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Journal of the Physical Society of Japan

Accurate Estimation of the Dynamical Critical Exponent of the Two-Dimensional Kinetic Ising Model Based on the Ising Machinem-TIS

Nobuyasu Ito, Makoto Taiji, Masuo Suzuki

Published Dec 15, 1987• 24 citations
Publisher

Abstract

The dynamical critical exponent, z , of the two-dimensional kinetic Ising model is estimated by the Monte Carlo method using finite-size scaling. The Monte Carlo simulalion is performed on a special-purpose computer system, m -TIS, designed and constructed by the present authors. The obtained value of z is 2.132±0.008. This result supports the conjecture by Racz and Collins that z is equal to 17/8 in the two-dimensional kinetic Ising model (Racz and Collins; Rhys. Rev. B 13 (1976) 3074).

DOI
10.1143/jpsj.56.4218

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The dynamical critical exponent, z , of the two-dimensional kinetic Ising model is estimated by the Monte Carlo method using finite-size scaling. The Monte Carlo simulalion is performed on a special-purpose computer system, m -TIS, designed and constructed by the present authors. The obtained value of z is 2.132±0.008. This result supports the conjecture by Racz and Collins that z is equal to 17/8 in the two-dimensional kinetic Ising model (Racz and Collins; Rhys. Rev. B 13 (1976) 3074).
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.
The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.
We consider power spectra for fluctuating quantities at a critical point, P( omega ) varies as omega - phi , where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered. From the magnetization spectra obtained with simulations, z is found to be 2.13+or-0.03 for the two-dimensional Glauber kinetic Ising model, hence in accord with recent Monte Carlo relaxational studies, but at odds with those obtained by series expansion and damage spreading methods. Excellent dynamic finite-size scaling for system sizes L=10 to 128 supports our estimate.