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arXiv:cond-mat/0005076 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Cluster algorithm for Potts models with fixed spin densities

R. P. Bikker, G. T. Barkema

Published Oct 1, 2000• 3 citations
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Abstract

A cluster algorithm is presented for the simulation of the q-state Potts models in which the number of spins is conserved in each state. The algorithm constructs Fortuin-Kasteleyn cluster configurations from spin configurations, in a way identical to the Swendsen-Wang algorithm; the spin assignment to these clusters is, however, different, and conserves the number of spins for each state. Compared to traditional nonlocal spin-exchange algorithms, the cluster algorithm presented here suffers less from critical slowing down, and consequently is more efficient near the critical temperature.

DOI
10.1103/physreve.62.5830

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A cluster algorithm is presented for the simulation of the q-state Potts models in which the number of spins is conserved in each state. The algorithm constructs Fortuin-Kasteleyn cluster configurations from spin configurations, in a way identical to the Swendsen-Wang algorithm; the spin assignment to these clusters is, however, different, and conserves the number of spins for each state. Compared to traditional nonlocal spin-exchange algorithms, the cluster algorithm presented here suffers less from critical slowing down, and consequently is more efficient near the critical temperature.
The simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down. A number of cluster algorithms, usually based on the Fortuin-Kasteleyn representation of the Potts model, and suitable generalizations for continuous-spin models have been used to increase simulation efficiency. The first algorithm making use of this representation, suggested by Sweeny in 1983, has not found widespread adoption due to problems in its implementation. However, it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well-known algorithm due to Swendsen and Wang. Here, we present an efficient implementation of Sweeny's approach for the random-cluster model using recent algorithmic advances in dynamic connectivity algorithms.
We propose a new effective cluster algorithm of tuning the critical point automatically, which is an extended version of the Swendsen-Wang algorithm. We change the probability of connecting spins of the same type, p = 1-e(-J/k(B)T), in the process of the Monte Carlo spin update. Since we approach the canonical ensemble asymptotically, we can use the finite-size scaling analysis for physical quantities near the critical point. Simulating the two-dimensional Potts models to demonstrate the validity of the algorithm, we have obtained the critical temperatures and critical exponents which are consistent with the exact values; the comparison has been made with the invaded cluster algorithm.
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.
Potts spin systems play a fundamental role in statistical mechanics and quantum field theory and can be studied within the spin, the Fortuin-Kasteleyn (FK) bond or the q-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with q-flow variables and formulate an LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the q-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model but also brings new insights into the rich geometric structures of the FK clusters.
A recently introduced cluster acceleration algorithm for spin models [J. Machta et al., Phys. Rev. Lett. 75, 2792 (1995)] based on invasion percolation is generalized to temperatures away from the critical point using concepts of random bond percolation. The generalized algorithm is equivalent to an "optimized" version of the Swendsen-Wang cluster algorithm [R. H. Swendsen and J. S. Wang, Phys. Rev. Lett. 58, 86 (1987)], and we demonstrate its success and speed at all temperatures for the d=2d=2d=2 Ising model. We argue that the previously discussed connection to self-organized criticality may be viewed as one limit of the fixed-cluster algorithm.