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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
$$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i)$$
for \\(Z\\) invariant under rotations \\(r\\): \\(Z(r(s),r(t))=Z(s,t)\\).
<table style="border-collapse: collapse; table-layout: fixed; width: 80%; margin: auto;">
<thead style="border-bottom: 2px solid #000;">
<tr>
<td class="first"></td><td>\(s\)<td>\(r\)</td><td>\(Z(s_i,s_j)\)</td><td>\(H(s)\)</td>
</tr>
</thead>
<tbody>
<tr>
<td class="first">Ising model</td><td>\(\{-1,1\}\)</td><td>\(s\mapsto-s\)</td><td>\(s_is_j\)</td><td>\(Hs\)</td>
</tr>
<tr>
<td class="first">Order-\(n\) model</td><td>\(S^n\)</td><td>\(\mathop{\mathrm{SO}}(n)\) (rotation)</td><td>\(s_i\cdot s_j\)</td><td>\(H\cdot s\)</td>
</tr>
<tr>
<td class="first">Potts model</td><td>\(\mathbb Z/q\mathbb Z\)</td><td>addition mod \(q\)</td><td>\(\delta(s_i,s_j)\)</td><td>\(\sum_iH_i\delta(i,s)\)</td>
</tr>
<tr>
<td class="first">Clock model</td><td>ℤ/<i>q</i>ℤ</td><td>addition mod \(q\)</td><td>\(\cos(2\pi\frac{s_i-s_j}q)\)</td><td>\(\sum_iH_i\cos(2\pi\frac{s-i}q)\)</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\\).
3. Compute change in energy \\(\Delta\mathcal H\\) resulting from taking \\(s\\) to \\(R(s)\\).
4. Take \\(s\\) to \\(R(s)\\) with probability \\(\max\\{1,e^{-\beta\Delta\mathcal H}\\}\\).
Problem: Scales very poorly near phase transitions.
Correlation time *τ* at critical point, *t* <sup>– *z/ν*</sup> `\(\tau\sim t^{-z/\nu}\)`
approaching it.
`\(z\)` takes large integer values for Ising, order-`\(n\)`, Potts model critical
]
.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[
1. Pick random spin, add to cluster.
2. Pick random rotation `\(R\)`.
3. For every neighboring spin, add to cluster with probability
`\(\min\{0,1-e^{-\beta(Z(R(s),R(t))-Z(R(s),t))}\}\)`.
4. Repeat 3 for every spin added to cluster.
5. Transform entire cluster with rotation `\(R\)`.
Relies on symmetry of *Z*
Fast near the critical point: early studies thought `\(z\)` was zero, actually
0.1–0.4.
]
.column[
<video width="320" height="320"><source src="figs/wolff_ising_zerofield.webm" type="video/webm"></video>
]
---
# We want to apply an external field, though
The external field `\(H\)` is not invariant under global rotations!
Let's make it that way: introduce an extra spin `\(s_0\)`, let `\(R_s\)` be the rotation that takes `\(s\)` to the
identity
`\[
\tilde Z(s_i,s_j)=
\begin{cases}
Z(s_i,s_j) & \text{if $i,j\neq0$}\\
H(R_{s_0}s_i) & \text{if $j=0$}\\
H(R_{s_0}s_j) & \text{if $i=0$}
\end{cases}
\]`
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
`\[
\langle\tilde
A\rangle=\mathop{\mathrm{Tr}}\nolimits_s\mathop{\mathrm{Tr}}\nolimits_{s_0}\tilde
A(s_0,s)=\mathop{\mathrm{Tr}}\nolimits_sA(s)=\langle A\rangle
\]`
---
---
---
# 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: 40pt; overflow: hidden; height: 6.5em; text-align: center; align-items: center; display: inline-flex; position: static; float: center"> <i>R</i> <br>→</span>
---
# Why is the extended method useful?
<img src="figs/potts.svg" alt="order-n"/>
---
# Correlation time scaling
Correlation time scales consistently in the whole phase space!
<img src="figs/autocorr-scaling-isotherm.png" style="float: left; width: 48%" />
<img src="figs/autocorr-scaling-temp.png" style="float: right; width: 48%" />
---
# Metastable state direct measurement
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