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Physical Review E

Monte Carlo study of the competition between long-range and short-range correlated disorder in a second-order phase transition

Manuel I. Marqués

Published May 21, 2009• 1 citation
Publisher

Abstract

The influence of coexisting correlated and noncorrelated impurities on the critical behavior of the three-dimensional Ising model is studied using Monte Carlo numerical simulations and finite-size scaling. The amount of correlated and noncorrelated vacancies is modified and controlled during the simulations. The long-range correlated (LRC) critical behavior is always found for any value of the concentration of correlated vacancies. The smaller the amount of correlated vacancies the larger the system size needed to detect the LRC universality class. This result explains why critical values measured in xerogel liquid-vapor experiments, where the concentration of correlated vacancies is marginal, seem to correspond to a short-range correlated disorder.

DOI
10.1103/physreve.79.052103

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The influence of correlated impurities on the critical behavior of the three-dimensional (3D) Ising model is studied using Monte Carlo simulations. Spins are confined into the pores of simulated aerogels (diffusion-limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of this magnetic system. Finite-size scaling is used to estimate critical couplings and exponents. Long-range correlated disorder does not affect the critical behavior. Asymptotic exponents differ from those of the pure 3D Ising model, but it is impossible, with our precision, to distinguish them from the randomly diluted Ising model.
We present results of large-scale Monte Carlo simulations for a three-dimensional Ising model with short-range interactions and planar defects, i.e., disorder perfectly correlated in two dimensions. We show that the phase transition in this system is smeared, i.e., there is no single critical temperature, but different parts of the system order at different temperatures. This is caused by effects similar to but stronger than Griffiths phenomena. In an infinite-size sample there is an exponentially small but finite probability to find an arbitrary large region devoid of impurities. Such a rare region can develop true long-range order while the bulk system is still in the disordered phase. We compute the thermodynamic magnetization and its finite-size effects, the local magnetization, and the probability distribution of the ordering temperatures for different samples. Our Monte-Carlo results are in good agreement with a recent theory based on extremal statistics.
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.
Monte Carlo simulations of the short-time dynamic behaviour are reported for three-dimensional Ising model and XY-model with long-range spatially correlated disorder at criticality, in the case corresponding to linear defects. The static and dynamic critical exponents are computed with the use of the corrections to scaling. The obtained values of the exponents are in a good agreement with results of the field-theoretic description of the critical behaviour of this model in the two-loop approximation and with our results of Monte Carlo simulations of three-dimensional Ising model in equilibrium state.
Encouraged by experiments on 4He in aerogels, we confine planar spins in the pores of simulated aerogels (diffusion limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of the three-dimensional XY model. Monte Carlo simulations and finite-size scaling are used to determine critical couplings K(c) and exponents. In agreement with experiments, clear evidence of change in the thermal critical exponents nu and alpha is found at nonzero volume fractions of impurities. These changes are explained in terms of hidden long-range correlations within disorder distributions.