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diff --git a/monte-carlo.tex b/monte-carlo.tex index 70cc952..7f1e042 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -72,17 +72,20 @@ distribution of systems states. These Monte Carlo algorithms are better the faster they arrive at a statistically independent sample. This typically becomes a problem near critical points, where critical slowing down \cite{wolff_critical_1990} results in power-law divergences of dynamic -timescales. Celebrated cluster algorithms largely addressed this in the -absence of symmetry-breaking fields by using nonlocal updates -\cite{janke_nonlocal_1998} whose clusters undergo a percolation transition at -the critical point of the system \cite{coniglio_clusters_1980}. These result -in relatively small dynamic exponents for many spin systems -\cite{wolff_comparison_1989, du_dynamic_2006, liu_dynamic_2014, -wang_cluster_1990}, including the Ising, $\mathrm O(n)$ -\cite{wolff_collective_1989}, and Potts \cite{swendsen_nonuniversal_1987, -baillie_comparison_1991} models. These algorithms rely on the natural symmetry -of the systems in question under symmetry operations on the spins. Some -success has been made in extending these algorithms to systems in certain +timescales. + +Celebrated cluster algorithms largely addressed this in the absence of +symmetry-breaking fields by using nonlocal updates \cite{janke_nonlocal_1998} +whose clusters undergo a percolation transition at the critical point of the +system \cite{coniglio_clusters_1980}. These result in relatively small +dynamic exponents for many spin systems \cite{wolff_comparison_1989, +du_dynamic_2006, liu_dynamic_2014, wang_cluster_1990}, including the Ising, +$\mathrm O(n)$ \cite{wolff_collective_1989}, and Potts +\cite{swendsen_nonuniversal_1987, baillie_comparison_1991} models. These +algorithms rely on the natural symmetry of the systems in question under +symmetry operations on the spins. + +Some success has been made in extending these algorithms to systems in certain external fields by adding a `ghost site' \cite{coniglio_exact_1989} that returns global rotation invariance to spin Hamiltonians at the cost of an extra degree of freedom, allowing the method to be used in a subcategory of @@ -95,16 +98,18 @@ using replica methods \cite{redner_graphical_1998, chayes_graphical_1998, machta_replica-exchange_2000}. Monte Carlo techniques that involve cluster updates at fixed magnetization have been used to examine quantities at fixed field by integrating the associated thermodynamic functions -\cite{martin-mayor_cluster_2009, martin-mayor_tethered_2011}. We show that -the scaling of correlation time near the critical point of several models -suggests that the `ghost' approach is a natural one, e.g., that it extends the -celebrated scaling of dynamics in these algorithms at zero field to various -non-symmetric perturbations. We also show, by a redefinition of the spin--spin -coupling in a generic class of spin systems, \emph{arbitrary} external fields -can be treated using cluster methods. Rather than the introduction of a `ghost -spin,\!' our representation relies on introducing a `ghost transformation.\!' - -\section{Introduction} +\cite{martin-mayor_cluster_2009, martin-mayor_tethered_2011}. + +We show that the scaling of correlation time near the critical point of +several models suggests that the `ghost' approach is a natural one, e.g., that +it extends the celebrated scaling of dynamics in these algorithms at zero +field to various non-symmetric perturbations. We also show, by a redefinition +of the spin--spin coupling in a generic class of spin systems, +\emph{arbitrary} external fields can be treated using cluster methods. Rather +than the introduction of a `ghost spin,\!' our representation relies on +introducing a `ghost transformation.\!' + +\section{Generalized Clusters Without a Field} We will pose the problem in a general way, but several specific examples can be found in Table~\ref{table:models} for concreteness. Let $G=(V,E)$ be a graph, where the set of vertices $V=\{1,\ldots,N\}$ |