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Physical Review A

Mean-field study of the Swendsen-Wang dynamics

Tane S. Ray, Pablo Tamayo, W. Klein

Published Jun 1, 1989• 44 citations
Publisher

Abstract

Nonuniversal dynamics introduced by Swendsen and Wang [Phys. Rev. Lett. 58, 86 (1987)] is studied in a mean-field Ising model. Numerical simulation data indicate that z, the dynamical critical exponent, is 0.98\ifmmode\pm\else\textpm\fi{}0.08. A modified version of this dynamics in which only large clusters are flipped apparently gives the same exponent, although the amplitude is reduced by a factor of 1/2. In addition, a theoretical argument is presented which gives z=1 and provides insight into the dynamics.

DOI
10.1103/physreva.39.5949

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Nonuniversal dynamics introduced by Swendsen and Wang [Phys. Rev. Lett. 58, 86 (1987)] is studied in a mean-field Ising model. Numerical simulation data indicate that z, the dynamical critical exponent, is 0.98\ifmmode\pm\else\textpm\fi{}0.08. A modified version of this dynamics in which only large clusters are flipped apparently gives the same exponent, although the amplitude is reduced by a factor of 1/2. In addition, a theoretical argument is presented which gives z=1 and provides insight into the dynamics.
Using a cluster-flipping Monte Carlo algorithm combined with a generalization of the histogram reweighting scheme of Ferrenberg and Swendsen [Phys. Rev. Lett. 61, 2635 (1988); 63, 1195 (1989)], we have studied the equilibrium properties of the thermal random-field Ising model on a cubic lattice in three dimensions. We have equilibrated systems of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L spins, with values of L up to 32, and for these systems the cluster-flipping method appears to a large extent to overcome the slow equilibration seen in single-spin-flip methods. From the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling. For the exponents we find \ensuremath{\nu}=1.02\ifmmode\pm\else\textpm\fi{}0.06, \ensuremath{\beta}=0.06\ifmmode\pm\else\textpm\fi{}0.07, \ensuremath{\gamma}=1.9\ifmmode\pm\else\textpm\fi{}0.2, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2.9\ifmmode\pm\else\textpm\fi{}0.2, where \ensuremath{\nu}, \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} govern the critical singularities in the correlation length, magnetization, and connected and disconnected susceptibilities, respectively. \textcopyright{} 1996 The American Physical Society.
In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained isβc=0. 075?. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.
In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.