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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" style="float: right;"><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" style="float: right;"><source src="figs/wolff_ising_zerofield.webm" type="video/webm"></video>
]
---
---
---
class: split-40
.column[
# Applying an *arbitrary* field
Introduce an extra “spin” `\(s_0\in R\)` 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
`\[
\tilde Z(s,t)=
\begin{cases}
Z(s,t) & \text{if $s,t\in X$}\\
B(t^{-1}\cdot s) & \text{if $t\in R$}\\
B(s^{-1}\cdot t) & \text{if $s\in R$}
\end{cases}
\]`
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, `\(\tilde
A(s_0,\{s\})=A(\{s_0^{-1}\cdot s\})\)` has the property
`\[
\langle\tilde
A\rangle=\mathop{\mathrm{Tr}}\nolimits_{s_0\in R}\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}\tilde
A(s_0,\{s\})=\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}A(\{s\})=\langle A\rangle
\]`
]
.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"> `\[\xrightarrow{r\in \mathrm O(2)}\]`</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).
Near-constant runtime as field is varied at high, low temperature, strictly faster than zero field at critical point.
]
.column[
<img src="time_compare.png" style="float: right; width: 32%" />
]
---
# Correlation time scaling
Correlation time scales consistently in the whole phase space: a natural extension of the algorithm's dynamics!
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[
#### Symmetry-breaking perturbations of XY model
For various integer values `\(p\)`, external fields of the form
`\[
B_p(s)=H\cos(p\theta(s))
\]`
have various effects on the criticality of the XY model.
Theory from Joeé, Kadanoff, Kirkpatrick, & Nelson: perturbations are relevant *except* for `\(p=6\)`.
We can test it!
]
.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: 47%;" />
]
---
# Questions?
<video style="margin-left: auto; margin-right: auto; margin: 0 auto; display: block; width: 75%;"><source src="figs/wolff_xy_field.webm" type="video/webm"></video>
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