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 $M, {\mathrm{MSD}}_{M}$, 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 ${\mathrm{MSD}}_{M}$ features a first crossover at time ${\ensuremath{\tau}}_{1}\ensuremath{\sim}{L}^{{z}_{1}}$, from ordinary diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}t$, to anomalous diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$. Purely on numerical grounds, we obtain the values ${z}_{1}=0.45(5)$ and $\ensuremath{\alpha}=0.752(5)$ 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 ${\ensuremath{\tau}}_{2}\ensuremath{\sim}{L}^{{z}_{2}}$. Here, ${\mathrm{MSD}}_{M}$ saturates to its late-time value $\ensuremath{\sim}{L}^{2+\ensuremath{\gamma}/\ensuremath{\nu}}$, 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 ${\mathrm{MSD}}_{M}$ requires that $\ensuremath{\alpha}({z}_{2}\ensuremath{-}{z}_{1})=\ensuremath{\gamma}/\ensuremath{\nu}\ensuremath{-}{z}_{1}$. We speculate that ${z}_{1}=1/2$ and $\ensuremath{\alpha}=3/4$, values that indeed lead to the expected ${z}_{2}=13/6$ result. A complementary analysis for the three-dimensional Ising model provides the estimates ${z}_{1}=1.35(2), \ensuremath{\alpha}=0.90(2)$, 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.

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