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Physical review. B, Condensed matter

Monte Carlo study of dynamic universality in two-dimensional Potts models

S. Tang, David Paul Landau

Published Jul 1, 1987• 91 citations
Publisher

Abstract

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 ${T}_{c}$, \ensuremath{\tau}\ensuremath{\propto}${L}^{z}$, 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.

DOI
10.1103/physrevb.36.567

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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.
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 study the dynamic critical behavior of the two-dimensional random-bond three-state Potts model using large-scale Monte Carlo simulations with the Wolff and Swendsen-Wang cluster algorithms. At the disorder-dependent critical temperature, we compute integrated and exponential autocorrelation times and extract the dynamic critical exponent z via finite-size scaling. Critical slowing down is significantly weakened by bond randomness, with the dynamic exponent decreasing from z≈0.54 in the pure system to z≈0.24 at strong disorder. From the finite-size scaling of the Wolff cluster size, we obtain γ/ν≈1.73, where γ and ν are the susceptibility and correlation-length exponents, respectively, consistent with the universality class of the two-dimensional three-state Potts model. These results indicate that bond randomness strongly affects the critical dynamics but leaves the underlying static universality class unchanged. Furthermore, both static and dynamic observables, including the specific heat and the autocorrelation times, show a lack of self-averaging.
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 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.