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Physical Review Letters

Critical clusters and efficient dynamics for frustrated spin models

V. Cataudella, Giancarlo Franzese, Mario Nicodemi, Antonio Scala A, Antonio Coniglio A

Published Mar 7, 1994• 45 citations
Publisher

Abstract

A general method to find, in a systematic way, efficient Monte Carlo cluster dynamics among the vast class of dynamics introduced by Kandel et al. [Phys. Rev. Lett. 65, 941 (1990)] is proposed. The method is successfully applied to a class of frustrated two-dimensional Ising systems. In the case of the fully frustrated model, we also find the intriguing result that critical clusters consist of self-avoiding walk at the \ensuremath{\theta} point.

DOI
10.1103/physrevlett.72.1541

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A general method to find, in a systematic way, efficient Monte Carlo cluster dynamics among the vast class of dynamics introduced by Kandel et al. [Phys. Rev. Lett. 65, 941 (1990)] is proposed. The method is successfully applied to a class of frustrated two-dimensional Ising systems. In the case of the fully frustrated model, we also find the intriguing result that critical clusters consist of self-avoiding walk at the \ensuremath{\theta} point.
We have developed a cluster algorithm for Monte Carlo simulations of the fully frustrated Ising model on the square lattice. The method does not suffer from problems of metastability, and is extremely efficient even at T=0, as we demonstrated previously. Here we describe results of more extensive simulations of the model at T=0 which allow us to extract the effective dynamical critical exponent of the algorithm, z\ensuremath{\approxeq}0.55, as well as the static exponent \ensuremath{\eta}=0.5. We also provide an argument that explains why our method works.
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.
A new class of Monte Carlo methods is introduced. The approach is devised to be especially useful for studying critical phenomena and phase transitions. The method can sample the critical point without a priori knowledge of the critical temperature. The method is first verified for Ising and Potts models. It is shown that the method also efficiently simulates the coexistence region near first-order transitions. The method is applied to dilute Ising models for studying disorder effects on critical phenomena. The dynamic properties of the method are investigated numerically. Finally a parallel version of the method is described and high-precision numerical results for the critical point of three-dimensional Ising models are presented.
Monte Carlo simulations of the short-time critical dynamics are reviewed. The short-time universal scaling behavior of the dynamic Ising model and Potts model are discussed in detail, while extension and application to more complex systems as the XY model, the fully frustrated XY model and other dynamic systems are also presented. The investigation of the universal behavior of the short-time dynamics not only enlarges the fundamental knowledge on critical phenomena but also, more interestingly, provides possible new ways to determine not only the new critical exponents θ and θ1, but also the traditional dynamic critical exponent z as well as all static critical exponents.