Title

# An efficient cluster algorithm for spin systems in a symmetry-breaking field

## Jaron Kent-Dobias & James Sethna

### Cornell University

## 9 March 2018

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# 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!

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# 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 *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>
]

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# Wolff: wow, what a solution

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>
]

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# 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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![scooped](figs/wolff-scoop_title.png)

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![scoop details](figs/wolff-scoop_explanation.png)

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# Why is the extended method useful?

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# Why is the extended method useful?

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# Correlation time scaling

Correlation time scales consistently in the whole phase space!

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# Metastable state direct measurement