\documentclass[aps,reprint,longbibliography,floatfix]{revtex4-2} \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, citecolor=purple, filecolor=purple, linkcolor=purple ]{hyperref} % ref and cite links with pretty colors \usepackage{amsmath, graphicx, xcolor} % standard packages \begin{document} \title{Analytic continuation over complex landscapes} \author{Jaron Kent-Dobias} \author{Jorge Kurchan} \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} \date\today \begin{abstract} \setcitestyle{authoryear,round} In this paper we follow up the study of `complex complex landscapes' [\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 theory. These trades, which occur at Stokes points, proliferate when the spectrum includes marginal directions and are exponentially suppressed otherwise. This gives a direct interpretation of the `threshold' energy---which in the real case separates saddles from minima---where the spectrum of typical critical points develops a gap. This leads to different consequences for the analytic continuation of real landscapes with different structures: the global minima of ``one step replica-symmetry broken'' landscapes lie beyond a threshold and are locally protected from Stokes points, whereas those of ``many step replica-symmetry broken'' lie at the threshold and Stokes points immediately proliferate. \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 $J$. 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. Unfortunately the study is not so relevant for low-dimensional `rugged' landscapes, which are typically constructed from the limits of series or integrals of analytic functions which are not themselves analytic \cite{Cavagna_1999_Energy}. \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 \begin{equation} Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\int_{\mathcal J_\sigma}du\,e^{-\beta H(u)} \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 $n_\sigma$. \section{Gradient descent dynamics} For a holomorphic Hamiltonian $H$, dynamics are defined by gradient descent on $\operatorname{Re}H$. 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^*_{\beta}\operatorname{Re}H)h^{\beta\alpha}\partial_\alpha z^i =-\tfrac12(\partial_\beta H)^*h^{\beta\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 $h^{\beta\alpha}=h^{-1}_{\beta\alpha}$ for $h_{\beta\alpha}=(\partial_\beta z^i)^*\partial_\alpha z^i=(J^\dagger J)_{\beta\alpha}$ 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 \begin{equation} \label{eq:flow} \dot z=-\tfrac12(\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T =-\tfrac12(\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 $\operatorname{Re}H$ 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 shown using \eqref{eq:flow} and the holomorphic property of $H$: \begin{equation} \begin{aligned} \frac d{dt}&\operatorname{Im}H =\dot z\partial\operatorname{Im}H+\dot z^*\partial^*\operatorname{Im}H \\ &=\frac i4\left( (\partial H)^\dagger P\partial H-(\partial H)^TP^\dagger(\partial H)^* \right) \\ &=\frac i4\left( (\partial H)^\dagger P\partial H-[(\partial H)^\dagger P\partial H]^* \right) \\ &=\frac i4\left( \|\partial H\|-\|\partial H\|^* \right)=0. \end{aligned} \end{equation} As a result of this conservation law, surfaces of constant $\operatorname{Im}H$ will be important when evaluting the possible endpoints of 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 $\mathbb R^{2N}$. Now, `place' a generic (nondegenerate) critical point in the function at $u_0$. In the vicinity of the critical point, the flow is locally \begin{equation} \begin{aligned} \dot z &\simeq-\frac12(\partial\partial H)^\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 manifold. 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. 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{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 \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 $z$. \subsection{2-spin} The Hamiltonian of the $2$-spin model is defined for $z\in\mathbb C^N$ by \begin{equation} H_0=\frac12z^TJz. \end{equation} $J$ is generically diagonalizable by a complex orthogonal matrix. In a diagonal basis, $J_{ij}=\lambda_i\delta_{ij}$. Then $\partial_i H=\lambda_iz_i$. We will henceforth assume to be working in this basis. To find the critical points, we use the method of Lagrange multipliers. Introducing $\epsilon$, the constrained Hamiltonian is \begin{equation} H=H_0+\epsilon(N-z^2) \end{equation} As usual, $\epsilon$ is equivalent to the energy per spin at any critical point. Critical points must satisfy \begin{equation} 0=\partial_iH=(\lambda_i-2\epsilon)z_i \end{equation} which is only possible for $z_i=0$ or $\epsilon=\frac12\lambda_i$. Generically the $\lambda_i$ will all differ, so this can only be satisfied for one $\lambda_i$ at a time, and to be a critical point all other $z_j$ must be zero. In the direction in question, \begin{equation} \frac1N\frac12\lambda_iz_i^2=\epsilon=\frac12\lambda_i, \end{equation} whence $z_i=\pm\sqrt{N}$. Thus there are $2N$ critical points, each corresponding to $\pm$ the cardinal directions in the diagonalized basis. Suppose that two critical points have the same imaginary energy; without loss 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} 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} \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. Since they sit at the corners of a simplex, 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. \subsection{Pure \textit{p}-spin} \begin{equation} H_p=\frac1{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p} \end{equation} \subsection{(2 + 4)-spin} \begin{equation} H_2+H_4 \end{equation} \section{Numerics} To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically. \begin{equation} \mathcal L(z(t), z'(t)) = 1-\frac{\operatorname{Re}(\dot z^*_iz'_i)}{|\dot z||z'|} \end{equation} \begin{equation} \mathcal C[\mathcal L]=\int dt\,\mathcal L(z(t), z'(t)) \end{equation} $\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ whose tangent is everywhere parallel to the direction of the dynamics. Therefore, finding functions which reach this minimum will find time-reparameterized Stokes lines. For the sake of numeric stability, we look for functions whose tangent has constant norm. \begin{acknowledgments} MIT mathematicians have been no help \end{acknowledgments} \bibliography{stokes} \appendix \section{Geometry} The surface $M\subset\mathbb C^N$ defined by $N=f(z)=z^2$ 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}. 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 is a holomorphic vector field given by \begin{equation} \operatorname{grad}\phi=\bar\partial^\sharp\phi \end{equation} 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} Let $z=u^{-1}$. \begin{equation} \frac\partial{\partial u^\alpha}=\frac{\partial z^i}{\partial u^\alpha}\frac\partial{\partial z^i} \end{equation} \begin{equation} \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} 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}. \end{equation} The Jacobian is \begin{equation} \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}=\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} h^{\bar\beta\alpha} =\delta^{\bar\beta\alpha}-\frac{\bar u^{\bar\beta}u^\alpha}{|N-u^2|+|u|^2}. \end{equation} Putting these pieces together, we find \begin{equation} \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} \begin{equation} \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} \end{document}