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arXiv:cond-mat/9806127 · International Journal of Modern Physics C

PARALLEL INVADED CLUSTER ALGORITHM FOR THE ISING MODEL

Y. S. CHOI, Jonathan L. Machta, Pablo Tamayo, L. Chayes

Published Feb 1, 1999• 16 citations
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Abstract

A parallel version of the invaded cluster algorithm is described. Results from large scale (up to 40962 and 5123) simulations of the Ising model are reported. No evidence of critical slowing down is found for the three-dimensional Ising model. The magnetic exponent is estimated to be 2.482±0.001(β/ν=0.518±0.001) for the three-dimensional Ising model.

DOI
10.1142/s0129183199000024

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A parallel version of the invaded cluster algorithm is described. Results from large scale (up to 40962 and 5123) simulations of the Ising model are reported. No evidence of critical slowing down is found for the three-dimensional Ising model. The magnetic exponent is estimated to be 2.482±0.001(β/ν=0.518±0.001) for the three-dimensional Ising model.
The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 20002 and 1203, respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are Kc=0.45412(2) for the three-dimensional XY model and η=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are Kc=1.120(1) and η=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.
The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000(2) and 120(3), respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K(c)=0.454 12(2) for the three-dimensional XY model and eta=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are K(c)=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.
We present Monte Carlo simulation results for the dynamical critical exponent z of the two- and three-dimensional kinetic Ising model. The z-values were calculated from the magnetization relaxation from an ordered state into the equilibrium state at Tc for very large systems with up to (169984)2 and (3072)3 spins. To our knowledge, these are the largest Ising-systems simulated todate. We also report the successful simulation of very large lattices on a massively parallel MIMD computer with high speedups of approximately 1000 and an efficiency of about 0.93.
The invaded cluster algorithm, a method for simulating phase transitions, is described in detail. Theoretical, albeit nonrigorous, justification of the method is presented and the algorithm is applied to Potts models in two and three dimensions. The algorithm is shown to be useful for both first-order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities. The dynamic properties of the invaded cluster algorithm are studied. Numerical evidence suggests that the algorithm has no critical slowing for Ising models. \textcopyright{} 1996 The American Physical Society.