From 1ba9d7113ba6548416ea70d4a977e71d27bfe8fb Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 8 Mar 2018 15:57:05 -0500 Subject: changes --- aps_mm_2018.html | 155 ++++++++++++++++++++++++++++++++++++++----------------- 1 file changed, 109 insertions(+), 46 deletions(-) (limited to 'aps_mm_2018.html') diff --git a/aps_mm_2018.html b/aps_mm_2018.html index b9746c1..a3f6d91 100644 --- a/aps_mm_2018.html +++ b/aps_mm_2018.html @@ -19,7 +19,9 @@ -

+
+  <textarea id="source">
+
 
 class: center, middle
 
@@ -35,30 +37,30 @@ class: center, middle
 
 # Spin systems
 
-Described by Hamiltonians
+Described by Hamiltonians whose degrees of freedom are spins `$s\in X$`
 
 $$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i)$$
 
-for \$Z\$ invariant under rotations \$r\$: \$Z(r(s),r(t))=Z(s,t)\$.
+for \$Z\$ invariant under the action of rotations \$r\in R\$, or \$Z(r\cdot s,r\cdot t)=Z(s,t)\$.
 
-<table style="border-collapse: collapse; table-layout: fixed; width: 80%; margin: auto;">
+<table style="border-collapse: collapse; width: 100%; 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><i>X</i><td><i>R</i></td><td>Group Action</td><td>$Z(s,t)$</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>
+      <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>$\mathop{\mathrm{SO}}(n)$ (rotation)</td><td>$s_i\cdot s_j$</td><td>$H\cdot s$</td>
+      <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>$H\cdot s$</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>
+      <td class="first">Potts model</td><td>ℤ/<i>q</i>ℤ</td><td>ℤ/<i>q</i>ℤ</td><td>$m\cdot s\mapsto{(s+m)}\pmod q$</td><td>$\delta(s,t)$</td><td>$\sum_mH_m\delta(m,s)$</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>
+      <td class="first">Clock model</td><td>ℤ/<i>q</i>ℤ</td><td>ℤ/<i>q</i>ℤ</td><td>$m\cdot s\mapsto{(s+m)}\pmod q$</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>
@@ -76,17 +78,17 @@ class: split-40
   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}\\}\$.
+  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 *τ* at critical point, *t* <sup>– *z/ν*</sup> `$\tau\sim t^{-z/\nu}$`
+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
+  `$z$` takes large integer values for Ising, order-`$n$`, Potts model critical regions.
   ]
 
   .column[
@@ -100,18 +102,19 @@ class: split-40
 # Wolff: wow, what a solution
 
 .column[
+Non-local updates!
 
- 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 `$R$`.
+  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*
+Relies on symmetry of *Z* under group of rotations.
 
-Fast near the critical point: early studies thought `$z$` was zero, actually
-  0.1–0.4.
+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.
 
   ]
 
@@ -121,36 +124,50 @@ Fast near the critical point: early studies thought `$z$` was zero, actually
 
 ---
 
-# We want to apply an external field, though
+  ![scooped](figs/wolff-scoop_title.png)
+
+---
+
+  ![scoop details](figs/wolff-scoop_explanation.png)
+
+---
+
+class: split-40
+
+.column[
 
-  The external field `$H$` is not invariant under global rotations!
+# Applying an *arbitrary* field
 
-  Let's make it that way: introduce an extra spin `$s_0$`, let `$R_s$` be the rotation that takes `$s$` to the
-  identity
+  Introduce an extra “spin” `$r_0\in R$` 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
   `\[
-    \tilde Z(s_i,s_j)=
+    \tilde Z(s,t)=
     \begin{cases}
-      Z(s_i,s_j) & \text{if $i,j\neq0$}\\
-      H(R_{s_0}s_i) & \text{if $j=0$}\\
-      H(R_{s_0}s_j) & \text{if $i=0$}
+      Z(s,t) & \text{if $s,t\in X$}\\
+      H(t^{-1}\cdot s) & \text{if $t\in R$}\\
+      H(s^{-1}\cdot t) & \text{if $s\in R$}
     \end{cases}
   \]`
+
+  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, `$\tilde
-  A(s_0,s)=A(R_{s_0}s)$` has the property
+  models: if `$A(\{s\})$` is an observable on old model in field, `$\tilde
+  A(r_0,\{s\})=A(\{r_0^{-1}\cdot s\})$` has the property
   `\[
     \langle\tilde
-    A\rangle=\mathop{\mathrm{Tr}}\nolimits_s\mathop{\mathrm{Tr}}\nolimits_{s_0}\tilde
-    A(s_0,s)=\mathop{\mathrm{Tr}}\nolimits_sA(s)=\langle A\rangle
+    A\rangle=\mathop{\mathrm{Tr}}\nolimits_{r_0\in R}\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}\tilde
+    A(r_0,\{s\})=\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}A(\{s\})=\langle A\rangle
   \]`
 
----
-
-  ![scooped](figs/wolff-scoop_title.png)
-
----
+  ]
 
-  ![scoop details](figs/wolff-scoop_explanation.png)
+  .column[
+<video width="320" height="640" style="float:right;"><source src="figs/wolff_ising_field.webm" type="video/webm"></video>
+]
 
 ---
 
@@ -158,13 +175,32 @@ Fast near the critical point: early studies thought `$z$` was zero, actually
 
 <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">  <i>R</i> <br>→</span>
+<span style="font-size: 30pt; overflow: hidden; height: 7em; text-align: center; align-items: center; display: inline-flex; position: static; float: center">  `\[\xrightarrow{r\in O(2)}\]`</span>
 
 ---
 
-# Why is the extended method useful?
+class: split-40
+
+.column[
+
+# Example: Clock Potts
+
+Consider the clock Potts model with field
+`\[
+H(m)=0.01\times\begin{cases}
+0 & \text{if $m=0$ (black)}\\
+\cos(2\pi/6) & \text{if $m=1$ (grey}\\
+-\cos(2\pi/6) & \text{if $m=2$ (white)}
+\end{cases}
+\]`
 
-<img src="figs/potts.svg" alt="order-n"/>
+<img src="figs/potts.svg" style="width:300px;"/>
+
+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>
+]
 
 ---
 
@@ -172,14 +208,41 @@ Fast near the critical point: early studies thought `$z$` was zero, actually
 
 Correlation time scales consistently in the whole phase space!
 
+Pictured: scaling collapses of 2D Ising model correlation time.
+
 <img src="figs/autocorr-scaling-isotherm.png" style="float: left; width: 48%" />
 <img src="figs/autocorr-scaling-temp.png" style="float: right; width: 48%" />
 
 ---
+class: split-50
+# Applications & future work
+.column[ #### Direct measurements in metastable states
+
+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: 80%; float: left;" />
+]
+
+.column[
+
+#### Symmetry-breaking perturbations of XY model
+
+For various values of `$p$`, external fields of the form
+
+`\[
+H_p(s)=\cos(p\theta(s))
+\]`
+
+have various effects on the criticality of the XY model.
+]
+
+---
+
+# Questions?
 
-# Metastable state direct measurement
+<video><source src="figs/wolff_xy_field.webm" type="video/webm"></video>
 
-

+ -- cgit v1.2.3-70-g09d2