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Physics of the Solid State

Monte Carlo Study of the Effect of Initial States and Structural Defects on Nonequilibrium Critical Behavior of the Three-Dimensional Ising Model

Vladimir V. Prudnikov, Pavel Vladimirovich Prudnikov, P. N. Malyarenko

Published Jun 1, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1134/s1063783418060306

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Progress of Theoretical and Experimental Physics•May 1, 2015•18 citations
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.
AbstractThe effect of various initial magnetizations m _0 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 m _0 These subclasses correspond to the system evolution from the high-temperature ( m _0 = 0) and low-temperature ( m _0 = 1) initial states with limit FDRs characteristic of these states.
Abstract The influence of various initial magnetizations m0 and structural defects on the nonequilibrium critical behavior of the two-dimensional Ising model is numerically simulated by Monte Carlo methods. Based on analysis of the time dependence of magnetization and the two-time dependences of autocorrelation function and dynamic susceptibility, we revealed the influence of logarithmic corrections and the crossover phenomena of percolation behavior on the nonequilibrium characteristics and the critical exponents. Violation of the fluctuation–dissipation theorem is studied, and the limiting fluctuation–dissipation ratio is calculated for the case of high-temperature initial state. The influence of various initial states on the limiting fluctuation–dissipation ratio is investigated. The nonequilibrium critical dynamics of weakly disordered systems with spin concentrations p ≥ 0.9 is shown to belong to the universality class of the nonequilibrium critical behavior of the pure model and to be characterized by the same critical exponents and the same limiting fluctuation–dissipation ratios. The nonequilibrium critical behavior of systems with p ≤ 0.85 demonstrates that the universal characteristics of the nonequilibrium critical behavior depend on the defect concentration and the dynamic scaling is violated, which is related to the influence of the crossover effects of percolation behavior.
Abstract A numerical Monte Carlo study has been performed on the influence of various initial states, easy-axis-type magnetic anisotropy, and structural defects on a nonequilibrium critical behavior of the classical three-dimensional Heisenberg model. An analysis of the time dependence of the magnetization and the autocorrelation function for the isotropic Heisenberg model shows substantial influence of the initial states on the relaxation of the magnetization and the aging effects in the behavior of the autocorrelation function, which are characterized by anomalous slowing down of the relaxation and correlation in the system as the waiting time increases. The study of the anisotropic Heisenberg model shows that the behavior of the magnetization and the autocorrelation function in a long-time regime is characterized by the critical exponents of the three-dimensional Ising model with a faster time decay of the autocorrelation function than that for the isotropic model. It has been revealed that the existence of structural defects in the case of the evolution of the system from the low-temperature initial state leads to an anomalously significant slowing down of the autocorrelation function. These features in the behavior of the autocorrelation function are characterized by the “superaging” effect with “superaging” exponent μ = 2.6(1) and related to the pinning of domain walls on structural defects during a nonequilibrium change in the domain structure of the system. In the case of evolution from the high-temperature initial state, the structural defects enhance the aging effect in the aging regime but their influence is irrelevant in the long-time regime.
The Monte Carlo study of the nonequilibrium critical evolution of structurally disordered anisotropic magnetic materials from the low-temperature initial state with the reduced magnetization m 0 = 1 is performed within the broad range of spin densities, p = 1.0, 0.95, 0.8, 0.6, and 0.5. It is shown that, in such systems, the pinning of domain walls by structural defects occurring when the evolution starts from the low-temperature state leads to significant changes in the nonequilibrium “aging” and “memory” effects in comparison to those characteristic of the “pure” system. As a result, in the long-term regime at times t–t w » t w » 1, an anomalously strong slowing down in the correlation effects is revealed. It is shown that a decrease in the autocorrelation function with time occurs according to a power law typical of the critical relaxation of the magnetization in contrast to a usual scaling dependence. Eventually, the limiting value of the fluctuation–dissipation ratio X∞ for the structurally disordered systems with p < 1 vanishes, whereas for the pure system, we have X∞ = 0.784(5). The nonequilibrium critical “superaging” stage is found. This stage is characterized by the critical exponent µ = 2.30(6) for weakly disordered systems and by µ = 2.80(7) for the systems with strong disorder.
A Monte Carlo numerical simulation of the specific features of nonequilibrium critical behavior is carried out for the two-dimensional structurally disordered XY model during its evolution from a low-temperature initial state. On the basis of the analysis of the two-time dependence of autocorrelation functions and dynamic susceptibility for systems with spin concentrations of p = 1.0, 0.9, and 0.6, aging phenomena characterized by a slowing down of the relaxation system with increasing waiting time and the violation of the fluctuation–dissipation theorem (FDT) are revealed. The values of the universal limiting fluctuation–dissipation ratio (FDR) are obtained for the systems considered. As a result of the analysis of the two-time scaling dependence for spin–spin and connected spin autocorrelation functions, it is found that structural defects lead to subaging phenomena in the behavior of the spin–spin autocorrelation function and superaging phenomena in the behavior of the connected spin autocorrelation function.
We investigate the non-equilibrium critical evolution of statistical systems and describe of its some features, such as aging and violation of the fluctuation-dissipation theorem. We consider some theoretical results of computations for universal quantities that have been obtained in recent years, such as the exponents determining the scaling behavior of dynamic response and correlation functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics. We derive our original Monte Carlo simulation results for 3D pure and diluted Ising models with Glauber and Metropolis dynamics. For these models, we analyze the influence of critical fluctuations, different non-equilibrium initial states, and the presence of nonmagnetic impurities on the two-time dependence of correlation and response functions on characteristic time variables, such as waiting time |twt_{\rm w}tw​| and time of observation |t−twt - t_{\rm w}t−tw​| with |t>twt>t_{\rm w}t>tw​|⁠. We discuss the obtained values of the non-equilibrium exponents for the autocorrelation and response functions and values of the universal long-time limit of the fluctuation-dissipation ratio |X∞X^{\infty }X∞|⁠. Our simulation results demonstrate that the insertion of disorder leads to new values of |X∞X^{\infty }X∞| with |Xdiluted∞>Xpure∞X_{\rm diluted}^{\infty }>X_{\rm pure}^{\infty }Xdiluted∞​>Xpure∞​|⁠.