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arXiv:cond-mat/9411110 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Dual Monte Carlo and cluster algorithms

Naoki Kawashima, James E. Gubernatis

Published Feb 1, 1995• 45 citations
arXivHTMLPDF

Abstract

We discuss the development of cluster algorithms from the viewpoint of probability theory and not from the usual viewpoint of a particular model. From the viewpoint of probability theory, we detail the nature of a cluster algorithm, make explicit the assumptions embodied in all clusters of which we are aware, and define the construction of free-cluster algorithms. We also illustrate these procedures by rederiving the Swendsen-Wang algorithm, presenting the details of the loop algorithm for a worldline simulation of a quantum S=1/2 model, and proposing a free-cluster version of the Swendsen-Wang replica method for the random Ising model. How the principle of maximum entropy might be used to aid the construction of cluster algorithms is also discussed.

DOI
10.1103/physreve.51.1547

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We discuss the development of cluster algorithms from the viewpoint of probability theory and not from the usual viewpoint of a particular model. From the viewpoint of probability theory, we detail the nature of a cluster algorithm, make explicit the assumptions embodied in all clusters of which we are aware, and define the construction of free-cluster algorithms. We also illustrate these procedures by rederiving the Swendsen-Wang algorithm, presenting the details of the loop algorithm for a worldline simulation of a quantum S=1/2 model, and proposing a free-cluster version of the Swendsen-Wang replica method for the random Ising model. How the principle of maximum entropy might be used to aid the construction of cluster algorithms is also discussed.
The purpose of this lecture is to discuss in detail the generalized approach of Kawashima and Gubernatis for the construction of cluster algorithms. We first present a brief refresher on the Monte Carlo method, describe the Swendsen-Wang algorithm, show how this algorithm follows from the Fortuin-Kastelyn transformation, and re=interpret this transformation in a form which is the basis of the generalized approach. We then derive the essential equations of the generalized approach. This derivation is remarkably simple if done from the viewpoint of probability theory, and the essential assumptions will be clearly stated. These assumptions are implicit in all useful cluster algorithms of which we are aware. They lead to a quite different perspective on cluster algorithms than found in the seminal works and in Ising model applications. Next, we illustrate how the generalized approach leads to a cluster algorithm for world-line quantum Monte Carlo simulations of Heisenberg models with S = 1/2. More succinctly, we also discuss the generalization of the Fortuin- Kasetelyn transformation to higher spin models and illustrate the essential steps for a S = 1 Heisenberg model. Finally, we summarize how to go beyond S = 1 to a general spin, XYZ model.
This talk will review the history of both cluster and histogram methods for Monte Carlo simulations. Cluster methods are based on the famous exact mapping by Fortuin and Kasteleyn from general Potts models onto a percolation representation. I will discuss the Swendsen‐Wang algorithm, as well as its improvement and extension to more general spin models by Wolff. The Replica Monte Carlo method further extended cluster simulations to deal with frustrated systems. The history of histograms is quite extensive, and can only be summarized briefly in this talk. It goes back at least to work by Salsburg et al. in 1959. Since then, it has been forgotten and rediscovered several times. The modern use of the method has exploited its ability to efficiently determine the location and height of peaks in various quantities, which is of prime importance in the analysis of critical phenomena. The extensions of this approach to the multiple histogram method and multicanonical ensembles have allowed information to be obtained over a broad range of parameters. Histogram simulations and analyses have become standard techniques in Monte Carlo simulations.
We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point.By combining the replica-exchange algorithm with cluster updates and an adaptive routine to find the range of interest, we introduce a new flexible and powerful method for systematic investigations of critical phenomena.As a result, we gain two further orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.