Critical dynamical behavior of the Ising model

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.

arXiv comment6 pages, 6 figures, version published in Physical Review E

References in corpus (13)

• Precision Islands in the Ising and $O(N)$ Models
• Ising Universality in Three Dimensions: A Monte Carlo Study
• Glauber dynamics in a single-chain magnet: From theory to real systems
• Pushing the Limits of Monte Carlo Simulations for the 3d Ising Model
• The Dynamic Exponent of the Two-Dimensional Ising Model and Monte Carlo Computation of the Sub-Dominant Eigenvalue of the Stochastic Matrix
• Critical Exponents of the 3D Ising Universality Class From Finite Size Scaling With Standard and Improved Actions
• The dynamic critical exponent $z$ of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model
• Nonequilibrium critical dynamics of the ferromagnetic Ising model with Kawasaki dynamics
• Metastable states of the Ising chain with Kawasaki dynamics
• Relaxational dynamics in 3D randomly diluted Ising models
• Super slowing down in the bond-diluted Ising model
• Critical Dynamical Exponent of the Two-Dimensional Scalar ${\varphi }^4$ Model with Local Moves
• Approximate dynamical eigenmodes of the Ising model with local spin-exchange moves

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