From 2dcfbc92d8cdc9b174e630d8e30381d5bee49e13 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 31 Jan 2022 15:52:11 +0100
Subject: Some more writing, now on gradient descent.

---
 stokes.tex | 105 +++++++++++++++++++++++++++++++++++++------------------------
 1 file changed, 64 insertions(+), 41 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 6a8c35d..cfec1f2 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -69,6 +69,8 @@ analytic continuation under these conditions?
 
 \section{Thimble integration and analytic continuation}
 
+\subsection{Decomposition of the partition function into thimbles}
+
 Consider an action $\mathcal S$ defined on the (real) phase space $\Omega$. A
 typical calculation stems from the partition function
 \begin{equation} \label{eq:partition.function}
@@ -94,7 +96,7 @@ phase space $\tilde\Omega$ will be $\mathbb C^N$, while for the sphere
 $S^{N-1}$ it takes a little more effort. $S^{N-1}$ can be defined by all points
 $x\in\mathbb R^N$ such that $x^Tx=1$. A complex extension of the sphere is made
 by extending this constraint: all points $z\in \mathbb C^N$ such that $z^Tz=1$.
-Both cases are complex manifolds and moreover K\"ahler manifolds, since they
+Both cases are complex manifolds and moreover Kähler manifolds, since they
 are defined by holomorphic constraints, and therefore admit a hermitian
 metric and a symplectic structure. In the extended complex phase space, the
 action potentially has more stationary points. We'll call $\Sigma$ the set of
@@ -119,10 +121,10 @@ in the introduction for performing analytic continuation in the first place: we
 want our partition function to be well-defined, e.g., for the phase space
 integral to converge, and we want to avoid oscillations in the phase of the
 integrand. The first condition, convergence, necessitates that the real part of
-the action $\operatorname{Re}\mathcal S$ be bounded from below, and that it
+the action $\operatorname{Re}\beta\mathcal S$ be bounded from below, and that it
 approach infinity in any limiting direction along the contour. The second,
 constant phase, necessitates that the imaginary part of the action
-$\operatorname{Im}\mathcal S$ be constant.
+$\operatorname{Im}\beta\mathcal S$ be constant.
 
 Remarkably, there is an elegant recipe for accomplishing both these criteria at
 once, courtesy of Morse theory. For a more thorough review, see
@@ -131,11 +133,11 @@ of a collection of pieces called \emph{Lefschetz thimbles}, or just thimbles.
 There is one thimble $\mathcal J_\sigma$ associated with each of the stationary
 points $\sigma\in\Sigma$ of the action, and it is defined by all points that
 approach the stationary point $z_\sigma$ under gradient descent on
-$\operatorname{Re}\mathcal S$.
+$\operatorname{Re}\beta\mathcal S$.
 
 Thimbles guarantee convergent integrals by construction: the value of
-$\operatorname{Re}\mathcal S$ is bounded from below on the thimble $\mathcal J_\sigma$ by its value
-$\operatorname{Re}\mathcal S(z_\sigma)$ at the stationary point,
+$\operatorname{Re}\beta\mathcal S$ is bounded from below on the thimble $\mathcal J_\sigma$ by its value
+$\operatorname{Re}\beta\mathcal S(z_\sigma)$ at the stationary point,
 since all other points on the thimble must descend to reach it. And, as we will
 see in a moment, thimbles guarantee constant phase for the integrand as well, a
 result of the underlying complex geometry of the problem.
@@ -148,7 +150,7 @@ take one continuously from left to right, perhaps with detours to well-behaved
 places at infinity (see Fig.~\ref{fig:1d.thimble}). The less simply stated versions follows.
 
 Let $\tilde\Omega_T$ be the set of all points $z\in\tilde\Omega$ such that
-$\operatorname{Re}\mathcal S(z)\geq T$, where we will take $T$ to be a very,
+$\operatorname{Re}\beta\mathcal S(z)\geq T$, where we will take $T$ to be a very,
 very large number. $\tilde\Omega_T$ is then the parts of the manifold where it
 is safe for any contour to end up if it wants its integral to converge, since
 these are the places where the real part of the action is very large and the
@@ -166,14 +168,15 @@ And, thankfully for us, Morse theory on our complex manifold $\tilde\Omega$
 implies that the set of all thimbles produces a basis for this relative
 homology group, and therefore any contour can be represented by some
 composition of thimbles! There is even a systematic way to determine the
-contribution from each thimble: for the critical point $\sigma\in\Sigma$, let
+contribution from each thimble: for the stationary point $\sigma\in\Sigma$, let
 $\mathcal K_\sigma$ be its \emph{antithimble}, defined by all points brought to
 $z_\sigma$ by gradient \emph{ascent} (and representing an element of the
 relative homology group $H_N(\tilde\Omega,\tilde\Omega_{-T})$). Then each
 thimble $\mathcal J_\sigma$ contributes to the contour with a weight given by
 its intersection pairing $n_\sigma=\langle\mathcal C,\mathcal K_\sigma\rangle$.
 
-With these tools in hands, we can finally write the partition function as a sum over contributions over thimbles, or
+With these tools in hands, we can finally write the partition function as a sum
+over contributions from each thimble, or
 \begin{equation} \label{eq:thimble.integral}
   Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\oint_{\mathcal J_\sigma}dz\,e^{-\beta\mathcal S(z)}.
 \end{equation}
@@ -182,65 +185,85 @@ generically persists. When the relative homology of the thimbles is unchanged
 by the continuation, the integer weights are likewise unchanged, and one can
 therefore use the knowledge of these weights in one regime to compute the
 partition function in the other. However, their relative homology can change,
-and when this happens the integer weights can be traded between critical
+and when this happens the integer weights can be traded between stationary
 points. These trades occur when two thimbles intersect, or alternatively when
 one stationary point lies in the gradient descent of another. These places are
 called \emph{Stokes points}, and the gradient descent trajectories that join
-two stationary points are called \emph{Stokes lines}.
+two stationary points are called \emph{Stokes lines}. An example of this
+behavior can be seen in Fig.~\ref{fig:1d.stokes}.
 
 The prevalence (or not) of Stokes points in a given continuation, and whether
 those that do appear affect the weights of critical points of interest, is a
 concern for the analytic continuation of theories. If they do not occur or
 occur order-one times, one could reasonably hope to perform such a procedure.
-If they occur exponentially often, there is little hope of keeping track of the
-resulting weights.
+If they occur exponentially often in the system size, there is little hope of
+keeping track of the resulting weights, and analytic continuation is intractable.
 
-\section{Gradient descent dynamics}
+\subsection{Gradient flow and the structure of thimbles}
 
 The `dynamics' describing thimbles is defined by gradient descent on the real
-part of the action.
+part of the action, with a given thimble incorporating all trajectories which
+asymptotically flow to its associated stationary point. Since our phase space
+is not necessary flat (as for the \emph{spherical} $p$-spin models), we will
+have to do a bit of differential geometry to work out their form. Gradient
+descent on a complex (Kähler) manifold is given by
 \begin{equation} \label{eq:flow.coordinate.free}
   \dot s
-  =-\operatorname{grad}\operatorname{Re}\mathcal S
-  =-\left(\frac\partial{\partial s^*}\operatorname{Re}\mathcal S\right)^\sharp
-  =-\frac12\frac{\partial\mathcal S^*}{\partial s^*}g^{-1}\frac\partial{\partial s},
+  =-\operatorname{grad}\operatorname{Re}\beta\mathcal S
+  =-\left(\frac\partial{\partial s^*}\operatorname{Re}\beta\mathcal S\right)^\sharp
+  =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial s^*}g^{-1}\frac\partial{\partial s}
 \end{equation}
 where $g$ is the metric and the holomorphicity of the action was used to set
-$\partial^*\mathcal S=0$.
-
-We will be dealing with actions where it is convenient to refer to coordinates
-in a higher-dimensional embedding space. Let $z:\tilde\Omega\to\mathbb C^N$ be
-an embedding of phase space into complex euclidean space. This gives
+$\partial\mathcal S/\partial s^*=0$. If the complex phase space is $\mathbb C^N$ and the
+metric is diagonal, this means that the flow is proportional to the conjugate
+of the gradient, or $\dot s\propto-\beta^*(\partial S/\partial s)^*$.
+
+In the cases we will consider here (namely, that of the spherical models), it
+will be more convenient to work in terms of coordinates in a flat embedding
+space than in terms of local coordinates in the curved space, e.g., in terms of
+$z\in\mathbb C^N$ instead of $s\in S^{N-1}$. Let $z:\tilde\Omega\to\mathbb C^N$
+be an embedding of complex phase space into complex euclidean space. The
+dynamics in the embedding space is given by
 \begin{equation}\label{eq:flow.raw}
   \dot z
-  =-\frac12\frac{\partial\mathcal S^*}{\partial z^*}(Dz)^* g^{-1}(Dz)^T\frac\partial{\partial z}
+  =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial z^*}(Dz)^* g^{-1}(Dz)^T\frac\partial{\partial z}
 \end{equation}
 where $Dz=\partial z/\partial s$ is the Jacobian of the embedding.
 The embedding induces a metric on $\tilde\Omega$ by $g=(Dz)^\dagger Dz$.
 Writing $\partial=\partial/\partial z$, this gives
 \begin{equation} \label{eq:flow}
-  \dot z=-\frac12(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T
+  \dot z=-\frac{\beta^*}2(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T
   =-\frac12(\partial \mathcal S)^\dagger P
 \end{equation}
 which is nothing but the projection of $(\partial\mathcal S)^*$ into the
-tangent space of the manifold, with $P=(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T$.
-Note that $P$ is hermitian.
-
-Gradient descent on $\operatorname{Re}\mathcal S$ is equivalent to Hamiltonian
-dynamics with the Hamiltonian $\operatorname{Im}\mathcal S$. This is because
-$(\tilde\Omega, g)$ is Kähler and therefore admits a symplectic structure, but
-that the flow conserves $\operatorname{Im}\mathcal S$ can be shown using
-\eref{eq:flow} and the holomorphic property of $\mathcal S$:
+tangent space of the manifold, with the projection operator
+$P=(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T$. Note that $P$ is hermitian. For the spherical models, where $\tilde\Omega$ is the complex phase spaced defined by all points $z\in\mathbb C^N$ such that $z^Tz=1$, the projection operator is given by
+\begin{equation}
+  P=I-\frac{zz^\dagger}{|z|^2}
+\end{equation}
+something that we be worked out in detail in a following section. One can
+quickly verify that this operator indeed projects the dynamics onto the
+manifold: its tangent at any point $z$ is given by $\partial(z^Tz)=z$, and
+$Pz=z-z|z|^2/|z|^2=0$. For any vector $u$ perpendicular to $z$, i.e.,
+$z^\dagger u=0$, $Pu=u$.
+
+Gradient descent on $\operatorname{Re}\beta\mathcal S$ is equivalent to
+Hamiltonian dynamics with the Hamiltonian $\operatorname{Im}\beta\mathcal S$
+and conjugate coordinates given by the real and imaginary parts of each complex
+coordinate. This is because $(\tilde\Omega, g)$ is Kähler and therefore admits
+a symplectic structure, but that the flow conserves
+$\operatorname{Im}\beta\mathcal S$ can be shown using \eref{eq:flow} and the
+holomorphic property of $\mathcal S$:
 \begin{eqnarray}
-  \frac d{dt}\operatorname{Im}\mathcal S
-  &=\dot z\partial\operatorname{Im}\mathcal S+\dot z^*\partial^*\operatorname{Im}\mathcal S \\
+  \frac d{dt}\operatorname{Im}\beta\mathcal S
+  &=\dot z\partial\operatorname{Im}\beta\mathcal S+\dot z^*\partial^*\operatorname{Im}\beta\mathcal S \\
   &=\frac i4\left(
-    (\partial \mathcal S)^\dagger P\partial\mathcal S-(\partial\mathcal S)^TP^*(\partial\mathcal S)^*
+    (\beta\partial \mathcal S)^\dagger P\beta\partial\mathcal S-(\beta\partial\mathcal S)^TP^*(\beta\partial\mathcal S)^*
   \right) \\
-  &=\frac i4\left(
+  &=\frac{i|\beta|^2}4\left(
     (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^*
   \right) \\
-  &=\frac i4\left(
+  &=\frac{i|\beta|^2}4\left(
     \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2
   \right)=0.
 \end{eqnarray}
@@ -251,11 +274,11 @@ form:
 \begin{equation}
   \dot{\mathcal S}
   =\dot z\partial\mathcal S
-  =-\frac12(\partial\mathcal S)^\dagger P\partial\mathcal S
-  =-\frac12\|\partial\mathcal S\|^2.
+  =-\frac{\beta^*}2(\partial\mathcal S)^\dagger P\partial\mathcal S
+  =-\frac{\beta^*}2\|\partial\mathcal S\|^2.
 \end{equation}
 In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in
-the negative real direction.
+a direction set by the argument of $\beta$.
 
 Let us consider the generic case, where the critical points of $\mathcal S$ have
 distinct energies. What is the topology of the $C=\operatorname{Im}\mathcal S$ level
-- 
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