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authorJaron Kent-Dobias <jaron@kent-dobias.com>2018-03-09 01:52:39 -0500
committerJaron Kent-Dobias <jaron@kent-dobias.com>2018-03-09 01:52:39 -0500
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1 files changed, 26 insertions, 20 deletions
diff --git a/aps_mm_2018.html b/aps_mm_2018.html
index 3be9597..a12c96e 100644
--- a/aps_mm_2018.html
+++ b/aps_mm_2018.html
@@ -92,7 +92,7 @@ Correlation time `\(\tau\sim L^z\)` at critical point, `\(\tau\sim t^{-z/\nu}\)`
]
.column[
-<video width="320" height="320"><source src="figs/metropolis_ising_zerofield.webm" type="video/webm"></video>
+<video width="320" height="320" style="float: right;"><source src="figs/metropolis_ising_zerofield.webm" type="video/webm"></video>
]
---
@@ -119,7 +119,7 @@ Fast near the critical point: early studies thought `\(z\)` was zero for 2D Isin
]
.column[
-<video width="320" height="320"><source src="figs/wolff_ising_zerofield.webm" type="video/webm"></video>
+<video width="320" height="320" style="float: right;"><source src="figs/wolff_ising_zerofield.webm" type="video/webm"></video>
]
---
@@ -138,7 +138,7 @@ class: split-40
# Applying an *arbitrary* field
- 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.
+ Introduce an extra &ldquo;spin&rdquo; `\(s_0\in R\)` 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.
@@ -147,8 +147,8 @@ class: split-40
\tilde Z(s,t)=
\begin{cases}
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 $s\in R$}
+ B(t^{-1}\cdot s) & \text{if $t\in R$}\\
+ B(s^{-1}\cdot t) & \text{if $s\in R$}
\end{cases}
\]`
@@ -156,11 +156,11 @@ class: split-40
Exact correspondence between expectation values of operators in old and new
models: if `\(A(\{s\})\)` is an observable on old model in field, `\(\tilde
- A(r_0,\{s\})=A(\{r_0^{-1}\cdot s\})\)` has the property
+ A(s_0,\{s\})=A(\{s_0^{-1}\cdot s\})\)` has the property
`\[
\langle\tilde
- A\rangle=\mathop{\mathrm{Tr}}\nolimits_{r_0\in R}\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}\tilde
- A(r_0,\{s\})=\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}A(\{s\})=\langle A\rangle
+ A\rangle=\mathop{\mathrm{Tr}}\nolimits_{s_0\in R}\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}\tilde
+ A(s_0,\{s\})=\mathop{\mathrm{Tr}}\nolimits_{\{s\}\in X^N}A(\{s\})=\langle A\rangle
\]`
]
@@ -175,7 +175,7 @@ class: split-40
<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">&nbsp;&nbsp;`\[\xrightarrow{r\in O(2)}\]`</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;`\[\xrightarrow{r\in \mathrm O(2)}\]`</span>
---
@@ -213,6 +213,8 @@ class: split-40
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.
]
.column[
@@ -223,7 +225,7 @@ Pictured: 2D Ising (`\(128\times128\)`) correlation time as a function of field
# Correlation time scaling
-Correlation time scales consistently in the whole phase space!
+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.
@@ -233,31 +235,35 @@ Pictured: scaling collapses of 2D Ising model correlation time.
---
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
+For various integer values `\(p\)`, external fields of the form
`\[
-H_p(s)=\cos(p\theta(s))
+B_p(s)=H\cos(p\theta(s))
\]`
have various effects on the criticality of the XY model.
+
+Theory from JoeƩ, Kadanoff, Kirkpatrick, & Nelson: perturbations are relevant *except* for `\(p=6\)`.
+
+We can test it!
+]
+
+.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: 47%;" />
]
---
# Questions?
-<video><source src="figs/wolff_xy_field.webm" type="video/webm"></video>
+<video style="margin-left: auto; margin-right: auto; margin: 0 auto; display: block; width: 75%;"><source src="figs/wolff_xy_field.webm" type="video/webm"></video>
</textarea>
<script src="https://remarkjs.com/downloads/remark-latest.min.js">