From 8c007b686a12321924a784d12f63a24ca6d47f3a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 22 Feb 2018 16:08:26 -0500
Subject: new slides

---
 aps_mm_2018.html | 41 ++++++++++++++++++++++++++++++++++-------
 1 file changed, 34 insertions(+), 7 deletions(-)

(limited to 'aps_mm_2018.html')

diff --git a/aps_mm_2018.html b/aps_mm_2018.html
index 8412808..b9746c1 100644
--- a/aps_mm_2018.html
+++ b/aps_mm_2018.html
@@ -39,12 +39,12 @@ 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)\\).
+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>
+      <td class="first"></td><td>\(s\)<td>\(r\)</td><td>\(Z(s_i,s_j)\)</td><td>\(H(s)\)</td>
     </tr>
   </thead>
   <tbody>
@@ -58,7 +58,7 @@ for \\(Z\\) invariant under rotations \\(R\\): \\(Z(R(s),R(t))=Z(s,t)\\).
       <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>
+      <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>
@@ -76,14 +76,14 @@ class: split-40
   Standard approach to modelling arbitrary stat mech system: metropolis.
   
   1. Pick random spin.
-  2. Pick random rotation \\(R\\).
+  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, `\(\tau\sim t^{-z/\nu}\)`
+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
@@ -104,11 +104,11 @@ class: split-40
  1. Pick random spin, add to cluster.
  2. Pick random rotation `\(R\)`.
  3. For every neighboring spin, add to cluster with probability
-      `\(\min\{0,1-e^{-\beta(Z(R(s),t)-Z(R(s),R(t)))}\}\)`.
+      `\(\min\{0,1-e^{-\beta(Z(R(s),R(t))-Z(R(s),t))}\}\)`.
  4. Repeat 3 for every spin added to cluster.
  5. Transform entire cluster with rotation `\(R\)`.
 
-Relies on symmetry of `\(Z\)`
+Relies on symmetry of *Z*
 
 Fast near the critical point: early studies thought `\(z\)` was zero, actually
   0.1–0.4.
@@ -152,6 +152,33 @@ Fast near the critical point: early studies thought `\(z\)` was zero, actually
 
   ![scoop details](figs/wolff-scoop_explanation.png)
 
+---
+
+# 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">&nbsp;&nbsp;<i>R</i>&nbsp;<br>&rarr;</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
+
     </textarea>
     <script src="https://remarkjs.com/downloads/remark-latest.min.js">
     </script>
-- 
cgit v1.2.3-70-g09d2