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Physical review. B, Condensed matter

Calculation of the dynamical exponent z for the three-state Potts model

Jan Tobochnik, Ciriyam Jayaprakash

Published Apr 1, 1982• 19 citations
Publisher

Abstract

We have found using the dynamic Monte Carlo renormalization-group method that the dynamic critical exponent $z$ for the three-state Potts model on a square lattice has the value 2.7\ifmmode\pm\else\textpm\fi{}0.4.

DOI
10.1103/physrevb.25.4893

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We have found using the dynamic Monte Carlo renormalization-group method that the dynamic critical exponent zzz for the three-state Potts model on a square lattice has the value 2.7\ifmmode\pm\else\textpm\fi{}0.4.
We employ a successive Monte Carlo renormalization group procedure in the presence of a linearly varying temperature to study the three-state (q=3) Potts model on square lattices. By matching correlation functions of different lattice sizes at different renormalization levels, a rate exponent r associated with the temperature sweep rate R and the correlation length exponent nu are obtained. The dynamic critical exponent z is then obtained by the scaling law z=r-1/nu derived in our method. The results of z=2.171(62) for q=3 in this work and the previously obtained z=2.15(13) for q=2 seem to support the extension of the "weak universality" hypothesis to dynamic critical behavior. With these calculated exponents, the dynamic scaling forms of both specific heat and order parameter at various R are presented and all the other static critical exponents are determined, which verify our method. Discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent.
Values of dynamic critical exponents are numerically estimated for various models with the nonequilibrium relaxation method to test the dynamic universality hypothesis. The dynamics used here are single-spin update with Metropolis-type transition probabities. The estimated values of nonequilibrium relaxation exponent of magnetization λ m (=β/ z ν) of Ising models on bcc and fcc lattices are estimated to be 0.251(3) and 0.252(3), respectively, which are consistent with the value of the model on simple-cubic lattice, 0.250(2). The dynamic critical exponents of three-states Potts models on square, honeycomb and triangular lattices are also estimated to be 2.193(5), 2.198(4), and 2.199(3), respectively. They are consistent within the error bars. It is also confirmed that Ising models with regularly modulated coupling constants on square lattice have the same dynamic critical exponents with the uniformly ferromagnetic Ising model.