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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>\(\mathbb Z/q\mathbb Z\)</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!
---
# Local Monte Carlo: Not Great
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 `\(\tau\sim L^z\)` at critical point, `\(\tau\sim t^{-z/\nu}\)`
approaching it.
`\(z\)` takes large integer values for Ising, order-`\(n\)`, Potts model critical
---
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
`\(\min\{0,1-e^{-\beta(Z(R(s),t)-Z(R(s),R(t)))}\}\)`.
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/test.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
`\[
\langle\tilde
A\rangle=\mathop{\mathrm{Tr}}\nolimits_s\mathop{\mathrm{Tr}}\nolimits_{s_0}\tilde
A(s_0,s)=\mathop{\mathrm{Tr}}\nolimits_sA(s)=\langle A\rangle
\]`
---
---
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