From 4025fcaf3995c976d4959d82d91e7a8914ee268e Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 9 Mar 2018 01:52:39 -0500 Subject: DONE --- aps_mm_2018.html | 46 ++++++++++++++++++++++++++-------------------- 1 file 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[ - + ] --- @@ -119,7 +119,7 @@ Fast near the critical point: early studies thought `\(z\)` was zero for 2D Isin ] .column[ - + ] --- @@ -138,7 +138,7 @@ class: split-40 # Applying an *arbitrary* field - Introduce an extra “spin” `\(r_0\in R\)` whose possible configurations come from the set of *rotations*, not the set of spin states. + Introduce an extra “spin” `\(s_0\in R\)` 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 order-n order-n -  `\[\xrightarrow{r\in O(2)}\]` +  `\[\xrightarrow{r\in \mathrm O(2)}\]` --- @@ -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 & 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. - - -] - .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 Joeé, 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. + + ] --- # Questions? - +