Critical dynamical behavior of the Ising model

TL;DR

This paper studies the dynamical critical behavior of the 2D and 3D Ising models, focusing on the mean-squared deviation of magnetization and autocorrelation functions, revealing multiple crossover regimes and estimating key dynamic exponents.

Contribution

It introduces a detailed analysis of MSD and autocorrelation crossovers in the Ising model, providing new estimates for dynamic exponents and highlighting the importance of the early-time exponent z1.

Findings

Identified two crossover times with distinct diffusion behaviors.

Estimated dynamic exponents z1, z2, and alpha for 2D and 3D Ising models.

Showed z1 is more relevant for practical simulations.

Abstract

We investigate the dynamical critical behavior of the two- and three-dimensional Ising model with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization $M$, MSD$_M$, 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 MSD$_M$ features a first crossover at time $\tau_1 \sim L^{z_{1}}$, from ordinary diffusion with MSD$_M$ $\sim t$, to anomalous diffusion with MSD$_M$ $\sim t^\alpha$. Purely on numerical grounds, we obtain the values $z_1=0.45(5)$ and $\alpha=0.752(5)$ 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 $\tau_2 \sim L^{z_{2}}$. Here, MSD$_M$ saturates to its late-time value $\sim L^{2+\gamma/\nu}$, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value $z_{2}=2.1665(12)$, earlier reported as the single dynamic exponent. Continuity of MSD$_M$ requires that $\alpha(z_{2}-z_{1})=\gamma/\nu-z_1$. We speculate that $z_{1} = 1/2$ and $\alpha = 3/4$, values that indeed lead to the expected $z_{2} = 13/6$ result. A complementary analysis for the three-dimensional Ising model provides the estimates $z_{1} = 1.35(2)$, $\alpha=0.90(2)$, and $z_{2} = 2.032(3)$. While $z_{2}$ has attracted significant attention in the literature, we argue that for all practical purposes $z_{1}$ is more important, as it determines the number of statistically independent measurements during a long simulation.