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+%
+%  research_midsummer.tex - Research Presentation for the Topaz lab.
+%
+%  Created by Jaron Kent-Dobias on Tue Mar 20 20:57:40 PDT 2012.
+%  Copyright (c) 2012 pants productions. All rights reserved.
+%
+
+\documentclass[fleqn,aspectratio=169]{beamer}
+
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath,amssymb,latexsym,graphicx,multimedia}
+\usepackage{concmath}
+\usepackage[T1]{fontenc}
+
+\usecolortheme{beaver}
+\usefonttheme{serif}
+\setbeamertemplate{navigation symbols}{}
+
+\title{Direct Measurement of Metastable Properties Near Critical Points}
+\author{ Jaron~Kent-Dobias \and James~Sethna}
+\institute{Cornell University}
+\date{9 March 2018}
+
+\begin{document}
+
+\def\H{\mathcal H}
+\def\Z{\mathbb Z}
+
+\begin{frame}
+  \titlepage
+\end{frame}
+
+\begin{frame}
+  \frametitle{Outline}
+  \begin{itemize}
+    \item Simulating equilibrium spin systems
+    \item Local updates: it's got problems
+    \item Solution: cluster flips!
+    \item Cluster flips\dots in an external field???
+    \item \dots{}yes!
+    \item Analysis of runtime, efficiency
+    \item Formal redefinition of magnetization
+    \item Use: direct measurement of the metastable state
+  \end{itemize}
+  \vfill
+\end{frame}
+
+\begin{frame}
+  \frametitle{Spin systems: we love them}
+  
+  Described by Hamiltonians
+  \[
+    \H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i)
+  \]
+  for $Z$ invariant under rotations $R$: $Z(R(s),R(t))=Z(s,t)$
+
+  \begin{table}
+  \renewcommand{\tabcolsep}{7pt}
+  \begin{tabular}{l||cccc}
+    & $s$ & $R$ & $Z(s_i,s_j)$ & $H(s)$ \\
+    \hline\hline
+    Ising model & $\{-1,1\}$ & $s\mapsto-s$ & $s_is_j$ & $Hs$ \\
+    Order-$n$ model & $S^n$ & $\mathop{\mathrm{SO}}(n)$ (rotation) & $s_i\cdot s_j$ & $H\cdot s$ \\
+    Potts model & $\Z/q\Z$ & addition mod $q$  & $\delta(s_i,s_j)$ &
+    $\sum_iH_i\delta(i,s)$ \\
+    Clock model & $\Z/q\Z$ & addition mod $q$  & $\cos(2\pi\frac{s_i-s_j}q)$ &
+    $\sum_iH_i\cos(2\pi\frac{s-i}q)$
+  \end{tabular}
+  \end{table}
+
+  Relatively simple with extremely rich behavior, phase transitions galore!
+\end{frame}
+
+\begin{frame}
+  \frametitle{Local Monte Carlo: Not Great}
+
+  Standard approach to modelling arbitrary stat mech system: metropolis.
+  
+  \begin{enumerate}
+    \item Pick random spin.
+    \item Pick random rotation $R$.
+    \item Compute change in energy $\Delta\H$ resulting from taking $s$ to
+      $R(s)$.
+    \item Take $s$ to $R(s)$ with probability $\max\{1,e^{-\beta\Delta\H}\}$.
+  \end{enumerate}
+
+  Problem: Scales very poorly near phase transitions.
+
+  Correlation time $\tau\sim L^z$ at critical point, $\tau\sim t^{-z/\nu}$
+  approaching it.
+
+  $z$ takes large integer values for Ising, order-$n$, Potts model critical
+  points.
+\end{frame}
+
+\begin{frame}
+  \frametitle{Wolff: wow, what a solution}
+
+  \begin{enumerate}
+    \item Pick random spin, add to cluster.
+    \item Pick random rotation $R$.
+    \item For every neighboring spin, add to cluster with probability
+      $\min\{0,1-e^{-\beta(Z(R(s),t)-Z(R(s),R(t)))}\}$.
+    \item Repeat 3 for every spin added to cluster.
+    \item Transform entire cluster with rotation $R$.
+  \end{enumerate}
+  Relies on symmetry of $Z$
+
+  Fast near the critical point: early studies thought $z$ was zero, actually
+  $0.1$--$0.4$.
+\end{frame}
+
+\begin{frame}
+  \frametitle{We want to apply an external field, though}
+
+  The external field $H$ is not invariant under global rotations!
+
+  Let's make it that way: $R_s$ is the rotation that takes $s$ to the
+  identity
+  (1, first basis vector, etc)
+  \[
+    \tilde Z(s_i,s_j)=
+    \begin{cases}
+      Z(s_i,s_j) & \text{if $i,j\neq N$}\\
+      H(R_{s_0}s_i) & \text{if $j=0$}\\
+      H(R_{s_0}s_j) & \text{if $i=0$}
+    \end{cases}
+  \]
+  Exact correspondence between expectation values of operators in old and new
+  models: if $A(s)$ is an observable on old model, $\tilde
+  A(s_0,s)=A(R_{s_0}s)$ has the property
+  \[
+    \langle\tilde
+    A\rangle=\mathop{\mathrm{Tr}}\nolimits_s\mathop{\mathrm{Tr}}\nolimits_{s_0}\tilde
+    A(s_0,s)=\mathop{\mathrm{Tr}}\nolimits_sA(s)=\langle A\rangle
+  \]
+\end{frame}
+
+\begin{frame}
+  \centering
+  \includegraphics[height=0.8\textheight]{figs/wolff-scoop_title}
+\end{frame}
+
+\begin{frame}
+  \centering
+  \includegraphics[height=0.8\textheight]{figs/wolff-scoop_explanation}
+\end{frame}
+
+\begin{frame}
+  \frametitle{But does it actually work well? (yes)}
+  \movie[height=.4\textwidth,width=0.4\textwidth]{}{figs/test.avi}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Measuring direction-dependant quantities}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Metastable state!}
+\end{frame}
+
+\begin{frame}
+\end{frame}
+
+\end{document}
+