From 4c1e9aea8011aa6185a351a407594ce020d63ac7 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 11 Mar 2021 12:26:28 +0100
Subject: Lots of writing, and a figure!

---
 stokes.tex | 164 ++++++++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 120 insertions(+), 44 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 7dab4da..41878b7 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -62,55 +62,38 @@ The surface $M\subset\mathbb C^N$ defined by $N=f(z)=z^2$ is an $N-1$ dimensiona
 holomorphic function \cite{Forstneric_2017_Stein}. One can define a Hermitian
 metric $h$ on $M$ by taking the restriction of the standard metric of $\mathbb
 C^N$ to vectors tangent along $M$. For any smooth function $\phi:M\to\mathbb
-R$, its gradient $\nabla\phi$ is a holomorphic vector field given by
+R$, its gradient is a holomorphic vector field given by
 \begin{equation}
-  \nabla^\alpha\phi=h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\phi
+  \operatorname{grad}\phi=\bar\partial^\sharp\phi
 \end{equation}
-Dynamics consists of gradient descent on $\operatorname{Re}H$, or
-\begin{equation} \label{eq:flow}
-  \dot u^\alpha=-\nabla^\alpha\operatorname{Re}H=-\tfrac12h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\bar H
+Suppose $u:M\to\mathbb C^{N-1}$ is a coordinate system. Then
+\begin{equation}
+  \operatorname{grad}\phi=h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\phi\frac{\partial}{\partial u^\alpha}
 \end{equation}
-Gradient descent on $\operatorname{Re}H$ is equivalent to Hamiltonian dynamics
-with the Hamiltonian $\operatorname{Im}H$. This is because $(M, h)$ is Kähler
-and therefore admits a symplectic structure, but that the flow conserves
-$\operatorname{Im}H$ can be seen from the Cauchy--Riemann equations and
-\eqref{eq:flow}:
+Let $z=u^{-1}$.
 \begin{equation}
-  \begin{aligned}
-    \frac d{dt}\operatorname{Im}H
-    &=\dot u^\alpha\partial_\alpha\operatorname{Im}H+\dot{\bar u}^{\bar\alpha}\bar\partial_{\bar\alpha}\operatorname{Im}H \\
-    &=\tfrac i4\left(\bar\partial_{\bar\beta}\bar Hh^{\bar\beta\alpha}\partial_\alpha H-\partial_\beta H\bar h^{\beta\bar\alpha}\bar\partial_{\bar\alpha}\bar H  \right)\\
-    &=0
-  \end{aligned}
+  \frac\partial{\partial u^\alpha}=\frac{\partial z^i}{\partial u^\alpha}\frac\partial{\partial z^i}
 \end{equation}
-since $h$ is a Hermitian operator with $\bar h=h^T$.
-
-Working with a particular map is inconvenient, and we would like to develop a
-map-independent dynamics. Suppose that $z:\mathbb C^{N-1}\to M$ is a map. Using
-the chain rule, one finds
 \begin{equation}
-  \begin{aligned}
-    \dot z^i
-    &=\dot u^\alpha\partial_\alpha z^i
-    =-\tfrac12\bar\partial_{\bar\beta}\bar Hh^{\bar\beta\alpha}\partial_\alpha z^i
-    =-\tfrac12\bar\partial_j\bar H\partial_{\bar\beta}\bar z^{\bar\jmath}h^{\bar\beta\alpha}\partial_\alpha z^i \\
-    &=-\tfrac12(J^\dagger h^{-1}J)^{\bar\jmath i}\bar\partial_{\bar\jmath}\bar H\\
-  \end{aligned}
+  \bar\partial_{\bar\beta}\phi=\frac{\partial\bar z^i}{\partial\bar u^{\bar\beta}}\frac{\partial\phi}{\partial\bar z^i}
+\end{equation}
+\begin{equation}
+  \operatorname{grad}\phi
+  =\frac{\partial\bar z^{\bar\jmath}}{\partial\bar u^{\bar\beta}}h^{\bar\beta\alpha}\frac{\partial z^i}{\partial u^\alpha}\frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i}
 \end{equation}
-where $J$ is the Jacobian of the coordinate map and $h$ is the Hermitian
-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
+At any point $z_0\in M$, $z_0$ is parallel to the normal to $M$. Without loss of generality, take $z_0=e^N$. In a neighborhood of $z_0$, the map $u^\alpha=z^\alpha$ is a coordinate system.
+its inverse is $z^i=u^i$ for $1\leq i\leq N-1$ and
 \begin{equation}
-  z^N=\sqrt{N-u^2}
+  z^N=\sqrt{N-u^2}.
 \end{equation}
 The Jacobian is
 \begin{equation}
-  J_\alpha^{\hphantom\alpha i}=\partial_\alpha z^i=\delta_\alpha^{\hphantom\alpha i}-\delta_N^{\hphantom Ni}\frac{u_\beta}{\sqrt{N-u^2}}
+  \frac{\partial z^i}{\partial u^\alpha}=\delta^i_\alpha-\delta_N^{i}\frac{u^\alpha}{\sqrt{N-u^2}}
 \end{equation}
 and therefore the Hermitian metric induced by the map is
 \begin{equation}
-  h_{\alpha\bar\beta}=\bar J_{\alpha}^{\hphantom\alpha i}J_{\bar\beta}^{\hphantom\beta\bar\jmath}\delta_{i\bar\jmath}
-  =\delta_{\bar\alpha\beta}+\frac{\bar u^{\alpha}u^{\bar\beta}}{|N-u^2|}
+  h_{\alpha\bar\beta}=\frac{\partial\bar z^i}{\partial\bar u^\alpha}\frac{\partial z^{\bar\jmath}}{\partial u^{\bar\beta}}\delta_{i\bar\jmath}
+  =\delta_{\alpha\bar\beta}+\frac{\bar u^{\alpha}u^{\bar\beta}}{|N-u^2|}
 \end{equation}
 The metric can be inverted explicitly:
 \begin{equation}
@@ -119,26 +102,119 @@ The metric can be inverted explicitly:
 \end{equation}
 Putting these pieces together, we find
 \begin{equation}
-  (J^\dagger h^{-1}J)^{\bar\jmath i}
+  \frac{\partial\bar z^{\bar\jmath}}{\partial\bar u^{\bar\beta}}h^{\bar\beta\alpha}\frac{\partial z^i}{\partial u^\alpha}
   =\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2}
 \end{equation}
-which is just the projector onto the constraint manifold \cite{Morrow_2006_Complex}.
-Therefore, a map-independent dynamics for $z\in M$ is given by
 \begin{equation}
-  \dot z
+  \operatorname{grad}\phi
+  =\left(\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2}\right)
+  \frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i}
+\end{equation}
+
+\section{Dynamics}
+
+For a holomorphic Hamiltonian $H$, dynamics are defined by gradient descent on $\operatorname{Re}H$, or
+\begin{equation} \label{eq:flow}
+  \dot z=-\operatorname{grad}\operatorname{Re}H
   =-\frac12(\partial H)^\dagger\left(I-\frac{zz^\dagger}{|z|^2}\right)
+  =-\tfrac12(\partial H)^\dagger P,
+\end{equation}
+where $P=I-\hat z\hat z^\dagger$ is the projection matrix on to the tangent space of $M$.
+Gradient descent on $\operatorname{Re}H$ is equivalent to Hamiltonian dynamics
+with the Hamiltonian $\operatorname{Im}H$. This is because $(M, h)$ is Kähler
+and therefore admits a symplectic structure, but that the flow conserves
+$\operatorname{Im}H$ can be seen from the Cauchy--Riemann equations and
+\eqref{eq:flow}:
+\begin{equation}
+  \begin{aligned}
+    \frac d{dt}\operatorname{Im}H
+    &=\dot z_i\partial_i\operatorname{Im}H+\dot{\bar z}_i\bar\partial_i\operatorname{Im}H \\
+    &=\frac i4\left(
+      \bar\partial_j\bar HP_{ji}\partial_i H-\partial_j H\bar P_{ji}\bar\partial_i\bar H
+    \right)
+    =0
+  \end{aligned}
+\end{equation}
+since $P$ is a Hermitian operator. This conservation law indicates that surfaces of constant $\operatorname{Im}H$ will be important when evaluting the possible endpoints of dynamic trajectories.
+
+Let us consider the generic case, where the critical points of $H$ have
+distinct energies. What is the topology of the $C=\operatorname{Im}H$ level
+set? We shall argue its form by construction. Consider initially the situation
+in the absence of any critical point. In this case the level set consists of a
+single simply connected surface, locally diffeomorphic to $\mathbb R^{2(N-1)-1}$. Now, `place' a generic
+(nondegenerate) critical point in the function at $z_0$. In the vicinity of the critical
+point, the flow is locally
+\begin{equation}
+  \begin{aligned}
+    \dot z_i
+    &\simeq-\frac12\left[
+      \partial_j\left(
+        H(z_0)+\frac12\partial_k\partial_\ell H(z_0)\Delta z_k\Delta z_\ell
+      \right)
+    \right]^* P_{ji} \\
+    &=-\frac12\left(
+      \partial_j\partial_kH(z_0)\Delta z_k
+    \right)^* P_{ji} \\
+    &=-\frac12\Delta z_k^*(\partial_k\partial_jH(z_0))^*P_{ji}
+  \end{aligned}
+\end{equation}
+The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of
+$\partial\partial H$ save a single marginal direction corresponding to $z_0$,
+the normal to the constraint surface. Assuming we are working in a diagonal basis, we find
+\begin{equation}
+  \dot z_i=-\frac12\lambda_i\Delta z_i^*+O(\Delta z^2)
 \end{equation}
+Breaking into real and imaginary parts gives
+\begin{equation}
+  \begin{aligned}
+    \frac{d\Delta x_i}{dt}&=-\frac12\left(
+      \operatorname{Re}\lambda_i\Delta x_i+\operatorname{Im}\lambda_i\Delta y_i
+    \right) \\
+    \frac{d\Delta y_i}{dt}&=-\frac12\left(
+      \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\lambda_i\Delta y_i
+    \right)
+  \end{aligned}
+\end{equation}
+Therefore, in the complex plane defined by each eigenvector of $(\partial\partial H)^\dagger P$ there is a separatrix flow of the form in Figure \ref{fig:local_flow}. The effect of these separatrices in each complex direction of the tangent space $T_{z_0}M$ is to separate that space into four quadrants: two disconnected pieces with greater imaginary part than the critical point, and two with lesser imaginary part. This partitioning implies that the level set of $\operatorname{Im} H=C$ for $C\neq\operatorname{Im}H(z_0)$ is splint into two disconnected pieces, one lying in each of two quadrants corresponding with its value relative to that at the critical point.
+
+\begin{figure}
+  \includegraphics[width=\columnwidth]{figs/local_flow.pdf}
+  \caption{
+    Gradient descent in the vicinity of a critical point, in the $z$--$z*$
+    plane for an eigenvector $z$ of $(\partial\partial H)^\dagger P$. The flow
+    lines are colored by the value of $\operatorname{Im}H$.
+  } \label{fig:local_flow}
+\end{figure}
+
+Continuing to `insert' critical points whose imaginary energy differs from $C$,
+one repeatedly partitions the space this way with each insertion. Therefore,
+for the generic case with $\mathcal N$ critical points, with $C$ differing in
+value from all critical points, the level set $\operatorname{Im}H=C$ has
+$\mathcal N+1$ connected components, each of which is simply connected,
+diffeomorphic to $\mathbb R^{2(N-1)-1}$ and connects two sectors of infinity to
+each other.
+
+When $C$ is brought to the same value as the imaginary part of some critical
+point, two of these disconnected surfaces pinch grow nearer and pinch together
+at the critical point when $C=\operatorname{Im}H$, as in the black lines of
+Figure \ref{fig:local_flow}. The unstable manifold of the critical point, which
+corresponds with the portion of this surface that flows away, is known as a
+\emph{Lefshetz thimble}.
 
 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.
+must have the same imaginary energy if they are to be connected by a Stokes
+line. This is not a generic phenomena, but will happen often as one model
+parameter is continuously varied. When two critical points do have the same
+imaginary energy and $C$ is brought to that value, the level set
+$C=\operatorname{Im}H$ sees formally disconnected surfaces pinch together in
+two places. We shall argue that generically, a Stokes line will exist whenever
+the two critical points in question lie on the same connected piece of this
+surface.
 
+What are the ramifications of this for disordered Hamiltonians? When some process brings two critical points to the same imaginary energy, whether a Stokes line connects them depends on whether the points are separated from each other by the separatrices of one or more intervening critical points. Therefore, we expect that in regions where critical points with the same value of $\operatorname{Im}H$ tend to be nearby, Stokes lines will proliferate, while in regions where critical points with the same value of $\operatorname{Im}H$ tend to be distant compared to those with different $\operatorname{Im}H$, Stokes lines will be rare.
 
 
 \section{2-spin}
-- 
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