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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} |