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\title{Rejection-free cluster Monte Carlo in arbitrary external fields}
\subtitle{Phys Rev E \textbf{98}, 063306 (2018)}
\author{Jaron Kent-Dobias \and James P Sethna}
\institute{Cornell University}
\date{}

\begin{document}

\def\tr{\mathop{\mathrm{Tr}}\nolimits}

\begin{frame}
  \maketitle
\end{frame}

\begin{frame}
  \frametitle{Simulating critical lattice models is slow}

  \begin{columns}
    \begin{column}{0.5\textwidth}
      \alert<2>{Critical timescales diverge like $L^z$.}

      \vspace{1em}

      \alert<3-4>{2D Ising local algorithms have $z\simeq2$--4.}

      \vspace{1em}

      \alert<5-9>{Cluster methods have $z\simeq0.3$!}

      \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<1-3>\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{Lattice models}

  Consider spins $s\in X$ with symmetry group $G$ and 
  \[
    \mathcal H=-\sum_{\langle ij\rangle}J(s_i,s_j)-\sum_iB(s_i)
  \]
  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{Cluster methods without potentials}

  \begin{columns}
    \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>{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}
    \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{The ghost site representation}

  \begin{columns}
    \begin{column}{0.5\textwidth}
      \includegraphics[width=\textwidth]{figs/ghost_site}

      \vspace{1em}
      \tiny{Coniglio, de Liberto, Monroy, \& Peruggi. J Phys A \textbf{22} (1989) L837}
    \end{column}
    \begin{column}{0.5\textwidth}
      \alert<2>{Introduce new site $0$ adjacent to all others.}

      \vspace{1em}

      \alert<3>{Draw object $s_0\in G$ on site from symmetry group $G$, \emph{not} spin space $X$.}

      \vspace{1em}

      \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{Cluster methods with potentials}
  \framesubtitle{Ising model with uniform $B(s)=Hs$}

  \begin{columns}
    \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}{0.5\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{Cluster methods with potentials}
  \framesubtitle{XY model with $B(s)=h_5\cos(5\theta)$}

  \begin{columns}
    \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<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}
    \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{Cluster methods with potentials}
  \framesubtitle{XY model with $B(s)=h_n\cos(n\theta)$}

  \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} 

      Jos\'e, Kadanoff, Kirkpatrick \& Nelson (1977) predict 
      relevance for $n\leq4$.

      \vspace{1em}

      Ala-Nissila et al.\ (1994) used hybrid metropolis and Wolff with cluster rejection to study.

      \vspace{1em}

      New method reveals different phenomena for $n=4,6$ faster and without rejection.

    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Is it efficient?}
  \begin{columns}
    \begin{column}{0.6\textwidth}
      Yes! Extension is fast and natural.

      \vspace{1.5em}

      Dynamic scaling works in entire $t$--$h$ plane for every model we've looked at.

      \vspace{1.5em}

      \alert<2>{Universal scaling functions decay with predictable power law $h^{-z\nu/\beta\delta}$ with Wolff or Swendsen--Wang $z$.}

      \vspace{1.5em}

      Distribution self-inverse $r\in G$ are sampled from affects performance far from criticality.

    \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}

  Generic and fast extension to cluster Monte Carlo with arbitrary on-site potentials.

  \vspace{1em}

  Demonstrated efficient for canonical fields, symmetry-breaking potentials.

  \vspace{1em}

  Using now with spins on sites and bonds that act as effective fields for each other.

  \vspace{1em}

  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}

