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Journal of Physics A Mathematical and General

The dynamic critical exponent of the q=3 and 4 state Potts model

L. de Arcangelis, Naeem Jan

Published Dec 21, 1986• 14 citations
Publisher

Abstract

The dynamic Monte Carlo renormalisation group method introduced by Jan, Moseley and Stauffer (1983) is used to determine the dynamic critical exponent, z, for the q=3 and q=4 state Potts model in two dimensions. The authors find that z=2.43+or-0.15 and 2.36+or-0.20 for the q=3 and q=4 state Potts models. These results are in disagreement with the recent conjecture of Domany (1984).

DOI
10.1088/0305-4470/19/18/012

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The dynamic Monte Carlo renormalisation group method introduced by Jan, Moseley and Stauffer (1983) is used to determine the dynamic critical exponent, z, for the q=3 and q=4 state Potts model in two dimensions. The authors find that z=2.43+or-0.15 and 2.36+or-0.20 for the q=3 and q=4 state Potts models. These results are in disagreement with the recent conjecture of Domany (1984).
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.
The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as ${\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01$, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.
Applying the dynamic Monte Carlo renormalisation group to the Glauber kinetic Ising model, the dynamical critical exponent z is found by simulation of up to 8192 2 and 512 3 spins on the vector computer CDC Cyber 205, using the new 'Method of 2 d colours' for the Monte Carlo part (update speed 22 megaspins/s). The two-dimensional result z=2.14-or+0.02 disagrees with Domany's hypothesis (1984). For three dimensions, a systematic trend in z with increasing blocksize leads to an extrapolated value z=1.965-or+0.010, which is consistent with a theoretically expected value 2.02.
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.