Finite-size scaling relations for a four-dimensional Ising model on Creutz cellular automatons

Abstract

The four-dimensional Ising model is simulated on Creutz cellular automatons using finite lattices with linear dimensions 4 ≤ L ≤ 8. The temperature variations and finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature for 7, 14, and 21 independent simulations. Approximate values for the critical temperature of the infinite lattice of Tc(∞) = 6.6965(35), 6.6961(30), 6.6960(12), 6.6800(3), 6.6801(2), 6.6802(1) and 6.6925(22) (without the logarithmic factor), 6.6921(22) (without the logarithmic factor), 6.6909(2) (without the logarithmic factor), 6.6822(13) (with the logarithmic factor), 6.6819(11) (with the logarithmic factor), and 6.6808(8) (with the logarithmic factor) are obtained from the intersection points of the specific heat curves, the Binder parameter curves, and straight line fits of specific heat maxima for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the results, 6.6802(1) and 6.6808(8), are in very good agreement with the results of a series expansion of Tc(∞), 6.6817(15) and 6.6802(2), the dynamic Monte Carlo value Tc(∞) = 6.6803(1), the cluster Monte Carlo value Tc(∞) = 6.680(1), and the Monte Carlo value using the Metropolis-Wolff cluster algorithm Tc(∞) = 6.6802632 ± 5 · 10−5. The average values calculated for the critical exponent of the specific heat are α =− 0.0402(15), − 0.0393(12), − 0.0391(11) with 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the result, α =− 0.0391(11), agrees with the series expansions result, α =− 0.12 ± 0.03 and the Monte Carlo result using the Metropolis-Wolff cluster algorithm, α ≥ 0 ± 0.04. However, α =− 0.0391(11) is inconsistent with the renormalization group prediction of α = 0.