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Journal of Physics A Mathematical and General

Critical dynamics of disordered two-dimensional Ising systems: a Monte Carlo study

Vladimir V. Prudnikov, O. N. Markov

Published Mar 23, 1995• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/28/6/011

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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.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
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.
We report extensive Monte Carlo simulations of the site-disordered 2-dimensional Ising system. For concentrations between p = 0.6 and p = 1.0 ( p c = 0.59) we have studied the magnetic and caloric properties as well as the cumulants of the magnetization distribution in the critical region. We have found that disorder induces a strong increase of γ eff (γ max = 2.16 for p = 0.7) in the usual experimental temperature range ( T - T c )/ T c = 10 -2 ÷ 10 0 . Asymptotically near T c , γ approaches the theoretical value γ = 1.75 independent of the concentration. We propose a model of weakly coupled compact clusters of spins to describe this crossover behaviour of disordered spin systems.
The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.
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.