From 61802fe09ed222c69983fe8fef5ce107ba3aa223 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 5 Mar 2021 17:01:39 +0100
Subject: Initial commit.

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+\documentclass[aps,reprint,longbibliography,floatfix]{revtex4-2}
+
+\usepackage[utf8]{inputenc} % why not type "Stokes" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[
+  colorlinks=true,
+  urlcolor=purple,
+  citecolor=purple,
+  filecolor=purple,
+  linkcolor=purple
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{amsmath, graphicx, xcolor} % standard packages
+
+\begin{document}
+
+\title{}
+
+\author{Jaron Kent-Dobias}
+\author{Jorge Kurchan}
+
+\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
+
+\date\today
+
+\begin{abstract}
+\end{abstract}
+
+\maketitle
+
+Consider a thermodynamic calculation involving the (real) $p$-spin model for a
+particular instantiation of the coupling tensor $J$
+\begin{equation} \label{eq:partition.function}
+  Z_J(\beta)=\int_{S^{N-1}}ds\,e^{-\beta H_J(s)}
+\end{equation}
+where $S^{N-1}$ is the $N-1$-sphere of radius $N$. Quantities of interest are
+usually related to the quenched free energy, produced by averaging over the
+$J$s the sample free energy $F_J$
+\begin{equation}
+  \overline F(\beta,\kappa)=-\beta^{-1}\int dJ\,p_\kappa(J)\log Z_J
+\end{equation}
+which can depend in general on the inverse temperature $\beta$ and on some
+parameter $\kappa$ which governs the distribution of $J$s. For most
+applications, $\beta$ is taken to be real and positive, and the distribution
+$p_\kappa$ is taken to be Gaussian or discrete on $\pm1$.
+
+We are interested in analytically continuing expressions like $\overline F$
+into the region of complex $\beta$ or distributions $p_\kappa$ involving
+complex $J$. The former has been considered extensively for the Gaussian
+$p$-spin in the past \cite{complex_energy}, while the latter is largely
+unexplored.
+
+When the argument of the exponential integrand in \eqref{eq:partition.function}
+acquires an imaginary component, various numeric and perturbative schemes for
+approximating its value can face immediate difficulties due to the emergence of
+a sign problem, resulting from rapid oscillations coinciding with saddles.
+
+The surface $M\subset\mathbb C^N$ defined by $z^2=N$ is an $N-1$ dimensional
+\emph{Stein manifold}, a type of complex manifold defined by the level set of a
+holomorphic function \cite{Forstneric_2017_Stein}. Suppose that $z:\mathbb
+C^{N-1}\to M$ is a holomorphic map. The Jacobian $J$ of the map is
+\begin{equation}
+  J_{i\alpha}=\frac{\partial z_i}{\partial u_\alpha}=\partial_\alpha z_i
+\end{equation}
+where Greek coefficients run from $1$ to $N-1$ and Latin coefficients from $1$
+to $N$. The hermitian metric is $g=J^\dagger J$. For any smooth function
+$\phi:M\to\mathbb R$, its gradient $\nabla\phi$ is a holomorphic vector field
+given by
+\begin{equation}
+  \nabla\phi=(\partial^*\phi)^\sharp=(\partial^*\phi)g^{-1}
+\end{equation}
+
+For
+coordinates $u\in\mathbb C^{N-1}$, dynamics consists of gradient descent on
+$\operatorname{Re}H$, or
+\begin{equation}
+  \dot u=-\nabla\operatorname{Re}H=-\tfrac12(\partial H)^\dagger g^{-1}
+\end{equation}
+These dynamics preserve $\operatorname{Im}H$ and in fact correspond to
+Hamiltonian dynamics, with the real and imaginary parts of the coordinates
+taking the role of conjugate variables. \cite{Morrow_2006_Complex}
+
+Working with a particular map is inconvenient, and we would like to develop a map-independent dynamics. Using the chain rule, one finds
+\begin{equation}
+  \begin{aligned}
+    \dot z_i
+    &=\dot u_\alpha\partial_\alpha z_i
+    =-\tfrac12(\partial_\beta H)^*g^{-1}_{\beta\alpha}\partial_\alpha z_i\\
+    &=-\tfrac12(\partial_j H)^*(\partial_\beta z_j)^*g^{-1}_{\beta\alpha}\partial_\alpha z_i
+    =-\tfrac12(\partial H)^\dagger(J^\dagger g^{-1}J)\\
+  \end{aligned}
+\end{equation}
+where $J$ is the Jacobian of the coordinate map and $g$ is the metric. In stereographic coordinates this can be worked out directly.
+Consider the coordinates $z_i=u_i$ for $1\leq i\leq N-1$ and
+\begin{equation}
+  z_N=\sqrt{N-u_\alpha u_\alpha}
+\end{equation}
+The Jacobian is
+\begin{equation}
+  J_{\alpha i}=\partial_\alpha z_i=\delta_{\alpha i}-\delta_{Ni}\frac{u_\alpha}{\sqrt{N-u_\beta u_\beta}}
+\end{equation}
+and the corresponding hermitian metric is
+\begin{equation}
+  g_{\alpha\beta}=J_{i\alpha}^*J_{i\beta}
+  =\delta_{\alpha\beta}+\frac{u_\alpha^*u_\beta}{|N-u_\gamma u_\gamma|}
+\end{equation}
+The metric can be inverted explicitly:
+\begin{equation}
+  g^{-1}_{\alpha\beta}
+  =\delta_{\alpha\beta}-\frac{u_\alpha^*u_\beta}{|N-u_\gamma u_\gamma|+|u|^2}.
+\end{equation}
+Putting these pieces together, we find
+\begin{equation}
+  (J^\dagger g^{-1}J)_{ij}
+  =\delta_{ij}-\frac{z_iz_j^*}{|z|^2}
+\end{equation}
+which is just the projector onto the constraint manifold.
+
+Therefore, a map-independent dynamics is given by
+\begin{equation}
+  \dot z
+  =-\frac12(\partial H)^\dagger\left(I-\frac{zz^\dagger}{|z|^2}\right)
+\end{equation}
+
+Stokes lines are trajectories that approach distinct critical points as time
+goes to $\pm\infty$. From the perspective of dynamics, these correspond to
+\emph{heteroclinic orbits}. What are the conditions under which Stokes lines
+appear? Because the dynamics conserves imaginary energy, two critical points
+must have the same imaginary energy if they are to be connected by a Stokes line.
+
+The level sets defined by $\operatorname{Im}H=c$ for $c\in\mathbb R$ are surfaces of
+$2(N-1)-1$ real dimensions. They must be simply connected, since gradient
+descent in $\operatorname{Re}H$ cannot pass the same point twice.
+
+\section{2-spin}
+
+\begin{equation}
+  H_0=\frac12z^TJz
+\end{equation}
+$J$ is generically diagonalizable by a complex orthogonal matrix $P$. With
+$z\mapsto Pz$, $J\mapsto D$ for diagonal $D$. Then $\partial_i H=d_iz_i$.
+Suppose that two critical points have the same imaginary energy; without loss
+of generality, assume these are the first and second components. Since the gradient is proportional to $z$, any components that are zero at some time will be zero at all times. The dynamics for the components of interest assuming all others are zero are
+\begin{equation}
+  \begin{aligned}
+    \dot z_1
+    &=-z_1^*\left(d_1^*-\frac{d_1^*z_1^*z_1+d_2^*z_2^*z_2}{|z_1|^2+|z_2|^2}\right) \\
+    &=-(d_1-d_2)^*z_1^*\frac{|z_2|^2}{|z_1|^2+|z_2|^2}
+  \end{aligned}
+\end{equation}
+and the same for $z_2$ with all indices swapped.  Since $\Delta=d_1-d_N$ is
+real, if $z_1$ begins real it remains real, with the same for $z_2$. Since the critical points are at real $z$, we make this restriction, and find
+\begin{equation}
+  \begin{aligned}
+    \frac d{dt}(z_1^2+z_2^2)=0\\
+    \frac d{dt}\frac{z_2}{z_1}=\Delta\frac{z_2}{z_1}
+  \end{aligned}
+\end{equation}
+Therefore $z_2/z_1=e^{\Delta t}$, with $z_1^2+z_2^2=N$ as necessary.  Depending on the sign of $\Delta$, $z$ flows
+from one critical point to the other over infinite time. This is a Stokes line,
+and establishes that any two critical points in the 2-spin model with the same
+imaginary energy will possess one.
+
+The critical points of the 2-spin model are all adjacent: no critical point is separated from another by the separatrix of a third. This means that when the imaginary energies of two critical points are brought to the same value, their surfaces of constant imaginary energy join.
+
+\section{p-spin}
+
+\section{(2 + 4)-spin}
+
+\begin{acknowledgments}
+  MIT mathematicians have been no help
+\end{acknowledgments}
+
+\bibliography{stokes}
+
+\end{document}
-- 
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