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Fractals

THERMALLY DILUTED ISING SYSTEMS

Manuel I. Marqués, Julio A. Gonzalo, Jorge Íñiguez

Published Feb 1, 2003• 0 citations
Publisher

Abstract

In this paper finite size scaling techniques are used to study the universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality. The critical temperature, the critical exponents and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short range correlated disordered systems. This result is in agreement with theoretical predictions previously made by Weinrib and Halperin for systems with long range correlated disorder.

DOI
10.1142/s0218348x03001719

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In this paper finite size scaling techniques are used to study the universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality. The critical temperature, the critical exponents and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short range correlated disordered systems. This result is in agreement with theoretical predictions previously made by Weinrib and Halperin for systems with long range correlated disorder.
The universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality, is investigated by finite size scaling techniques using the Monte Carlo method. We find that the critical temperature, the critical exponents, and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short-range correlated disordered systems. Our results agree fairly well with theoretical predictions previously made by Weinrib and Halperin for systems with long-range correlated disorder.
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 equilibrium ensemble approach to disordered systems is used to investigate the critical behavior of the two-dimensional Ising model in the presence of quenched random site dilution. The numerical transfer matrix technique in semi-infinite strips of finite width, together with phenomenological renormalization and conformal invariance, is particularly suited to putting the equilibrium ensemble approach to work. A method by which to extract with great precision the critical temperature of the model is proposed and applied. A more systematic finite-size scaling analysis than in previous numerical studies has been performed. A parallel investigation, along the lines of the two main scenarios currently under discussion, namely, the logarithmic corrections scenario (with critical exponents fixed in the Ising universality class) versus the weak universality scenario (critical exponents varying with the degree of disorder), is carried out. In interpreting our data, maximum care is constantly taken to be open to both possibilities. A critical discussion shows that an unambiguous discrimination between the two scenarios is still not possible on the basis of the available finite-size data.
We use large-scale Monte Carlo simulations to test the Weinrib-Halperin criterion that predicts new universality classes in the presence of sufficiently slowly decaying power-law correlated quenched disorder. While new universality classes are reasonably well established, the predicted exponents are controversial. We propose a method of growing such correlated disorder using the three-dimensional Ising model as a benchmark system for both generating disorder and studying the resulting phase transition. Critical equilibrium configurations of a disorder-free system are used to define the two-value distributed random bonds with a small power-law exponent given by the pure Ising exponent. Finite-size scaling analysis shows a new universality class with a single phase transition, but the critical exponents ${\ensuremath{\nu}}_{d}=1.13(5),{\ensuremath{\eta}}_{d}=0.48(3)$ differ significantly from theoretical predictions. We find that depending on the details of the disorder generation, disorder-averaged quantities can develop peaks at two temperatures for finite sizes. Finally, a layer model with the two values of bonds spatially separated in halves of the system genuinely has multiple phase transitions, and thermodynamic properties can be flexibly tuned by adjusting the model parameters.
We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.