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Europhysics Letters (EPL)

Critical Dynamics of the q -State Potts Model in Two Dimensions

O. F. de Alcantara Bonfim

Published Aug 1, 1987• 18 citations
Publisher

Abstract

The Monte Carlo method is used to determine the nonlinear dynamic critical exponent z ( q ) for the q -state Potts model in two dimensions. From the power-law relaxation behaviour of the magnetization we find z (2) = 2.16 ± 0.05, z (3) = 2.16 ± 0.04 and z (4) = 2.18 ± 0.03, suggesting that the critical exponent z is independent of q , as opposed to previous calculations.

DOI
10.1209/0295-5075/4/3/020

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The Monte Carlo method is used to determine the nonlinear dynamic critical exponent z ( q ) for the q -state Potts model in two dimensions. From the power-law relaxation behaviour of the magnetization we find z (2) = 2.16 ± 0.05, z (3) = 2.16 ± 0.04 and z (4) = 2.18 ± 0.03, suggesting that the critical exponent z is independent of q , as opposed to previous calculations.
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 determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.
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.