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Mathematical Models and Computer Simulations

Computation of Relative Variances of Magnetization and Susceptibility in a Disordered Ising Model: The Results of Computer Simulation

Albert Babaevich Babaev, Akai Kurbanovich Murtazaev

Published Jul 1, 2019• 15 citations
Publisher

Abstract

Abstract Based on the Monte Carlo method, the relative variances of magnetization Rm and susceptibility Rχ in the disordered spin lattice Ising model are computed as a function of the degree of the dilution of a disorder. It is shown that the introduction of a disorder in the form of nonmagnetic impurities in the three-dimensional Ising model produces nonzero values for Rm and Rχ at the critical point.

DOI
10.1134/s2070048219040021

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Abstract Based on the Monte Carlo method, the relative variances of magnetization Rm and susceptibility Rχ in the disordered spin lattice Ising model are computed as a function of the degree of the dilution of a disorder. It is shown that the introduction of a disorder in the form of nonmagnetic impurities in the three-dimensional Ising model produces nonzero values for Rm and Rχ at the critical point.
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.
The relative variances of the magnetization Rm, heat capacity Rc, and susceptibility Rχ are calculated by the Monte Carlo method in the spin lattice four-component weakly diluted Potts model on a square lattice at a spin density p = 0.80. It is shown that introducing a disorder in the form of nonmagnetic impurities into the 2D Potts model leads to nonzero values of Rm, Rc, and Rχ at the critical point. It has been found that these values decrease noticeably for systems with linear sizes L ≥ 120 interatomic distances.
Abstract The relative variances of the magnetization Rm, heat capacity Rc, and susceptibility Rχ are calculated by the Monte Carlo method in the spin lattice four-component weakly diluted Potts model on a square lattice at a spin density p = 0.80. It is shown that introducing a disorder in the form of nonmagnetic impurities into the 2D Potts model leads to nonzero values of Rm, Rc, and Rχ at the critical point. It has been found that these values decrease noticeably for systems with linear sizes L ≥ 120 interatomic distances.
Simulation of critical relaxation of the magnetization in a three-dimensional Ising model diluted by quenched nonmagnetic impurity atoms is carried out. The cubic systems with linear size L = 144 and spin concentrations p = 0.95, 0.8, 0.6, and 0.5 are studied. The dynamical critical exponent z is determined by the Monte Carlo method combined with the dynamical renormalization group method. The obtained results are: z (p): z (0.95) = 2.19 ± 0.07, z (0.8) = 2.29 ± 0.06, z (0.6) = 2.72 ± 0.08, and z (0.5) = 2.75 ± 0.08.
In this work we study a disordered binary Ising model on the square lattice. The model system consists of two different particles with spin-1/2 and spin-1, which are randomly distributed on the lattice. It has been considered only spin nearest-neighbor exchange interactions with J>0J>0J>0. This system can represent a disordered magnetic binary alloy AxB1−xA_{x}B_{1-x}Ax​B1−x​, obtained from the high temperature quenching of a liquid mixture. The results were obtained by the use of Monte Carlo simulations for several lattice sizes LLL, temperature TTT and concentration xxx of ions AAA with spin-1/2. We found its critical temperature, through the reduced fourth-order Binder cumulant for the several values of the concentration xxx of particles (spin-1/2, spin-1), and also the magnetization, the susceptibility and the specific heat as a function of temperature TTT.