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Herald of Omsk University

Calculation of logarithmic corrections for the correlation length and its manifestation in the nonequilibrium critical behavior of the autocorrelation function of the disordered two-dimensional Ising model

Vladimir V. Prudnikov, Pavel Vladimirovich Prudnikov, P. N. Malyarenko, Щур Лев Николаевич

Published Jan 1, 2019• 0 citations
Publisher

Abstract

The results of a numerical Monte Carlo study of features of defects influence on nonequilibrium critical behavior of the two-dimensional Ising model are presented. The logarithmic corrections for the correlation length of disordered systems with different spin concentrations are calculated. Analysis of correlation length time dependence and the two-time dependences of the autocorrelation function has revealed an influence of logarithmic corrections and crossover phenomena of percolation behavior on nonequilibrium characteristics and critical exponents.

DOI
10.24147/1812-3996.2019.24(4).16-24

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The results of a numerical Monte Carlo study of features of defects influence on nonequilibrium critical behavior of the two-dimensional Ising model are presented. The logarithmic corrections for the correlation length of disordered systems with different spin concentrations are calculated. Analysis of correlation length time dependence and the two-time dependences of the autocorrelation function has revealed an influence of logarithmic corrections and crossover phenomena of percolation behavior on nonequilibrium characteristics and critical exponents.
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Abstract A Monte Carlo simulation of the nonequilibrium critical behavior is carried out for the two-dimensional structurally disordered XY model during its evolution from a high-temperature initial state. The features of the effect of structural disorder on the time dependence of the correlation length ξ(t) of the system and the dynamical scaling of the spin-spin autocorrelation function CSS(t, tw) are revealed. A direct calculation of the dynamic behavior of the correlation length ξ(t) of the two-dimensional structurally disordered XY model is carried out, and it is shown that this model, just as the pure model, exhibits a time dependence with a logarithmic correction ξ(t) ∝ (t/lnt)1/2, which is associated with the nonequilibrium annihilation of vortices and antivortices in the forming vortex pairs. Based on the analysis of the time dependence of the correlation length ξ(t) and the magnetization cumulant g2(t), it is shown that the two-dimensional XY model with the spin concentration p = 0.7 is so close to the spin percolation threshold pc that the influence of the attraction of the percolation fixed point becomes crucial for the relaxation properties of the system. However, the features of the critical dynamics of systems with spin concentrations of p = 0.9 and p = 0.8 are determined by the attraction of the pure fixed point. Temperature and concentration dependences of the Fisher critical exponent η(p, T) are determined using the scaling properties of the calculated two-time dependence of the spin-spin autocorrelation function CSS(t, tw). The scaling functions of the two-time dependence of the spin-spin autocorrelation function CSS(t, tw) are calculated using the dynamic dependence of the correlation length ξ(t) obtained as a result of simulation, and the values of the decay exponents λC(p, T) of the scaling functions in the long-time regime are determined, which are in good agreement with the relation λC(p, T) = 1 + η(p, T) and prove that the dynamical scaling is realized for the nonequilibrium characteristics of structurally disordered systems.
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.
The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.
We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.