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authorJaron Kent-Dobias <jaron@kent-dobias.com>2019-09-04 21:54:19 -0400
committerJaron Kent-Dobias <jaron@kent-dobias.com>2019-09-04 21:54:19 -0400
commite51722156e7d93490cb0c2b1374eb132fea26027 (patch)
treed0d231469dcd473a429308d5493a510657ae471c
parent736908493db28e892d19d7b6a63bdef1372dd196 (diff)
downloadstatphys_27-e51722156e7d93490cb0c2b1374eb132fea26027.tar.gz
statphys_27-e51722156e7d93490cb0c2b1374eb132fea26027.tar.bz2
statphys_27-e51722156e7d93490cb0c2b1374eb132fea26027.zip
many changesHEADmaster
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-rw-r--r--figs/harmonic_susceptibilities.pngbin0 -> 135219 bytes
-rw-r--r--figs/ising_hl_00_0.pdfbin0 -> 5736 bytes
-rw-r--r--figs/ising_hl_00_1.pdfbin0 -> 5733 bytes
-rw-r--r--figs/ising_hl_00_2.pdfbin0 -> 5894 bytes
-rw-r--r--figs/ising_hl_00_3.pdfbin0 -> 5725 bytes
-rw-r--r--figs/ising_hl_0_1.pdfbin5735 -> 5690 bytes
-rw-r--r--figs/ising_hl_0_2.pdfbin5931 -> 5698 bytes
-rw-r--r--figs/ising_hl_0_3.pdfbin5722 -> 7183 bytes
-rw-r--r--figs/ising_hl_0_4.pdfbin5722 -> 7186 bytes
-rw-r--r--figs/ising_hl_0_5.pdfbin0 -> 7186 bytes
-rw-r--r--figs/ising_hl_0_6.pdfbin0 -> 7186 bytes
-rw-r--r--figs/ising_hl_0_7.pdfbin0 -> 5701 bytes
-rw-r--r--figs/ising_hl_0_8.pdfbin0 -> 5942 bytes
-rw-r--r--figs/ising_hl_0_9.pdfbin0 -> 5647 bytes
-rw-r--r--figs/ising_hl_h_1.pdfbin5789 -> 5816 bytes
-rw-r--r--figs/ising_hl_h_10.pdfbin0 -> 5494 bytes
-rw-r--r--figs/ising_hl_h_11.pdfbin0 -> 5759 bytes
-rw-r--r--figs/ising_hl_h_2.pdfbin5729 -> 5822 bytes
-rw-r--r--figs/ising_hl_h_3.pdfbin5468 -> 7312 bytes
-rw-r--r--figs/ising_hl_h_4.pdfbin5763 -> 7320 bytes
-rw-r--r--figs/ising_hl_h_5.pdfbin0 -> 7317 bytes
-rw-r--r--figs/ising_hl_h_6.pdfbin0 -> 7318 bytes
-rw-r--r--figs/ising_hl_h_7.pdfbin0 -> 7304 bytes
-rw-r--r--figs/ising_hl_h_8.pdfbin0 -> 5828 bytes
-rw-r--r--figs/ising_hl_h_9.pdfbin0 -> 5798 bytes
-rw-r--r--figs/times_3dising.pngbin0 -> 125966 bytes
-rw-r--r--figs/times_3dising_new.pdfbin0 -> 18240 bytes
-rw-r--r--figs/times_3dising_new_alert.pdfbin0 -> 18246 bytes
-rw-r--r--figs/times_3do2.pngbin0 -> 105316 bytes
-rw-r--r--figs/times_3dxy_new.pdfbin0 -> 17251 bytes
-rw-r--r--figs/times_3dxy_new_alert.pdfbin0 -> 17257 bytes
-rw-r--r--ising_visualization.nb66219
-rw-r--r--plots.nb6923
-rw-r--r--statphys27.tex412
34 files changed, 73321 insertions, 233 deletions
diff --git a/figs/harmonic_susceptibilities.png b/figs/harmonic_susceptibilities.png
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diff --git a/figs/ising_hl_00_0.pdf b/figs/ising_hl_00_0.pdf
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diff --git a/figs/ising_hl_0_1.pdf b/figs/ising_hl_0_1.pdf
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diff --git a/figs/ising_hl_0_2.pdf b/figs/ising_hl_0_2.pdf
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diff --git a/figs/ising_hl_0_3.pdf b/figs/ising_hl_0_3.pdf
index d091bc7..edd850f 100644
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diff --git a/figs/ising_hl_0_4.pdf b/figs/ising_hl_0_4.pdf
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diff --git a/figs/ising_hl_0_6.pdf b/figs/ising_hl_0_6.pdf
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diff --git a/figs/ising_hl_0_8.pdf b/figs/ising_hl_0_8.pdf
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diff --git a/figs/ising_hl_0_9.pdf b/figs/ising_hl_0_9.pdf
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diff --git a/figs/ising_hl_h_1.pdf b/figs/ising_hl_h_1.pdf
index 4f7e278..11126f4 100644
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diff --git a/figs/ising_hl_h_10.pdf b/figs/ising_hl_h_10.pdf
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diff --git a/figs/ising_hl_h_11.pdf b/figs/ising_hl_h_11.pdf
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diff --git a/figs/ising_hl_h_2.pdf b/figs/ising_hl_h_2.pdf
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diff --git a/figs/ising_hl_h_3.pdf b/figs/ising_hl_h_3.pdf
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diff --git a/figs/ising_hl_h_4.pdf b/figs/ising_hl_h_4.pdf
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diff --git a/figs/ising_hl_h_5.pdf b/figs/ising_hl_h_5.pdf
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diff --git a/figs/ising_hl_h_6.pdf b/figs/ising_hl_h_6.pdf
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diff --git a/figs/ising_hl_h_7.pdf b/figs/ising_hl_h_7.pdf
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diff --git a/figs/times_3dising.png b/figs/times_3dising.png
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diff --git a/figs/times_3dising_new.pdf b/figs/times_3dising_new.pdf
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diff --git a/figs/times_3dising_new_alert.pdf b/figs/times_3dising_new_alert.pdf
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diff --git a/figs/times_3do2.png b/figs/times_3do2.png
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diff --git a/figs/times_3dxy_new_alert.pdf b/figs/times_3dxy_new_alert.pdf
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+ AxesOrigin->{0, 0},
+ CoordinatesToolOptions:>{"DisplayFunction" -> ({
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+ "Effects" -> {
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+ "placement" -> {"x" -> "All", "y" -> "None"}}}},
+ "DefaultMeshStyle" -> AbsolutePointSize[6], "DefaultPlotStyle" -> {
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diff --git a/statphys27.tex b/statphys27.tex
index 2524aef..048f7b3 100644
--- a/statphys27.tex
+++ b/statphys27.tex
@@ -22,343 +22,289 @@
\end{frame}
\begin{frame}
- \frametitle{Monte Carlo is too slow}
+ \frametitle{Simulating critical lattice models is slow}
\begin{columns}
\begin{column}{0.5\textwidth}
- Critical timescales diverge like $L^z$.
+ \alert<2>{Critical timescales diverge like $L^z$.}
- \vspace{2em}
+ \vspace{1em}
- For 2D Ising local algorithms have $z\simeq2$--4.
+ \alert<3-4>{2D Ising local algorithms have $z\simeq2$--4.}
- \vspace{2em}
+ \vspace{1em}
- Cluster methods have $z\simeq0.3$!
+ \alert<5-9>{Cluster methods have $z\simeq0.3$!}
- \vspace{2em}
+ \vspace{1em}
+ \alert<10-11>{Don't naturally work with on-site potentials like external fields.}
+
+ \vspace{1em}
+
+ \alert<12>{Our extension admits arbitrary on-site potentials for most lattice models.}
\end{column}
\begin{column}{0.5\textwidth}
\begin{overprint}
- \onslide<1>\includegraphics[width=\columnwidth]{figs/ising_hl_0_0}
- \onslide<2,4>\includegraphics[width=\columnwidth]{figs/ising_hl_0_1}
- \onslide<5>\includegraphics[width=\columnwidth]{figs/ising_hl_0_2}
- \onslide<3,6>\includegraphics[width=\columnwidth]{figs/ising_hl_0_3}
+ \onslide<1-3>\includegraphics[width=\columnwidth]{figs/ising_hl_00_0}
+ \onslide<4-5,7>\includegraphics[width=\columnwidth]{figs/ising_hl_00_1}
+ \onslide<8,11->\includegraphics[width=\columnwidth]{figs/ising_hl_00_2}
+ \onslide<6,9-10>\includegraphics[width=\columnwidth]{figs/ising_hl_00_3}
\end{overprint}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
- \frametitle{Monte Carlo is too slow}
-
- Monte Carlo useful for lattice models, but near critical points suffers from \emph{critical slowing down}, power-law divergence of timescales.
-
- \vspace{1em}
-
- Often alleviated with cluster algorithms, but many applications lack a clean solution.
-
- \vspace{1em}
-
- We introduce a generic, natural, efficient way to extend models with existing cluster algorithms to operate in arbitrary external fields.
-
- \vspace{1em}
-
- \begin{enumerate}
- \item Introduction: The Ising Model
- \begin{enumerate}
- \item The Fortuin--Kasteleyn representation \& related algorithm
- \item The ghost spin Hamiltonian \& extension to a field
- \end{enumerate}
- \item Our work: Other lattice models
- \begin{enumerate}
- \item Fortuin--Kasteleyn representations \& algorithms via Ising embeddings
- \item The ghost transformation Hamiltonian \& clusters in arbitrary fields
- \end{enumerate}
- \end{enumerate}
-
-\end{frame}
+ \frametitle{Lattice models}
-
-\begin{frame}
- \frametitle{Introduction: The Ising Model}
- \framesubtitle{The Fortuin--Kasteleyn representation}
-
- The Ising model
- $
- \mathcal H=-\sum_{\langle ij\rangle}J_{ij}s_is_j
- $
- for $s_i=\pm1$ can be written
+ Consider spins $s\in X$ with symmetry group $G$ and
\[
- Z=\tr_se^{-\beta\mathcal H}\propto\tr_f\tr_s\prod_{\langle ij\rangle}\big[\delta_{f_{ij},0}(1-p_{ij})+\delta_{f_{ij},1}\delta_{s_i,s_j}p_{ij}\big]
+ \mathcal H=-\sum_{\langle ij\rangle}J(s_i,s_j)-\sum_iB(s_i)
\]
- for $f_{ij}\in\{0,1\}$ on the bonds and $p_{ij}=1-e^{-2\beta J_{ij}}$.
-
- \vspace{1em}
-
- This gives conditional probabilities
- \begin{align*}
- P(f_{ij}=1\mid s_i,s_j)=\begin{cases}p_{ij} & s_i=s_j \\ 0 & s_i\neq s_j\end{cases}
- &&
- P(s_i=s_j\mid f)=\begin{cases}1 & \text{$i$, $j$ in same cluster} \\ \frac12 & \text{otherwise}\end{cases}
- \end{align*}
+ Cluster method for $B=0$ if self-inverse $r^{-1}=r\in G$ are transitive.
+
+ \vspace{2em}
+
+ \small
+ \begin{tabular}{l|ccl}
+ \hline
+ & spin space ($X$) & symmetry group ($G$) & self-inverse elements ($r$'s) \\
+ \hline
+ \alert<2>{Ising} & \alert<2>{$\pm1$} & \alert<2>{$\mathbb Z_2$} & \alert<2>{spin flips} \\
+ \alert<3>{$n$-vector} & \alert<3>{$(n-1)$ sphere} & \alert<3>{$\mathrm O(n)$} & \alert<3>{reflections through origin} \\
+ Potts & $\{1,\ldots,q\}$ & Symmetric ($S_q$)& transpositions \\
+ Clock & regular $q$-gon vertices & Dihedral ($D_q$) & reflections through origin\\
+ Roughening & $\mathbb Z$ & Infinite Dihedral ($D_\infty$)& subtraction by integer\\
+ \alert<4>{Chiral Potts} & \alert<4>{$\{1,\ldots,q\}$} & \alert<4>{$\mathbb Z/q\mathbb Z$} & \alert<4>{none}
+ \end{tabular}
\end{frame}
\begin{frame}
- \frametitle{Introduction: The Ising Model}
- \framesubtitle{From representation to algorithm}
+ \frametitle{Cluster methods without potentials}
\begin{columns}
- \begin{column}{0.55\textwidth}
- \begin{overprint}
- \onslide<1-2>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_1}
- \onslide<3>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_2}
- \onslide<4>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_3}
- \onslide<5>\includegraphics[height=0.8\textheight]{figs/ising_fk_0_4}
- \end{overprint}
- \end{column}
- \begin{column}{0.45\textwidth}
- The joint probabilities imply algorithm based on switching back an forth:
+ \begin{column}{0.5\textwidth}
+ With $\Delta\mathcal H_{ij}=J(r\cdot s_i, s_j)-J(s_i, s_j)$ and
+ \[
+ p_{ij}=\begin{cases}1-e^{\beta\Delta\mathcal H_{ij}} & \Delta\mathcal H_{ij}>0 \\ 0 & \text{otherwise,}\end{cases}
+ \]
\begin{enumerate}
- \item\alert<2>{Take a spin configuration.}
- \item\alert<3>{Conditionally sample a configuration of bonds.}
- \item\alert<4>{Gather sites connected by bonds into clusters.}
- \item\alert<5>{Conditionally sample a configuration of spins.}
+ \item \alert<2>{Pick self-inverse $r\in G$.}
+ \item \alert<3>{Pick a random site, add to cluster.}
+ \item \alert<4-7>{Add neighbors to cluster with probability $p_{ij}$.}
+ \item \alert<8-9>{Repeat for all sites added to cluster.}
+ \item \alert<10>{Apply $r$ to cluster.}
\end{enumerate}
-
- \vspace{1em}
-
- \tiny\raggedleft {Swendsen \& Wang, Phys Rev Lett \textbf{58} (1987) 56.}
+ \end{column}
+ \begin{column}{0.5\textwidth}
+ \begin{overprint}
+ \onslide<1-2>\includegraphics[width=\columnwidth]{figs/ising_hl_0_1}
+ \onslide<3>\includegraphics[width=\columnwidth]{figs/ising_hl_0_2}
+ \onslide<4>\includegraphics[width=\columnwidth]{figs/ising_hl_0_3}
+ \onslide<5>\includegraphics[width=\columnwidth]{figs/ising_hl_0_4}
+ \onslide<6>\includegraphics[width=\columnwidth]{figs/ising_hl_0_5}
+ \onslide<7>\includegraphics[width=\columnwidth]{figs/ising_hl_0_6}
+ \onslide<8>\includegraphics[width=\columnwidth]{figs/ising_hl_0_7}
+ \onslide<9>\includegraphics[width=\columnwidth]{figs/ising_hl_0_8}
+ \onslide<10>\includegraphics[width=\columnwidth]{figs/ising_hl_0_9}
+ \end{overprint}
\end{column}
\end{columns}
+
\end{frame}
+
\begin{frame}
- \frametitle{Introduction: The Ising Model}
- \framesubtitle{The ghost spin representation}
+ \frametitle{The ghost site representation}
\begin{columns}
- \begin{column}{0.4\textwidth}
- A field means clusters flip with probability that depends on size.
-
- \vspace{1em}
-
- But, Fortuin--Kasteleyn doesn't care about lattice topology! Adding a ghost spin coupled to every site with $\tilde J_{0i}=H_i$ gives
- \[
- \begin{aligned}
- \tilde{\mathcal H}&=-\sum_{\langle ij\rangle}J_{ij}s_is_j-s_0\sum_iH_is_i \\
- &=-\sum_{\langle ij\rangle}\tilde J_{ij}s_is_j
- \end{aligned}
- \]
- \end{column}
- \begin{column}{0.6\textwidth}
+ \begin{column}{0.5\textwidth}
\includegraphics[width=\textwidth]{figs/ghost_site}
\vspace{1em}
- \tiny\raggedleft {Coniglio, de Liberto, Monroy, \& Peruggi. J Phys A: Math Gen \textbf{22} (1989) L837.}
- \end{column}
- \end{columns}
-\end{frame}
-\begin{frame}
- \frametitle{Introduction: The Ising Model}
- \framesubtitle{The ghost spin algorithm}
- \begin{columns}
- \begin{column}{0.55\textwidth}
- \begin{overprint}
- \onslide<1>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_1}
- \onslide<2>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_2}
- \onslide<3>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_3}
- \onslide<4>\includegraphics[height=0.8\textheight]{figs/ising_fk_h_4}
- \end{overprint}
+ \tiny{Coniglio, de Liberto, Monroy, \& Peruggi. J Phys A \textbf{22} (1989) L837}
\end{column}
- \begin{column}{0.45\textwidth}
- Same algorithm can be run on new Hamiltonian without modification.
+ \begin{column}{0.5\textwidth}
+ \alert<2>{Introduce new site $0$ adjacent to all others.}
\vspace{1em}
- If the cluster containing $s_0$ is flipped, flip it too!
+ \alert<3>{Draw object $s_0\in G$ on site from symmetry group $G$, \emph{not} spin space $X$.}
\vspace{1em}
- Properties of the original spins must be taken after ``unflipping'' the external field, or $s_0\times s$.
-
+ \alert<4>{Take the new Hamiltonian
+ \[
+ \begin{aligned}
+ \tilde{\mathcal H}
+ &=-\sum_{\langle ij\rangle}J(s_i,s_j)-\sum_iB(s_0^{-1}\cdot s_i) \\
+ &=-\sum_{\langle ij\rangle'}\tilde J(s_i,s_j)
+ \end{aligned}
+ \]
+ for $\tilde J(s_0,s_i)=B(s_0^{-1}\cdot s_i)$.}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
- \frametitle{Other lattice models}
- \framesubtitle{Fortuin--Kasteleyn via embeddings}
+ \frametitle{Cluster methods with potentials}
+ \framesubtitle{Ising model with uniform $B(s)=Hs$}
\begin{columns}
- \begin{column}{0.4\textwidth}
- Cluster methods also known for models whose spins live in more complicated spaces $X$ and have
- \[
- \mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)
- \]
- If $G$ is the symmetry group of the spins, then a self-inverse element $r\in G$ can embed an Ising model
- \[
- J_{ij}(r,s)=\frac12|Z(s_i, s_j)-Z(s_i, r\cdot s_j)|
- \]
+ \begin{column}{0.5\textwidth}
+ With $\Delta\tilde{\mathcal H}_{ij}=\tilde J(r\cdot s_i,s_j)-\tilde J(s_i,s_j)$ and
+ \[
+ \tilde p_{ij}=\begin{cases}1-e^{\beta\Delta\tilde{\mathcal H}_{ij}} & \Delta\tilde{\mathcal H}_{ij}>0 \\ 0 & \text{otherwise,}\end{cases}
+ \]
+ \begin{enumerate}
+ \item \alert<2>{Pick self-inverse $r\in G$.}
+ \item \alert<3>{Pick a random site, add to cluster.}
+ \item \alert<4-8>{Add neighbors to cluster with probability $\tilde p_{ij}$.}
+ \item \alert<9-10>{Repeat for all sites added to cluster.}
+ \item \alert<11-12>{Apply $r$ to cluster.}
+ \end{enumerate}
\end{column}
- \begin{column}{0.6\textwidth}
- \centering
- \begin{tabular}{l|cc}
- & $X$ & $G$ \\
- \hline
- Ising & $\pm1$ & $\mathbb Z/2\mathbb Z$ \\
- $n$-vector & $(n-1)$ sphere & $\mathrm O(n)$ \\
- Potts & $\{1,\ldots,q\}$ & Symmetric \\
- Clock & $\{1,\ldots,q\}$ & Dihedral \\
- Roughening & $\mathbb Z$ & Infinite Dihedral
- \end{tabular}
-
- \vspace{1em}
-
- \includegraphics[width=0.9\textwidth]{figs/clocks}
+ \begin{column}{0.5\textwidth}
+ \begin{overprint}
+ \onslide<1-2>\includegraphics[width=\columnwidth]{figs/ising_hl_h_1}
+ \onslide<3>\includegraphics[width=\columnwidth]{figs/ising_hl_h_2}
+ \onslide<4>\includegraphics[width=\columnwidth]{figs/ising_hl_h_3}
+ \onslide<5>\includegraphics[width=\columnwidth]{figs/ising_hl_h_4}
+ \onslide<6>\includegraphics[width=\columnwidth]{figs/ising_hl_h_5}
+ \onslide<7>\includegraphics[width=\columnwidth]{figs/ising_hl_h_6}
+ \onslide<8>\includegraphics[width=\columnwidth]{figs/ising_hl_h_7}
+ \onslide<9>\includegraphics[width=\columnwidth]{figs/ising_hl_h_8}
+ \onslide<10>\includegraphics[width=\columnwidth]{figs/ising_hl_h_9}
+ \onslide<11>\includegraphics[width=\columnwidth]{figs/ising_hl_h_10}
+ \onslide<12>\includegraphics[width=\columnwidth]{figs/ising_hl_h_11}
+ \end{overprint}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
- \frametitle{Other lattice models}
- \framesubtitle{From embedding to algorithm\dots again}
+ \frametitle{Cluster methods with potentials}
+ \framesubtitle{XY model with $B(s)=h_5\cos(5\theta)$}
+
\begin{columns}
- \begin{column}{0.55\textwidth}
- \begin{overprint}
- \onslide<1-2>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_1}
- \onslide<3>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_2}
- \onslide<4>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_3}
- \onslide<5>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_4}
- \onslide<6>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_5}
- \onslide<7>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_6}
- \onslide<8>\includegraphics[height=0.8\textheight]{figs/ising_fks_0_7}
- \end{overprint}
- \end{column}
- \begin{column}{0.45\textwidth}
+ \begin{column}{0.5\textwidth}
+ With $\Delta\tilde{\mathcal H}_{ij}=\tilde J(r\cdot s_i,s_j)-\tilde J(s_i,s_j)$ and
+ \[
+ \tilde p_{ij}=\begin{cases}1-e^{\beta\Delta\tilde{\mathcal H}_{ij}} & \Delta\tilde{\mathcal H}_{ij}>0 \\ 0 & \text{otherwise,}\end{cases}
+ \]
\begin{enumerate}
- \item\alert<2>{Take a spin configuration.}
- \item\alert<3>{Draw a self-inverse $r\in G$.}
- \item\alert<4>{Infer Ising $J_{ij}$.}
- \item\alert<5>{Sample bonds as before.}
- \item\alert<6>{Gather sites into clusters.}
- \item\alert<7>{Sample spins by applying $r$ to clusters.}
+ \item \alert<2>{Pick self-inverse $r\in G$.}
+ \item \alert<3>{Pick a random site, add to cluster.}
+ \item \alert<3>{Add neighbors to cluster with probability $\tilde p_{ij}$.}
+ \item \alert<3>{Repeat for all sites added to cluster.}
+ \item \alert<4-6>{Apply $r$ to cluster.}
\end{enumerate}
-
- \vspace{1em}
-
- \tiny\raggedleft{Wolff, Phys Rev Lett \textbf{62} (1989) 361.}
+ \end{column}
+ \begin{column}{0.5\textwidth}
+ \begin{overprint}
+ \onslide<1>\includegraphics[width=\columnwidth]{figs/vector_hl_h5_0}
+ \onslide<2>\includegraphics[width=\columnwidth]{figs/vector_hl_h5_1}
+ \onslide<3>\includegraphics[width=\columnwidth]{figs/vector_hl_h5_2}
+ \onslide<4>\includegraphics[width=\columnwidth]{figs/vector_hl_h5_3}
+ \onslide<5>\includegraphics[width=\columnwidth]{figs/vector_hl_h5_4}
+ \onslide<6>\includegraphics[width=\columnwidth]{figs/vector_hl_h5_5}
+ \end{overprint}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
- \frametitle{Other lattice models}
- \framesubtitle{The ghost\dots something representation}
+ \frametitle{Cluster methods with potentials}
+ \framesubtitle{XY model with $B(s)=h_n\cos(n\theta)$}
- Can we add an external field with a ghost spin as before? Yes, but only for fields whose interaction is like that of another spin.
-
- \vspace{1em}
+ \begin{columns}
+ \begin{column}{0.38\textwidth}
+ \includegraphics[width=\columnwidth]{figs/harmonic_susceptibilities}
+ \end{column}
+ \begin{column}{0.62\textwidth}
+ Symmetry breaking fields $B(\theta)=h_n\cos(n\theta)$ expected due to lattice anisotropies.
- Rules out novel fields like harmonic lattice anisotropies, cubic potentials, around Potts first-order lines, etc.
+ \vspace{1em}
- \vspace{1em}
+ Jos\'e, Kadanoff, Kirkpatrick \& Nelson (1977) predict
+ relevance for $n\leq4$.
- Need to track the full array of transformations that have included the ghost\dots
+ \vspace{1em}
- \vspace{1em}
+ Ala-Nissila et al.\ (1994) used hybrid metropolis and Wolff with cluster rejection to study.
- \dots which is precisely what elements of the symmetry group do!
-\end{frame}
+ \vspace{1em}
-\begin{frame}
- \frametitle{Other lattice models}
- \framesubtitle{The ghost transformation representation}
+ New method reveals different phenomena for $n=4,6$ faster and without rejection.
- For a lattice model with spins with symmetry group $G$ and
- \[
- \mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iB(s_i)
- \]
- for any function $B$, we introduce a ghost \emph{transformation} $s_0$ and modified Hamiltonian
- \[
- \tilde{\mathcal H}=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iB(s_0^{-1}\cdot s_i)
- =-\sum_{\langle ij\rangle}\tilde Z(s_i,s_j)
- \]
- for $\tilde Z(s_0,s_i)=B(s_0^{-1}\cdot s_i)$. Both Hamiltonians yields the same statistics for $s_i$ if accumulated transformations are undone first with $s_0^{-1}\cdot s_i$.
-\end{frame}
-
-\begin{frame}
- \frametitle{Other lattice models}
- \framesubtitle{Ghost transformation in action}
- \begin{columns}
- \begin{column}{0.55\textwidth}
- \begin{overprint}
- \onslide<1-2>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_1}
- \onslide<3>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_2}
- \onslide<4>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_3}
- \onslide<5>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_4}
- \onslide<6>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_5}
- \onslide<7>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_6}
- \onslide<8>\includegraphics[height=0.8\textheight]{figs/potts_fk_h_7}
- \end{overprint}
- \end{column}
- \begin{column}{0.45\textwidth}
- Example: 5-spin clock model with a field favoring the two states to the bottom right.
- \begin{enumerate}
- \item\alert<2>{Take a spin configuration.}
- \item\alert<3>{Draw a self-inverse $r\in G$.}
- \item\alert<4>{Infer Ising $J_{ij}$.}
- \item\alert<5>{Sample bonds as before.}
- \item\alert<6>{Gather sites into clusters.}
- \item\alert<7>{Sample spins by applying $r$ to clusters.}
- \end{enumerate}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
- \frametitle{Other lattice models}
- \framesubtitle{The method is good}
+ \frametitle{Is it efficient?}
+ \begin{columns}
+ \begin{column}{0.6\textwidth}
+ Yes! Extension is fast and natural.
- Results generalize to arbitrary bond and site dependence.
+ \vspace{1.5em}
- \vspace{0.5em}
+ Dynamic scaling works in entire $t$--$h$ plane for every model we've looked at.
- Models already efficient at zero field are more efficient with a field.
+ \vspace{1.5em}
- \vspace{0.5em}
+ \alert<2>{Universal scaling functions decay with predictable power law $h^{-z\nu/\beta\delta}$ with Wolff or Swendsen--Wang $z$.}
- Extension appears natural in the scaling sense.
+ \vspace{1.5em}
- \centering
+ Distribution self-inverse $r\in G$ are sampled from affects performance far from criticality.
- \includegraphics[width=0.85\textwidth]{figs/timescales}
-
+ \end{column}
+ \begin{column}{0.4\textwidth}
+ \begin{overprint}
+ \onslide<1,3->\includegraphics[width=\textwidth]{figs/times_3dising_new}
+ \onslide<2>\includegraphics[width=\textwidth]{figs/times_3dising_new_alert}
+ \end{overprint}
+ \begin{overprint}
+ \onslide<1,3->\includegraphics[width=\textwidth]{figs/times_3dxy_new}
+ \onslide<2>\includegraphics[width=\textwidth]{figs/times_3dxy_new_alert}
+ \end{overprint}
+ \end{column}
+ \end{columns}
\end{frame}
+
\begin{frame}
- \frametitle{Summary \& Extensions}
+ \frametitle{Summary}
+
+ Generic and fast extension to cluster Monte Carlo with arbitrary on-site potentials.
- Introduced a generic method for running cluster Monte Carlo on lattice systems with any external field.
-s-
\vspace{1em}
- Already used to efficiently show relevance/irrelevance of various harmonic perturbations to the XY model.
+ Demonstrated efficient for canonical fields, symmetry-breaking potentials.
\vspace{1em}
- Presently being used to model novel lattice models with coupled spins on sites and bonds which act as effective fields for each other.
+ Using now with spins on sites and bonds that act as effective fields for each other.
\vspace{1em}
- Currently working on using machine learning techniques to maximize efficiency related to the choice of the distribution of self-inverse group elements, i.e., Ising embeddings.
+ Developing a generic way to optimize distributions self-inverse $r\in G$ are drawn from.
\vspace{1em}
Phys Rev E \textbf{98}, 063306 (2018) or contact Jaron Kent-Dobias (\texttt{jpk247@cornell.edu}).
+ \vspace{2em}
+
+ \centering
+
+ \Large
+
+ Questions?
+
\end{frame}
\end{document}