Liu, Zihua and Vatansever, Erol and Barkema, Gerard T and Fytas, Nikolaos G (2023) Critical dynamical behavior of the Ising model. Physical Review E, 108 (3-1). 034118-. DOI https://doi.org/10.1103/physreve.108.034118
Liu, Zihua and Vatansever, Erol and Barkema, Gerard T and Fytas, Nikolaos G (2023) Critical dynamical behavior of the Ising model. Physical Review E, 108 (3-1). 034118-. DOI https://doi.org/10.1103/physreve.108.034118
Liu, Zihua and Vatansever, Erol and Barkema, Gerard T and Fytas, Nikolaos G (2023) Critical dynamical behavior of the Ising model. Physical Review E, 108 (3-1). 034118-. DOI https://doi.org/10.1103/physreve.108.034118
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, 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 τ_{1}∼L^{z_{1}}, from ordinary diffusion with MSD_{M}∼t, to anomalous diffusion with MSD_{M}∼t^{α}. Purely on numerical grounds, we obtain the values z_{1}=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 τ_{2}∼L^{z_{2}}. Here, MSD_{M} saturates to its late-time value ∼L^{2+γ/ν}, 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 α(z_{2}-z_{1})=γ/ν-z_{1}. We speculate that z_{1}=1/2 and α=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), α=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.
Item Type: | Article |
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Divisions: | Faculty of Science and Health Faculty of Science and Health > Mathematics, Statistics and Actuarial Science, School of |
SWORD Depositor: | Unnamed user with email elements@essex.ac.uk |
Depositing User: | Unnamed user with email elements@essex.ac.uk |
Date Deposited: | 30 Oct 2023 16:22 |
Last Modified: | 16 May 2024 22:02 |
URI: | http://repository.essex.ac.uk/id/eprint/36448 |
Available files
Filename: PhysRevE.108.034118.pdf
Licence: Creative Commons: Attribution 4.0