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Journal of Physics A Mathematical and General

Critical crossover phenomena in disordered Ising systems

Hans-O Heuer

Published Mar 21, 1993• 70 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/26/6/007

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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.
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).
Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.
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.
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.
We report about extensive Monte Carlo simulations of disordered Ising systems with concentrations of magnetic sites between p = 1.0 and p = 0.5. We have measured the magnetization, the susceptibility and the caloric properties in the critical region. In addition the magnetization-energy correlation function has been investigated and the critical exponent ζ = 1 - β = (α + γ)/2 has been determined. We find that power law behaviour and scaling is fulfilled in the whole concentration range. The effective critical exponents γ, β and ζ are shown to be concentration dependent. In the range of strong disorder γ and β increase until they reach a kind of asymptotic behaviour at β = 0.335 ± 0.01 and γ = 1.49 ± 0.02 for p = 0.5. In accord with the exponent relation ζ = 1 - β the critical exponent ζ decreases with dilution to ζ = 0.675 ± 0.01.
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.