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diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..9edb78b --- /dev/null +++ b/.gitignore @@ -0,0 +1,14 @@ +*.aux +*.bbl +*.blg +*.fdb_latexmk +*.fls +*.log +*Notes.bib +*.out +/*.pdf +*.dvi +*.synctex.gz +*.synctex(busy) +*.bcf +*.run.xml diff --git a/stokes.bib b/stokes.bib new file mode 100644 index 0000000..a1148bc --- /dev/null +++ b/stokes.bib @@ -0,0 +1,25 @@ +@book{Forstneric_2017_Stein, + author = {Forstnerič, Franc}, + title = {{Stein} Manifolds and Holomorphic Mappings}, + publisher = {Springer International Publishing}, + year = {2017}, + volume = {56}, + url = {https://doi.org/10.1007%2F978-3-319-61058-0}, + doi = {10.1007/978-3-319-61058-0}, + edition = {2}, + isbn = {978-3-319-61058-0}, + series = {Ergebnisse der Mathematik und ihrer Grenzgebiete}, + subtitle = {The Homotopy Principle in Complex Analysis} +} + +@book{Morrow_2006_Complex, + author = {Morrow, James and Kodaira, Kunihiko}, + title = {Complex manifolds}, + publisher = {AMS Chelsea Publishing: An Imprint of the American Mathematical Society}, + year = {1971}, + url = {https://bookstore.ams.org/chel-355-h}, + address = {Providence, RI}, + isbn = {9780821840559} +} + + diff --git a/stokes.tex b/stokes.tex new file mode 100644 index 0000000..13138f7 --- /dev/null +++ b/stokes.tex @@ -0,0 +1,176 @@ +\documentclass[aps,reprint,longbibliography,floatfix]{revtex4-2} + +\usepackage[utf8]{inputenc} % why not type "Stokes" with unicode? +\usepackage[T1]{fontenc} % vector fonts plz +\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{} + +\author{Jaron Kent-Dobias} +\author{Jorge Kurchan} + +\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} + +\date\today + +\begin{abstract} +\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 $J$. The former has been considered extensively for the Gaussian +$p$-spin in the past \cite{complex_energy}, while the latter is largely +unexplored. + +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. + +The surface $M\subset\mathbb C^N$ defined by $z^2=N$ 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}. Suppose that $z:\mathbb +C^{N-1}\to M$ is a holomorphic map. The Jacobian $J$ of the map is +\begin{equation} + J_{i\alpha}=\frac{\partial z_i}{\partial u_\alpha}=\partial_\alpha z_i +\end{equation} +where Greek coefficients run from $1$ to $N-1$ and Latin coefficients from $1$ +to $N$. The hermitian metric is $g=J^\dagger J$. For any smooth function +$\phi:M\to\mathbb R$, its gradient $\nabla\phi$ is a holomorphic vector field +given by +\begin{equation} + \nabla\phi=(\partial^*\phi)^\sharp=(\partial^*\phi)g^{-1} +\end{equation} + +For +coordinates $u\in\mathbb C^{N-1}$, dynamics consists of gradient descent on +$\operatorname{Re}H$, or +\begin{equation} + \dot u=-\nabla\operatorname{Re}H=-\tfrac12(\partial H)^\dagger g^{-1} +\end{equation} +These dynamics preserve $\operatorname{Im}H$ and in fact correspond to +Hamiltonian dynamics, with the real and imaginary parts of the coordinates +taking the role of conjugate variables. \cite{Morrow_2006_Complex} + +Working with a particular map is inconvenient, and we would like to develop a map-independent dynamics. Using the chain rule, one finds +\begin{equation} + \begin{aligned} + \dot z_i + &=\dot u_\alpha\partial_\alpha z_i + =-\tfrac12(\partial_\beta H)^*g^{-1}_{\beta\alpha}\partial_\alpha z_i\\ + &=-\tfrac12(\partial_j H)^*(\partial_\beta z_j)^*g^{-1}_{\beta\alpha}\partial_\alpha z_i + =-\tfrac12(\partial H)^\dagger(J^\dagger g^{-1}J)\\ + \end{aligned} +\end{equation} +where $J$ is the Jacobian of the coordinate map and $g$ is the metric. In stereographic coordinates this can be worked out directly. +Consider the coordinates $z_i=u_i$ for $1\leq i\leq N-1$ and +\begin{equation} + z_N=\sqrt{N-u_\alpha u_\alpha} +\end{equation} +The Jacobian is +\begin{equation} + J_{\alpha i}=\partial_\alpha z_i=\delta_{\alpha i}-\delta_{Ni}\frac{u_\alpha}{\sqrt{N-u_\beta u_\beta}} +\end{equation} +and the corresponding hermitian metric is +\begin{equation} + g_{\alpha\beta}=J_{i\alpha}^*J_{i\beta} + =\delta_{\alpha\beta}+\frac{u_\alpha^*u_\beta}{|N-u_\gamma u_\gamma|} +\end{equation} +The metric can be inverted explicitly: +\begin{equation} + g^{-1}_{\alpha\beta} + =\delta_{\alpha\beta}-\frac{u_\alpha^*u_\beta}{|N-u_\gamma u_\gamma|+|u|^2}. +\end{equation} +Putting these pieces together, we find +\begin{equation} + (J^\dagger g^{-1}J)_{ij} + =\delta_{ij}-\frac{z_iz_j^*}{|z|^2} +\end{equation} +which is just the projector onto the constraint manifold. + +Therefore, a map-independent dynamics is given by +\begin{equation} + \dot z + =-\frac12(\partial H)^\dagger\left(I-\frac{zz^\dagger}{|z|^2}\right) +\end{equation} + +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. + +The level sets defined by $\operatorname{Im}H=c$ for $c\in\mathbb R$ are surfaces of +$2(N-1)-1$ real dimensions. They must be simply connected, since gradient +descent in $\operatorname{Re}H$ cannot pass the same point twice. + +\section{2-spin} + +\begin{equation} + H_0=\frac12z^TJz +\end{equation} +$J$ is generically diagonalizable by a complex orthogonal matrix $P$. With +$z\mapsto Pz$, $J\mapsto D$ for diagonal $D$. Then $\partial_i H=d_iz_i$. +Suppose that two critical points have the same imaginary energy; without loss +of generality, assume these are the first and second components. 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 $\Delta=d_1-d_N$ 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. + +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. + +\section{p-spin} + +\section{(2 + 4)-spin} + +\begin{acknowledgments} + MIT mathematicians have been no help +\end{acknowledgments} + +\bibliography{stokes} + +\end{document} |