Finite-size scaling relations of the four-dimensional Ising model on the Creutz cellular automaton

Abstract

The four-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4 ≤ L ≤ 8. The temperature variations and the finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature for the 7, 14, and 21 independent simulations. The approximate values for the critical temperature of the infinite lattice, Tc(∞) = 6.6965(35), 6.6961(30), 6.6960(12), 6.6800(3), 6.6801(2), 6.6802(1) and 6.6925(22) (without logarithmic factor), 6.6921(22) (without logarithmic factor), 6.6909(2) (without logarithmic factor), 6.6822(13) (with logarithmic factor), 6.6819(11) (with logarithmic factor), 6.6808(8) (with logarithmic factor) are obtained from the intersection points of specific heat curves, the Binder parameter curves and the straight line fit of specific heat maxima for the 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results, 6.6802(1) and 6.6808(8), are in very good agreement with the series expansion results of Tc(∞) = 6.6817(15), 6.6802(2), the dynamic Monte Carlo result of Tc(∞) = 6.6803(1), the cluster Monte Carlo result of Tc(∞) = 6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm of Tc(∞) = 6.6802632 ± 5⋅10⁻⁵. The average values obtained for the critical exponent of the specific heat are calculated as α = –0.0402(15), –0.0393(12), –0.0391(11) for the 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained result, α = –0.0391(11), is agreement with the series expansions results of α = –0.12 ± 0.03 and the Monte Carlo using Metropolis and Wolff-cluster algorithm of α ≥ 0±0.04. However, α = –0.0391(11) isn’t consistent with the renormalization group prediction of α = 0.