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Journal of Physics Conference Series

Numerical investigation of the critical behavior of the three-dimensional Ising model near the percolation threshold

Maria Shlyakhtich, Pavel Vladimirovich Prudnikov, A S Papushin

Published Feb 1, 2019• 0 citations
Publisher

Abstract

In this work we investigate the critical behavior of the disordered three-dimensional Ising model with an impurity concentration close to the threshold of impurity percolation on high-performance computing systems. Starting from initial configurations, the system was updated with Metropolis algorithm and invaded cluster algorithm. The values of dynamic critical exponent z and static critical exponent β / ν were obtained in this paper for both algorithm.

DOI
10.1088/1742-6596/1163/1/012029

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In this work we investigate the critical behavior of the disordered three-dimensional Ising model with an impurity concentration close to the threshold of impurity percolation on high-performance computing systems. Starting from initial configurations, the system was updated with Metropolis algorithm and invaded cluster algorithm. The values of dynamic critical exponent z and static critical exponent β / ν were obtained in this paper for both algorithm.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
In this work we performed numerical simulations for the Ising model on three dimensional lattices with coordination number equal 5. With Monte Carlo simulations in the static case we evaluated the critical temperature and the static critical exponents ννν, γγγ and βββ. Once that we have the critical temperature value we investigated the dynamical critical behavior with Glauber dynamics, starting from disordered states. From our simulations we obtained the values for the dynamic exponent z=2.037(8)z=2.037(8)z=2.037(8), the exponent for the autocorrelation λ/z=1.364(5)λ/z=1.364(5)λ/z=1.364(5), the exponent of the critical initial increase θ′=0.109(8)θ'=0.109(8)θ′=0.109(8) and the asymptotic value of the fluctuation-dissipation ratio X∞=0.433(1)X^\infty=0.433(1)X∞=0.433(1). All of these results are in good agreement to the previous values reported for the 3D Ising model universality class.
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.
A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.