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%  research_midsummer.tex - Research Presentation for the Topaz lab.
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%  Created by Jaron Kent-Dobias on Tue Mar 20 20:57:40 PDT 2012.
%  Copyright (c) 2012 pants productions. All rights reserved.
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\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}

