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Journal of Physics A Mathematical and General

Dynamic scaling of disordered Ising systems

Hans-O Heuer

Published Mar 21, 1993• 37 citations
Publisher

Abstract

The critical dynamics of site-disordered three-dimensional Ising systems has been studied by conventional local dynamics simulations investing 10 15 single spin flips. The author has verified that the correlation functions of disordered Ising systems obey dynamic scaling behaviour. The dynamic scaling functions have been calculated. The critical exponent z appears to be strongly dependent on the degree of disorder. The author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour. The dynamical exponent has been estimated to be z=2.4+or-0.1 which is surprisingly near to the renormalization group result in O( in 1/2).

DOI
10.1088/0305-4470/26/6/008

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The critical dynamics of site-disordered three-dimensional Ising systems has been studied by conventional local dynamics simulations investing 10 15 single spin flips. The author has verified that the correlation functions of disordered Ising systems obey dynamic scaling behaviour. The dynamic scaling functions have been calculated. The critical exponent z appears to be strongly dependent on the degree of disorder. The author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour. The dynamical exponent has been estimated to be z=2.4+or-0.1 which is surprisingly near to the renormalization group result in O( in 1/2).
The author has investigated the critical behaviour of site-disordered Ising systems over a large range of dilution (0.6<or=p<or=1.0) investing about 10 15 single spin flips with conventional local dynamics. Critical temperatures have been determined with high-precision using the cumulant method. The finite-size simulations lead to concentration-dependent exponents which seem to violate universality. The author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region. The author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder.
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 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.
We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size dependence of suitably defined autocorrelation times at the critical point provides the estimate z = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to T c . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behaviour gives an estimate of z that is perfectly consistent with the equilibrium estimate z = 2.35(2).
The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.
We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.