From 6539131ba2e5652f6cb70f83a9144eb0b99254a6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 11 Nov 2021 14:50:08 +0100
Subject: Writing.

---
 stokes.tex | 300 +++++++++++++++++++++++++++++++++----------------------------
 1 file changed, 163 insertions(+), 137 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index af66e3a..91f7f18 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -2,7 +2,6 @@
 
 \usepackage[utf8]{inputenc} % why not type "Stokes" with unicode?
 \usepackage[T1]{fontenc} % vector fonts
-\usepackage{newtxtext,newtxmath} % Times for PR
 \usepackage[
   colorlinks=true,
   urlcolor=purple,
@@ -10,7 +9,8 @@
   filecolor=purple,
   linkcolor=purple
 ]{hyperref} % ref and cite links with pretty colors
-\usepackage{amsmath, graphicx, xcolor} % standard packages
+\usepackage{amsmath, amssymb, graphicx, xcolor} % standard packages
+\usepackage{newtxtext,newtxmath} % Times for PR
 \usepackage[subfolder]{gnuplottex} % need to compile separately for APS
 
 \begin{document}
@@ -44,31 +44,38 @@
 
 \maketitle
 
-
-Consider a thermodynamic calculation involving the (real) $p$-spin model for a
-particular instantiation of the coupling tensor $J$
+Consider an action $\mathcal S_\lambda$ defined on the phase space $\Omega$ and
+depending on parameters $\lambda$. In the context of statistical mechanics,
+$\mathcal S_{\beta,J}=-\beta H_J$ for some hamiltonian $H_J$ with quenched
+parameters $J$ at inverse temperature $\beta$. A typical calculation stems from
+the partition function
 \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 $J$. 
-
-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.
+  Z(\lambda)=\int_\Omega ds\,e^{\mathcal S_\lambda(s)}.
+\end{equation}
+This integral is often dominated by its behavior near stationary points of the
+action, and understanding these points is usually important to evaluate the
+partition function.
+
+Recent developments have found that stationary points of the action are
+important for understanding another aspect of the partition function: its
+analytic continuation. The integral \eqref{eq:partition.function} is first
+interpreted as a contour in a larger complex phase space, then deformed into a
+linear combination of specially constructed contours each enumerated by a
+stationary point. Analytic continuation of parameters preserves this
+decomposition except at nongeneric points where contours intersect.
+
+We investigate the plausibility of analytic continuation in complex models
+where the action has a macroscopic number of stationary points. Such actions
+are common in studies of glasses, spin glass, machine learning, black holes,
+\dots We find that the geometry of the landscape, and in particular the
+relative position and spectrum of stationary points, is key.
+
+Analytic continuation of partition functions is useful for many reasons. Some
+physically motivated theories have actions whose partition function is formally
+infinite, but can be defined by continuing from a set of parameters where it
+converges. Other theories have oscillatory actions that lead to a severe sign
+problem in estimating the partition function, which can be addressed by taking
+advantage of a deformed phase space where the phase of the action varies slowly.
 
 Unfortunately the study is not so relevant for low-dimensional `rugged'
 landscapes, which are typically constructed from the limits of series or
@@ -77,56 +84,51 @@ integrals of analytic functions which are not themselves analytic
 
 \section{Integration by Lefschetz thimble}
 
-Consider an $N$-dimensional hermitian manifold $M$ and a Hamiltonian $H:M\to\mathbb C$. The partition function
-\begin{equation}
-  Z(\beta)=\int_S du\,e^{-\beta H(u)}
-\end{equation}
-for $S$ a submanifold (not necessarily complex) of $M$. For instance, the
-$p$-spin spherical model can be defined on the complex space $M=\{z\mid
-z^2=N\}$, but typically one is interested in the subspace $S=\{z\mid
-z^2=N,z\in\mathbb R\}$.
-
-If $S$ is orientable, then the integral can be converted to one over a contour
-corresponding to $S$. In this case, the contour can be freely deformed without
-affecting the value of the integral. Two properties of this deformed contour
-would be ideal. First, that as $|u|\to\infty$ the real part of $-\beta H(u)$
-goes to $-\infty$. This ensures that the integral is well defined. Second, that
-the contours piecewise correspond to surfaces of constant phase of $-\beta H$,
-so as to ameliorate sign problems.
-
-Remarkably, there is a recipe for accomplishing both these criteria at once,
-courtesy of Morse theory. For a more thorough review, see
-\citet{Witten_2011_Analytic}. Consider a critical point of $H$. The union of
-all gradient descent trajectories on the real part of $-\beta H$ that terminate
-at the critical point as $t\to-\infty$ is known as the \emph{Lefschetz thimble}
-corresponding with that critical point. Since each point on the Lefschetz
-thimble is a descent from a critical point, the value of
-$\operatorname{Re}(-\beta H)$ is bounded from above by its value at the
-critical point. Likewise, we shall see that the imaginary part of $\beta H$ is
-preserved under gradient descent on its real part.
-
-Morse theory provides the universal correspondence between contours and
-thimbles: one must produce an integer-weighted linear combination of thimbles
-such that the homology of the combination is equivalent to that of the contour.
-If $\Sigma$ is the set of critical points and for each $\sigma\in\Sigma$
-$\mathcal J_\sigma$ is its Lefschetz thimble, then this gives
+We return to the partition function \eqref{eq:partition.function}.  If
+the action can be continued to a holomorphic function on the Kähler
+manifold $\tilde\Omega\supset\Omega$ and $\Omega$ is orientable in $\tilde\Omega$,
+then \eqref{eq:partition.function} can be considered a contour integral. In
+this case, the contour can be freely deformed without affecting the value of
+the integral. Two properties of this deformed contour would be ideal. First,
+that as $|s|\to\infty$ the real part of the action goes to $-\infty$, to ensure
+the integral converges. Second, that the contours piecewise correspond to
+surfaces of slowing vary phase of the action, so as to ameliorate sign
+problems.
+
+Remarkably, there is an elegant recipe for accomplishing both these criteria at
+once, courtesy of Morse theory. For a more thorough review, see
+\citet{Witten_2011_Analytic}. Consider a stationary point of the action. The
+union of all gradient descent trajectories on the real part of the action that
+begin at the stationary point is known as a \emph{Lefschetz thimble}. Since
+each point on the Lefschetz thimble is found through descent from the
+stationary point, the real part of the action is bounded from above by its
+value at the stationary point. Likewise, we shall see that the imaginary part
+of the action is constant on a thimble.
+
+Morse theory provides a universal correspondence between contours and thimbles.
+For any contour $\Omega$, there exists a linear combination of thimbles such
+that the relative homology of the combination with respect to decent int he
+action is equivalent to that of the contour.  If $\Sigma$ is the set of
+stationary points of the action and for each $\sigma\in\Sigma$ the set
+$\mathcal J_\sigma\subset\tilde\Omega$ is its thimble, then this gives
 \begin{equation} \label{eq:thimble.integral}
-  Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\int_{\mathcal J_\sigma}du\,e^{-\beta H(u)}
+  Z(\lambda)=\sum_{\sigma\in\Sigma}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{\mathcal S_\lambda(s)}.
 \end{equation}
 Each of these integrals is very well-behaved: convergent asymptotic series
-exist for their value about the critical point $\sigma$, for example. One must
-know the integer weights $n_\sigma$.
-
-Under analytic continuation of, say, $\beta$, the form of
-\eqref{eq:thimble.integral} persists. When the 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 other. However, their homology can change, and when
-this happens the integer weights can be traded between critical points. These
-trades occur when two thimbles intersect, or alternatively when one critical
-point lies on the gradient descent of another. These places are called
-\emph{Stokes points}, and the gradient descent trajectories that join two
-critical points are called \emph{Stokes lines}.
+exist for their value about each critical point. The integer weights $n_\sigma$
+are fixed by comparison with the initial contour. For a real action, all maxima
+in $\Omega$ contribute in equal magnitude.
+
+Under analytic continuation, the form of \eqref{eq:thimble.integral}
+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
+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}.
 
 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
@@ -137,105 +139,114 @@ resulting weights.
 
 \section{Gradient descent dynamics}
 
-For a holomorphic Hamiltonian $H$, dynamics are defined by gradient descent on
-$\operatorname{Re}\beta H$. In hermitian geometry, the gradient is given by raising
-an index of the conjugate differential, or
-$\operatorname{grad}f=(\partial^*f)^\sharp$. This implies that, in terms of
-coordinates $u:M\to\mathbb C^N$, gradient descent follows the dynamics
-\begin{equation} \label{eq:flow.raw}
-  \dot z^i
-  =-(\partial^*_{\gamma}\operatorname{Re}\beta H)h^{\gamma\alpha}\partial_\alpha z^i
-  =-\tfrac12(\beta\partial_\gamma H)^*h^{\gamma\alpha}\partial_\alpha z^i
-\end{equation}
-where $h$ is the Hermitian metric and we have used the fact that, for holomorphic $H$, $\partial^*H=0$.
-
-This can be simplied  by noting that $h^{\gamma\alpha}=h^{-1}_{\gamma\alpha}$ for
-$h_{\gamma\alpha}=(\partial_\gamma z^i)^*\partial_\alpha z^i=(J^\dagger
-J)_{\gamma\alpha}$ where $J$ is the Jacobian of the coordinate map. Writing
-$\partial H=\partial H/\partial z$ and inserting Jacobians everywhere they
-appear, \eqref{eq:flow.raw} becomes
+The `dynamics' describing thimbles is defined by gradient descent on the real
+part of the action.
+\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},
+\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
+\begin{equation}\label{eq:flow.raw}
+  \dot z
+  =-\frac12\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=-\tfrac12(\beta\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T
-  =-\tfrac12\beta^*(\partial H)^\dagger P
-\end{equation}
-which is nothing but the projection of $(\partial H)^*$ into the tangent space of the manifold, with $P=J^*[J^\dagger J]^{-1}J^T$. Note that $P$ is hermitian: $P^\dagger=(J^*[J^\dagger J]^{-1}J^T)^\dagger=J^*[J^\dagger J]^{-1}J^T=P$.
-
-Gradient descent on $\operatorname{Re}\beta H$ is equivalent to Hamiltonian dynamics
-with the Hamiltonian $\operatorname{Im}\beta H$. This is because $(M, h)$ is Kähler
-and therefore admits a symplectic structure, but that the flow conserves
-$\operatorname{Im}\beta H$ can be shown using \eqref{eq:flow} and the holomorphic property of $H$:
+  \dot z=-\tfrac12(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T
+  =-\tfrac12(\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
+\eqref{eq:flow} and the holomorphic property of $\mathcal S$:
 \begin{equation}
   \begin{aligned}
-    \frac d{dt}&\operatorname{Im}\beta H
-    =\dot z\partial\operatorname{Im}\beta H+\dot z^*\partial^*\operatorname{Im}\beta H \\
+    \frac d{dt}&\operatorname{Im}\mathcal S
+    =\dot z\partial\operatorname{Im}\mathcal S+\dot z^*\partial^*\operatorname{Im}\mathcal S \\
     &=\frac i4\left(
-      \beta^*\beta(\partial H)^\dagger P\partial H-\beta\beta^*(\partial H)^TP^\dagger(\partial H)^*
+      (\partial \mathcal S)^\dagger P\partial\mathcal S-(\partial\mathcal S)^TP^*(\partial\mathcal S)^*
     \right) \\
-    &=\frac i4|\beta|^2\left(
-      (\partial H)^\dagger P\partial H-[(\partial H)^\dagger P\partial H]^*
+    &=\frac i4\left(
+      (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^*
     \right) \\
-    &=\frac i4|\beta|^2\left(
-      \|\partial H\|^2-(\|\partial H\|^*)^2
+    &=\frac i4\left(
+      \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2
     \right)=0.
   \end{aligned}
 \end{equation}
-As a result of this conservation law, surfaces of constant $\operatorname{Im}\beta H$
-will be important when evaluting the possible endpoints of trajectories. A consequence of this conservation is that the flow in the energy takes a simple form:
+As a result of this conservation law, surfaces of constant imaginary action
+will be important when evaluting the possible endpoints of trajectories. A
+consequence of this conservation is that the flow in the action takes a simple
+form:
 \begin{equation}
-  \dot H
-  =\dot z\partial H
-  =-\frac12\beta^*(\partial H)^\dagger P H
-  =-\frac12\beta^*\|\partial H\|^2.
+  \dot{\mathcal S}
+  =\dot z\partial\mathcal S
+  =-\frac12(\partial\mathcal S)^\dagger P\partial\mathcal S
+  =-\frac12\|\partial\mathcal S\|^2.
 \end{equation}
-In the complex-$H$ plane, dynamics is occurs along straight lines whose
-direction is the same as $\arg \beta^*$.
+In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in
+the negative real direction.
 
-Let us consider the generic case, where the critical points of $H$ have
-distinct energies. What is the topology of the $C=\operatorname{Im}\beta H$ level
+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
 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 $\mathbb R^{2N}$. Now, `place' a generic
-(nondegenerate) critical point in the function at $u_0$. In the vicinity of the critical
+single simply connected surface, locally diffeomorphic to $\mathbb R^{2D-1}$. 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
-    &\simeq-\tfrac12\beta^*(\partial\partial H)^\dagger P(z-z_0)^*
+    &\simeq-\tfrac12(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^*
   \end{aligned}
 \end{equation}
-The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of
-$(\partial\partial H)^\dagger$ save marginal directions corresponding to the normals to
+The matrix $(\partial\partial\mathcal S)^\dagger P$ has a spectrum identical to that of
+$(\partial\partial\mathcal S)^\dagger$ save marginal directions corresponding to the normals to
 manifold. Assuming we are working in a diagonal basis, this becomes
 \begin{equation}
-  \dot z_i=-\tfrac12\beta^*\lambda_i\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2)
+  \dot z_i=-\tfrac12\lambda_i\Delta z_i^*+O(\Delta z^2,(\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}\beta\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i
+      \operatorname{Re}\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i
     \right) \\
     \frac{d\Delta y_i}{dt}&=-\frac12\left(
-      \operatorname{Im}\beta\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i
+      \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\beta\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
+$(\partial\partial\mathcal S)^\dagger P$ 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 $\operatorname{Im}\beta H=C$ for
-$C\neq\operatorname{Im}\beta H(z_0)$ is split into two disconnected pieces, one
+that the level set of $\operatorname{Im}\mathcal S=C$ for
+$C\neq\operatorname{Im}\mathcal S(z_0)$ is split 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
+    Gradient descent in the vicinity of a critical point, in the $z$--$z^*$
+    plane for an eigenvector $z$ of $(\partial\partial\mathcal S)^\dagger P$. The flow
     lines are colored by the value of $\operatorname{Im}H$.
   } \label{fig:local_flow}
 \end{figure}
@@ -243,14 +254,14 @@ the critical point.
 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 $\operatorname{Im}\beta H=C$ has
+value from all critical points, the level set $\operatorname{Im}\mathcal S=C$ has
 $\mathcal N+1$ connected components, each of which is simply connected,
-diffeomorphic to $\mathbb R^{2N-1}$ and connects two sectors of infinity to
+diffeomorphic to $\mathbb R^{2D-1}$ 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 $C=\operatorname{Im}\beta H$, as in the black lines of
+at the critical point when $C=\operatorname{Im}\mathcal S$, 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, produce the
 Lefschetz thimble associated with that critical point.
@@ -263,7 +274,7 @@ 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
-$C=\operatorname{Im}\beta H$ sees formally disconnected surfaces pinch together in
+$C=\operatorname{Im}\mathcal S$ 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.
@@ -286,15 +297,30 @@ those with different energies, Stokes lines will be rare.
 
 \section{p-spin spherical models}
 
-For $p$-spin spherical models, one is constrained to the manifold $M=\{z\mid
-z^2=N\}$. The normal to this manifold at any point $z\in M$ is always in the
-direction $z$. The projection operator onto the tangent space of this manifold
-is given by
+The $p$-spin spherical models are statistical mechanics models defined by the
+action $\mathcal S=-\beta H$ for the Hamiltonian
+\begin{equation} \label{eq:p-spin.hamiltonian}
+  H(x)=\sum_{p=2}^\infty\frac{a_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p}
+\end{equation}
+where the $x\in\mathbb R^N$ are constrained to lie on the sphere $x^2=N$.  The tensor
+components $J$ are complex normally distributed with zero mean and variances
+$\overline{|J|^2}=p!/2N^{p-1}$ and $\overline{J^2}=\kappa\overline{|J|^2}$, and
+the numbers $a$ define the model. The pure real $p$-spin model has
+$a_i=\delta_{ip}$ and $\kappa=1$.
+
+The phase space manifold $\Omega=\{x\mid x^2=N, x\in\mathbb R^N\}$ has a
+complex extension $\tilde\Omega=\{z\mid z^2=N, z\in\mathbb C^N\}$. The natural
+extension of the hamiltonian \eqref{eq:p-spin.hamiltonian} to this complex manifold is
+holomorphic. The normal to this manifold at any point $z\in\tilde\Omega$ is
+always in the direction $z$.  The projection operator onto the tangent space of
+this manifold is given by
 \begin{equation}
   P=I-\frac{zz^\dagger}{|z|^2},
 \end{equation}
 where indeed $Pz=z-z|z|^2/|z|^2=0$ and $Pz'=z'$ for any $z'$ orthogonal to $z$.
 
+To find critical points, we use the method of Lagrange multipliers. Introducing $\mu\in\mathbb C$, 
+
 \subsection{2-spin}
 
 The Hamiltonian of the $2$-spin model is defined for $z\in\mathbb C^N$ by
-- 
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