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@@ -1,4 +1,4 @@ -\documentclass[aps,reprint,longbibliography,floatfix]{revtex4-2} +\documentclass[]{iopart} \usepackage[utf8]{inputenc} % why not type "Stokes" with unicode? \usepackage[T1]{fontenc} % vector fonts @@ -9,24 +9,21 @@ filecolor=purple, linkcolor=purple ]{hyperref} % ref and cite links with pretty colors -\usepackage{amsmath, amssymb, graphicx, xcolor} % standard packages -\usepackage{newtxtext,newtxmath} % Times for PR +\usepackage{amsopn, amssymb, graphicx, xcolor} % standard packages \usepackage[subfolder]{gnuplottex} % need to compile separately for APS \begin{document} \title{Analytic continuation over complex landscapes} -\author{Jaron Kent-Dobias} -\author{Jorge Kurchan} +\author{Jaron Kent-Dobias and Jorge Kurchan} -\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} +\address{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} +\ead{jaron.dobias@phys.ens.fr} -\date\today - -\begin{abstract} \setcitestyle{authoryear,round} +\begin{abstract} In this paper we follow up the study of `complex complex landscapes' - [\cite{Kent-Dobias_2021_Complex}], rugged landscapes of many complex + \cite{Kent-Dobias_2021_Complex}, rugged landscapes of many complex variables. Unlike real landscapes, there is no useful classification of saddles by index. Instead, the spectrum at critical points determines their tendency to trade topological numbers under analytic continuation of the @@ -58,7 +55,7 @@ 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 +analytic continuation. The integral \eref{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 @@ -84,10 +81,10 @@ integrals of analytic functions which are not themselves analytic \section{Integration by Lefschetz thimble} -We return to the partition function \eqref{eq:partition.function}. If +We return to the partition function \eref{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 +then \eref{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 @@ -97,7 +94,7 @@ 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 +\cite{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 @@ -119,7 +116,7 @@ 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} +Under analytic continuation, the form of \eref{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 @@ -161,8 +158,8 @@ 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(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T - =-\tfrac12(\partial \mathcal S)^\dagger P + \dot z=-\frac12(\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 $P=(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T$. @@ -172,22 +169,20 @@ 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}\mathcal S - =\dot z\partial\operatorname{Im}\mathcal S+\dot z^*\partial^*\operatorname{Im}\mathcal S \\ - &=\frac i4\left( - (\partial \mathcal S)^\dagger P\partial\mathcal S-(\partial\mathcal S)^TP^*(\partial\mathcal S)^* - \right) \\ - &=\frac i4\left( - (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^* - \right) \\ - &=\frac i4\left( - \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2 - \right)=0. - \end{aligned} -\end{equation} +\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 i4\left( + (\partial \mathcal S)^\dagger P\partial\mathcal S-(\partial\mathcal S)^TP^*(\partial\mathcal S)^* + \right) \\ + &=\frac i4\left( + (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^* + \right) \\ + &=\frac i4\left( + \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2 + \right)=0. +\end{eqnarray} 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 @@ -209,28 +204,24 @@ single simply connected surface, locally diffeomorphic to $\mathbb R^{2D-1}$. No (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(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* - \end{aligned} + \dot z + \simeq-\frac12(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* \end{equation} 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\lambda_i\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2) + \dot z_i=-\frac12\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}\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i - \right) \\ - \frac{d\Delta y_i}{dt}&=-\frac12\left( - \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i - \right) - \end{aligned} -\end{equation} +\begin{eqnarray} + \frac{d\Delta x_i}{dt}&=-\frac12\left( + \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}\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i + \right) +\end{eqnarray} Therefore, in the complex plane defined by each eigenvector of $(\partial\partial\mathcal S)^\dagger P$ there is a separatrix flow of the form in Figure \ref{fig:local_flow}. The effect of these separatrices in each complex @@ -243,7 +234,7 @@ 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} + \includegraphics{figs/local_flow.pdf} \caption{ Gradient descent in the vicinity of a critical point, in the $z$--$z^*$ plane for an eigenvector $z$ of $(\partial\partial\mathcal S)^\dagger P$. The flow @@ -288,14 +279,17 @@ energies tend to be nearby, Stokes lines will proliferate, while in regions where critical points with the same energies tend to be distant compared to those with different energies, Stokes lines will be rare. -\textcolor{teal}{ - Here we make a generic argument that, for high-dimensional landscapes with - exponentially many critical points, the existence of exponentially many - Stokes points depends on the spectrum of the Hessian $\partial\partial H$ of - critical points. -} +\section{Analytic continuation} -\section{p-spin spherical models} +\begin{eqnarray} + Z(\beta) + &=\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)} \\ + &=\sum_{\sigma\in\Sigma_0}(-1)^{k_\sigma}\int_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)} \\ + &=\sum_{\sigma\in\Sigma_0}(-1)^{k_\sigma}\left(\frac{2\pi}{\beta}\right)^{N/2}(\det\partial\partial\mathcal S(s_\sigma))^{-N/2}e^{-\beta\mathcal S_(s_\sigma)} \\ + &=\sum_k(-1)^k\int d\epsilon\,\mathcal N_k(\epsilon)\left(\frac{2\pi}{\beta}\right)^{N/2}\exp\left\{-\beta N\epsilon-\frac N2\int_0^\infty d\lambda\,\rho(\lambda\mid\epsilon)\log\lambda\right\} +\end{eqnarray} + +\section{The \textit{p}-spin spherical models} The $p$-spin spherical models are statistical mechanics models defined by the action $\mathcal S=-\beta H$ for the Hamiltonian @@ -310,7 +304,7 @@ $a_i=\delta_{ip}$ 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 +extension of the hamiltonian \eref{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 @@ -319,7 +313,7 @@ this manifold is given by \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 $\mu\in\mathbb C$, +To find critical points, we use the method of Lagrange multipliers. Introducing $\mu\in\mathbb C$, \subsection{2-spin} @@ -347,21 +341,17 @@ of generality, assume these are associated with the first and second cardinal directions. 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} +\begin{eqnarray} + \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{eqnarray} and the same for $z_2$ with all indices swapped. Since $\Delta=d_1-d_2$ 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} + \frac d{dt}(z_1^2+z_2^2)=0 \qquad + \frac d{dt}\frac{z_2}{z_1}=\Delta\frac{z_2}{z_1} \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 @@ -392,19 +382,17 @@ 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. -\begin{equation} - \begin{aligned} - Z(\beta) - &=\int_{S^{N-1}}dx\,e^{-\beta H(x)} - =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\ - &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\ - &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\ - &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\det(\beta J+\lambda I)}}e^{\lambda N} \\ - &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\prod_i(\beta\lambda_i+\lambda)}}e^{\lambda N} \\ - &=(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac12\sum_i\log(\beta\lambda_i+\lambda)} \\ - &\simeq(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac N2\int d\lambda'\,\rho(\lambda')\log(\beta\lambda'+\lambda)} \\ - \end{aligned} -\end{equation} +\begin{eqnarray} + Z(\beta) + &=\int_{S^{N-1}}dx\,e^{-\beta H(x)} + =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\ + &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\ + &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\ + &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\det(\beta J+\lambda I)}}e^{\lambda N} \\ + &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\prod_i(\beta\lambda_i+\lambda)}}e^{\lambda N} \\ + &=(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac12\sum_i\log(\beta\lambda_i+\lambda)} \\ + &\simeq(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac N2\int d\lambda'\,\rho(\lambda')\log(\beta\lambda'+\lambda)} \\ +\end{eqnarray} \subsection{Pure \textit{p}-spin} @@ -467,10 +455,7 @@ $\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ whos -\begin{acknowledgments} - MIT mathematicians have been no help -\end{acknowledgments} - +\bibliographystyle{unsrt} \bibliography{stokes} \appendix |