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 4 ≤ L ≤ 8. The exponents in the finitesize scaling relations for the order parameter and the magnetic susceptibility at the finitelattice critical temperature are computed to be β = 0.49(7), β = 0.49(5), β = 0.50(1) and γ = 1.04(4), γ = 1.03(4), γ = 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 β = 0.5 and γ = 1. The values for the critical temperature of the infinite lattice T c (∞) = 6.6788(65), T c (∞) = 6.6798(69), T c (∞) = 6.6802(70) are obtained from the straight-line fit of the magnetic susceptibility maxima using 4 ≤ L ≤ 8 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 (∞) = 6.6817(15), T c (∞) = 6.6802(2), the dynamic Monte Carlo result of T c (∞) = 6.6803(1), the cluster Monte Carlo result of T c (∞) = 6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm result of T c (∞) = 6.6802632 ± 5 × 10 −5 .