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 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!

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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\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>
]

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# Wolff: wow, what a solution

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>
]

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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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# 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>

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# Example: Clock Potts

Consider the clock Potts model with field
`\[
H(m)=0.01\times\begin{cases}
0 & \text{if $m=0$ (blue)}\\
\cos(2\pi/6) & \text{if $m=1$ (green}\\
-\cos(2\pi/6) & \text{if $m=2$ (yellow)}
\end{cases}
\]`

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>
]

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# 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.
]

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# 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.

---
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!
]

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%;" />
]

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# Questions?