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Journal of Physics Conference Series

Nonequilibrium Critical Relaxation in the 2D Random Field Ising Model

László Környei

Published May 1, 2006• 3 citations
Publisher

Abstract

A particular type of nonequilibrium critical dynamics is discussed in this paper, in which the initial state is generated in the square lattice Ising model with random fields of zero mean and variance h. Thus we interpolate between the ordered (h = 0) and disordered (h → ∞) initial states. In particular we consider the square lattice Ising model in which for weak enough random fields the clusters of parallel spins are percolating in the initial state. We aim to study the effect of correlations in the initial state on the nonequilibrium dynamics, so we measure the nonequilibrium relaxation of the magnetization and the autocorrelation function. The relaxation of the magnetization, characterized by the initial slip exponent, θ, is not altered by the percolating nature of the initial state. However, the starting part of the magnetization curve is affected by a cluster dissolution effect leading to a reentrance in time.

DOI
10.1088/1742-6596/40/1/004

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A particular type of nonequilibrium critical dynamics is discussed in this paper, in which the initial state is generated in the square lattice Ising model with random fields of zero mean and variance h. Thus we interpolate between the ordered (h = 0) and disordered (h → ∞) initial states. In particular we consider the square lattice Ising model in which for weak enough random fields the clusters of parallel spins are percolating in the initial state. We aim to study the effect of correlations in the initial state on the nonequilibrium dynamics, so we measure the nonequilibrium relaxation of the magnetization and the autocorrelation function. The relaxation of the magnetization, characterized by the initial slip exponent, θ, is not altered by the percolating nature of the initial state. However, the starting part of the magnetization curve is affected by a cluster dissolution effect leading to a reentrance in time.
We consider nonequilibrium critical dynamics of the two-dimensional Ising model for which the initial state is prepared by switching on random fields with zero mean and variance H . In the initial state there is no magnetic order but the clusters of parallel spins have a percolation transition for small enough H . Using heat-bath dynamics we measure the relaxation of the magnetization which shows a re-entrance in time. Due to cluster dissolution in the early time regime there is a decrease of the magnetization, followed by an increase due to nonequilibrium domain growth which itself turns to a decrease due to equilibrium relaxation. The power law decay of the nonequilibrium autocorrelation function is not influenced by the percolation in the initial state.
We investigate the critical relaxation of dilute Ising systems starting from a macroscopically prepared initial state with short-range correlations. Using the methods of renormalized field theory we calculate the exponent theta ' which describes the initial increase of the magnetization to second order in square root epsilon , where epsilon =4-d. Since computer simulations of the dilute Ising model have shown that a large part of the critical region is governed by crossover phenomena, we also discuss the influence of the slow crossover on the relaxation.
The effect of various initial magnetizations m0 and structural defects the nonequilibrium critical behavior of the three-dimensional Ising model is numerically studied. Based on an analysis of the time dependence of the magnetization and the two-time dependence of the autocorrelation function and dynamic susceptibility, the significant effect of initial states on relaxation magnetizations and aging effects characterized by anomalous relaxation inhibition and correlation in the system with increasing waiting time was revealed. The fluctuation–dissipation theorem violation was studied, and the values of the limit fluctuation–dissipation ratio (FDR) are calculated. It is shown that two universality subclasses can be distinguished in the nonequilibrium critical behavior of the three-dimensional Ising model with random initial magnetization m0 These subclasses correspond to the system evolution from the high-temperature (m0 = 0) and low-temperature (m0 = 1) initial states with limit FDRs characteristic of these states.
We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength h = HTk B , as well as the Villain model on the square lattice. After a quench from a disordered initial state to T = 0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, L , we define and measure sample-dependent ‘first passage time’, t r , which is the number of Monte Carlo steps until the energy is relaxed to the ground state value. The distribution of t r , in particular its mean value, ⟨ t r ( L )⟩, is shown to obey the scaling relation, ⟨ t r ( L )⟩∼ L 2 ln( L / L 0 ), for both models. Scaling of the autocorrelation function of the antiferromagnetic triangular model is shown to involve logarithmic corrections, both at H = 0 and at the field-induced Kosterlitz–Thouless transition: however, the autocorrelation exponent is found to be H -dependent.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
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.