Critical dynamical behavior of the Ising model

Abstract

We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization M, MSDM , as a function of time, as well as on the autocorrelation function of M. These two functions are distinct but closely related. We find that MSDM features a first crossover at time τ1 ∼ Lz1 , from ordinary diffusion with MSDM ∼ t, to anomalous diffusion with MSDM ∼ t α. Purely on numerical grounds, we obtain the values z1 = 0.45(5) and α = 0.752(5) for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time τ2 ∼ Lz2 . Here, MSDM saturates to its late-time value ∼L2+γ /ν , while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value z2 = 2.1665(12), earlier reported as the single dynamic exponent. Continuity of MSDM requires that α(z2 − z1 ) = γ /ν − z1. We speculate that z1 = 1/2 and α = 3/4, values that indeed lead to the expected z2 = 13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates z1 = 1.35(2), α = 0.90(2), and z2 = 2.032(3). While z2 has attracted significant attention in the literature, we argue that for all practical purposes z1 is more important, as it determines the number of statistically independent measurements during a long simulation