From 9a0ac87220d3eeff00362ba6b9981a4be6c6d406 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 2 Jul 2019 11:53:36 -0400 Subject: first draft of talk --- statphys27.tex | 310 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 310 insertions(+) create mode 100644 statphys27.tex (limited to 'statphys27.tex') diff --git a/statphys27.tex b/statphys27.tex new file mode 100644 index 0000000..6230887 --- /dev/null +++ b/statphys27.tex @@ -0,0 +1,310 @@ + +\documentclass[fleqn,aspectratio=169]{beamer} + + +\setbeamerfont{frametitle}{family=\bf} +\setbeamerfont{normal text}{family=\rm} +\setbeamertemplate{navigation symbols}{} + +\usepackage{textcomp,rotating} + +\title{Rejection-free cluster Monte Carlo in arbitrary external fields} +\subtitle{Phys Rev E \textbf{98}, 063306 (2018)} +\author{Jaron Kent-Dobias \and James P Sethna} +\institute{Cornell University} +\date{} + +\begin{document} + +\def\tr{\mathop{\mathrm{Tr}}\nolimits} + +\begin{frame} + \maketitle +\end{frame} + +\begin{frame} + \frametitle{Monte Carlo is too slow} + + Critical phenomena are often studied on lattice models using Monte Carlo, but near critical points it suffers from \emph{critical slowing down}, power-law divergence of timescales. + + \vspace{1em} + + Slowing down has been alleviated in many models using cluster algorithms and their derivatives, but many applications lack a clean solution. + + \vspace{1em} + + We introduce a generic, natural, and efficient way to extend models with existing cluster algorithms to operate in arbitrary external fields. + + \vspace{1em} + + \begin{enumerate} + \item Introduction: The Ising Model + \begin{enumerate} + \item The Fortuin--Kasteleyn representation \& related algorithm + \item The ghost spin Hamiltonian \& extension to a field + \end{enumerate} + \item Our work: Other lattice models + \begin{enumerate} + \item Fortuin--Kasteleyn representations \& algorithms via Ising embeddings + \item The ghost transformation Hamiltonian \& clusters in arbitrary fields + \end{enumerate} + \end{enumerate} + +\end{frame} + + +\begin{frame} + \frametitle{Introduction: The Ising Model} + \framesubtitle{The Fortuin--Kasteleyn representation} + + The Ising model + \[ + \mathcal H=-\sum_{\langle ij\rangle}J_{ij}s_is_j + \] + for $s_i=\pm1$ on the lattice sites has a representation + \[ + Z=\tr_se^{-\beta\mathcal H}\propto\tr_f\tr_s\prod_{\langle ij\rangle}\big[\delta_{f_{ij},0}(1-p_{ij})+\delta_{f_{ij},1}\delta_{s_i,s_j}p_{ij}\big] + \] + for $f_{ij}\in\{0,1\}$ on the lattice bonds and $p_{ij}=1-e^{-2\beta J_{ij}}$. This gives joint probability distributions + \begin{align*} + P(f_{ij}=1\mid s_i,s_j)=\begin{cases}p_{ij} & s_i=s_j \\ 0 & s_i\neq s_j\end{cases} + && + P(s_i=s_j\mid f)=\begin{cases}1 & \text{$i$, $j$ in same cluster} \\ \frac12 & \text{otherwise}\end{cases} + \end{align*} +\end{frame} + +\begin{frame} + \frametitle{Introduction: The Ising Model} + \framesubtitle{From representation to algorithm} + + \begin{columns} + \begin{column}{0.55\textwidth} + \begin{overprint} + \onslide<1-2>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_1} + \onslide<3>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_2} + \onslide<4>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_3} + \onslide<5>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_4} + \end{overprint} + \end{column} + \begin{column}{0.45\textwidth} + The joint probabilities imply algorithm based on switching back an forth: + \begin{enumerate} + \item\alert<2>{Take a spin configuration.} + \item\alert<3>{Conditionally sample a configuration of bonds.} + \item\alert<4>{Gather sites connected by bonds into clusters.} + \item\alert<5>{Conditionally sample a configuration of spins.} + \end{enumerate} + + \vspace{1em} + + \tiny\raggedleft {Swendsen \& Wang, Phys Rev Lett \textbf{58} (1987) 56.} + \end{column} + \end{columns} +\end{frame} +\begin{frame} + \frametitle{Introduction: The Ising Model} + \framesubtitle{The ghost spin representation} + + \begin{columns} + \begin{column}{0.4\textwidth} + A field means clusters flip with probability that depends on size. + + \vspace{1em} + + But, Fortuin--Kasteleyn doesn't care about lattice topology! Adding a ghost spin coupled to every site with $\tilde J_{0i}=H_i$ gives + \[ + \begin{aligned} + \tilde{\mathcal H}&=-\sum_{\langle ij\rangle}J_{ij}s_is_j-s_0\sum_iH_is_i \\ + &=-\sum_{\langle ij\rangle}\tilde J_{ij}s_is_j + \end{aligned} + \] + \end{column} + \begin{column}{0.6\textwidth} + \includegraphics[width=\textwidth]{figs/ghost_site} + + \vspace{1em} + \tiny\raggedleft {Coniglio, de Liberto, Monroy, \& Peruggi. J Phys A: Math Gen \textbf{22} (1989) L837.} + \end{column} + \end{columns} +\end{frame} +\begin{frame} + \frametitle{Introduction: The Ising Model} + \framesubtitle{The ghost spin algorithm} + \begin{columns} + \begin{column}{0.55\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_1} + \onslide<2>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_2} + \onslide<3>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_3} + \onslide<4>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_4} + \end{overprint} + \end{column} + \begin{column}{0.45\textwidth} + Same algorithm can be run on new Hamiltonian without modification. + + \vspace{1em} + + If the cluster containing $s_0$ is flipped, flip it too! + + \vspace{1em} + + Properties of the original spins must be taken after ``unflipping'' the external field, or $s_0\times s$. + + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Other lattice models} + \framesubtitle{Fortuin--Kasteleyn via embeddings} + + \begin{columns} + \begin{column}{0.4\textwidth} + Cluster methods also known for models whose spins live in more complicated spaces $X$ and have + \[ + \mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j) + \] + If $G$ is the symmetry group of the spins, then a self-inverse element $r\in G$ can embed an Ising model + \[ + J_{ij}(r,s)=\frac12|Z(s_i, s_j)-Z(s_i, r\cdot s_j)| + \] + \end{column} + \begin{column}{0.6\textwidth} + \centering + \begin{tabular}{l|cc} + & $X$ & $G$ \\ + \hline + Ising & $\pm1$ & $\mathbb Z/2\mathbb Z$ \\ + $n$-vector & $(n-1)$ sphere & $\mathrm O(n)$ \\ + Potts & $\{1,\ldots,q\}$ & Symmetric \\ + Clock & $\{1,\ldots,q\}$ & Dihedral \\ + Roughening & $\mathbb Z$ & Infinite Dihedral + \end{tabular} + + \vspace{1em} + + \includegraphics[width=0.9\textwidth]{figs/clocks} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Other lattice models} + \framesubtitle{From embedding to algorithm\dots again} + \begin{columns} + \begin{column}{0.55\textwidth} + \begin{overprint} + \onslide<1-2>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_1} + \onslide<3>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_2} + \onslide<4>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_3} + \onslide<5>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_4} + \onslide<6>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_5} + \onslide<7>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_6} + \onslide<8>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_7} + \end{overprint} + \end{column} + \begin{column}{0.45\textwidth} + \begin{enumerate} + \item\alert<2>{Take a spin configuration.} + \item\alert<3>{Draw a self-inverse $r\in G$.} + \item\alert<4>{Infer Ising $J_{ij}$.} + \item\alert<5>{Sample bonds as before.} + \item\alert<6>{Gather sites into clusters.} + \item\alert<7>{Sample spins by applying $r$ to clusters.} + \end{enumerate} + + \vspace{1em} + + \tiny\raggedleft{Wolff, Phys Rev Lett \textbf{62} (1989) 361.} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Other lattice models} + \framesubtitle{The ghost\dots something representation} + + Can we add an external field with a ghost spin as before? Yes, but only for fields whose interaction is like that of another spin. + + \vspace{1em} + + Rules out novel fields like harmonic lattice anisotropies, cubic potentials, around Potts first-order lines, etc. + + \vspace{1em} + + Need to track the full array of transformations that have included the ghost\dots + + \vspace{1em} + + \dots which is precisely what elements of the symmetry group do! +\end{frame} + +\begin{frame} + \frametitle{Other lattice models} + \framesubtitle{The ghost transformation representation} + + For a lattice model with spins with symmetry group $G$ and + \[ + \mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iB(s_i) + \] + for any function $B$, we introduce a ghost \emph{transformation} $s_0$ and modified Hamiltonian + \[ + \tilde{\mathcal H}=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iB(s_0^{-1}\cdot s_i) + =-\sum_{\langle ij\rangle}\tilde Z(s_i,s_j) + \] + for $\tilde Z(s_0,s_i)=B(s_0^{-1}\cdot s_i)$. Both Hamiltonians yields the same statistics for $s_i$ if accumulated transformations are undone first with $s_0^{-1}\cdot s_i$. +\end{frame} + +\begin{frame} + \frametitle{Other lattice models} + \framesubtitle{Ghost transformation in action} + \begin{columns} + \begin{column}{0.55\textwidth} + \begin{overprint} + \onslide<1-2>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_1} + \onslide<3>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_2} + \onslide<4>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_3} + \onslide<5>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_4} + \onslide<6>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_5} + \onslide<7>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_6} + \onslide<8>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_7} + \end{overprint} + \end{column} + \begin{column}{0.45\textwidth} + \begin{enumerate} + \item\alert<2>{Take a spin configuration.} + \item\alert<3>{Draw a self-inverse $r\in G$.} + \item\alert<4>{Infer Ising $J_{ij}$.} + \item\alert<5>{Sample bonds as before.} + \item\alert<6>{Gather sites into clusters.} + \item\alert<7>{Sample spins by applying $r$ to clusters.} + \end{enumerate} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Summary} + + Introduced a generic method for running cluster Monte Carlo on lattice systems with any external field. + + \vspace{1em} + + Results generalize to arbitrary bond, site dependence. + + \vspace{1em} + + Dynamic scaling works as expected with Wolff or Swendsen--Wang exponents: models efficient at zero field are more efficient with a field, extension appears natural in the scaling sense. + + \vspace{1em} + + Currently working on using machine learning techniques to maximize efficiency related to the choice of the distribution of self-inverse group elements, i.e., Ising embeddings. + + \vspace{1em} + + Phys Rev E \textbf{98}, 063306 (2018) or contact Jaron Kent-Dobias (\texttt{jpk247@cornell.edu}). + +\end{frame} + +\end{document} + -- cgit v1.2.3-54-g00ecf