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authorJaron Kent-Dobias <jaron@kent-dobias.com>2018-03-08 15:57:05 -0500
committerJaron Kent-Dobias <jaron@kent-dobias.com>2018-03-08 15:57:05 -0500
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-rw-r--r--aps_mm_2018.html155
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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 @@
<link rel="stylesheet" type="text/css" href="fonts/Concrete/cmun-concrete.css">
</head>
<body>
- <textarea id="source">
+
+ <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 &ldquo;spin&rdquo; `\(r_0\in R\)` 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">&nbsp;&nbsp;<i>R</i>&nbsp;<br>&rarr;</span>
+<span style="font-size: 30pt; overflow: hidden; height: 7em; text-align: center; align-items: center; display: inline-flex; position: static; float: center">&nbsp;&nbsp;`\[\xrightarrow{r\in O(2)}\]`</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 &amp; 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>
- </textarea>
+ </textarea>
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