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International Journal of Modern Physics C

Critical Dynamics Behavior of the Wolff Algorithm in the Site-Bond-Correlated Ising Model

Paulo R. A. Campos, Roberto Nicolau Onody

Published Sep 1, 1998• 0 citations
Publisher

Abstract

Here we apply the Wolff single-cluster algorithm to the site-bond-correlated Ising model and study its critical dynamical behavior. We have verified that the autocorrelation time diminishes in the presence of dilution and correlation, showing that the Wolff algorithm performs even better in such situations. The critical dynamical exponents are also estimated.

DOI
10.1142/s0129183198000807

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Here we apply the Wolff single-cluster algorithm to the site-bond-correlated Ising model and study its critical dynamical behavior. We have verified that the autocorrelation time diminishes in the presence of dilution and correlation, showing that the Wolff algorithm performs even better in such situations. The critical dynamical exponents are also estimated.
Recently we reported some interesting features of the Wolff's algorithm behavior when applied to the site-bond-correlated Ising model.Our main results were that a stronger correlation diminishes the autocorrelation time but it does not change or affectthe critical dynamic exponent.In this paper, we analyse the Wolff's cluster structure and how it varies with the spatial correlation. The fractal dimensions are determined for several values of the magnetic atoms concentration and spacial correlation, giving an estimative of the critical dynamic exponent.
Recently we reported some interesting features of the Wolff algorithm behaviour when applied to the site-bond-correlated Ising model. Our main results were that a stronger correlation diminishes the autocorrelation time but it does not change or affect the critical dynamic exponent. In this work, we analyse the Wolff cluster structure and how it varies with the spatial correlation. The fractal dimensions are determined for several values of the magnetic atom concentration and spatial correlation, giving an estimation of the Wolff critical dynamic exponent.
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.
Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).
We calculate the dynamic critical exponent for the Niedermayer algorithm applied to the two-dimensional Ising and XY models, for various values of the free parameter E 0 . For E 0 = − 1 we regain the Metropolis algorithm and for E 0 = 1 we regain the Wolff algorithm. For − 1 < E 0 < 1, we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, L , but eventually saturates at a given lattice size , which depends on E 0 . For , the Niedermayer algorithm is equivalent to the Metropolis one, i.e., they have the same dynamic exponent. For E 0 > 1, the autocorrelation time is always greater than for E 0 = 1 (Wolff) and, more important, it also grows faster than a power of L . Therefore, we show that the best choice of cluster algorithm is the Wolff one, when comparing against the Niedermayer generalization. We also obtain the dynamic behavior of the Wolff algorithm: although not conclusively, we propose a scaling law for the dependence of the autocorrelation time on L .