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@@ -19,7 +19,6 @@ \author{Jaron Kent-Dobias and Jorge Kurchan} \address{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} -\ead{jaron.dobias@phys.ens.fr} \begin{abstract} In this paper we follow up the study of `complex complex landscapes' @@ -41,6 +40,8 @@ \maketitle +\section{Introduction} + Analytic continuation of physical theories is sometimes useful. Some theories have a well-motivated hamiltonian or action that nevertheless results in a divergent partition function, and can only be properly defined by continuation @@ -49,81 +50,133 @@ in oscillatory phase space measures that spoil the use of Monte Carlo or saddle point techniques, but can be treated in a regime where the measure does not oscillated and the results continued to the desired model \cite{}. -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 +In any case, the nicest modern technique (which we will describe in some detail +later) consists of deforming the phase space integral into a complex phase +space and then breaking it into pieces associated with stationary points of the +action. Each of these pieces, known as \emph{thimbles}, has wonderful +properties that guarantee convergence and prevent oscillations. Once such a +decomposition is made, analytic continuation is mostly easy, save for instances +where the thimbles interact, which must be accounted for. + +When your action has a manageable set of stationary points, this process is +usually tractable. However, many actions of interest are complex, having many +stationary points with no simple symmetry relating them, far too many to +individually monitor. Besides appearing in classical descriptions of structural +and spin glasses, complex landscapes are recently become important objects of +study in the computer science of machine learning, the condensed matter theory +of strange metals, and the high energy physics of black holes. What becomes of +analytic continuation under these conditions? + +\section{Thimble integration and analytic continuation} + +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} - 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 \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 -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 -integrals of analytic functions which are not themselves analytic -\cite{Cavagna_1999_Energy}. - -\section{Integration by Lefschetz thimble} - -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 \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 -the integral converges. Second, that the contours piecewise correspond to -surfaces of slowing vary phase of the action, so as to ameliorate sign -problems. + Z(\beta)=\int_\Omega ds\,e^{-\beta\mathcal S(s)} +\end{equation} +We've defined $Z$ in a way that strongly suggests application in statistical +mechanics, but everything here is general: the action can be complex- or even +imaginary-valued, and $\Omega$ could be infinite-dimensional. In typical +contexts, $\Omega$ will be the euclidean real space $\mathbb R^N$ or some +subspace of this like the sphere $S^{N-1}$ (as in the $p$-spin spherical models +we will study later). We will consider only the analytic continuation of the +parameter $\beta$, but any other would work equally well, e.g., of some +parameter inside the action. The action will have some stationary points, e.g., +minima, maxima, saddles, and the set of those points in $\Omega$ we will call +$\Sigma_0$, the set of real stationary points. + +In order to analytically continue \eref{eq:partition.function} by the method we +will describe, $\mathcal S$ must have an extension to a holomorphic function on +a larger complex phase space $\tilde\Omega$ containing $\Omega$. In many cases +this is accomplished by simply noticing that the action is some sum or product +of holomorphic functions, e.g., polynomials. For $\mathbb R^N$ the complex +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 +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 +\emph{all} stationary points of the action, which naturally contains the set of +\emph{real} stationary points $\Sigma_0$. + +Assuming $\mathcal S$ is holomorphic (and that the phase space $\Omega$ is +orientable, which is usually true) the integral in \eref{eq:partition.function} +can be considered an integral over a contour in the complex phase space $\tilde\Omega$, +or +\begin{equation} \label{eq:contour.partition.function} + Z(\beta)=\oint_\Omega dz\,e^{-\beta\mathcal S(z)} +\end{equation} +For the moment this translation has only changed some of our symbols from +\eref{eq:partition.function}, but conceptually it is very important: contour +integrals can have their contour freely deformed (under some constraints) +without changing their value. This means that we are free to choose a nicer +contour than our initial phase space $\Omega$. + +What contour properties are desirable? Consider the two main motivations cited +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 +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. Remarkably, there is an elegant recipe for accomplishing both these criteria at once, courtesy of Morse theory. For a more thorough review, see -\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 -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 +\cite{Witten_2011_Analytic}. We are going to construct our deformed contour out +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 +$\operatorname{Re}\mathcal S$. + +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, +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. + +What thimbles are necessary to reproduce our original contour, $\Omega$? The +answer is, we need the minimal set which produces a contour between the same +places. Simply stated, if $\Omega=\mathbb R$ produced a phase space integral +running along the real line from left to right, then our contour must likewise +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, +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 +integrand vanishes exponentially. The relative homology group +$H_N(\tilde\Omega,\tilde\Omega_T)$ describes the homology of cycles which begin +and end in $\Omega_T$, i.e., are well-behaved. Therefore, any well-behaved +cycle must represent an element of $H_N(\tilde\Omega,\tilde\Omega_T)$. In order +for our collection of thimbles to produce the correct contour, the composition +of the thimbles must represent the same element of this relative homology +group. + +Each thimble represents an element of the relative homology, since each thimble +is a contour on which the real part of the action diverges in any direction. +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 +$\mathcal K_\sigma$ 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 \begin{equation} \label{eq:thimble.integral} - Z(\lambda)=\sum_{\sigma\in\Sigma}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{\mathcal S_\lambda(s)}. + Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\oint_{\mathcal J_\sigma}dz\,e^{-\beta\mathcal S(z)}. \end{equation} -Each of these integrals is very well-behaved: convergent asymptotic series -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 \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 |