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Journal of the Physical Society of Japan

Dynamic Critical Exponents of Three-Dimensional Ising Models and Two-Dimensional Three-States Potts Models

Yohsuke Murase, Nobuyasu Ito

Published Dec 25, 2007• 17 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.77.014002

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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.
The static and dynamic critical properties of the ferromagnetic q -state Potts models on a square lattice with q=2 and 3 are numerically studied via the nonequilibrium relaxation method. The relaxation behavior of both the order parameter and energy as well as that of the second moments are investigated, from which static and dynamic critical exponents can be obtained. We find that the static exponents thus obtained from the relaxation of the order parameter and energy together with the second moments of the order parameter exhibit a close agreement with the exact exponents, especially for the case of the q=2 (Ising) model, when care is taken in the choice of the initial states for the relaxation of the second moments. As for the case of q=3 , the estimates for the static exponents become less accurate, but still exhibit reasonable agreement with the exactly known static exponents. The dynamic critical exponent for the q=2 (Ising) model is estimated from the relaxation of the second moments of the order parameter with mixed initial conditions to give z(q=2) approximately 2.1668(19) .
Critical scaling and universality in short-time dynamics for spin models on a twodimensional triangular lattice are investigated by using Monte Carlo simulation. Emphasis is placed on the dynamic evolution from fully ordered initial states to show that universal scaling exists already in the short-time regime in form of power-law behavior of the magnetization and Binder cumulant. The results measured for the dynamic and static critical exponents, θ, z, β and ν, confirm explicitly that the Potts models on the triangular lattice and square lattice belong to the same universality class. Our critical scaling analysis strongly suggests that the simulation for the dynamic relaxation can be used to determine numerically the universality.
The kinetic Ising model on a square lattice is studied by means of the finite-size scaling with the help of the Monte Carlo method. The results obtained are consistent with the universality hypothesis that the value of the dynamic critical expotent is universal for certain types of transition probabilities. The present results also suggest that the dynamic critical exponents for the relaxation times of the magnetization and energy are the same. Our estimation of the dynamic critical exponent z gives that z \cong2.2 ±0.1.
We investigate the dynamical critical behavior of the two-dimensional three-state Potts model with single spin-flip dynamics in equilibrium. We focus on the mean-squared deviation of the magnetization M (MSD_{M}) as a function of time, as well as on the autocorrelation function of M. Our simulations reveal the existence of two crossover behaviors at times τ_{1}∼L^{z_{1}} and τ_{2}∼L^{z_{2}}, separating three dynamical regimes. MSD_{M} appears to shift from ordinary diffusion in the first regime, to anomalous diffusion in the second, and finally to be constant in the third regime. The magnetization autocorrelation function, on the other hand, is found to fluctuate between exponential decay, stretched-exponential decay, and then again exponential decay along these three regimes. This behavior is in agreement with the one reported recently for the two-dimensional Ising ferromagnet [Phys. Rev. E 108, 034118 (2023)2470-004510.1103/PhysRevE.108.034118], indicating that the existence of two dynamic critical exponents is not a peculiarity of the Ising model itself. A comparison of both MSD_{M} and the magnetization's autocorrelation function suggests that within our numerical accuracy the exponents z_{1} and z_{2} are shared between the Ising and three-state Potts models at least for the particular case of single spin-flip dynamics studied here, even though their equilibrium universality classes are clearly distinct. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}, in which α is the anomalous exponent in the intermediate regime. Since the ratio γ/ν is not shared between the two models, it follows that α is not shared either, an aspect well verified in our simulations. Finally, we also discuss the relevance of our main findings using another useful observable, namely the line magnetization M_{l}.
The ferromagnetic critical exponents are estimated along the phase boundary for the gauge glass model in three dimensions. The nonequilibrium relaxation method is applied to estimate the transition temperature and critical exponents. Together with the previously obtained results for the ± J Ising model in two and three dimensions, universality classes are examined for these disordered models. The result reveals a direct numerical confirmation of the Harris criterion for typical models with positive, zero and negative values of α 0 , the exponent for the specific heat in the regular case. In addition, we show a nonuniversal behavior of the dynamical exponent z with respect to the randomness for these models irrespective of α 0 .