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From 05725b9fa4843d19561b026567979709cc6b62a2 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 21 Feb 2018 11:40:28 -0500 Subject: initial commit --- aps_mm_2018.html | 194 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 194 insertions(+) create mode 100644 aps_mm_2018.html (limited to 'aps_mm_2018.html') diff --git a/aps_mm_2018.html b/aps_mm_2018.html new file mode 100644 index 0000000..afa519b --- /dev/null +++ b/aps_mm_2018.html @@ -0,0 +1,194 @@ + + + + Title + + + + + +

+
+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
+  `\[
+    \langle\tilde
+    A\rangle=\mathop{\mathrm{Tr}}\nolimits_s\mathop{\mathrm{Tr}}\nolimits_{s_0}\tilde
+    A(s_0,s)=\mathop{\mathrm{Tr}}\nolimits_sA(s)=\langle A\rangle
+  \]`
+
+---
+
+  ![scooped](figs/wolff-scoop_title.png)
+
+---
+
+  ![scoop details](figs/wolff-scoop_explanation.png)
+
+

+ + + + + + + + -- cgit v1.2.3-70-g09d2