welposedhh\] Let $(\rho_h^{n-1},m_h^{n-1}) \in Q_h \times V_h$ with $\rho_h^{n-1} > 0$. Then for $\tau>0$ sufficiently small, the system for determining $(\rho_h^{n},m_h^{n})$ in Problem \[prob:full\] has a unique solution with $\rho_h^{n} > 0$. If in addition $a \ge \underline a > 0$, then a solution exists for any choice of the time step.
The first assertion follows from the implicit function theorem. To show the second assertion, we can proceed with similar arguments as in the proof of Lemma \[lem:wellposedh\] to obtain a uniform bound $\|\frac{1}{\rho_h^n}\|_{L^\infty(\E)} \le D_h e^{C_h n} \|\frac{1}{\rho_h^0}\|_{L^\infty(\E)}$ for any solution $(\rho_h^n,m_h^n)$. Via the discrete energy estimate of Lemma \[lem:identitieshh\], we then also obtain uniform a-priori bounds for $\|m_h^n\|_{L^2(\E)}$ and $\|p(\rho_h)\|_{L^1(\E)}$, and by equivalence of norms on finite dimensional spaces also for $\|m_h^n\|_{L^\infty(\E)}$ and $\|\rho_h^n\|_{L^\infty(\E)}$. Existence of a solution for the next time step then follows by a homotopy argument and Brouwer’s fixed point theorem.
\[rem:wellposedhh\] The previous lemma guarantees uniqueness of the solution $(\rho_h^n,m_h^n)$ only for a sufficiently small time step $\tau$. The size of the admissible time step will in general depend on $\rho_h^{n-1}$ and $m_h^{n-1}$, in particular on the lower bounds for $\rho_h^{n-1}$. As long as the density $\rho_h^{n-1}$ stays well away from zero, the time step problem will be uniquely solvable for reasonably large time steps, which we also observe in our numerical experiments.
Solution of the nonlinear problems {#sec:implementation}
==================================
Before we proceed to numerical tests, let us discuss in some more detail the actual implementation of the fully discrete scheme stated in Problem \[prob:full\]. For the solution of the nonlinear system in the $n$th time step, we consider the following fixed-point iteration.
\[prob:fixed\] $ $\
Set $\widetilde \rho_h = \rho_h^{n-1}$ and $\widetilde m_h = m_h^{n-1}$. For $k=1,2,\ldots$ solve for $(\rho_h,m_h)$ the system $$\begin{aligned}
%
&\frac{1}{\tau}\left(\rho_h,q_h\right)_\E + \left(m_h,q_h\right)_\E = \frac{1}{\tau}\left(\rho_h^{n-1},q_h\right)_\E, \hspace*{5em} \\
%
&\frac{1}{\tau}\left(\frac{1}{\rho_h^{n-1}} m_h ,v_h\right)_{\!\!\E}
%
- \left(\frac{\widetilde m_h}{2 \widetilde \rho_h^2} m_h + \frac{P'(\widetilde\rho_h)}{\widetilde \rho_h} \rho_h - \frac{a}{|\widetilde \rho_h|^2} \dx' m_h , \dx' v_h\right)_{\!\!\E} \hspace*{5em} \\
&\qquad \qquad \
+ \left(\frac{\widetilde m_h}{2 \widetilde \rho_h^2} \dx' m_h + \frac{b|\widetilde m_h|}{\widetilde \rho_h^2} m_h, v_h\right)_{\!\!\E}
%
%
=\frac{1}{\tau}\left(\frac{1}{\rho_h^{n-1}} m_h^{n-1} + \frac{\widetilde m_h}{2\widetilde \rho_h^2} \left(\widetilde \rho_h - \rho_h^{n-1}\right),v_h\right)_{\!\!\E}.\end{aligned}$$ Then set $\widetilde \rho_h = \rho_h$ and $\widetilde m_h=m_h$, and repeat with $k=k+1$.
Note that the systems to be solved in every step of the fixed-point iteration are linear now. The iteration yields reasonable approximations for the nonlinear system to be solved.
Any fixed-point $(\rho_h,m_h)$ of the iteration stated in Problem \[prob:fixed\] with $\rho_h>0$ is also a solution $(\rho_h^n,m_h^n)$ of the nonlinear system in Problem \[prob:full\].
Setting $\rho_h=\widetilde \rho_h=\rho_h^n$ and $m_h=\widetilde m_h=m_h^n$ in the above iteration directly leads to the nonlinear system of Problem \[prob:full\].
Due to the special structure of the linear systems characterizing the fixed-point iteration of Problem \[prob:fixed\], we are able to guarantee that the iteration is well-defined.
Let $\rho_h^{n-1}$, $m_h^{n-1}$, $\widetilde \rho_h$, and $\widetilde w_h$ be given and $\rho_h^{n-1},\widetilde \rho_h>0$. Then there exists a unique solution $(\rho_h,m_h)$ of the linear system in Problem \[prob:fixed\].
Note that the system for determining $(\rho_h,m_h)$ in iteration $k$ is linear and finite. Testing with $q_h=\frac{P'(\widetilde \rho_h)}{\widetilde \rho_h} \rho_h$ and $v_h=m_h$ and summing up the two equations leads to $$\begin{aligned}
\frac{1}{\tau} \left(\frac{P'(\widetilde \rho_h)}{\widetilde \rho_h} \rho_h,\rho_h \right)_{\!\!\E}
+ \frac{1}{\tau} \left(\frac{1}{\rho_h^n} m_h,m_h\right)_{\!\!\E} + \left(\frac{a}{\widetilde \rho_h^2} \dx' m_h,\dx' m_h\right)_{\!\!\E} + \left(\frac{b|\widetilde m_h|}{\widetilde \rho_h^2} m_h, m_h\right)_{\!\!\E}
%
=
\widetilde C,\end{aligned}$$ where $\widetilde C$ only depends on the known quantities $\rho_h^{n-1}$, $\widetilde \rho_h$, and $\widetilde m_h$. Observe that many of the terms dropped out because of antisymmetry of the particular linearization defining the fixed-point iteration. Since $\rho_h^{n-1}$, $\widetilde \rho_h$ and $P'(\widetilde \rho_h)$ are positive by assumption, the above identity already yields the uniqueness and hence also the existence of the solution.
By a homotopy argument, one can again show that for $\tau>0$ sufficiently small, the solution provided by the previous lemma satisfies $\rho_h>0$. In order to guarantee the well-posedness for arbitrary step size $\tau$ and all iterates, one can replace the terms $\widetilde \rho_h$ in the denominators of the fixed point iteration by $\max\{\underline \rho,\widetilde \rho_h\}$ for some fixed value $\underline \rho>0$. If in addition $a \ge \underline{a}>0$, we actually expect positivity of $\rho_h$ and a unique fixed-point for reasonably large time step; see also Remark \[rem:wellposedhh\]. For the solution of the nonlinear systems of Problem \[prob:full\], one may alternatively also utilize a Newton-type iteration.
Numerical tests {#sec:numerics}
===============
We now complement our theoretical investigations with some computational results. For the numerical solution of all test problems, we consider the fully discrete scheme defined in Problem \[prob:full\]. For the solution of the nonlinear systems in every time step, we utilize the fixed point iteration given in Problem \[prob:fixed\]. All computations are carried out on uniform meshes with mesh size $h$ and a fixed time step $\tau$. We report on the particular choices below.
A shock tube problem
--------------------
As a first test problem, we consider the undamped system – with $a=b=0$ on a single pipe $e=[-5,5]$ directed from left to right. For the state equation , we use $c=1/2$ and $\gamma=2$, and as initial conditions, we choose $$\begin{aligned}
\rho_0(x)=2-\text{sgn}(x) \qquad \text{and} \qquad m_0(x) = 0.\end{aligned}$$ This setting amounts to the dam-break test problem for the shallow water equations discussed in [@LeVeque02 Example 13.4]. In Figure \[fig:shocktube\], we depict the numerical solutions obtained with our method with mesh size $h=0.01$ and time step size $\tau=0.005$. Two fixed point iterations were used for the solution of the nonlinear problems in every time step.
![Solution for the shock tube problem: A shock wave is propagating to the right and a rarefaction wave to the left. The results are in very good agreement with the ones presented in [@LeVeque02 Fig. 13.4]\[fig:shocktube\].](shocktube_rho00 "fig:"){width="41.00000%"} ![Solution for the shock tube problem: A shock wave is propagating to the right and a rarefaction wave to the left. The results are in very good agreement with the ones presented in [@LeVeque02 Fig. 13.4]\[fig:shocktube\].](shocktube_m00 "fig:"){width="41.00000%"}\
![Solution for the shock tube problem: A shock wave is propagating to the right and a rarefaction wave to the left. The results are in very good agreement with the ones presented in [@LeVeque02 Fig. 13.4]\[fig:shocktube\].](shocktube_rho05 "fig:"){width="41.00000%"} ![Solution for the shock tube problem: A shock wave is propagating to the right and a rarefaction wave to the left. The results are in very good agreement with the ones presented in [@LeVeque02 Fig. 13.4]\[fig:shocktube\].](shocktube_m05 "fig:"){width="41.00000%"}\
![Solution for the shock tube problem: A shock wave is propagating to the right and a rarefaction wave to the left. The results are in very good agreement with the ones presented in [@LeVeque02 Fig. 13.4]\[fig:shocktube\].](shocktube_rho20 "fig:"){width="41.00000%"} ![Solution for the shock tube problem: A shock wave is propagating to the right and a rarefaction wave to the left. The results are in very good agreement with the ones presented in [@LeVeque02 Fig. 13.4]\[fig:shocktube\].](shocktube_m20 "fig:"){width="41.00000%"}
The solution to the problem here consists of a shock wave propagating to the right and a rarefaction wave propagating to the left. The approximations obtained with our numerical method are in very good agreement with the ones reported in [@LeVeque02 Fig. 13.4]. Some slight smoothing of the discontinuity and the kinks can be observed due to the dissipative nature of the implicit time stepping scheme mentioned after Lemma \[lem:identitieshh\]. The effect of smoothing slightly increases when the time step $\tau$ is further enlarged. The total mass of the system is conserved exactly while the total energy is slightly decreasing over time due to numerical dissipation. For the numerical solution depicted in Figure \[fig:shocktube\], the energy at the final time is $E(2.0)=0.983 \times E_0$, which amounts to an energy loss of about 1.7% due to numerical dissipation. The loss of energy could be further reduced by decreasing the time step size.
Convergence towards steady states in the presence of friction
-------------------------------------------------------------
As a second test scenario, we consider the flow of gas in a single pipe $e=[-5,5]$. As a model for the propagation, we here use the system – with $a=0$ and $b=100$, which amounts to the inviscid case with large damping typically observed in gas pipelines [@BrouwerGasserHerty11; @Osiadacz84]. The parameters in the pressure law are again set to $c=1/2$ and $\gamma=2$. As initial conditions, we here use $$\begin{aligned}
\rho_0=11 \qquad \text{and} \qquad m_0=0. \end{aligned}$$ The value for the initial density $\rho_0$ is chosen large enough in order to guarantee that the density stays positive for all $t \to \infty$. As boundary conditions, we consider $$\begin{aligned}
m(-5,t)=m(5,t)=1 \qquad \text{for } t > 0.\end{aligned}$$ This means that we start to inject gas at time $t=0$ with a constant rate at the left end of the pipe and we extract the same amount of gas on the right end. The total mass of the system is therefore conserved for all time. In Figure \[fig:gas\], we depict the numerical solutions obtained with our method with mesh size $h=0.01$ and time step size $\tau=0.005$. Two fixed point iterations are used again for the solution of the nonlinear problems in every time step.
![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_rho000 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_m000 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_rho001 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_m001 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_rho010 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_m010 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_rho100 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_m100 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_rho1000 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_m1000 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_rho10000 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport problem: Due to the strong damping, the discontinuities stemming from sudden change of boundary conditions decay rapidly and the system approaches steady state.\[fig:gas\]](gas_m10000 "fig:"){width="41.00000%"}
Due to the sudden change of the boundary conditions, discontinuities are generated at the pipe ends at time $t=0$, but they rapidly decay as an effect of the strong damping. Another consequence of the damping is that the system converges quickly towards a steady state which is the typical situation observed in gas pipelines. For the test problem under consideration, the stationary solution is governed by the system $$\begin{aligned}
\bar m &= 1, \\
\dx \left( \frac{1}{\bar \rho} + \frac{1}{2} \bar \rho^2 \right) &= 100 \frac{1}{\rho} \qquad \text{and} \qquad \int_{-5}^5 \bar \rho dx = 110,\end{aligned}$$ where we already simplified the momentum equation using the condition $\bar m=1$ and the specific form of the pressure law. The last condition comes from the fact that the total mass is conserved. The problem of determining $\bar \rho$ using the equations of the second line can be solved numerically by a shooting method. The approximation for the exact stationary solution obtained in this manner is shown in Figure \[fig:gasstat\].
![Stationary solution of the gas transport problem computed by the method of Heun and bisection to determine the initial value $\bar \rho(-5)$.\[fig:gasstat\]](gas_rhostat "fig:"){width="41.00000%"} ![Stationary solution of the gas transport problem computed by the method of Heun and bisection to determine the initial value $\bar \rho(-5)$.\[fig:gasstat\]](gas_mstat "fig:"){width="41.00000%"}
An inspection of the plots reveals the convergence of the numerical solution towards the correct steady state. Note that the exact conservation of mass on the discrete level is crucial here to get the correct steady state.
Gas flow across a junction
--------------------------
As a last test case, we consider the flow of gas through the simple pipe network depicted in Figure \[fig:graph\]. All pipes are chosen to be of unit length. The model parameters are set to $a=0$ and $b=100$, and we take $c=1/2$ and $\gamma=2$ in the pressure law as before. The initial conditions are now chosen as $$\begin{aligned}
m_0=0 \qquad \text{and} \qquad \rho_{0,e_1}=5, \ \rho_{0,e_2}=3, \ \rho_{0,e_3}=1.\end{aligned}$$ These initial values correspond to a specific Riemann problem at the junction [@Reigstad15]. We further assume that the ports of the network are closed, such that $$\begin{aligned}
m(v,t)=0 \qquad \text{for all } v \in \Vb, \ t>0.\end{aligned}$$ Due to the dissipation of energy caused by friction, we again expect convergence to a steady flow. As can be verified easily, the stationary solution is here given by $$\begin{aligned}
\bar m = 0 \qquad \text{and} \qquad \bar \rho = 3.\end{aligned}$$ The numerical solutions obtain in our experiments are depicted in Figure \[fig:network\].
![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_rho000 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_m000 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_rho001 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_m001 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_rho010 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_m010 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_rho100 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_m100 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_rho1000 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_m1000 "fig:"){width="41.00000%"}\
![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_rho10000 "fig:"){width="41.00000%"} ![Numerical solutions for the gas transport on the network depicted in Figure \[fig:graph\]. The x-axis displays the path length along the flow and the junction is located at $x=1$. The different solutions for pipe $1$ to $3$ are depicted with straight, dashed, and dotted lines, respectively.\[fig:network\]](valve_m10000 "fig:"){width="41.00000%"}
The initial value for the density is discontinuous across the junction, but this discontinuity is smeared out quickly due to the damping, and the system approaches a steady state on the long run. The fluxes are discontinuous across the junction for all $t>0$ but as a result of the coupling condition , which is satisfied exactly at every point in time, they always sum up to zero.
Further remarks on computational tests
--------------------------------------
For the problems under investigation, the numerical diffusion resulting from the implicit time stepping scheme was sufficiently large to yield unique global solutions in all our computations. The stability of the scheme was not affected when refining the mesh and convergence was observed for all test cases. Some additional artificial viscosity can be added to further regularize the numerical solutions. In fact, very similar results were obtained for all test cases, when the viscosity parameter $a$ was chosen positive and sufficiently small. Numerical computations were carried out also for other model and discretization parameters. In these tests we observed a very robust behavior of the scheme and the expected convergence with mesh and time step going to zero. Our method and analysis is applicable to general finite networks. Computations for more complicated network topologies lead to very similar results and these were therefore not presented here.
Discussion {#sec:discussion}
==========
In this paper, we proposed a novel variational characterization of solutions to compressible flow problems on networks that allowed us to establish conservation of mass and energy in a very direct manner and to apply conforming Galerkin approximations for the discretization in space. As a particular space discretization, we considered a mixed finite element method. We proved the well-posedness of the resulting semi-discrete scheme and established the conservation of mass and energy. We additionally proposed an implicit time stepping leading to a fully discrete scheme that exactly conserves mass and slightly dissipates energy as an effect of numerical dissipation. Well-posedness of the fully discrete scheme was established and a fixed-point iteration for the solution of the nonlinear problems arising in every time step was proposed. The stability, accuracy, and conservation properties of our numerical method was demonstrated for some numerical test problems. The method proved to be extremely robust concerning the choice of model and discretization parameters, which can be explained by the inherent energy stability of the approximation scheme.
The energy estimates provided in this paper may serve as a first step towards a complete convergence analysis for the method. A more precise characterization of the numerical dissipation provided by the implicit time stepping scheme may lead to sharper statements about well-posedness of the fully discrete scheme. The extension of our arguments to higher order approximations and more general cases including multi-dimensional problems might be interesting and seems feasible. These topics are left for future research.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank for support by the German Research Foundation (DFG) via grants IRTG 1529 and TRR 154, and by the “Excellence Initiative” of the German Federal and State Governments via the Graduate School of Computational Engineering GSC 233 at Technische Universität Darmstadt.
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---
abstract: |
A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$ on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of $\mathfrak {so}^*(4n)$ on the Poisson manifold $T^*\mathbb H_*^n/\mathrm{Sp}(1)$. (Here $\mathbb H_*^n:=\mathbb H^n\backslash \{0\}$ and the Hamiltonian action of $\mathrm{Sp}(1)$ on $T^*\mathbb H_*^n$ is induced from the natural right action of $\mathrm{Sp}(1)$ on $\mathbb H_*^n$. )
**Keywords.** Kepler problem, Jordan algebra, dynamic symmetry, Laplace-Runge-Lenz vector, Weinstein’s universal phase space.
address:
- 'Division of Science and Mathematics, New York University Abu Dhabi, Po Box 129188, Abu Dhabi, United Arab Emirates.'
- 'Department of Mathematics, Hong Kong Univ. of Sci. and Tech., Clear Water Bay, Kowloon, Hong Kong.'
author:
- Sofiane Bouarroudj
- Guowu Meng
title: 'The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems'
---
[^1]
Introduction
============
The Kepler problem is a textbook example of super integrable models. Its hamiltonian is invariant under the Lie group $\mathrm {SO}(4)$, larger than the manifest symmetry group $\mathrm{SO}(3)$. A remarkable fact about the Kepler problem is that the real non-compact Lie algebra $\frak{so}(4,2)$ has a nontrivial Poisson realization on its phase space. This Poisson realization, more precisely its quantized form, was initially discovered by I. A. Malkin and V. I. Man’ko [@Malkin66] in 1966. (For the prehistory of this important discovery about the Kepler problem, one may consult Footnote 2 in Ref. [@Barut67].) In the literature, the real non-compact Lie algebra $\frak{so}(4,2)$ is referred to as the *dynamical symmetry algebra* for the Kepler problem and its afore-mentioned Poisson realization is referred to as the *classical dynamical symmetry* for the Kepler problem.
The Kepler problem has magnetized versions, under the name of MICZ-Kepler problems. The work of A. Barut and G. Bornzin [@Barut71], extends the study of dynamical symmetry to these magnetized Kepler problems at the quantum level. Later, the higher dimensional analogue of MICZ-Kepler problems, under the name of generalized MICZ-Kepler problems, were found[@meng2007] and the dynamical symmetry for these models was studied as well, at both the quantum level [@mengzhang] and the classical level [@meng2013b].
About six years ago the second author [@meng2011; @meng2013] discovered that the Kepler problem has a vast generalization based on simple euclidean Jordan algebra, for which the conformal algebra of the Jordan algebra is the dynamical symmetry algebra. He also made the following observation [@meng2014']: for a generalized Kepler problem, its hamiltonian and its Laplace-Runge-Lenz vector can all be derived from its dynamical symmetry.
Recently, we exhibited in Ref. [@BM2015] the classical dynamical symmetry for the $\mathrm U(1)$-Kepler problems (see Ref. [@meng2014a] ). These Kepler-type problems are naturally associated with the euclidean Jordan algebras of complex hermitian matrices. In this article we shall exhibit the classical dynamical symmetry for the $\mathrm {Sp}(1)$-Kepler problems [@meng2014b], i.e. the quaternionic analogues of the $\mathrm U(1)$-Kepler problems. As a result, we obtain the Laplace-Runge-Lenz vector for each $\mathrm {Sp}(1)$-Kepler problem as well as a formulae of expressing the total energy in terms of the angular momentum and the Laplace-Runge-Lenz vector. For the convenience of readers, we end this introduction with
[**A list of symbols**]{}
$$\begin{aligned}
\begin{array}{ll}
\mathbb H & \quad\text{the set of quaternions}\cr
\mathrm H_n(\mathbb H)&\quad\text{the Jordan algebra of quaternionic hermitian matrices of order $n$}\cr
\mathpzc{Im}\,\mathbb H & \quad\text{the set of imaginary quaternions}\cr
\mathrm i, \mathrm j, \mathrm k & \quad \text{the standard orthonormal basis for $\mathpzc{Im}\,\mathbb H $ such that $\mathrm i\mathrm j=\mathrm k$}\cr
\bar q &\quad\text{the quaternionic conjugate of quaternion $q$, e.g., $\bar{\mathrm k}=-{\mathrm k}$} \cr
\mathpzc{Re}\, q & \quad {1\over 2}(q+\bar q)\cr
\mathpzc{Im}\, q & \quad {1\over 2}(q-\bar q)\cr
\lrcorner & \quad\mbox{the interior product of vectors with forms}\cr
\wedge & \quad\mbox{the wedge product of forms}\cr
{\mathrm d} & \quad\mbox{the exterior derivative operator}\cr
\pi_X:\, T^*X\to X &\quad\mbox{the cotangent bundle projection}\cr
G & \quad\mbox{a compact connected Lie group} \cr
\mathfrak g, \mathfrak g^* & \quad\mbox{the Lie algebra of $G$ and its dual} \cr
\xi & \quad\mbox{an element in $\mathfrak g$}\cr
\langle\, ,\,\rangle & \quad\mbox{either the paring of vectors with co-vectors or inner product}\cr
\langle\, \mid\,\rangle & \quad\mbox{the inner product on the Jordan algebra $\mathrm H_n(\mathbb H)$}\cr
\mathrm{Ad}_a & \quad\mbox{the adjoint action of $a\in G$ on $\mathfrak g$}\cr
P\to X & \quad\mbox{a principal $G$-bundle} \cr
\Theta & \quad\mbox{a $\mathfrak g$-valued differential one-form on $P$ that}\cr
&\quad\mbox{defines a principal connection on $P\to X$} \cr
R_a & \quad\mbox{the right action on $P$ by $a\in G$}\cr
X_\xi & \quad\mbox{the vector field on $P$ which represents the}\cr
&\quad\mbox{infinitesimal right action on $P$ by $\xi\in \mathfrak g$}\cr
F & \quad\mbox{a hamiltonian $G$-space}\cr
\mathcal F:=P\times_G F& \quad\mbox{the quotient of $P\times F$ by the action of $G$}\cr
\Phi: F\to \mathfrak g^* & \quad\mbox{the $G$-equivariant moment map}\cr
\mathcal F^\sharp &\quad \text{Sternberg phase space}\cr
\mathcal W &\quad \text{Weinstein's universal phase space}\cr
\end{array}\nonumber\end{aligned}$$
The dynamical symmetry algebra
==============================
The dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler model at level $n$ with magnetic charge $\mu$ that we shall exhibit is a Poisson realization of the dynamic symmetry algebra $\frak{so}^*(4n)$ on its phase space. Note that $\frak{so}^*(4n)$, being the conformal algebra of the simple euclidean Jordan algebra $\mathrm H_n(\mathbb H)$ of quaternionic hermitian matrices of order $n$, can be understood naturally in the language of Jordan algebra [@PJordan33]. The details are given in the next two paragraphs (see [@FK94] for more details).
For each $u\in V:= \mathrm H_n(\mathbb H)$, we use $L_u$ to denote the Jordan multiplication by $u$, and for each $u, v\in V$, we let $S_{uv}=[L_u, L_v]+L_{uv}$ where $[L_u, L_v]$ stands for the commutator: $L_uL_v-L_vL_u$ and $uv$ in $L_{uv}$ means $L_u (v)$, i.e., the symmetrized matrix product of $u$ with $v$. We use $\{uvw\}$ to denote $S_{uv}(w)$. Then we have $$[S_{uv}, S_{zw}]=S_{\{uvz\}w}- S_{z\{vuw\}}\quad \mbox{for any $u, v, z, w$ in $V$. }$$ So these $S_{uv}$ span a real Lie algebra. This Lie algebra is denoted by $\frak{str}$, and is referred to as the structure algebra of $V$. In fact $\frak{str}=\frak{su}^*(2n)\oplus \mathbb R$ where the center $\mathbb R$ is generated by $L_e$ — the Jordan multiplication by the Jordan identity element $e$.
The conformal algebra $\frak{co}$ is an extension of the structure algebra $\frak{str}$. As a real vector space we have $$\frak{co}=V\oplus \frak{str}\oplus V^*.$$ An element $z$ in $V$, rewritten as $X_z$, behaves like a vector: $$[S_{uv}, X_z]=X_{\{uvz\}}$$ and an element in $V^*$ behaves like a co-vector. Via the inner product on $V$, we can identify this element in $V^*$ with an element $w$ in $W$, which is rewritten as $Y_w$, then $$[S_{uv}, Y_w]=-Y_{\{vuw\}}.$$ The remaining commutation relations are $$[X_u, Y_v]=0, \quad [Y_u, Y_v]=0, \quad [X_u, Y_v]=-2S_{uv}\quad \mbox{for any $u, v$ in $V$. }$$ One can verify that, indeed, $\frak{co} = \frak{so}^*(4n)$.
The Phase Space
===============
When the magnetic charge is not zero, the phase space, being a Sternberg phase space [@Sternberg77], is a bit involved, so the Poisson realization of the dynamic symmetry algebra on the the phase space is a bit complicated and the verification of various Poisson relations becomes quite tedious, as evidenced already in simpler models such as the $\mathrm U(1)$-Kepler problems [@BM2015].
To circumvent this complication, we resort to an insight of A. Weinstein into the Sternberg phase space. As we shall see the Sternberg phase spaces form a bundle of symplectic manifolds over the affine space of principal connections. Since its base space is contractible, this fiber bundle must be topologically trivial. Indeed, A. Weinstein observed that [@Weinstein78] this bundle of symplectic manifolds is canonically isomorphic to a product bundle whose fiber is a fixed symplectic manifold, i.e., Weinstein’s *universal phase space*.
Review of the work by S. Sternberg and A. Weinstein
---------------------------------------------------
The goal of this subsection is to review the work by S. Sternberg [@Sternberg77] and A. Weinstein [@Weinstein78]. Let us start with the setup for Sternberg phase space:
(i) A compact Lie group $G$ and a principal $G$-bundle $P\to X$ with a principal connection form $\Theta$,
(ii) A Hamiltonian $G$-space $F$ with symplectic form $\Omega$ and a $G$-equivariant moment map $\Phi$: $F\to \frak g^*$. Here $\frak g$ is the Lie algebra of the Lie group $G$.
Note that, a co-adjoint orbit of $G$ with the Kirillov-Kostant-Souriau symplectic form is a typical example of Hamiltonian $G$-space.
For the convenience of readers, let us recall that a principal connection $\Theta$ on the principal $G$-bundle $P\to X$ is a $\frak g$-valued differential one-form on $P$ satisfying the following two conditions:
1\) ${R_{a^{-1}}}^*\,\Theta ={\mathrm {Ad}}_a\Theta$ for any $a\in G$, 2) $\Theta(X_\xi)=\xi$ for any $\xi\in \mathfrak g$.
Here $a\in G$, $R_{a^{-1}}$ is the right multiplication of $a^{-1}$ on $P$, ${\mathrm {Ad}}_a$ is the adjoint action of $a$ on $\frak g$, and $X_\xi$ is the vector field on $P$ which represents the induced action of $\xi\in \frak g$ on $P$, i.e., for any $f\in C^\infty(P)$, we have the Lie derivative $$\begin{aligned}
{\mathcal L} _{X_\xi}f \left|_{p} =\frac{\mathrm d}{\mathrm dt} \right|_{t=0}f(p\cdot \exp (t\xi)).\end{aligned}$$ It is easy to see that $[\mathcal L _{X_\xi}, \mathcal L _{X_\eta}] = \mathcal L _{X_{[\xi, \eta]}}$, or equivalently $[X_\xi, X_\eta] = X_{[\xi, \eta]}$.
It is a tautology that a smooth (right) $G$-action on a smooth manifold $P$ yields a hamiltonian $G$-action on the symplectic manifold $T^*P$ with the $G$-equivariant moment map $\rho$: $T^*P\to \frak g^*$. Indeed, if we use $\pi_P$ to denote the bundle projection $T^*P\to P$, then $\rho$ is defined via equation $$\langle \rho(z), \xi\rangle =\langle z, X_\xi (\pi_P(z))\rangle.$$ Another way to see it is this: for each point $p\in P$, the dual of linear map $$\begin{aligned}
\frak g &\to& T_pP\cr
\xi &\mapsto & X_\xi(p)\end{aligned}$$ is a linear map from $T_p^*P$ to $\frak g^*$. Assembling these dual maps together, we get the map $\rho$: $T^*P\to \frak g^*$.
The $G$-equivariance of $\rho$ is reflected by the fact that $(R_g)_*(X_\xi)=X_{\mathrm{Ad}_{g^{-1}}(\xi)}$ and the fact that $\mathrm{Ad}_g^*(\alpha)=\alpha\circ \mathrm{Ad}_{g^{-1}}$ for any $\alpha\in \frak g^*$. The map $\rho$ is a moment map, i.e., $\mathrm d \langle \rho, \xi\rangle=-\hat X_\xi\;\lrcorner\; \omega_P$. Here $\omega_P$ is the canonical symplectic form on $T^*P$, and the vector field $\hat X_\xi$ on $T^*P$ is the cotangent lift of the vector field $X_\xi$. In local coordinates, we have $$z =p_i(z)\,\mathrm d x^i|_{\pi_P(z)}, \quad \omega_P=\mathrm d p_i\wedge \mathrm d x^i, \quad X_\xi= X_\xi^i\frac{\partial}{\partial x^i}, \quad \hat X_\xi= X_\xi^i\frac{\partial}{\partial x^i} -p_i\frac{\partial X_\xi^i}{\partial x^j}\frac{\partial }{\partial p_j}$$ and $\langle \rho, \xi\rangle = p_iX^i_\xi$. Here $x^i$ is a system of local coordinates on $P$.
Combining $\rho$ and $\Phi$, we obtain a $G$-equivariant moment map $$\psi:\quad T^*P\times F\to \frak g^*.$$ which maps $(x, y)$ to $-\rho(x)+\Phi(y)$.
Since $P\to X$ is a principal $G$-bundle, $\rho|_{T^*_pP}$: $T_p^*P\to \frak g^*$ is a diffeomorphism for each $p\in P$, then $\rho$: $T^*P\to \frak g^*$ is a submersion. Consequently $\psi$ is a submersion as well. In particular, this means that $\psi^{-1}(0)$ is a submanifold of $T^*P\times F$. Since the isotropic group of $0\in \frak g^*$, being the Lie group $G$, is compact, its free action on $\psi^{-1}(0)$ is proper, Theorem 1 in Ref. [@Marsden74] applies, so there is a unique symplectic structure $\omega$ on $\psi^{-1}(0)/G$ such that $\pi^*\omega=\iota^*(\omega_P+\Omega)$ where $\omega_P$ is the tautological symplectic form on $P$, $\pi$ is the projection and $\iota$ is the inclusion: $$\begin{aligned}
\begin{tikzcd}[column sep=small]
\psi^{-1}(0) \arrow{d}[swap]{\pi} \arrow{r}{\iota} & T^*P\times F\\
\psi^{-1}(0)/G&
\end{tikzcd}\nonumber\end{aligned}$$ In the literature this reduced phase space $(\psi^{-1}(0)/G, \omega)$ is called the Weinstein’s *universal phase space* $\mathcal W$. Note that, no connection is required for the existence of this universal phase space.
To understand the meaning of the word “universal", let us suppose that a principal connection $\Theta$ on $P\to X$ is given. For each point $p\in P$, let $x$ be the image of $p$ under the bundle projection map, then the equivariant horizontal lifting of tangent vectors on $X$ (provided by $\Theta$) defines a linear map $T_xX\to T_pP$. By assembling the dual of these linear maps together, we arrive at the commutative square $$\begin{aligned}
\begin{tikzcd}[column sep=small]
T^*P \arrow{d}[swap]{\pi_P}\arrow{r} & T^*X\arrow{d}{\pi_X} \\
P\arrow{r} & X
\end{tikzcd}\nonumber\end{aligned}$$ where the top arrow, fiber-wise speaking, is the dual of the horizontal lifting of tangent vectors of $X$ to tangent vectors of $P$. Let $\tilde P$ be the pullback of $$\begin{aligned}
\begin{tikzcd}[column sep=small]
& T^*X\arrow{d}{\pi_X} \\
P\arrow{r} & X
\end{tikzcd}\nonumber\end{aligned}$$ then we have a $G$-equivariant map $T^*P\to \tilde P$, hence, by taking its product with the identity map on $F$, we obtain a $G$-equivariant map $$\alpha_\Theta: T^*P\times F\to \tilde P\times F.$$ Next, Weinstein observed that the restriction of $\alpha_\Theta$ to $\psi^{-1}(0)$ is a diffeomorphism; then, passing to the quotient by the action of $G$, one obtains a diffeomorphism $$\overline \alpha_\Theta: \quad \psi^{-1}(0)/G\to \tilde P\times_GF.$$ So there is a unique symplectic structure $\omega_\Theta$ on $\mathcal F^\sharp:= \tilde P\times_GF$ such that ${\overline\alpha_\Theta}^*(\omega_\Theta)=\omega$. Then $(\mathcal F^\sharp, \omega_\Theta)$ is a symplectic manifold, which is precisely the Sternberg phase space of $\Theta$ described in Ref. [@Sternberg77]. In other words, the Sternberg phase spaces form a bundle of symplectic manifolds over the space of principal connections, and this bundle is canonically isomorphic to the product bundle whose fiber is Weinstein’s universal phase space $\mathcal W$.
Sternberg phase space for $\mathrm {Sp}(1)$-Kepler problems
-----------------------------------------------------------
The classical $\mathrm {Sp}(1)$-Kepler problems (or models) are indexed by integer (called *level*) $n\ge 2$ and real number (called *magnetic charge*) $\mu\ge 0$. For the model at level $n$ and magnetic charge $\mu$, its phase space is the Sternberg phase space $\mathcal F_\mu^\sharp$ with the following data [@meng2014b]:
(i) The compact Lie group $G$ is $\mathrm{Sp}(1)$(i.e. $\mathrm{SU}(2)$) and the principal $G$-bundle is $$\begin{aligned}
\mathbb H^n_*&\to & \mathcal C_1\cr
Z&\mapsto &n ZZ^\dag \end{aligned}$$ and the principal connection form is $$\begin{aligned}
\label{principalConnection}
\Theta = {\mathpzc{Im}(\bar Z\cdot \mathrm{d}Z)\over |Z|^2}.\end{aligned}$$ Here $\mathcal C_1$ is the rank-one Kepler cone for the simple Euclidean Jordan algebra $\mathrm H_n(\mathbb H)$ of quarternonic hermitian matrices of order $n$. As a submanifold of the Euclidean space $\mathrm H_n(\mathbb H)$, $\mathcal C_1$ consists of all rank one semi-positive definite elements of $\mathrm H_n(\mathbb H)$. However, $\mathcal C_1$ is not a Riemannian submanifold of the Euclidean space $\mathrm H_n(\mathbb H)$ because the Riemannian metric on $\mathcal C_1$, called the Kepler metric, does not come from the Euclidean metric via restriction, see Ref. [@meng2014b] for the details.
(ii) For simplicity, we shall identify $\mathfrak g^*$ with $\mathfrak g:= \mathpzc{Im}\, \mathbb H $ via the standard invariant inner product $\langle, \rangle$, i.e. the one such that the imaginary units $\mathrm i$, $\mathrm j$ and $\mathrm k$ form an orthonormal basis for $\mathfrak g$. Then the Hamiltonian $G$-space is $$\begin{aligned}
F:=\{\xi\in \mathpzc{Im}\, \mathbb H \mid \xi\bar \xi =\mu^2\}\end{aligned}$$ whose symplectic form $\Omega_\mu$, being the Kirillov-Kostant-Souriau symplectic form, is given by the formulae $$\begin{aligned}
\Omega_\mu ={\langle\xi, \mathrm{d}\xi\wedge \mathrm{d}\xi \rangle\over 2|\xi|^2}. \end{aligned}$$ Note that, if we write $\xi = \xi^1\mathrm i+ \xi^2\mathrm j+\xi^3\mathrm k$, then this symplectic form yields the following basic Poisson relations on $F$: $$\begin{aligned}
\label{poissonrelation}
\{\xi^1, \xi^2\}=\xi^3, \quad \{\xi^2, \xi^3\}=\xi^1, \quad\{\xi^3, \xi^1\}=\xi^2.\end{aligned}$$ (Note: In our convention, $\mathrm i\mathrm j=\mathrm k$.)
(iii) The $G$-equivariant moment map is $$\begin{aligned}
\label{phi}
\Phi: F&\to& \mathpzc{Im}\, \mathbb H \cr
\xi &\mapsto & 2\xi \end{aligned}$$ (Here the $G$-action on both $F$ and $ \mathpzc{Im}\, \mathbb H$ is the adjoint action.) Indeed, $\Phi$ is (obviously) $G$-equivariant; moreover, with a seemingly odd factor of $2$ included in Eq. , one can verify that $$\mathrm d \langle \Phi, \eta\rangle = X_\eta\;\lrcorner\; \Omega_\mu.$$ Here $X_\eta$ is the vector field on $F$ that represents the adjoint action on $F$ by the Lie algebra element $\eta$, so $X_\eta(\xi)=(\xi, [\eta, \xi])\in T_\xi F$.
Weinstein’s Universal Phase Space for $\mathrm {Sp}(1)$-Kepler problems
-----------------------------------------------------------------------
The right action of $\mathrm{Sp}(1)$ on $\mathbb H_*^n$ is the map that sends $(Z, q)\in \mathbb H_*^n\times \mathrm{Sp}(1)$ to $Z\,[q]$ — the matrix multiplication of the column matrix $Z$ with the $1\times 1$-matrix $[q]$. The action induces a hamiltonian $\mathrm{Sp}(1)$-action on $T^*\mathbb H_*^n$ with a tautological moment map.
The canonical trivialization of $T{\mathbb H}_*^n$ yields two $\mathbb H$-valued function on $T{\mathbb H}_*^n$, i.e. the position vector $Z$ and velocity vector $W$. Via the standard inner product on $\mathbb H^n$: $$\langle U, V\rangle =\mathpzc{Re}(U^\dag V),$$ one can identify the total cotangent space $T^*{\mathbb H}^n_*$ with the total tangent space $T{\mathbb H}^n_*$, then $T{\mathbb H}^n_*$ become a Poisson manifold with the following basic Poisson relation: for any $U, V\in \mathbb H^n$, $$\{\langle U, Z\rangle, \langle V, W\rangle\}= \langle U, V\rangle, \quad \{\langle U, Z\rangle, \langle V, Z\rangle\}=\{\langle U, W\rangle, \langle V, W\rangle\} =0.$$
With the aforementioned identification of $T^*\mathbb H_*^n$ with $T\mathbb H_*^n$ and $\mathfrak g^*$ with $\mathfrak g$ as well, one can check that the moment map $\rho$ is identified with the map from $T{\mathbb H}_*^n$ to $\frak g$ that maps $(Z, W)$ to $-\mathpzc{Im} (W^\dag Z)$. Therefore, the moment map $\psi$: $T\mathbb H_*^n\times F\to \mathfrak g$ is $$\begin{aligned}
\psi(Z, W, \xi)=\mathpzc{Im} (W^\dag Z)+2\xi.\end{aligned}$$ Note that the action of $g$ maps $(Z, W, \xi)$ to $(Z\cdot g^{-1}, W\cdot g^{-1}, g\xi g^{-1})$.
The map $\psi$ has a natural extension to $\tilde \psi$: $T\mathbb H_*^n\times \mathfrak g\to \mathfrak g$ which is defined by the same formulae: $
\tilde \psi(Z, W, \xi)=\mathpzc{Im} (W^\dag Z)+2\xi.
$ Then $\tilde \psi^{-1}(0)$ is the graph of the map $\xi =-\frac{1}{2} \mathpzc{Im} (W^\dag Z)$. This map is clearly $G$-equivariant. Moreover, it is a Poisson map. Indeed, for example, $$\begin{aligned}
\{ \langle \mathrm i, W^\dag Z\rangle, \langle \mathrm j, W^\dag Z\rangle\} &=& \{ \langle W\mathrm i, Z\rangle, \langle W\mathrm j, Z\rangle\} \cr
&=& \{ \langle W\mathrm i, \contraction{}{Z}{\rangle, \langle }{W}Z\rangle, \langle W\mathrm j, Z\rangle\} -<\mathrm i \leftrightarrow \mathrm j>\cr
&=& - \{ \langle W\mathrm i, \contraction{}{Z}{\rangle, \langle }{W}Z\rangle, \langle W, Z\mathrm j\rangle\} -<\mathrm i \leftrightarrow \mathrm j> \cr
&=& - \langle W\mathrm i, Z\mathrm j\rangle-<\mathrm i \leftrightarrow \mathrm j>\cr
&=& - 2\langle \mathrm k, W^\dag Z\rangle.\nonumber\end{aligned}$$ So, in view of the fact that $\xi^1=-\frac{1}{2} \langle \mathrm i, W^\dag Z\rangle$, $\xi^2=-\frac{1}{2} \langle \mathrm j, W^\dag Z\rangle$, $\xi^3=-\frac{1}{2} \langle \mathrm k, W^\dag Z\rangle$, we have $\{\xi^1, \xi^2\}=\xi^3$, cf. Eq. .
Therefore, the $\mathrm{Sp}(1)$-equivariant projection map $T\mathbb H_*^n\times \mathfrak g\to T\mathbb H_*^n$, when restricted to $\tilde \psi^{-1}(0)$, yields a $\mathrm{Sp}(1)$-equivariant Poisson isomorphism of $\tilde \psi^{-1}(0)$ with $T\mathbb H_*^n$. Since $\psi^{-1}(0)$ is a $\mathrm{Sp}(1)$-equivariant submanifold of $\tilde\psi^{-1}(0)$, the universal phase space $$\mathcal W_\mu:=\psi^{-1}(0)/\mathrm{Sp}(1) =\left\{\mathrm{Sp}(1)\cdot (Z, W, \xi)\mid \xi =-\frac{1}{2} \mathpzc{Im} (W^\dag Z), |\xi|=\mu \right \}$$ is naturally identified with a submanifold of $T\mathbb H_*^n/\mathrm{Sp}(1)$. Indeed, as a symplectic manifold, $\mathcal W_\mu$ is naturally identified with the Poisson leave $$\left\{\mathrm{Sp}(1)\cdot (Z, W) \mid |\mathpzc{Im} (W^\dag Z)|=2\mu \right \}$$ of the Poisson manifold $T\mathbb H_*^n/\mathrm{Sp}(1)$. In other words, in view of the fact that the Sternberg phase space $\mathcal F_\mu^\sharp$ can be identified with the Weinstein’s universal phase space $\mathcal W_\mu$, *the totality of Sternberg phase space can be identified with the Poisson manifold $T\mathbb H_*^n/\mathrm{Sp}(1)$*.
Part (ii) of Lemma \[key lemma\] in the next section says that *the conformal algebra of $\mathrm H_n(\mathbb H)$ has a Poisson realization on the Poisson manifold $T\mathbb H_*^n/\mathrm{Sp}(1)$*, hence on both the universal phase space $\mathcal W_\mu$ and the Sternberg phase space $\mathcal F_\mu^\sharp$.
Dynamical Symmetry
==================
Let us fix a $\mathrm{Sp}(1)$-Kepler problem, say at level $n$ and with magnetic charge $\mu$. Its phase space, being the Sternberg phase space $\mathcal F_\mu^\sharp$, fibers over $T^*\mathcal C_1$. Via the metric on the Euclidean space $V:=\mathrm H_n(\mathbb H)$, $T^*\mathcal C_1$ can be identified with $T\mathcal C_1$, so $\mathcal F_\mu^\sharp$ fibers over $T\mathcal C_1$ as well. Let functions $x$ and $\pi$ (taking value in vector space $V$) be defined via diagram $$\begin{aligned}
\begin{tikzcd}[column sep=small]
&T\mathcal C_1 \arrow{d}[description]{\iota}\arrow{ldd}[swap]{x}\arrow{rdd}{\pi} & \\
&TV\arrow{ld}[description]{\tau_V}\arrow{rd} [description]{t}& \\
V&&V
\end{tikzcd}\nonumber\end{aligned}$$ where $\iota$ is the inclusion map and $\tau_V$ is the tangent bundle projection and $t$ is the natural trivialization map of the tangent bundle of the affine space $V$. The pullback of $x$ and $\pi$ under the bundle map $\mathcal F_\mu^\sharp\to T\mathcal C_1$ shall still be denoted by $x$ and $\pi$. Recall that the inner product on the Jordan algebra $\mathrm H_n(\mathbb H)$ is denoted by $\langle \,\mid\, \rangle$, so if $u\in \mathrm H_n(\mathbb H)$, then $\langle x\mid u\rangle$ is a real function on $\mathcal F_\mu^\sharp$.
\[main theorem\] For the conformal algebra of the Jordan algebra $\mathrm H_n(\mathbb H)$, there is a unique Poisson realization on the Sternberg phase space $\mathcal F^\sharp_\mu$ such that $Y_u$ is realized as function $\mathcal Y_u:=\langle x\mid u\rangle$ and $X_e$ is realized as function $\mathcal X_e:=\langle x\mid \pi^2\rangle +\frac{\mu^2}{ \langle e\mid x\rangle}$. Moreover, if $(e_\alpha)$ is an orthonormal basis for $\mathrm H_n(\mathbb H)$ and $\mathcal L_u$ represents $L_u$ in this Poisson realization, we have the following primary quadratic relation: $$\begin{aligned}
{2\over n}\sum_\alpha \mathcal L_{e_\alpha}^2 =\mathcal L_e^2 + \mathcal X_e \mathcal Y_e -\mu^2.\end{aligned}$$
The uniqueness of this Poisson realization (if it exists) is clear, that is because $X_e$ and $Y_u$ generate the conformal algebra.
As for the existence of this Poisson realization, the direct verification is very complicated. Since Sternberg phase space can be identified with Weinstein’s universal phase space via diffeomorphism $$\begin{aligned}
\overline \alpha_\Theta: \quad \mathcal W_\mu:=\psi^{-1}(0)/\mathrm{Sp}(1)\to \mathcal F_\mu^\sharp:=\tilde P\times_{\mathrm{Sp}(1)} F, \end{aligned}$$ one just needs to verify the existence of the corresponding Poisson realization on Weinstein’s universal phase space $\mathcal W_\mu$, a task which turns to be much simpler.
\[key lemma\]
1. Let $\mathrm{Sp}(1)\cdot (Z, W, \xi)$ be an element of $\mathcal W_\mu$, and let $\mathcal Y_u$ and $\mathcal X_e$ be the functions defined in Theorem \[main theorem\] . Then $$\mathcal Y_u\circ \overline \alpha_\Theta (\mathrm{Sp}(1)\cdot (Z, W, \xi)) = \langle Z, uZ\rangle, \quad \mathcal X_e\circ \overline \alpha_\Theta (\mathrm{Sp}(1)\cdot (Z, W, \xi)) =\frac{1}{4}|W|^2.$$
2. For any vectors $u$, $v$ in $V:=\mathrm{H}_n(\mathbb H)$, define $\mathrm{Sp}(1)$-invariant functions $$\begin{aligned}
\left\{
\begin{array}{rcl}
\mathscr X_u &:=& \frac{1}{4} \langle W, uW\rangle \\[2mm]
\mathscr Y_v &:=& \langle Z, vZ\rangle\\[2mm]
\mathscr S_{uv} &:=& \frac{1}{2} \langle W, (u\cdot v) Z \rangle
\end{array}\right.\end{aligned}$$ on $T{\mathbb H}_*^n$. Here $u\cdot v$ means the matrix multiplication of $u$ with $v$. Then, for any vectors $u$, $v$, $z$, $w$ in $V$, the following Poisson bracket relations hold: $$\begin{aligned}
\left\{
\begin{matrix}
\{\mathscr X_u, \mathscr X_v\} =0, \quad \{\mathscr Y_u, \mathscr Y_v\}=0, \quad \{\mathscr X_u,
\mathscr Y_v\} = -2\mathscr S_{uv},\cr\\
\{\mathscr S_{uv}, \mathscr X_z\}=\mathscr X_{\{uvz\}}, \quad \{\mathscr S_{uv}, \mathscr Y_z\} = -\mathscr Y_{\{vuz\}},\cr\\
\{\mathscr S_{uv}, \mathscr S_{zw}\} = \mathscr S_{\{uvz\}w}-\mathscr S_{z\{vuw\}}.
\end{matrix}\right.\nonumber\end{aligned}$$ Consequently, we have a Poisson realization on the Poisson manifold $T^*{\mathbb H}_*^n/\mathrm{Sp}(1)$ for the conformal algebra of the Jordan algebra $\mathrm H_n(\mathbb H)$.
3. Let $e$ be the identity element of $V$, $(e_\alpha)$ be an orthonormal basis for $V$, and $\mathscr L_u=\mathscr S_{eu}$ for any $u\in V$. Then $$\begin{aligned}
{2\over n}\sum_\alpha \mathscr L_{e_\alpha}^2=\mathscr L_e^2 + \mathscr X_e \mathscr Y_e -\mu^2.\end{aligned}$$
<!-- -->
1. To understand the map $\overline \alpha_\Theta$, we need to figure out the horizontal lift induced from the connection $\Theta$. For $Z\in \mathbb H_*^n$, we let $x=nZZ^\dag$. Suppose that $(x, \dot x)$ is a tangent vector of $\mathcal C_1$ at $x$, and $(Z, \dot Z)$ is its horizontal lift to point $Z$ in $\mathbb H_*^n$. Then, in view of Eq. , we have equations $$n(\dot Z Z^\dag + Z\dot Z^\dag) =\dot x, \quad \mathpzc{Im} (Z^\dag \dot Z) =0.$$ By solving these equations jointly, we obtain $$\dot Z = \frac{1}{n|Z|^2}(\dot x Z -\frac{{{\mathrm}{tr}\,}\dot x}{2}Z).$$
Suppose that the $\Theta$-induced map $T^*\mathbb H_*^n\to T^*\mathcal C_1$ maps $(Z, \langle W, \, \rangle)$ to $(x, \langle \pi\mid\;\rangle)$, then $x=nZZ^\dag$ and $$\begin{aligned}
\label{hlift}
\langle \pi\mid \dot x\rangle = \frac{1}{n|Z|^2}\left\langle W, \dot x Z -\frac{{{\mathrm}{tr}\,}\dot x}{2}Z\right\rangle.\end{aligned}$$ In particular, in view of the fact that $(x, ux)\in T_x\mathcal C_1$, we have $$\begin{aligned}
\label{simpleformula}
\langle \pi\mid ux\rangle &= & \frac{1}{n|Z|^2}\left\langle W, (ux) Z -\frac{{{\mathrm}{tr}\,}(ux)}{2}Z\right\rangle\quad \text{$ux$ is the Jordan product of $u$ with $x$}\cr
&=& \frac{1}{2|Z|^2}\left\langle W, uZZ^\dag Z+ZZ^\dag uZ - \mathpzc{Re}\,{{\mathrm}{tr}\,}(uZZ^\dag)Z\right\rangle\cr
&=&\frac{1}{2}\langle W, uZ\rangle.\end{aligned}$$ Then $$\begin{aligned}
\overline \alpha_\Theta (G\cdot (Z, W, \xi))=G\cdot (Z, nZZ^\dag, \pi, \xi).\end{aligned}$$ Consequently $$\mathcal Y_u\circ \overline \alpha_\Theta (G\cdot (Z, W, \xi))=\langle x\mid u\rangle =\langle nZZ^\dag\mid u\rangle =\mathpzc{Re}\, {{\mathrm}{tr}\,}(ZZ^\dag u)=\langle Z, uZ\rangle$$ and $$\begin{aligned}
\mathcal X_e\circ \overline \alpha_\Theta (G\cdot (Z, W, \xi)) &= & \langle x\mid \pi^2\rangle +\frac{\mu^2}{ \langle e\mid x\rangle}=\langle \pi\mid \pi x\rangle +\frac{n\mu^2}{ {{\mathrm}{tr}\,}x}\cr
&=& \frac{1}{2}\langle W, \pi Z\rangle+\frac{\mu^2}{|Z|^2}\quad{\text{using Eq.}\, \eqref{simpleformula}}\cr
&=& \frac{n}{2}\langle \pi\mid (ZW^\dag)_+\rangle+\frac{\mu^2}{|Z|^2}. \nonumber\end{aligned}$$ Here $(ZW^\dag)_+=\frac{1}{2}(ZW^\dag+WZ^\dag)$. One can check that $(x, (ZW^\dag)_+)$ is a tangent vector of $\mathcal C_1$ at $x$. Therefore, in view of Eq. , we have $$\begin{aligned}
\mathcal X_e\circ \overline \alpha_\Theta (G\cdot (Z, W, \xi))
&=&\frac{1}{2|Z|^2}\left\langle W, (ZW^\dag)_+ Z -\frac{{{\mathrm}{tr}\,}(ZW^\dag)_+}{2}Z\right\rangle+\frac{\mu^2}{|Z|^2}\cr
&=&\frac{1}{2|Z|^2}\left\langle W, (ZW^\dag)_+ Z -\frac{\langle W, Z\rangle}{2}Z\right\rangle+\frac{\mu^2}{|Z|^2} \cr
&=&\frac{1}{2|Z|^2}\left\langle W, (ZW^\dag)_+ Z\right\rangle - \frac{1}{4|Z|^2}\langle W, Z\rangle^2+\frac{\mu^2}{|Z|^2}\cr
&=&\frac{1}{4|Z|^2}\left(|W|^2|Z|^2+ \mathpzc{Re}\, (W^\dag Z)^2 -\langle W, Z\rangle^2\right)+\frac{\mu^2}{|Z|^2}\cr
&=&\frac{1}{4|Z|^2}\left(|W|^2|Z|^2+ (\mathpzc{Im} (W^\dag Z) )^2\right)+\frac{\mu^2}{|Z|^2}\cr
&=&\frac{1}{4|Z|^2}\left(|W|^2|Z|^2 - |\mathpzc{Im} (W^\dag Z)|^2\right)+\frac{\mu^2}{|Z|^2}\cr
&=&\frac{1}{4}|W|^2.\nonumber
\end{aligned}$$
2. It is clear that $\{\mathscr X_u, \mathscr X_v\} =0$ and $\{\mathscr Y_u, \mathscr Y_v\}=0$. Next, we have $$\begin{aligned}
\{\mathscr X_u, \mathscr Y_v\}& =&\frac{1}{4} \left \{ \langle W, uW\rangle, \langle Z, vZ\rangle\right \}\cr
&=&\left \{\langle \contraction{}{W}{, uW\rangle, \langle} {Z}W, uW\rangle, \langle Z, vZ\rangle\right \} = -\langle uW, vZ\rangle \cr
&=& -2\mathscr S_{uv}\nonumber\end{aligned}$$ and $$\begin{aligned}
\{\mathscr S_{uv}, \mathscr S_{zw}\} & =&\frac{1}{4} \left \{ \langle W, (u\cdot v) Z \rangle, \langle W, (z \cdot w) Z \rangle\right \}\cr
&=& \frac{1}{4} \left \{ \langle \contraction{}{W}{, (u\cdot v) Z \rangle, \langle W, (z\cdot w) }{Z}W, (u\cdot v) Z \rangle, \langle W, (z\cdot w) Z \rangle\right \} -<(u\cdot v)\leftrightarrow (z\cdot w)>\cr
&=& -\frac{1}{4}\langle (z\cdot w)^\dag W, (u\cdot v)Z\rangle + \frac{1}{4} \langle (u\cdot v)^\dag W, (z\cdot w)Z\rangle \cr
&=&\frac{1}{4}\langle W, [u\cdot v, z\cdot w]Z\rangle\cr
&=& \mathscr S_{\{uvz\}w}-\mathscr S_{z\{vuw\}}\nonumber\end{aligned}$$ because $\{uvz\}=\frac{1}{2}(u\cdot v\cdot z+z\cdot v\cdot u)$ and $\{vuw\}=\frac{1}{2}(v\cdot u\cdot w+w\cdot u\cdot v)$.
Thirdly, $$\begin{aligned}
\{\mathscr S_{uv}, \mathscr X_{z}\} & =&\frac{1}{8} \left \{ \langle W, (u\cdot v) Z \rangle, \langle W, z W \rangle\right \} = \frac{1}{4} \left \{ \langle W, (u\cdot v) \contraction{}{Z}{\rangle, \langle }{W}Z \rangle, \langle W, z W \rangle\right \} \cr
&=& \frac{1}{4}\langle W, (u\cdot v\cdot z)W\rangle = \frac{1}{4}\langle W, (z\cdot v\cdot u)W\rangle \cr
&=&\frac{1}{8}\langle W,(z\cdot v\cdot u+u\cdot v\cdot z)W\rangle\cr
&=& \mathscr X_{\{uvz\}}.\nonumber\end{aligned}$$
Finally, we have $$\begin{aligned}
\{\mathscr S_{uv}, \mathscr Y_{z}\} & =&\frac{1}{2} \left \{ \langle W, (u\cdot v) Z \rangle, \langle Z, z Z \rangle\right \} = \left \{ \langle \contraction{}{W}{, (u\cdot v)Z \rangle, \langle }{Z}W, (u\cdot v)Z \rangle, \langle Z, z Z \rangle\right \} \cr
&=& - \langle Z, (z\cdot u\cdot v)Z \rangle = - \langle Z, (v\cdot u\cdot z)Z \rangle\cr
&=&-\frac{1}{2}\langle Z, (z\cdot u\cdot v+v\cdot u\cdot z)Z\rangle\cr
&=& -\mathscr Y_{\{vuz\}}.\nonumber\end{aligned}$$
Since functions $\mathscr S_{uv}$, $\mathscr X_z$ and $\mathscr Y_w$ on $T{\mathbb H}_*^n$ (actually on $T^*{\mathbb H}_*^n$) are $\mathrm{Sp}(1)$-invariant, and the action of $\mathrm{Sp}(1)$ on $T^*{\mathbb H}_*^n$ is symplectic, we have a Poisson realization on the Poisson manifold $T^*{\mathbb H}_*^n/\mathrm{Sp}(1)$ for the conformal algebra of the Jordan algebra $\mathrm H_n(\mathbb H)$.
3. Since $\mathscr L_u=\frac{1}{2}\langle W, uZ\rangle$, we have $$\begin{aligned}
{2\over n}\sum_\alpha \mathscr L_{e_\alpha}^2& =&\frac{1}{2n} \sum_\alpha \langle W, e_\alpha Z\rangle^2 = \frac{n}{2} \sum_\alpha \langle (ZW^\dag)_+\mid e_\alpha \rangle^2 \cr
&= & \frac{n}{2} \langle (ZW^\dag)_+\mid (ZW^\dag)_+ \rangle = \frac{1}{2} {{\mathrm}{tr}\,}\left ((ZW^\dag)_+\right )^2\cr
&=& \frac{1}{8} {{\mathrm}{tr}\,}\left (ZW^\dag+WZ^\dag\right )^2\cr
&=& \frac{1}{8} {{\mathrm}{tr}\,}(Z(W^\dag ZW^\dag)+(WZ^\dag W)Z^\dag+|W|^2 ZZ^\dag+|Z|^2WW^\dag)\cr
&=& \frac{1}{8} ( (W^\dag Z)^2+(Z^\dag W)^2)+\frac{1}{4}|W|^2 |Z|^2\cr
&=& \frac{1}{16} (W^\dag Z-Z^\dag W)^2+ \frac{1}{16} (W^\dag Z+Z^\dag W)^2+\frac{1}{4}|W|^2 |Z|^2\cr
&=& -\frac{1}{4} |\mathpzc{Im} (W^\dag Z)|^2+\frac{1}{4}\langle W, Z\rangle^2+\frac{1}{4}|W|^2 |Z|^2\cr
&=& -\mu^2 +\mathscr L_e^2+\mathscr X_e\mathscr Y_e.\nonumber\end{aligned}$$
. As we remarked early that the Poisson realization is obviously unique, if it exists. From part (ii) of the above lemma, we know that there is a Poisson realization for $\frak{so}^*(4n)$ on the Poisson manifold $T^*{\mathbb H}_*^n/\mathrm{Sp}(1)$, hence on each of its symplectic leave. Since a Sternberg phase space is symplectic equivalent to a symplectic leave of $T^*{\mathbb H}_*^n/\mathrm{Sp}(1)$, in view of part (i) of the above lemma, one obtains the first part of Theorem \[main theorem\]. The remaining part of Theorem \[main theorem\] follows trivially from part (iii) of the above lemma.
The Poisson realization in Theorem \[main theorem\] is called the dynamical symmetry for the $\mathrm{Sp}(1)$-Kepler problem at level $n$ and with magnetic charge $\mu$.
The natural identification of Sternberg phase spaces with symplectic leaves of $T^*{\mathbb H}_*^n/\mathrm{Sp}(1)$ is a bijection, so the totality of Sternberg phase spaces is naturally equivalent to the Poisson manifold $T^*{\mathbb H}_*^n/\mathrm{Sp}(1)$. This really indicates that $\mathrm{Sp}(1)$ Kepler problems can be obtained via symplecic-reduction from a model whose phase space is $T^*{\mathbb H}_*^n$ and whose hamiltonian is $\mathrm{Sp}(1)$-invariant, namely the $n$th quaternionic conformal Kepler problem in section 6 of Ref. [@meng2014b].
In view of Ref. [@meng2014'], Theorem \[main theorem\] implies that the corresponding $\mathrm{Sp}(1)$-Kepler problem is the Hamiltonian system with phase space $T\mathcal C_1$, Hamiltonian $$H={1\over 2} {\mathcal X_e\over \mathcal Y_e} - {1\over \mathcal Y_e}$$ and Laplace-Runge-Lenz vector $$\mathcal A_u={1\over 2}\left(\mathcal X_u - \mathcal Y_u {\mathcal X_e\over \mathcal Y_e}\right)+{\mathcal Y_u\over \mathcal Y_e}$$ where $\mathcal X_u$ and $\mathcal Y_u$ are the functions that represent $X_u$ and $Y_u$ respectively in the Poisson realization in Theorem \[main theorem\]. Indeed, a simple computation yields $$H=\frac{1}{2}\frac{\langle x|\pi^2\rangle}{r} +{\mu^2\over 2r^2} -\frac{1}{r},$$ i.e., the Hamiltonian in Definition 1.1 of Ref. [@meng2014b]. Here $r=\frac{{{\mathrm}{tr}\,}x}{n}$.
Quadratic Relations and Energy Formulae
=======================================
For the Poisson realization in Theorem \[main theorem\], let us assume that the elements of the conformal algebra such as $S_{uv}$, $X_z$ and $Y_w$ are realized as functions $$\mathcal S_{u, v}, \quad \mathcal X_z, \quad \mathcal Y_w$$ respectively. We shall use $\mathcal L_u$ to denote $\mathcal S_{ue}$ and $\mathcal L_{u, v}$ to denote $\frac{1}{2}(\mathcal S_{uv}+\mathcal S_{vu})$.
The main purpose of this section is to list two corollaries of Theorem \[main theorem\], one concerning the secondary quadratic relations and one concerning a formula connecting the Hamiltonian to the angular momentum and the Laplace-Runge-Lenz vector. The proof can be taken verbatim from the last section of Ref. [@BM2015].
\[QRelations\] Let $e_\alpha$ be an orthonormal basis for ${\mathrm H}_n(\mathbb H)$. In the following we hide the summation sign over $\alpha$ or $\beta$. For the Poisson realization in Theorem \[main theorem\], we have the following secondary quadratic relations:
(i) $\mathcal X_{e_\alpha} \mathcal L_{e_\alpha}=n\mathcal X_e \mathcal L_e$, $\mathcal Y_{e_\alpha} \mathcal L_{e_\alpha} = n\mathcal Y_e \mathcal L_e$,\
(ii) ${4\over n}\mathcal L_{e_\alpha, u} \mathcal L_{e_\alpha}=-\mathcal X_u\mathcal Y_e+\mathcal X_e \mathcal Y_u$,\
(iii) $\mathcal X_{e_\alpha}^2 =n\mathcal X_e^2$, $\mathcal Y_{e_\alpha}^2 =n\mathcal Y_e^2$,\
(iv) ${2\over n} \mathcal L_{e_\alpha, u} \mathcal X_{e_\alpha}=-\mathcal X_u \mathcal L_e+\mathcal L_u \mathcal X_e$, ${2\over n}\mathcal L_{e_\alpha,u} \mathcal Y_{e_\alpha}=\mathcal Y_u \mathcal L_e-\mathcal L_u\mathcal Y_e$,\
(v) $\mathcal X_{e_\alpha} \mathcal Y_{e_\alpha}= n (\mathcal L_e^2 + \mu^2),
$\
(vi) $\frac{4}{n^3}\mathcal L_{e_\alpha, e_\beta}^2=\mathcal X_e \mathcal Y_e-\mathcal L_e^2+\frac{n-2}{n}\mu^2$.
Let $e_\alpha$ be an orthonormal basis for ${\mathrm H}_n(\mathbb H)$, $L^2={1\over 2}\sum_{\alpha, \beta} \mathcal L_{e_\alpha, e_\beta}^2$, and $A^2=-1+\sum_\alpha \mathcal A_{e_\alpha}^2$. Then the Hamiltonian $H$ satisfies the relation $$\begin{aligned}
\label{HLA}
-2H\left (L^2- \frac{n^2(n-1)}{4}\mu^2\right ) = \left(\frac{n}{2}\right)^2(n-1-A^2).\end{aligned}$$
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[^1]: The authors were supported by the Hong Hong Research Grants Council under RGC Project No. 16304014; SB was also supported by the grant NYUAD-065.
|
---
author:
- 'Linpeng Cheng , Yongheng Zhao and Jianyan Wei'
date: CJAA
title: ' Variability of Soft X-ray Spectral Shape in Blazars Observed by ROSAT'
---
Introduction
============
Blazars, including BL Lac objects, highly polarized and optically violently variable quasars, and flat-spectrum radio quasars (FSRQs), are characterized by highly variable non-thermal emission which dominates their characteristics from the radio through the $\gamma$-rays. The mechanism believed to be responsible for their broadband emission is synchrotron radiation followed by inverse Compton (IC) scattering at higher energies (e.g. Blandford & Konigl [@Blandford1979]). Relativistic beaming of a jet viewed at very small angles is the most natural explanation for the extreme properties of the class, including violent variability (up to 1-5 magnitudes in the optical; see Wagner & Witzel [@Wagner1995]), high $\gamma$-ray luminosities in some cases (Mukherjee et al. [@Mukherjee1997]), superluminal motion (Vermeulen & Cohen [@Vermeulen1994]), and high optical and radio polarization, sometimes extending up to $\rm 10\%$ (Catanese & Sambruna [@Catanese2000]). In addition, the multiwavelength spectra of blazars usually show two peaks. The first one peaks at infrared to X-ray energies and is most probably from synchrotron radiation, which originates from electrons in a relativistic jet pointing close to the line of sight. The second peaks at $\gamma$-ray band from GeV to TeV energies and is dominated by inverse Compton emission from low-frequency seed photons (Georganopoulos [@Georganopoulos2000]), which may be the synchrotron photons themselves (Sychrotron self-Compton radiation (SSC)), UV photons coming from a nearby accretion disk or from the broad-line region (Sambruna et al. [@Sambruna1995]). However, the origin of the high energy emission is still a matter of considerable debate (e.g. Buckley [@Buckley1998]).
In X-rays, blazars not only exhibit large amplitude variability, but also show significant spectral variations with respect to intensity changes. For example, the spectrum of BL Lac PKS 2005-489 by ROSAT observation softens with decreasing flux (Sambruna et al. [@Sambruna1995]). EXOSAT observations also indicate its spectral steepening during flux decrease, a behavior often displayed by other X-ray strong BL Lac objects (Sambruna et al. [@Sambruna1994a]). What is more, a similar X-ray spectral variability trend is that their spectrum becomes harder as overall flux increases, especially during a flare state. This trend has been consistently found by Chiappeiti et al. ([@Chiappeiti1999]), Perlman et al. ([@Perlman1999]), Brinkmann ([@Brinkmann2001]) and Romerto et al. ([@Romerto2000]), although in different energy bands. A possible explanation based on an inhomogeneous jet model is that spectral hardening with rising intensity is caused by either the ejection of particles into the jet or particle acceleration, and that the spectral steepening was the result of synchrotron cooling (Perlman et al. [@Perlman1999]; Sambruna et al. [@Sambruna1995]).
The aim of this paper is to show what spectral variation is typical of blazars and to give some discussion or interpretation for their spectral variability. Here we present a complete spectral shape variability analysis of blazars observed by ROSAT/PSPC through the same analysis method as paper I (Cheng et al. [@Cheng2001], hereafter paper I). The observations and data reduction are described briefly in Sec. 2, and our results are presented in Sec. 3. In Sec. 4 we discuss possible interpretation for different spectral variations. \[sect:intro\]
The Observations and data reduction {#sect:Obs}
===================================
All the blazars were observed by ROSAT/PSPC mode during the periods from days to years. Besides 9 sources selected by the criteria in paper I, we selected some more blazars, including BL Lac objects with optical polarization $\rm < 3\%$ (BLs) and high optical polar ization ($\rm > 3\%$) blazars (HPs), from cross identification of veron (2000)’s AGN catalogue with ROSAT pointed catalog. Applying the ROSAT public archive of PSPC observations, the sources with average Count Rates (CTs) more than 0.05 $\rm s^{-1}$ are selected so that the error of data points is moderate. This yielded 25 blazars. The datasets are then processed for instrumental corrections and background subsection using the EXSAS/MIDAS software.
The light curve for each blazar is obtained from original ROSAT datasets with time binning of 400 seconds in three energy bands: 0.1-2.4 keV (overall band), 0.1-0.4 keV ( A band), 0.5-2.0 keV (B band). Then we pick up nine of twenty-five objects by the following criteria: 1) for each source the ratio of maximal CTs to minimal CTs is greater than 2, which assures that the range of CTs variability is large enough; 2) the data points are not too scarce ($>5$) and they distribute in one diagram consecutively; 3) HR1 error is small ($<$ 40$\%$). These sources include five BLs and four HPs. Thus 18 blazars are included in our sample.
Results of spectral shape variability Analysis
==============================================
All these X-ray count rates in 0.1-2.4 keV band were gained from original ROSAT observations with time binning of 400 s. In addition, 4 energy bands are shown: A 0.1-0.4 keV, B 0.5-2.0 keV, C 0.5-0.9 keV, D 0.9-2.0 keV. The standard hardness ratios, HR1 and HR2, for ROSAT-PSPC data are defined as: $$HR1 = \frac{B-A}{B+A}, HR2 = \frac{C-D}{C+D}$$ In order to describe the spectral shape variability, we present the HR1-CTs correlation for 18 blazars, as shown in Figure 1, and the results are listed in Table 1. To distinguish different variation trend of each object we fit the data through a linear formula (HR1=a+b$\times$CTs): when the slope b is a positive or negative value and its relative error is less than 50%, we think it has a positive or negative correlation; the other instances are of random or no clear correlation. These correlations are summarized as the following:
1. For 18 blazars in our sample, ten of them which include 5 BLs and 5 HPs, show a positive HR1-CTs correlation in the sense that the spectrum hardens as the overall flux increases, in common with most of previous observations in blazars.
2. There are 6 objects, 3 BLs and 3 HPs, displaying random variation of the HR1 versus CTs relation. In other words, their spectra do not exhibit a clear softening or hardening trend with increasing flux.
3. Two exceptional sources, HP S5 1803+78 and possible BL Lac object 1207+39W4, indicate an anti-correlation of the HR1 versus CTs, implying that their spectrum steepens with rising intensity, a behavior rarely observed in blazars.
4. Considering the HPs and BLs separately in Table 1, we can see that the overall photon index decreases from the BLs with a positive HR1-CTs correlation to those BLs showing random relation of HR1-CTs. On the other hand, the HPs show an opposite trend of the photon index change to the BLs: the HPs with a positive correlation to those displaying random HR1-CTS correlation exhibit a sequence of increasing soft X-ray slope. The average photon index of different subgroups is 2.70$\pm$0.21, 2.38$\pm$0.40, 2.00$\pm$0.23, 2.58$\pm$0.37 for the BLs with a positive HR1-CTs and random correlation, and the HPs showing a positive and random relation of HR1 versus CTs, respectively. It appears that the two groups, HPs and BLs, though attributed to the same class blazars, behave differently.
----------------- ----------------- ------------ ------------- ------- ------ ---------------------- ------------- -- -- -- --
Name ROSAT name RA DEC z Type $\rm \Gamma_{rosat}$ HR1-CTs
( 1RXPJ) (2000) (2000) correlation
RX J0916$+$52 091648$+$5239.3 09 16 53.5 52 38 28 0.190 BL 2.82 Positive
1E S1212$+$078 121510$+$0732.0 12 15 10.9 07 32 02 0.136 BL 2.61 Positive
PKS 2005$-$489 200924$-$4849.7 20 09 24.8 $-$48 49 45 0.071 BL 2.92 Positive
MS 03313$-$3629 033312$-$3619.8 03 33 12.3 $-$36 19 50 0.308 BL 2.38 Positive
S5 0716$+$71 072152$+$7120.4 08 41 24.4 70 53 41 0.000 BL 2.77 Positive
MS 1332$-$2935 133531$-$2950.5 13 35 30.3 $-$29 50 42 0.250 BL 2.10 None
2E 0336$-$2453 033813$-$2443.6 03 38 13.2 $-$24 43 42 0.251 BL 2.21 None
1631.9$+$3719 163338$+$3713.3 16 33 38.2 37 13 13 0.115 BL 2.84 None
1207$+$39W4 121026$+$3929.0 12 10 26.7 39 29 10 0.610 BL? 2.11 Negative
PG 1218$+$304 122120$+$3010.1 12 21 20.7 30 10 10 0.182 HP 2.21 Positive
3A 1218$+$303 122122$+$3010.5 12 21 21.9 30 10 36 0.000 HP 2.28 Positive
1E1552$+$2020 155424$+$2011.2 15 54 24.6 20 11 47 0.222 HP 1.89 Positive
3C 454.3 225357$+$1608.7 22 53 57.6 16 08 53 0.859 HP 1.73 Positive
3C 345.0 164258$+$3948.5 16 42 58.7 39 48 37 0.594 HP 1.89 Positive
B2 1215$+$30 121752$+$3006.7 12 17 52.1 30 07 00 0.000 HP 3.00 None
MS 12218$+$2452 122422$+$2436.1 12 24 22.9 24 36 11 0.218 HP 2.46 None
2E 1415$+$2557 141757$+$2543.5 14 17 57.5 25 43 35 0.237 HP 2.2 None
S5 1803$+$78 180042$+$7827.9 18 00 42.4 78 27 57 0.680 HP 2.26 Negative
----------------- ----------------- ------------ ------------- ------- ------ ---------------------- ------------- -- -- -- --
DISCUSSION AND CONCLUSIONS
==========================
The variation in the spectral index during the overall flux change can provide insights into the emission mechanism and physical conditions of the group blazars. As found previously, BL Lacs show a general hardening of the spectrum during their flares and a spectral steepening with fading intensity (Perlman et al. [@Perlman1999]; Sambruna et al [@Sambruna1995]). Instead of soft X-ray photon index, here we have presented the correlation of hardness ratio versus count rates to describe the spectral variability. Among our sample, ten of eighteen blazars show a hardening spectrum during total flux enhance and 6 objects do not exhibit evident spectral variance trend or have random variations. The only two exceptions, 1207+39W4 and S5 1803+78, soften with increasing flux. These results are consistent with what have been described above. The fact that two particular objects indicate softening spectrum during the intensity increase was also observed in PKS 2155-304 by Sembay et al. ([@Sembay1993]). Next we will discuss the implications and possible interpretations with respect to different spectral variations.
There is a general consensus that the multifrequency continuum from blazars at least up to the UV band is due to synchrotron radiation from high-energy electrons within a relativistic jet (e.g. Konigl 1989). The “curved” shape of the continuum may be due to the superposition of different emission regions with different particle spectra (inhomogeneous models), or to curvature of the particle spectrum within a single emission location or both (Ghisellini, Maraschi & Treves [@Ghisellini1985]). On the other hand, the $\gamma$-ray band is widely accepted to come from inverse Compton scattering emission based on inhomogeneous or homogenous models (Georganopoulos [@Georganopoulos2000]). In terms of the X-ray band, two different radiation components share and the relative contribution varies with various blazars and different energy states (Cappi et al. [@Cappi1994]).
As stated above, most objects in our sample exhibit a hardening soft X-ray spectrum with increasing intensity, a behavior often displayed in other X-ray band. That might be a typical feature of the class blazars. In the framework of the inhomogeneous SSC model, Sambruna et al. ([@Sambruna1995]) have given a good fit to the broadband energy distribution of the normal BL Lac object PKS 2005-489 in its high and low state, respectively. In Fig. 5 of Sambruna et al. ([@Sambruna1995]) it is evident that the soft X-ray spectrum could be fitted well by a single synchrotron emission and becomes steeper in the low state than in the high state. In addition, the similar spectral flattening with increasing intensity can be well fitted and explained by a single IC radiation (Madejeski et al. [@Madejeski1999]). From these, we can see that the X-ray energy spectrum of BL Lac objects consistently becomes harder during intensity rise when the energy band is dominated either by a single synchrotron emission or by IC radiation, which exactly explained the spectral hardening during the overall flux increase. Assuming the model applicable to other blazars, the main spectral variation in this paper could be well interpreted. At the same time, it suggest that the variable slope and flux of the X-rays may be due to a change in the electron distribution function in the inner part of the jet. Possible mechanism to change electron distribution is the injection of particle into the jet or in situ particle acceleration.
Besides, there are 6 objects, which show random spectral variability in the sense that the spectrum does not indicate a clear change trend with varying intensity. Two possible explanations have been proposed. The first one is that the observational span time is not suitable, which would constrain the flux and spectral variation analysis. The more probable interpretation is that the soft X-ray energy distribution of these blazars is shared by two radiative components, the synchrotron and inverse Compton emissions. According to radiation theories of blazars, two components can present different spectral steepness and variation with changing flux, and thus the blended spectrum might display a complex spectral variability though the overall intensity increases significantly. To our surprise, two particular objects in our sample showed a spectral steepening with rising intensity, a phenomenon rarely observed in blazars. Up to now there are only a few cases: spectral softening during flux increase has been seen twice in PKS 2155-304 (Sembay et al. [@Sembay1993]); S5 0716+714 (Giommi et al. [@Giommi1999b]) and AO 0235+164 (Madejeski et al. [@Madejeski1996]) exhibited X-ray spectral steepening in their flare states. As indicated in Table 1, one of the two blazars, 1207+39W4, is still a possible BL Lac object and further identification should determine if it is a peculiar blazar. The rest one S5 1803+78, a high optical polarization source, displays the spectral variation similar to the intermediate-energy peaked BL Lacs (IBLs) S5 0716+714 and AO 0235+164. Correspondingly, as described by Perlman et al. ([@Perlman1999]) the spectral steepening is probably because the X-ray spectrum is dominated by the very flat inverse Compton scattering radiation in the low state, but by the soft “tail” of the steep synchrotron emission in the high state.
It is interesting to note that the BLs displaying a steepening spectrum with increasing flux statistically have a steeper soft X-ray spectrum than those showing no clear spectral variation trend while the HPs indicate a trend opposite to the BLs. For blazars it is well accepted that the slope from the synchrotron radiation is much steeper than that of the IC emission, and that the X-ray energy distribution of high energy-peaked blazars is dominated by the synchrotron emission while the IC radiation prepondered the energy band for low frequency-peaked blazars (Perlman et al. [@Perlman1999]). Moreover, it is revealed that the blazars with low optical polarization generally show relatively high peaked frequency (Scarpa & Falomo [@Scarpa1997]; Padovani & Gliommi [@Padovani1996]). As described above, the slope difference between the BLs and HPs could be interpreted below: the soft X-ray spectra of the BLs with a hardening spectrum during the overall intensity enhances are dominated by steep synchrotron radiation, in contrast, those of the HPs indicating the same spectral flattening are mainly attributed to relatively flat IC emission; for the BLs and HPs showing random spectral variations and a softening spectrum with rising intensity, the energy band at 0.1 $<$ E $<$ 2.4 keV should be dominated by the combination of the synchrotron and IC radiation. Thus, the photon index of the BLs varies differently from that of the HPs from the objects exhibiting a hardening spectrum to random spectral variation during overall intensity increase. Further broad-band energy distribution analysis would give a detailed description to the two dichotomous properties.
From the discussions above, it appears that the three groups of blazars represent relative dominance of the synchrotron and IC radiation in the soft X-ray band. The BLs exhibiting a positive HR1-CTs correlation in our sample may be of the synchrotron emission preponderance and usually high-energy peaked blazars. On the contrary, the spectrum of those HPs showing an positive correlation of the HR1 versus CTs could be dominated by the flat IC radiation and they might be of low-energy peaked blazars. What is more, the soft X-ray energy distribution of the blazars whose spectrum varies randomly with rising flux is probably dominated by the mergence of the synchrotron and IC radiation, while for softening spectra that is induced by an alternation between the synchrotron and the IC radiation and their synchrotron peaked frequency would be intermediate. Consequently, it seems that from the BLs indicating a hardening spectrum through the blazars exhibiting random spectral variation or a spectral softening to the HPs showing a hardening spectrum, their synchrotron peaked frequency shifts from high to low.
In conclusion, in this paper we analyzed a complete spectral shape variability of blazars by ROSAT/PSPC observations. Most of the blazars in our sample exhibit a hardening soft X-ray spectrum with increasing flux, a typical behavior of the subclass; there are also 6 blazars which do not show any clear spectral variation trend and 2 objects display a steepening spectrum with rising intensity, a behavior rarely revealed in blazars. Based on the properties of the synchrotron and IC emissions we argue that different spectral variations might represent the relative contribution of the two components: the soft X-ray spectrum of the BLs with a hardening spectrum are dominated by the synchrotron emission while for the HPs it is dominated by the IC radiation instead; those showing random spectral variation are prepondered by the combination of the two radiations, and the steepening are prevailed by the alternation between the IC and synchrotron emissions. Thus, different soft X-ray spectral variations might correspond to a sequence of shifting synchrotron peaked frequency.
We thank Dr. Luo Ali and other members of LAMOST Group for helping us with data reduction and some software applications. Supports under Chinese NSF (19973014), the Pandeng Project, and the 973 Project (NKBRSF G19990754) are also gratefully acknowledged.
[99]{}
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---
abstract: 'We reply to arXiv:1508.00203 ‘Comment on “Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets” (arXiv:1507.01537v2)’.'
author:
- Dibyendu Mandal
- 'Alexander B. Boyd'
- 'James P. Crutchfield'
bibliography:
- 'chaos.bib'
title: 'Memoryless Thermodynamics? A Reply'
---
Introduction
============
Several years ago, Chris Jarzynski and one of us (DM) introduced a solvable model of a thermodynamic ratchet that leveraged information to convert thermal energy to work [@Mand012a; @Lu14a]. Our hope was to give a new level of understanding of the Second Law of Thermodynamics and one of its longest-lived counterexamples—Maxwell’s Demon. As it reads in “bits” from an input string $Y$, a detailed-balance stochastic multistate controller raises or lowers a mass against gravity, writing “exhaust” bits to an output string $Y'$.
A complete understanding of the ratchet’s thermodynamics requires exactly accounting for all of the information embedded the input and output strings and how that information is changed by the ratchet. To simplify, we assumed the input bits came from a biased coin and so the input information could be measured using the *single-bit Shannon entropy* ${\operatorname{H}}[Y_0]$. The information in the output string was much more challenging to quantify, since correlations are necessarily introduced by the action of the memoryful ratchet. Unfortunately, due to mathematical complications arising from this, we could only estimate the single-bit entropy ${\operatorname{H}}[Y'_0]$ of the output. Which, it must be said, is only an upper bound on the actual information per output bit. Nonetheless, the estimate of the change $\Delta {\operatorname{H}}= {\operatorname{H}}[Y'_0] - {\operatorname{H}}[Y_0]$ from input to output was good enough to show that the ratchet was quite functional, operating as an “engine” in some regimes and an “eraser” in others.
Following in this spirit, the three of us here recently introduced a similar memoryful ratchet for which all of the informational correlations in the output bit string can be calculated exactly and in closed form [@Boyd15a]. As a result, one of its contributions is that we could then show that the change $d\hmu = \hmu[Y'] - \hmu[Y]$ in the *Shannon entropy rate* $\hmu[X] =
\lim_{\ell \to \infty} {\operatorname{H}}[X_0 X_{1} \ldots X_{\ell}]/\ell$ allowed one to identify all of the ratchet’s thermodynamic functionality. We emphasized, in particular, that using single-bit Shannon entropy $\Delta {\operatorname{H}}$ would miss much of that functionality, as ${\operatorname{H}}[Y'_0] \geq \hmu[Y']$. And, as such, we generalized Refs. [@Bara2014b; @Bara2014a] single-bit $\Delta {\operatorname{H}}$ “Second Law” to use the Shannon entropy rate $d\hmu$. The underlying methods leveraged a new way to account for the information storage and transformation induced by memoryful channels [@Barn13a]. A similarly complete analytical treatment of a companion Demon—Szilard’s Engine—was recently given by two of us (AB and JPC) [@Boyd14b].
Special Case of the Memoryless Transducer
=========================================
A recent arXiv post [@Merh15b] complained that our work [@Boyd15a] is misleading in certain aspects. It also claims priority over our entropy-rate Second Law [@Boyd15a Eq. (4)], stating that Eq. (24) of Ref. [@Merh15a] is the same. This is mathematically incorrect. Moreover, our Ref. [@Boyd15a] is very clear about its contributions. In short, our treatment is more general, since it considers the much broader class of Demons with arbitrary memory. Such Demons, as we describe in our manuscript, can be represented as memoryful channels, otherwise known as transducers [@Barn13a]. In stark contrast, Ref. [@Merh15a]’s treatment is sufficient only for describing memoryless channels; a highly restricted, markedly simpler case. More to the point, its methods are inapplicable to our memoryful channel setup. This error occurs in the proof of Ref. [@Merh15a]’s Eq. (24) as it contains a statement that can be violated by memoryful channels. We provide two counterexamples to this erroneous statement in our response below. Finally, the case of memoryless Demons violates the spirit of Refs. [@Mand012a; @Lu14a]’s original work. The mathematical errors and misinterpretation of physical relevance subvert the arXiv post’s claims. We now turn to respond to its three specific comments in greater detail.
Comment 1 {#comment-1 .unnumbered}
---------
In his first comment, the arXiv post’s author mentions that the following sentences in our paper give a [*[“very strong misleading impression that the paper above is the **first** to incorporate correlations successfully in general"]{}*]{} (using his own words). This is simple misreading, as our text makes clear:
> We introduce a family of Maxwellian Demons for which correlations among information bearing degrees of freedom can be calculated exactly and in compact analytical form. This allows one to precisely determine Demon functional thermodynamic operating regimes, when previous methods either misclassify or simply fail due to the approximations they invoke.
Note that this explicitly mentions the solvable aspect of our model—that correlations can be calculated exactly in a compact, analytical form. We stand by the claim that ours is the first such solvable model. We did not claim to be the first to consider correlations. More pointedly, the author’s actual article [@Merh15a] *does not have any model with calculable correlations*. We justify the second sentence quoted above on identifying functional thermodynamics through explicit calculations and diagrams in Sec. V of our paper. The arXiv post ignores these.
The author claims that the [*“main result in Section 4 of \[1\] was exactly the same as the above mentioned upper bound on the extracted work in terms of the change in the joint entropy ... ."*]{} In this, he refers to Eq. (4) of our paper and claims that he had derived it before as Eq. (24) of his paper [@Merh15a]. While we agree that our equation superficially looks like the infinite-time limit of the author’s equation, their relationship is different than a glance suggests:
- The author’s proof of Eq. (24) [@Merh15a] does not apply to our setup. This is because the author considered the much simpler case of memoryless channels, whereas we considered the much more mathematically challenging case of memoryful channels. (We return to this point again in context of the $3^\text{rd}$ comment.)
- Appendix A in our paper clearly shows that Eq. (4) there is valid only in the asymptotic limit of stationary input bits for a finite-state Demon. (These are standard assumptions in the field.) In absence of these assumptions, we have a more general form of the Second Law discussed in detail in Appendix A [@Boyd15a]. The arXiv post neglects these discussions.
Comment 2 {#comment-2 .unnumbered}
---------
The arXiv post quotes the following from our paper:
> In effect, they account for Demon information-processing by replacing the Shannon information of the components as a whole by the sum of the components’ individual Shannon informations. Since the latter is larger than the former \[19\], these analyses lead to weak bounds on the Demon performance.
And, then goes on to claim that the second assertion may not be true if the incoming bits $\{Y_i\}$ are correlated. This *is* the case in the author’s Ref. [@Merh15a], where the sum of individual entropy differences is actually stronger, under the additional assumption that the Demon is memoryless. We agree. But, as the author himself points out, our claim is true if the incoming bits are uncorrelated. We explicitly state that we are considering this case, where the input is uncorrelated, in the paragraph following Eq. (4). And, this happens to be the case for all the exactly solvable models of Maxwell’s Demon developed so far (referred to by “they" in the above quote). (We reiterate, the author has not given any exactly solvable model of Maxwell’s Demon with calculable correlations in [@Merh15a].)
The author did not sufficiently consider the remainder of our development before expressing his criticism in public. After Eq. (4), we explicitly mention the sufficient condition of uncorrelated incoming bits for Eq. (4) to be stronger than Eq. (2).
According to the author *“the point in second law and its extensions ... should be to provide, first and foremost, an extended version of the second law in a faithful manner, namely, to show the increase of the real entropy of the entire system, including that of the information reservoir. In the correlated case, the latter is given by the change in the joint entropy of the symbols, regardless of whether or not this is smaller or larger than the sum of individual entropy differences."* We disagree. This is nothing more than an attempt to rewrite the history of physics.
The primary emphasis of the Second Law from its very inception has been on the strongest possible bounds. When Sadi Carnot formulated the Second Law, it was **all** about maximum efficiency of heat engines—the maximum possible work that can be extracted [@Carn97a]. Entropy was a derived concept, entering through the works of Clausius and Thompson [^1].
The author mentions that *“bounds are useful when they are easier to calculate than the real quantity of interest, which is not quite the case in this context. Quite the contrary, joint entropies (especially of long blocks) are much harder to calculate."* He fails to notice that we attained precisely this “hard" task by calculating exactly the entropy rate $\hmu[Y'] = \lim_{\ell \rightarrow \infty} {\operatorname{H}}[Y'_{0:\ell}]/\ell$. (We might, at this point, recommend the review of correlations and information in random-variable blocks presented by Ref. [@Crut01a].) And, the entropy rate is smaller than the individual entropy difference, which in our case has **observable consequences**, as discussed in detail in Sec. V of our paper. Even the later part of his comment *“work itself ... depends only on the input and output marginals"* is not true in a generic memoryful situation. We have explicit examples (unpublished) where the extracted work also depends on correlations.
Comment 3 {#comment-3 .unnumbered}
---------
Here, the author claims that the *“bound in \[57\] ... is exactly the same as in eq. (4) of 1507.01537v2, except that in \[57\], no limit on is taken over the normalized entropies (but this is because even stationarity is not assumed there, so the limit might not exist). Moreover, while it is true that in the model of \[57\] the channel was memoryless, the derivation itself of this very same bound (in Section 4 of \[57\]) was not sensitive to the channel memorylessness assumption."* (Citation \[57\] corresponds to Ref. [@Merh15a] here.) We agree that Eq. (4) in our paper appeared in a somewhat different form than in his paper, as Eq. (24). This is moot, however. His derivation does not apply to our case nor to the original solvable Maxwell’s demon [@Mand012a]. In his justification, the author says that the *“crucial step in \[57\] ... was the equality* $$\begin{aligned}
{\operatorname{H}}(Y'_i | Y_1 , \ldots, Y_{i-1}, Y'_1, \ldots, Y'_{i-1})
= {\operatorname{H}}(Y'_i | Y_1 , \ldots, Y_{i-1})
~,
\label{eq:FirstStep}\end{aligned}$$ *which is the case when* $$\begin{aligned}
Y'_i \rightarrow (Y_1 , \ldots, Y_{i-1}) \rightarrow (Y'_1 , \ldots, Y'_{i-1})
\label{eq:MarkovChain}\end{aligned}$$ *forms a Markov chain, and this happens not only for a memoryless channel, but for any causal channel without feedback, namely, $$\begin{aligned}
P(Y'_1 , \ldots, Y'_n | Y_1 , \ldots, Y_n )
= \Pi_{i = 1}^n P (Y'_i | Y_1 , . . . , Y_i)
~.
\label{eq:CondlProb}\end{aligned}$$ In physical terms, this actually means full generality."*
This analysis is incorrect. Equation (\[eq:FirstStep\]) above is not sufficiently general, since it does not consider the case in which the Demon is a memoryful channel. When the Demon has memory, its internal state can depend on both the input past $Y_1 , \ldots, Y_{i-1}$ and output past $Y'_1 , \ldots,
Y'_{i-1}$. The Demon’s internal states store information about the past of $Y$ or $Y'$ and can communicate it to the outgoing bits of $Y'$. The author’s assertion of “full generality” is false. In fact, the memoryless assumption is violated for the original solvable model of Maxwell’s Demon [@Mand012a] in which the Demon has three internal states. For a memoryful Demon Eq. (\[eq:MarkovChain\]) above is not a Markov chain. See Ref. [@Barn13a]’s discussion of memoryful transduction.
To see how Eq. (\[eq:FirstStep\]) can be violated, consider the case of a memoryful Demon that simply ignores the input bits and outputs a period-$2$ process. This means that there are two possible output words: $$\begin{aligned}
\Pr(Y'_1Y'_2...=010101...) = \Pr(Y'_1Y'_2...=101010...) = 1/2
~.\end{aligned}$$ In this case the uncertainty of the $i$th output given the history of inputs is ${\operatorname{H}}[Y'_i | Y_{1:i}]=1$, since we are completely uncertain as to whether or not the $i$th bit is a zero or one. Note that we used the notational shorthand $Y_{1:i}$ to represent the random variables $Y_1,Y_2, \ldots Y_{i-1}$. When we also condition on the history of output bits, we find that we are completely certain of the next bit, since we know the output’s phase, and ${\operatorname{H}}[Y'_i | Y_{1:i}, Y'_{1:i}]=0$. The most general relation for the uncertainty of the output is: $$\begin{aligned}
{\operatorname{H}}[Y'_i | Y_{1:i}, Y'_{1:i}]
\leq {\operatorname{H}}[Y'_i | Y_{1:i}]
~.\end{aligned}$$ This inequality, replacing Eq. (\[eq:FirstStep\]) above, renders the proof in the author’s paper inapplicable to our situation.
This reflects the fact that a memoryful ratchet can and typically does create correlations among the outgoing bits even though the incoming bits may not be correlated. In fact, we can exactly calculate the uncertainty in the next output bit conditioned on the infinite length input and output histories of the memoryful ratchet we describe in our Ref. [@Boyd15a]. When the ratchet is driven by a fair coin input process, the two quantities of interest are: $$\begin{aligned}
\lim_{i \rightarrow \infty}{\operatorname{H}}[Y'_i | Y_{1:i}]
= \frac{1}{2}\left( {\operatorname{H}}\left(\frac{p}{2}\right)
+{\operatorname{H}}\left(\frac{q}{2}\right)\right)
~,\end{aligned}$$ where ${\operatorname{H}}(b)$ is the binary entropy function for a coin of bias $b$ [@Cove06a], and: $$\begin{aligned}
\lim_{i \rightarrow \infty} {\operatorname{H}}[Y'_i | Y_{1:i}, Y'_{1:i}]
=\frac{1}{4}\left( {\operatorname{H}}\left(p\right)
+ {\operatorname{H}}\left(q\right)\right),
\end{aligned}$$ and their difference: $$\begin{aligned}
\lim_{i \rightarrow \infty} & ({\operatorname{H}}[Y'_i | Y_{1:i}]
- {\operatorname{H}}[Y'_i | Y_{1:i}, Y'_{1:i}]) \\
& = \frac{1}{2}\left( {\operatorname{H}}\left( \frac{p}{2}\right) -\frac{{\operatorname{H}}(p)}{2}\right)
+ \frac{1}{2}\left( {\operatorname{H}}\left( \frac{q}{2}\right) -\frac{{\operatorname{H}}(q)}{2}\right) \\
& \geq 0
~,\end{aligned}$$ by the concavity of ${\operatorname{H}}(\cdot)$. This is only zero when p=q=0. Thus, the assumption made in Eq. (1) is not just insufficiently general, but it is explicitly violated in the physical memoryful ratchet considered in our work.
On a more conceptual level, Eq. (4) in our paper is valid only in the asymptotic limit of a stationary input with a finite-state Demon. Otherwise, there would be natural generalizations incorporating the Demon’s entropy and its correlations with the bits, as is amply discussed in Appendix A of our paper.
Summing Up
==========
As our response to the arXiv post’s Comment 3 just made plain, the essential issue reduces to the post’s author misapplying results for memoryless channels. Most directly, the post’s claim to priority for our entropy-rate Second Law is invalid. Perhaps the simple memoryless channel case, one very broadly adopted in elementary information theory [@Cove06a], prevented the post’s author from appreciating this and related technical points. Whatever the motivation, it led to the post’s public airing of a series of grievances—grievances that derive not from misleading text, but from the author’s misinterpretation. That said, we do appreciate the opportunity to emphasize the central role of memory and structure in thermodynamics.
Acknowledgments {#acknowledgments .unnumbered}
===============
As an External Faculty member, JPC thanks the Santa Fe Institute for its hospitality during visits. This work was supported in part by the U. S. Army Research Laboratory and the U.S. Army Research Office under contracts W911NF-13-1-0390 and W911NF-12-1-0234.
[^1]: Furthermore, entropy rate in the information-theoretic sense is not always applicable in the thermodynamic sense. This is seen in context of nonlinear, chaotic dynamics where information (about the initial state) is continuously produced without any need for thermodynamic irreversibility [@Dorf99a]. For thermodynamics, one must consider the time-reversed description.
|
---
abstract: 'Given a symplectic manifold $M$, we may define an operad structure on the the spaces ${{\mathcal O}}^k$ of the Lagrangian submanifolds of $(\overline{M})^k\times M$ via symplectic reduction. If $M$ is also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semi-classical part of Kontsevich’s deformation of $C^\infty({\mathbb R}^d)$ is a deformation of the trivial symplectic groupoid structure of $T^*{\mathbb R}^d$.'
address:
- 'Institut für Mathematik, Universität Zürich–Irchel, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland'
- 'D-MATH, ETH-Zentrum, CH-8092 Zürich, Switzerland'
- 'D-MATH, ETH-Zentrum, CH-8092 Zürich, Switzerland'
author:
- 'Alberto S. Cattaneo'
- Benoit Dherin
- Giovanni Felder
title: '**Formal Lagrangian Operad**'
---
Introduction
============
Symplectic groupoids, in the extended symplectic category, may be thought as the analog of associative algebras in the category of vector spaces. For the latter, a deformation theory exists and is well known. In this article, we will present a conceptual framework as well as an explicit deformation of the trivial symplectic groupoid over ${\mathbb R}^d$. In fact, rephrased appropriately, most constructions of the deformation theory of algebras can be extended to symplectic groupoids, at least for the trivial one over ${\mathbb R}^d$. Our guide line will be the Kontsevich deformation of the usual algebra of functions over ${\mathbb R}^d$, $\big(C^\infty({\mathbb R}^d),\cdot\big)$. Namely, the usual point-wise product of functions $S_0^2(f,g) = fg$ generates a suboperad, the product suboperad, ${{\mathcal O}}_S^n=\big\{S_0^n \big\},$ of the endomorphism operad ${{\mathcal O}}$ of $C^\infty({\mathbb R}^d)$, where $S_0^n$ is the $n$-multilinear map defined by $S_0^n(f_1,\dots,f_n) = f_1f_2\dots f_n.$ For each $n$ one may choose the vector subspace ${{{\mathcal O}}_{\mathrm{def}}^{n}}\subset {{\mathcal O}}^n$ of $n$-multidifferential operators. The operad structure of ${{\mathcal O}}$ induces an operad structure on ${{\mathcal O}}_S+{{{\mathcal O}}_{\mathrm{def}}^{}}$, which in turns generates an operad structure on ${{{\mathcal O}}_{\mathrm{def}}^{}}$ which is, however, non-linear. Then, $\gamma$ is a deformation of the usual product $S_0^2$, i.e., an element $\gamma\in{{{\mathcal O}}_{\mathrm{def}}^{2}}$ such that $S_0^2+\gamma$ is still an associative product, iff $\gamma$ is a product in the induced deformation operad ${{{\mathcal O}}_{\mathrm{def}}^{}}$. We may also consider the formal version by replacing ${{{\mathcal O}}_{\mathrm{def}}^{}}$ by the formal power series in $\epsilon$, $\epsilon {{{\mathcal O}}_{\mathrm{def}}^{}}[[\epsilon]]$. M. Kontsevich in [@kontsevich1997] gives an explicit formal deformation of the product of functions over ${\mathbb R}^d$, $$S_\epsilon = S_0^2+\sum_{n=1}^\infty\epsilon^n\sum_{\Gamma\in G_{n,2}}W_\Gamma B_\Gamma,$$ where the $W_\Gamma$’s are the Kontsevich weights and the $B_\Gamma$’s the Kontsevich bidifferential operators associated to the Kontsevich graphs of type $(n,2)$ (see [@CDF2005] for a brief introduction ).
If we consider the trivial symplectic groupoid ${\mathrm{T}^*{\mathbb R}^{d}}$ over ${\mathbb R}^d$, we see that the multiplication space $$\Delta_2^n:=\Big\{(p_1,x),(p_2,x),(p_1+p_2,x):p_1,p_2\in {\mathbb R}^{d*},
x\in {\mathbb R}^d\Big\}$$ generates an operad ${{{\mathcal O}}_{\Delta}^{n}} = \big\{\Delta_n\big\},$ where $$\Delta_n := \Big\{(p_1,x),\dots,(p_n,x),(p_1+\dots+p_n,x):p_i\in{\mathbb R}^{d*},x\in{\mathbb R}^d\Big\}.$$ $\Delta_2$ is a product in this operad. The compositions are given by symplectic reduction as the $\Delta_n$’s are Lagrangian submanifolds of $\overline{({\mathrm{T}^*{\mathbb R}^{d}})^n}\times {\mathrm{T}^*{\mathbb R}^{d}}$. The main difference with the vector space case is that there is no “true” endomorphism operad where ${{{\mathcal O}}_{\Delta}^{}}$ would naturally embed into. Thus, the question of finding a deformation operad for ${{{\mathcal O}}_{\Delta}^{}}$ must be taken with more care. The first remark is that the $\Delta_n$ may be expressed in terms of generating functions $$S_0^n(p_1,\dots,p_n,x) = (p_1+\dots+p_n)x.$$ Namely, $\Delta_n = {\operatorname{graph}}dS_0^n$. The idea is to look at the operad structure induced on the generating functions by symplectic reduction. In fact it is possible to find a vector space of special functions ${{{\mathcal O}}_{\mathrm{def}}^{n}}$ for each $n$ such that ${{{\mathcal O}}_{\Delta}^{}}+{{{\mathcal O}}_{\mathrm{def}}^{}}$ remains an operad. The formal version of it gives a surprising result. Namely, we may find an explicit deformation of the trivial generating function $S_0^2$ , it is given by the formula $$S_\epsilon = S_0^2 +\sum_{n=1}^\infty \epsilon^n\sum_{\Gamma\in T_{n,2}}W_\Gamma \hat B_\Gamma,$$ where the $W_\Gamma$ are the Kontsevich weights and the $\hat B_\Gamma$ are the symbols of the Kontsevich bidifferential operators and the sum is taken over all Kontsevich trees $T_{n,2}$. This formula may be seen as the semi-classical part of Kontsevich deformation quantization formula.
As a last comment, note that Kontsevich derives its star product formula from a more general result. In fact, he shows that $U = \sum_{n} \epsilon^n U_n$ where $$U_n(\xi_1,\dots,\xi_n) = \sum_{\Gamma\in G_n} W_\Gamma B_\Gamma(\xi_1,\dots,\xi_n)$$ for $\xi_i\in\Gamma(\wedge^{d^i}TM)$, $i=1,\dots,d$ is an $L_\infty$-morphism from the multivector fields to the multidifferential operators on ${\mathbb R}^d$. In our perspective, we may still write $$\tilde U_n(\xi_1,\dots,\xi_n) = \sum_{\Gamma\in T_n} W_\Gamma \hat B_\Gamma(\xi_1,\dots,\xi_n)$$ summing over Kontsevich trees instead of Kontsevich graphs and replacing multidifferential operators by their symbols. Exactly, as in Kontsevich case, $$S_\epsilon = S_0^2 +\sum_{n\geq 1} \epsilon^n \tilde U_n(\alpha,\dots,\alpha)$$ is an associative deformation of the generating function of the trivial symplectic groupoid $T^*{\mathbb R}^d$. However, it is still not completely clear how to define “semi-classical $L_\infty$-morphisms”.
### Organization of the article {#organization-of-the-article .unnumbered}
In Section \[prodinext\], we describe the endomorphism operad ${{\mathcal O}}(M) = {\operatorname{Hom}}(M^{\otimes^n},M)$ associated to any object $M$ in a monoidal category. We explain what is an associative product $S$ on $M$ in an monoidal category and we define the product suboperad ${{\mathcal O}}_S(M)$ of ${{\mathcal O}}(M)$. If the category is further associative, we may choose a deformation operad for $S$, which is a choice, for each $n\in {\mathbb N}$ of a vector subspace ${{{\mathcal O}}_{\mathrm{def}}^{n}}$ such that ${{\mathcal O}}_S+{{{\mathcal O}}_{\mathrm{def}}^{}}$ is still an operad. We describe the deformations of $S$ in terms of products in ${{{\mathcal O}}_{\mathrm{def}}^{}}$. As an example of this construction, we expose Kontsevich product deformation in this language. At last, we show that the extended symplectic category, although not being a true category, exhibits monoidal properties allowing us to carry the precedent construction up to a certain point. Then, we focus on the trivial symplectic groupoid over ${\mathbb R}^d$ case and define the product operad associated to its multiplications space. We give a deformation operad on a local form, the local deformation operad. In particular, we show that any local deformation of the trivial product gives rise to a local symplectic groupoid over ${\mathbb R}^d$. We conclude this Section by defining equivalence between deformations of the trivial generating function and we show that two equivalent deformations induce the same local symplectic groupoid.
In Section \[combinatorics\], we describe the combinatorial tools needed to give a formal version of the local Lagrangian operad. As the problem consists mainly in taking Taylor’s series of some implicit equations we need devices to keep track of all terms to all orders. The crucial point is that these implicit equations, describing the composition in the local Lagrangian operad, have a form extremely close to a special Runge-Kutta method: the partitioned implicit Euler method. We borrow then some techniques form numerical analysis of ODEs to make the expansion at all orders.
In the last Section, we describe the formal Lagrangian operad, which is the perturbative version of the local one, in terms of composition of bipartite trees. We give in particular the product equation in the formal deformation operad in terms of these trees. At last, we restate the main Theorem of [@CDF2005] in this language. This tells us that the semi-classical part of Kontsevich star product on ${\mathbb R}^d$ is a product in the formal deformation operad of the cotangent Lagrangian operad in $d$ dimensions.
This article is inspired for a large part from unpublished notes [@Cattaneo2002] of one of the authors, in which the notion of Lagrangian operad first appeared, and from the PhD thesis [@dherin2004] of an another author. It is a natural development of results presented in [@CDF2005].
We thank Domenico Fiorenza for useful comments and suggestions.
Product in the extended symplectic category {#prodinext}
===========================================
Basic constructions and Kontsevich deformation
----------------------------------------------
In this Section, we describe, in any monoidal category, a natural generalization of an associative algebra structure over a vector space. It is the notion of product in the endomorphism operad ${{\mathcal O}}(M)$ of an object $M$ in the category. If the category is further additive, we explain what is a deformation of a product $S\in{{\mathcal O}}^2(M)$ and construct a non-linear operad, the deformation operad ${{\mathcal O}}_{\mathrm{def}}(M,S)$ associated to $S$ in which any product is equivalent to a deformation of $S$. We present the well-known Kontsevich deformation of the usual product of functions over ${\mathbb R}^d$ in this language. At last, we see that most parts of this construction, can be applied to the extended symplectic category, leading to the notion of Lagrangian operad.
An operad ${{\mathcal O}}$ consists of
1. a collection of sets ${{\mathcal O}}^n, n \geq 0$
2. composition laws $$\begin{aligned}
{{\mathcal O}}^n \times {{\mathcal O}}^{k_1} \times \dots \times {{\mathcal O}}^{k_n}
& \longrightarrow &
{{\mathcal O}}^{k_1 + \dots +k_n}\\
(F,G_1,\dots,G_n) & \mapsto & F(G_1,\dots,G_n)\end{aligned}$$ satisfying the following associativity relations, $$\begin{gathered}
F(G_1,\dots,G_n)(H_{11},\dots,H_{1k_1},\dots,H_{n1},\dots,H_{nk_n})
= \\
F(G_1(H_{11},\dots,H_{1k_1}),\dots,G_n(H_{n1},\dots,H_{nk_n}))\end{gathered}$$
3. a unit element $I \in {{\mathcal O}}^1$ such that $F(I,\dots,I) = F \textrm{ for all } F \in {{\mathcal O}}^n$
It is usually also required some equivariant action of the symmetric group. We do not require this here.
The structure we have just defined should then be called more correctly “non symmetric operad”. However, we will simply keep using the term “operad” instead of “non symmetric operad” in the sequels.
### Product in a monoidal category {#product-in-a-monoidal-category .unnumbered}
We consider here a monoidal category $\mathcal{C}$. We denote by $\otimes : \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$ the product bifunctor and by ${\textbf{e}}\in \mathcal{C}$ the neutral object. Let us recall that we have the following canonical isomorphisms $$(A \otimes B )\otimes C \simeq A \otimes (B \otimes C)\quad\textrm{ and }\quad
{\textbf{e}}\otimes A \simeq A \otimes {\textbf{e}}\simeq A$$ for all $A,B,C\in{\operatorname{Obj}}\mathcal C$.
Let $\mathcal{C}$ be a monoidal category and an object $M \in {\operatorname{Obj}}\mathcal{C}$. We define the endomorphism operad of $M$ in the following way:
1. ${{\mathcal O}}^n(M):={\operatorname{Hom}}(M^{\otimes n},M), \: {{\mathcal O}}^0(M) :=
{\operatorname{Hom}}({\textbf{e}},M)$
2. $F(G_1,\dots,G_n):= F \circ (G_1 \otimes \dots \otimes G_n)$
3. the unit is given by ${\operatorname{id}}_M \in {{\mathcal O}}^1(M)$.
The operad axioms follow directly from the bifunctoriality of $\otimes$, i.e, $$\begin{aligned}
(f \otimes g) \circ (\psi \otimes \phi) & = &
(f \circ \psi)\otimes (g \circ \phi)\\
{\operatorname{id}}_M \otimes \dots \otimes {\operatorname{id}}_M & = & {\operatorname{id}}_{M \otimes \dots \otimes M}.\end{aligned}$$ If $M$ is an object of a monoidal category $\mathcal{C}$, we may define a product on $M$.
An associative product [^1]on an operad ${{\mathcal O}}$ is an element $S \in {{\mathcal O}}^2$ such that $S(I,S)=S(S,I)$. An associative product on $M$ is an associative product in the endomorphism operad ${{\mathcal O}}(M)$. In the sequels, we will constantly use the term product to mean in fact associative product.
Given a product $S\in{{\mathcal O}}^2$, the associativity of the operad implies that, for any $F \in {{\mathcal O}}^k$, $G \in {{\mathcal O}}^l$ and $H \in {{\mathcal O}}^m$ we have, $$\begin{aligned}
S(F,S(G,H)) & = & S (I,S)(F,G,H) \\
& = & S(S,I)(F,G,H) \\
& = & S(S(F,G),H).\end{aligned}$$
This notion is the natural generalization of an associative product on a vector space. Namely, if $M$ is a vector space, ${{\mathcal O}}^2(M)$ is the set of bilinear map on $M$. As in this case ${{\mathcal O}}^0(M)={\operatorname{Hom}}({\mathbb C},M)=M$, we have that $S : {{\mathcal O}}^0(M) \times {{\mathcal O}}^0(M) \longrightarrow {{\mathcal O}}^0(M)$ is an associative product on $M$.
### Product deformation in a monoidal additive category {#product-deformation-in-a-monoidal-additive-category .unnumbered}
Suppose we have a product $S\in{{\mathcal O}}^2(M)$, where $M$ is an object of a monoidal category $\mathcal C$. If the category $\mathcal{C}$ is further additive, we may try to deform $S$, i.e., to find an element $\gamma \in {{\mathcal O}}^2(M)$ such that $S+\gamma$ is still a product.
At this point, we may follow the main trend and introduce the Hochschild complex of the linear operad ${{\mathcal O}}(M)$, define the bilinear Gerstenhaber bracket and the Hochschild differential associated to the product $S$. A deformation of $S$ would then be a solution of the Maurer-Cartan equation written in the Hochschild Differential Graded Lie Algebra controlling the deformations of $S$.
We will however rephrase slightly this deformation theory in a way allowing us to include afterward the case when the morphisms of the category are still vector spaces but the composition of morphism is no longer bilinear. That will be exactly the case we will have to deal with in the next Sections.
The first step is to notice that a product $S\in {{\mathcal O}}^2(M)$ generate a sub-operad ${{\mathcal O}}_S(M)$, which we call a [**product operad**]{}, in ${{\mathcal O}}(M)$ with only one point in each degrees: $$\begin{gathered}
{{\mathcal O}}_S^0(M) := \emptyset,\quad {{\mathcal O}}_S^1(M) := \Big\{I\Big\},\quad {{\mathcal O}}_S^2(M) := \Big\{S \Big\},\\
{{\mathcal O}}_S^3(M) := \Big\{S(S,I) \Big\},\quad {{\mathcal O}}_S^4(M) := \Big\{S(S(S,I),I) \Big\},\dots \textrm{etc}\end{gathered}$$ To simplify the notation we will denote by $S_0^n$ the unique element in ${{\mathcal O}}_S^n(M)$.
Let $M$ be an object of an additive monoidal category $\mathcal C$ and $S\in{{\mathcal O}}^2(M)$ a product. A **deformation operad**, ${{\mathcal O}}_{\mathrm{def}}(M,S)$ for $S$ is a choice for each $n\in{\mathbb N}$ of a vector space space ${{{\mathcal O}}_{\mathrm{def}}^{n}}(M,S)\subset{{\mathcal O}}^n(M),$ such that ${{\mathcal O}}_S+{{\mathcal O}}_{\mathrm{def}},$ is still an operad with respect to the compositions of ${{\mathcal O}}(M)$, i.e., $$\begin{aligned}
\label{operadeq}
(S_0^n+\gamma)(S_0^{k_1}+\gamma^{1},\dots,S_0^{k_n}+\gamma^{n})
& = & S_0^{k_1+\dots +k_n} +R(\gamma;\gamma^{1},\dots,\gamma^{n}),\end{aligned}$$ with $$R(\gamma;\gamma^{1},\dots,\gamma^{n})\in {{{\mathcal O}}_{\mathrm{def}}^{k_1+\dots+k_n}}(M,S),$$ for $\gamma\in{{{\mathcal O}}_{\mathrm{def}}^{n}}(M,S)$ and $\gamma^{i}\in{{{\mathcal O}}_{\mathrm{def}}^{k_i}}(M,S)$ for $i=1,\dots,n$.
We say that an element $\gamma\in{{\mathcal O}}_{\mathrm{def}}^2(M,S)$ is a [**deformation of the product**]{} $S$ w.r.t. the deformation operad ${{\mathcal O}}_{\mathrm{def}}$ if $S+\gamma$ is still a product in ${{\mathcal O}}_S+{{\mathcal O}}_{\mathrm{def}}$.
Remark that the $R$s are not multilinear.
Let ${{\mathcal O}}_{\mathrm{def}}(M,S)$ be a deformation operad for a product $S\in{{\mathcal O}}^2(M)$. Then the compositions $$\gamma(\gamma^{1},\dots,\gamma^{n}):= R(\gamma;\gamma^{1},\dots,\gamma^{n}),$$ defined by equation (\[operadeq\]) gives ${{\mathcal O}}_{\mathrm{def}}(M,S)$ together with the unit $0\in {{\mathcal O}}_{\mathrm{def}}^1(M,S)$ the structure of an operad.
The proof is direct using only equation (\[operadeq\]) and the operad structure of the endomorphism operad ${{\mathcal O}}(M)$.
Let $S\in {{\mathcal O}}^2(M)$ be a product. Take an element $\gamma\in {{{\mathcal O}}_{\mathrm{def}}^{2}}(M,S)$. Then, $\gamma$ is a deformation of the product $S$ iff $\gamma$ is a product in ${{\mathcal O}}_{\mathrm{def}}(M,S)$. In particular, $0\in {{{\mathcal O}}_{\mathrm{def}}^{2}}(M,S)$ is always a product in the deformation operad of $S$.
$\gamma$ is a deformation of $S$ iff $$(S+\gamma)(S+\gamma,I) = (S+\gamma)(I,S+\gamma),$$ which is equivalent to $$S_0^3+R(\gamma;\gamma, 0) = S_0^3+R(\gamma;0,\gamma).$$
From now on, we will write $0_1$ for the identity element of the deformation operad which is the zero of ${{{\mathcal O}}_{\mathrm{def}}^{1}}$ and $0_2$ for the trivial product of the deformation operad which is the $0$ element in ${{{\mathcal O}}_{\mathrm{def}}^{2}}(M,S)$.
Notice that neither ${{\mathcal O}}_S(M)$ nor ${{\mathcal O}}_S(M) +{{\mathcal O}}_{\mathrm{def}}(M,S)$ is a linear operad in the sense that, although the compositions are multilinear, the spaces for each degrees are not vector spaces but affine spaces. On the other hand the spaces for each degrees of the deformation operad ${{\mathcal O}}_{\mathrm{def}}(M,S)$ are vector spaces but the induced operad compositions are not linear in general.
We may however introduce the Gerstenhaber bracket of the deformation operad $$\begin{gathered}
[,]:{{{\mathcal O}}_{\mathrm{def}}^{k}}(M,S) \times {{{\mathcal O}}_{\mathrm{def}}^{l}}(M,S) \longrightarrow {{{\mathcal O}}_{\mathrm{def}}^{k+l-1}}(M,S)\end{gathered}$$ defined by $$\begin{gathered}
\label{Gerstenhaber}
[F,G] = F \circ G - (-1)^{(k-1)(l-1)} G \circ F \end{gathered}$$ where $$F \circ G = \sum_{i=1}^k (-1)^{(i-l)(l-1)}
R(F;0_1,\dots,0_1,\underbrace{G}_{i^{\textrm{th}}},0_1,\dots,0_1).$$ This bracket is not bilinear. An important fact concerning this bracket is that, $$\begin{gathered}
\frac{1}{2}[\gamma,\gamma]=R(\gamma;\gamma,0_1)-R(\gamma;0_1,\gamma),\end{gathered}$$ which means that $\gamma$ is a product in the deformation operad iff $$\begin{aligned}
\label{prodeq}\frac12[\gamma,\gamma] & = & 0.\end{aligned}$$
Moreover, we may define an equivalent of the Hochschild differential $$\begin{gathered}
d : {{{\mathcal O}}_{\mathrm{def}}^{n}}(M,S) \longrightarrow {{{\mathcal O}}_{\mathrm{def}}^{n+1}}(M,S),\end{gathered}$$ $$\begin{gathered}
\label{hochdiff}
dF := [0_2,F]
= R(0_2;F,0_1)+(-1)^{n-1}R(0_2;0_1,F)-\\
-(-1)^{n-1}\sum_{i=1}^n(-1)^{i-1}R(F;0_1,\dots,0_1,\underbrace{0_2}_{i^{th}},0_1,\dots,0_1).\end{gathered}$$
It turn out that $d$ is still a coboundary operator.
$d$ defined by equation (\[hochdiff\]) is a coboundary operator, i.e., $d^2= 0$. Moreover, $\gamma\in{{{\mathcal O}}_{\mathrm{def}}^{2}}(M,S)$ satisfies product equation $\frac12 [\gamma,\gamma] = 0,$ in ${{\mathcal O}}_{\mathrm{def}}(M,S)$ iff $$\begin{aligned}
\label{nonlinMCeq}
d\gamma +\gamma(\gamma,S_0^1,)-\gamma(S_0^1,\gamma) & =& 0.\end{aligned}$$
Using equation (\[operadeq\]) we obtain $d$ in terms of the endomorphism compositions $$\begin{gathered}
dF = S_0^2(F,S_0^1)+(-1)^{n-1}S_0^2(S_0^1,F)-\\
-(-1)^{n-1}\sum_{i=1}^n(-1)^{i-1}F(S_0^1,\dots,\underbrace{S_0^2}_{i^{th}},\dots,S_0^1).\end{gathered}$$ The result follows directly from the linearity of the compositions in the endomorphism operad. Using again equation (\[operadeq\]) we get, $$\begin{gathered}
\frac12[\gamma,\gamma]
= R(\gamma;\gamma,0_1)-R(\gamma;0_1,\gamma)
= S_0^2(\gamma,S_0^1)+\\
+\gamma(S_0^2,S_0^1)+\gamma(\gamma,S_0^1) -S_0^2(S_0^1,\gamma)-\gamma(S_0^1,S_0^2)-\gamma(S_0^1,\gamma),\end{gathered}$$ which gives equation (\[nonlinMCeq\]).
A formal deformation $S_\epsilon$ of $S$ is a formal power series $$S_{\epsilon} = \epsilon S_1 + \epsilon^2 S_2 + \dots\in
{{\mathcal O}}^n_{\textrm{form}}(M,S)
:= \epsilon{{\mathcal O}}_{\textrm{def}}^n(M,S) \otimes k [[\epsilon]],\quad n\in{\mathbb N}_*,$$ where $\epsilon$ is a formal parameter and ${{\mathcal O}}_{\textrm{def}}(M,S)$ is a deformation operad for $S$, such that $S+S_\epsilon$ is a product in ${{\mathcal O}}_S(M) + {{\mathcal O}}_{\mathrm{form}}(M,S)$.
Equivalently, one may say that $S_\epsilon$ must satisfy $$[S_{\epsilon},S_{\epsilon}]=0,$$ or, thanks to equation (\[nonlinMCeq\]) that the $S_i$’s satisfy at each order $n\in {\mathbb N}_*$ the following recursive equation: $$\begin{aligned}
\label{recursive} dS_n +H_n(S_{n-1},\dots,S_1) & = & 0,\end{aligned}$$ where $$H_n(S_{n-1},\dots,S_1) = \sum_{n= i+j} S_i(S_j,S_0^1)-S_i(S_0^1,S_i).$$
### The Kontsevich product deformation {#the-kontsevich-product-deformation .unnumbered}
Consider the category of real vector spaces. In this category we take the real vector space $M = C^{\infty}({\mathbb R}^d)$ of smooth functions on ${\mathbb R}^d$. The endomorphism operad of $C^{\infty}({\mathbb R}^d)$ is $${{\mathcal O}}^n(M) = \Big\{\textrm{ $n$-multilinear maps from $C^{\infty}({\mathbb R}^d)^{\otimes n}$ to $C^\infty({\mathbb R}^d)$}\Big\}.$$ The usual product of functions induces a product in ${{\mathcal O}}(M)$, namely $$S_0^2(F,G)(f_1,\dots,f_k,g_1,\dots,g_l) = F(f_1,\dots,f_k)G(g_1,\dots,g_l),$$ for $F\in {{\mathcal O}}^k(M)$ and $G\in{{\mathcal O}}^l(M)$.
The induced product operad is $${{\mathcal O}}_S^n(M) =\Big\{S_0^n\Big\},$$ where $$S_0^n(f_1,\dots,f_n) = f_1f_2\dots f_n.$$ As deformation operad, we take $${{{\mathcal O}}_{\mathrm{def}}^{n}}(M,S) :=\Big\{\textrm{ $n$-multidifferential operators on $C^\infty({\mathbb R}^d)$} \Big\}.$$ The induced coboundary operator on ${{\mathcal O}}_{\mathrm{def}}(M,S)$ is the Hochschild coboundary operator, $$\begin{gathered}
dF(f_1,\dots,f_n) = F(f_1,\dots,f_n)f_{n+1}+(-1)^{n-1}f_1F(f_2,\dots,f_{n+1})-\\
-(-1)^{n-1}\sum_{i=1}^n(-1)^{(i-1)}F(f_1,\dots,f_{i-1},f_if_{i+1},f_{i+2},\dots,f_{n+1}).\end{gathered}$$ and the product equation $$d\gamma +\gamma(\gamma,S_0^1,)-\gamma(S_0^1,\gamma) = 0,$$ is nothing but the usual Maurer-Cartan equation.
Kontsevich in [@kontsevich1997] shows that there exits a formal deformation $$S \in {{\mathcal O}}_S^2(M)+ \epsilon {{\mathcal O}}^2_{def}(M)[[\epsilon]]$$ of $S_0^2$. He provides the explicit formula for this deformation $$\begin{gathered}
S= S_0^2 + \sum_{n = 1}^\infty \epsilon^n \sum_{\Gamma \in G_{n,2}}
W_{\Gamma} B_{\Gamma}, \end{gathered}$$ where the $G_{n,2}$ are the Kontsevich graphs of type $(n,2)$, $W_\Gamma$ their associated weight and $B_\Gamma$ their associated bidifferential operator ( and [@kontsevich1997] for more precisions).
Monoidal structure of $\mathcal{SYM}$
-------------------------------------
Let us recall that the extended symplectic “category” $\mathcal{SYM}$ is given by $$\begin{aligned}
{\operatorname{Obj}}& = & \Big\{\textrm{symplectic manifolds}\Big\}\\
{\operatorname{Hom}}(M,N) & = & \Big\{L\subset \overline M\times N:L\textrm{ is Lagrangian}\Big\},\end{aligned}$$ where $\overline M$ denotes the symplectic manifold $M$ with opposite symplectic structure $-\omega$. The identity morphism of ${\operatorname{Hom}}(M,M)$ is the diagonal $${\operatorname{id}}_M := \Delta_M=\Big\{(m,m)\subset\overline M\times M\Big\}.$$ The composition of two morphisms $L\subset{\operatorname{Hom}}(M,N)$ and $\tilde L\subset{\operatorname{Hom}}(N,P)$ is given by the composition of canonical relations, $$\tilde L\circ L := \pi_{M\times P}\Big(
(L\times \tilde L)\cap (M\times\Delta_N\times P)
\Big)\subset \overline M\times P.$$ Everything works fine except the fact that the composition $\tilde L\circ L$ may fail to be a Lagrangian submanifold of $\overline M\times P$. It is always the case when $L\times \tilde L$ intersects $M\times\Delta_N\times P$ cleanly (see [@dherin2005] for more precisions).
Let us pretend for a while that $\mathcal{SYM}$ is a true category or, better, that we have selected special symplectic manifolds and special arrows between them such that the composition is always well-defined.
We define the tensor product between two objects $M$ and $N$ of $\mathcal{SYM}$ as the Cartesian product $$M\otimes N:= M\times N,$$ and the tensor product between morphisms as $$\begin{aligned}
L_1\otimes L_2 & := & \Big\{ (m,a,n,b):(m,n)\in L_1\\
&& \textrm{ and } (a,b)\in L_2 \Big\}\subset {\operatorname{Hom}}(M\otimes A,N\otimes B),\end{aligned}$$ for $L_1\in {\operatorname{Hom}}(M,N)$ and $L_2\in{\operatorname{Hom}}(A,B)$.
The neutral object is $\{*\}$, the one-point symplectic manifold. The following proposition tells us that $\mathcal{SYM}$ would be a monoidal category if it were a true category.
The following statements hold:
1. Consider $L_1\in{\operatorname{Hom}}(M,A)$, $L_2\in{\operatorname{Hom}}(N,B)$, $L_3\in{\operatorname{Hom}}(A,X)$ and $L_4\in{\operatorname{Hom}}(B,Y)$. Then we have the following equality of sets $$(L_3\otimes L_4)\circ(L_1\otimes L_2) = (L_3\circ L_1)\otimes (L_4\circ L_2).$$
2. ${\operatorname{id}}_M\otimes {\operatorname{id}}_N = {\operatorname{id}}_{M\otimes N}$ for any object $M$ and $N$.
3. $(M\otimes A)\otimes X = M\otimes (A\otimes X)$ for any objects $M$, $A$ and $X$
4. $(L_1\otimes L_2)\otimes L_3 = L_1\otimes (L_2\otimes L_3)$ for any arrows $L_1\in{\operatorname{Hom}}(M,A)$, $L_2\in{\operatorname{Hom}}(N,B)$ and $L_3\in{\operatorname{Hom}}(P,C)$.
5. $\{*\}\otimes A \simeq A\simeq A\otimes \{*\}$ for all object $A$ and ${\operatorname{id}}_{\{*\}}\otimes L\simeq L\simeq L\otimes {\operatorname{id}}_{\{*\}}$ for all arrows $L$, where $A\simeq B$ means that the two sets $A$ and $B$ are in bijection.
\(1) $$\begin{aligned}
I & = & (L_3\otimes L_4)\circ(L_1\otimes L_2) \\
& = & \pi\Big( \big((L_1\otimes L_2)\times(L_3\otimes L_4)\big)\cap (\overline{N\times M}\times\Delta_{A\times B}\times X\times Y)\Big)\\
& = & \Big\{(m,n,\tilde x,\tilde y): \exists (a,b)\in A\times B\textrm{ s.t. }\quad (m,n,a,b)\in L_1\otimes L_2\textrm{ and }\\
& & (a,b,x,y)\in L_3\otimes L_4\Big\}\\
& = & \Big\{ (m,n,\tilde x,\tilde y):\exists a\in A,\quad (m,a)\in L_1\textrm{ and }(a,x)\in L_3\\
& & \quad \exists b\in B,\quad (n,b)\in L_2\textrm{ and }(b,y)\in L_4 \Big\}\\
& = & (L_3\circ L_1)\otimes (L_4\circ L_2)\end{aligned}$$
\(2) $\Delta_M\otimes\Delta_N = \Big\{(m,n,m,n):m\in M\textrm{ and } n\in N\Big\} = \Delta_{M\otimes N}.$
\(3) The associativity between objects is trivial.
\(4) For morphisms, we have,
$$\begin{aligned}
L_1\otimes L_2 & = & \Big\{(m,n,a,b):(m,a)\in L_1\textrm{ and }(n,b)\in L_2\Big\}\\
(L_1\otimes L_2)\otimes L_3 & = & \Big\{(m,n,p,a,b,c):(m,a)\in L_1,(n,b)\in L_2,\\
& & (p,c)\in L_3\Big\}\end{aligned}$$
and, $$\begin{aligned}
L_2\otimes L_3 & = & \Big\{(n,p,b,c):(n,b)\in L_2\textrm{ and }(p,c)\in L_3\Big\}\\
L_1\otimes (L_2\otimes L_3) & = & \Big\{(m,n,p,a,b,c):(m,a)\in L_1,(n,b)\in L_2, \\
&& (p,c)\in L_3\Big\}.\end{aligned}$$
\(5) is trivial.
Lagrangian operads
------------------
If $\mathcal{SYM}$ were a true category, we could consider the endomorphism operad of a symplectic manifold $M$. However, we may be able to restrict to a subset of Lagrangian submanifolds ${{\mathcal O}}_{\operatorname{rest}}^n(M) \subset {{\mathcal O}}^n(M)$ for each $n\geq 0$ such that the composition $$L_n(L_{k_1},\dots, L_{k_n}) : = L_n\circ(L_{k_1}\otimes\dots\otimes L_{k_n}),$$ yields always a Lagrangian submanifold in ${{\mathcal O}}_{\operatorname{rest}}^{k_1+\dots+k_n}(M)$ for every $L_n \in
{{\mathcal O}}_{\operatorname{rest}}^n(M)$ and $L_{k_i} \in {{\mathcal O}}_{\operatorname{rest}}^{k_i}(M)$, $i=1,\dots,n$. For instance, there is alway the trivial choice $${{\mathcal O}}_{\mathrm{rest}}^1(M) = \Big\{\Delta_M\Big\},\quad {{\mathcal O}}_{\mathrm{rest}}^n(M)=\emptyset,\quad n\neq 1.$$
In this way, we may get a true operad ${{\mathcal O}}_{\operatorname{rest}}(M)$.
The next natural question to ask is the following.
**What is a product in a Lagrangian operad over M?**
As a first hint, take the situation where the symplectic manifold is a symplectic groupoid $G$. In this case, we may generate an operad from the multiplication space $G^m\in {{\mathcal O}}^2(G)$ and the base $G^{(0)}\in {{\mathcal O}}^0(G)$, the identity being the diagonal $\Delta_G\in {{\mathcal O}}^1(G)$. Remark that $G^m$ is a product in this operad, i.e., that $G^m(G^m,\Delta_G) = G^m(\Delta_G, G^m)$. Notice that the inverse of the symplectic groupoid does not play any role in this construction.
We will answer this question completely for the case were the symplectic manifold is $T^*{\mathbb R}^d$ and will try to develop a deformation theory for the product in this case.
### Local cotangent Lagrangian operads {#local-cotangent-lagrangian-operads .unnumbered}
Remember that ${\mathrm{T}^*{\mathbb R}^{d}}$ has always a structure of a symplectic groupoid over ${\mathbb R}^d$: the trivial one. The multiplication space is given in this case by $$\Delta_2 =\Big\{(p_1,x),(p_2,x),(p_1+p_2,x):p_1,p_2\in {\mathbb R}^{d*},\quad x\in {\mathbb R}^d\Big\}.$$ The base is $$\Delta_0 = \Big\{(0,x):x\in {\mathbb R}^d\Big\}.$$ If we set further $$\Delta_n := \Big\{(p_1,x),\dots,(p_n,x),(p_1+\dots+p_n,x):p_i\in {\mathbb R}^{d*},x\in {\mathbb R}^d\Big\},$$ it it immediate to see that the operad generated by $\Delta_0$ and $\Delta_2$ is exactly $${{{\mathcal O}}_{\Delta}^{n}}(T^*{\mathbb R}^d) = \Big\{\Delta_n\Big\},$$ and that $\Delta_2$ is a product in it.
Following [@Cattaneo2002], we will call this operad the **cotangent Lagrangian operad** over $T^*{\mathbb R}^d$. It is the exact analog of the product operad in a monoidal category, the only difference is that there is no true endomorphism operad to embed ${{\mathcal O}}_\Delta({\mathrm{T}^*{\mathbb R}^{d}})$ into. The idea now is to enlarge the cotangent Lagrangian operad, i.e., by considering Lagrangian submanifolds close enough to $\Delta_n$ for each $n\in {\mathbb N}$ in order to have still an operad.
Notice at this point that the $\Delta_n$’s are given by generating functions. Namely, we may identify $\overline{(T^*{\mathbb R}^d)^n}\times T^*{\mathbb R}^d$ with $T^*B_n$, where $B_n := ({\mathbb R}^{d*})^n\times {\mathbb R}^d$. Then, $$\Delta_n = \left\{
\left(\left(p_1,\frac{\partial S_0^n}{\partial p_1}(z)\right)
,\dots,
\left(p_n,\frac{\partial S_0^n}{\partial p_n}(z)\right),
\left(\frac{\partial S_0^n}{\partial x}(z),x \right)\right)
:z=(p_1,p_2,x)\in B_n
\right\}$$ where $S_0^n$ is the function on $B_n$ defined by [^2] $$S_0^n(p_1,\dots,p_n,x) = \sum_{i=1}^d(p_1^i+\dots+p_n^i) x_i.$$ The cotangent Lagrangian operad may then be identified with $${{{\mathcal O}}_{\Delta}^{n}} =\Big\{S_0^n\Big\},\quad {{{\mathcal O}}_{\Delta}^{0}} = \Big\{0\Big\}.$$ In order to define a deformation operad for $S$, a natural idea would be to consider Lagrangian submanifolds whose generating functions are of the form $$F = S_0^n +\tilde F,$$ where $\tilde F\in C^\infty(B_n)$. The Lagrangian submanifold associated to $F$ is $$L_F := {\operatorname{graph}}dF.$$ As such, the idea does not work in general. In fact, we have to consider generating functions only defined in some neighborhood. Let us be more precise.
We introduce the following notation, $$B_n^0 = \{0\}\times {\mathbb R}^d\subset B_n,$$ $V(B_n^0)$ will stand for the set of all neighborhoods of $B_n^0$ in $B_n$.
We define ${\mathcal O^{n}_{\mathrm{loc}}({\mathrm{T}^*{\mathbb R}^{d}})}$ to be the space of germs at $B_n^0$ of smooth functions $\tilde F$ (defined on an open neighborhood $U_{\tilde F}\subset B_n$ of $B_n^0$) which satisfy $\tilde F(0,x) = 0$ and $\nabla_p \tilde F(0,x) = 0$. Note that the composition will always be understood in terms of composition of germs.
\[prop:comp\] Let be $F\in{{{\mathcal O}}_{\Delta}^{n}}+{{{\mathcal O}}_{\mathrm{loc}}^{n}}$ and $G_i\in{{{\mathcal O}}_{\Delta}^{k_i}} + {{{\mathcal O}}_{\mathrm{loc}}^{k_i}}$ for $i=1,\dots,n$. Consider the function $\phi$ defined by the formula $$\begin{gathered}
\Phi(p_G,x_F) = G_1\cup\dots\cup G_n(p_G,x_G) + F(p_F,x_F) - x_G p_F\label{phi-func}\end{gathered}$$ $$\begin{aligned}
p_F & = & \nabla_x G_1\cup\dots\cup G_n(p_G,x_G)\label{eq1},\\
x_G & = & \nabla_p F(p_F,x_F)\label{eq2}, \end{aligned}$$ where $$G_1\cup\dots\cup G_n(p_G,x_G):= G_1(p_{G_1},x_{G_1})+ \dots + G_n(p_{G_n},x_{G_n})$$ and $p_G = (p_{G_1},\dots,p_{G_n}),$ $p_{G_i} \in ({\mathbb R}^{d*})^{k_i}$, $x_{G_i}\in {\mathbb R}^d$ and $(p_{G_i},x_{G_i})\in U_{G_i}$, for $i=1,\dots,n$.
Then, $$\phi \in {{{\mathcal O}}_{\Delta}^{k_1+\dots+k_n}}+{{{\mathcal O}}_{\mathrm{loc}}^{k_1+\dots+k_n}},\quad \textrm{and}\quad L_\phi = L_F(L_{G_1},\dots,L_{G_n}).$$ In other words, ${{{\mathcal O}}_{\Delta}^{}} +{{{\mathcal O}}_{\mathrm{loc}}^{}}$ together with the product $$\phi = F(G_1,\dots,G_n)$$ is an operad.
Moreover, the induced operad structure on ${{{\mathcal O}}_{\mathrm{loc}}^{}}$ is given by $$R(\tilde F;\tilde G_1,\dots,\tilde G_n) = H,$$ where $H$ is the function $H\in {{{\mathcal O}}_{\mathrm{loc}}^{k_1+\dots+k_n}}$ defined by $$H(p_G,x_F) = \tilde G(p_G,x_G) +\tilde F(p_F,x_F) -\nabla_p\tilde F(p_F,x_F)\nabla_x\tilde G(p_G,x_G),$$ $$\begin{aligned}
p_F & = & p_F^0 + \nabla_x\tilde G(p_G,x_G),\quad p_F^0:= (p_{G_1}^\Sigma,\dots,p_{G_n}^\Sigma),\\
x_G & = & x_G^0 +\nabla_p\tilde F(p_F,x_F),\quad x_G^0 := (x_F,\dots,x_F).\end{aligned}$$
Formula for $\Phi$ can be interpreted in terms of saddle point evaluation for $\hbar\to0$ of the following integral: $$\begin{gathered}
\int {{\mathrm{e}}}^{\frac i\hbar
\left[F(p^1,\dots,p^k,x)+\sum_{i=1}^k \left(G_i(\pi^{i1},\dots,\pi^{il_i},y_i)
-p^i\cdot y_i\right)\right]}\;
\prod_{i=1}^k\frac{{{\mathrm{d}}}^np^i\,{{\mathrm{d}}}^ny_i}{(2\pi\hbar)^n} =\\
= {{\mathrm{e}}}^{\frac{{\mathrm{i}}}\hbar\Phi(\pi^{11},\dots,\pi^{1l_1},\pi^{21},\dots,\pi^{2l_2},\dots\dots,
\pi^{k1},\dots,\pi^{kl_k},x)}\,(C+O(\hbar)),\end{gathered}$$ where $C$ is some constant.
To simplify the computations, we identify $(T^*{\mathbb R}^d)^n$ with $T^*({\mathbb R}^{dn})$ and $(T^*{\mathbb R}^d)^{k_i}$ with $T^*({\mathbb R}^{dk_i})$. With this identifications the graphs of $F$ and $G_i$, $i=1,\dots,n$ may be written as $$\begin{aligned}
L_F & = & \Big \{ \Big(\big(p_F, \nabla_p F(p_F,x_F)\big), \big(\nabla_xF(p_F,x_F), x_F \big) \Big):\\
& & (p_F,x_F)\in U_F\Big\}\subset T^*({\mathbb R}^{dn})\times T^*{\mathbb R}^d,\\
L_{G_i} & = & \Big \{ \Big(\big(p_{G_i}, \nabla_p {G_i}(p_{G_i},x_{G_i})\big), \big(\nabla_x{G_i}(p_{G_i},x_{G_i}), x_{G_i}\big) \Big):\\
& & (p_{G_i},x_{G_i})\in U_{G_i}\Big\}\subset T^*({\mathbb R}^{dk_i})\times T^*{\mathbb R}^d,\\\end{aligned}$$ where $U_F\in V(B_n^0)$ and $U_{G_i}\in V(B_{k_i}^0)$ for $i=1,\dots,n$.
Consider now the composition, $$L_F(L_{G_1},\dots,L_{G_n}) = L_F \circ (L_{G_1} \otimes \dots \otimes L_{G_n}).$$ First of all, observe that, $$\begin{aligned}
L_G & := & L_{G_1} \otimes \dots \otimes L_{G_n}\\
& = & \Big\{\Big(\big(p_G, \nabla_p G(p_G,x_G)\big),\big(\nabla_x G(p_G,x_G), x_G \big)\Big)
:(p_{G_i},x_{G_i})\in U_{G_i}\Big\}\\
L_G & \subset & T^*({\mathbb R}^{d(k_1+\dots+k_n)})\times T^*({\mathbb R}^{dn}).\end{aligned}$$ Thus, $$\begin{aligned}
L_F \circ L_G & = & \pi \Big( (L_G \times L_F) \cap ( T^*{\mathbb R}^{d(k_1+\dots+k_n)} \times \Delta_{T^*{\mathbb R}^{dn}} \times T^*{\mathbb R}^d)\Big)\\
& = & \Big \{ \Big (\big( p_G, \nabla_p G(p_G,x_G)\big) ,\big( \nabla_x F(p_F,x_F),x_F\big)\Big):\\
& & :\quad x_G = \nabla_p F(p_F,x_F),\quad p_F = \nabla_x G(p_G,x_G), \quad(p_G,x_F)\in \tilde U \Big \}\\
L_F \circ L_G & \subset & T^*({\mathbb R}^{d(k_1+\dots+k_n)})\times T^*{\mathbb R}^d, \end{aligned}$$
where $\tilde U$ is the subset of $(p_G,x_F)\in B_{k_1+\dots+k_n}$ such that the system, $$\begin{aligned}
p_F & = & \nabla_x G(p_G,x_G),\\
x_G & = & \nabla_p F(p_F,x_F),\end{aligned}$$ has a unique solution $(p_F,x_G)$ and such that $(p_{G_i},x_{G_i})\in U_{G_i}$, $i=1,\dots,n$, and $(p_F,x_F)\in U_F$. Let us check that $\tilde U$ always exists and is a neighborhood of $B_{k_1+\dots+k_n}^0$. To begin with, observe that for any $(0,x_F)\in B_n^0$ this system has the unique solution $(0,\nabla_pF(0,x_F))$. Set now, $$H(p_G,x_F,p_F,x_G) = \left(\begin{array}c
p_F-\nabla_x G(p_G,x_G)\\
x_F-\nabla_p F(p_F,x_F)
\end{array}\right).$$ Thanks to the fact that $G(0,x) = \sum_{i=1}^n G_i(0,x) = 0 $ we get that the Jacobi matrix $$D_{p_F,x_G}H\big((0,x_f,0,\nabla_pF(0,x_F)\big) = \left(\begin{array}{cc}
{\operatorname{id}}& 0\\
-\nabla_p\nabla_pF(0,x_F) & {\operatorname{id}}\end{array}\right)$$ is invertible.
Thus, the implicit function theorem gives us the desired neighborhood $\tilde U$ of $B_{k_1+\dots+k_n}^0$.
Now, take $\phi$ as defined in (\[phi-func\]). The previous considerations tell us that $\phi$ is exactly defined on $\tilde U$. Let us compute its graphs, $$L_\phi = \Big \{ \Big(\big(p_G, \nabla_p \Phi(p_G,x_F)\big), \big(\nabla_x \Phi(p_G,x_F),x_F \big)
:(p_G,x_F)\in \tilde U\Big\}.$$ We have that $$\begin{gathered}
\nabla_p\phi(p_G,x_F) = \nabla_p G(p_G,x_G)+\nabla_xG(p_G,x_G)\frac{dx_G}{dp}+\\
+\nabla_pF(p_F,x_F)\frac{dp_F}{dp}-p_F\frac{dx_G}{dp}-\frac{dp_F}{dp}x_G
= \nabla_pG(p_G,x_G).\end{gathered}$$ Similarly, $\nabla_x\phi(p_G,x_F) = \nabla_xF(p_F,x_F)$. Thus, $L_\phi = L_F\circ L_G$.
At last, let us check that $\phi\in{{{\mathcal O}}_{\mathrm{loc}}^{k_1+\dots+k_n}}$. First of all, remember that $$\begin{aligned}
F(p_F,x_F) & = & p_F^\Sigma x_F+\tilde F(p_F,x_F) \\
G(p_G,x_F) & = & \sum_{i=1}^n p_{G_i}^\Sigma x_{G_i} +\tilde G(p_G,x_G).\end{aligned}$$ Thus, we obtain immediately that $$\phi(p_G,x_F) = p_G^\Sigma x_F+H(p_G,x_F),$$ where $H$ is a function only defined on $\tilde U$ by the equations, $$H(p_G,x_F) = \tilde G(p_G,x_G) +\tilde F(p_F,x_F) -\nabla_p\tilde F(p_F,x_F)\nabla_x\tilde G(p_G,x_G),$$ $$\begin{aligned}
p_F & = & p_F^0 + \nabla_x\tilde G(p_G,x_G),\quad p_F^0:= (p_{G_1}^\Sigma,\dots,p_{G_n}^\Sigma),\\
x_G & = & x_G^0 +\nabla_p\tilde F(p_F,x_F),\quad x_G^0 := (x_F,\dots,x_F).\end{aligned}$$ But now, if we set $p_G = 0$ then $p_F = 0$, $x_G = x_G^0+\nabla_p\tilde F(0,x_F)$ and $H(0,x_F) = 0$. Similarly, one easily checks that $\nabla_p H(0,x_F) = 0 $.
We will call the operad ${{{\mathcal O}}_{\Delta}^{}}+{{{\mathcal O}}_{\mathrm{loc}}^{}}$ **local cotangent Lagrangian operad** over ${\mathrm{T}^*{\mathbb R}^{d}}$ or for short the local Lagrangian operad when no ambiguities arise. The induced operad ${{{\mathcal O}}_{\mathrm{loc}}^{}}$ will be called the **local deformation operad of ${{{\mathcal O}}_{\Delta}^{}}$**.
### Associative products in the local deformation operad {#associative-products-in-the-local-deformation-operad .unnumbered}
We say that a generating function $S\in C^\infty(B_2)$ satisfies the **Symplectic Groupoid Associativity equation** if for a point $(p_1,p_2,p_3,x)\in B_3$ sufficiently close to $B_3^0$ the following implicit system for $\bar x,\bar p,\tilde x$ and $\tilde p$, $$\bar x =\nabla_{p_1}S(\bar p ,p_3,x) ,\quad \bar p =\nabla_{x}S(p_1 ,p_2,\bar x),$$ $$\tilde x =\nabla_{p_2}S(p_1,\tilde p,x) ,\quad \tilde p =\nabla_{x}S(p_2 ,p_3,\tilde x),$$ has a unique solution and if the following additional equation holds $$S(p_1,p_2,\bar x) + S(\bar p,p_3,x)-\bar x\bar p = S(p_2,p_3,\tilde x)+S(p_1,\tilde p,x)-\tilde x\tilde p .$$
If $S$ also satisfies the **Symplectic Groupoid Structure conditions**, i.e., if $$S(p,0,x)=S(0,p,x)=px \qquad\textrm{ and }\qquad S(p,-p,x) =0$$ then $S$ generates a Poisson structure $$\alpha(x) = 2\big(\nabla_{p_k^1}\nabla_{p_l^2}S(0,0,x)\big)_{k,l=1}^d$$ on ${\mathbb R}^d$ together with a local symplectic groupoid integrating it, whose structure maps are given by $$\begin{array}{cccc}
\epsilon(x) & = & (0,x) &\textbf{unit map}\\
i(p,x) & = & (-p,x) &\textbf{inverse map}\\
s(p,x) & = & \nabla_{p_2}S(p,0,x) & \textbf{source map}\\
t(p,x) & = & \nabla_{p_1}S(0,p,x) & \textbf{target map}.
\end{array}$$
In this case, we call $S$ a generating function of the Poisson structure $\alpha$ or a generating function of the local symplectic groupoid. See [@CDF2005], [@dherin2004] and [@dherin2005] for proofs and explanations about generating functions of Poisson structures.
The following Proposition explains what is a product in the local cotangent Lagrangian operad.
$\tilde S\in{{{\mathcal O}}_{\mathrm{loc}}^{2}}$ is a product in ${{{\mathcal O}}_{\mathrm{loc}}^{}}$ iff $S = S_0^2+\tilde S$ satisfies the Symplectic Groupoid Associativity equation.
We know that $\tilde S$ is a product in ${{{\mathcal O}}_{\mathrm{loc}}^{}}$ iff $S = S_0^2+\tilde S$ is a product in ${{{\mathcal O}}_{\Delta}^{}}+ {{{\mathcal O}}_{\mathrm{loc}}^{}}$, i.e., iff $S(S,I) = S(I,S)$. Let us compute. $$\begin{aligned}
S(S,I)(p_1,p_2,p_3,x) & = & S\cup I(p_1,p_2,p_3,\bar x_1,\bar x_2)+S(\bar p_1,\bar p_2,x) -\bar x_1\bar p_1-\bar p_2\bar x_2\\
& = & S(p_1,p_2,\bar x_1)+p_3\bar x_2 + S(\bar p_1,\bar p_2,x)-\bar p_1\bar x_1-\bar p_2\bar x_2,\end{aligned}$$ with $$\begin{aligned}
\bar p_1 & = & \nabla_{x_1}S\cup I(p_1,p_2,p_3,\bar x_1,\bar x_2) = \nabla_xG(\bar x)\\
\bar p_2 & = & \nabla_{x_2}S\cup I(p_1,p_2,p_3,\bar x_1,\bar x_2) = p_3\\
\bar x_1 & = & \nabla_{p_1}S(\bar p_1,\bar p_2,x)\\
\bar x_2 & = & \nabla_{p_2}S(\bar p_1,\bar p_2,x).\end{aligned}$$ Then we get $$S(S,I) = S(p_1,p_2,\bar x)+S(\bar p,p_3,x)-\bar p\bar x,$$ $$\begin{aligned}
\bar x & = & \nabla_{p_1}S(\bar p,p_3,x)\\
\bar p & = & \nabla_xS(p_1,p_2,\bar x).\end{aligned}$$ Similarly, we get $$S(I,S) = S(p_2,p_3,\tilde x)+S(p_1,\tilde p,x)-\tilde p\tilde x,$$ $$\begin{aligned}
\tilde x & = & \nabla_{p_2}S(p_1,\tilde p,x)\\
\tilde p & = & \nabla_xS(p_2,p_3,\tilde x).\end{aligned}$$ Hence, $\tilde S\in{{{\mathcal O}}_{\mathrm{loc}}^{2}}(T^*{\mathbb R}^d)$ is a product iff $S_0^2+\tilde S$ satisfies the SGA equation.
At this point, we may still introduce the Gerstenhaber bracket as in $(\ref{Gerstenhaber})$ and the product equation in terms of the bracket would still be $\frac12[\tilde S,\tilde S] = 0$. We may also still write a formula for the coboundary operator. But, as this time the compositions in ${{{\mathcal O}}_{\Delta}^{}}+{{{\mathcal O}}_{\mathrm{loc}}^{}}$ are not multilinear, we cannot develop the expression $\frac12[\tilde S,\tilde S]$ in terms of the coboundary operator. Nevertheless, in Section \[FormalOperad\], we will develop the bracket with help of Taylor’s expansion and recover a form very close to Equations (\[recursive\]) in the additive category case.
### Equivalence of associative products {#equivalence-of-associative-products .unnumbered}
To each $F\in {{{\mathcal O}}_{\Delta}^{1}} +{{{\mathcal O}}_{\mathrm{loc}}^{1}}$, we may associate a symplectomorphism $\psi_F$ which is defined only on a neighborhood $U_F$ of $B_1^0$ in $T^*{\mathbb R}^d$ and which fixes $B_0^1$. The composition of two such $\psi_G$ and $\psi_F$, which may always be defined on a possibly smaller neighborhood $\tilde U\subset U_G$ of $B_1^0$ , is exactly $\psi_{F(G)}$ where $F(G)$ is the composition of $F$ by $G$ in the local Lagrangian operad.
We denote by $F^{-1} \in {{{\mathcal O}}_{\Delta}^{1}}+{{{\mathcal O}}_{\mathrm{loc}}^{1}}$ the generating function of the $(\psi_F) ^{-1}$, i.e., the generating function such that $F(F^{-1}) = F^{-1}(F) = I$. Two associative products $S$ and $\tilde S$ will be called equivalent if $$\tilde S = F(S)(F^{-1},F^{-1})$$ for a certain $F\in{{{\mathcal O}}_{\Delta}^{1}} +{{{\mathcal O}}_{\mathrm{loc}}^{1}}$. It is clear that if $S\in {{{\mathcal O}}_{\Delta}^{1}} + {{{\mathcal O}}_{\mathrm{loc}}^{1}}$ is an associative product, then $\tilde S $ also is. The following questions naturally arises.
In fact, two equivalent associative products, which are also generating functions of local symplectic groupoids, induce isomorphic local symplectic groupoids. The isomorphism is given explicitly by $\psi_F$. As a consequence the induced Poisson structures on the base are the same, i.e., $$\alpha(x) = \nabla_{p_1}\nabla_{p_2} S(0,0,x) = \nabla_{p_1}\nabla_{p_2} \tilde S(0,0,x).$$
The following two Propositions prove these statements.
Let be $F\in {{{\mathcal O}}_{\Delta}^{1}}+ {{{\mathcal O}}_{\mathrm{loc}}^{1}}$. The following implicit equations, $$\begin{aligned}
\label{mapeq1}
x_1 & = & \nabla_p F(p_1,x_2)\\
\label{mapeq2} p_2 & = &\nabla_x F(p_1,x_2),\end{aligned}$$ define a symplectomorphism $\psi_F(p_1,x_1) = (p_2,x_2)$ on a neighborhood $U_F$ of $B_1^0 =\left\{(0,x):x\in{\mathbb R}^d\right\}$ in $T^*{\mathbb R}^d$ which fixes $B_1^0$ and which is close to the identity in the sense that $F(p,x)= px+\tilde F(p,x)$ induces the identity if $\tilde F = 0$. Consider now $\psi_F$ and $\psi_G$ defined respectively on $U_F$ and $U_G$ for $F,G\in{{{\mathcal O}}_{\Delta}^{1}} + {{{\mathcal O}}_{\mathrm{loc}}^{1}}$. Then we have that $\psi_G\circ\psi_F = \psi_{F(G)}$ on $U_{F(G)}$.
\(1) Let us check that the system (\[mapeq1\]) and (\[mapeq2\]) generates a diffeomorphism around $B_1^0$. Namely one verifies that $(\bar p_1,\bar x_1,\bar p_2,\bar x_2):= (0, \nabla_p F(0,x_2),0,x_2)$ is a solution of the system. Set now $$H(p_1,x_1,p_2,x_2) := \left(\begin{array}{c}x_1-\nabla_pF(p_1,x_2)\\
p_2-\nabla_xF(p_1,x_2)\end{array}\right).$$ As $$D_{p_1,x_1}H(\bar p_1,\bar x_2,\bar p_2,\bar x_2) =
\left(\begin{array}{cc}-\nabla_p\nabla_pF(0,\bar x_2)& {\operatorname{id}}\\
\nabla_x\nabla_pF(0,\bar x_2) & 0
\end{array}\right)$$ and $$D_{p_2,x_2}H(\bar p_1,\bar x_2,\bar p_2,\bar x_2) =
\left(\begin{array}{cc}0 & \nabla_x\nabla_pF(0,\bar x_2)\\
{\operatorname{id}}& 0 \end{array}\right),$$ the implicit function theorem gives us the result. Let us call $\tilde U$ the neighborhood of $B_1^0$ where $\psi_F$ is defined.
\(2) We check now that $\psi_F$ is symplectic. From equations (\[mapeq1\]) and (\[mapeq2\]) we get the relation $$\frac{\partial p_l^2}{\partial p_k^1}
= \frac{\partial x_k^1}{\partial x_l^2},$$ which directly implies that $d\psi_F \operatorname J (d\psi_F)^* = \operatorname J$ where $$\operatorname J = \left(\begin{array}{cc}0 & {\operatorname{id}}\\
-{\operatorname{id}}& 0 \end{array}\right).$$
\(3) Let us see that $\psi_F(0,x) = (0,x)$. We have already noticed that $(0,\nabla_pF(0,x_2),0,x_2)$ is a solution of the system $(\ref{mapeq1})$ and $(\ref{mapeq2})$. But $F(p,x) = px + \tilde F(p,x)$ with $\nabla_p\tilde F(0,p) = 0$ and then $\nabla_x\nabla_p F(0,x_2) = x_2$.
\(4) Clearly $F(p,x) = px$ generates the identity.
\(5) Recall that $$\begin{aligned}
L_G & = & \Big \{ \Big(p_1, \nabla_p G(p_1,x_2), \nabla_xG(p_1,x_2), x_2 \Big): (p_1,x_2)\in U_G\Big\},\\
L_F & = & \Big \{ \Big(p_2, \nabla_p F(p_2,x_3), \nabla_xF(p_2,x_3), x_3 \Big):(p_2,x_3)\in U_F\Big\}.\end{aligned}$$ Thus, $L_G = {\operatorname{graph}}\psi_G$ and $L_F = {\operatorname{graph}}\psi_F$. The composition of these two canonical relations yields that $L_F\circ L_G ={\operatorname{graph}}\psi_F\circ \psi_G$. On the other hand, $L_F\circ L_G = L_{F(G)} = {\operatorname{graph}}\psi_{F(G)}$. Taking care on the domain of definitions, we have that $\psi_F\circ \psi_G = \psi_{F(G)}$ on $U_{F(G)}$.
Let $S\in{{{\mathcal O}}_{\Delta}^{2}} + {{{\mathcal O}}_{\mathrm{loc}}^{2}}$ be a generating function of a symplectic groupoid, i.e., $$S(S,I) = S(I,S),\quad S(p,0,x) = S(0,p,x) = px\quad \textrm{ and }\quad S(p,-p,x) = 0.$$ Let $F \in {{{\mathcal O}}_{\Delta}^{1}} + {{{\mathcal O}}_{\mathrm{loc}}^{1}}$ such that $F(-p,x) = -F(p,x)$. Then, $$\tilde S := (F(S))(F^{-1},F^{-1})$$ is also a generating function of a symplectic groupoid. The subset of odd function in $p$ forms a subgroup of ${{{\mathcal O}}_{\Delta}^{1}} + {{{\mathcal O}}_{\mathrm{loc}}^{1}}$. Moreover, $\psi_F$ is a groupoid isomorphism between the local symplectic groupoid generated by $S$ and the one generated by $\tilde S$. As a consequence $S$ and $\tilde S$ induce the same Poisson structure on the base.
To simplify the notation, we set $G=F^{-1}$. A straightforward computation gives that $$\begin{gathered}
F(S)(G,G)(p_1,p_2,x) =
S(\bar p,\tilde p,\dot x)+\\
+ F(\dot p,x) + G(p_1,\bar x) + G(p_2,\tilde x) -\bar p\bar x-\tilde p\tilde x-\dot x\dot p\end{gathered}$$
$$\begin{aligned}
\label{myeq}
\dot x & = \nabla_p F(\dot p,x) & \bar x & = \nabla_{p_1}S(\bar p,\tilde p,\dot x) & \tilde x & = \nabla_{p_2}S(\bar p,\tilde x,\dot x) \\
\dot p & = \nabla_x S(\bar p,\tilde p, \dot x) & \bar p & = \nabla_xG(p_1,\bar x) &\tilde p & = \nabla_x G(p_2,\tilde x)\end{aligned}$$
\(1) Setting $p_1 = p$ and $p_2 = 0$, we have immediately $$F(S)(G,G)(p,0,x) = G(p,\dot x) + F(\dot p,x) -\dot x\dot p$$ with $\dot x = \nabla_{p} F(\dot p,x)$ and $\dot p = \nabla_x G(p,\dot x).$ We recognize then that $$F(S)(G,G)(p,0,x) = F(G)(p,x) = I(p,x) = px.$$ The case $p_1 = 0$ and $p_2 = p$ is analog.
\(2) One reads directly from the equation $$px = F^{-1}(p,\dot x) + F(\dot p,x) -\dot x\dot p$$ where $\dot x = \nabla_p F(\dot p,x)$ and $\dot p = \nabla_x F^{-1}(p,\dot x)$, that if $F$ is odd in $p$ then is also $F^{-1}$ and reciprocally. Similarly, we check directly from the composition formula that $F(G)$ is odd in $p$ if $F$ and $G$ both are. Thus, the odd functions form a subgroup of ${{{\mathcal O}}_{\Delta}^{1}} + {{{\mathcal O}}_{\mathrm{loc}}^{1}}$.
\(3) Suppose now that $p_1 = p$ and $p_2 = -p$. $G$ odd in $p$ implies that $\bar p = -\tilde p$. As $S(p,-p,0) = 0$, we get immediately that $\tilde x =\bar x$ and $\dot p =0$ which in turns implies that $\dot x = x$. Putting everything together, we get that $(F(S))(G,G)(p,-p,x) =0$
\(4) Let us prove now that $\psi_F$ is also a groupoid isomorphism. Consider the multiplication space of the symplectic groupoid generated by an generating function $S$, i.e, $$G^{(m)}( S) = \left\{
(p_1,\nabla_{p_1} S), (p_2,\nabla_{p_2} S),(\nabla_x S,x):\quad
p_1,p_2\in ({\mathbb R}^d)^*,\quad x \in {\mathbb R}^d
\right\},$$ where the partial derivative are evaluated in $(p_1,p_2,x)$.
We have to show that $(\psi_F\times\psi_F\times\psi_F) \left(G^{(m)}(S)\right) = G^{(m)}(\tilde S)$.
A straightforward computation gives that $$\begin{aligned}
\nabla_{p_1}\tilde S(p_1,p_2,x) & = \nabla_{p}G(p_1,\bar x) \\
\nabla_{p_2}\tilde S(p_1,p_2,x) & = \nabla_{p}G(p_2,\tilde x)\\
\nabla_x \tilde S(p_1,p_2,x) & = \nabla_xF(\dot p, x).\end{aligned}$$
From this, we check immediately that $$\begin{aligned}
\psi_G\left(\left(p_1,\nabla_{p_1}\tilde S(p_1,p_2,x)\right)\right) = \left(\bar p,\nabla_{p_1}S(\bar p, \tilde p, \dot x)\right)\\
\psi_G\left(\left(p_2,\nabla_{p_2}\tilde S(p_1,p_2,x)\right)\right) = \left(\tilde p,\nabla_{p_2}S(\bar p, \tilde p, \dot x)\right)\\
\psi_F\left(\left(\nabla_x S(\bar p, \tilde p, \dot x),\dot x\right)\right) = \left(\nabla_x \tilde S(p_1,p_2,x), x\right)\end{aligned}$$ which ends the proof.
Suppose that $S$ is a generating function of a local symplectic groupoid. Let $F\in{{{\mathcal O}}_{\Delta}^{1}} + {{{\mathcal O}}_{\mathrm{loc}}^{1}}$ act on $S$, i.e., $\tilde S =(F(S))(F^{-1},F^{-1})$. Then, the condition $S(p,0,x) = S(0,p,x) = px$ is preserved by any $F\in{{{\mathcal O}}_{\Delta}^{1}}+{{{\mathcal O}}_{\mathrm{loc}}^{1}}$. However, the condition $S(p,-p,0)$ is only preserved by the odd $F$s. Observe now that we have imposed the inverse map to be $i(p,x) = (-p,x)$. This implies that $$\left(
\left(-p_2,\nabla_{p_2}S(p_2,p_1,x)\right),
\left(-p_1,\nabla_{p_1}S(p_2,p_1,x)\right),
\left(-\nabla_x S(p_2,p_1,x),x\right)
\right)\in G^{(m)}(S),$$ and thus, that $S(p_1,p_2,x) = - S(-p_1,-p_2,x)$. From this last equation, we get that $S$ must satisfy $S(p,-p,x) = 0$ and that the induced local symplectic groupoid is a [**s**ymmetric]{} one, i.e., $t(p,x) = s(-p,x)$. Thus, odd transformations map symmetric groupoids to symmetric groupoids. However, they are not the only ones.
The combinatorics {#combinatorics}
=================
In this Section, we present some tools which will allow us to write down at all orders the perturbative version of the composition, Equation (\[phi-func\]), in the local cotangent operad. All these compositions have essentially the same form. We will first give an abstract version of the equations describing the compositions, then we will introduce some trees which will help us to keep track of the terms involved in the computations and, at last, we will perform the expansion in the general case.
The tools and methods presented here are essentially the same as those used in the Runge–Kutta theory of ODEs to determine the order conditions of a particular numeric method. We follows approximatively the notations of [@GeomInt].
The equation {#equation}
------------
Let $F:{\mathbb R}^{n*}\rightarrow {\mathbb R}^n$ and $G:{\mathbb R}^n\rightarrow {\mathbb R}^{n*}$ be two smooth functions. Consider the point $\phi\in{\mathbb R}$ defined by $$\begin{aligned}
\label{phi-eq}\phi & := & G(\bar x)+F(\bar p)-\bar p\bar x,\end{aligned}$$ where $\bar x$ and $\bar p$ are defined by the implicit equations, $$\begin{aligned}
\label{p-eq}
\bar p & = & \nabla_xG(\bar x)\\ \label{x-eq}
\bar x & = & \nabla_pF(\bar p).\end{aligned}$$ Without any assumptions on $F$ and $G$, equations (\[p-eq\]) and (\[x-eq\]) may not have a solution at all or the solution may be not unique. Hence, the value $\phi$ is not always defined. However, if we assume that $F$ and $G$ are formal power series of the form $$G(x) = p_0x+\sum_{i=1}^\infty \epsilon^i G^{(i)}(x),\quad\textrm{ and }\quad
F(p) = x_0p+\sum_{i=1}^\infty \epsilon^i F^{(i)}(p),$$
equations (\[p-eq\]) and (\[x-eq\]) become, $$\bar p = p_0+\sum_{i=1}^n\epsilon^i\nabla_x G^{(i)}(\bar x),\quad\textrm{ and }\quad
\bar x = x_0+\sum_{i=1}^n\epsilon^i\nabla_p F^{(i)}(\bar p),$$ which are always recursively uniquely solvable.
Let us compute the first terms of $\bar p$, $\bar x$ and $\phi$ to get a feeling of what is happening: $$\begin{aligned}
\bar p & = & p_0+\epsilon\nabla_x G^{(1)}(x_0)+\epsilon^2 \nabla_x^{(2)}G^{(1)}(x_0)\nabla_pF^{(1)}(p_0)+\cdots \\
\bar x & = & x_0+\epsilon\nabla_p F^{(1)}(x_0)+\epsilon^2 \nabla_p^{(2)}F^{(1)}(x_0)\nabla_xG^{(1)}(x_0)+\cdots \\
\phi & = &p_0x_0+\epsilon(G^{(1)}(x_0)+F^{(1)}(p_0))+\epsilon^2 2\nabla_p F^{(1)}(p_0)\nabla_x G^{(1)}(x_0)+\cdots\end{aligned}$$
As we continue the expansion, the terms get more and more involved and, very soon, expressions as such become untractable. One common strategy in physics as in numeric analysis is to introduce some graphs to keep track of the fast growing terms. Let us present these graphs. We mainly take our inspiration from the book [@GeomInt].
The trees {#Cayleytree}
---------
-
1. [A **graph** $t$ is given by a set of vertices $V_t = \{1,\dots,n)$ and a set of edges $E_t$ which is a set of pairs of elements of $V_t$. We denote the number of vertices by $|t|$. An **isomorphism** between two graphs $t$ and $t'$ having the same number of vertices is a permutation $\sigma\in S_{|t|}$ such that $\{\sigma(v),\sigma(w)\}\in E_{t'}$ if $\{v,w\}\in E_{t}$. Two graphs are called **equivalent** if there is an isomorphism between them. The **symmetries** of a graph are the automorphisms of the graph. We denote the group of symmetries of a graph $t$ by $sym(t)$.]{}
2. [A **tree** is a graph which has no cycles. Isomorphisms and symmetries are defined the same way as for graphs ]{}
3. [A **rooted tree** is a tree with one distinguished vertex called root. An **isomorphism** of rooted trees is an isomorphism of graphs which sends the root to the root. Symmetries and equivalence are defined correspondingly.]{}
4. [A **bipartite graph** is a graph $t$ together with a map $\omega:V_t\rightarrow
\{\circ,\bullet\}$ such that $\omega(v)\neq\omega(w)$ if $\{v,w\}\in E_t$. An isomorphism of bipartite trees is an isomorphism of graphs which respects the coloring, i.e., $\omega(\sigma
(v))=\omega(v)$.]{}
5. A **weighted graph** is a graph $t$ together with a weight map $L:V_t\rightarrow {\mathbb N}\backslash\{0\}$. An isomorphism of weighted graph is an isomorphism of graph $\sigma$ which respects the weights, i.e., $\sigma(L(v)) = L(\sigma(v))$. We denote by $\|t\|$ the sum of the weights on all vertices of $t$.
The following table summarizes some notations we will use in the sequel.
-------------- ---------------------------------------------
$T$ the set of bipartite trees
$RT$ the set of rooted bipartite trees
$RT_\circ$ the set of elements of $RT$ with white root
$RT_\bullet$ the set of elements of $RT$ with black root
-------------- ---------------------------------------------
We will give the name [**Cayley trees**]{} to trees in $T$.
We denote by $[A]$ the set of equivalence classes of graphs in $A$ (ex: $[RT]$). They are called **topological** “$A$” trees. Moreover, we denote by $A_\infty$ the weighted version of graphs in $A$. Notice that we will use the notation $[A]_\infty$ instead of the more correct $[A_\infty]$.
The elements of $[RT]_\infty$ can be described recursively as follows:
1. $\circ_i,\bullet_j\in [RT]_\infty$ where $i = L(\circ_i)$ and $j= L(\bullet_j)$
2. if $t_1,\dots,t_m\in [RT_\circ]_\infty$, then the tree $[t_1,\dots,t_m]_{\bullet_i}\in [RT]_\infty$ where $[t_1,\dots,t_m]_{\bullet_i}$ is defined by connecting the roots of $t_1,\dots,t_m$ with the weighted vertex $\bullet_i$ and declaring that $\bullet_i$ is the new root. And the same if we interchange $\circ$ and $\bullet$.
Now, let us describe in terms of trees the expressions arising in the expansions of Subsection \[equation\].
Given two collections of functions $F=\{F_i)_{i=1}^\infty$ and $G=\{G_j)_{j=1}^\infty$, where $F_i:{\mathbb R}^{n*}\rightarrow {\mathbb R}^d$ and $G_j:{\mathbb R}^n\rightarrow {\mathbb R}^{d*}$ are smooth functions, we may associate to any rooted tree $t\in[RT]_\infty$ a vector field on $T^*{\mathbb R}^d$, $DC_t(F,G)\in {\operatorname{Vect}}(T^*{\mathbb R}^d)$, called the **elementary differential** and a function on $T^*{\mathbb R}^d$, $C_t(F,G)\in C^\infty (T^*{\mathbb R}^d)$, called the **elementary function**.
1. The **elementary differential** $DC_t(F,G)$ is recursively defined as follows:
1. $DC_{\circ_i}(F,G)(p,x) = \nabla_x G^{(i)}(x)$ , $DC_{\bullet_j}(F,G)(p,x) = \nabla_p F^{(j)}(p)$
2. $DC_{t}(F,G) = \nabla_x^{(m+1)}G^{(i)}(DC_{t_1}(F,G),\dots,DC_{t_m}(F,G))$ if $t= [t_1,\dots,t_m]_{\circ_i}$
3. $DC_{t}(F,G) = \nabla_p^{(m+1)} F^{(j)}(DC_{t_1}(F,G),\dots,DC_{t_m}(F,G))$ if $t= [t_1,\dots,t_m]_{\bullet_j}$.
2. The **elementary function** $C_t(F,G)$, are recursively defined as follows:
1. $C_{\circ_i}(F,G)(p,x) = G^{(i)}(x)$ ,$C_{\bullet_j}(F,G)(p,x) = F^{(j)}(p)$
2. $C_{t}(F,G) = \nabla_x^{(m)}G^{(i)}(DC_{t_1}(F,G),\dots,DC_{t_m}(F,G))$ if $t= [t_1,\dots,t_m]_{\circ_i}$.
3. $C_{t}(F,G) = \nabla_p^{(m)} F^{(j)}(DC_{t_1}(F,G),\dots,DC_{t_m}(F,G))$ if $t= [t_1,\dots,t_m]_{\bullet_j}$.
The notation $\nabla_x^{(m)}$ (resp. $\nabla_p^{(m)}$) stands for the $m^{th}$ derivative in the direction $x$ (resp. $p$).
Some examples are given in the following table:
--------- ------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------
Diagram Elementary Differential Elementary Function
$\nabla_x^{(2)}G^{(i)}\nabla_p F^{(j)} $ $\nabla_x G^{(i)}\nabla_p F^{(j)}$
$ \nabla_p^{(3)}F^{(i)}(\nabla_x G^{(j)}, \nabla_x G^{(k)}) $ $\nabla_p^{(2)}F^{(i)}(\nabla_x G^{(j)}, \nabla_x G^{(k)})$
$ \nabla_x^{(3)}G^{(i)}(\nabla_p F^{(j)},\nabla_p^{(2)}F^{(k)}\nabla_x G^{(l)} ) $ $ \nabla_x^{(2)}G^{(i)}(\nabla_p F^{(j)},\nabla_p^{(2)}F^{(k)}\nabla_x G^{(l)} ) $
--------- ------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------
Remark that for elementary functions it is not important which vertex is the root. This is not the case for elementary differentials.
Let $u = [u_1,\dots,u_k], v = [v_1,\dots,v_l]\in [RT]$ (resp. $\in [RT]_\infty$). We denote by $$\begin{aligned}
u\circ v &=& [u_1,\dots,u_k,v]\\
v\circ u &=& [v_1,\dots,v_l,u]\end{aligned}$$ the Butcher product. We have not written the obvious conditions on the $u_i$ and $v_i$ so that the product remains bipartite (resp. weighted bipartite).
Recall that an equivalence relation on a set $A$ is a special subset $R$ of $A\times A$. The equivalence relations on $A$ are moreover ordered by inclusion. It makes then sense to consider the minimal equivalence on $A$ containing a certain subset $U\subset A$.
We consider here the minimal equivalence relation on $[RT]$ (resp. on $[RT]_\infty)$) such that $u\circ v \sim v\circ u$.
**Properties of this relation:**
It is clear that
1. Two topological rooted trees are equivalent if it is possible to pass from one to the other by changing the root. More precisely: $t,t'\in[RT]_{(\infty)}$, $t\sim t'$ iff there exists a representative $(E,V,r)$ of $t$ and a representative $(E',V',r')$ of $t'$ and a vertex $r''\in V$ such that $(E,V,r'')$ and $(E',V',r')$ are isomorphic (weighted) rooted trees.
2. The quotient of $[RT]_{(\infty)}$ by this equivalence relation is exactly $[T]_{(\infty)}$.
3. It follows immediately from the definition that $C_{t}(F,G) = C_{t'}(F,G)$ if $t\sim t'$ for $i=1,2$.
Then, it makes sense to define the elementary functions on bipartite trees.
At last, we introduce some important functions on trees: the symmetry coefficients.
Let $t = [t_1,\dots,t_m]\in [RT]_{\infty}$. Consider the list $\tilde t_1,\dots,\tilde t_k$ of all non isomorphic trees appearing in $t_1,\dots,t_m$. Define $\mu_i$ as the number of time the tree $\tilde t_i$ appears in $t_1,\dots,t_m$. Then we introduce the **symmetry coefficient** $\sigma(t)$ of $t$ by the following recursive definition:
$$\sigma(t) = \mu_1!\mu_2!\dots\sigma(\tilde t_1)\dots\sigma(\tilde t_k)$$ and initial condition $\sigma(\circ_i)=\sigma(\bullet_j)= 1$.
It is clear that $\sigma(t)$ is the number of symmetries for each representative of $t$ (i.e. $\sigma(t) = |Sym(t')|$ for all $t'\in t$).
The expansion
-------------
We give now a power series expansion for equation (\[phi-eq\]).
\[prop:expansion\] Suppose we are given the following formal power series in $\epsilon$, $$G(x) = p_0x+\sum_{i=1}^\infty \epsilon^i G^{(i)}(x),\quad \textrm{ and }\quad
F(p) = x_0p+\sum_{j=1}^\infty \epsilon^j F^{(j)}(p),$$
where $G^{(i)}:{\mathbb R}^n\rightarrow {\mathbb R}^{n*}$ and $F^{(j)}:{\mathbb R}^{n*}\rightarrow {\mathbb R}^n$ are smooth functions for $i,j> 0$.
Define $\phi(p_0,x_0)\in {\mathbb R}[[\epsilon]]$ as $$\phi(p_0,x_0) := G(\bar x)+F(\bar p)-\bar p\bar x,$$ where the formal power series $\bar x(\epsilon)$ and $\bar p(\epsilon)$ are uniquely determined by the implicit equations, $$\bar p = p_0+\sum_{i=1}^\infty \epsilon^i\nabla_x G^{(i)}(\bar x),\quad\textrm{ and }\quad
\bar x = x_0+\sum_{j=1}^\infty \epsilon^j\nabla_p F^{(j)}(\bar p).$$ Then, we have that $$\phi(p_0,x_0) = p_0x_0 +\sum_{t\in T_\infty} \frac{\epsilon^{\|t\|}}{|t|!}C_t(F,G)(p_0,x_0).$$
The proof of Proposition \[prop:expansion\] is broken into several lemmas.
The method used is essentially the same as in numerical analysis when one wants to express the Taylor series of the numerical flow of a Runge–Kutta method. Namely, the defining equations for $\bar p(\epsilon)$ and $\bar x(\epsilon)$ have a form very close to the partioned implicit Euler method(see [@GeomInt]).
There exist unique formal power series for $\bar x(\epsilon)$ and for $\bar p(\epsilon)$ which satisfy equations (\[p-eq\]) and (\[x-eq\]). They are given by $$\begin{aligned}
\label{x-sol}\bar{x}(\epsilon) &=&x_0+\sum_{t\in [RT_{\bullet}]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)}DC_{t}(F,G),\\
\label{p-sol}\bar{p}(\epsilon) &=&p_0+\sum_{t\in [RT_{\circ}]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)}DC_{t}(F,G).
\end{aligned}$$
Uniqueness is trivial. Let us check that we have the right formal series. We only check equation (\[x-sol\]). The other computation is similar. $$\begin{split}
\bar{x}(\epsilon) & = x_0+\sum_{i\geq 1}\epsilon^i\nabla_pF^{(i)}(\bar p) \\
& = x_0+\sum_{i\geq 1}\epsilon^i\sum_{m\geq 0}\frac{1}{m!}
\nabla_p^{(m+1)}F^{(i)}\bigg(\sum_{t\in [RT_{\circ}]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)}DC_{t}(F,G),\dots\\
& \qquad \dots,\sum_{t\in [RT_{\circ}]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)}DC_{t}(F,G)\bigg)\\
& = x_0+\sum_{i\geq 1}\sum_{m\geq 0}\sum_{t_1\in [RT_\circ]_\infty}\dots\sum_{t_m\in [RT_\circ]_\infty}\frac{\epsilon^{i+\|t_1\|+\dots+\|t_m\|}}{m!\sigma(t_1) \dots\sigma(t_m)}\times\\
& \qquad\times\nabla_p^{(m+1)}F^{(i)}(DC_{t_1}(F,G),\dots,DC_{t_m}(F,G))\\
& = x_0 +\sum_{i\geq 1}\sum_{m\geq 0}\sum_{t_1}\dots\sum_{t_m}\frac{\epsilon^{\|t\|}} {m!\sigma(t)}(\mu_1!\mu_2!\dots) DC_{t}(F,G),\\
& \quad \textrm{with }t=[t_1,\dots,t_m]_{\bullet_i}\\
& = x_0 + \sum_{t\in[RT_\bullet]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)} DC_{t}(F,G).
\end{split}$$
We have the following expansion for $\phi(p_0,x_0)$: $$\begin{gathered}
\phi(p_0,x_0) = p_0x_0+ \sum_{t\in [RT]_\infty} \frac{\epsilon^{\|t\|}}{\sigma(t)} C_{t}(F,G)-\\
-\Big(\sum_{t\in [RT_\circ]_\infty}
\frac{\epsilon^{\|t\|}}{\sigma(t)} DC_{t}(F,G) \Big) \Big(\sum_{t\in [RT_\bullet]_\infty}
\frac{\epsilon^{\|t\|}}{\sigma(t)} DC_{t}(F,G) \Big).\end{gathered}$$
We compute the different terms arising in $G(\bar x)+F(\bar p)-\bar p\bar x$ in terms of trees. $$\begin{split}
G(\bar x) & = p_0\bar x + \sum_{i\geq 1}\epsilon^i\sum_{m\geq 0}\frac{1}{m!}\nabla_x^{(m)}G^{(i)} \bigg(\sum_{t\in [RT_{\bullet}]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)}DC_{t}(F,G) ,\dots\\
& \qquad ,\dots,\sum_{t\in [RT_{\bullet}]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)}DC_{t}(F,G)\bigg)\\
& = p_0\bar x + \sum_{i\geq 1}\sum_{m\geq 0}\sum_{t_1\in [RT_\bullet]_\infty}\dots \sum_{t_m\in [RT_\bullet]_\infty}\frac{\epsilon^{\|t\|}}{m!\sigma(t)}(\mu_1!\mu_2!\dots)\times\\
& \quad \times\nabla_x^{(m)}G^{(i)}(DC_{t_1}(F,G),\dots,DC_{t_m}(F,G)),\\
& \quad\qquad \textrm{with }t=[t_1,\dots,t_m]_{\bullet_i}\\
& = p_0\bar x + \sum_{t\in [RT_\circ]_\infty}\frac{\epsilon^{\|t\|}}{\sigma(t)} C_{t}(F,G)
\end{split}$$
By the same sort of computations we obtain, $$F(\bar p) = x_0\bar p+\sum_{t\in [RT_\bullet]_\infty} \frac{\epsilon^{\|t\|}}{\sigma(t)} C_{t}(F,G).$$ Finally, we get the desired result as, $$\begin{gathered}
p_0\bar x+x_0\bar p-\bar p\bar x = p_0x_0 -\\
-\Big(\sum_{t\in [RT_\circ]_\infty}
\frac{\epsilon^{\|t\|}}{\sigma(t)} DC_{t}(F,G) \Big) \Big(\sum_{t\in [RT_\bullet]_\infty}
\frac{\epsilon^{\|t\|}}{\sigma(t)} DC_{t}(F,G) \Big).\end{gathered}$$
Thus, $\phi(p_0,x_0)$ is expressed as sums over topological weighted rooted bipartite trees. We would like now to regroup the terms of the formula in the previous Lemma. To do so, we express all terms in terms of topological trees (no longer rooted).
Let $u\in[RT_\circ]_\infty$ and $v\in [RT_\bullet]_\infty$. Then, $$DC_{u}(F,G)DC_{v}(F,G) = C_{u\circ v}(F,G) = C_{v\circ u}(F,G).$$
Suppose $u=[u_1,\dots,u_m]_{\circ_i}$, $v=[v_1,\dots,v_l]_{\bullet_j}$, then we get $$\begin{aligned}
A & = & DC_{u}(F,G)DC_{v}(F,G)\\
& = & \nabla_x^{(m+1)}G^{(i)}(DC_{u_1}(F,G),\dots,DC_{u_m}(F,G)).DC_{v}(F,G)\\
& = & \nabla_x^{(m+1)}G^{(i)}(DC_{u_1}(F,G),\dots,DC_{u_m}(F,G),DC_{v}(F,G))\\
& = & C_{u\circ v}(F,G).\\\end{aligned}$$
\[Lemma:sym\] Let $t=(V_t,E_t)\in T_\infty$. For all $v\in V_t$ let $t_v$ be the bipartite rooted tree $(V_t,E_t,v)\in RT_\infty$. For $v\in V_t$ and $e= \{u,v\}\in E_t$ we have
$$\begin{aligned}
\frac{|sym(t)|}{|sym(t_v)|} & = & |\{v'\in V_t /t_{v'}\textrm{is isomorphic to } t_v\}|\\
\frac{|sym(t)|}{|sym(t_u)||sym(t_v)|} & = & |\{e'\in E_t /t_{u'}\sqcup t_{v'}\textrm{is isomorphic to } t_u\sqcup t_v\}|\end{aligned}$$
Consider the induced action of the symmetry group of the tree on the set of vertices. Notice that two vertices $v$ and $w$ are in the same orbit iff $t_v$ is isomorphic to $t_w$. Then the number of vertices of $t$ which lead to rooted tree isomorphic to $t_v$ is exactly the cardinality of the orbit of $v$, which is exactly $|sym(t)|$ divided by the cardinality of the isotropy subgroup which fixes $v$. But the latter is $|sym(t_v)|$ by definition. We then get the first statement.
For the second statement we have to consider the induced action on the edges and apply the same type of argument.
We get $$\phi(p_0,x_0) = p_0x_0+\sum_{t\in T_\infty}\frac{\epsilon^{\|t\|}}{|t|!}C_t(F,G).$$
Let us perform the last computation. $$\begin{split}
\phi(p_0,x_0) & = p_0x_0+ \sum_{t\in [RT]_\infty} \frac{\epsilon^{\|t\|}}{\sigma(t)}C_{t}(F,G)-\\
&\qquad -\sum_{u\in[RT_\circ]_\infty}\sum_{v\in[RT_\bullet]_\infty} \frac{\epsilon^{\|u\|+\|v\|}}{\sigma(u)\sigma(v)}DC_{u}(F,G)DC_{v}(F,G)\\
& = p_0x_0+ \sum_{\bar{t}\in [T]_\infty} \epsilon^{|\bar{t}|}C_{\bar{t}}(F,G)
\Big\{ \sum_{t\in\bar{t}} \frac{1}{|sym(t)|}-\\
&\qquad-\sum_{\substack{u\in [RT_\bullet]_\infty,v\in [RT_\circ]_\infty\\ u\circ v \in\bar{t}}} \frac{1}{|sym(u)||sym(v)|} \Big\}\\
& = p_0x_0+ \sum_{t\in T_\infty}\frac{\epsilon^{\|t\|}}{|t|!}C_{t}(F,G)\Big\{ \sum_{v\in V_t} \frac{|sym(t)|}{|sym(t_v)|}\frac{1}{k(t,v)}-\\
&\qquad -\sum_{e = \{u,v)\in E_t} \frac{|sym(t)|}{|sym(t_u)||sym(t_v)|}\frac{1}{l(t,e)} \Big\}
\end{split}$$ where $k(t,v) = |\{v'\in V_t /t_{v'}\textrm{is isomorphic to } t_v\}|$ and $l(t,e) = |\{e'\in E_t /t_{u'}\sqcup t_{v'}\textrm{is isomorphic to } t_u\sqcup t_v)|$. Using Lemma \[Lemma:sym\] and the fact that for a tree the difference between the number of vertices and the number of edges is equal to 1 we get the desired result.
Using now the fact that $S$ is a formal power series we immediately get Proposition \[prop:expansion\].
Deformation of a non-linear structure {#FormalOperad}
=====================================
The formal cotangent Lagrangian operad
--------------------------------------
The formal cotangent Lagrangian operad on $T^*{\mathbb R}^d$ is the perturbative/formal version of the local cotangent operad on $T^*{\mathbb R}^d$. Recall that in the latter the product for $F \in {{{\mathcal O}}_{\Delta}^{n}}+{{{\mathcal O}}_{\mathrm{loc}}^{n}}$ and $G_i \in {{{\mathcal O}}_{\Delta}^{n}}+ {{{\mathcal O}}_{\mathrm{loc}}^{k_i}}$, $i= 1,\dots n$ was expressed as in Proposition \[prop:comp\]: $$F(G_1,\dots,G_n)(p_G,x_F) = G_1 \cup \dots \cup G_n (p_G,x_G) + F(p_F,x_F) - p_F \cdot x_G,$$ $$\begin{aligned}
p_F & = & \nabla_x G_1 \cup \dots \cup G_n (p_G,x_G) , \\
x_F & = & \nabla_p F(p_F,x_F). \end{aligned}$$ If we consider $p_G$ and $x_F$ as parameters in the previous equations, we have then that, $$G(p_G, \cdot) : {\mathbb R}^{nd} \longrightarrow {\mathbb R},\quad\textrm{ and } F( \cdot, x_F) : ({\mathbb R}^{nd})^* \longrightarrow {\mathbb R},$$ Suppose now that the $F$ and $G_i$, $i=1,\dots,n$, are formal series of the form $$\begin{aligned}
F(p_F,x_F) & = & p_F^{\Sigma} \cdot x_F + \sum_{i=1}^{\infty} \epsilon^i F^{(i)}(p_F,x_F)\\
G_l(p_{G_l},x_{G_l}) & = & p_{G_l}^{\Sigma} \cdot x_{G_l} + \sum_{i=1}^{\infty}\epsilon^i G_l^{(i)}(p_{G_l},x_{G_l})\end{aligned}$$ where $$p^{\Sigma} := \sum_{i=1}^n p_i\quad\textrm{ for }\quad p=(p_1,\dots,p_n)\in({\mathbb R}^{dn})^*.$$ We may rewrite $F$ and $G$ as, $$\begin{gathered}
F(p_F,x_F) = x_0^Fp_F + \sum_{i=1}^{\infty} \epsilon^i F^{(i)}(p_F,x_F)\end{gathered}$$ $$\begin{gathered}
G(p_G,x_G) = p_0^G x_G + \sum_{i=1}^{\infty} \epsilon^i G^{(i)}(p_G,x_G)\end{gathered}$$ where $x_0^F = (x_F,\dots,x_F) \in {\mathbb R}^{dn}$ and $p_0^G = (p_{G_1}^\Sigma,\dots,p_{G_n}^\Sigma) \in ({\mathbb R}^{nd})^*$ for $x_G \in {\mathbb R}^{dn}$ and $p_F \in ({\mathbb R}^{dn})^*.$
Applying now Proposition \[prop:expansion\], we obtain for the compositions the following expansion: $$\begin{gathered}
\label{treecomp}
F(G_1,\dots,G_n)(p_G,x_G) = p_G^{\Sigma} \cdot x_F +\\ +\sum_{t \in T_{\infty}} \frac{\epsilon^{||t||}}{|t|!} C_t \Big(
F(\cdot,x_F),G_1 \cup \dots \cup G_n (p_G,\cdot)\Big) (p_0^G,x_0^F).\end{gathered}$$
This motivates to define the formal deformation space of the cotangent Lagrangian operad ${{{\mathcal O}}_{\Delta}^{}}({\mathrm{T}^*{\mathbb R}^{d}})$ as $${{{\mathcal O}}_{\mathrm{form}}^{n}}({\mathrm{T}^*{\mathbb R}^{d}},\Delta) := \Big\{ \sum_{i=1}^{\infty}\epsilon^i F^{(i)} : F^{(i)} \in P_i^n(T^*{\mathbb R}^d) \Big\},$$ where $P_i^n(T^*{\mathbb R}^d)$ stands for the vector space of functions $F:B_n \longrightarrow {\mathbb R}$ such that
1. $F(p,x)$ is a polynomial in the variables $p=(p_1,\dots,p_n)$,
2. $F(\mu p,x) = \mu^{i+1}F(p,x).$
One may think of ${{{\mathcal O}}_{\mathrm{loc}}^{}} +{{{\mathcal O}}_{\mathrm{form}}^{}}$ as the Taylor series of functions in ${{{\mathcal O}}_{\Delta}^{}}+{{{\mathcal O}}_{\mathrm{loc}}^{}}$ The compositions are given by formula (\[treecomp\]), which also tells us that ${{{\mathcal O}}_{\Delta}^{}}+{{{\mathcal O}}_{\mathrm{loc}}^{}}$ is an operad. The unit is $$I(p,x)= px,\quad I \in {{{\mathcal O}}_{\Delta}^{}} + {{{\mathcal O}}_{\mathrm{form}}^{1}}.$$
The induced operad structure on ${{{\mathcal O}}_{\mathrm{form}}^{}}$ is then given by, $$\begin{gathered}
I \in {{{\mathcal O}}_{\mathrm{form}}^{1}},\quad I(p,x)=0,\\
{{{\mathcal O}}_{\mathrm{form}}^{n}} = \Big\{ \sum_{i=1}^{\infty} \epsilon^i F^{(i)} : F^{(i)} \in P_i^n(T^*{\mathbb R}^d)\Big \}\\
F(G_1,\dots,G_n)(p_G,x_F) = \sum_{t \in T_{\infty}}\frac{\epsilon^{||t||}}{|t|!} C_t \big(
F, G_1 \cup \dots \cup G_n\big).\end{gathered}$$ This operad will be called **the formal deformation operad** of the cotangent Lagrangian operad ${{{\mathcal O}}_{\Delta}^{}}$.
Product in the formal deformation operad
----------------------------------------
Exactly as for the local deformation operad, $S_\epsilon$ is a product in ${{{\mathcal O}}_{\mathrm{form}}^{}}$ iff $S_0^2+S_\epsilon$ satisfies formally the SGA equation. Moreover, if $S_0^2+S_\epsilon$ satisfies the SGS conditions, then $S_0^2+S_\epsilon$ is the generating function of a formal symplectic groupoid over ${\mathbb R}^d$.
Again, the zero of ${{{\mathcal O}}_{\mathrm{form}}^{2}}$ is a product in ${{{\mathcal O}}_{\mathrm{form}}^{}}$. We will stick to the conventions introduced for ${{{\mathcal O}}_{\mathrm{loc}}^{}}$. Namely, $0_1$ will stand for the zero of ${{{\mathcal O}}_{\mathrm{form}}^{1}}$, which is also the identity of the operad and $0_2$ will stand for the zero of ${{{\mathcal O}}_{\mathrm{form}}^{2}}$, which is the trivial product of the operad.
Thanks to the composition formula (\[treecomp\]), we are now able to rewrite the product equation in ${{{\mathcal O}}_{\mathrm{form}}^{}}$ as a cohomological equation, exactly as the deformation equation of a product in an additive category. Note that the Taylor expansion plays the same role as the linear expansion played in the additive case.
Let us define the Gerstenhaber bracket in ${{{\mathcal O}}_{\mathrm{form}}^{}}$ as follows: $$[F,G] = F\circ G-(-1)^{(k-1)(l-1)}G\circ F,$$ where $$F\circ G = \sum_{i=1}^k (-1)^{(i-l)(l-1)}F\{0_1,\dots,0_1,\underbrace{G}_{i^{th}},0_1\,\dots,0_1),$$ for $F\in {{{\mathcal O}}_{\mathrm{form}}^{k}}$ and $G\in {{{\mathcal O}}_{\mathrm{form}}^{l}}$.
We are now able to define a true coboundary operator.
Consider $d:{{{\mathcal O}}_{\mathrm{form}}^{n}}\rightarrow {{{\mathcal O}}_{\mathrm{form}}^{n+1}}$ $$dF := [0_2,F].$$ Then, $d$ may be written as $$\begin{gathered}
\label{Sdiff}
dF(p_1,\dots,p_{n+1}) = F(p_1,\dots,p_n,x)+\\
+\sum_{j=1}^n(-1)^{n+j-1}F(p_1,\dots, p_j+p_{j+1},\dots, p_n,x) + (-1)^{n-1}F(p_2,\dots,p_{n+1},x)\end{gathered}$$ Moreover, $d$ is linear and $d^2 = 0$
For more clarity, let us break our convention and write $\tilde I$ instead of $0_1$ and $\tilde S$ instead of $0_2$. We have that $[\tilde S,F] = \tilde S\circ F-(-1)^{n-1}F\circ\tilde S$. As $\tilde S = 0$, only the trees $\circ_i$ and $\bullet_j$ will contribute to the product. Then we have, $$\begin{aligned}
I_1 & = & \tilde S\circ F(p_1,\dots,p_{n+1},x) \\
&=& \sum_{i\geq 1}\epsilon^i \Bigg(C_{\bullet_i}\bigg(\tilde S(\cdot,x),F\cup \tilde I(p,\cdot)\bigg)\Big((\sum_1^n p_l,p_{n+1}),(x,x)\Big)+ \\
&& +(-1)^{n-1}C_{\bullet_i}\bigg(\tilde S(\cdot,x),\tilde I\cup F(p,\cdot)\bigg)\Big((p_1,\sum_2^{n+1} p_l),(x,x)\Big)\Bigg)\\
&=& \sum_{i\geq 1}\epsilon^i\Big(F^{(i)}(p_1,\dots,p_n,x)+ (-1)^{n-1}F^{(i)}(p_2,\dots,p_{n+1},x)\Big)\end{aligned}$$ and $$\begin{aligned}
I_2 & = & F\circ\tilde S(p_1,\dots,p_{n+1}) \\
& = & \sum_{j=1}^n(-1)^{j-1}\sum_{i\geq 1}\epsilon^i C_{\circ_i}\bigg(F(\cdot,x),( \tilde I\cup\dots \tilde I \cup \underbrace{\tilde S}_{j^{th}}\cup\tilde I\dots\\
& & \dots \cup \tilde I)(p,\cdot) \bigg)\Big((p_1,\dots,p_j+p_{j+1},\dots,p_{n+1}),(x,\dots,x)\Big)\\
& = & \sum_{j=1}^n (-1)^{j-1}\sum_{i\geq 1} \epsilon^i F^{(i)}(p_1,\dots,p_j+p_{j+1},\dots,p_{n+1},x),\end{aligned}$$ which gives the desired formula. The check that $d^2 = 0$ is straightforward.
We have then a complex $$\Big(C^\bullet= \oplus_{n\geq 0}{{{\mathcal O}}_{\mathrm{form}}^{n}}, d\Big).$$ This complex is exactly the Hochschild complex of (formal) multi-differential operators lifted on the level of symbols ( see for instance [@CahenGutt1980]). This remark gives us the cohomology of the complex, $$\operatorname H^n(C^\bullet) \simeq \mathcal \epsilon \mathcal V^n({\mathbb R}^d)[[\epsilon]],$$ where $\mathcal V^n({\mathbb R}^d)$ is the space of $n$-multi-vector fields on ${\mathbb R}^d$.
We come now to the question of finding a product $S_\epsilon$ in the formal deformation operad of ${{{\mathcal O}}_{\Delta}^{}}$ This is exactly the same problem as deforming the trivial generating function $S_0^2$ in ${{{\mathcal O}}_{\Delta}^{}}+{{{\mathcal O}}_{\mathrm{form}}^{}}$. We are thus looking for an element $ S_\epsilon\in{{{\mathcal O}}_{\mathrm{form}}^{2}}$ of the form $$S_\epsilon = \epsilon S_1+\epsilon^2 S_2+\dots$$ such that $$\begin{aligned}
\label{MaurerCartan} [ S_\epsilon, S_\epsilon] = 0. \end{aligned}$$ Equation (\[MaurerCartan\]) becomes, on the level of trees, $$\begin{aligned}
\label{operadSGA}\sum_{t\in T_\infty}\frac{\epsilon^{\|t\|}}{|t|!}
\Big( C_t(S_\epsilon,S_\epsilon\cup I)-C_t(S_\epsilon,I\cup S_\epsilon) \Big)
& = & 0.\end{aligned}$$
One sees immediately that this equation is equivalent to the following infinite set of recursive equations, $$\begin{aligned}
\label{cohomeq} d S_n + H_n(S_{n-1},\dots,S_1) & = & 0,\end{aligned}$$ where $$H_n(S_{n-1},\dots,S_1) = \sum_{\substack{t\in T_\infty^{k,n}\\2\leq |k|\leq n}}\frac1{|t|!}
\Big( C_t(S_\epsilon,S_\epsilon\cup I)-C_t(S_\epsilon,I\cup S_\epsilon) \Big),$$ where $T_\infty^{k,n}$ is the subset of trees in $T_\infty^{k,n}$ with $k$ vertices and such that $\|t\| = n$. These recursive equations are the exact analog of Equations (\[recursive\]).
Formal symplectic groupoid generating function {#proof}
----------------------------------------------
We restate now the main Theorem of [@CDF2005], Theorem 1, in terms of the new structures defined in this article.
For each Poisson structure $\alpha$ on $\mathbb R^d$, we have that $$S_\epsilon(\alpha) = \sum_{n=1}^\infty \frac{\epsilon^n}{n!}\sum_{\Gamma\in T_{n,2}}W_\Gamma \hat B_\Gamma(\alpha)$$ is a product in the formal deformation operad ${{{\mathcal O}}_{\mathrm{form}}^{}}({\mathrm{T}^*{\mathbb R}^{d}},\Delta)$ of the cotangent Lagrangian operad ${{{\mathcal O}}_{\Delta}^{}}({\mathrm{T}^*{\mathbb R}^{d}})$. Moreover, $S_\epsilon(\alpha)$ is the unique natural product in ${{{\mathcal O}}_{\mathrm{form}}^{}}({\mathrm{T}^*{\mathbb R}^{d}},\Delta)$ whose first order is $\epsilon \alpha$.
In the above Theorem, the $T_{n,2}$ stand for the set of Kontsevich trees of type $(n,2)$, $W_\Gamma$ is the Kontsevich weight of $\Gamma$ and $\hat B_\Gamma$ is the symbol of the bidifferential operator $B_\Gamma$ associated to $\Gamma$. We refer the reader to [@CDF2005] for exact definitions of Kontsevich trees, weights, operators and naturallity.
We called $S_\epsilon(\alpha)$ the (formal) [**symplectic groupoid generating function**]{} because, as shown in [@CDF2005], it generates a “geometric object”, a (formal) symplectic groupoid over ${\mathbb R}^d$ associated to the Poisson structure $\alpha$ whose structure maps are explicitly given by $$\begin{array}{cccc}
\epsilon_\epsilon(x) & = & (0,x) &\textbf{unit map}\\
i_\epsilon(p,x) & = & (-p,x) &\textbf{inverse map}\\
s_\epsilon(p,x) & = & x + \nabla_{p_2}S_\epsilon(\alpha)(p,0,x) & \textbf{source map}\\
t_\epsilon(p,x) & = & x + \nabla_{p_1}S_\epsilon(\alpha)(0,p,x) & \textbf{target map}.
\end{array}$$
This exhibits a strong relationship between star products and symplectic groupoids already foreseen by Costes, Dazord, Weinstein, Karasev, Maslov and Zakrzewski in respectively [@CDW1987], [@karasev1989] and [@zakrzewski1990] . Recently and from a completely different point of view, Karabegov in [@karabegov2004bis] went still a step further by showing how to associate a kind of “formal symplectic groupoid” to any star product.
In [@dherin2004] and [@dherin2005], we prove that the product $S_\epsilon(\alpha)$ has a non-zero convergence radius provided that the Poisson structure $\alpha$ is analytic. In this case, the generated formal symplectic groupoid is the local one. We also compared compared this local symplectic groupoid with the one constructed by Karasev and Maslov in [@karasev1989] and we proved that this two local symplectic groupoids are not only isomorphic as they should but exactly identical.
[9]{}
Cahen, M., Gutt, S., Local cohomology of the algebra of $C^\infty$ functions on a connected manifold, Letters in Mathematical Physics 4(1980) 157-167.
Cattaneo, A. S. The Lagrangian operad, unpublished notes, <http://www.math.unizh.ch/reports/05_05.pdf>
A. S. Cattaneo; B. Dherin; G. Felder. [*Formal symplectic groupoid.*]{} Comm. Math. Phys. 253 (2005), no. 3, 645–674.
Coste, A. ; Dazord, P. ; Weinstein, A. Groupoïdes symplectiques. (French) \[Symplectic groupoids\] Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, i–ii, 1–62, Publ. Dép. Math. Nouvelle Sér. A, 87-2, Univ. Claude-Bernard, Lyon, 1987.
B. Dherin. [*Star Products and Symplectic Groupoids.*]{} Ph. D. Thesis, ETH Zürich, 2004. (Advisor: Prof. G. Felder, Co-advisor: Prof. A. S. Cattaneo). <http://www.math.ethz.ch/~dherin>
B. Dherin. [*The Universal Generating Function of Analytical Poisson Structures.* ]{} In preparation.
Gerstenhaber, Murray; Voronov, Alexander A. Homotopy $G$-algebras and moduli space operad. Internat. Math. Res. Notices 1995, no. 3, 141–153 (electronic).
Hairer, E. ; Lubich, C. ; Wanner, G. Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2002. xiv+515 pp. ISBN: 3-540-43003-2
Karabegov, Alexander V. Formal symplectic groupoid of a deformation quantization.[math.QA/0408007](math.QA/0408007) 41 pages. QA.
Karasëv, M. V. The Maslov Quantization Conditions in Higher Cohomology and Analogs of Notions Developed in Lie Theory for Canonical Fibre Bundles of Symplectic Manifolds. II Selecta Mathematica Sovietica. Vol. 8 (1989), no. 3, 235-257.
Kontsevich, M. Deformation quantization of Poisson manifolds, I. [q-alg/9709040 (1997)](q-alg/9709040 (1997)).
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[^1]: In [@GV1995], Gerstenhaber and Voronov call it a multiplication.
[^2]: In the sequels, we will use the shorter notation $(p_1+\dots+p_n)x$ instead of $\sum_{i=1}^d(p_1^i+\dots+p_n^i) x_i$.
|
---
abstract: 'For a finite positive Borel measure $\mu$ on $\R$ its exponential type, $T_\mu$, is defined as the infimum of $a>0$ such that finite linear combinations of complex exponentials with frequencies between 0 and $a$ are dense in $L^2(\mu)$. The definition can be easily extended from finite to broader classes of measures. In this paper we prove a new formula for $T_\mu$ and use it to study growth and additivity properties of measures with finite positive type. As one of the applications, we show that Frostman measures on $\R$ may only have type zero or infinity.'
address: |
Texas A&M University\
Department of Mathematics\
College Station, TX 77843, USA\
and\
Department of Mathematics and Mechanics\
St. Petersburg State University\
St. Petersburg, Russia
author:
- 'A. Poltoratski'
title: Type alternative for Frostman measures
---
Introduction {#sInt}
============
For $a>0$ denote by $\EE_a$ the family of complex exponential functions on $\R$ with frequencies between $0$ and $a$: $$\EE_a=\{e^{isx}|\ s\in [0,a]\}.$$ If $\mu$ is a positive finite measure on $\R$ its exponential type $T_\mu$ is defined as $$T_\mu=\inf\{a|\ \EE_a \textrm{ is complete in }L^2(\mu)\},$$ if the set on the right-hand side is non-empty and as infinity otherwise. Recall that a family of vectors in a Banach space is complete in the space if finite linear combinations of vectors from the family are dense in the space. The type problem, the problem of finding $T_\mu$ in terms of $\mu$, has many connections in Fourier analysis, spectral analysis of differential operators and related fields, see for instance [@BS; @Krein1; @Krein2; @Krein3; @Khabibullin; @Koosis; @Type; @CBMS]. A formula for $T_\mu$ was recently obtained in [@Type] and further developed in [@CBMS]. One of the implications of these results is that $L^2$ in the above definition of $T_\mu$ may be replaced with any $L^p,\ p>1,$ without changing the value of $T_\mu$, i.e., that the $p$-type of $\mu$ for any $p>1$ is equal to the $2$-type.
Although the type formula from [@Type] has improved existing results and gave new examples in the area of the type problem, some of the natural questions remained open. This note originated from one of such questions. Consider the Poisson measure $\Pi$ on the real line, $$d\Pi(x)=\frac{dx}{1+x^2}.$$ Its type $T_\Pi$ can be easily shown to be equal to infinity. On the opposite end of the scale, if one considers any measure with ’long’ gaps in its support (see Section \[sGap\]), the type is equal to zero, as follows from another classical result, Beurling’s gap theorem. The simplest examples of measures of zero type are measures with semi-bounded (or bouneded) support, for instance, a restriction of $\Pi$ to a half-line. The sharpness of Beurling’s theorem was demonstrated by M. Benedicks in [@Benedicks] where further examples of absolutely continuous measures of zero and infinite type were constructed. Examples from [@Benedicks] focused on restrictions of the Poisson measure to unions of unit intervals.
To obtain an example of an absolutely continuous measure with finite positive type one needs to utilize further results on the type problem. Define the density $f$ to be equal to $e^{|n|}/(1+n^2)$ on each interval $[n,n+e^{-|n|}),\ n\in\Z$, and to zero elsewhere. Then the type of $\mu,\ d\mu(x)=f(x)dx$, is equal to $2\pi$, as follows from the results of Borichev and Sodin [@BS] or the results of [@Type].
Notice that in the last example the density is extremely unbounded, while in the examples with more regular densities the type always comes out to be zero or infinity. This pattern persists over all known examples giving raise to the following natural question: Can an absolutely continuous measure with bounded density have finite positive type?
In this note we give a negative answer to this question extending the result to a slightly wider class of measures. A positive measure $\mu$ on $\R$ is Poisson-finite if $$\int\frac{d\mu(x)}{1+x^2}<\infty.$$ The definition of type given above can be easily extended from finite to Poisson-finite and even wider classes of measures, see Section \[sPrelim\].
A positive measure $\mu$ on $\R$ is a Frostman measure if there exist positive constants $C$ and $\alpha$ such that for any interval $I\subset\R$, $$\mu(I)<C|I|^\alpha,
\label{eF}$$ where $|I|$ denotes the length of $I$. It is not difficult to see that every Frostman measure is Poisson-finite. Any absolutely continuous measure $\mu, \ d\mu(x)=f(x)dx$ with bounded density is a Frostman measure. Moreover, if $f\in L^p(\R),\ 1<p< \infty$ then the measure is Frostman since $$\mu(I)\leqslant ||f||_p|I|^{1/p}.$$
A measure is doubling if there exists a constant $C>0$ such that $$\mu((x-2r,x+2r))<C\mu((x-r,x+r))$$ for any $x\in \R$ and any $r>0$. Under an additional restriction that $\mu(I)<D$ for all intervals $I$ of fixed (unit) length and some $D>0$, a doubling measure is a Frostman measure. Among singular measures, a standard Cantor measure on $[0,1]$ periodically extended to the rest of $\R$ is a relevant example of a singular Frostman measure.
As we show in Theorem \[tFrost\], Section \[sTypeF\], the type of any Frostman measure can only be equal to zero or infinity. Our result relates local properties of a measure to its type, whereas previously known results only explored the relations between the type and asymptotic properties of the measure near infinity.
To prove Theorem \[tFrost\] we first obtain a new version of the type formula in Section \[sTypeL\], Theorem \[tLp\]. Unlike the previous versions, the formula does not include a $\mu$-finite weight inherited from Bernstein’s version of the type problem, see Section \[sTypeB\]. Such an improvement may be useful in applications, such as the problems discussed here.
If a measure with bounded density cannot have a finite positive type, it is natural to ask how fast the density of a measure of such a type must grow. We give an answer to this question in Section \[sGrowth\], Theorem \[tGrowth\]. In Section \[sAdd\] we investigate additivity properties of type.
Preliminaries {#sPrelim}
=============
If $f$ is a function from $L^1(\R)$ we denote by $\hat f$ its Fourier transform
$$\hat f(z)=\int_\R f(t)e^{- i z t}dt.\label{FTdef}$$
Let $M$ be a set of all finite Borel complex measures on the real line. Similarly, for $\mu\in M$ we define
$$\hat \mu(z)=\int_\R e^{- i z t}d\mu(t).$$
Via Parseval’s theorem, the Fourier transform may be extended to be a unitary operator from $L^2(\R)$ onto itself and can be defined for even broader classes of distributions.
If one extends all functions from $L^2(-a,a)$ as $0$ to the rest of the line, one can apply the Fourier transform to all such functions and obtain the Paley-Wiener space of entire functions $$\PWa=\{\hat f|\ f\in L^2(-a,a)\}.$$
The definition of type $T_\mu$, given in the introduction for finite measures $\mu$, can be naturally extended to broader classes of measures on $\R$: $$T_\mu=2\inf\{a|\ \PWa\cap L^2(\mu) \textrm{ is dense in }L^2(\mu)\},$$ if the set of such $a$ is non-empty and as infinity otherwise. Via this definition one can consider the type problem in the class of polynomially growing measures, i.e., measures $\nu$ such that $\nu=(1+|x|^n)\mu$ for some $n>0$ and some finite measure $\mu$. Poisson-finite measures are polynomially growing measures with $n=2$. We will denote the set of all polynomially growing Borel complex measures on the real line by $M_p$. The notation $M^+_p$ will be used for the subset of positive measures.
Note that by duality, a family $F$ of functions is not dense in $L^q(\mu),\ q>1,$ iff there exists $$f\in L^p(\mu),\ \frac 1p +\frac 1q =1,$$ annihilating the family, i.e., such that $$\int f\bar g d\mu=0$$ for every $g$ from the family. In such cases we write $f\perp F$. We denote by $T^p_\mu$ the $p$-type of $\mu$ defined as $$T^p_\mu=2\inf\{a|\ \exists f\in L^p(\mu),\ f\not\equiv 0, \ f\perp(\PWa\cap L^q(\mu)) \},$$ Note that $T_\mu=T^2_\mu$. Cases $p\neq 2$ were considered in several papers, see for instance articles by Koosis [@Koosis2] or Levin [@Levin2] for the case $p=\infty$.
It is well-known that if $\nu=(1+|x|^n)\mu$ then $T^p_\nu=T^p_\mu$, which reduces the problem for polynomially growing measures to its original settings of finite measures. The equality of $T^p_\mu,\ p>1$ to $T_\mu$ remains true for polynomially growing measures.
Suppose that $T^p_\mu>a$ for some finite measure $\mu, \ p>1$ and $a>0$. By our definitions this means that there exists a function $$f\in L^q(\mu),\ \frac 1p + \frac 1q =1,$$ which annihilates $\EE_a$, i.e., such that $$\hat f(s)=\int e^{-isx}f(x)dx=0\textrm{ for all }s\in [0,a].$$ An alternative way to extend the definition of type is to say that $T_\mu\geqslant a$ if $\widehat{f\mu}$, understood in the sense of distributions for infinite measures, has no support on $[0,a]$ for some $f\in L^p(\mu),\ f\not\equiv 0$.
Hence the type problem becomes a version of the gap problem, which studies measures whose Fourier transform has a non-trivial gap in its support. As we can see, to find $T^p_\mu$ is the same as to find what non-zero density $f\in L^q(\mu)$ gives the largest spectral gap for the measure $f\mu$. It turns out that to approach the type problem it is beneficial to first solve the gap problem in the case $q=1$, which is no longer a dual version of a $p$-type problem.
Gap formula {#sGap}
===========
We say that a polynomially growing measure $\mu\in M_p$ annihilates a Paley-Wiener space $\PWa$, and write $\mu\perp \PW_a$ if for all functions $f\in \PW_a\cap L^1(|\mu|)$, $$\int f d\mu =0.$$ Note that $\PWa$ contains a dense set of fast-decreasing functions belonging to $L^1(|\mu|)$ for any polynomially growing $\mu$. If $X$ is a closed subset of the real line we denote by $M_p(X)$ the set of polynomially growing measures supported on $X$. We denote by $\GG_X$ the gap characteristic of $X$ defined as $$\GG_X=2\sup\{a\ |\ \exists\ \mu\in M_p(X) ,\ \mu\not\equiv 0,\text{ such that}\ \mu\perp\PWa\}.$$ when the set is non-empty and zero otherwise. Alternatively, one could define $\GG_X$ as the supremum of the size of the gap in the support of $\hat\mu$, taken over all non-zero finite measures $\mu$ supported on $X$ (which explains the name).
The problem of finding $\GG_X$ in terms of $X$ has many connections and applications, see for instance [@MiPo; @CBMS] for results and further references. It was recently solved in [@GAP]. This version of the gap problem is related to the version mentioned in the last section via the following observation.
For $\mu\in M^+_p$, $$T^1_\mu=\GG_{\supp\mu}.$$
As we discussed in the last section, for $p>1$ the type $T^p_\mu=T_\mu$ is different from $T^1_\mu$. Further formulas for the type will be discussed in the next two sections.
To give the formula for $\GG_X$ [@GAP] we will need the following definitions.
A sequence of disjoint intervals $\{I_n\}$ on the real line is called long (in the sense of Beurling and Malliavin) if $$\sum_n\frac{|I_n|^2}{1+\dist^2(0,I_n)}=\infty\label{long}$$ where $|I_n|$ stands for the length of $I_n$. If the sum is finite we call $\{I_n\}$ short.
Let $$...<a_{-2}<a_{-1}<a_0=0<a_1<a_2<...$$ be a two-sided sequence of real points. We say that the intervals $I_n=(a_n,a_{n+1}]$ form a short partition of $\R$ if $|I_n|\to\infty$ as $ n\to \pm\infty$ and the sequence $\{I_n\}$ is short, i.e. the sum in is finite.
Let $\Lambda=\{\lambda_1,...,\lambda_n\}$ be a finite set of points on $\R$. Consider the quantity
$$E(\Lambda)=\sum_{\lambda_k,\lambda_j\in\L,\ \lambda_k\neq\lambda_j} \log|\lambda_k-\lambda_l|.\label{electrons}$$
As usual, a sequence of points $\L=\{\lan\}\subset\C$ is called discrete if it has no finite accumulation points.
Now we are ready to give the definition of $d$-uniform sequences used in our type formulas.
Let $\L=\{\lan\}$ be a discrete sequence of distinct real points and let $d$ be a positive number. We say that $\L$ is a $d$-uniform sequence if there exists a short partition $\{I_n\}$ such that $$\Delta_n=\#(\L\cap I_n)= d|I_n|+o(|I_n|)\ \text{as} \ n\to\pm\infty \ \textrm{(density condition)}\label{density}$$ and $$\sum_n \frac{\Delta_n^2\log|I_n|-E_n}{1+\dist^2(0,I_n)}<\infty\ \ \ \ \ \textrm{(energy/work condition)},\label{energy}$$ where
$$E_n=E(\Lambda\cap I_n).$$
The following formula for the gap characteristic of a closed set was obtained in [@GAP], see also [@CBMS].
\[MAINGAP\]
$$\GG_X=2\pi\sup\{ d |\ X\textrm{ contains a $d$-uniform sequence}\},$$
if the set on the right is non-empty and $\GG_X=0$ otherwise.
Type formula in Bernstein’s settings {#sTypeB}
====================================
We first approach the type problem in the settings of Bernstein’s weighted uniform approximation.
Consider a weight $W$, i.e. a lower semicontinuous function $W:\R\to [1,\infty]$ that tends to $\infty$ as $x\to\pm\infty$. The space $C_W$ is the space of all continuous functions on $\R$ satisfying $$\lim_{x\to\pm\infty} \frac{f(x)}{W(x)}=0.$$ We define the semi-norm in $C_W$ as $$||f||_W=|| fW^{-1}||_\infty.$$ Finding conditions on $W$ ensuring completeness of polynomials or exponentials in $C_W$ is a classical problem, see for instance [@Bernstein; @Lub; @Mergelyan; @CBMS] for further discussion and references.
For a weight $W$ we define $$\GG_W=\inf \{a|\ \EE_a\textrm{ is complete in }C_W\}.$$ We put $\GG_W=0$ if the last set is empty.
\[tBmain\] $$\GG_W=2\pi\sup \left\{d\ |\ \sum\frac{\log W(\lan)}{1+\lan^2}<\infty\textrm{ for some $d$-uniform sequence }\L \right\},$$ if the set is non-empty, and $0$ otherwise.
As was shown by A. Bakan [@Bakan], $\EE_a$ is complete in $L^p(\mu),\ 1\leqslant p\leqslant \infty,$ iff there exists a weight $W\in L^p(\mu)$ such that $\EE_a$ is complete in $C_W$. This result yields the following corollary from the last theorem. For $\mu\in M^+_p$, we call a weight $W$ a $\mu$-weight if $$\int Wd\mu<\infty.$$
\[Typemain\] Let $\mu$ be a finite positive measure on the line. Let $1<p\leqslant\infty$ and $d>0$ be constants.
Then $T^p_\mu\geqslant 2\pi d$ if and only if for any $\mu$-weight $W$ there exists a $d$-uniform sequence $\L=\{\lan\}\subset \supp \mu$ such that $$\sum\frac{\log W(\lan)}{1+\lan^2}<\infty.\label{ur4}$$
Note that this statement implies $T^p_\mu=T_\mu$ for all $p>1$, the property mentioned in previous sections. We will need these statements to obtain a new version of the type formula in the next section.
Type formula in $L^p$-settings {#sTypeL}
==============================
If $\L=\{\lan\}\subset\R$ is a discrete sequence of distinct points we denote by $\L^*=\{\lan^*\}$ the sequence of closed intervals such that each $\lan^*$ is centered at $\lan$ and has the length equal to one-third of the distance from $\lan$ to the rest of $\L$. Note that then the intervals $\lan^*$ are pairwise disjoint.
If $\L$ is a discrete sequence we will write that $D_1(\L)=d$ if there exists a short partition on which $\L$ satisfies .
Note that if $\L$ satisfies with some $d$ on some short partition then the asymptotic density of $\L$ is $d$ and therefore $\L$ cannot satisfy with any other $d$ on a different short partition, which implies correctness of the last definition.
\[tLp\] Let $\mu\in M^+_p$. Suppose that $T_\mu<\infty$. Then $$T_\mu=2\pi\max\{ d|\ \exists\ d\textrm{-uniform }\L=\{\lan\}\textrm{ such that }\sum\frac{\log\mu(\lan^*)}{1+n^2}>-\infty\}$$ if the set of such $d$ is non-empty and $T_\mu=0$ otherwise.
Note that he maximum on the right-hand side exists whenever the set is non-empty. The maximal sequence can always be chosen inside the support of the measure, which will be useful for us in applications.
The statement covers the case of finite type $T_\mu$, which is needed for Theorem \[tFrost\] in the next section. We would like to leave it as an open problem whether the formula can be extended to the infinite case, i.e., if it is true that the equation (with supremum in place of the maximum) holds when $T_\mu=\infty$. It is not difficult to see that if the supremum of the set on the right is infinite then $T_\mu=\infty$. It therefore remains to check the opposite implication.
As follows from our discussion in Section \[sPrelim\], it is enough to prove the theorem for finite $\mu$.
Let $T_\mu=2\pi d<\infty$. One can show that then there exists a $\mu$-weight $W$ such that is satisfied for some $d$-uniform sequence $\L$ but is not satisfied for any $d+\e$-uniform sequence. Define $$s=\max\{ d|\ \exists\ d\textrm{-uniform }\{\l_{n_k}\}\subset\L,\ \sum\frac{\log\mu(\l_{n_k}^*)}{1+{n_k}^2}>-\infty\}$$ (note that $\max$ can be used in the last formula instead of $\sup$). Suppose that $s<d$. Let $\G$ be the maximal subsequence in the last formula.
Then the subsequence $\L_1=\{\l_{n_k}\}= \L\setminus \G$ satisfies $D_1(\L_1)=\e>0$ and the corresponding sequence of intervals $\L^*_1=\{\l^*_{n_k}\}$ has the property that for any subsequence $ \L_2=\{\l_{n_{k_l}}\}\subset\L_1$, $D_1(\L_2)>0$, $$\sum_l\frac{\log\mu(\l_{n_{k_l}}^*)}{1+{n_{k_l}}^2}=-\infty.$$ Indeed, if there existed $$\L_2\subset \L_1,\ D_1(\L_2)>0$$ for which the last sum were finite, then by Lemma \[lsubuniform\], $\G\cup\L_2$ would have an $(s+\e)$-uniform subsequence with finite sum, which would contradict maximality of $\G$.
Now one can obtain a contradiction in the following way. Define a new weight $W_1$ to be equal to $$\frac{\max(W(\l_{n_k}),1/\mu(\l^*_{n_k}))}{1+{n_k}^2}$$ on each interval $\l^*_{n_k}$ from $\L^*_1$ and equal to $W$ elsewhere. Then $W_1$ is a $\mu$-weight and therefore must be satisfied with some $d$-uniform sequence $\Phi$. Notice that then $\Phi$ intersects the intervals from $\L_1^*$ only for a subsequence of density zero, i.e., there exists a subsequence $\Theta=\{\theta_n\}\subset \L_1$ such that $D_1(\Theta)=D_1(\L_1)=\e$ and $$\Phi\setminus\cup_{\l_{n_k}\in\Theta}\ \l^*_{n_k}=\emptyset.$$ Since $\L_1$ is an $\e$-uniform sequence, $\Theta$ is an $\e$-uniform sequence. By Lemma \[l1\], $\Phi\cup \Theta$ is a $(d+\e)$-uniform sequence on which the original weight $W$ satisfies , which contradicts our choice of $W$. Hence, $s\geqslant d$.
In the opposite direction, suppose that there exists a $d$-uniform sequence $\L$ such that $$\sum\frac{\log\mu(\lan^*)}{1+n^2}>-\infty.$$ Note that every $\mu$-weight $W$ satisfies $$\min_{\lan^*} W(x)<\frac {C}{\mu(\lan^*)}$$ on every $\lan^*$, with $$C=\int Wd\mu.$$ The sequence of points where the minima occur will give us a $d$-uniform sequence on which $W$ satisfies . Thus $T_\mu\geqslant 2\pi d$.
Type of Frostman measures {#sTypeF}
=========================
In this section we solve the problem of type for measures with bounded densities discussed in the introduction. As it turns out, our result can be formulated for a broader class of Frostman measures, which seems to be the right class for such a statement due to the elementary property that $\log|I|^\alpha=\alpha\log|I|$.
Recall that a positive measure $\mu$ on $\R$. is a Frostman measure if there exist positive constants $\alpha$ and $C$ such that $$\mu((x-\epsilon,x+\epsilon))<C\epsilon^\alpha\label{eFrost}$$ for all $\epsilon>0, x\in \R$. It easily follows that Frostman measures are Poisson-finite.
\[tFrost\] If $\mu$ is a Frostman measure then $T_\mu$ equals either 0 or $\infty$.
If $T_\mu=2\pi d,\ 0<d<\infty,$ then there exists a $d$-uniform sequence $\L$ such that $\L^*$ satisfies $$\sum\frac{\log\Delta_n}{1+n^2}>-\infty,\label{eTypeL}$$ where $\Delta_n=\mu(\lan^*)$. WLOG $\mu$ satisfies with $C=1$ and some $\alpha,\ 0<\alpha\leqslant 1$. Divide the interval $\lan^*$ into $M_n$ equal subintervals, so that $$\frac 12 [\Delta_n/6]^{1/\alpha}\leqslant |\lan^*|/M_n \leqslant [\Delta_n/6]^{1/\alpha}.$$ Then the mass of each subinterval is at most $\Delta_n/6$ (and hence $M_n\geqslant 6$). Since the total mass is $\Delta_n$, there exist at least $3$ subintervals of mass at least $$\Delta_n/2M_n\asymp \Delta_n^{1+\frac 1\alpha}/|\lan^*|.\label{emass}$$ Let $l^n_1$ and $l^n_2$ be two of these intervals, not adjacent to each other. Then the distance $d_n$ between the intervals is greater or equal to the length of each interval and its logarithm can be estimated from below by $$\log d_n\geqslant \log |\lan^*|/M_n\geqslant C_1\log\Delta_n +C_2,$$ which together with implies $$\sum\frac{\log d_n}{1+n^2}>-\infty.\label{e00001}$$ Consider the sequence $\G=\{\gamma_m\}$ composed of the centers of all such intervals $l^n_1, l^n_2$ for all $n$. Since the sequence $\L$ was $d$-uniform and because of , $\G$ is a $2d$-uniform sequence. Our construction implies that each of the intervals $\gamma^*_m$ contains one of the intervals $l^n_1, l^n_2$, whose mass is at least the expression in . Since $\Delta_n=\mu(\lan^*)$ satisfy , the sequence of intervals $\{\gamma^*_m\}$ satisfies $$\sum\frac{\log\mu(\gan^*)}{1+n^2}\gtrsim\sum\frac{\log\Delta_n}{1+n^2}+\const>-\infty$$ which implies $T_\mu\geqslant 4\pi d$ contradicting our initial assumption.
As was discussed in the introduction, the last theorem has the following corollaries.
\[c1\] Let $\mu=fm$, where $f\in L^p(\R),\ p>1$. Then $T_\mu$ is either 0 or infinity
If $\mu$ is a doubling measure such that $$\sup_{x\in\R}\mu((x,x+1))<\infty$$ then $T_\mu$ is either 0 or infinity.
Additivity properties of type {#sAdd}
=============================
It is natural to ask if the type of a measure satisfies any additivity conditions, in general or in special cases. Our first observation in this direction is that the inequality $T_{\mu+\nu}\lesssim T_\mu +T_\nu$ fails in general. Indeed, let $\mu$ and $\nu$ be the restrictions of Lebesgue measure $m$ to $\R_+$ and $\R_-$ respectively. Then $T_\mu=T_\nu=0$, because if $f\in L^2(\nu)\ (\in L^2(\mu))$ then $\widehat{f\nu}\ (\widehat{f\mu})$ belongs to the Hardy space $H^2\ (\bar H^2)$ and cannot vanish on a set of positive measure, unless $f\equiv 0$. On the other hand $T_{\mu+\nu}=T_{m}=\infty$.
Further examples of this type, without semi-bounded supports, can be obtained using Beurling’s theorem and choosing $\mu$ to be the restriction of $m$ to the union of odd dyadic intervals $[2^{2n+1},2^{2n+2}],\ n\in \Z$ and $\nu$ as the restriction to even intervals. Then once again $T_\mu=T_\nu=0$ but $T_{\mu+\nu}=T_{m}=\infty$.
In the opposite direction, we obviously have $T_{\mu+\nu}\geqslant \max(T_\mu,T_\nu)$. However, the inequality $T_{\mu+\nu}\geqslant T_\mu +T_\nu$ does not hold in general. To construct an example, consider $$\mu=\sum_{n=-\infty}^{\infty} \delta_n\textrm{ and }\nu=\sum_{n=-\infty}^{\infty} \delta_{n+e^{-|n|}}.$$ Since both measures are supported by $1$-uniform sequences, $T_\mu=T_\nu=2\pi$ by Theorem \[tLp\] (in this simple case the type can be calculated directly without any advanced results). Since the support of $\mu+\nu$ does not contain any $d$-uniform sequences with $d>1$, $T_{\mu+\nu}=2\pi$ by Theorem \[MAINGAP\] or \[tLp\].
Nonetheless, the following ’splitting’ property holds for measures of finite positive type.
Let $\mu\in M^+_p$, $T_\mu=2\pi d<\infty$. Let $c_1,\ c_2$ be non-negative constants such that $c_1+c_2=d$. Then there exists a closed set $X\subset\R$ such that the measures $\mu_1=\mu |_X$, $\mu_2=\mu-\mu_1$ satisfy $$T_{\mu_1}=2\pi c_1, \ \ \ \ T_{\mu_2}=2\pi c_2.$$
Note that the analog of the above statement for measures of infinite type is false. If $\mu$ is a measure with bounded density such that $T_\mu=\infty$, $c_1$ is finite positive and $c_2$ is infinite then any restriction of $\mu$ will again be a measure with bounded density and therefore, by Theorem \[tFrost\], will not have its type equal to $2\pi c_1$.
By Theorem \[tLp\] there exists a $d$-uniform sequence $\L$ such that the sequence of intervals $\L^*$ satisfies $$\sum\frac{\log\mu(\lan^*)}{1+n^2}>-\infty.\label{esum}$$ If $\{I_n\}$ is a short partition corresponding to $\L$ from the definition of $d$-uniform sequences, choose a subsequence $\Gamma\subset\Lambda$ satisfying on $\{I_n\}$ with $c_1$ in place of $d$. Note that the energy condition for $\Gamma$ will then be satisfied on the same partition and therefore $\Gamma$ is a $c_1$-uniform sequence.
Put $X=\cup_{\lan\in\Gamma} \lan^*$ and $\mu_1=\mu |_X$. We claim that $\mu_1$ and $\mu_2=\mu-\mu_1$ are the desired measures. First, notice that $T_{\mu_1}=c_1$. Indeed, since $\Gamma$ is a $c_1$-uniform sequence, $T_{\mu_1}\geqslant 2\pi c_1$ by Theorem \[tLp\]. If $T_{\mu_1}=2\pi p>2\pi c_1$ then there exists a $p$-uniform sequence $\Phi$ satisfying for $\mu_1$. As was remarked after Theorem \[tLp\], we can choose $\Phi$ from $X$. Moreover, we can assume that each interval $\lan^*,\ \lan\in \Gamma$ contains at least one point from $\Phi$. Then the intervals from $\Phi^*$ do not intersect the intervals $\lan^*,\ \lan\in \L\setminus\Gamma$.
Divide each interval $\lan^*,\ \lan\in \L\setminus\G$ into two equal subintervals. Note that at least one of them has mass of at least one half of the original interval, with respect to $\mu$. For each $n$ choose the half-interval with the larger mass and denote the centers of these half-intervals by $\psi_n,\ \psi_n\in\lan^*$. Note that $\Psi=\{\psi_n\}$ is a $c_2$-uniform sequence. By Lemma \[l1\], the sequence $\Phi\cup\Psi$ is a $(p+c_2)$-uniform sequence. By our construction, the sequence of intervals $(\Phi\cup\Psi)^*$ satisfies , which implies that $T_\mu\geqslant 2\pi (p+c_2)>2\pi d$, a contradiction.
Similarly, $T_{\mu_2}=2\pi c_2$.
Growth of density in the case of finite positive type {#sGrowth}
=====================================================
As we saw from Corollary \[c1\], a measure of finite positive type cannot have bounded density. It is natural to ask how fast should its density grow. To this account we prove the following statement.
\[tGrowth\] Let $\mu\in M^+_p$, be absolutely continuous, $\mu=fm$. Suppose that $T_\mu=2\pi d$, $0<d<\infty$. For $x>0$ denote $$\MM_f(x)= \textrm{ess sup }_{[-x,x]} f(x).$$ Then $$\int_{0}^{\infty}\frac{\log(1+\MM_f(x))}{1+x^2}dx=\infty.$$
Suppose that the integral from the statement is finite. Consider the measure $\nu(x)=\mu(x)/(1+\MM_f(|x|))$. This measure has bounded density and therefore has type 0 or infinity. Note that for any $\nu$-weight $W$, $U=W/(1+\MM_f) + 1$ is a $\mu$-weight. Hence there exists a $d$-uniform sequence $\L$ such that $U$ satisfies . But $$\log U(\lan)=\log W(\lan) - \log (1+\MM_f(|\lan|))$$ and $$\sum_n \frac{\log(1+\MM_f(|\lan|))}{1+\lan^2}\lesssim \int_0^\infty\frac{\log(1+\MM_f(|x|))}{1+x^2}<\infty.$$ Therefore $W$ satisfies on the same sequence. Hence, by Corollary \[Typemain\], $T_\nu\geqslant 2\pi d$, which implies that $T_\nu=\infty$.
Therefore, for any $\nu$-weight $U$ there exists a $2d$-uniform sequence $\L$ which satisfies . Since any $\mu$-weight $W$ is equal to $U(1+\MM_f(|x|)$ for a $\nu$-weight $U$, similar to above we conclude that $W$ satisfies for a $2d$-uniform sequence $\L$. Thus by Corollary \[Typemain\], $T_\mu\geqslant 4\pi d$, a contradiction.
Let us show that the condition of the last theorem is sharp in its scale. Let $M:\R_+\to [1,\infty)$ be an increasing function such that $$\int_{0}^{\infty}\frac{\log M(x)}{1+x^2}dx=\infty.$$ Then there exists $\mu=fm$, $\MM_f(x)\leqslant M(x)$ such that $T_\mu$ is finite positive. Indeed, put $f$ to be equal to $M(n)$ on every interval $$I_n=\left[n,n + \frac 1{M(n)}\right]$$ and zero elsewhere. One can show that then every sequence $S\subset \Z$ such that $$\sum_{n\in S}\frac{\log |I_n|}{1+n^2}>-\infty$$ has density zero. Hence, $\Z$ is the maximal $d$-uniform sequence from the statement of Theorem \[tLp\] and $T_\mu=2\pi$.
Additivity properties of uniform sequences {#sLem}
==========================================
The following lemma was proved in [@GAP].
\[l-energy\] Let $\L$ be a sequence of real points and let $\{I_n\}$ be a short partition such that $\L$ satisfies $$a|I_n|<\#(\L\cap I_n)$$ for all $n$ with some $a>0$ and the energy condition on $\{I_n\}$. Then for any short partition $\{J_n\}$, there exists a subsequence $\G\subset\L$ which satisfies $$\#(\L\sm\G)\cap J_n=o(|J_n|)$$ as $n\to\pm\infty$, and the energy condition on $\{J_n\}$.
All $d$-uniform sequences satisfy the following simple properties.
\[lsubuniform\] Let $\L$ be a $d$-uniform sequence.
1\) If $\Gamma\subset\L$ is a $c$-uniform sequence then $\L\setminus \Gamma$ contains a $(d-c)$-uniform subsequence.
2\) if $\Gamma_1\subset\L$ is a $c_1$-uniform sequence and $\Gamma_2\subset\L$ is a $c_2$-uniform sequence, $\G_1\cap \G_2=\emptyset$, then $\G_1\cup \G_2$ contains a $(c_1+c_2)$-uniform subsequence.
1\) Let $\{I_n\}$ and $\{J_n\}$ be the short partitions from the definition of $d$-uniform sequences in Section \[sGap\] for the sequences $\L$ and $\G$ correspondingly. Let $L_n$ be a third short partition with the property that $$\max \{|I_m|\ :\ I_m\cap L_n\neq \emptyset\} + \max \{|J_m|\ :\ J_m\cap L_n\neq \emptyset\}=$$ $$=o(|L_n|)\textrm{ as }n\to\infty.\label{e3part}$$ By Lemma \[l-energy\] there is a subsequence $\L'\subset\L$ such that $$\#((\L\setminus\L')\cap L_n)=o(|L_n|)$$ and $\L'$ satisfies the energy condition on $\{L_n\}$. Notice that then $\L'\setminus\G$ satisfies the energy condition and the density condition with the constant $d-c$ on $\{L_n\}$.
2\) Similarly to the last part, if $\{I_n\}$ and $\{J_n\}$ are short partitions corresponding to $\G_1$ and $\G_2$, choose a short partition $\{L_n\}$ to satisfy . Choose a subsequence $\L'\subset \L$ like in the last part. Note that since $\L'$ satisfies the energy condition on $L_n$, so does $\L'\cap (\G_1\cup \G_2)$. The last sequence also satisfies the density condition with the constant $c_1+c_2$ on $L_n$.
A union of uniform sequences is a uniform sequence, up to a sequence of density zero, if the original sequences are separated from each other in the following precise sense.
\[l1\] Let $\L=\{\lan\}$ be a $c$-uniform sequence and let $\G=\{\gan\}$ be a $d$-uniform sequence. Let $\{I_n\}$ be a sequence of disjoint intervals such that $I_n$ is centered at $\lan$, $$|I_n|\leq\frac 13\dist (\lan,\L\setminus\{\lan\})$$ and $$\sum\frac{\log|I_n|}{1+n^2}>-\infty.$$ Suppose that $\G\cap(\cup I_n)=\emptyset$. Then $\L\cup\G$ contains a $(c+d)$-uniform subsequence.
Denote by $p_n$ the intervals $$p_n=\frac 12 I_n.$$ Choose the intervals $q_n$ to be centered at $\gan$ and such that $$|q_n|=\frac 13 \dist (\gan, \cup I_n\cup\G\setminus\{\gan\}).$$ Note that then the intervals $q_n,p_n$ are disjoint and satisfy $$\sum \frac{\log |p_n|}{1+n^2}>-\infty,\ \sum \frac{\log |q_n|}{1+n^2}>-\infty.\label{e100}$$
As follows from Lemma \[l-energy\], one can choose a short partition $\{S_n\}$ on which both sequences $\L$ and $\G$ have subsequences which satisfy and with constants $c$ and $d$ correspondingly. WLOG we will assume that the subsequences are equal to $\L$ and $\G$. We will also assume that the endpoints of $S_n$ do not fall into any of the intervals $p_n,q_n$ and that $$|S_n|=\frac 1c\#\L\cap S_n=\frac1d\#\G\cap S_n$$ (omitting the $o(|\cdot|)$ term in subsequent formulas).
For each $k\in\Z$ consider the functions $u_k$ and $v_k$ defined as follows. Each function $u_k$ is continuous and piecewise linear. It is zero on $(-\infty,a_k)$, where $S_k=(a_k,b_k)$ and its derivative is zero outside of $(\cup p_n)\cap S_k$. On each $p_n, p_n\subset S_k$, $u_k$ grows linearly by one. It follows that $u_k=c|S_k|$ on $(b_k,\infty)$. Repeat the same construction for $v_k$ with the intervals $q_n$ in place of $p_n$.
Now define the functions $\phi_k$ as $u_k(x)-cx$ on $S_k$ and as 0 outside of $S_k$. Define $\psi_k$ as $v_k(x)-dx$ on $S_k$ and as 0 outside of $S_k$. Employing some of the techniques used in [@GAP] we notice the following.
The functions $\phi_k$ and $\psi_k$ belong to the Dirichlet class $\DD$ with $$||\phi_k||_{\DD}^2= \frac 1\pi (c^2|S_k|^2\log|S_k|-E_{\L\cap S_k})-\sum_{p_n\subset S_k} \log |p_n|+ O(|S_k|^2) ,$$ $$||\psi_k||_{\DD}^2= \frac 1\pi (d^2|S_k|^2\log|S_k|-E_{\G\cap S_k})-\sum_{q_n\subset S_k} \log |q_n|+ O(|S_k|^2) .$$ and $$||\phi_k+\psi_k||_{\DD}^2= \frac 1\pi ((c+d)^2|S_k|^2\log|S_k|-E_{(\L\cup\G)\cap S_k})-$$$$\sum_{p_n\subset S_k} \log |p_n|-
\sum_{q_n\subset S_k} \log |q_n|+O(|S_k|^2) ,$$ as $k\to\infty$.
The function $\phi=\phi_k$ is a function with bounded harmonic conjugate and compactly supported on $S_k=S$ derivative. Hence it belongs to $\DD$ and its norm can be calculated as $$||\phi ||_{\DD}^2=\int_\R \phi d\ti\phi= -\int_{\R}\ti \phi d\phi.$$ Integrating by parts, $$-\int_{\R}\ti \phi d\phi=\frac 1\pi\int_{S}\phi'\left[\int_{S} \frac {\phi(t)dt}{t-x} \right] dx=$$ $$-\frac 1\pi\int_{S}\phi'\left[\int_{S} \log|t-x| \phi'(t)dt\right] dx.$$ Let $p_1,p_2, ... p_n$ be the intervals from our construction above inside $S$. Recall that by our construction of $u=u_k$, $u'=1/|p_l|$ on each $p_l$ and to zero elsewhere. Since $\phi=u-cx$ we obtain $$-\int_{S}\phi'\left[\int_{S} \log|t-x| \phi'(t)dt\right] dx=
-c^2\int_S\int_S \log|t-x| dtdx$$$$+2c^2\int_S\sum_{m=1}^n\frac 1{|p_m|}\int_{p_m}\log|t-x|dtdx$$$$-\sum_{m=1}^n\sum_{k=1}^m \frac 1{|p_m||p_k|}\int_{p_m}\int_{p_k}\log|t-x|dtdx=$$ $$-I+II-III.$$ Recall that the points $\l_1,\l_2,...\lan$ are the centers of the intervals $p_1,p_2, ... p_n$. Because the distance between adjacent intervals $p_l,p_{l+1}$ is at least $\max(|p_l|,|p_{l+1}|)$, for the last term we have $$III= \sum_{1\leqslant m,k\leqslant n,\ m\neq k} \log|\l_m-\l_k| + \sum_{p_n\subset S_k}\log |p_n|+O(|S|^2).$$ Similarly, $$II= 2c^2|S|^2\log |S| + O(|S|^2).$$ Finally, via elementary calculations, $$I=c^2|S|^2\log|S|+O(|S|^2).$$ Combining the last three equations we obtain the statement for $\phi=\phi_k$. The equations for $\psi_k$ and $\phi_k+\psi_k$ can be proved similarly.
To finish the proof of the lemma, it remains to notice that the claim implies $$\sum_k\frac{(c+d)^2|S_k|^2\log|S_k|-E_{(\L\cup\G)\cap S_k}}{1+k^2} +\const$$$$\lesssim \sum_k ||\phi_k+\psi_k||_{\DD}^2+\const\lesssim \sum_k\left(
||\phi_k||_{\DD}^2+||\phi_k||_{\DD}^2\right)+\const
\lesssim$$$$\sum_k\frac{(c^2|S_k|^2\log|S_k|- E_{\L\cap S_k})+(d^2|S_k|^2\log|S_k|-E_{\G\cap S_k})}{1+k^2}+\const.$$ Since the sequences $\L$ and $\G$ are $c$- and $d$-uniform respectively on the partition $\{S_n\}$, the last sum is finite. Therefore the sequence $\L\cup\G$ satisfies the energy condition on the partition $\{S_n\}$. Since it also satisfies the density condition with $a=c+d$ on $\{S_n\}$, it is a $(c+d)$-uniform sequence.
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abstract: 'Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing, algorithms with sublinear round complexity for triangle finding and listing have recently been developed in the powerful CONGEST clique model, where communication is allowed between any two nodes of the network. In this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the communication topology is the same as the topology of the network. More precisely, we give randomized algorithms for triangle finding and listing with round complexity $O(n^{2/3}(\log n)^{2/3})$ and $O(n^{3/4}\log n)$, respectively, where $n$ denotes the number of nodes of the network. We also show a lower bound $\Omega(n^{1/3}/\log n)$ on the round complexity of triangle listing, which also holds for the CONGEST clique model.'
author:
- |
Taisuke Izumi[^1]\
Graduate School of Engineering\
Nagoya Institute of Technology\
[[email protected]]([email protected])
- |
Fran[ç]{}ois Le Gall[^2]\
Graduate School of Informatics\
Kyoto University\
[[email protected]]([email protected])
title: Triangle Finding and Listing in CONGEST Networks
---
Introduction
============
#### Background.
The most standard way of modeling distributed networks is to use graphs, where vertices and edges respectively correspond to computing entities (e.g., nodes or processes) and communication channels. The graphical structure of networks naturally motivates, both practically and theoretically, the study of graph algorithms, i.e., in-network solving of combinatorial graph problems for the own topology of the network. In the last twenty years, research on distributed complexity theory has successfully investigated many fundamental graph problems such as independent set [@ABI86; @BEPS16], dominating set [@FR05; @KMW16], coloring [@Linial92; @BE13], minimum spanning trees [@GHS83; @Elkin04; @Elkin06; @GKP93; @PR99; @LPPP05] shortest paths [@LP13; @LP15; @Nanongkai14; @HW12], max-flow and min-cut [@GK13; @NS14; @GKKLP15]. The seminal textbook by Peleg [@Peleg00] provides a good overview of those achievements.
A high-level perspective on distributed graph algorithms classifies problems into two categories according to their locality characteristics. Problems like minimum spanning trees, shortest paths or max-flow/min-cut mentioned above are classified as global problems because some node must (indirectly) communicate with a node $D$-hop away, where $D$ denotes the diameter of the network, which requires $\Omega(D)$ rounds. Since any problem can be solved within $O(D)$ rounds by a naive centralized approach if the communication bandwidth of each channel is unlimited, technical challenges on global problems appear only in the model of limited-bandwidth channels, the so-called CONGEST communication model where each channel has only $O(\log n)$-bit bandwidth. On the other hand, many local problems, like independent set, dominating set or coloring allow distributed (approximate) solutions within $o(D)$ rounds. Besides the limited bandwidth, a significant difficulty when designing distributed algorithms for local problems often consists in breaking the symmetry between nearby nodes.
#### Triangle finding.
Triangle finding is an important counterpart of the perspective explained above. Since the distance-two information for a node $i$ is sufficient to find all triangles around $i$, it is a purely local task. Aggregating at each node the set of its 2-hop neighbors, however, takes $\Theta(d_{\mathit{max}})$ rounds in the CONGEST model, where $d_{\mathit{max}}$ denotes the maximum degree of the nodes. In the case of dense graphs, such an approach gives linear round complexity. In this sense, despite being a local problem, the underlying difficulty of triangle finding is similar to the difficulty encountered when working with global problems in the CONGEST model, that is, the lack of communication bandwidth. These particularities make triangle finding a difficult problem to handle with current techniques. As summarized in Table \[table:comparison\], recently some non-trivial upper bounds have been obtained by Dolev et al. [@DLP12] and then improved by Censor-Hillel et al. [@CKKLPS15], but these upper bounds hold only in the much stronger CONGEST clique model (where at each round messages can even be sent between non-adjacent nodes) for which bandwidth management is significantly easier. No nontrivial upper bound is known in the standard CONGEST model. As far as lower bounds are concerned, the locality of triangle finding rules out many standard approaches used to obtain lower bounds on the complexity of global problems (e.g., the approaches from [@SHKKANPPW12; @FHW12]). The only known lower bounds have been obtained recently by Drucker et al. [@DKO14], but hold only for the much weaker broadcast CONGEST model, where at each round the nodes can broadcast only a single common message to all other nodes, under a (reasonable) conjecture in communication complexity theory. No nontrivial lower bound is known for triangle finding in the standard CONGEST model.
Another, more practical, motivation for considering triangle finding is that for several graph problems faster algorithms are known over triangle-free graphs (we refer to [@Hirvonen+14; @Pettie+15] for some examples in the distributed setting). The ability to efficiently check if the network is triangle-free, and more generally detect which part of the network is triangle-free, is essential when considering such algorithms in practice.
[cccc]{} Paper & Time Bound & Problem & Model\
& $O(n^{1/3}(\log n)^{2/3})$& &\
&$O(d_{\mathit{max}}^{3}/n)$ &&\
Censor-Hillel et al. [@CKKLPS15] & $O(n^{0.1572})$ & Finding & CONGEST clique\
This paper (Theorem \[th:UB-finding\]) & $O(n^{2/3}(\log n)^{2/3})$ & Finding & CONGEST\
This paper (Theorem \[th:UB-listing\])& $O(n^{3/4}\log n)$ & Listing & CONGEST\
Drucker et al. [@DKO14] & $\Omega\big(\frac{n}{e^{\sqrt{\log n}}\log n}\big)$ (conditional) & Finding & CONGEST broadcast\
Pandurangan et al. [@Pandurangan+16]& $\Omega(\frac{n^{1/3}}{\log^3 n})$ & Listing & CONGEST clique\
This paper (Theorem \[thm:lowerBound\])& $\Omega(\frac{n^{1/3}}{\log n})$ & Listing & CONGEST clique\
#### Technical contribution.
In this paper, in addition to triangle finding we also consider the triangle listing problem, which requires that each triangle of the network should be output by at least one node (we refer to Section \[sec:prelim\] for the formal definition). The triangle listing problem can be seen as a special case of motif finding, which is a popular problem in the context of network data analysis. It is obvious that triangle finding is not harder than triangle listing, since finding reduces to listing. In the literature mentioned above, triangle finding and triangle listing are not explicitly distinguished, but the algorithms by Dolev et al. [@DLP12] are actually for the listing version, while the algorithms by Censor-Hillel et al. [@CKKLPS15] are based on an algebraic approach and only support finding. The lower bound by Drucker et al. [@DKO14] applies to triangle finding.
Our results are summarized in Table \[table:comparison\]. Our main contributions are a $O(n^{2/3}(\log n)^{2/3})$-round algorithm for triangle finding and a $O(n^{3/4}\log n)$-round algorithm for triangle listing, both in the CONGEST model. These two algorithms are the first algorithms with sublinear round complexity for triangle finding or listing applicable to the standard CONGEST model. The existence of sublinear-time algorithms shows, from an algorithmic perspective as well, that these two problems are indeed easier than global problems like diameter computation, which has a nearly linear-time lower bound in the CONGEST model [@FHW12]. Interestingly, this result contrasts with the known relations between triangle finding and diameter computation in the centralized setting, in which the best known algorithms for both problems rely on matrix multiplication and have the same complexity.
We also show that there exist networks of $n$ nodes for which any triangle listing algorithm requires $\Omega(n^{1/3}/\log n)$ rounds in the CONGEST model. Actually we show a stronger result: any triangle listing algorithm requires $\Omega(n^{1/3}/\log n)$ rounds even in the CONGEST clique model. This lower bound slightly improves the $\Omega(n^{1/3}/\log^3 n)$-round lower bound for triangle listing in the CONGEST clique model obtained recently by Pandurangan et al. [@Pandurangan+16]. Note that since the algorithm by Dolev et al. [@DLP12] has round complexity $O(n^{1/3}(\log n)^{2/3})$, these lower bounds are tight up to possible polylogarithmic factors in the CONGEST clique model. These lower bounds also have the following interesting consequence. They shows that triangle listing is strictly harder than triangle finding in the CONGEST clique model since, as already mentioned, $(n^{0.1572})$-round algorithms for triangle finding have been shown by Censor-Hillel et al. [@CKKLPS15]. Actually, since the algorithms in [@CKKLPS15] also count the number of triangles, these lower bounds thus even show that triangle listing is harder than counting in this model.
Our lower bound is shown by a simple information-theoretic argument on random graphs. By similar arguments (see Proposition \[cor\] in Section \[sec:LB\]), we can also show a $\Omega(n/\log n)$-round lower bound for local listing algorithms, where each node $i$ is required to output all the triangles containing $i$. This corollary implies that any triangle listing algorithm with sublinear complexity inherently requires some counter-intuitive mechanism that enables a triangle $t$ to be output by a node not contained in $t$. This is precisely how our $O(n^{3/4}\log n)$-round algorithm works.
#### Other related works.
Distributed algorithms for triangle-freeness have recently also been considered in the setting of property testing [@Censor-Hillel+DISC16] (see also [@Fraigniaud+DISC16]). Note that in property testing the goal is to decide whether either the network is triangle-free or it contains a large number triangles. Since there is no need to consider the case where the network may have only a small number of triangles, this property testing version is significantly easier than the problems studied in the present paper.
Triangle finding is a fundamental problem in the field of centralized algorithms as well. It has been known for a long time that this problem is not harder than Boolean matrix multiplication [@Itai+SICOMP78]. A few years ago Vassilevska Williams and Williams showed a converse reduction [@Williams+FOCS10]: they proved that a subcubic-time algorithm for triangle finding can be used, in a combinatorial way, to construct a subcubic-time algorithm for Boolean matrix multiplication. This result, combined with recent developments [@Bansal+ToC12; @Czumaj+SICOMP09; @PatrascuSTOC10; @Vassilevska+SICOMP13], have put triangle finding as a central problem in the recent theory of fine-grained complexity.
Preliminaries {#sec:prelim}
=============
In this paper we consider undirected graphs. We will use $G = (V, E)$ to denote the graph considered, and write $n=|V|$ and $m=|E|$. In this section we present some of our notations, give details of our model and present some facts about hashing functions that will be used in Section \[sec:UB\].
#### Graph-theoretic notations.
For any vertex $i\in V$, we denote $\mathcal{N}(i)$ the set of neighbors of $i$. For any finite set $X$, we will use the notation ${\mathcal{E}}(X)$ to represent the set of unordered pairs of elements in $X$, and use $\mathcal{T}(X)$ to represent the set of unordered triples of elements in $X$. We will write ${\mathcal{E}}={\mathcal{E}}(V)$ and $\mathcal{T}=\mathcal{T}(V)$. Given a pair of vertices $\{j,k\}\in{\mathcal{E}}$, we write $${\ensuremath{\#(\{j,k\})}}=
\Big|\left\{
l\in V\:|\:
\{j,l\}\in E \textrm{ and } \{k,l\}\in E
\right\}\Big|.$$ Given $t = \{j,k,l\} \in \mathcal{T}$, and $e \in \mathcal{E}$, we will write $e \in t$ when $e = \{j, k\}$, $e = \{j, l\}$, or $e = \{k, l\}$. For any $R \subseteq \mathcal{T}$, we denote by ${\mathcal{P}}(R)\subseteq E$ the set of edges $e\in E$ such that $e\in t$ for some triple $t \in R$.
We define a triangle $t = \{j, k, l\}\in \mathcal{T}$ to be an unordered triple of vertices where any pair corresponds to an edge in $E$. Note that if $e$ is an edge of the graph, then ${\ensuremath{\#(e)}}$ represents the number of triangles containing $e$. We write $T(G)$ the set of all triangles contained in graph $G$.
#### Communication Model.
In the paper we mainly consider the CONGEST communication model. The graph $G = (V, E)$ represents the topology of the network, executions proceed with round-based synchrony and each node can transfer one $O(\log n)$-bit message to each adjacent node per round. All links and nodes (corresponding to the edges and vertices of $G$, respectively) are reliable and suffers no faults. Each node has a distinct identifier from a domain ${\mathcal{I}}$. For simplicity we will assume ${\mathcal{I}}=V = [0, n-1]$, but this assumption is not essential and easy to remove as long as $|{\mathcal{I}}| = \mathrm{poly}(n)$. It is also assumed that each node can access infinite sequence of local random bits, that is, the algorithm can be randomized. Initially, each node knows nothing about the topology of the network except the set of edges incident to itself and the value of $n$. All our upper bounds given in Section \[sec:UB\] hold for the CONGEST model.
The CONGEST clique model is a powerful variant of the CONGEST model. It allows an algorithm to transfer a $O(\log n)$-bit message per round between any two nodes not necessarily adjacent in $G$ at each round. This means that in this model the communication topology is the complete graph on the $n$ nodes, and the graph $G$ only takes the role of input instances to the algorithm. Except for the communication topology, all other features are common with the CONGEST model described in the above paragraph. The CONGEST clique model will be considered only in Section \[sec:LB\] when proving our lower bound (note that a lower bound for this model immediately holds for the CONGEST model as well).
#### Triangle finding and listing in the CONGEST model.
In the triangle finding problem, at least one node must output a triangle if $T(G)$ is not empty, or all the nodes must output “not found” otherwise. In the triangle listing problem, the goal is that for each triangle in $T(G)$ at least one node in $V$ outputs it.
Formally, we describe the output of an algorithm for triangle finding or listing by an $n$-tuple $T = (T_0, T_1, \dots, T_{n-1})$, where $T_i \subseteq T(G)$ is the output by node $i$. The algorithm can be deterministic or randomized. In the latter case, note that the condition on the $T_i$’s implies that any triple output at a node should corresponds to a triangle of $G$, and thus the algorithm should be one-sided error. Note also that we do not require that the $T_i$’s are mutually disjoint. We often use notation $T$ as the union of $T_i$ over all nodes if there is no ambiguity. We say that the algorithm solves the triangle finding problem if its output $T$ satisfies $T\cap T(G)\neq \emptyset$. We say that it solves the triangle listing problem if $T=T(G)$.
#### Hash functions.
Let $\mathcal{X}$ and $\mathcal{Y}$ be two finite sets. For any integer $s\ge 1$, a family of hash functions ${\mathcal{H}}= \{h_1, h_2, \dots, h_p\}$, where each $h_i$ is a function from $\mathcal{X}$ to $\mathcal{Y}$, is called *s-wise independent* if for any distinct $x_1, x_2, \dots, x_s \in \mathcal{X}$ and any $y_1, y_2, \dots y_s
\in \mathcal{Y}$, a function $h$ sampled from ${\mathcal{H}}$ uniformly at random satisfies $\Pr[\bigwedge_{1 \leq i \leq s} h(x_i)
= y_i] = 1 / |\mathcal{Y}|^s$. We will use the following lemma, which follows almost immediately from the definition of $s$-wise independence.
\[l:hashfunction\] Let ${\mathcal{H}}$ be any $3$-wise independent family of hash functions from $\mathcal{X}$ to $\mathcal{Y}$, and $h$ be a function sampled from ${\mathcal{H}}$ uniformly at random. Let $H(y) = \{x'' \in \mathcal{X}\:|\:h(x'')=y\}$. For any triple $(x,x',y)\in \mathcal{X}\times \mathcal{X}\times \mathcal{Y}$ we have $$\Pr\left[h(x)=h(x')=y \bigwedge
\Big|H(y)\Big| \leq 4\left(2 + \frac{|\mathcal{X}| - 2}{|\mathcal{Y}|}\right)\right] \geq \frac{3}{4|\mathcal{Y}|^2}.$$
Let us write $Z=\{x''\in \mathcal{X}\:|\:h(x'')=y\}$. For any $a\in \mathcal{X}$, let $Z_{a}$ be the indicator random variable corresponding to the event $a \in Z$. By the 3-wise independence property of $h$, we have $$\begin{aligned}
\lefteqn{E\Big[|Z|\:|\:Z_x=Z_{x'}=1\Big]} \hspace{10mm} \\
&= 2+\sum_{a \in \mathcal{X} \setminus \{x, {x'}\}} E[Z_a\:|\:Z_x=Z_{x'}=1] \\
&= 2+\frac{|\mathcal{X}| - 2}{|\mathcal{Y}|}.
$$ By Markov’s inequality, we have $$\begin{aligned}
\Pr
\left[
|Z| > 4\left(2+\frac{|\mathcal{X}| - 2}{|\mathcal{Y}|}\right) \ \middle| \ Z_{x}=Z_{{x'}}=1
\right]
&\leq \frac{1}{4}.\end{aligned}$$ Since $\Pr[Z_x = Z_{x'} = 1] = 1/|\mathcal{Y}|^2$ holds, we obtain $$\begin{aligned}
\lefteqn{\Pr\left[Z_x = Z_{x'} = 1 \bigwedge |Z| \le 4\left(2+\frac{|\mathcal{X}| - 2}{|\mathcal{Y}|}\right)\right]} \hspace{10mm} \\
&= \Pr\left[|Z| \le 4\left(2+\frac{|\mathcal{X}| - 2}{|\mathcal{Y}|}\right) \middle| Z_x = Z_{x'} = 1 \right] \\
& \hspace{40mm} \times
\Pr[Z_x = Z_{x'} = 1] \\
&\ge
\frac{3}{4} \times \frac{1}{|\mathcal{Y}|^2},\end{aligned}$$ as claimed.
Note that using the standard construction by Wegman and Carter [@WC81] we can encode $k$-wise independent hash functions using only $O(k\log |\mathcal{Y}|)$ bits.
Upper Bounds {#sec:UB}
============
In this section we prove the following two theorems.
\[th:UB-finding\] Let $\delta>0$ be any constant. In the CONGEST model there exists a randomized algorithm for triangle finding with round complexity $O(n^{2/3}(\log n)^{2/3})$, where $n$ denotes the size of the network, and success probability at least $1-\delta$.
\[th:UB-listing\] In the CONGEST model there exists a randomized algorithm for triangle listing with round complexity $O(n^{3/4}\log n)$ and success probability at least $1-1/n$, where $n$ denotes the size of the network.
Theorems \[th:UB-finding\] and \[th:UB-listing\] follow from three algorithms described in Propositions \[prop:tri-heavy-finding\]–\[prop:tri-light\] below, which rely on the following concept of heavy triangles. Let $\varepsilon$ be any real number such that $0\le \varepsilon\le 1$. We say that a triangle $t \in T(G)$ is *$\varepsilon$-heavy* if there exists an edge $e \in t$ such that ${\ensuremath{\#(e)}} \geq n^{\varepsilon}$. We define ${\ensuremath{{T}_{\varepsilon}(G)}} \subseteq T(G)$ as the set of all $\varepsilon$-heavy triangles in $T(G)$. The first proposition is almost trivial, and shows how to find an $\varepsilon$-heavy triangle by a straightforward sampling strategy:
\[prop:tri-heavy-finding\] There exists a $O(n^{1-\varepsilon})$-round randomized algorithm ${\mathcal{A}}_1$ returning a set $T\subseteq T(G)$ such that, if ${\ensuremath{{T}_{\varepsilon}(G)}}\neq \emptyset$, then $T\cap{\ensuremath{{T}_{\varepsilon}(G)}}\ \neq \emptyset$ with probability $\Omega(1)$.
Algorithm ${\mathcal{A}}_1$ is as follows: First, each node $j\in V$ constructs a random set $S_j\subseteq\mathcal{N}(j)$ by including in $S_j$ each element of $\mathcal{N}(j)$ with probability $n^{-\varepsilon}$. If $|S_j|> 4n^{1-\varepsilon}$, node $j$ then does not send anything to its neighbors. Otherwise (i.e., if $|S_j|\le 4n^{1-\varepsilon}$), node $j$ then sends $S_j$ to each neighbor $k\in\mathcal{N}(j)$, and each such neighbor $k$ checks if there is a triangle containing $j$, $k$ and a third node in $S_j$, which can be done by computing locally the set $\mathcal{N}(k)\cap S_j$.
Observe that if ${\ensuremath{{T}_{\varepsilon}(G)}}\neq \emptyset$ then at least one $\varepsilon$-heavy triangle will be found with constant probability. Indeed, for each edge $\{j,k\}\in E$ such that ${\ensuremath{\#(\{j,k\})}}\ge n^\varepsilon$, with constant probability at least one node adjacent to both $j$ and $k$ is included in $S_j$, and $S_j$ is small enough.
The second proposition is about finding each $\varepsilon$-heavy triangle:
\[prop:tri-heavy\] There exists a $O(n^{1-\varepsilon/2})$-round randomized algorithm ${\mathcal{A}}_2$ returning a set $T\subseteq T(G)$ such that, for any triangle $t\in {\ensuremath{{T}_{\varepsilon}(G)}}$, this set $T$ contains $t$ with probability $\Omega(1)$.
The third proposition is about finding each triangle that is not $\varepsilon$-heavy:
\[prop:tri-light\] There exists a $O(n^{1-\varepsilon}+n^{(1+\varepsilon)/2}\log n)$-round randomized algorithm ${\mathcal{A}}_3$ returning a set $T\subseteq T(G)$ such that, for any triangle $t\in T(G)\setminus{\ensuremath{{T}_{\varepsilon}(G)}}$, this set $T$ contains $t$ with probability $\Omega(1)$.
Proposition \[prop:tri-heavy\] and especially Proposition \[prop:tri-light\] are the main technical contributions of this section. Their proofs are given in Subsections \[sub:heavy\] and \[sub:light\], respectively.
We now explain how Theorems \[th:UB-finding\] and \[th:UB-listing\] immediately follow from these three propositions.
The triangle finding algorithm simply applies Algorithm ${\mathcal{A}}_1$ and then Algorithm ${\mathcal{A}}_3$. From Propositions \[prop:tri-heavy-finding\] and \[prop:tri-light\] we know that a triangle will thus be found with at least constant probability (if $G$ contains a triangle). For any constant $\delta$ the success probability can be amplified to $1-\delta$ by repeating this process $c$ times for a large enough constant $c$.
The round complexity of this algorithm is $
O(n^{1-\varepsilon} + n^{1-\varepsilon} + \allowbreak n^{(1+\varepsilon)/2}\log n).
$ Choosing $\varepsilon$ such that $n^\varepsilon =\frac{n^{1/3}}{(\log n)^{2/3}}$ gives the claimed upper bound.
The triangle listing algorithm repeats ${\left\lceil c \log n \right\rceil}$ times the following process (here $c$ is a large constant): apply Algorithm ${\mathcal{A}}_2$ and then Algorithm ${\mathcal{A}}_3$.
We now show that this algorithm lists all the triangles of $G$. Let $t$ be a triangle in $T(G)$. From Propositions \[prop:tri-heavy\] and \[prop:tri-light\] we know that $t$ is found with at least constant probability at each step. If $c$ is large enough, repeating this process ${\left\lceil c\log n \right\rceil}$ times guarantees that $t$ is found with probability at least $1-1/n^4$. From the union bound, we can conclude that all triangles are found with probability at least $1-1/n$.
The round complexity of this algorithm is $
O(((n^{1-\varepsilon/2} + n^{1-\varepsilon} \allowbreak + n^{(1+\varepsilon)/2}\log n)\log n).
$ Choosing $\varepsilon$ such that $n^\varepsilon =\frac{n^{1/2}}{(\log n)^{2}}$ gives the claimed upper bound.
Listing all $\varepsilon$-heavy triangles: Proof of Proposition \[prop:tri-heavy\] {#sub:heavy}
----------------------------------------------------------------------------------
The goal of this subsection is to prove Proposition \[prop:tri-heavy\].
The main idea to find $\varepsilon$-heavy triangles is that each node $j$ will decide which information about $\mathcal{N}(j)$ it will send to each neighbor $a$ according to a hash function $h_a$ generated by $a$. More precisely, the function $h_a$ will be taken (by node $a$, and then distributed to all its neighbors) from a 3-wise independent family ${\mathcal{H}}$ of hash functions from $V$ to $\{0,1,\ldots,\lfloor n^{\varepsilon/2}\rfloor-1\}$, and $j$ will send an edge $\{j,l\}$ to $a$ if $h_a(l)=0$. The complete algorithm, which is Algorithm ${\mathcal{A}}_2$ claimed in Proposition \[prop:tri-heavy\], is described in Figure \[fig:algorithmheavylisting\] and analyzed below.
We first prove the correctness of Algorithm ${\mathcal{A}}_2$. Let $t = \{j, k, l\}$ be any $\varepsilon$-heavy triangle of $G$. Let $\{j, k\}$ be the edge shared by at least $n^{\varepsilon}$ triangles and $A$ be the set of nodes such that $\{j, k, a\}$ forms a triangle for each $a \in A$. Consider the function $h_a\in\mathcal{H}$ chosen by node $a$. Applying Lemma \[l:hashfunction\] with $|\mathcal{X}|=n$ and $|\mathcal{Y}|=\lfloor n^{\varepsilon/2}\rfloor$, we obtain the inequality $$\Pr\left[h_a(k)=h_a(l)=0 \bigwedge |E_a^j\cup E_a^k|\leq 8+\frac{4n}{\lfloor n^{\varepsilon/2}\rfloor}\right] \geq
\frac{3}{4n^{\varepsilon}}.$$ This bound implies that with probability at least $\frac{3}{4n^{\varepsilon}}$ node $a$ receives $\{j,k\}$ from $j$, $\{j,l\}$ from $j$ and $\{k,l\}$ from $k$ at Step 2, and thus $T_a$ (i.e., the output of node $a$) includes $t$ with the same probability. Since the events $t \not\in T_a$ are mutually independent over all $a \in A$, we can bound the probability that no node in $A$ finds $t$ as follows: $$\Pr\left[\bigcap_{a \in A} t \not\in T_a\right]
\leq \left(1 - \frac{3}{4n^{\varepsilon}}\right)^{|A|}
\leq \left(1 - \frac{3}{4n^{\varepsilon}}\right)^{n^{\varepsilon}}
\leq e^{-\Omega(1)}.$$
The round complexity of Algorithm ${\mathcal{A}}_2$ is $O(n^{1-\varepsilon/2})$ since the hash function $h_i$ sent at Step 1 can be encoded using $O(\log n)$ bits, as explained in Section \[sec:prelim\].
This concludes the analysis of Algorithm ${\mathcal{A}}_2$ and the proof of Proposition \[prop:tri-heavy\].
Finding triangles that are not heavy: Proof of Proposition \[prop:tri-light\] {#sub:light}
-----------------------------------------------------------------------------
The goal of this subsection is to prove Proposition \[prop:tri-light\]. We first give a brief description of the main ideas underlying our algorithm before giving the full description of Algorithm ${\mathcal{A}}_3$.
#### Key definition and outline of our approach.
For any set ${X}\subseteq V$, define the set $${\Delta({X})}={\mathcal{E}}(V)\setminus \bigcup_{x\in {X}}{\mathcal{E}}(\mathcal{N}(x)),$$ which is the set of pairs of nodes that do not have a common neighbor in $X$. The following easy lemma shows how the set ${\Delta({X})}$ is related to finding triangles which are not $\varepsilon$-heavy.
\[lma:not-heavy-1\] Let $\varepsilon$ be any real number such that $0\le\varepsilon\le 1$. Suppose that ${X}$ is a set obtained by including each node of $V$ into ${X}$ with probability $\frac{1}{9n^\varepsilon}$. Then, for any triangle $t\in T(G)\setminus{\ensuremath{{T}_{\varepsilon}(G)}}$, the three edges of $t$ are in ${\Delta({X})}$ with probability at least $2/3$.
Let us consider any triangle $t\in T(G)\setminus{\ensuremath{{T}_{\varepsilon}(G)}}$. For each edge $e\in t$ we have ${\ensuremath{\#(e)}}< n^\varepsilon$. From the union bound, we obtain the inequality $
\Pr_{X}\left[
e\in
\bigcup_{x\in {X}}{\mathcal{E}}(\mathcal{N}(x))
\right]<
\frac{1}{9}.
$ Thus, again from the union bound, $t$ has its three edges in ${\Delta({X})}$ with probability at least $2/3$.
In view of Lemma \[lma:not-heavy-1\], our strategy will be to take a set ${X}$ as in the lemma and then find the triangles with three edges in ${\Delta({X})}$ using the following approach: Each node $i\in V$ sends to all its neighbors the set $\mathcal{N}(i)\cap {X}$, in $O(|{X}|)$ rounds. Then each node $k\in V$ constructs for each $j\in \mathcal{N}(k)$ the set containing all nodes $l\in \mathcal{N}(k)$ such that $\{j,l\}\in{\Delta({X})}$, which can be done locally. Let us call this set $\mathcal{S}(j,k)$ for now – later it will be called $\mathcal{S}_V^X(j,k)$. Node $k$ then sends $\mathcal{S}(j,k)$ to neighbor $j$, who can report all triangles of the form $\{j,k,l\}$ with $\{j,l\}\in {\Delta({X})}$.
This approach works, but its round complexity depends on the size of the sets $\mathcal{S}(j,k)$. The crucial point is that we can show from a combinatorial argument that with high probability (on the choice of $X$) the average size of these sets is small. The main remaining issue is that, even if the average size is small, in general there exist pairs $(j,k)$ for which $\mathcal{S}(j,k)$ is large. We solve this issue by sending $\mathcal{S}(j,k)$ only when its size is close to the average and using a different strategy when its size exceeds the average. This latter strategy is based on a notion of “good” nodes. Roughly speaking, a node $j$ is good if it has only a small number of neighbors $k$ such that $\mathcal{S}(j,k)$ is large (see Definition \[def\] for the formal definition). The same combinatorial argument as above guarantees that most nodes are good (this is shown in Lemma \[lma:not-heavy-2\] below). Triangles containing at least one good node are fairly easy to deal with, and we can deal with triangle containing three bad nodes by applying the same approach recursively on the subgraph induced by the bad nodes. Our final algorithm will thus be recursive: it will keep a set $U\subseteq V$, with initially $U=V$, and successively search for triangles in the subgraph of $G$ induced by $U$, removing the good nodes from $U$ at each step, until $U=\emptyset$.
#### Further definitions and full description of Algorithm ${\mathcal{A}}_3$.
For any set $U\subseteq V$ and any nodes $j,k\in U$ such that $\{j,k\}\in E$, we define the set ${\mathcal{S}_U^{{X}}(j,k)}\subseteq U$ as follows: $$\begin{aligned}
{\mathcal{S}_U^{{X}}(j,k)}&=
\Big\{l\in U\:|\: \{j,l\}\in {\Delta({X})}\textrm { and } \{k,l\}\in E\Big\}.\end{aligned}$$ Note that this definition is asymmetric: we have ${\mathcal{S}_U^{{X}}(j,k)}\neq {\mathcal{S}_U^{{X}}(k,j)}$ in general. Let ${r}$ be any positive real number. For any node $j\in U$, we define the set $${\mathcal{V}_{U,{r}}^{{X}}(j)}=
\big\{
k\in U\:|\:
\{j,k\}\in E
\textrm{ and }
\left|
{\mathcal{S}_U^{{X}}(j,k)}
\right|
>
{r}$$ }. Finally, we will use the following concept of good nodes.
\[def\] Let $U$ and ${X}$ be any subsets of $V$, and ${r}$ be any positive real number. A node $j\in U$ is ${r}$-good for $(U,{X})$ if $
\big|
{\mathcal{V}_{U,{r}}^{{X}}(j)}
\big|
\le {r}.
$
The following crucial lemma shows an upper bound on the number of nodes which are not good, when ${X}$ is chosen as in Lemma \[lma:not-heavy-1\].
\[lma:not-heavy-2\] Let $\varepsilon$ be any real number such that $0\le\varepsilon\le 1$ and ${X}$ be a set chosen at random as in Lemma \[lma:not-heavy-1\]. Then, for any real number ${r}\ge \sqrt{54n^{1+\varepsilon}\log n}$, the following statement holds with probability at least $1-1/n$: $$\label{eq1}
\parbox{75mm}{
For any set $U\subseteq V$ there are at most $|U|/2$ nodes in $U$ that are not ${r}$-good for $(U,{X})$. }$$
Consider any pair $\{j,l\}\in{\mathcal{E}}$ such that $${\ensuremath{\#(\{j,l\})}}\ge 27n^\varepsilon\log n.$$ We have $$\Pr_{X}\left[
\{j,l\}\notin
\bigcup_{x\in {X}}{\mathcal{E}}(\mathcal{N}(x))
\right]\le
\left(1-\frac{1}{9n^\varepsilon}\right)^{27n^\varepsilon\log n}
\le
\frac{1}{n^3}.$$ From the union bound we can thus conclude that with probability at least $1-1/n$ the following statement holds: $$\label{eq1b}
{\ensuremath{\#(\{j,l\})}}
< 27n^\varepsilon\log n
\textrm{
for any pair $\{j,l\}\in {\Delta({X})}$.
}$$
We now show that Statement (\[eq1b\]) implies Statement (\[eq1\]). Let us consider the number of ordered triples $(j,k,l)\in U\times U\times U$ such that $\{j,l\}\in {\Delta({X})}$, $\{j,k\}\in E$ and $\{k,l\}\in E$. If Statement (\[eq1b\]) holds then this number is at most $27|U|^2n^\varepsilon \log n$. This number of triples can also be computed in another way: it is equal to $$\sum_{\begin{subarray}{c}(j,k)\in U\times U\\\textrm{s.t. } \{j,k\}\in E\end{subarray}}\left|{\mathcal{S}_U^{{X}}(j,k)}\right|.$$ Thus if Statement (\[eq1b\]) holds we obtain the inequality $$27|U|^2n^\varepsilon \log n\ge \sum_{\begin{subarray}{c}(j,k)\in U\times U\\\textrm{s.t. } \{j,k\}\in E\end{subarray}}\left|{\mathcal{S}_U^{{X}}(j,k)}\right|
\ge \sum_{j\in U} \left|{\mathcal{V}_{U,{r}}^{{X}}(j)}\right|{r},$$ which implies that the number of nodes that are not ${r}$-good for $(U,{X})$ is at most $$27|U|^2n^\varepsilon \log n\times \frac{1}{{r}^2}\le \frac{27|U|n^{1+\varepsilon} \log n}{{r}^2}.$$ For any ${r}\ge \sqrt{54n^{1+\varepsilon}\log n}$ we obtain the claimed upper bound.
We now present an algorithm that lists all triangles of $G$ with three edges in ${\Delta({X})}$. The algorithm is denoted ${\mathcal{A}}({X},{r})$ and described in Figure \[fig:algorithm\]. We analyze it in the next proposition.
\[prop:tri-delta\] Assume that the set ${X}$ satisfies Statement (\[eq1\]). Then Algorithm ${\mathcal{A}}({X},{r})$ terminates after $O(\log n)$ iterations of the while loop, has overall round complexity $O(|{X}|+{r}\log n)$, and its output $T\subseteq T(G)$ includes all the triangles of $G$ with three edges in ${\Delta({X})}$.
Let us first show that Algorithm ${\mathcal{A}}({X},{r})$ terminates after $O(\log n)$ iterations of the while loop. The assumption on the set ${X}$ implies that $
|U\setminus U'|\le |U|/2
$ at Step 4.2 of each iteration of the while loop. Since we start with $|U| = n$, after at most $\log_2(n)+1$ iterations of the while loop the set $U$ becomes empty, at which point we exit the while loop.
We now analyze the overall round complexity of the algorithm by considering the complexity of each step. Step 1 requires only one round. Note that after Step 1 each node $k$ can locally compute the set $\mathcal{N}(k)\cap {X}$. Step 2 can then be implemented using at most $|{X}|$ rounds, since $|\mathcal{N}(k)\cap {X}|\le |{X}|$. Step 4.1 can be implemented using at most ${r}$ rounds. Step 4.2 does not require any communication since the enough information has been obtained at Step 4.1 to compute ${\mathcal{V}_{U,{r}}^{{X}}(k)}$ locally. Step 4.3 only requires at most ${r}$ rounds, since $|{\mathcal{V}_{U,{r}}^{{X}}(j)}|\le {r}$ for any $j\in U'$. Finally, Step 4.5 obviously requires only one round. The overall round complexity is thus at most $$|{X}|+1+(\log_2(n)+1)({r}+{r}+1)= O(|{X}|+{r}\log n).$$
We finally show that the algorithm lists all triangles of $G$ with three edges in ${\Delta({X})}$. First observe that for any set $U\subseteq V$, any triangle of $G$ with three nodes in $U$ and three edges in ${\Delta({X})}$ satisfies at least one of the following three properties (where $U'\subseteq U$ denotes the set of nodes that are ${r}$-good for $(U,{X})$):
- it contains three distinct nodes $j,k,l\in U$ such that $\left|{\mathcal{S}_U^{{X}}(j,k)}
\right|
\le
{r}$ and $l\in {\mathcal{S}_U^{{X}}(j,k)}$;
- it contains two distinct nodes $j,k\in U$ such that $j\in U'$ and $k \in{\mathcal{V}_{U,{r}}^{{X}}(j)}$;
- all its three nodes are in $U\setminus U'$.
Indeed, if such a triangle does not satisfy Property (a) then the inequality $|{\mathcal{S}_U^{{X}}(j,k)}|>{r}$ holds for any two nodes $j$ and $k$ of the triangle (which means that $k\in{\mathcal{V}_{U,{r}}^{{X}}(j)}$). In that case it should satisfy either Property (b) or Property (c).
Algorithm ${\mathcal{A}}({X},{r})$ starts with $U=V$ and decreases the set $U$ until it is empty. Let us consider what happens at each execution of the while loop, with the current set $U$. All triangles of type (a) are found at Step 4.1 and all triangles of type (b) are found at Step 4.3. The only remaining triangles are triangles of type (c). They are considered at the next execution of the while loop, where $U$ is replaced by $U\setminus U'$.
We are now ready to present Algorithm ${\mathcal{A}}_3$ and give the proof of Proposition \[prop:tri-light\].
Algorithm ${\mathcal{A}}_3$ is as follows. First each node selects itself with probability $\frac{1}{9n^{\varepsilon}}$. Then the nodes run Algorithm ${\mathcal{A}}({X},{r})$ with $X$ being the set of selected nodes (note that each node $i$ of the network knows whether $i\in {X}$ or not, as required) and ${r}=\sqrt{54n^{1+\varepsilon}\log n}$, but stop as soon as the round complexity exceeds $c\times (n^{1-\varepsilon}+n^{(1+\varepsilon)/2}\log n)$ for some large enough constant $c$.
By construction, the round complexity of Algorithm ${\mathcal{A}}_3$ is $$O(n^{1-\varepsilon}+n^{(1+\varepsilon)/2}\log n).$$
Let $t$ be any triangle in $T(G)\setminus{\ensuremath{{T}_{\varepsilon}(G)}}$, and let us show that Algorithm ${\mathcal{A}}_3$ finds $t$ with constant probability. Note that the expectation of the value $|{X}|$ is $\frac{1}{9}n^{1-\varepsilon}$. By Chernoff bound, we know that the probability that $|{X}|> \frac{2}{9}n^{1-\varepsilon}$ is negligible. Combining this observation with Lemmas \[lma:not-heavy-1\] and \[lma:not-heavy-2\], we conclude (using the union bound) that with probability $\Omega(1)$ the following three conditions hold: ${X}$ has size at most $\frac{2}{9}n^{1-\varepsilon}$, ${X}$ satisfies Statement (\[eq1\]), and $t$ has its three edges in ${\Delta({X})}$. From Proposition \[prop:tri-delta\] we know that in this case algorithm ${\mathcal{A}}(X,{r})$ stops within $O(n^{1-\varepsilon}+n^{(1+\varepsilon)/2}\log n)$ rounds and finds $t$. By choosing the constant $c$ large enough we can thus guarantee that the output of Algorithm ${\mathcal{A}}_3$ contains the triangle $t$ with probability $\Omega(1)$.
Lower Bounds {#sec:LB}
============
This section proves the following lower-bound theorem.
\[thm:lowerBound\] Let $\mathcal{A}$ be any triangle listing algorithm with error probability less than $1/32$. Then there exists a probability distribution on inputs such that the expected round complexity of $\mathcal{A}$ is $\Omega(n^{1/3}/\log n)$.
In this section we will write $V=\{0,1,\ldots,n-1\}$. Without loss of generality, the run of any algorithm $\mathcal{A}$ for triangle listing on any given instance $G = (V, E)$ can be described by two following steps:
1. Each node $i$ locally constructs its initial state ${\boldsymbol{\rho}}_i$ according to $G$. This state depends on the set of edges incident to $i$, but is independent of all other edges.
2. Each node $i$ construct its output ${\boldsymbol{T}}_i$ from ${\boldsymbol{\rho}}_i$ and the transcript ${\boldsymbol{\pi}}_i$ of the communication received it receives during the execution of the algorithm.
To prove Theorem \[thm:lowerBound\], we consider as input the random graph $G(n, 1/2)$, which is the graph on $n$ nodes where each pair of nodes is independently taken as an edge with probability $1/2$. Since ${\boldsymbol{\rho}}_i$, ${\boldsymbol{\pi}}_i$, and ${\boldsymbol{T}}_i$ depend only on the algorithm $\mathcal{A}$ (and its random bits if $\mathcal{A}$ is randomized) and the input $G$, we see all of them as random variables. For any pair $\{j,k\}\in {\mathcal{E}}$, let ${\boldsymbol{e}}_{\{j,k\}}$ be the random variable with value 1 if $\{j,k\}$ is an edge, and value 0 otherwise. Let ${\boldsymbol{E}}=({\boldsymbol{e}}_{\{0,1\}}, {\boldsymbol{e}}_{\{0,2\}}, \dots, {\boldsymbol{e}}_{\{n-2,n-1\}})$ be the concatenation of these $|{\mathcal{E}}|$ random variables. The proof idea is to bound the amount of information in ${\boldsymbol{\pi}}_i$ necessary to construct ${\boldsymbol{T}}_i$ locally. We write ${\boldsymbol{T}}=({\boldsymbol{T}}_0,{\boldsymbol{T}}_1,\ldots,{\boldsymbol{T}}_{n-1})$ and use the notation $w({\boldsymbol{T}})$ to represent the node identifier $i$ such that $|T_i|$ is maximum (i.e., the index of the node that outputs the maximum number of triangles).
Our proof relies on the following graph-theoretic lemma, which is shown in [@Rivin02].
\[lma:triangleEdge\] If a graph $G$ contains $t$ triangles, then $G$ has at least $\frac{\sqrt{2}}{3}t^{2/3}$ edges.
We first recall the definition of the entropy. For a random variable ${\boldsymbol{X}}$ with domain $\mathcal{X}$ and probability distribution $p$, its entropy $H({\boldsymbol{X}})$ is defined as $H({\boldsymbol{X}}) = - \sum_{x \in \mathcal{X}} p(x)\log p(x)$. The conditional entropy can be defined similarly. For any random variables ${\boldsymbol{X}}$ and ${\boldsymbol{Y}}$ the mutual information of ${\boldsymbol{X}}$ and ${\boldsymbol{Y}}$ is defined as $I({\boldsymbol{X}}; {\boldsymbol{Y}}) = H({\boldsymbol{X}})-H({\boldsymbol{X}}|{\boldsymbol{Y}})$. We will use in our proofs the following standard results from information theory about the mutual information.
\[fact:mutualInfo\] For any random variables ${\boldsymbol{X}}$, ${\boldsymbol{Y}}$, and ${\boldsymbol{Z}}$, the following three properties hold:
- $I({\boldsymbol{X}}; {\boldsymbol{Y}}) = I({\boldsymbol{Y}}; {\boldsymbol{X}})$,
- $I({\boldsymbol{X}}; {\boldsymbol{Y}}) \leq H({\boldsymbol{X}})$ (and $I({\boldsymbol{X}}; {\boldsymbol{Y}}) \leq H({\boldsymbol{Y}})$), and
- $I({\boldsymbol{X}}; {\boldsymbol{Y}}) \geq I({\boldsymbol{X}}; ({\boldsymbol{Y}}, {\boldsymbol{Z}})) -
I( {\boldsymbol{X}}; {\boldsymbol{Z}})$.
\[fact:DataProcessing\] For any three random variables ${\boldsymbol{X}}, {\boldsymbol{Y}}, {\boldsymbol{Z}}$ such that ${\boldsymbol{X}}$ and ${\boldsymbol{Z}}$ are conditionally independent given ${\boldsymbol{Y}}$, $I({\boldsymbol{X}}; {\boldsymbol{Y}})
\geq I({\boldsymbol{X}}; {\boldsymbol{Z}})$.
The key technical ingredient of our proof is the following lemma (the notation ${\mathcal{P}}$ is defined in Section \[sec:prelim\]).
\[lma:infoLowerBoundGeneral\] For any node $i$, the inequality $I({\boldsymbol{E}}; {\boldsymbol{T}}_i) \geq E[|{\mathcal{P}}({\boldsymbol{T}}_i)|]$ holds.
The proof of Lemma \[lma:infoLowerBoundGeneral\] is based on an information-theoretic argument. The intuition is very simple: since the information ${\boldsymbol{T}}_i$ completely reveals the information on the edges forming triangles in ${\boldsymbol{T}}_i$, it must contain all the information on ${\mathcal{P}}({\boldsymbol{T}}_i)$.
We will write $M = |{\mathcal{E}}| = \frac{n(n - 1)}{2}$ and $N = |\mathcal{T}| = \frac{n(n-1)(n-2)}{6}$.
Since the random variables ${\boldsymbol{e}}_{\{j,k\}}$ are independent of each other, we have $$\begin{aligned}
\lefteqn{I({\boldsymbol{E}}; {\boldsymbol{T}}_i)} \\
&= H({\boldsymbol{E}}) - H({\boldsymbol{E}}|{\boldsymbol{T}}_i) \\
&= H({\boldsymbol{E}}) - \sum_{\{j,k\} \in \mathcal{E}} H({\boldsymbol{e}}_{\{j,k\}} | {\boldsymbol{T}}_i) \\
&= H({\boldsymbol{E}}) - \sum_{\{j,k\} \in \mathcal{E}} \sum_{R \subseteq \mathcal{T}}
H({\boldsymbol{e}}_{\{j,k\}} | {\boldsymbol{T}}_i = R) \cdot \Pr[{\boldsymbol{T}}_i = R] \\
&= M - \sum_{R \subseteq \mathcal{T}} \sum_{\{j,k\} \in \mathcal{E}}
H({\boldsymbol{e}}_{\{j,k\}} | {\boldsymbol{T}}_i = R) \cdot \Pr[{\boldsymbol{T}}_i = R]\\
&= M - \sum_{R \subseteq \mathcal{T}}\:\: \sum_{\{j,k\} \in{\mathcal{E}}\setminus{\mathcal{P}}(R)}
H({\boldsymbol{e}}_{\{j,k\}} | {\boldsymbol{T}}_i = R) \cdot \Pr[{\boldsymbol{T}}_i = R],\end{aligned}$$ where the last equality comes from the assumption that Algorithm $\mathcal{A}$ never outputs a triple that is not a triangle (thus $\Pr[{\boldsymbol{T}}_i = R]\neq 0$ only if $R\subseteq T(G)$, and conditioned to this event we have ${\boldsymbol{e}}_{\{j,k\}}=1$ with probability $1$ for any $\{j,k\}\in{\mathcal{P}}(R)$). Since $H({\boldsymbol{e}}_{\{j,k\}} | {\boldsymbol{T}}_i = R) \leq 1$ always holds, we conclude that $$\begin{aligned}
I({\boldsymbol{E}}; {\boldsymbol{T}}_i) &\geq M - \sum_{R \subseteq \mathcal{T}} (M - |{\mathcal{P}}(R)|) \cdot \Pr[{\boldsymbol{T}}_i = R] \\
&\geq \sum_{R \subseteq \mathcal{T}} |{\mathcal{P}}(R)| \cdot \Pr[{\boldsymbol{T}}_i = R] = E[|{\mathcal{P}}({\boldsymbol{T}}_i)|],\end{aligned}$$ as claimed.
Theorem \[thm:lowerBound\] is proved by combining Lemma \[lma:triangleEdge\] and Lemma \[lma:infoLowerBoundGeneral\]. The intuition is again fairly simple: In expectation, an instance of $G(n, 1/2)$ contains $\Omega(n^3)$ triangles, and thus the average number of triangles output per node node is $\Omega(n^2)$. Thus node $w({\boldsymbol{T}})$ outputs at least $\Omega(n^2)$ triangles and, from Lemma \[lma:triangleEdge\], we obtain $E[|{\mathcal{P}}({\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}})|] = \Omega(n^{4/3})$. Lemma \[lma:infoLowerBoundGeneral\] then gives the same lower bound for the mutual information between ${\boldsymbol{E}}$ and ${\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}$, which then gives the lower bound $\Omega(n^{4/3})$ on the amount of communication received by node $w({\boldsymbol{T}})$ during the execution of Algorithm ${\mathcal{A}}$. This implies the claimed $\Omega(n^{1/3}/\log n)$-round lower bound for triangle listing since $w({\boldsymbol{T}})$ can receive at most $O(n \log n)$ bits of information per round.
From Lemma \[lma:infoLowerBoundGeneral\], we have $$I({\boldsymbol{E}}; {\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}) \geq E[|P({\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}})|].$$ By Lemma \[lma:triangleEdge\], for any $R\subseteq T(G)$ the inequality $|{\mathcal{P}}(R)| \geq \frac{\sqrt{2}}{3}|R|^{2/3}$ holds. We thus have $$\begin{aligned}
\lefteqn{I({\boldsymbol{E}}; {\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}) \geq E[|{\mathcal{P}}({\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}})|]} \hspace{5mm} \\
&= \sum_{R \subseteq \mathcal{T}} |P(R)| \cdot \Pr[{\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}= R] \\
&\geq \sum_{R \subseteq \mathcal{T}, |R| \geq \frac{N}{16n}}
|{\mathcal{P}}(R)| \cdot \Pr[{\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}= R]. \\
&\geq \sum_{R \subseteq \mathcal{T}, |R| \geq \frac{N}{16n}}
\frac{\sqrt{2}}{3}\left(\frac{N}{16n}\right)^{2/3} \cdot \Pr[{\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}= R]\\
&\geq \frac{\sqrt{2}}{3}\left(\frac{N}{16n}\right)^{2/3} \cdot \Pr\left[|{\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}| \geq \frac{N}{16n} \right].\end{aligned}$$ The expected number of triangles in $G(n,1/2)$ is $N/8$ and thus, since there cannot be more than $N$ triangles, with probability at least $1/15$ the number of triangles exceeds $N/16$. We conclude that the inequality $|{\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}}| \geq N/(16n)$ necessarily holds if the number of triangles is at least $N/16$ and Algorithm $\mathcal{A}$ correctly lists all the triangles of the graph, which occurs with overall probability probability at least $1/15 - 1/32$. Consequently we obtain $$\label{eq0}
I({\boldsymbol{E}}; {\ensuremath{{\boldsymbol{T}}_{w({\boldsymbol{T}})}}})\geq \frac{\sqrt{2}}{3}\left(\frac{N}{16n}\right)^{2/3} \cdot \left(\frac{1}{15}-\frac{1}{32}\right) =
\Omega(n^{4/3}).$$
From Fact \[fact:mutualInfo\] we have $$\begin{aligned}
H({\boldsymbol{\pi}}_{w({\boldsymbol{T}})})
&\geq I({\boldsymbol{E}}; {\boldsymbol{\pi}}_{w({\boldsymbol{T}})})\nonumber \\
&\geq
I({\boldsymbol{E}}; ({\boldsymbol{\pi}}_{w({\boldsymbol{T}})}, {\boldsymbol{\rho}}_{w({\boldsymbol{T}})}))
- I({\boldsymbol{E}}; {\boldsymbol{\rho}}_{w({\boldsymbol{T}})}) \nonumber \\
&\geq I({\boldsymbol{E}}; ({\boldsymbol{\pi}}_{w({\boldsymbol{T}})}, {\boldsymbol{\rho}}_{w({\boldsymbol{T}})}))
- H({\boldsymbol{\rho}}_{w({\boldsymbol{T}})}).\label{eq1c}\end{aligned}$$ Since the initial knowledge of each node $i$ is only the set of edges incident to itself, the inequality $$\label{eq2}
H({\boldsymbol{\rho}}_{i})\le \sum_{j\neq i} H({\boldsymbol{e}}_{i,j}) = n-1$$ holds for each node $i\in V$. Since the output ${\boldsymbol{T}}_{w({\boldsymbol{T}})}$ is computed locally only from the transcript ${\boldsymbol{\pi}}_{w({\boldsymbol{T}})}$ and the initial state ${\boldsymbol{\rho}}_{w({\boldsymbol{T}})}$, the random variables ${\boldsymbol{E}}$ and ${\boldsymbol{T}}_{w({\boldsymbol{T}})}$ are conditionally independent given $({\boldsymbol{\rho}}_{w({\boldsymbol{T}})}, {\boldsymbol{\pi}}_{w({\boldsymbol{T}})})$. We can thus use Fact \[fact:DataProcessing\] in Inequality (\[eq1c\]), which combined with Inequalities (\[eq0\]) and (\[eq2\]) gives $$\begin{aligned}
H({\boldsymbol{\pi}}_{w({\boldsymbol{T}})})
&\geq I({\boldsymbol{E}}; {\boldsymbol{T}}_{w({\boldsymbol{T}})}) - H({\boldsymbol{\rho}}_{w({\boldsymbol{T}})}) = \Omega(n^{4/3}).\end{aligned}$$ Since $H({\boldsymbol{\pi}}_i)$ lower bounds the average length of the transcript ${\boldsymbol{\pi}}_i$, this implies that node ${w({\boldsymbol{T}})}$ receives messages of average total length $\Omega(n^{4/3})$ bits. Since node ${w({\boldsymbol{T}})}$ can receive only $O(n\log n)$ bits per round, we conclude that the expected round complexity of Algorithm $\mathcal{A}$ is $\Omega(n^{1/3} / \log n)$.
The information-theoretic arguments of Lemma \[lma:infoLowerBoundGeneral\] can also be used to derive the following stronger lower bound mentioned in the introduction for local triangle listing, the setting where each node $i$ is required to output all the triangles containing $i$.
\[cor\] Let $\mathcal{A}$ be any local triangle listing algorithm with error probability less than $1/32$. Then there exists a probability distribution on inputs such that the expected round complexity of $\mathcal{A}$ is $\Omega(n/\log n)$.
The proof of Proposition \[cor\] is similar to the proof of Theorem \[thm:lowerBound\] but does not even requires Lemma \[lma:triangleEdge\]: we can immediately obtain the stronger bound $E[|{\mathcal{P}}({\boldsymbol{T}}_i)|] = \Omega(n^{2})$ since for local triangle listing ${\boldsymbol{T}}_i$ should include all the triangles containing $i$.
The proof is almost similar to the proof of Theorem \[thm:lowerBound\], except that we can derive the stronger lower bound $$\label{eq4}
I({\boldsymbol{E}}; {\boldsymbol{T}}_i) =
\Omega(n^{2})
\hspace{2mm}
\textrm{ for any $i\in V$},$$ instead of the lower bound (\[eq0\]). This new lower bound implies that $H({\boldsymbol{\pi}}_{w({\boldsymbol{T}})})=\Omega(n^2)$ and gives the claimed lower bound on the round complexity of the local triangle listing algorithm ${\mathcal{A}}$.
We now explain how to derive the lower bound (\[eq4\]). From Lemma \[lma:infoLowerBoundGeneral\] we get $$\begin{aligned}
\lefteqn{I({\boldsymbol{E}};{\boldsymbol{T}}_i) \geq E[|{\mathcal{P}}( {\boldsymbol{T}}_i)|]} \\
&= \sum_{R \subseteq \mathcal{T}} |{\mathcal{P}}(R)| \cdot \Pr[ {\boldsymbol{T}}_i = R] \\
&\geq \frac{M}{16}
\cdot \Pr\big[ {\boldsymbol{T}}_i \textrm{ includes at least $M/6$ triangles containing $i$} \big], \end{aligned}$$ since the quantity $|{\mathcal{P}}(R)|$ is lower bounded by the number of triangles in $R$ containing node $i$. The expected number of triangles in $G(n,1/2)$ containing any fixed node is $M/8$ and thus, since there cannot be more than $M$ such triangles, with probability at least $1/15$ the number of triangles containing node $i$ exceeds $M/16$. Since in Algorithm $\mathcal{A}$ node $i$ correctly lists all the triangles containing $i$ with probability at least $1-1/32$, we conclude that ${\boldsymbol{T}}_i$ includes at least $M/6$ triangles including $i$ with overall probability at least $1/15 - 1/32$. We thus obtain $$\begin{aligned}
I({\boldsymbol{E}};{\boldsymbol{T}}_i)&\geq \frac{M}{16}
\cdot \left(\frac{1}{15}-\frac{1}{32}\right) =
\Omega(n^2),\end{aligned}$$ as claimed
Acknowledgments {#acknowledgments .unnumbered}
===============
FLG was partially supported by MEXT KAKENHI (24106009) and JSPS KAKENHI (15H01677, 16H01705, 16H05853). TI was partially supported by JSPS KAKENHI (15H00852, 16H02878) and JST-SICORP Japan-Israel Bilateral Program “ICT for a resilient society”.
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[^1]: Postal address: Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555, Japan. Tel: (+81)-52-735-5567.
[^2]: Postal address: Department of Communications and Computer Engineering, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan. Tel: (+81)-75-753-5382.
|
---
author:
- 'Yujun Shen, Ceyuan Yang, Xiaoou Tang, Bolei Zhou'
bibliography:
- 'ref.bib'
title: '[InterFaceGAN: Interpreting the Disentangled Face Representation Learned by GANs]{}'
---
{width="1.0\linewidth"}
years have witnessed the great success of Generative Adversarial Networks (GANs) [@gan] in high-fidelity face synthesis [@pggan; @stylegan; @stylegan2]. Based on adversarial training, GANs learn to map a random distribution to the real data observation and then produce photo-realistic images from randomly sampled latent codes.
Despite the appealing synthesis quality, it remains much less explored about what knowledge GANs actually learn in the latent representation and how we can reuse such knowledge to control the generation process. For example, given a latent code, how does GAN determine the facial attributes of the output face, *e.g.*, an old man or a young woman? How are different attributes organized in the latent space and are they entangled or disentangled? Can we manipulate the attributes of the synthesized face as we want? How does the attribute manipulation affect the face identity? Can we apply a well-trained GAN model for real image editing?
To answer these questions, we propose a novel framework, termed as *InterFaceGAN*, to *Inter*pret the latent *Face* representation learned by *GAN* models. For this purpose, we employ some off-the-shelf classifiers to predict semantic scores for the images synthesized by GANs. In this way, we are able to bridge the latent space and the semantic space and further utilize such connection for representation analysis. In particular, we analyze how an individual semantic is encoded in the latent space *both theoretically and empirically*. It turns out that a true-or-false facial attribute actually aligns with a linear subspace of the latent space. Based on this discovery, we study the entanglement between different semantics emerging in the latent representation and manage to disentangle them via subspace projection.
Besides finding the latent semantics, InterFaceGAN proposes an effective pipeline for face editing. By simply modulating the latent code, we can successfully manipulate the gender, age, expression, presence of eyeglasses, and even facial pose of the synthesized image, as shown in Fig.\[fig:teaser\](a). In addition, thanks to our disentanglement analysis, we propose conditional manipulation to alter one attribute without affecting others, as shown in Fig.\[fig:teaser\](b). More importantly, InterFaceGAN achieves high-quality face manipulation by reusing the semantic knowledge learned by GANs *without* any retraining.
To get a better interpretation of the semantics in the GAN representation, we conduct thorough analysis on the editing results made by InterFaceGAN. First, we compare the semantic scores before and after the manipulation to quantitatively verify whether the semantics identified by InterFaceGAN are indeed manipulable. Then, we make layer-wise analysis on StyleGAN, whose generator is with per-layer stochasticity [@stylegan], to explore how semantics originate from the latent representation layer by layer. Finally, considering the importance of the identity information for faces, we make in-depth identity analysis to see how identity is preserved in the manipulation process as well as how identity is sensitive to different facial attributes.
How to apply a GAN model to real image editing is another important issue since GANs commonly lack the inference ability. In this work, we integrate two approaches to extend InterFaceGAN to real face manipulation. One is to combine with GAN inversion, which is able to invert a target image back to a latent code, and then directly vary the inverted code. The other is to use InterFaceGAN to build a synthetic dataset which contains the pairs of synthetic images before and after manipulation and then train a pixel-to-pixel model on this dataset. We compare these approaches and evaluate their strengths and weaknesses.
The preliminary result of this work is published at [@interfacegan]. Compared to the previous conference paper, we include following new contents: (i) a detailed analysis on the face representation learned by StyleGAN [@stylegan] as well as its comparison to the representation learned by PGGAN [@pggan]; (ii) a comparison between the entanglement of latent semantics and the attribute distribution of training data, which sheds light on how GANs learn to encode various semantics during the training process; (iii) quantitative evaluation on the editing results achieved by InterFaceGAN; (iv) layer-wise analysis on the per-layer representation learned by StyleGAN [@stylegan]; (v) identity analysis on the images before and after manipulation; and (vi) a new method to apply InterFaceGAN to real face editing, which is to train feed-forward models on the synthetic data collected by InterFaceGAN.
We summarize our contributions as follows:
- We propose InterFaceGAN to deeply interpret the face representation learned by state-of-the-art GAN models, *i.e.*, PGGAN [@pggan] and StyleGAN [@stylegan], and observe that GANs spontaneously encode various semantics in the latent representation.
- We study the entanglement between different semantics and compare it with the attribute distribution of the training data. This sheds light on how GANs learn to organize the semantic knowledge in the training process. We also find that these semantics become disentangled after some linear transformations in the latent space.
- We show that InterFaceGAN enables semantic face editing with *fixed* pre-trained GAN models. We can precisely control the generation process even with an unconditional GAN. Some results are shown in Fig.\[fig:teaser\]. Besides gender, age, expression, and the presence of eyeglasses, we can noticeably also vary the face pose and correct some artifacts produced by GANs.
- We analyze the face identity before and after the manipulation using InterFaceGAN to verify whether the latent semantics learned by GANs are able to preserve identity. With the help of a *generative* model, we can in turn evaluate how a particular facial attribute affects the performance of a *discriminative* face verification model.
- We perform layer-wise analysis on StyleGAN model which employs multiple latent codes for face synthesis. This provides us some insights into how different semantics originate from the per-layer face representation.
- We extend InterFaceGAN with GAN inversion methods to facilitate real image editing. By mapping a real face back to the latent space of a GAN model, we can successfully manipulate its attributes by simply varying the inverted code, even the GAN model is not specifically designed for this task.
- We utilize the manipulation capability of InterFaceGAN to build paired dataset, *i.e.*, one is the raw synthesis while the other is the result by manipulating a particular attribute. Training with such synthetic data, we get feed-forward real face editing models with fast inference speed.
Related Work {#sec:related-work}
============
**Generative Adversarial Networks.** Due to the great potential of GAN [@gan] in producing photo-realistic images, it has been widely applied to image editing [@lample2017fader; @bau2019semantic], super-resolution [@ledig2017photo; @wang2018esrgan], image inpainting [@yeh2017semantic; @yu2019free], video synthesis [@wang2018video; @wang2019few], *etc*. Many attempts have been made to improve GANs by introducing new criteria [@wgan; @wgan_gp; @sngan], designing novel network structures [@sagan; @began; @stylegan], or optimizing the training pipeline [@pggan; @biggan]. The recent StyleGAN2 [@stylegan2] model achieves state-of-the-art face synthesis results with incredible image quality. Despite this tremendous success, little work has been done on understanding how GANs learn to connect the latent representation with the semantics in the real visual world.
**Study on Latent Space of GANs.** Latent space of GANs is generally treated as Riemannian manifold [@gan_metrics; @latent_oddity; @kuhnel2018latent]. Prior work focused on exploring how to make the output image vary smoothly from one synthesis to another through interpolation in the latent space [@feature_based_metrics; @riemannian_geometry]. Bojanowski *et al.* [@glo] optimized the generator and latent code simultaneously to learn a better representation. However, the studies on how a well-trained GAN is able to encode different semantics in the latent space as well as how to reuse these semantic knowledge to control the generation process are still missing. Bau *et al.* [@bau2019gandissect] found that some units from intermediate layers of the GAN generator are specialized to synthesize certain visual concepts, such as sofa and TV for living room synthesis. Some work [@dcgan; @feature_interpolation] observed the vector arithmetic property in the latent space. Beyond that, this work provides detailed analysis on how semantics are encoded in the face representation learned by GANs from both the property of a single semantic and the disentanglement of multiple semantics. Some concurrent work also explores the latent semantics in GANs: Goetschalckx *et al.* [@goetschalckx2019ganalyze] improves the memorability of the output image. Jahanian *et al.* [@gansteerability] studies the steerability of GANs concerning camera motion and image color tone. Yang *et al.* [@yang2019semantic] observes the semantic hierarchy emerging in the scene synthesis models. Unlike them, we focus on interpreting the face representation by theoretically and empirically studying how various semantics originate from and are organized in the latent space. We further extend our method to *real image* editing.
**Semantic Face Editing with GANs.** Semantic face editing aims at manipulating facial attributes of a given image. Compared to unconditional GANs which can generate image arbitrarily, semantic editing expects the model to only change the target attribute but maintain other information of the input face. To achieve this goal, current methods required carefully designed loss functions [@acgan; @infogan; @drgan], introduction of additional attribute labels [@lample2017fader; @ffgan; @opensetgan; @elegant; @facefeatgan], or special architectures [@sdgan; @faceidgan] to train new models. Different from previous learning-based methods, this work explores the interpretable semantics inside the latent space of *fixed* GAN models. By reusing the semantic knowledge spontaneously learned by GANs, we are able to unleash its manipulation capability and *turn unconstrained GANs to controllable GANs* by simply varying the latent code.
**GAN Inversion.** GAN is typically formulated as a two-player game, where a generator takes a sampled latent code as the input and outputs an image synthesis while a discriminator differentiates real domain from synthesized domain. Hence, it leaves no space for making inference on real images. To enable real image editing with fixed GAN models [@zhu2016generative; @bau2019semantic], a common practice is to get the reverse mapping from the image space to the latent space, which is also known as GAN Inversion [@perarnau2016invertible; @lipton2017precise; @creswell2018inverting]. Prior work either performed instance-level optimization [@invertibility; @image2stylegan; @image2stylegan++] or explicitly learned an encoder corresponding to the generator [@ali; @bigan; @lia]. Some methods combined these two ideas by using the encoder to produce a good starting point for optimization [@bau2019seeing; @bau2019inverting]. Recently, the GAN inversion task is significantly advanced: Gu *et al.* [@gu2020image] proposes to increase the number of latent codes for a better image reconstruction. Pan *et al.* [@pan2020exploiting] optimizes the latent code together with the model weights. Zhu *et al.* [@zhu2020indomain] takes semantic information into account besides recovering the pixel values. Being orthogonal to these approaches, our work interprets the representation learned by GANs and then utilizes GAN inversion as a tool to enable real image editing by reusing the latent knowledge.
**Image-to-Image Translation.** Image-to-Image translation, aiming at learning a deterministic model to transfer images from one domain to another, is another way to manipulate real images. Existing work used image-to-image translation models to generate photo-realistic images from scene layouts [@karacan2016learning; @park2019semantic], sketches [@pix2pix], or segmentation labels [@pix2pixhd]. This idea is further freed from the requirement of paired training data, resulting in an unsupervised learning manner [@liu2017unsupervised; @cyclegan]. There are also some attempts that increase the diversity of the translated images by introducing stochasticity [@huang2018multimodal; @bicyclegan] or translate images among multiple domains [@stargan; @stargan2]. However, all these models rely on paired data or domain labels, which are not that easy to obtain. In this work, we manage to leverage the semantics learned by GANs to create unlimited synthetic data pairs. By training image-to-image translation networks with such synthetic data, we are able to apply the knowledge encoded in the latent representation to feed-forward real image editing.
Framework of InterFaceGAN {#sec:interfacegan}
=========================
In this section, we introduce the framework of InterFaceGAN. We first provide rigorous analysis on several properties of the semantic attributes emerging in the latent space of well-trained GAN models, and then construct a pipeline of utilizing the identified semantics in latent code for face editing.
Semantics in Latent Space {#subsec:semantics-interpretation}
-------------------------
Given a well-trained GAN model, the generator can be formulated as a deterministic function $g: {\mathcal{Z}}\rightarrow{\mathcal{X}}$. Here, ${\mathcal{Z}}\subseteq{\mathbb{R}}^{d}$ denotes the $d$-dimensional latent space, for which Gaussian distribution ${\mathcal{N}}({{\rm\bf 0}}, {{\rm\bf I}}_d)$ is commonly used [@sngan; @pggan; @biggan; @stylegan]. ${\mathcal{X}}$ stands for the image space, where each sample ${{\rm\bf x}}$ possesses certain semantic information, like gender and age for face model. Suppose we have a semantic scoring function $f_S: {\mathcal{X}}\rightarrow{\mathcal{S}}$, where ${\mathcal{S}}\subseteq{\mathbb{R}}^m$ represents the semantic space with $m$ semantics. We can bridge the latent space ${\mathcal{Z}}$ and the semantic space ${\mathcal{S}}$ with ${{\rm\bf s}}= f_S(g({{\rm\bf z}}))$, where ${{\rm\bf s}}$ and ${{\rm\bf z}}$ denote semantic scores and the sampled latent code respectively.
**Single Semantic.** It has been widely observed that when linearly interpolating two latent codes ${{\rm\bf z}}_1$ and ${{\rm\bf z}}_2$, the appearance of the corresponding synthesis changes continuously [@dcgan; @biggan; @stylegan]. It implicitly means that the semantics contained in the image also change gradually. According to ***Property 1***, the linear interpolation between ${{\rm\bf z}}_1$ and ${{\rm\bf z}}_2$ forms a direction in ${\mathcal{Z}}$, which further defines a hyperplane. We therefore make an assumption[^1] that for any binary semantic (*e.g.*, male *v.s.* female), there exists a hyperplane in the latent space serving as the separation boundary. Semantic remains the same when the latent code walks within one side of the hyperplane yet turns into the opposite when across the boundary.
Given a hyperplane with unit normal vector ${{\rm\bf n}}\in{\mathbb{R}}^d$, we define the “distance” from a sample ${{\rm\bf z}}$ to this hyperplane as $$\begin{aligned}
{{\rm d}}({{\rm\bf n}}, {{\rm\bf z}})= {{\rm\bf n}}^T{{\rm\bf z}}. \label{eq:distance}\end{aligned}$$ Here, ${{\rm d}}(\cdot,\cdot)$ is not a strictly defined distance since it can be negative. When ${{\rm\bf z}}$ lies near the boundary and is moved toward and across the hyperplane, both the “distance” and the semantic score vary accordingly. And it is just at the time when the “distance” changes its numerical sign that the semantic attribute reverses. We therefore expect these two to be linearly dependent with $$\begin{aligned}
f(g({{\rm\bf z}})) = \lambda{{\rm d}}({{\rm\bf n}},{{\rm\bf z}}), \label{eq:linear-dependency}\end{aligned}$$ where $f(\cdot)$ is the scoring function for a particular semantic, and $\lambda > 0$ is a scalar to measure how fast the semantic varies along with the change of “distance”. According to ***Property 2***, random samples drawn from ${\mathcal{N}}({{\rm\bf 0}},{{\rm\bf I}}_d)$ are very likely to locate close enough to a given hyperplane. Therefore, the corresponding semantic can be modeled by the linear subspace that is defined by ${{\rm\bf n}}$.
***Property 1** Given ${{\rm\bf n}}\in{\mathbb{R}}^d$ with ${{\rm\bf n}}\neq{{\rm\bf 0}}$, the set $\{{{\rm\bf z}}\in{\mathbb{R}}^d:{{\rm\bf n}}^T{{\rm\bf z}}=0\}$ defines a hyperplane in ${\mathbb{R}}^d$, and ${{\rm\bf n}}$ is called the normal vector. All vectors ${{\rm\bf z}}\in{\mathbb{R}}^d$ satisfying ${{\rm\bf n}}^T{{\rm\bf z}}>0$ locate from the same side of the hyperplane.*
***Property 2** Given ${{\rm\bf n}}\in{\mathbb{R}}^d$ with ${{\rm\bf n}}^T{{\rm\bf n}}=1$, which defines a hyperplane, and a multivariate random variable ${{\rm\bf z}}\sim{\mathcal{N}}({{\rm\bf 0}},{{\rm\bf I}}_d)$, we have ${{\rm P}}(|{{\rm\bf n}}^T{{\rm\bf z}}|\leq2\alpha~\sqrt{\frac{d}{d-2}})\geq(1-3e^{-c d})(1-\frac{2}{\alpha}e^{-\alpha^2/2})$ for any $\alpha\geq1$ and $d\geq4$. Here, ${{\rm P}}(\cdot)$ stands for probability and $c$ is a fixed positive constant.[^2]*
**Multiple Semantics.** When the case comes to $m$ different semantics, we have $$\begin{aligned}
{{\rm\bf s}}\equiv f_S(g({{\rm\bf z}}))=\Lambda{{\rm\bf N}}^T{{\rm\bf z}}, \label{eq:multiple-semantics}\end{aligned}$$ where ${{\rm\bf s}}= [s_1, \dots, s_m]^T$ denotes the semantic scores, $\Lambda = \text{diag}(\lambda_1, \dots, \lambda_m)$ is a diagonal matrix containing the linear coefficients, and ${{\rm\bf N}}= [{{\rm\bf n}}_1, \dots, {{\rm\bf n}}_m]$ indicates the separation boundaries. Aware of the distribution of random sample ${{\rm\bf z}}$, which is ${\mathcal{N}}({{\rm\bf 0}},{{\rm\bf I}}_d)$, we can easily compute the mean and covariance matrix of the semantic scores ${{\rm\bf s}}$ as $$\begin{aligned}
\bm{\mu}_{{\rm\bf s}}&= {\mathbb{E}}(\Lambda{{\rm\bf N}}^T{{\rm\bf z}}) = \Lambda{{\rm\bf N}}^T{\mathbb{E}}({{\rm\bf z}}) = {{\rm\bf 0}}, \label{eq:score-mean} \\
\bm{\Sigma}_{{\rm\bf s}}&= {\mathbb{E}}(\Lambda{{\rm\bf N}}^T{{\rm\bf z}}{{\rm\bf z}}^T{{\rm\bf N}}\Lambda^T) = \Lambda{{\rm\bf N}}^T{\mathbb{E}}({{\rm\bf z}}{{\rm\bf z}}^T){{\rm\bf N}}\Lambda^T \notag \\
&= \Lambda{{\rm\bf N}}^T{{\rm\bf N}}\Lambda. \label{eq:score-cov}\end{aligned}$$
We therefore have ${{\rm\bf s}}\sim{\mathcal{N}}({{\rm\bf 0}},\bm{\Sigma}_{{\rm\bf s}})$, which is a multivariate normal distribution. Different entries of ${{\rm\bf s}}$ are disentangled if and only if $\bm{\Sigma}_{{\rm\bf s}}$ is a diagonal matrix, which requires $\{{{\rm\bf n}}_1, \dots, {{\rm\bf n}}_m\}$ to be orthogonal with each other. If this condition does not hold, some semantics will entangle with each other. ${{\rm\bf n}}_i^T{{\rm\bf n}}_j$ can be used to measure the entanglement between the $i$-th and $j$-th semantics to some extent.
Manipulation in Latent Space {#subsec:semantics-manipulation}
----------------------------
In this part, we introduce how to use the semantics found in the latent space for image editing.
**Single Attribute Manipulation.** According to Eq., to manipulate the attribute of a synthesized image, we can easily edit the original latent code ${{\rm\bf z}}$ with ${{\rm\bf z}}_{edit}={{\rm\bf z}}+ \alpha {{\rm\bf n}}$. It will make the synthesis look more positive on such semantic with $\alpha>0$ since the score becomes $f(g({{\rm\bf z}}_{edit}))=f(g({{\rm\bf z}})) + \lambda\alpha$ after editing. Similarly, $\alpha < 0$ will make the synthesis look more negative.
**Conditional Manipulation.** When there is more than one attribute, editing one may affect another since some semantics can be coupled with each other. To achieve more precise control, we propose *conditional manipulation* by manually forcing ${{\rm\bf N}}^T{{\rm\bf N}}$ in Eq. to be diagonal. In particular, we use projection to orthogonalize different vectors. As shown in Fig.\[fig:subspace\], given two hyperplanes with normal vectors ${{\rm\bf n}}_1$ and ${{\rm\bf n}}_2$, we find a projected direction ${{\rm\bf n}}_1 - ({{\rm\bf n}}_1^T{{\rm\bf n}}_2){{\rm\bf n}}_2$, such that moving samples along this new direction can change “attribute 1” without affecting “attribute 2”. If there are multiple attributes to be conditioned on, we just subtract the projection from the primal direction onto the plane that is constructed by all conditioned directions.
![ Illustration of the **conditional manipulation via subspace projection**. The projection of ${{\rm\bf n}}_1$ onto ${{\rm\bf n}}_2$ is subtracted from ${{\rm\bf n}}_1$, resulting in a new direction ${{\rm\bf n}}_1 - ({{\rm\bf n}}_1^T{{\rm\bf n}}_2){{\rm\bf n}}_2$. []{data-label="fig:subspace"}](figures/subspace.pdf){width="0.75\linewidth"}
**Real Image Manipulation.** InterFaceGAN enables semantic editing from the latent space of a *fixed* GAN model. Therefore, to manipulate real images, a straightforward way is to infer the best latent code that can be used to reconstruct the target image, *i.e.*, GAN inversion. For this purpose, both optimization-based [@zhu2020indomain] and learning-based [@lia] approaches can be used. We thoroughly evaluate their strengths and weaknesses in Sec.\[sec:real-image-manipulation\]. We also use InterFaceGAN to prepare synthetic data pairs and then train image-to-image translation models [@pix2pixhd] to achieve real face editing. This kind of approach is also analyzed in Sec.\[sec:real-image-manipulation\].
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Implementation Details {#subsec:implementation-details}
----------------------
We choose five key facial attributes for analysis, including pose, smile (expression), age, gender, and eyeglasses. The corresponding positive directions are defined as turning right, laughing, getting old, changing to male, and wearing eyeglasses. Note that we can always easily plug in more attributes as long as the attribute predictor (scoring function) is available.
To better predict these attributes from synthesized images, we train an auxiliary attribute prediction model using the annotations from the CelebA dataset [@celeba] with ResNet-50 network [@resnet]. This model is trained with multi-task losses to simultaneously predict smile, age, gender, eyeglasses, as well as the 5-point facial landmarks (left eye, right eye, nose, left corner of mouth, right corner of mouth). Here, the facial landmarks are used to compute yaw pose, which is also treated as a binary attribute (left or right) in further analysis. Besides the landmarks, all other attributes are learned as a bi-classification problem with softmax cross-entropy loss, while landmarks are optimized with $l_2$ regression loss. As images produced by PGGAN and StyleGAN are with $1024\times1024$ resolution, we resize them to $224\times224$ before feeding them to the attribute model.
Given the pre-trained GAN model, we synthesize $500K$ images by randomly sampling from the latent space. There are mainly two reasons in preparing such large-scale dataset: (i) to eliminate the randomness caused by sampling and make sure the distribution of the sampled code is as expected, and (ii) to get enough wearing-glasses samples, which are really rare in PGGAN model.
To find the semantic boundaries in latent space, we use the pre-trained attribute prediction model to assign attribute scores for all $500K$ synthesized images. For each attribute, we sort the corresponding scores and choose $10K$ samples with the highest scores and $10K$ with the lowest ones as candidates. The reason in doing so is that the prediction model is not absolutely accurate and may produce wrong predictions for ambiguous samples, *e.g.*, middle-aged person for age attribute. We then randomly choose 70% samples from the candidates as the training set to learn a linear SVM, resulting in a decision boundary. Recall that, normal directions of all boundaries are normalized to unit vectors. Remaining 30% samples are used for verifying how the linear classifier behaves. Here, for SVM training, the inputs are the $512d$ latent codes, while the binary labels are assigned by the auxiliary attribute prediction model.
Interpreting Face Representation {#sec:interpretation}
================================
In this section, we apply InterFaceGAN to interpreting the face representation learned by state-of-the-art GAN models, *i.e.*, PGGAN [@pggan] and StyleGAN [@stylegan], both of what are able to produce high-quality faces with $1024\times1024$ resolution. PGGAN is a representative of traditional generator where the latent code is only fed into the very first convolutional layer. By contrast, StyleGAN proposed a style-based generator, which first maps the latent code from latent space ${\mathcal{Z}}$ to a disentangled latent space ${\mathcal{W}}$ before applying it for generation. In addition, the disentangled latent code is fed to all convolutional layers.
Separability of Latent Space {#subsec:latent-space-separation}
----------------------------
As mentioned in Sec.\[subsec:semantics-interpretation\], our framework is based on an assumption that for any binary attribute, there exists a hyperplane in the latent space such that all samples from the same side are with the same attribute. In this part, we would like to first evaluate the correctness of this assumption to make the remaining analysis considerable.
![ Synthesized samples by PGGAN [@pggan] with the distance near to (middle row) and extremely far away from (top and bottom rows) the separation boundary. Each column corresponds to a particular attribute. []{data-label="fig:pggan-separation"}](figures/separation.pdf){width="1.0\linewidth"}
### PGGAN
For each attribute, we will get a latent boundary after the training of the linear SVM classifier. We evaluate the classification performance on the validation set ($3K$ positive testing samples and $3K$ negative testing samples) as well as the entire set (remaining $480K$ samples besides the $20K$ candidates with high confidence level). Fig.\[fig:separation-accuracy\] shows the results. We find that all linear boundaries of PGGAN achieve over 95% accuracy on the validation set and over 75% on the entire set, suggesting that for a binary attribute, there indeed exists a linear hyperplane in the latent space that can well separate the data into two groups.
We also visualize some samples in Fig.\[fig:pggan-separation\] through ranking them by the “distance” to the decision boundary. Note that those extreme cases (top and bottom rows) are very unlikely to be directly sampled, instead constructed by moving a latent code towards the normal direction “infinitely”. From Fig.\[fig:pggan-separation\], we can tell that the positive samples and negative samples are distinguishable to each other with respect to the corresponding attribute. This further demonstrates that the latent space is linearly separable and InterFaceGAN is able to successfully find the proper separation hyperplane.
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![ Examples on **fixing the artifacts** that PGGAN [@pggan] has generated. First row shows some bad generation results, while the following two rows present the gradually corrected synthesis by moving the latent codes along the positive “quality” direction. []{data-label="fig:pggan-correction"}](figures/artifact.pdf){width="1.0\linewidth"}
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### StyleGAN
Compared to PGGAN, StyleGAN employs two latent spaces, which are the native latent space ${\mathcal{Z}}$ and the mapped latent space ${\mathcal{W}}$. We analyze the separability of both spaces and the results are shown in Fig.\[fig:separation-accuracy\]. Note that ${\mathcal{W}}$ space is not subject to normal distribution like ${\mathcal{Z}}$ space, but we can conduct a similar analysis on ${\mathcal{W}}$ space, demonstrating the generalization ability of InterFaceGAN.
We mainly have three observations from Fig.\[fig:separation-accuracy\]. (i) Boundaries from both ${\mathcal{Z}}$ space and ${\mathcal{W}}$ space separate the validation set well. Even though the performances of ${\mathcal{Z}}$ boundaries on the entire set are not satisfying, it may be caused by the inaccurate attribute prediction on the ambiguous samples. (ii) ${\mathcal{W}}$ space shows much stronger separability than ${\mathcal{Z}}$ space. That is because ${{\rm\bf w}}\in{\mathcal{W}}$, instead of ${{\rm\bf z}}\in{\mathcal{Z}}$, is the code finally fed into the generator. Accordingly, the generator tends to learn various semantics based on ${\mathcal{W}}$ space. (iii) The accuracy of StyleGAN ${\mathcal{W}}$ space is higher than the PGGAN ${\mathcal{Z}}$ space, both of which are the immediate input space of the generator. The reason is that the semantic attributes may not be normally distributed in real data. Compared to ${\mathcal{Z}}$ space, which is subject to normal distribution, ${\mathcal{W}}$ space has no constraints and hence is able to better fit the underlying real distribution.
Semantics in Latent Space for Face Manipulation {#subsec:latent-space-manipulation}
-----------------------------------------------
In this part, we verify whether the semantics found by InterFaceGAN are manipulable.
### PGGAN
**Manipulating Single Attribute.** Fig.\[fig:pggan-manipulation\] plots the manipulation results on five different attributes. It suggests that our manipulation approach performs well on all attributes in both positive and negative directions. Particularly on *pose* attribute, we observe that even the boundary is searched by solving a bi-classification problem, moving the latent code can produce continuous changing. Furthermore, although there lacks enough data with extreme poses in the training set, GAN is capable of imagining how profile faces should look like. The same situation also happens on eyeglasses attribute. We can manually create a lot of faces wearing eyeglasses despite the inadequate data in the training set. These two observations provide strong evidence that GAN does not produce images randomly, but learns some interpretable semantics in the latent space.
**Distance Effect of Semantic Subspace.** When manipulating the latent code, we observe an interesting distance effect, which is that the samples will suffer from severe changes in appearance if being moved too far from the boundary, and finally tend to become the extreme cases shown in Fig.\[fig:pggan-separation\]. Fig.\[fig:pggan-limitation\] illustrates this phenomenon by taking gender editing as an instance. Near-boundary manipulation works well. When samples go beyond a certain region[^3], however, the editing results are no longer like the original face anymore. But this effect does not affect our understanding of the disentangled semantics in the latent space. That is because such extreme samples are very unlikely to be directly drawn from a standard normal distribution, which is pointed out in *Property 2* in Sec.\[subsec:semantics-interpretation\]. Instead, they are constructed manually by keeping moving a normally sampled latent code along a certain direction.
**Artifacts Correction.** We further apply our approach to fix the artifacts that sometimes occurring in the synthesis. We manually labeled $4K$ bad synthesis and then trained a linear SVM to find the separation hyperplane, same as other attributes. We surprisingly find that GAN also encodes such quality information in the latent space. Based on this discovery, we manage to correct some mistakes GAN has made in the generation process by moving the latent code towards the positive “quality” direction, as shown in Fig.\[fig:pggan-correction\].
### StyleGAN
We further apply InterFaceGAN to StyleGAN model by manipulating the latent codes in both ${\mathcal{Z}}$ space and ${\mathcal{W}}$ space. We have the following observations from Fig.\[fig:stylegan-manipulation\]. (i) Besides the conventional generator (*e.g.*, PGGAN), InterFaceGAN all works well on the style-based generator. We can successfully edit the attributes by moving the latent code along the corresponding directions in either ${\mathcal{Z}}$ space or ${\mathcal{W}}$ space. (ii) By learning from a more diverse dataset, FF-HQ [@stylegan], StyleGAN learns various semantics more thoroughly. For example, StyleGAN can even generate children when making people younger (third example). This is beyond the ability of PGGAN, which is trained on CelebA-HQ [@pggan]. Also, StyleGAN is capable of producing faces with extreme poses. (iii) ${\mathcal{W}}$ space shows better performance than ${\mathcal{Z}}$ space, especially for long-distance manipulation. In other words, when the latent code locates near the separation boundary (between the two dashed lines), manipulations in ${\mathcal{Z}}$ space and ${\mathcal{W}}$ space have similar effect. However, when the latent code goes further from the boundary, manipulating one attribute in ${\mathcal{Z}}$ space might affect another. Taking gender editing (fourth example) as an instance, the person in the red box takes off his eyeglasses when moving along the gender direction. By contrast, ${\mathcal{W}}$ space shows stronger robustness. (iv) Some attributes are correlated to each other. For example, people are wearing eyeglasses when turning old (third example), and people are tending to become happier when being feminized (fourth example). More detailed analysis about this phenomenon will be discussed in Sec.\[subsec:conditional-manipulation\].
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Disentanglement Analysis and Conditional Manipulation {#subsec:conditional-manipulation}
-----------------------------------------------------
In this section, we study the disentanglement between different semantics encoded in the latent representation and evaluate the conditional manipulation approach.
### Disentanglement Measurement
Different from Karras *et al.* [@stylegan] which introduced perceptual path length and linear separability to measure the disentanglement property of the latent space, we focus more on the relationship between different attributes and study how they are entangled with each other. In other words, besides the problem that how one semantic can be well encoded in the latent space, we also explore the correlations between multiple semantics and try to decouple them. In particular, we use following metrics for analysis.
1. *Attribute correlation of real data*. We use the prepared predictors to predict the attribute scores of real data on which the GAN model is trained. Then we compute the correlation coefficient $\rho$ between any two attributes with $\rho_{A_1,A_2}=\frac{Cov(A_1,A_2)}{\sigma_{A_1}\sigma_{A_2}}$. Here, $A_1$ and $A_2$ represent two random variables with respect to these two attributes. $Cov(\cdot,\cdot)$ and $\sigma$ denote covariance and standard deviation respectively.
2. *Attribute correlation of synthesized data*. We also compute the attribute correlation score on the $500K$ synthesized data. By comparing this score to that of the real data, we can get some clues on how GANs learn to encode such semantic knowledge in the latent representation.
3. *Semantic boundary correlation*. Given any two semantics, with the latent boundaries ${{\rm\bf n}}_1$ and ${{\rm\bf n}}_2$, we compute the cosine similarity between these two directions with $\cos({{\rm\bf n}}_1, {{\rm\bf n}}_2)={{\rm\bf n}}_1^T {{\rm\bf n}}_2$. Here, ${{\rm\bf n}}_1$ and ${{\rm\bf n}}_2$ are both unit vectors. This metric is used to evaluate how the above attribute correlation is reflected in our InterFaceGAN framework.
![ **Conditional manipulation results with more than one conditions** using PGGAN [@pggan]. Left: Original synthesis. Middle: Manipulations along single boundary. Right: Conditional manipulations. Green arrows indicate the primal direction and red arrows represent the directions to condition on. []{data-label="fig:pggan-multiple-conditions"}](figures/pggan_multiple_conditions.pdf){width="1.0\linewidth"}
\
{width="1.0\linewidth"}
### PGGAN
**Disentanglement Analysis.** Tab.\[tab:pggan-disentanglement\] reports the correlation metrics of PGGAN model trained on CelebA-HQ dataset [@pggan]. By comparing the attribution correlation of real data (Tab.\[tab:pggan-disentanglement\](a)) and that of synthesized data (Tab.\[tab:pggan-disentanglement\](b)), we can tell that they are very close to each other. For example, “pose” and “smile” are almost independent to other attributes, while “gender”, “age”, and “eyeglasses” are highly correlated with each other from both real data and synthesized data, which means old men are more likely to wear eyeglasses. This implies that GANs actually learn the underlying semantic distribution from the real observation when trained to synthesize images. Then, by comparing such attribution correlation with the boundary correlation (Tab.\[tab:pggan-disentanglement\](c)), we also find that they behave similarly. This suggests that InterFaceGAN is able to not only accurately identify the semantics encoded in that latent representation but also capture the entanglement among them.
**Conditional Manipulation.** As shown in Tab.\[tab:pggan-disentanglement\](c), if two boundary are not orthogonal to each other, modulating the latent code along one direction will definitely affect the other. Hence, we propose conditional manipulation via subspace projection to eliminate this entanglement as much as possible. Details are described in Sec.\[subsec:semantics-manipulation\]. Fig.\[fig:pggan-condition\] shows the discrepancies between unconditional manipulation and conditional manipulation. Taking the top-left example in Fig.\[fig:pggan-condition\] as an instance, the results tend to become male when being edited to get old (top row). We fix this problem by subtracting its projection onto the gender direction from the age direction, resulting in a new direction. By moving latent codes along this projected direction, we can make sure the gender component is barely affected in the editing process (bottom row). Fig.\[fig:pggan-multiple-conditions\] shows a more complex case where we perform manipulation with multiple constraints. Taking “eyeglasses” attribute as an example, in the beginning, adding eyeglasses is entangled with changing both age and gender. But we manage to disentangle eyeglasses from age and gender by manually forcing the eyeglasses direction to be orthogonal to the other two. These two experiments demonstrate that our proposed conditional approach helps to achieve independent and precise attribute control.
### StyleGAN
**Disentanglement Analysis.** We conduct similar analysis on the StyleGAN model trained in FF-HQ dataset [@stylegan]. As mentioned above, StyleGAN introduces a disentangled latent space ${\mathcal{W}}$ beyond the original latent space ${\mathcal{Z}}$. Hence, we analyze the boundary correlation from both of these two spaces. Results are shown in Tab.\[tab:stylegan-disentanglement\]. Besides the conclusions from PGGAN, we have three more observations. (i) “Smile” and “gender” are not correlated in CelebA-HQ (Tab.\[tab:pggan-disentanglement\](a)), but entangled in FF-HQ (Tab.\[tab:stylegan-disentanglement\](a)). This phenomenon is also reflected in the synthesized data (Tab.\[tab:stylegan-disentanglement\](b)). (ii) ${\mathcal{W}}$ space (Tab.\[tab:stylegan-disentanglement\](d)) is indeed more disentangled than ${\mathcal{Z}}$ space (Tab.\[tab:stylegan-disentanglement\](c)), as pointed out by Karras *et al.* [@stylegan]. In ${\mathcal{W}}$ space, almost all attributes are orthogonal to each other. (iii) The boundary correlation from ${\mathcal{W}}$ space no longer aligns with the semantic distribution from real data (Tab.\[tab:stylegan-disentanglement\](a)). In other words, ${\mathcal{W}}$ space may “over-disentangle” these semantics and encode some entanglement as a new “style”. For example, in the training data, men are more like to wear eyeglasses than women. ${\mathcal{W}}$ space may capture such information and learn “man with eyeglasses” as a coupled style.
**Conditional Manipulation.** We also evaluate the proposed conditional manipulation on the style-based generator to verify its generalization ability. In particular, given a sample, we manipulate its attribute from both ${\mathcal{Z}}$ space and ${\mathcal{W}}$ space, and then perform conditional manipulation in ${\mathcal{Z}}$ space. Note that such conditional operation is *not* applicable to ${\mathcal{W}}$ space. As shown in Tab.\[tab:stylegan-disentanglement\](d), all boundaries are already orthogonal to each other. Projection barely changes the primal direction, which we call “over-disentanglement” problem. Fig.\[fig:stylegan-condition\] gives an example about the entanglement between “age” and “eyeglasses”. In Fig.\[fig:stylegan-condition\], manipulating from ${\mathcal{Z}}$ space and ${\mathcal{W}}$ space produces similar results when the latent code still locates near to the boundary. For long-distance manipulation, ${\mathcal{W}}$ space (first row) shows superiority over ${\mathcal{Z}}$ space, *e.g.*, hair length and face shape do not change in the first row. Even so, “age” and “eyeglasses” are still entangled with each other in both spaces. However, we can use the proposed conditional manipulation to decorrelate “eyeglasses” from “age” in ${\mathcal{Z}}$ space (third row), resulting in more appealing results.
Quantitative Analysis on Manipulation {#sec:quantitative-analysis}
=====================================
We show plenty of qualitative results in Sec.\[subsec:latent-space-manipulation\] and Sec.\[subsec:conditional-manipulation\] to evaluate the controllable disentangled semantics identified by InterFaceGAN. In this part, we would like to quantitatively analyze the properties of the disentangled semantics and the editing process, including (i) whether the manipulation can indeed increase or decrease the attribute score, and how manipulating one attribute affects the scores of other attributes (Sec.\[subsec:rescoring-analysis\]); (ii) how GANs learn the face representation layer by layer (Sec.\[subsec:layerwise-analysis\]); and (iii) how the attribute manipulation affects the face identity (Sec.\[subsec:identity-analysis\]).
Evaluating Editing Performance with Re-scoring {#subsec:rescoring-analysis}
----------------------------------------------
Re-scoring means to re-predict the attribute scores from the faces after manipulation. Then, we compute the score change by comparing with the scores before manipulation. This is used to verify whether the manipulation happens as what we want. For example, when we move the latent code towards “male” direction (*i.e.*, the positive direction of “gender” boundary), we would expect the “gender” score to increase. This metric can also be used to evaluate the disentanglement between different semantics. For example, if we want to see how “gender” and “age” correlate with each other, we can move the latent code along the “gender” boundary and see how the “age” score varies.
We use $2K$ synthesis for re-scoring analysis on PGGAN [@pggan], StyleGAN [@stylegan] ${\mathcal{Z}}$ space, and StyleGAN ${\mathcal{W}}$ space, whose results are reported in Tab.\[tab:rescoring\] We have three major conclusions. (i) InterFaceGAN is able to convincingly increase the target semantic scores by manipulating the appropriate attributes (see diagonal entries). (ii) Manipulating one attribute may affect the scores of other attributes. Taking PGGAN (Tab.\[tab:rescoring\](a)) as an example, when manipulating “age”, “gender” score also increases. This is consistent with the observation from Sec.\[subsec:conditional-manipulation\]. Actually, we can treat this as an another disentanglement measurement. Under this metric, we also see that ${\mathcal{W}}$ space (Tab.\[tab:rescoring\](c)) is more disentangled than ${\mathcal{Z}}$ space (Tab.\[tab:rescoring\](b)) in StyleGAN. (iii) This new metric is asymmetry. Taking PGGAN (Tab.\[tab:rescoring\](a)) as an example, when we manipulate “age”, “eyeglasses” is barely affected. But when we manipulate “eyeglasses”, “age” score increase a lot. Same phenomenon also happens to “gender” and “eyeglasses”. This provides us more adequate information about the entanglement between the semantics learned in the latent representation.
[cc\*[5]{}[C[24pt]{}]{}]{} & & & & & &\
Pose & 0.51 & 0.42 & 0.20 & 0.03 & 0.01 & 0.00\
Smile & 0.50 & 0.02 & 0.32 & 0.24 & 0.08 & 0.01\
Age & 0.54 & 0.09 & 0.20 & 0.23 & 0.19 & 0.04\
Gender & 0.45 & 0.05 & 0.44 & 0.10 & 0.02 & 0.00\
Glasses & 0.41 & 0.23 & 0.28 & 0.01 & 0.00 & 0.00\
{width="1.0\linewidth"}
[c|\*[1]{}[C[50pt]{}]{}|\*[1]{}[C[50pt]{}]{}|\*[6]{}[C[30pt]{}]{}]{} & & &\
& & & & & & & &\
Pose & 0.48 & 0.41 & 0.46 & 0.39 & 0.28 & 0.07 & 0.03 & 0.01\
Smile & 0.24 & 0.31 & 0.21 & 0.04 & 0.20 & 0.10 & 0.04 & 0.01\
Age & 0.53 & 0.47 & 0.28 & 0.12 & 0.18 & 0.09 & 0.06 & 0.01\
Gender & 0.61 & 0.51 & 0.40 & 0.11 & 0.37 & 0.13 & 0.03 & 0.01\
Glasses & 0.55 & 0.49 & 0.37 & 0.21 & 0.29 & 0.09 & 0.06 & 0.01\
Per-Layer Representation Learned by GANs {#subsec:layerwise-analysis}
----------------------------------------
Different from the traditional generator, the style-based generator in StyleGAN [@stylegan] feeds the latent code to all convolutional layers. This enables us to study the per-layer representation. Given a particular boundary, we can use it to only modulate the latent codes that are fed into a subset of layers. In practice, we manually divide the 18 layers into 5 groups, *i.e.*, 00-01, 02-03, 04-05, 06-07, and 08-17. Then we conduct the same experiment as in re-scoring analysis.
Tab.\[tab:layerwise\] and Fig.\[fig:layerwise\] show the quantitative and qualitative results respectively. From Tab.\[tab:layerwise\], we can see that “pose” is mostly controlled at layer 00-01, “smile” is controlled at layer 02-05, “age” is controlled at layer 02-07, “gender” is controlled at layer 02-03, and “eyeglasses” is controlled at layer 00-03. All attributes are barely affected by editing layer 08-17. Visualization results in Fig.\[fig:layerwise\] also gives the same conclusion. It implies that GANs actually learn different representation at different layers. This provide us some insights into a better understanding of the learning mechanism of GANs.
{width="1.0\linewidth"}
{width="1.0\linewidth"}
Effect of Learned Semantics on Face Identity {#subsec:identity-analysis}
--------------------------------------------
Identity is very important for face analysis. Accordingly, we perform identity analysis to see how the identity information varies during the manipulation process of InterFaceGAN. Similar to the re-scoring analysis, we employ a face recognition engine to extract the identity features from the faces before and after semantic editing. Cosine distance is used as the metric to evaluate the discrepancy.
Tab.\[tab:identity\] shows the results corresponding to different latent spaces from different models, from which we have following observations. (i) “Gender” affects the identity most and “smile” affects the identity least. Actually, this can be used to verify how sensitive the face identity is to a particular attribute. For example, “pose” and “eyeglasses” seem to also affect the identity a lot. This makes sense since large pose is always the obstacle in face recognition task and eyeglasses are commonly used to disguise identity in real world. We may use InterFaceGAN to synthesize more hard samples to in turn improve the face recognition model. (ii) StyleGAN ${\mathcal{W}}$ space best preserves the identity information due to its disentanglement property. That is because identity is much more complex that other semantics. A more disentangled representation is helpful in identity control. (iii) As for the layer-wise results, we can get similar conclusion to the layer-wise analysis in Sec.\[subsec:layerwise-analysis\].
{width="1.0\linewidth"}
{width="1.0\linewidth"}
Real Image Manipulation {#sec:real-image-manipulation}
=======================
In this section, we apply the semantics implicitly learned by GANs to real face editing. Since most of the GAN models lack the inference function to deal with real images, we try two different approaches. One is based on GAN inversion, which inverts any given image to the latent code so that we can manipulate. The other uses InterFaceGAN to generate synthetic image pairs and trains additional feed-forward pixel-to-pixel models.
Combining GAN Inversion with InterFaceGAN {#subsec:inversion-based}
-----------------------------------------
Recall that InterFaceGAN achieves semantic face editing by moving the latent code along a certain direction in the latent space. Accordingly, for real image editing, one straightforward way is to invert the target face back to a latent code. It turns out to be a non-trivial task because GANs do not fully capture all the modes as well as the diversity of the true distribution, which means it is extremely hard to perfectly recover any real image with a finitely dimensional latent code. To invert a pre-trained GAN model, there are two typical approaches. One is optimization-based approach, which directly optimizes the latent code with the fixed generator to minimize the pixel-wise reconstruction error [@invertibility; @zhu2020indomain]. The other is encoder-based, where an extra encoder network is trained to learn the inverse mapping [@zhu2016generative]. We tested the two baseline approaches on PGGAN [@pggan] and StyleGAN [@stylegan].
Results are shown in Fig.\[fig:real-image-manipulation-inversion\]. We can tell that both optimization-based (first row) and encoder-based (second row) methods show poor performance when inverting PGGAN. This can be imputed to the strong discrepancy between training and testing data distributions. For example, the model tends to generate Western people even the input is an Easterner (see the right example in Fig.\[fig:real-image-manipulation-inversion\]). However, even unlike the inputs, the inverted images can still be semantically edited with InterFaceGAN. Compared to PGGAN, the results on StyleGAN (third row) are much better. Here, we treat the layer-wise styles (*i.e.*, ${{\rm\bf w}}$ for all layers) as the optimization target following prior work [@image2stylegan; @image2stylegan++; @zhu2020indomain]. Such over-parameterization significantly improves the inversion quality, and hence leads to better manipulation results. In this way, we are able to *turn conditional GANs, like StyleGAN, to controllable GANs*.
We also test InterFaceGAN on LIA [@lia], which is a generative model with encoder-decoder structure. It trains an encoder together with the generator and therefore has the inference ability. The manipulation result is shown in Fig.\[fig:real-image-manipulation-encoder\] where we successfully edit the input faces with various attributes, like age and face pose. It suggests that the latent code in the encoder-decoder based generative models also supports semantic manipulation, demonstrating the generalization ability of InterFaceGAN. More results on StyleGAN inversion and LIA are shown in Fig.\[fig:real-image-manipulation\]. We can tell that the optimization-based method better recovers the input images and hence better preserves the identity information. But for both methods, the interpretable semantics inside the latent representation are capable of faithfully editing the corresponding facial attributes of the reconstructed face.
Training with Paired Synthetic Data Collected from InterFaceGAN {#subsec:learning-based}
---------------------------------------------------------------
Another way to apply InterFaceGAN to real image editing is to train additional models. Different from existing face manipulation models [@faceidgan; @lample2017fader; @stargan] that are trained on real dataset, we use InterFaceGAN to build synthetic dataset for training. There are two advantages: (i) With recent advancement, GANs are already able to produce high-quality image [@stylegan; @stylegan2], significantly narrowing the domain gap. (ii) With the strong manipulation capability of InterFaceGAN, we can easily create unlimited paired data, which is hard to collect in the real world. Taking eyeglasses editing as an example, we can sample numerous latent codes, move them along the “eyeglasses” direction, and re-score them to select the ones with highest score change. With paired data as input and supervision, we train pix2pixHD [@pix2pixhd] to achieve face editing. However, pix2pixHD has its own shortcoming which is that we cannot manipulate the attribute gradually [@viazovetskyi2020stylegan2]. To solve this problem, we introduce DNI [@wang2019deep] by first training an identical mapping network and then fine-tuning it for a particular attribute. We summarize the training pipeline as:
1. Prepare $10K$ synthetic pairs for each attribute.
2. Learn an identical pix2pixHD model.
3. Fine-tune the model to transfer a certain attribute.
4. Interpolate the model weights for gradual editing.
We choose “gender”, “eyeglasses”, and “smile” as the target attributes. Fig.\[fig:real-image-manipulation-learn\] shows the editing results. We can conclude that: (i) The pix2pixHD models trained on synthetic dataset can successfully manipulate the input face with respect to the target attribute. This suggests that the data generated by InterFaceGAN can well support model training, which may lead to more applications. (ii) For “gender” attribute, we only use female as the input and use male as the supervision. However, after the model training, we can even use this model to add mustache onto male faces as shown in Fig.\[fig:real-image-manipulation-learn\] (bottom two examples). (iii) Following DNI [@wang2019deep], we interpolate the weight of the identical model and the weight of the fine-tuned model. By doing so, we can gradually manipulate the attributes of the input face, same as the inversion-based method described in Sec.\[subsec:inversion-based\]. The main advantage of learning additional feed-forward model is its fast inference speed. (iv) During the weight interpolation process, we find that “smile” attribute does not perform as well as “gender” and “eyeglasses”. That is because smiling is not a simple pixel-to-pixel translation task but requires the reasonable movement of lips. This is also the reason why pix2pixHD can not be applied to learning pose rotation, which requires larger movement. Accordingly, the major limitation of this kind of approach is that it can only transfer some easy-to-map semantics, such as “eyeglasses” and “age”.
Discussion and Conclusion {#sec:conclusion}
=========================
Interpreting the representation learned by GANs is vital for understanding the internal mechanism of the synthesis process, which yet remains less explored. In this work we provide some pilot results in this direction by taking face synthesis as an example. There are many future works to be done. As our visual world is far more complex than faces, looking into the generative models trained to synthesize other generic objects and scenes would be one of them. For example, for scene generation, besides learning the semantics for the entire image, the model should also learn to synthesize any individual object inside the scene as well as create a layout for different objects. From this point of view, we need a more general method to interpret other GAN models beyond faces. Even for face models, there are also some directions worth further exploring. On one hand, as we have already discussed in Sec.\[subsec:latent-space-manipulation\], our method may fail for long-distance manipulation. This is restricted by the linear assumption. New adaptive method of changing the semantic boundary based on the latent code to manipulate would solve this problem. On the other hand, we use off-the-shelf classifiers as predictors to interpret the latent representation. This limits the semantics we can find since sometimes we may not have the proper classifiers or the attribute is not well defined or annotated. Hence, how to identify the semantics emerging from synthesizing images in an unsupervised learning manner would be future work as well.
To conclude, in this work we interpret the disentangled face representation learned by GANs and conduct a thorough study on the emerging facial semantics. By leveraging the semantic knowledge encoded in the latent space, we are able to realistically edit the attributes in face images. Conditional manipulation technique is further introduced to decorrelate different semantics for more precise control of facial attributes. Extensive experiments suggest that InterFaceGAN can also be applied to real image manipulation.
Acknowledgments {#acknowledgments .unnumbered}
===============
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is supported in part by the Early Career Scheme (ECS) through the Research Grants Council of Hong Kong under Grant No.24206219, in part by RSFS grant from CUHK Faculty of Engineering, and in part by SenseTime Collaborative Grant.
In this part, we provide detailed proof of *Property 2* in the main paper. Recall this property as follows.
***Property 2** Given ${{\rm\bf n}}\in{\mathbb{R}}^d$ with ${{\rm\bf n}}^T{{\rm\bf n}}=1$, which defines a hyperplane, and a multivariate random variable ${{\rm\bf z}}\sim{\mathcal{N}}({{\rm\bf 0}},{{\rm\bf I}}_d)$, we have ${{\rm P}}(|{{\rm\bf n}}^T{{\rm\bf z}}|\leq2\alpha~\sqrt{\frac{d}{d-2}})\geq(1-3e^{-c d})(1-\frac{2}{\alpha}e^{-\alpha^2/2})$ for any $\alpha\geq1$ and $d\geq4$. Here ${{\rm P}}(\cdot)$ stands for probability and $c$ is a fixed positive constant.*
*Proof.*
Without loss of generality, we fix ${{\rm\bf n}}$ to be the first coordinate vector. Accordingly, it suffices to prove that ${{\rm P}}(|z_1|\leq2\alpha~\sqrt{\frac{d}{d-2}})\geq(1-3e^{-c d})(1-\frac{2}{\alpha}e^{-\alpha^2/2})$, where $z_1$ denotes the first entry of ${{\rm\bf z}}$.
{width="0.80\linewidth"}
\[appendix:fig:property-2\]
As shown in Fig.\[appendix:fig:property-2\], let $H$ denote the set $$\begin{aligned}
\{{{\rm\bf z}}\sim{{\rm\bf N}}({{\rm\bf 0}},{{\rm\bf I}}_d):||{{\rm\bf z}}||_2\leq2\sqrt{d}, |z_1|\leq2\alpha\sqrt{\frac{d}{d-2}}\}, \nonumber\end{aligned}$$ where $||\cdot||_2$ stands for the $l_2$ norm. Obviously, we have ${{\rm P}}(H)\leq{{\rm P}}(|z_1|\leq2\alpha\sqrt{\frac{d}{d-2}})$. Now, we will show ${{\rm P}}(H)\geq(1-3e^{-c d})(1-\frac{2}{\alpha}e^{-\alpha^2/2})$
Considering the random variable $R=||{{\rm\bf z}}||_2$, with cumulative distribution function $F(R\leq r)$ and density function $f(r)$, we have $$\begin{aligned}
{{\rm P}}(H) &= {{\rm P}}(|z_1|\leq2\alpha\sqrt{\frac{d}{d-2}}|R\leq2\sqrt{d}){{\rm P}}(R\leq2\sqrt{d}) \nonumber \\
&= \int_0^{2\sqrt{d}}{{\rm P}}(|z_1|\leq2\alpha\sqrt{\frac{d}{d-2}}|R=r)f(r)dr. \nonumber\end{aligned}$$
According to *Theorem 1* below, when $r\leq2\sqrt{d}$, we have $$\begin{aligned}
{{\rm P}}(H) &= \int_0^{2\sqrt{d}}{{\rm P}}(|z_1|\leq2\alpha\sqrt{\frac{d}{d-2}}|R=r)f(r)dr \nonumber \\
&= \int_0^{2\sqrt{d}}{{\rm P}}(|z_1|\leq\frac{2\sqrt{d}}{r}\frac{\alpha}{\sqrt{d-2}}|R=1)f(r)dr \nonumber \\
&\geq \int_0^{2\sqrt{d}}{{\rm P}}(|z_1|\leq\frac{\alpha}{\sqrt{d-2}}|R=1)f(r)dr \nonumber \\
&\geq \int_0^{2\sqrt{d}}(1-\frac{2}{\alpha}e^{-\alpha^2/2})f(r)dr \nonumber \\
&= (1-\frac{2}{\alpha}e^{-\alpha^2/2})\int_0^{2\sqrt{d}}f(r)dr \nonumber \\
&= (1-\frac{2}{\alpha}e^{-\alpha^2/2}){{\rm P}}(0\leq R\leq2\sqrt{d}). \nonumber\end{aligned}$$
Then, according to *Theorem 2* below, by setting $\beta=\sqrt{d}$, we have $$\begin{aligned}
{{\rm P}}(H) &= (1-\frac{2}{\alpha}e^{-\alpha^2/2}){{\rm P}}(0\leq R\leq2\sqrt{d}) \nonumber \\
&\geq (1-\frac{2}{\alpha}e^{-\alpha^2/2})(1-3e^{-c d}). \nonumber\end{aligned}$$
Q.E.D.
![ Diagram for ***Theorem 1***. []{data-label="appendix:fig:theorem-1"}](figures/theory_theorem1.pdf){width="0.70\linewidth"}
***Theorem 1** Given a unit spherical $\{{{\rm\bf z}}\in{\mathbb{R}}^d:||{{\rm\bf z}}||_2=1\}$, we have ${{\rm P}}(|z_1|\leq\frac{\alpha}{\sqrt{d-2}})\geq1-\frac{2}{\alpha}e^{-\alpha^2/2}$ for any $\alpha\geq1$ and $d\geq4$.*
*Proof.*
By symmetry, we just prove the case where $z_1\geq0$. Also, we only consider about the case where $\frac{\alpha}{\sqrt{d-2}} \leq 1$.
Let $U$ denote the set $\{{{\rm\bf z}}\in{\mathbb{R}}^d:||{{\rm\bf z}}||_2=1,z_1\geq\frac{\alpha}{\sqrt{d-2}}\}$, and $K$ denote the set $\{{{\rm\bf z}}\in{\mathbb{R}}^d:||{{\rm\bf z}}||_2=1,z_1\geq0\}$. It suffices to prove that the surface of $U$ area and the surface of $K$ area in Fig.\[appendix:fig:theorem-1\] satisfy $$\begin{aligned}
\frac{surf(U)}{surf(K)}\leq\frac{2}{\alpha}e^{-\alpha^2/2}, \nonumber\end{aligned}$$ where $surf(\cdot)$ stands for the surface area of a high dimensional geometry. Let $A(d)$ denote the surface area of a $d$-dimensional unit-radius ball. Then, we have $$\begin{aligned}
surf(U) &=\int_{\frac{\alpha}{\sqrt{d-2}}}^{1} (1-z_1^2)^{\frac{d-2}{2}}A(d-1)dz_1 \nonumber \\
&\leq \int_{\frac{\alpha}{\sqrt{d-2}}}^{1} e^{-\frac{d-2}{2}z_1^2}A(d-1)dz_1 \nonumber \\
&\leq \int_{\frac{\alpha}{\sqrt{d-2}}}^{1} \frac{z_1\sqrt{d-2}}{\alpha}e^{-\frac{d-2}{2}z_1^2}A(d-1)dz_1 \nonumber \\
&\leq \int_{\frac{\alpha}{\sqrt{d-2}}}^{\infty} \frac{z_1\sqrt{d-2}}{\alpha}e^{-\frac{d-2}{2}z_1^2}A(d-1)dz_1 \nonumber \\
&= \frac{A(d-1)}{\alpha\sqrt{d-2}}e^{-\alpha^2/2}. \nonumber\end{aligned}$$
Similarly, we have $$\begin{aligned}
surf(K) &=\int_{0}^{1} (1-z_1^2)^{\frac{d-2}{2}}A(d-1)dz_1 \nonumber \\
&\geq \int_{0}^{\frac{1}{\sqrt{d-2}}} (1-z_1^2)^{\frac{d-2}{2}}A(d-1)dz_1 \nonumber \\
&\geq \frac{1}{\sqrt{d-2}} (1-\frac{1}{d-2})^{\frac{d-2}{2}}A(d-1). \nonumber\end{aligned}$$
Considering the fact that $(1-x)^a\geq1-ax$ for any $a\geq1$ and $0\leq x\leq1$, we have $$\begin{aligned}
surf(K) &\geq \frac{1}{\sqrt{d-2}} (1-\frac{1}{d-2})^{\frac{d-2}{2}}A(d-1) \nonumber \\
&\geq \frac{1}{\sqrt{d-2}} (1-\frac{1}{d-2}\frac{d-2}{2})A(d-1) \nonumber \\
&= \frac{A(d-1)}{2\sqrt{d-2}}. \nonumber\end{aligned}$$
Accordingly, $$\begin{aligned}
\frac{surf(U)}{surf(K)} \leq \frac{\frac{A(d-1)}{\alpha\sqrt{d-2}}e^{-\alpha^2/2}}{\frac{A(d-1)}{2\sqrt{d-2}}}
= \frac{2}{\alpha}e^{-\alpha^2/2}. \nonumber\end{aligned}$$
Q.E.D.
***Theorem 2 (Gaussian Annulus Theorem [[@blum2020foundations]]{})** For a $d$-dimensional spherical Gaussian with unit variance in each direction, for any $\beta\leq\sqrt{d}$, all but at most $3e^{-c\beta^2}$ of the probability mass lies within the annulus $\sqrt{d}-\beta\leq||{{\rm\bf z}}||_2\leq\sqrt{d}+\beta$, where $c$ is a fixed positive constant.*
That is to say, given ${{\rm\bf z}}\sim{{\rm\bf N}}({{\rm\bf 0}}, {{\rm\bf I}}_d)$, $\beta\leq\sqrt{d}$, and a constant $c>0$, we have $$\begin{aligned}
{{\rm P}}(\sqrt{d}-\beta\leq||{{\rm\bf z}}||_2\leq\sqrt{d}+\beta)\geq(1-3e^{-c\beta^2}). \nonumber\end{aligned}$$
[^1]: This assumption is empirically demonstrated in Sec.\[sec:interpretation\].
[^2]: When $d=512$, we have $P(|{{\rm\bf n}}^T{{\rm\bf z}}|>5.0)<1e^{-6}$. It suggests that almost all sampled latent codes are expected to locate within 5 unit-length to the boundary. Proof can be found in **Appendix**.
[^3]: We choose 5.0 as the threshold.
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abstract: 'Maintaining literature databases and online bibliographies is a core responsibility of metadata aggregators such as digital libraries. In the process of monitoring all the available data sources the question arises which data source should be prioritized. Based on a broad definition of information quality we are looking for different ways to find the best fitting and most promising conference candidates to harvest next. We evaluate different conference ranking features by using a pseudo-relevance assessment and a component-based evaluation of our approach.'
bibliography:
- 'main.bib'
title: Prioritizing and Scheduling Conferences for Metadata Harvesting in dblp
---
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abstract: 'Recently several authors have developed multilinear and in particular quadratic extensions of the classical Morawetz inequality. Those extensions provide (among other results) an easy proof of asymptotic completeness in the energy space for nonlinear Schrödinger equations in arbitrary space dimension and for Hartree equations in space dimension greater than two in the noncritical cases. We give a pedagogical review of the latter results.'
address:
- 'Laboratoire de Physique Théorique, Université de Paris XI, Bâtiment 210, F-91405 ORSAY Cedex, France'
- 'Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Italy'
author:
- Jean Ginibre
- Giorgio Velo
title: Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations
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Introduction
============
This paper is devoted to the exposition of an elementary subset of some recent results bearing on scattering theory for the nonlinear Schrödinger (NLS) equation $$i \partial_t u = - (1/2) \Delta u + g(\rho ) u \eqno(1.1)$$
in $n$ space dimensions, where $u$ is a complex valued function defined in space time ${I\hskip-1truemm R}^{n+1}$, $\rho = |u|^2$ and $g$ is a real valued function of $\rho$, typically a sum of powers $$g(\rho ) = \lambda_1 \ \rho^{(p_1-1)/2} + \lambda_2 \ \rho^{(p_2-1)/2} \eqno(1.2)$$
with $1 < p_1 <p_2$ and $\lambda_1 , \lambda_2 \in {I\hskip-1truemm R}$. We also present a straightforward extension of those results to the Hartree equation (1.1) with $$g(\rho ) = V \star \rho \eqno(1.3)$$
where $V$ is a real even function of the space variable and $\star$ denotes the convolution in ${I\hskip-1truemm R}^n$. The first main question of scattering theory is the existence of the wave operators, namely the construction of solutions that behave asymptotically in time as solutions of the free Schrödinger equation, namely such that $$u(t) \sim U(t) u_+ \qquad \hbox{for $t \to \infty$} \eqno(1.4)$$
(and the analogue for $t \to - \infty )$, where $$U(t) = \exp (i (t/2)\Delta )\ . \eqno(1.5)$$
The second main question of scattering theory is asymptotic completeness (AC), simply called “scattering” in some of the recent literature, and consists in proving that all solutions of the relevant equation, in a suitable functional framework, behave asympotically as solutions of the free Schrödinger equation, namely satisfy (1.4) and the analogue for $t\to - \infty$. Of special interest is the case of finite energy solutions of (1.1), namely of solutions in $L^{\infty}({I\hskip-1truemm R} , H^1)$. An essential tool in the proof of AC for such solutions is the Morawetz inequality, first derived for the nonlinear Klein-Gordon (NLKG) equation [@13r] and then extended to the NLS equation [@12r]. That inequality was applied to prove AC first for the NLKG equation and then for the NLS equation in space dimension $n \geq 3$ in seminal papers by Morawetz and Strauss [@14r] and by Lin and Strauss [@12r], for slightly more regular solutions. The case of general finite energy solutions in space dimension $n \geq 3$ was treated later in [@10r] for NLS equations and in [@11r] for Hartree equations. The treatment was then improved for the NLS equation in [@15r] which covers in addition the more difficult cases $n = 1,2$, as well as the case of the NLKG equation, and for the Hartree equation in [@16r].
More recently, several groups of authors have studied the more difficult problem of extending some of the previous results, in particular the proof of existence of global solutions and the proof of AC (“scattering”), on the one hand to the case of critical interactions, and on the other hand to the case of subenergy solutions, namely of solutions of intermediate regularity between $L^2$ and $H^1$. An important tool in some of those works is a new version of the Morawetz inequality, of a multilinear and in particular bilinear or quadratic type. That inequality has appeared in various forms in the literature and seems to have now stabilized to a simple form ([@1r] [@3r]-[@9r] [@14newr] [@17r] [@19r]-[@25r] and references therein quoted). Leaving aside the difficult problems arising for critical interactions and/or for subenergy solutions, that new inequality provides a unified proof of AC for noncritical NLS in the energy space for all space dimensions, as well as for the Hartree equation for $n\geq 3$. That proof is much simpler than the previous ones. The present paper is devoted to an exposition of that new quadratic Morawetz inequality and of its application to the proof of AC for the NLS and Hartree equations in the energy space in noncritical situations. That result for noncritical NLS appears as a by product for $n = 1$ in [@4r], for $n=2$ in [@4newr] and for $n \geq 3$ in [@20r], which is mostly devoted to the critical cases. In Section 2, we first derive the quadratic Morawetz identity and we deduce therefrom the basic estimate that leads to the proof of AC. The formal proof of the identity is formulated in terms of conservation laws, in the spirit of [@17r]. In Section 3, we exploit the previous estimate to prove AC. We treat the case of the NLS equation in some detail, and we give the modifications needed for the Hartree equation. The result applies to $L^2$ supercritical and $H^1$ subcritical nonlinearities. Some peripheral results are collected in Appendices. In Appendix 1, we give an estimate which points to the usefulness of the Morawetz inequality at lower regularity levels than $H^1$, in particular at the level of $H^{1/2}$. In Appendix 2, we exploit the point of view of conservation laws to derive a quadratic identity for the NLKG equation. That identity however does not lead to estimates because of a lack of positivity. In Appendix 3, we rewrite the original Morawetz inequality for the NLS equation in a form which exhibits its relation to the quadratic identity derived in Section 2. In Appendix 4, we justify the formal computation of Section 2 by a suitable limiting procedure.
We conclude this introduction by giving some notation and estimates which will be used freely throughout this paper. For any integer $n \geq 1$, for any $r$, $1 \leq r \leq \infty$, we denote by $\parallel \cdot \parallel_r$ the norm in $L^r \equiv L^r ({I\hskip-1truemm R}^n)$, by $\overline{r}$ the conjugate exponent defined by $1/r + 1/\overline{r} = 1$, and we define $\delta (r) \equiv n/2 - n/r$. We denote by $<\cdot , \cdot >$ the scalar product in $L^2$. We shall use the Sobolev spaces $\dot{H}_r^{\sigma} \equiv \dot{H}_r^{\sigma}({I\hskip-1truemm R}^n)$ and $H_r^{\sigma} \equiv H_r^{\sigma}({I\hskip-1truemm R}^n)$ defined for $0 \leq \sigma < \infty$ and $1 < r < \infty$ by $$\dot{H}_r^{\sigma} = \left \{ u : \parallel u, \dot{H}_r^{\sigma}\parallel \ \equiv\ \parallel \omega^{\sigma} u\parallel_r < \infty \right \}$$ $$H_r^{\sigma} = \left \{ u : \parallel u, H_r^{\sigma}\parallel \ \equiv\ \parallel <\omega>^{\sigma} u\parallel_r < \infty \right \}$$
where $\omega = (-\Delta )^{1/2}$ and $<\cdot > = (1 + |\cdot |^2)^{1/2}$. The subscript $r$ will be omitted if $r=2$. For any interval $I$ of ${I\hskip-1truemm R}$, for any Banach space $X$, we denote by $\mathscr{C}(I, X)$ the space of continuous functions from $I$ to $X$ and, for $1 \leq q \leq \infty$, by $L^q (I, X)$ (resp. $L_{loc}^q(I,X)$) the space of measurable functions from $I$ to $X$ such that $\parallel u(\cdot ); X \parallel\ \in L^q(I)$ (resp. $\in L_{loc}^q(I)$).
We introduce the following definition. A pair of exponent $(q,r)$ is admissible if $0 \leq 2/q = \delta (r) = \delta$ and $\delta \leq 1/2$ for $n=1$, $\delta < 1$ for $n=2$ and $\delta \leq 1$ for $n\geq 3$. Then the well known Strichartz estimates take the form :\
[**Lemma 1.1.**]{}
*Let $U(t)$ be given by (1.5). Then*
\(1) For any admissible pair $(q,r)$ $$\parallel U(t) v; L^q ({I\hskip-1truemm R},L^r) \parallel\ \leq \ C \parallel v \parallel _2 \ . \eqno(1.6)$$
\(2) For any admissible pairs $(q_i,r_i)$, $i=1,2$, and for any interval $I$ of ${I\hskip-1truemm R}$
$$\parallel \int_{I \cap \{t':t'\leq t \}} dt' \ U(t-t') \ f(t') ; L^{q_1} (I, L^{r_1})\parallel \ \leq \ C \parallel f; L^{\overline{q}_2} (I, L^{\overline{r}_2})\parallel \eqno(1.7)$$
[*where the constant $C$ is independent of $I$*]{}.
3 truemm
Lemma 1.1 suggests to study the Cauchy problem for the equation (1.1) in spaces of the following type. Let $I$ be an interval of ${I\hskip-1truemm R}$. We define $$X_{(loc)} (I) = \left \{ u : u \in \mathscr{C}(I,L^2)\ {\rm and}\ u \in L_{loc}^q (I, L^r) \hbox{ for all admissible $(q,r)$}\right \}$$
and $$X_{(loc)}^1(I) = \left \{ u ; u, \nabla u \in X_{(loc)} (I) \right \}\ .$$
Quadratic Morawetz inequalities
===============================
In this section we derive the quadratic Morawetz identity for the NLS and Hartree equations and we deduce therefrom the basic estimates that lead to the proof of asymptotic completeness in the energy space for those equations. We begin with a formal derivation of the quadratic Morawetz identity for the NLS equation, assuming sufficient smoothness and decay at infinity of the solutions to give a meaning to the calculation and in particular to the integrations by parts. The underlying algebraic structure is a pair of related conservation laws $$\partial_t \rho + \nabla \cdot j = 0 \eqno(2.1)$$ $$\partial_t j + \nabla \cdot T = 0 \ . \eqno(2.2)$$
The first one is a scalar conservation law with scalar density $\rho$ and vector current $j$, the second one is a vector conservation law with vector density $j$ and second rank tensor current $T$, and the two laws are related by the fact that the current $j$ of the first one is at the same time the density of the second one. That situation occurs for the NLS equation and with a minor modification for the Hartree equation, as we shall review in this paper. It also occurs for any space time translation invariant system with a symmetric energy momentum tensor, with $\rho$ and $j$ being respectively the energy and momentum densities, and in particular for a class of NLKG equations, for which however it does not lead to useful estimates because of a lack of positivity (see Appendix 2 for the relevant formal calculation in that case). One could also consider the more general situation of two unrelated conservation laws, but that does not seem to be useful in the present case. Let now $h$ be a sufficiently regular real even function defined in ${I\hskip-1truemm R}^{n}$. The starting point is the auxiliary quantity (which will be mostly forgotten at the end) $$J = (1/2) <\rho , h \star \rho > \ . \eqno(2.3)$$
From (2.1) (2.2) and with two integrations by parts, it follows that $$M \equiv \partial_t J = - \ <\rho, h \star \nabla \cdot j>\ = - \ < \rho , \nabla h \star j > \ , \eqno(2.4)$$ $$\partial_t M = \partial_t^2 J = \ <\nabla \cdot j, \nabla h \star j >\ + \ <\rho, \nabla h \star \nabla \cdot T >$$ $$= - \ <j, \nabla^2 h \star j>\ + \ <\rho , \nabla^2 h \star T> \quad \ \ \eqno(2.5)$$
where $\nabla^2h$ is the second rank tensor $\nabla_k\nabla_{\ell} h$ and contractions are performed in the obvious way. The quadratic Morawetz identity is then the identity $$\partial_t M = - \partial_t \ < \rho , \nabla h \star j>\ = - \ < j, \nabla^2 h \star j>\ + \ <\rho, \nabla^2 h \star T > \ . \eqno(2.6)$$
We now consider the NLS equation $$i \partial_t u = - (1/2) \Delta u + gu \eqno(1.1)\equiv (2.7)$$
where $g = g(\rho )$ is a real function of $\rho = |u|^2$. That equation is the Euler-Lagrange equation with Lagrangian density $$L(u) = \ -\ {\rm Im}\ \overline{u} \partial_t u - (1/2) |\nabla u|^2 - G(\rho ) \eqno(2.8)$$
where $$G(\rho ) = \int_0^{\rho} d \rho ' \ g(\rho ')\ . \eqno(2.9)$$
The basic structure (2.1) (2.2) is realized with $\rho = |u|^2$ and $$j = \ {\rm Im} \ \overline{u} \nabla u \ , \eqno(2.10)$$
and (2.1) is the conservation law of the mass (or charge). The mass current $j$ turns out to be the momentum density, and (2.2) becomes the momentum conservation law. In fact the energy momentum tensor $\widetilde{T}$ is given by $$\left \{ \begin{array}{l} \widetilde{T}_{0\ell} = 2{\rm Re} \ \displaystyle{{\partial L \over \partial (\partial_t u)}} \nabla_{\ell} u = - \ {\rm Im} \ \overline{u} \nabla_{\ell} u = - j_{\ell} \\ \\ \widetilde{T}_{k\ell} = 2{\rm Re} \ \displaystyle{{\partial L \over \partial (\nabla_k u)}} \nabla_{\ell} u - \delta_{k\ell} L = -\ {\rm Re} \ \nabla_k \overline{u} \nabla_{\ell} u - \delta_{k\ell} L \end{array}\right . \eqno(2.11)$$ and (2.2) coincides (up to sign) with the conservation law $$\partial_t \ \widetilde{T}_{0\ell} + \nabla_k \ \widetilde{T}_{k\ell} = 0 \ . \eqno(2.12)$$
with $T_{k\ell} = - \widetilde{T}_{k\ell}$. For $u$ a solution of (2.7), $L(u)$ reduces to $$L(u) = - (1/4) \Delta \rho + \rho g(\rho ) - G(\rho ) \eqno(2.13)$$
so that $$T_{k\ell} = \ {\rm Re}\ \nabla_k \overline{u} \nabla_{\ell} u - \delta_{k\ell} \left ( (1/4) \Delta \rho - \rho g + G\right ) \ . \eqno(2.14)$$
The conservation law (2.2) then holds with $j$ and $T$ defined by (2.10) (2.14), namely $$\partial_t j = - \nabla \cdot {\rm Re} \nabla \overline{u} \nabla u + \nabla ((1/4) \Delta \rho - \rho g + G) \eqno(2.15)$$
which can of course be obtained by a direct computation using (2.7). Substituting (2.10) (2.14) or (2.15) into (2.6) yields $$\partial_t M = \ < \rho , \Delta h \star \left ( - (1/4) \Delta \rho + \rho g - G\right ) >\ - \ < j, \nabla^2 h \star j>$$ $$+ \ <\rho, \nabla^2 h \star \nabla \overline{u} \nabla u> \eqno(2.16)$$
where we have used the symmetry of $\nabla^2h$ to eliminate the real part condition in the last term. On the other hand $$<j, \nabla^2 h \star j>\ = \ <\overline{u} \nabla u, \nabla^2 h \star \overline{u} \nabla u>\ - \ <{\rm Re}\ \overline{u} \nabla u, \nabla^2 h \star \ {\rm Re} \ \overline{u} \nabla u>\qquad \qquad$$ $$= \ <\overline{u} \nabla u, \nabla^2 h \star \overline{u} \nabla u>\ - (1/4)\ <\nabla \rho, \nabla^2h \star \nabla \rho > \eqno(2.17)$$
so that $$\partial_t M = (1/2)\ <\nabla \rho , \Delta h \star \nabla \rho > \ + \ < \rho , \Delta h \star (\rho g - G)>\ + \ R \eqno(2.18)$$
where we have used the fact that $$-\ < \rho , \Delta h \star \Delta \rho >\ = \ <\nabla \rho, \nabla^2h \star \nabla \rho >\ = \ < \nabla \rho, \Delta h \star \nabla \rho > \eqno(2.19)$$
by integration by parts, and where $$R = \ <\overline{u}u, \nabla^2 h \star \nabla \overline{u} \nabla u >\ - \ < \overline{u} \nabla u , \nabla^2 h \star \overline{u}\nabla u >$$ $$= (1/2) \int dx \ dy \left ( \overline{u}(x) \nabla \overline{u}(y) - \overline{u}(y)\nabla \overline{u}(x)\right ) \nabla^2 h (x-y)$$ $$\left ( u(x) \nabla u (y) - u(y) \nabla u (x) \right ) \ . \eqno(2.20)$$
Integrating (2.18) over time in an interval $[t_1, t_2]$ yields $$\int_{t_1}^{t_2} dt \left \{ (1/2) \ < \nabla \rho , \Delta h \star \nabla \rho >\ + \ < \rho , \Delta h \star (\rho g - G) > + R \right \}$$ $$\left . = - \ < \rho , \nabla h \star\ {\rm Im} \ \overline{u} \nabla u > \right |_{t_1}^{t_2} \ . \eqno(2.21)$$
That identity will yield useful estimates if $\nabla h \in L^{\infty}$ and if $\nabla^2 h$ is nonnegative as a matrix. Under the latter assumption, $R$ is nonnegative, the first term in the integrand is positive, and the second term is nonnegative if $\rho g - G \geq 0$.
We now consider a representative situation where the previous formal computations can be made rigorous. We take $h(x) = |x|$, so that $$\left \{ \begin{array}{l} \nabla h = |x|^{-1} x \\ \\ \nabla^2 h = |x|^{-1} \left ( \1 - |x|^{-2} x \otimes x \right ) \ ,\ \Delta h = (n-1) |x|^{-1} \quad \hbox{for $n\geq 2$} \\ \\ \nabla^2 h = \Delta h = 2 \delta (x) \quad \hbox{for $n=1$}\ . \end{array}\right . \eqno(2.22)$$
In that case $$< \nabla \rho , \Delta h \star \nabla \rho >\ = c\ <\nabla \rho , \omega^{1-n} \nabla \rho >\ = \ c \parallel \rho ; \dot{H}^{(3 - n)/2} \parallel^2 \eqno(2.23)$$ $$< \rho , \Delta h \star (\rho g - G) >\ = c\ <\rho , \omega^{1-n} (\rho g-G) > \eqno(2.24)$$
where $\omega = (- \Delta )^{1/2}$ and $c$ is a constant depending only on $n$ [@18r].
We take for $g$ a sum of two powers $$g(\rho ) = \lambda_1\ \rho^{(p_1-1)/2} + \lambda_2 \ \rho^{(p_2-1)/2} \eqno(1.2) \equiv (2.25)$$
with $\lambda_1$, $\lambda_2 \in {I\hskip-1truemm R}$, so that $$G(\rho ) = 2 \lambda_1 \left ( p_1 + 1\right )^{-1} \ \rho^{(p_1+1)/2} + 2 \lambda_2 \left ( p_2 + 1\right )^{-1} \ \rho^{(p_2+1)/2} \eqno(2.26)$$ $$\rho g(\rho ) - G(\rho ) = \lambda_1\ {p_1 - 1 \over p_1 + 1} \ \rho^{(p_1+1)/2} + \lambda_2\ {p_2 - 1 \over p_2 + 1} \ \rho^{(p_2+1)/2} \eqno(2.27)$$
with $1 \leq p_1 < p_2$. More general $g$ can be easily accomodated. For $H^1$ subcritical powers, the Cauchy problem for NLS is well known to be locally well posed in $X_{loc}^1$ for initial data in $H^1$ and possibly globally well posed [@2r].\
[**Proposition 2.1.**]{}
*Let $h(x) = |x|$ and let $g$ be defined by (2.25) with $1 \leq p_1 < p_2$ and $p_2 < 1+4/(n-2)$ for $n \geq 3$. Let $I$ be an interval and let $u \in X_{loc}^1(I)$ be a solution of the NLS equation (2.7). Then*
\(1) The identity (2.21) holds for any $t_1$, $t_2 \in I$.
\(2) Let in addition $\lambda_1$, $\lambda_2 \geq 0$ (so that $u \in X_{loc}^1({I\hskip-1truemm R}) \cap L^{\infty} ({I\hskip-1truemm R}, H^1)$). Then $u$ satisfies the estimate $$\parallel \rho ; L^2({I\hskip-1truemm R}, \dot{H}^{(3-n)/2})\parallel^2 + \int dt\ < \rho , \omega^{1-n} (\rho g - G) > \ \leq\ C \parallel u \parallel_2^3 \ \parallel u ; L^{\infty} ({I\hskip-1truemm R} , H^1)\parallel \eqno(2.28)$$
In particular $\rho \in L^2({I\hskip-1truemm R} , \dot{H}^{(3-n)/2})$.
\
[**Sketch of proof.**]{} The proof of Part (1) consists in making the previous formal computation rigorous under the available regularity properties by introducing suitable cut offs and eliminating them by a limiting procedure. This is done in Appendix 4. At this level of regularity, one checks easily that all the terms in the identity are well defined already in the differential form (2.18). Actually by (2.23) and Sobolev inequalities $$\begin{array}{lll} < \nabla \rho , \Delta h \star \nabla \rho > &\leq \ C\parallel u \parallel_r^2\ \parallel \nabla u \parallel_2^2 &\hbox{with $n/2 - n/r = 1/2$}\\ \\
&\leq \ C \parallel u ; \dot{H}^{1/2}\parallel^2 \ \parallel \nabla u \parallel_2^2 &\hbox{for $n \geq 2$}\ . \end{array} \eqno(2.29)$$
Similarly, for $g$ a single power $p$ $$\left | < \rho , \Delta h \star (\rho g-G)> \right | \ \leq\ C\parallel u \parallel_{p+1}^{p+1} \ \parallel \omega^{1-n} |u|^2 \parallel_{\infty}$$ $$\leq \ C \parallel u \parallel_{p+1}^{p+1} \ \parallel u \parallel _{r_+} \ \parallel u \parallel_{r_-} \qquad \hbox{for $n \geq 2$} \eqno(2.30)$$
with $n/r_{\pm} = n/2 - 1/2 \pm \varepsilon$. Furthermore $R = 0$ for $n=1$, while for $n \geq 2$, $R$ is the sum of terms of the type $< \overline{u}\nabla u, \nabla^2 h \star \overline{u} \nabla u>$ which are estimated as in (2.29), and $< \overline{u} u, \nabla^2 h \star \nabla \overline{u} \nabla u>$ which are estimated by $$\left | < \overline{u} u, \nabla^2 h \star \nabla \overline{u} \nabla u>\right | \leq C <|u|^2, \omega^{1-n} | \nabla u|^2>$$ $$\leq \ C\parallel \nabla u \parallel_2^2 \ \parallel \omega^{1-n} |u|^2\parallel_{\infty}\ \leq\ C \parallel \nabla u \parallel_2^2\ \parallel u \parallel_{r_+} \ \parallel u \parallel_{r_-} \ . \eqno(2.31)$$
Finally the right hand-side of (2.21) is estimated by $$\left | < \rho , \nabla h \star {\rm Im}\ \overline{u} \nabla u > \right | \leq \ \parallel \nabla h\parallel_{\infty}\ \parallel u \parallel_2^3\ \parallel \nabla u\parallel_2 \ . \eqno(2.32)$$
Part (2) follows from (2.21) by taking the limit $t_1 \to - \infty$, $t_2 \to \infty$, from (2.23) (2.24), from the positivity of $\rho g - G$ and of $R$, and from (2.32).
$\sq$
We now sketch briefly some further developments along the previous lines. First the formal computation leading to (2.18) can easily be extended to yield a bilinear Morawetz identity for two solutions of the NLS equation. Actually the identity (2.18) can also be arrived at by applying the original Morawetz identity [@12r] to a suitable tensor product of two solutions of (2.7). Let therefore $u_i$, $i = 1,2$, be two solutions of (2.7), let $\rho_i$, $ j_i$, $T_i$ be the associated density, current and tensor $T$, and let $g_i = g(\rho_i)$, $G_i = G(\rho_i)$. We start from $$J = (1/2)\ <\rho_1 , h \star \rho_2> \eqno(2.33)$$
so that $$M \equiv \partial_t J = - (1/2) \left ( <\rho_1 , \nabla h \star j_2> + <\rho_2 , \nabla h \star j_1>\right ) \ , \eqno(2.34)$$ $$\partial_t M = \partial_t^2 J = - <j_1, \nabla^2 h \star j_2> + (1/2) \left ( <\rho_1, \nabla^2 h \star T_2> + <\rho_2, \nabla^2 h \star T_1>\right ) \ . \eqno(2.35)$$
Substituting (2.10) (2.15) into (2.35) and proceeding as before, we obtain $$\partial_t M = (1/2) \Big \{ <\nabla \rho_1 , \Delta h \star \nabla \rho_2> + <\rho_1, \Delta h \star (\rho_2 g_2 - G_2)>$$ $$+ <\rho_2, \Delta h \star (\rho_1 g_1 - G_1)>\Big \} + R \eqno(2.36)$$
where now $$R = (1/2) \left \{ <\overline{u}_1 u_1, \nabla^2 h \star \nabla \overline{u}_2 \nabla u_2> + (1 \leftrightarrow 2 )\right \} - <\overline{u}_1 \nabla u_1, \nabla^2 h \star \overline{u}_2 \nabla u_2>$$ $$= (1/2) \int dx\ dy \left ( \overline{u}_1(x) \nabla \overline{u}_2(y) - \overline{u}_2 (y) \nabla \overline{u}_1(x) \right ) \nabla^2 h (x-y)$$ $$\times \left ( u_1(x) \nabla u_2(y) - u_2 (y) \nabla u_1(x)\right ) \ .\eqno(2.37)$$
The identity (2.36) is the bilinear version of (2.18). Remarkably enough, $R$ is still nonnegative in that case if $\nabla^2h$ is a nonnegative matrix. On the other hand for a repulsive (defocusing) $g$, the terms in (2.36) containing $g$ are also nonnegative, while the first term in the bracket is positive for $n\geq 3$ (but in general not for $n = 1,2$), so that in that case (2.36) yields some bilinear estimates. Whether such estimates can be useful remains to be seen.
A second further development of the previous calculation consists in using for $h$ other functions than $|x|$. For instance one can take $h(x) = |\theta \cdot x|$ for $\theta \in S^{n-1}$ and more generally $h(x) = |Px|$ for $P$ the orthogonal projection on a generic $k$ dimensional plane in ${I\hskip-1truemm R}^n$. The first choice leads naturally to an estimate of the Radon transform of $\rho$ [@17r]. One can also take advantage of the fact that the derivation of (2.18) involves mainly two integrations by parts from $h$ to $\nabla^2h$ in order to treat the case of a domain $\Omega \subset {I\hskip-1truemm R}^n$, typically the complement of a convex (or at least star-shaped) compact subset of ${I\hskip-1truemm R}^n$. One then obtains identities similar to (2.18) with additional surface terms, from which one can derive estimates of solutions in $\Omega$ [@17r].
A third possible development consists in extending the estimates of Proposition 2.1, part (2) to the case of attractive (focusing) interactions $g$ and of small solutions. We consider for illustration the case of a single power $$g(\rho ) = - \rho^{(p-1)/2} \eqno(2.38)$$
in dimension $n=1$. In that case, (2.18) becomes (remember that $R = 0$ for $n=1$) $$\partial_t M = \ \parallel \nabla \rho \parallel_2^2 - {p-1 \over p+1} \int dx \ \rho^{(p+3)/2}\ . \eqno(2.39)$$
By Sobolev inequalities, we estimate $$\parallel \rho \parallel_{(p+3)/2}^{(p+3)/2} \ \leq \ C \parallel \nabla \rho \parallel_2^2 \ \parallel \rho \parallel_{(p-1)/4}^{(p-1)/2}\ \leq \ C \parallel \nabla \rho \parallel_2^2 \ \parallel u; \dot{H}^{\sigma_c} \parallel^{p-1}\eqno(2.40)$$
where $\sigma_c = 1/2 - 2/(p-1)$ is the value of $\sigma$ for which $g$ given by (2.38) is $\dot{H}^{\sigma}$ critical, provided $\sigma_c \geq 0$, namely $p \geq 5$. Therefore $$\partial_t M \geq\ \parallel \nabla \rho \parallel_2^2 \left ( 1 - C \parallel u ; \dot{H}^{\sigma_c} \parallel^{p-1}\right ) \eqno(2.41)$$
so that (2.18) again yields an a priori estimate of $\rho$ in $L^2( {I\hskip-1truemm R}, \dot{H}^1)$ provided $M$ is controlled and provided $u$ is small in $L^{\infty}({I\hskip-1truemm R}, \dot{H}^{\sigma_c})$. The latter condition can be realized for energy solutions by taking some initial data $u_0$ small in $L^2$ if $\sigma_c = 0$, namely $p = 5$, and $u_0$ small in $H^1$ if $p > 5$. That smallness condition is of the same type as that occurring in the proof of boundedness of the $H^1$ norm from the energy conservation law which is used in the standard proof of globalization in $H^1$.
A similar situation can occur in higher space dimensions in so far as one can prove the estimate $$\left | <\rho , \omega^{1-n} \rho^{(p+1)/2}>\right | \leq \ C \parallel \rho ; \dot{H}^{(3-n)/2} \parallel^2 \ \parallel u; \dot{H}^{\sigma_c} \parallel^{p-1}\eqno(2.42)$$
where again $\sigma_c = n/2 - 2/(p-1)$ is the critical Sobolev exponent corresponding to $p$, provided $\sigma_c \geq 0$, namely $p \geq 1 + 4/n$, the $L^2$ critical value. The estimate (2.42) can be proved easily by the use of Sobolev inequalities for $n = 2,3$ and $p$ not too large. We leave the investigation of that estimate for general $n$ and $p$ as an open question.
A last possible development consists in using the Morawetz inequality to prove global wellposedeness and possibly AC (“scattering”) at a lower level of regularity than $H^1$, and that possibility has been extensively exploited. See for instance [@3r]-[@4r] [@6r]-[@9r] [@16newr] [@25r] and references therein quoted. In particular the right hand-side of (2.21) is controlled by the $H^{1/2}$ norm of $u$. For completeness we give a proof of that fact in Appendix 1 (see also [@6r] for the case $n \geq 3$).
We now turn to the Hartree equation (1.1) with $g$ given by (1.3). The formal computation is almost the same as for the NLS equation, except for the fact that, because of the nonlocality of the interaction, the equation is not Lagrangian. However the evolution equation of $j$ for the NLS equation takes the form $$\partial_t j = \ {\rm kinetic \ terms}\ - \rho \nabla g \eqno(2.43)$$
as follows in the same way as (2.15) from a computation which can be done without referring to the special form of $g$, so that (2.43) also holds for the Hartree equation (1.1) (1.3). Substituting (2.43) into $\partial_t M$ and using the fact that the kinetic terms are unchanged, we obtain $$\partial_t M = (1/2) \ <\nabla \rho , \Delta h \star \nabla \rho > + < \rho , \nabla h \star (\rho \nabla (V \star \rho ))> + \ R \eqno(2.44)$$
where $R$ is given by (2.20) as before. Integrating (2.44) over time in an interval $[t_1, t_2]$ yields $$\int_{t_1}^{t_2} dt \left \{ (1/2) \ <\nabla \rho , \Delta h \star \nabla \rho > + < \rho, \nabla h \star (\rho \nabla (V \star \rho ))> + \ R \right \}$$ $$\left . = -\ <\rho , \nabla h \star \ {\rm Im} \ \overline{u} \nabla u >\right |_{t_1}^{t_2} \ . \eqno(2.45)$$
As in the case of the NLS equation, that identity will yield useful estimates if $\nabla h \in L^{\infty}$ and if $\nabla^2 h$ is nonnegative as a matrix, so that $R$ is nonnegative, and if in addition the potential term in (2.45) is nonnegative. We now show that this is the case if $V$ is radial and nonincreasing. Assuming sufficient smoothness and decay at infinity for $V$, we obtain $$P \equiv \ <\rho , \nabla h \star (\rho \nabla (V \star \rho ))> \ = \int dx\ dy\ dz \ \rho (x) \nabla h (x-y) \rho (y) \nabla V (y-z) \rho (z)$$ $$= (1/2) \int dx\ dy\ dz\ \rho (x) \ \rho (y) \ \rho (z) \ \nabla V(y-z) (\nabla h(x-y) - \nabla h (x-z)) \eqno(2.46)$$
where we have used the fact that $\nabla V$ is an odd function. In order to prove the positivity of that integral, it suffices to prove that for all $x$, $y$ $$\nabla V(x) \cdot (\nabla h (x+y) - \nabla h (y)) \leq 0 \eqno(2.47)$$
where we have changed variables from $(y-z, x-y, x-z)$ to $(x,y,x+y)$. Let now $V(x) = v(|x|)$. The left hand-side of (2.47) can be written as $$\int_0^1 d\theta \ \nabla V (x) x \cdot \nabla^2 h (y + \theta x)$$ $$= \int_0^1 d\theta \ |x|^{-1} v'(|x|) (x \otimes x) \cdot \nabla^2 h (y + \theta x) \leq 0 \eqno(2.48)$$
for nonpositive $v'$ and nonnegative $\nabla^2 h$.
We now give a proposition where we assume sufficient regularity of $V$ to ensure wellposedness in $H^1$ and to make the previous formal computation rigorous.\
[**Proposition 2.2.**]{}
*Let $h = |x|$ and let $V \in L^{p_1} + L^{p_2}$ where $$p_2 \geq 1 \quad , \qquad n/4 < p_2 <p_1 \leq \infty\ . \eqno(2.49)$$*
Let $I$ be an interval and let $u \in X_{loc}^1(I)$ be a solution of the Hartree equation (1.1) (1.3). Then
\(1) The identity (2.45) holds for any $t_1, t_2 \in I$
\(2) Let in addition $V$ be radial non increasing (so that $V$ is non negative, possibly up to a harmless constant, and $u \in X_{loc}^1({I\hskip-1truemm R}) \cap L^{\infty}({I\hskip-1truemm R}, H^1)$). Then $u$ satisfies the estimate
$$\parallel \rho ; L^2({I\hskip-1truemm R}, \dot{H}^{(3-n)/2}) \parallel^2\ \leq \ C \parallel u \parallel_2^3 \ \parallel u; L^{\infty}({I\hskip-1truemm R}, H^1) \parallel \ . \eqno(2.50)$$ 3 truemm
[**Sketch of proof.**]{} The proof of Part (1) follows the same pattern as that of Proposition 2.1. Here we simply verify that the Hartree potential term $P$ in (2.45) is well defined at the available level of regularity. By the Hölder and Young inequalities, we estimate $$|P| \equiv \left | <\rho , \nabla h \star (\rho (V \star \nabla \rho ))> \right | \leq \ \parallel \rho \parallel_1 \ \parallel \nabla h \parallel_{\infty}\ \parallel \rho \parallel_{k/2}\ \parallel V \parallel_p \ \parallel u \parallel_k \ \parallel \nabla u \parallel_2$$ $$= \ C \parallel \rho \parallel_1\ \parallel \nabla u \parallel_2 \ \parallel u \parallel_k^3 \eqno(2.51)$$
with $\delta (k) \equiv n/2 - n/k = n/(3p)$. For the relevant values of $p$, one can take $\delta (k) \leq 1/4$ for $n=1$, $\delta (k) < 1$ for $n = 2$, $\delta (k) \leq 1$ for $n=3$ and for $n \geq 4$ if $p \geq n/3$, so that $ \parallel u \parallel_k$ is controlled by the $H^1$ norm of $u$ and $P$ is controlled in $L_{loc}^{\infty} (I)$, namely at the differential level. For $n \geq 4$ and $n/4 \leq p < n/3$, we use the fact that $u \in L_{loc}^q(I,L^k)$ with $2/q = \delta (k) - 1 = n/(3p)-1 \leq 1/3$, so that $u \in L_{loc}^6(I,L^k)$ and therefore $P \in L_{loc}^2 (I)$.
Part (2) follows from (2.45) by taking the limit $t_1 \to - \infty$, $t_2 \to \infty$, from (2.23) (2.32) and from the positivity of $P$ defined in (2.46) and of $R$ defined by (2.20).
$\sq$
Asymptotic completeness in the energy space
===========================================
In this section we exploit the Morawetz estimates of Propositions 2.1 and 2.2 to derive asymptotic completeness in $H^1$ for the NLS and Hartree equations. We begin with the NLS equation for which we restrict our attention to a single power interaction $$g(\rho ) = \lambda \ \rho^{(p-1)/2} \eqno(3.1)$$
We shall use the parameter $\sigma_c$ defined equivalently by $$\sigma_c = n/2 - 2/(p-1) \qquad {\rm or}\quad p-1 = 4/(n-2 \sigma_c) \eqno(3.2)$$
so that $g$ given by (3.1) is $\dot{H}^{\sigma_c}$ critical. We shall assume $0 < \sigma_c < 1$ so that $g$ is $L^2$ supercritical and $H^1$ subcritical. The treatment extends in a trivial way to a sum of such powers and to more general $g$. The case of critical powers is much more complicated and we refer to [@20r] for a treatment of that case in dimension $n \geq 3$. Some of the arguments can also be applied to solutions in $H^{\sigma}$ for $0 < \sigma \leq 1$.
The main tehnical step is the following proposition.\
[**Proposition 3.1.**]{} [*Let $g$ be defined by (3.1) with $0 < \sigma_c < 1$ ($\sigma_c < 1/2$ for $n = 1$). Let $u \in X_{loc}^1 ({I\hskip-1truemm R}) \cap L^{\infty} ({I\hskip-1truemm R}, H^1)$ be a solution of the NLS equation (1.1) (3.1) such that $\rho = |u|^2 \in L^2 ( {I\hskip-1truemm R}, \dot{H}^{(3-n)/2})$. Then $u \in X^1 ({I\hskip-1truemm R})$.*]{}\
[**Remark 3.1.**]{} For repulsive (defocusing) interaction $g$, namely for $\lambda > 0$, the Cauchy problem with initial data in $H^1$ is known to yield solutions satisfying the first assumption, and those solutions satisfy the condition on $\rho$ by Proposition 2.1. For attractive (focusing) interaction, the first assumption is satisfied for small data in $H^1$, and the assumption on $\rho$ can also be satisfied in some cases, for instance for $n = 1$, and for $n = 2,3$ and $p$ not too large, as discussed in the comments after Proposition 2.1 (see in particular (2.41)).\
[**Proof.**]{} Let $I = [t_0, t_1]$ be an interval and $u_0 = u(t_0)$. We start from the integral equation $$u(t) = U(t-t_0)u_0 - i \int_{t_0}^t dt' \ U(t-t') \ g(\rho (t')) \ u(t')\ . \eqno(3.3)$$
Using the Strichartz inequalities, we estimate in a standard way [@2r] $$\parallel u; X^1(I)\parallel \ \leq \ C\left ( \parallel u_0:H^1\parallel\ + \ \parallel g(\rho ) u; L^{\overline{q}} (I, H_{\overline{r}}^1)\parallel \right )$$ $$\leq C \left ( \parallel u_0;H^1\parallel\ + \ \parallel u; X^1(I)\parallel\ \parallel u; L^k(I,L^{\ell})\parallel^{p-1} \right ) \eqno(3.4)$$
where $1/\overline{r} + 1/r = 1/\overline{q} + 1/q = 1$, $(q,r)$ is an admissible pair, and $$\left \{ \begin{array}{l}2/k = \left ( n/2 - \sigma_c\right ) (1 - \delta )\\ \\ n/\ell = \left ( n/2 - \sigma_c \right ) \delta \end{array} \right . \eqno(3.5)$$
where $\delta \equiv \delta (r) = n/2 - n/r$.
The main step of the proof consists in estimating $u$ in $L^k(L^{\ell})$ by interpolation between the Morawetz quantity $\parallel \rho ; L^2(\dot{H}^{(3-n)/2}) \parallel$ and some norm which is controlled by$\parallel u; L^{\infty} (H^1)\parallel$, typically $\parallel u; L^{\infty} (\dot{H}^{\sigma})\parallel$ for some $\sigma$, $0 \leq\sigma \leq 1$. For orientation, we first consider the homogeneity degree of the various norms involved, where the degree of $\parallel u ; L^q (\dot{H}^{\sigma}_r)\parallel$ is defined as $\sigma + \delta (r) - 2/q$, so that it reduces to $\sigma$ for admissible $(q,r )$. In particular the degree of $\parallel u ;L^k(L^{\ell})\parallel$ is $\sigma_c$ by (3.5), that of $\parallel u ; L^{\infty} (\dot{H}^{\sigma})\parallel$ is $\sigma$ and the degree $\sigma_M$ of the Morawetz quantity is obtained by comparing from the point of view of dimension $$\parallel \rho ; L^2 (\dot{H}^{(3-n)/2})\parallel \ \sim \ \parallel u ; L^{\infty} ( \dot{H}^{\sigma_M})\parallel^2\ ,$$
which gives $$1 + n/2 + (n-3)/2 = 2\left ( n/2 - \sigma_M\right )$$
and therefore $\sigma_M = 1/4$.
We have to combine information on $u$ and on $\rho$, which can be transformed into information bearing only on $u$ or only on $\rho$. We consider separately the cases $n \geq 2$ and $n = 1$.\
[**The case n $\geq$ 2.**]{} Here we work with $u$. The information on $\rho$ implies the following information on $u$.
$$\left \{ \begin{array}{ll}\rho \in L^2 ({I\hskip-1truemm R}, \dot{H}^{1/2}) \subset L^2 ({I\hskip-1truemm R}, L^4) \Leftrightarrow u \in L^4 ({I\hskip-1truemm R}, L^8) &\hbox{for $n = 2$} \\ \\ \rho \in L^2 ({I\hskip-1truemm R}, L^{2})\Leftrightarrow u \in L^4 ({I\hskip-1truemm R}, L^4) &\hbox{for $n = 3$} \\ \\ \rho \in L^2 ({I\hskip-1truemm R}, \dot{H}^{(3-n)/2}) \Rightarrow u \in L^4 ({I\hskip-1truemm R}, \dot{H}_4^{(3-n)/4}) &\hbox{for $n \geq 4$} \end{array}\right . \eqno(3.6)$$
where the last result follows from Lemma 5.6 in [@20r]. We want to estimate $u$ in $L^k (I, L^{\ell})$ with $k$, $\ell$ satisfying (3.5) for some $k < \infty$ and some $\delta$ with $0 \leq \delta < 1$ (the value $\delta = 1$ is excluded a priori for $n=2$, and by the condition $k < \infty$ for $n \geq 3$). From (3.5), we obtain $$2/k + n/\ell = n/2 - \sigma_c \ . \eqno(3.7)$$
Conversely if $k$, $\ell$ satisfy (3.7) with $1 \leq k < \infty$ and $2 \leq \ell \leq \infty$, then $\delta$ defined by (3.5) satisfies $0 \leq \delta < 1$, so that it suffices to consider (3.7). We estimate by Sobolev inequalities and by (3.6) $$\parallel u; L^k(I, L^{\ell} )\parallel \ \leq \ C\ \parallel u;L^4(I, \dot{H}_4^{(3-n)/4}) \parallel^{\theta} \ \parallel u; L^{\infty} (I, \dot{H}^{\sigma })\parallel^{1 - \theta} \eqno(3.8)$$
(where $\dot{H}_4^{1/4}$ should be replaced by $L^8$ for $n = 2$ according to (3.6)) for some $\sigma$ and $\theta$ with $0 \leq \sigma \leq 1$ and $0 < \theta \leq 1$, such that $$\left \{ \begin{array}{l} 2/k = \theta /2 \\ \\ n/\ell = \theta (n/2 - 3/4) + (1 - \theta ) (n/2 - \sigma ) \end{array}\right . \eqno(3.9)$$
so that $$2/k + n/\ell = n/2 - \sigma_c = \theta (n/2 - 1/4) + (1 - \theta ) (n/2 - \sigma )$$
or equivalently $$\sigma_c = \theta /4 + (1 - \theta ) \sigma \eqno(3.10)$$
in accordance with the homogeneity argument given above. In addition for $n \geq 4$, the Sobolev inequality requires $$\theta (n-3)/4 \leq (1 - \theta ) \sigma \ . \eqno(3.11)$$
For a given $\sigma_c$ with $0 < \sigma_c < 1$, it is therefore sufficient to find $\sigma$ and $\theta$ with $0 \leq \sigma \leq 1$ and $0 < \theta \leq 1$, satisfying (3.10) and in addition (3.11) or equivalently $$\theta \leq 4\sigma /(n-3 + 4 \sigma ) \eqno(3.12)$$
for $n \geq 4$. One can make the following choices.\
[**Case n = 2, 3.**]{} For $\sigma_c = 1/4$, one can take $\theta = 1$ and the norm in $L^{\infty} (\dot{H}^{\sigma})$ is not needed. For $\sigma_c \not= 1/4$, the allowed values of $\sigma$ are given by $$0 \leq \sigma < \sigma_c < 1/4\qquad \hbox{or} \quad 1/4 < \sigma_c < \sigma \leq 1\ , \eqno(3.13)$$
with $\theta$ defined by (3.10).\
[**Case n $\geq$ 4.**]{} For $\sigma_c = 1/4$, one must take $\sigma = 1/4$ and one can take $\theta = (n-2)^{-1}$. For $\sigma_c \not= 1/4$, the allowed values of $\sigma$ are given by $$(0 <) \sigma_0 \leq \sigma < \sigma_c < 1/4\qquad \hbox{or} \quad 1/4 < \sigma_c < \sigma \leq \sigma_0 \wedge 1\ , \eqno(3.14)$$
where $\sigma_0$ is defined by (3.10) and (3.12) with equality, namely $$\sigma_0 = \sigma_c (n-3) /\left ( n-2-4 \sigma_c\right ) \ , \eqno(3.15)$$
with $\theta$ defined by (3.10).
We can now complete the proof of the proposition. Substituting (3.8) into (3.4) yields $$\parallel u; X^1(I)\parallel \left ( 1 - M_1 \parallel \rho ; L^2(I, \dot{H}^{(3-n)/2}) \parallel^{\theta (p-1)/2} \right ) \leq M_2 \eqno(3.16)$$
where $M_1$, $M_2$ depend only on $\parallel u ; L^{\infty} ({I\hskip-1truemm R}, H^1)\parallel$. By Proposition 2.1, one can partition ${I\hskip-1truemm R}$ into a finite number of intervals such that $$M_1 \parallel \rho ; L^2(I, \dot{H}^{(3-n)/2}) \parallel^{\theta (p-1)/2}\ \leq 1/2 \eqno(3.17)$$
and the number of intervals is also estimated in terms of $\parallel u; L^{\infty} ({I\hskip-1truemm R}, H^1)\parallel$. This yields an estimate of $\parallel u; X^1(I)\parallel$ for each such interval. Furthermore $u \in X^1({I\hskip-1truemm R})$ and $\parallel u; X^1({I\hskip-1truemm R})\parallel$ is estimated by a (computable) power of $\parallel u; L^{\infty} ({I\hskip-1truemm R}, H^1)\parallel$. This completes the proof for $n \geq 2$.\
[**Remark 3.2.**]{} For $n=2,3$, the argument is the same whether one uses $u$ or $\rho$. For $n \geq 4$, the argument can also be made by using $\rho$ and the fact that $$\parallel \rho ; \dot{H}_{n/(n- \sigma )}^{\sigma} \parallel \ \leq \ C \parallel u; \dot{H}^{\sigma}\parallel ^2 \eqno(3.18)$$
for $0 \leq \sigma \leq 1$ by Leibniz and Sobolev inequalities, and one ends up again with the condition (3.14) with however $$\sigma_0 = 2 \sigma_c (n-3)/\left ( 2n-5-4 \sigma_c \right ) \eqno(3.19)$$
which makes the restriction on $\sigma$ slightly stronger.\
[**The case n = 1.**]{} Here we work with $\rho$. For low values of $p$, we shall need the implication for $\rho$ of some Strichartz norms of $u$. We need the following lemma.\
[**Lemma 3.1.**]{}
*Let $0 \leq \sigma < 1/r \leq 1/2$. Then $$\parallel \rho ; \dot{H}_{(2/r- \sigma )^{-1}}^{\sigma} \parallel \ \leq \ C \parallel u; \dot{H}_r^{\sigma}\parallel ^2 \eqno(3.20)$$*
and therefore for $2/q = \delta (r)$ and for any interval $I$
$$\parallel \rho ; L^{q/2} (I, \dot{H}_{(2/r- \sigma )^{-1}}^{\sigma}) \parallel \ \leq \ C \parallel u; L^q (I, \dot{H}_r^{\sigma})\parallel ^2 \ . \eqno(3.21)$$ 5 truemm
[**Proof of Lemma 3.1.**]{} We estimate by fractional Leibniz and Sobolev inequalities $$\parallel \omega^{\sigma} \rho \parallel_{(2/r - \sigma )^{-1}} \ \leq \ C \parallel \omega^{\sigma} u \parallel_r \ \parallel u \parallel_{(1/r - \sigma )^{-1}} \ \leq \ C \parallel \omega^{\sigma} u \parallel_r^2\ .$$ $\sq$
We come back to the proof of the proposition. We start again from (3.4), so that we need to estimate $u$ in $L^k (I, L^{\ell})$ with $$\left \{ \begin{array}{l} 2/k = (1/2 - \sigma_c ) (1 - \delta )\\ \\ 1/\ell =(1/2 - \sigma_c ) \delta \end{array}\right . \eqno(3.22)$$
for some $\delta$ with $0 \leq \delta \leq 1/2$, or equivalently with $$2/k + 1/\ell = 1/2 - \sigma_c \ , \eqno(3.23)$$ $$0 \leq 1/\ell \leq (1/2 - \sigma_c )/2 \ . \eqno(3.24)$$
We estimate $$\parallel u; L^k(I, L^{\ell} )\parallel^2 \ = \ \parallel \rho ; L^{k/2} (I, L^{\ell /2}) \parallel \ \leq \ C\ \parallel \rho ;L^2(I, \dot{H}^1) \parallel^{\theta} \ \parallel \rho; L^{q/2} (I, \dot{H}_{(2/r - \sigma )^{-1}}^{\sigma })\parallel^{1 - \theta} \eqno(3.25)$$
by Sobolev inequalities, for some $\sigma$, $\theta$ and admissible $(q,r)$ satisfying $0 \leq \sigma < 1/r \leq 1/2$, $0 < \theta \leq 1$ and $$\left \{ \begin{array}{l} 2/k = \theta /2 + (1 - \theta )2/q \\ \\ 1/\ell =- \theta /4+ (1 - \theta ) (1/r - \sigma ) \ . \end{array}\right . \eqno(3.26)$$
Substituting (3.26) into (3.23) (3.24) yields $$\sigma_c = \theta /4 + (1 - \theta )\sigma \ , \eqno(3.27)$$ $$0 \leq - \theta /4 + (1 - \theta) (1/r- \sigma ) \leq (1/2 - \sigma_c )/2 \ . \eqno(3.28)$$
For $\sigma_c = 1/4$, namely $p = 9$, we must take $ \sigma = 1/4$ and we ensure (3.27) (3.28) by taking $r = 2$ and $\theta = 1/2$.
For $\sigma_c \not= 1/4$, we must take $$0 \leq \sigma < \sigma_c < 1/4 \qquad {\rm or}\qquad 1/4 < \sigma_c < \sigma < 1/2 \eqno(3.29)$$
and the elimination of $\theta$ betwen (3.27) (3.28) yields $$\sigma_c \leq {(4 \sigma_c - 1) \over (4 \sigma - 1)r} \leq (1 + 2 \sigma_c)/4 \eqno(3.30)$$
which implies the condition $ \sigma < 1/r$ since $${1 \over r} \geq {4 \sigma - 1 \over 4 \sigma_c -1} \ \sigma_c = \sigma + { \sigma - \sigma_c \over 4 \sigma_c - 1} > \sigma \eqno(3.31)$$
by (3.29). One can fulfill (3.30) with $r=2$ provided $$\sigma_+ \ \mathrel{\mathop >_{<}}\ \sigma \ \mathrel{\mathop >_{<}}\ \sigma_- (\mathrel{\mathop >_{<}}\ \sigma_c) \ \mathrel{\mathop >_{<}}\ 1/4 \eqno(3.32)$$
where $$\sigma_+ = \left ( 6 \sigma_c - 1\right )/8 \sigma_c\ , \qquad \sigma_- = (10 \sigma_c - 1) /(8 \sigma_c + 4)\ , \eqno(3.33)$$
which is compatible with (3.29) provided $ \sigma_- \geq 0$, namely $ \sigma_c \geq 1/10$ or $p \geq 6$. In that case we take $r=2$ and we can take $$\left \{ \begin{array}{ll} 0 \leq \sigma \leq \sigma_- (< \sigma_c) &\hbox{for $1/10 \leq \sigma_c \leq 1/6$} \\ \\ (0 \leq ) \sigma_+ \leq \sigma \leq \sigma_- (< \sigma_c) &\hbox{for $1/6 \leq \sigma_c < 1/4$} \\ \\ \sigma_+ = \sigma = \sigma_- = \sigma_c= 1/4 &\hbox{for $\sigma_c = 1/4$} \\ \\ (\sigma_c <) \sigma_- \leq \sigma \leq \sigma_+ (< 1/2) &\hbox{for $1/4 < \sigma_c < 1/2$} \end{array}\right . \eqno(3.34)$$
with $\theta$ defined by (3.27) for $\sigma_c \not= 1/4$. For such $(\sigma , \theta )$, one obtains $$\parallel u ; L^k(I, L^r)\parallel\ \leq \ C\parallel \rho ; L^2(I, \dot{H}^1)\parallel^{\theta /2} \ \parallel u ; L^{\infty}(I, \dot{H}^{\sigma})\parallel^{1 - \theta} \eqno(3.35)$$
which implies (3.16) with $n=1$.
For $0 < \sigma_c < 1/10$, namely $5 < p < 6$, one can take $\sigma = 0$ and take for $r$ the minimal value allowed by (3.30), namely $$4/r = (1 + 2\sigma_c )/(1 - 4\sigma_c ) \eqno(3.36)$$
and $\theta = 4 \sigma_c$. One then obtains $$\parallel u ; L^k(I, L^\ell)\parallel\ \leq \ C\parallel \rho ; L^2(I, \dot{H}^1)\parallel^{\theta /2} \ \parallel u ; X(I) \parallel^{1 - \theta} \eqno(3.37)$$
so that by (3.4) $$\parallel u ; X^1(I)\parallel\ \leq \ C\Big ( \parallel u ; L^{\infty} (I, H^1) \parallel \ + \ \parallel \rho ; L^2(I, \dot{H}^1)\parallel^{(p-1) \theta /2}$$ $$\times \ \parallel u ; X^1(I)\parallel^{1 + (p - 1)(1 - \theta)}\Big ) \eqno(3.38)$$
which gives again an estimate of $\parallel u, X^1(I) \parallel$ provided $\parallel \rho ; L^2(I, \dot{H}^1)\parallel$ is sufficiently small.
The end of the proof proceeds as in the case $n \geq 2$.
$\sq$
[**Remark 3.3.**]{} If one wants to use values of $\sigma$ arbitrarily close to $\sigma_c$ for $\sigma_c > 1/4$, one needs to take $r > 2$ in the region $\sigma_c < \sigma < \sigma_-$. The lowest possible value of $r$ is given by (see (3.30)) $$4/r = (1 + 2\sigma_c) (4\sigma - 1) /(4\sigma_c - 1) \ . \eqno(3.39)$$ 5 truemm
[**Remark 3.4.**]{} One could use $u$ instead of $\rho$ also in the case $n=1$. From the inequality $$\rho^{3/2} \leq (3/4) \int dx \ \rho^{1/2} |\rho '| \leq (3/4) \parallel \rho \parallel_1^{1/2}\ \parallel \rho '\parallel_2$$
we obtain $$\parallel u; L^6 ({I\hskip-1truemm R}, L^{\infty})\parallel^3\ = \ \parallel \rho ; L^3({I\hskip-1truemm R} , L^{\infty})\parallel^{3/2} \ \leq (3/4) \parallel u \parallel_2\ \parallel \rho ; L^2({I\hskip-1truemm R} , \dot{H}^1)\parallel \eqno(3.40)$$
and one can perform the estimates by using $\parallel u; L^6 ({I\hskip-1truemm R}, L^{\infty})\parallel$ instead of $\parallel \rho : L^2({I\hskip-1truemm R} , \dot{H}^1)\parallel$. The results are essentially the same with however stronger restrictions on $\sigma$.\
We now exploit Proposition 3.1 to prove AC in $H^1$ for the NLS equation (1.1) with interaction (3.1). We first recall some standard results on scattering for that equation [@2r].\
[**Proposition 3.2.**]{}
*Let $0 \leq \sigma_c < 1$, $\sigma_c < 1/2$ for $n=1$, or equivalently $p \geq 1 + 4/n$, $p < 1 + 4/(n-2)$ for $n \geq 3$, and $\lambda > 0$.*
\(1) Let $u_+ \in H^1$. Then the NLS equation (1.1) (3.1) has a unique solution $u \in X_{loc}^1({I\hskip-1truemm R}) \cap X^1 ({I\hskip-1truemm R}^+)$ such that $$\parallel U(-t) \ u(t) - u_+ ; H^1 \parallel \to 0 \eqno(3.41)$$
when $t \to \infty$.
\(2) Let $u \in X^1({I\hskip-1truemm R}^+)$ be a solution. Then there exists $u_+ \in H^1$ such that (3.41) holds.
\
[**Sketch of proof.**]{} The proof uses mainly Strichartz inequalities. In order to prove Part (1), one starts from the integral equation (3.3) with $u_+ = U(-t_0)u_0$ and $t_0 \to \infty$, namely $$u(t) = U(t)\ u_+ + i \int_t^{\infty} dt'\ U(t-t') \ g(\rho (t')) \ u(t') \eqno(3.42)$$
and one solves that equation locally in a neighborhood of infinity in time, namely in $I = [T, \infty )$ for $T$ sufficiently large. The proof uses the estimate (3.4) followed by $$\parallel u ; L^k(I, L^\ell )\parallel\ \leq \ C\parallel u ; L^{q_1}(I, \dot{H}_{r_1}^{\sigma_c})\parallel \ \leq \ C\parallel u ; X^1(I)\parallel \eqno(3.43)$$
for admissible $(q_1, r_1)$ with $k = q_1 < \infty$, namely $$0 < 2/k = (n/2 - \sigma_c ) (1 - \delta ) = 2/q_1 = \delta (r_1)$$
which can always be realized for suitable $\delta$.
In order to prove Part (2), one estimates $$\parallel U(-t_1) \ u(t_1) - U(-t_2)\ u(t_2); H^1 \parallel\ = \ \parallel \int_{t_1}^{t_2} dt'\ U(t-t')\ g(\rho (t')) u(t'); H^1 \parallel$$ $$\leq \ C \parallel u ; X^1(I)\parallel \ \parallel u ; L^{q_1}(I, \dot{H}_{r_1}^{\sigma_c})\parallel^{p-1} \eqno(3.44)$$
with $I = [t_1, t_2]$, and the last norm tends to zero when $t_1, t_2 \to \infty$ for $u \in X^1({I\hskip-1truemm R}^+)$ and $q_1 < \infty$.
The previous proposition yields the existence of the wave operators in Part (1) and the fact that AC holds for solutions in $X^1({I\hskip-1truemm R})$ in Part (2). Putting together Propositions 3.1 and 3.2 yields AC for finite energy solutions.\
[**Proposition 3.3.**]{} [*Let $0 < \sigma_c < 1$, $\sigma_c < 1/2$ for $n=1$, or equivalently $p > 1 + 4/n$, $p < 1 + 4/(n-2)$ for $n \geq 3$, and let $\lambda > 0$. Let $u$ be a finite energy solution of the NLS equation (1.1) (3.1), namely a solution $u \in X_{loc}^1({I\hskip-1truemm R})$. Then $u \in X^1({I\hskip-1truemm R})$ and there exist $u_\pm \in H^1$ such that $$\parallel U(-t) \ u(t) - u_\pm ; H^1 \parallel \to 0 \eqno(3.45)$$ when $t \to \pm \infty$.*]{}\
We now turn to the Hartree equation (1.1) with $g$ given by (1.3). We assume that $V \in L^p$ for suitable $p$, for which we shall use the parameter $\sigma_c$ defined by $$\sigma_c = n/2p-1 \ . \eqno(3.46)$$
The treatment extends in a trivial way to more general $V$ such as those considered in Proposition 2.2.
The main technical result is the following proposition.\
[**Proposition 3.4.**]{} [*Let $n \geq 3$. Let $0 < \sigma_c < 1$, $\sigma_c \leq 1/2$ for $n = 3$, or equivalently $n/4 < p < n/2$, $p \geq 1$ for $n=3$. Let $V \in L^p$ be real even and let $u \in X_{loc}^1({I\hskip-1truemm R}) \cap L^{\infty} ({I\hskip-1truemm R}, H^1)$ be a solution of the Hartree equation (1.1) (1.3) such that $\rho = |u|^2 \in L^2({I\hskip-1truemm R} , \dot{H}^{(3-n)/2})$. Then $u \in X^1({I\hskip-1truemm R})$*]{}.\
[**Remark 3.5.**]{} For nonnegative $V$, the Cauchy problem is globally well posed in $H^1$ and yields solutions $u \in X_{loc}^1({I\hskip-1truemm R}) \cap L^{\infty} ({I\hskip-1truemm R}, H^1)$.\
[**Sketch of proof.**]{} The proof follows the same pattern as that of Proposition 3.1. We start again from (3.3). Using the Strichartz estimates and the Young inequality, we estimate $$\parallel u;X^1(I)\parallel\ \leq \ C \left ( \parallel u_0 ; H^1\parallel + \parallel V \parallel_p \ \parallel u; X^1(I)\parallel \ \parallel u; L^k(I, L^{\ell})\parallel^2 \right ) \eqno(3.47)$$
where now $$\left \{ \begin{array}{l} 2/k = 1 - \delta \\ \\ n/\ell = n/2 - \sigma_c + \delta - 1\end{array} \right .$$
for some $\delta$, $0 \leq \delta \leq 1$. It is then sufficient to estimate $u \in L^k(I, L^{\ell})$ for $0 < 2/k \leq 1$ and $$2/k + n/\ell = n/2 - \sigma_c\ .$$
The proof then proceeds as for the NLS equation. In particular one uses the estimate (3.8) with $k$, $\ell$ satisfying (3.9) so that $0 < \theta = 2(1- \delta ) \leq 1$, which ensures the condition $0 < 2/k \leq 1$.
$\sq$
The analogue of Proposition 3.2 can be proved for the Hartree equation [@11r] and the final result follows therefrom and from Propositions 3.4 and 2.2.\
[**Proposition 3.5.**]{} [*Let $n \geq 3$. Let $0 < \sigma_c < 1$, $\sigma_c \leq 1/2$ for $n = 3$, or equivalently $n/4 < p < n/2$, $p \geq 1$ for $n=3$. Let $V \in L^p$ be real radial and non increasing (and therefore nonnegative). Let $u$ be a finite energy solution of the Hartree equation (1.1) (1.3), namely a solution $u \in X_{loc}^1({I\hskip-1truemm R})$. Then $u \in X^1 ({I\hskip-1truemm R})$ and there exist $u_{\pm} \in H^1$ satisfying (3.45) when $t \to \pm \infty$.*]{}
Appendix 1. Estimate of the RHS of (2.21) in H$^{\bf 1/2}$ for h = $|$x$|$ {#appendix-1.-estimate-of-the-rhs-of-2.21-in-hbf-12-for-h-x .unnumbered}
==========================================================================
$\ \ \ $
5 truemm
[**Lemma**]{} $$|<\rho, \nabla |x| \star {\rm Im} \ \overline{u} \nabla u > | \leq \ C \parallel u \parallel_2^2 \ \parallel u; \dot{H}^{1/2}\parallel^2 \ . \eqno({\rm A}1.1)$$ 3 truemm [**Proof.**]{} We estimate $$|<\rho , \nabla |x| \star {\rm Im}\ \overline{u} \nabla u >|\ \leq \ |<\nabla u, u (\nabla |x| \star \rho)>|$$ $$\leq \ \parallel \omega^{1/2} u \parallel_2\ \parallel \omega^{1/2}u(\nabla |x| \star \rho ) \parallel_2$$ $$\leq \ C \Big ( \parallel \omega^{1/2} u \parallel_2^2 \ \parallel \nabla |x| \star \rho \parallel_{\infty}$$ $$+ \ \parallel \omega^{1/2} u \parallel_2\ \parallel u \parallel_r \ \parallel \omega^{1/2} (\nabla |x| \star \rho ) \parallel _{n/\delta } \eqno({\rm A}1.2)$$
with $\delta = \delta (r) > 0$, by fractional Leibniz inequalities. Clearly $$\parallel \nabla |x| \star \rho \parallel _{\infty} \ \leq \ \parallel \rho \parallel_1 \ . \eqno({\rm A}1.3)$$
We then use the fact that $$F(\nabla |x|) = C\ P \ \xi |\xi|^{-(n+1)}$$
where $F$ is the Fourier transform and $P$ denotes the principal value ([@18r], Theorem 5, p. 73 with $k=1$) so that $$\omega^{1/2} (\nabla |x| \star \rho ) = C\ F^{-1} (\xi |\xi|^{-(n+1/2)} \widehat{\rho} (\xi )) = C\ x|x|^{-3/2} \star \rho$$
and therefore $$\parallel \omega^{1/2} (\nabla |x| \star \rho ) \parallel_{n/\delta} \ \leq \ C \parallel \rho \parallel_{s/2}\ = \ C \parallel u \parallel_s^2 \eqno({\rm A}1.4)$$
by the Hardy Littlewood Sobolev inequality ([@18r], Theorem 1, p. 119), where $$\delta + 2 \delta (s) = 1/2\ ,$$
provided $0 < \delta < n$ and $0 < \delta (s) < n/2$. The last term in (A1.2) is then estimated by $$C \parallel \omega^{1/2} u \parallel_2\ \parallel u \parallel_r \ \parallel u \parallel_s^2 \ \leq\ C \parallel \omega^{1/2} u \parallel_2^2 \ \parallel u \parallel_2^2 \eqno({\rm A}1.5)$$
by Sobolev inequalities, which together with (A1.2) and (A1.3) yields (A1.1). One can easily choose $r$ satisfying the required restrictions, for instance by taking $r=s$, which yields $\delta = 1/6$.
$\sq$
Appendix 2. A quadratic identity for the NLKG equation {#appendix-2.-a-quadratic-identity-for-the-nlkg-equation .unnumbered}
======================================================
As mentioned in Section 2, the algebraic structure (2.1) (2.2) is realized for any system with symmetric conserved energy momentum tensor $T_{\lambda \mu}$, namely $$\left \{ \begin{array}{l} \partial_t \sigma + \nabla \cdot j = 0 \\ \\ \partial_t j + \nabla \cdot T = 0 \end{array} \right . \eqno({\rm A}2.1)$$
where $\sigma = T_{00}$ is the energy density, $j_k = - T_{0k} = - T_{k0}$ is both the energy current and the momentum density, and $T = \{ T_{k \ell}\}$ is the space-space part of $T_{\lambda \mu}$. Here we use $\sigma$ instead of $\rho$ in order to keep the notation $\rho$ for $|u|^2$, greek (resp. latin) indices run from $0$ to $n$ (resp. from 1 to $n$), and the index $0$ refers to time. We shall also need the Minkowski metric $\eta_{\lambda \mu}$ with $\eta_{00} = - \eta_{jj} = 1$. We now consider the NLKG (or nonlinear wave NLW) equation $$\sq u + g(\rho ) u = 0 \eqno({\rm A}2.2)$$
where $g = g(\rho )$ is a real function of $\rho = |u|^2$. The Lagrangian density is $$L(u) = |\partial_t u |^2 - |\nabla u|^2 - G(\rho ) \eqno({\rm A}2.3)$$
with $G$ defined by (2.9). The energy momentum tensor is well known to be $$T_{\lambda \mu} = 2{\rm Re}\ \partial_{\lambda} \overline{u} \ \partial_{\mu} u - \eta_{\lambda \mu} L \ . \eqno({\rm A}2.4)$$
We define as before for real even $h$ $$J = (1/2) \ < \sigma , h\star \sigma > \eqno({\rm A}2.5)$$
so that $$M \equiv \partial_t J = - \ <\sigma, \nabla h \star j>\eqno({\rm A}2.6)$$ $$\partial_t M = \partial_t^2 J = - \ <j, \nabla^2h \star j>\ +\ < \sigma, \nabla^2 h \star T>\ . \eqno({\rm A}2.7)$$
Substituting $\sigma = T_{00}$, $j_k = - T_{0k}$ and $T = \{ T_{k \ell}\}$ given by (A2.4) into (A2.7) yields $$\partial_t M = - \ <2{\rm Re}\ \nabla \overline{u} \ \partial_t u, \nabla^2 h \star 2{\rm Re}\ \nabla \overline{u} \ \partial_t u>$$ $$+ \ < |\partial_t u|^2 + |\nabla u |^2 + G, \Delta h \star \left ( |\partial_t u|^2 - |\nabla u|^2 - G\right ) + \nabla^2 h \star 2 \nabla\overline{u} \nabla u>$$
and finally, ordering the terms by the powers of $G$ $$\partial_t M = - \ < G, \Delta h \star G>\ +\ <G, - 2 \Delta h \star |\nabla u|^2 + \nabla^2h \star 2\nabla\overline{u} \nabla u>$$ $$+ \ < |\partial_t u|^2 + |\nabla u |^2, \Delta h \star\left ( |\partial_t u|^2 - |\nabla u|^2\right ) + \nabla^2 h \star 2 \nabla\overline{u} \nabla u>$$ $$- \ <2{\rm Re}\ \nabla \overline{u}\ \partial_t u, \nabla^2 h \star 2{\rm Re}\ \nabla \overline{u} \ \partial_t u> \ . \eqno({\rm A}2.8)$$
In space dimension $n=1$, the linear term in $G$ vanishes and (A2.8) reduces to $$\partial_t M = - \ < G, h'' \star G>\ +\ < |\partial_t u|^2 + |\partial_x u|^2, h'' \star (|\partial_t u|^2 + |\partial_x u|^2>$$ $$- \ <2\ {\rm Re}\ \partial_x \overline{u}\ \partial_t u, h'' \star 2\ {\rm Re}\ \partial_x \overline{u}\ \partial_t u>$$ $$= \ < |\partial_t u - \partial_x u|^2, h'' \star |\partial_t u+ \partial_x u|^2>\ -\ < G, h'' \star G> \eqno({\rm A}2.9)$$
so that if $h''$ has a given sign, the kinetic and potential terms have opposite signs, which precludes a straightforward use of that identity to derive estimates.
Appendix 3. Relation between the original and the quadratic Morawetz identities {#appendix-3.-relation-between-the-original-and-the-quadratic-morawetz-identities .unnumbered}
===============================================================================
Here we rewrite the original Morawetz identity for the NLS equation in a form which exhibits its relation to the quadratic identity derived in Section 2. The original version starts from the quantity $<\nabla h, j>$ with $j = {\rm Im} \ \overline{u} \nabla u$. Using instead of $\nabla h$ a space translate thereof is then equivalent to consider the quantity $$M_0 (x) = - \nabla h \star j \eqno({\rm A}3.1)$$
so that $M$ defined in (2.4) is simply $M = <\rho , M_0>$. Using the evolution equation of $j$ given by (2.2) (2.15) yields
1.7 truecm $\partial_t M_0 = - \nabla^2 h \star T$ $$= \nabla^2 h \star ( \nabla \overline{u} \nabla u) + \Delta h \star (-(1/4) \Delta \rho + \rho g - G) \ . \eqno({\rm A}3.2)$$
For $h(x) = |x|$ and by the use of (2.22), we obtain $$\partial_t M_0 = \nabla^2 h \star ( \nabla \overline{u} \nabla u) - (1/4) \Delta^2 |x| \star \rho + (n-1) |x|^{-1} \star (\rho g - G) \eqno({\rm A}3.3)$$
for $n > 1$, which is the original Morawetz identity for NLS. The first term in the RHS is nonnegative. The second term is positive for $n \geq 3$ only since $$\Delta^2 |x| = \left \{\begin{array}{ll} - 8 \pi \delta (x) &\hbox{for $n=3$}\\ \\ - (n-1) (n-3)|x|^{-3} &\hbox{for $n\geq 4$}\end{array}\right .$$
Early applications of the method [@10r] [@12r] used (A3.3) to derive an estimate of the last term and were therefore restricted to space dimension $n \geq 3$. Taking the scalar product of $M_0$ with $\rho$ gives a useful quantity because $\rho$ satisfies the conservation law (2.1).
Appendix 4. Proof of (2.21) by regularization {#appendix-4.-proof-of-2.21-by-regularization .unnumbered}
=============================================
Let $u \in X_{loc}^1(I)$ be a solution of the NLS equation (2.7) and let $\varphi \in \mathscr{C}_0^{\infty} ({I\hskip-1truemm R}^n, {I\hskip-1truemm R}^+)$ with $\parallel \varphi \parallel _1 = 1$ denote a regularizing function of the space variable which will eventually converge to the measure $\delta$. We define $u_{\varphi} = \varphi \star u$, $\rho_\varphi = |u_\varphi |^2$ $j_\varphi = {\rm Im} \ \overline{u}_\varphi \nabla u_\varphi$, $f(u) = g(|u|^2)u$, $f_\varphi = \varphi \star f(u)$ and $f_{\not=} = f_\varphi - f(u_\varphi )$. Then $u_\varphi \in \displaystyle{\mathrel{\mathop \cap_{\sigma \geq 0}}}\ \mathscr{C}(I, H^{\sigma})$ and $u_\varphi$ satisfies the equation $$i \partial_t u_\varphi = - (1/2) \Delta u_\varphi + f_\varphi \eqno({\rm A}4.1)$$
which is the regularized form of (2.7). From (A4.1), we obtain the regularized form of (2.1) (2.2) by direct computation, namely $$\partial_t \ \rho_\varphi + \nabla \cdot j_\varphi = P_1 \ , \eqno({\rm A}4.2)$$ $$\partial_t \ j_\varphi + \nabla \cdot T(u_\varphi ) = P_2 \eqno({\rm A}4.3)$$
where $$P_1 = 2\ {\rm Im} \left ( \overline{u}_\varphi f_{\not=}\right ) \ , \eqno({\rm A}4.4)$$ $$T_{k\ell} (u_{\varphi}) = {\rm Re} \nabla_k \overline{u}_\varphi \nabla_{\ell} u_{\varphi} - \delta_{k\ell} \left ( (1/4) \Delta \rho_\varphi - \rho_\varphi g(\rho_\varphi ) + G(\rho_\varphi )\right ) \ , \eqno({\rm A}4.5)$$ $$P_2 = {\rm Re} \left ( f_{\not=} \nabla \overline{u}_\varphi - \overline{u}_\varphi \nabla f_{\not=} \right ) \ . \eqno({\rm A}4.6)$$
Using (A4.2) (A4.3) and the regularity properties of $u_\varphi$, we can derive the regularized version of (2.21) in the same way as in Section 2 (see (2.16)-(2.20)), namely $$\int_{t_1}^{t_2} dt \Big \{ (1/2) \ <\nabla \rho_{\varphi}, \Delta h \star \nabla \rho_{\varphi}>\ + \ < \rho_{\varphi}, \Delta h \star \left (\rho_{\varphi} g(\rho_{\varphi}) - G(\rho_{\varphi})\right ) >$$ $$\left . + R(u_{\varphi} ) - S_{\varphi}\Big \} = - \ < \rho_{\varphi}, \nabla h \star j_{\varphi}> \right |_{t_1}^{t_2}\eqno({\rm A}4.7)$$
where $$R(v) = \ < \overline{v}v, \nabla^2h \star \nabla \overline{v}\nabla v> \ - \ < \overline{v}\nabla v, \nabla^2 h \star \overline{v}\nabla v > \ , \eqno({\rm A}4.8)$$ 1.7 truecm $S_{\varphi} = -\ <j_{\varphi}, \nabla h \star P_1 >\ +\ < \rho_{\varphi} , \nabla h \star P_2>$ $$= 2 \ {\rm Re} \left \{ <u_\varphi \nabla \overline{u}_{\varphi} , \nabla h \star \overline{u}_\varphi f_{\not=}>\ + \ < \rho_{\varphi} , \nabla h \star f_{\not=} \nabla \overline{u}_\varphi> \right \} \eqno({\rm A}4.9)$$
by a straightforward rewriting.
We now remove the cutoff, and without loss of generality, we restrict our attention to the case of a single power interaction of the form (3.1). We restrict ourselves to proving that $$\lim_{\varphi \to \delta} \int_{t_1}^{t_2} dt\ S_{\varphi} = 0 \eqno({\rm A}4.10)$$
since the remaining terms in (A4.7) converge to the corresponding terms without $\varphi$ by estimates similar to (2.29) (2.31). We estimate $$|S_{\varphi}| \ \leq \ 4 \parallel u \parallel_2^2\ \parallel \nabla u \parallel_2 \ \parallel f_{\not=} \parallel_2 \eqno({\rm A}4.11)$$
since $\parallel \nabla h \parallel_\infty = 1$ by (2.22). From the identity $$f_{\not=} = \varphi \star f(u) - f(u) + f(u) - f(u_{\varphi})$$
and from the estimates $$|f(u)| \leq C|u|^p\ , \ |f(u) - f(u_{\varphi})| \leq C |u- u_{\varphi}| \left ( |u|^{p-1} + |u_{\varphi}|^{p-1} \right ) \ ,$$
we obtain $$\parallel f_{\not=} \parallel _2\ \leq \ \parallel \varphi \star f(u) - f(u) \parallel _2\ + \ C \parallel u \parallel _{2p}^{p-1} \ \parallel u - u_{\varphi}\parallel _{2p}\ . \eqno({\rm A}4.12)$$
We next estimate $\parallel u \parallel_{2p}$. For $n=1,2$ and for $n \geq 3$ with $p-1 \leq 2/(n-2)$, we estimate $$\parallel u \parallel_{2p} \ \leq\ C \parallel u; H^1\parallel\ \in L_{loc}^{\infty}(I)\ .$$
For $n \geq 3$ and $$2/(n-2) < p-1 < 4/(n-2)$$
we estimate $$\parallel u \parallel_{2p} \ \leq \ C \parallel u; \dot{H}_r^1\parallel \ \in L_{loc}^q(I)$$
where $$0 < 2/q = \delta (r) = \delta (2p) - 1 < (n-2)/(n+2)$$
so that $\parallel f(u) \parallel_2 \ = C \parallel u \parallel_{2p}^p \ \in L_{loc}^k(I)$ with $$0 < 2/k = (p-1)(n/2-1) - 1 < 1\ .$$
In both cases $\parallel f_{\not=} \parallel _2$ tends to zero for each $t \in I$ when $\varphi$ tends to $\delta$, and (A4.10) follows from (A4.11) and from the Lebesgue dominated convergence theorem.
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[^1]: Unité Mixte de Recherche (CNRS) UMR 8627
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