The Effect of the Number of Simulations on the Exponents Obtained by Finite-Size Scaling Relations of the Order Parameter and the Magnetic Susceptibility for the Four-Dimensional Ising Model on the Creutz Cellular Automaton

Abstract

The four-dimensional Ising model is simulated on the Creutz cellular automaton using finite-size lattices with linear dimension 4a parts per thousand currency signLa parts per thousand currency sign8. The exponents in the finite-size scaling relations for the order parameter and the magnetic susceptibility at the finite-lattice critical temperature are computed to be beta=0.49(7), beta=0.49(5), beta=0.50(1) and gamma=1.04(4), gamma=1.03(4), gamma=1.02(4) for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are consistent with the renormalization group predictions of beta=0.5 and gamma=1. The values for the critical temperature of the infinite lattice T (c) (a)=6.6788(65), T (c) (a)=6.6798(69), T (c) (a)=6.6802(70) are obtained from the straight-line fit of the magnetic susceptibility maxima using 4a parts per thousand currency signLa parts per thousand currency sign8 for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are in very good agreement with the series expansion results of T (c) (a)=6.6817(15), T (c) (a)=6.6802(2), the dynamic Monte Carlo result of T (c) (a)=6.6803(1), the cluster Monte Carlo result of T (c) (a)=6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm result of T (c) (a)=6.6802632 +/- 5x10(-5).