1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
|
%
% 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}
|
