many changesHEADmaster

1 files changed, 177 insertions, 231 deletions

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}
-
-
-\begin{frame}
-  \frametitle{Introduction: The Ising Model}
-  \framesubtitle{The Fortuin--Kasteleyn representation}
+  \frametitle{Lattice models}
 
-  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 $p_{ij}=1-e^{-2\beta J_{ij}}$.
+  Cluster method for $B=0$ if self-inverse $r^{-1}=r\in G$ are transitive.
 
-  \vspace{1em}
+  \vspace{2em}
 
-  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*}
+  \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}
-
-  Rules out novel fields like harmonic lattice anisotropies, cubic potentials, around Potts first-order lines, etc.
+  \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.
 
-  \vspace{1em}
+      \vspace{1em} 
 
-  Need to track the full array of transformations that have included the ghost\dots
+      Jos\'e, Kadanoff, Kirkpatrick \& Nelson (1977) predict 
+      relevance for $n\leq4$.
 
-  \vspace{1em}
+      \vspace{1em}
 
-  \dots which is precisely what elements of the symmetry group do!
-\end{frame}
+      Ala-Nissila et al.\ (1994) used hybrid metropolis and Wolff with cluster rejection to study.
 
-\begin{frame}
-  \frametitle{Other lattice models}
-  \framesubtitle{The ghost transformation representation}
+      \vspace{1em}
 
-  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}
+      New method reveals different phenomena for $n=4,6$ faster and without rejection.
 
-\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}