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Journal of Experimental and Theoretical Physics Letters

Damage spreading method in studies of the critical behavior of disordered systems

Vladimir V. Prudnikov, Andrey N. Vakilov, Д. В. Талашок

Published Jan 1, 2015• 5 citations
Publisher

Abstract

The computer simulations of the critical dynamics of the structurally disordered three-dimensional Ising model are performed by the damage spreading method. For the systems with spin densities p = 0.6 and 0.8, we calculate the critical temperature and dynamic critical exponent z characterizing the behavior of relaxation properties near the critical point. The analysis of the results demonstrates the nonuniversality of the critical dynamics in the disordered Ising model. To interpret such dynamics, we introduce the concept of two universal types of critical behavior corresponding to weak and strong disorder, respectively.

DOI
10.1134/s0021364014220147

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The computer simulations of the critical dynamics of the structurally disordered three-dimensional Ising model are performed by the damage spreading method. For the systems with spin densities p = 0.6 and 0.8, we calculate the critical temperature and dynamic critical exponent z characterizing the behavior of relaxation properties near the critical point. The analysis of the results demonstrates the nonuniversality of the critical dynamics in the disordered Ising model. To interpret such dynamics, we introduce the concept of two universal types of critical behavior corresponding to weak and strong disorder, respectively.
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.
The influence of nonequilibrium initial states on the evolution of anisotropic systems with quenched uncorrelated structural defects at the critical point is studied. The field-theoretical description of the nonequilibrium critical behavior of 3D systems is obtained for the first time, and the dynamic critical exponent of the short-time evolution in the two-loop approximation without the use of ɛ expansion is calculated. The values of dynamic critical exponents calculated using the series resummation methods are compared with the results of computer simulation of nonequilibrium critical behavior of the 3D disordered Ising model in the short-time regime. It is demonstrated that the values of the critical exponents calculated in this paper are in better agreement with the results of computer simulation than the results of application of ɛ expansion.
By studying damage spreading in spin models in a temperature gradient, it is possible to determine not only the critical temperature, and the correlation length exponent v, but also a second critical exponent beta with high precision. This is done by studying the dynamics of damage spreading, in particular the overhangs which develop in the damage front. The authors demonstrate the method for the two-dimensional Ising model using heat bath dynamics, and discuss differences with Metropolis dynamics.
Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations p=0.95 and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the use of the corrections to scaling for systems starting separately from ordered and disordered initial states. The obtained values of the exponents demonstrate a universal behavior of weakly site-diluted Ising model in the critical region. The values of the exponents are compared to results of numerical simulations which have been obtained in various works and, also, with results of the renormalization-group description of this model.
The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.