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Journal of Physics A Mathematical and General

Critical dynamics of the four-state Potts model

S. Jain

Published Jun 21, 1987• 1 citation
Publisher

Abstract

The critical relaxation of the four-state Potts model on L*L lattices with L<or=64 is studied by the Monte Carlo method. The decay of the magnetisation from the nonequilibrium initial state exhibits a crossover from a power-law relaxation for t<<t c (L) to an exponential one for t>>t c (L), where t c (L) is some crossover time. This crossover is discussed within the context of dynamic finite-size scaling theory. As a result, the dynamic critical exponent is estimated to be Z=2.94+or-0.29.

DOI
10.1088/0305-4470/20/9/010

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The critical relaxation of the four-state Potts model on L*L lattices with L<or=64 is studied by the Monte Carlo method. The decay of the magnetisation from the nonequilibrium initial state exhibits a crossover from a power-law relaxation for t<<t c (L) to an exponential one for t>>t c (L), where t c (L) is some crossover time. This crossover is discussed within the context of dynamic finite-size scaling theory. As a result, the dynamic critical exponent is estimated to be Z=2.94+or-0.29.
We review the current state of knowledge of dynamic critical phenomena in simple magnetic models and present high statistics Monte Carlo data for critical slowing down in Ising and Potts models in two and three dimensions. By fitting the relaxation functions to multi-exponential decays at the critical temperature and applying finite size scaling we are able to extract very accurate estimates for the dynamical critical exponent z. Our results are compared with previous numerical estimates and with the predictions of renormalization group theory.
We present an extensive Monte Carlo study of the critical relaxation for the two-dimensional square lattice random-bond Potts model using Swendsen-Wang cluster flipping. The integrated autocorrelation time \ensuremath{\tau} is calculated and the dynamic exponent z is estimated by analyzing the size dependence of \ensuremath{\tau}. We find that z\ensuremath{\approx}0 in agreement with estimates for the two-dimensional Ising model. We also present a study of the size dependence of the dynamic behavior of the pure eight-state Potts model which undergoes a first-order transition. The scaling is describable by the product of an exponential times a power law, a behavior which is quite different from that found in the random-bond case.
We have carried out extensive computer simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L q-state Potts models for q=2, 3, and 4. We fit the equilibrium relaxation functions for energy and order parameter to a sum of exponential decays and use the finite-size dependence at Tc{T}_{c}Tc​, \ensuremath{\tau}\ensuremath{\propto}Lz{L}^{z}Lz, to extract estimates for z. We find for the longest relaxation time for both energy and for order parameter that z=2.17\ifmmode\pm\else\textpm\fi{}0.04 describes all three values of q. Our results support the dynamic generalization of the ``weak universality'' hypothesis proposed by Suzuki.