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Communications in Theoretical Physics

The Universality of Dynamic Exponent θ′ Demonstrated by the Dynamical Two-Dimensional Ising Model

Xia-Wei Liu, Zhibing Li

Published Jun 15, 1997• 6 citations
Publisher

Abstract

By means of Monte–Carlo simulation, we calculate the dynamic exponent θ' , which governs the initial slip in the critical relaxation, in triangular lattice with both heat-bath and Metropolis dynamical algorithms. Our best estimation for θ' is 0.1934(6) with the heat-bath algorithm and 0.195(2) with Metropolis algorithm. Comparing also to the results obtained in square lattices, we conclude that the universality hypothesis is consistent in our calculation.

DOI
10.1088/0253-6102/27/4/511

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By means of Monte–Carlo simulation, we calculate the dynamic exponent θ' , which governs the initial slip in the critical relaxation, in triangular lattice with both heat-bath and Metropolis dynamical algorithms. Our best estimation for θ' is 0.1934(6) with the heat-bath algorithm and 0.195(2) with Metropolis algorithm. Comparing also to the results obtained in square lattices, we conclude that the universality hypothesis is consistent in our calculation.
Values of dynamic critical exponents are numerically estimated for various models with the nonequilibrium relaxation method to test the dynamic universality hypothesis. The dynamics used here are single-spin update with Metropolis-type transition probabities. The estimated values of nonequilibrium relaxation exponent of magnetization λ m (=β/ z ν) of Ising models on bcc and fcc lattices are estimated to be 0.251(3) and 0.252(3), respectively, which are consistent with the value of the model on simple-cubic lattice, 0.250(2). The dynamic critical exponents of three-states Potts models on square, honeycomb and triangular lattices are also estimated to be 2.193(5), 2.198(4), and 2.199(3), respectively. They are consistent within the error bars. It is also confirmed that Ising models with regularly modulated coupling constants on square lattice have the same dynamic critical exponents with the uniformly ferromagnetic Ising model.
Critical scaling and universality in short-time dynamics for spin models on a twodimensional triangular lattice are investigated by using Monte Carlo simulation. Emphasis is placed on the dynamic evolution from fully ordered initial states to show that universal scaling exists already in the short-time regime in form of power-law behavior of the magnetization and Binder cumulant. The results measured for the dynamic and static critical exponents, θ, z, β and ν, confirm explicitly that the Potts models on the triangular lattice and square lattice belong to the same universality class. Our critical scaling analysis strongly suggests that the simulation for the dynamic relaxation can be used to determine numerically the universality.
The kinetic Ising model on a square lattice is studied by means of the finite-size scaling with the help of the Monte Carlo method. The results obtained are consistent with the universality hypothesis that the value of the dynamic critical expotent is universal for certain types of transition probabilities. The present results also suggest that the dynamic critical exponents for the relaxation times of the magnetization and energy are the same. Our estimation of the dynamic critical exponent z gives that z \cong2.2 ±0.1.
The time evolution of the three-dimensional critical Ising model relaxing from a non-equilibrium initial state is studied by means of Monte Carlo simulation. We observe the characteristic initial increase of the (spatially) averaged magnetization predicted by Janssen et al. The exponent theta ' that governs the initial behaviour is determined, and the dependence of the long-time linear decay on the initial magnetization, m 0 , is analysed. Our simulation corroborates earlier results derived from continuum models.
We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.