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
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
|
<!DOCTYPE html>
<html>
<head>
<title>Title</title>
<meta charset="utf-8">
<style>
@import url(https://fonts.googleapis.com/css?family=Yanone+Kaffeesatz);
@import url(https://fonts.googleapis.com/css?family=Droid+Serif:400,700,400italic);
@import url(https://fonts.googleapis.com/css?family=Ubuntu+Mono:400,700,400italic);
body { font-family: 'Computer Modern Concrete'; }
h1, h2, h3 {
font-family: 'Computer Modern Concrete';
font-weight: normal;
}
.remark-code, .remark-inline-code { font-family: 'Ubuntu Mono'; }
</style>
<link rel="stylesheet" type="text/css" href="main.css">
<link rel="stylesheet" type="text/css" href="fonts/Concrete/cmun-concrete.css">
</head>
<body>
<textarea id="source">
class: center, middle
# An efficient cluster algorithm for spin systems in a symmetry-breaking field
## Jaron Kent-Dobias & James Sethna
### Cornell University
## 9 March 2018
---
# Spin systems
Described by Hamiltonians whose degrees of freedom are spins `\(s\in X\)`
$$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iB(s_i)$$
for \\(Z\\) invariant under the action of rotations \\(r\in R\\), or \\(Z(r\cdot s,r\cdot t)=Z(s,t)\\).
<table style="border-collapse: collapse; width: 100%; margin: auto;">
<thead style="border-bottom: 2px solid #000;">
<tr>
<td class="first"></td><td><i>X</i><td><i>R</i></td><td>Group Action</td><td>\(Z(s,t)\)</td><td>\(B(s)\)</td>
</tr>
</thead>
<tbody>
<tr>
<td class="first">Ising model</td><td>\(\{-1,1\}\)</td><td>ℤ/2ℤ</td><td>\(0\cdot s\mapsto s\)<br>\(1\cdot s\mapsto-s\)</td><td>\(s\times t\)</td><td>\(Hs\)</td>
</tr>
<tr>
<td class="first">Order-\(n\) model</td><td>\(S^n\)</td><td>\(\mathop{\mathrm{O}}(n)\)</td><td>\(O\cdot s\mapsto Os\)</td><td>\(s^{\mathrm T}t\)</td><td>\(H^{\mathrm T}s\)</td>
</tr>
<tr>
<td class="first">Potts model</td><td>{0,…,<i>q</i> – 1}</td><td><i>D<sub>q</sub></i></td><td>Rotation & reflection<br> of polygon vertices</td><td>\(\delta(s,t)\)</td><td>\(\sum_mH_m\delta(m,s)\)</td>
</tr>
<tr>
<td class="first">Clock model</td><td>{0,…,<i>q</i> – 1}</td><td><i>D<sub>q</sub></i></td><td>Rotation & reflection<br> of polygon vertices</td><td>\(\cos(2\pi\frac{s-t}q)\)</td><td>\(\sum_mH_m\cos(2\pi\frac{s-m}q)\)</td>
</tr>
</tbody>
</table>
Relatively simple with extremely rich behavior, phase transition galore!
---
class: split-40
# Local Monte Carlo: Not Great
.column[
Standard approach to modelling arbitrary stat mech system: metropolis.
1. Pick random spin.
2. Pick random rotation \\(r\in R\\).
3. Compute change in energy \\(\Delta\mathcal H\\) resulting from taking \\(s\\) to \\(r\cdot s\\).
4. Take \\(s\\) to \\(r\cdot s\\) with probability \\(\max\\{1,e^{-\beta\Delta\mathcal H}\\}\\).
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 regions.
]
.column[
<video width="320" height="320"><source src="figs/metropolis_ising_zerofield.webm" type="video/webm"></video>
]
---
class: split-40
# Wolff: wow, what a solution
.column[
Non-local updates!
1. Pick random spin `\(s\in X\)`, add to cluster.
2. Pick random rotation `\(r\in R\)`.
3. For every neighboring spin `\(t\)`, add to cluster with probability
`\(\min\{0,1-e^{-\beta(Z(r\cdot s,r\cdot t)-Z(r\cdot s,t))}\}\)`.
4. Repeat 3 for every new spin added to cluster.
5. Transform entire cluster by applying the action of rotation `\(r\)`.
Relies on symmetry of *Z* under group of rotations.
Fast near the critical point: early studies thought `\(z\)` was zero for 2D Ising, actually
0.1–0.4, below one for models listed here.
]
.column[
<video width="320" height="320"><source src="figs/wolff_ising_zerofield.webm" type="video/webm"></video>
]
---
![scooped](figs/wolff-scoop_title.png)
---
![scoop details](figs/wolff-scoop_explanation.png)
---
class: split-40
.column[
# Applying an *arbitrary* field
Introduce an extra “spin” `\(r_0\in R\)` whose possible configurations come from the set of *rotations*, not the set of spin states.
Mark this spin a neighbor of every spin on the lattice.
New effective coupling defined by
`\[
\tilde Z(s,t)=
\begin{cases}
Z(s,t) & \text{if $s,t\in X$}\\
H(t^{-1}\cdot s) & \text{if $t\in R$}\\
H(s^{-1}\cdot t) & \text{if $s\in R$}
\end{cases}
\]`
Preform ordinary Wolff algorithm on the extended phase space.
Exact correspondence between expectation values of operators in old and new
models: if `\(A(\{s\})\)` is an observable on old model in field, `\(\tilde
A(r_0,\{s\})=A(\{r_0^{-1}\cdot s\})\)` has the property
`\[
\langle\tilde
A\rangle=\mathop{\mathrm{Tr}}\nolimits_{r_0\in R}\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}\tilde
A(r_0,\{s\})=\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}A(\{s\})=\langle A\rangle
\]`
]
.column[
<video width="320" height="640" style="float:right;"><source src="figs/wolff_ising_field.webm" type="video/webm"></video>
]
---
# Why is the extended method useful?
<img src="figs/vector-1.svg" alt="order-n" style="float: left;"/>
<img src="figs/vector-2.svg" alt="order-n" style="float: right;"/>
<span style="font-size: 30pt; overflow: hidden; height: 7em; text-align: center; align-items: center; display: inline-flex; position: static; float: center"> `\[\xrightarrow{r\in O(2)}\]`</span>
---
class: split-40
.column[
# Example: Clock Potts
Consider the clock Potts model with field
`\[
H(m)=0.01\times\begin{cases}
0 & \text{if $m=0$ (blue)}\\
\cos(2\pi/6) & \text{if $m=1$ (green}\\
-\cos(2\pi/6) & \text{if $m=2$ (yellow)}
\end{cases}
\]`
<img src="figs/potts.svg" style="width:300px;"/>
No rotation `\(r\in\mathbb Z/3\mathbb Z\)` that takes field to the identity!
]
.column[
<video width="320" height="640" style="float:right;"><source src="figs/wolff_potts_field.webm" type="video/webm"></video>
]
---
class: split-40
.column[
# Correlation time comparison
The correlation time of this algorithm beats metropolis and hybrid Wolff/metropolis in whole phase space!
Pictured: 2D Ising (`\(128\times128\)`) correlation time as a function of field for three algorithms in high-temperature phase (top), at `\(T_c\)` (middle), and in the low temperature phase (bottom).
]
.column[
<img src="time_compare.png" style="float: right; width: 32%" />
]
---
# Correlation time scaling
Correlation time scales consistently in the whole phase space!
Pictured: scaling collapses of 2D Ising model correlation time.
<img src="figs/autocorr-scaling-isotherm.png" style="float: left; width: 48%" />
<img src="figs/autocorr-scaling-temp.png" style="float: right; width: 48%" />
---
class: split-50
# Applications & future work
.column[ #### Direct measurements in metastable states
Rapid sampling of states near the critical point of models in a small field occasionally yields microstates against the direction of the field.
<img src="figs/metastable-scaling.png" style="width: 80%; float: left;" />
]
.column[
#### Symmetry-breaking perturbations of XY model
For various values of `\(p\)`, external fields of the form
`\[
H_p(s)=\cos(p\theta(s))
\]`
have various effects on the criticality of the XY model.
]
---
# Questions?
<video><source src="figs/wolff_xy_field.webm" type="video/webm"></video>
</textarea>
<script src="https://remarkjs.com/downloads/remark-latest.min.js">
</script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS_HTML&delayStartupUntil=configured" type="text/javascript"></script>
<script src="https://code.jquery.com/jquery-3.3.1.min.js"></script>
<script src="vissense/lib/vissense.js"></script>
<script>
$(document).ready(function(){
var videos = $('video'), // All videos element
allVidoesVisenseObj = [];
var monitorVideo = function(video){ //Handler for each video element
var visibility = VisSense(video, { fullyvisible: 0.75 }),
visibility_monitor = visibility.monitor({
fullyvisible: function(e) {
video.play();
},
hidden: function(e) {
video.pause();
}
}).start();
return {
visMonitor : visibility_monitor,
monitorStop : function(){
this.visMonitor.stop();
},
monitorStart : function(){
this.visMonitor.start();
}
};
};
videos.each(function( index ) {
var monitorVids = monitorVideo(this);
allVidoesVisenseObj.push(monitorVids);
});
});
</script>
<script type="text/javascript">
var slideshow = remark.create({ ratio: "16:9" });
MathJax.Hub.Config({
tex2jax: {
skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
}
});
MathJax.Hub.Configured();
</script>
</body>
</html>
|