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Europhysics Letters (EPL)

Monte Carlo Simulation of Strongly Disordered Ising Ferromagnets

Hans-O Heuer

Published Jul 15, 1990• 27 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/12/6/014

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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.
We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio C+{\mathit{C}}_{+}C+​/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.
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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.
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We study the critical behavior of the Ising model in three dimensions on a lattice with site disorder by using Monte Carlo simulations. The disorder is either uncorrelated or long-range correlated with correlation function that decays according to a power law ${r}^{\ensuremath{-}a}$. We derive the critical exponent of the correlation length $\ensuremath{\nu}$ and the confluent correction exponent $\ensuremath{\omega}$ in dependence of aaa by combining different concentrations of defects $0.05\ensuremath{\le}{p}_{d}\ensuremath{\le}0.4$ into one global fit ansatz and applying finite-size scaling techniques. We simulate and study a wide range of different correlation exponents $1.5\ensuremath{\le}a\ensuremath{\le}3.5$ as well as the uncorrelated case $a=\ensuremath{\infty}$ and are able to provide a global picture not yet known from previous works. Additionally, we perform a dedicated analysis of our long-range correlated disorder ensembles and provide estimates for the critical temperatures of the system in dependence of the correlation exponent aaa and the concentration of defects pd{p}_{d}pd​. We compare our results to known results from other works and to the conjecture of Weinrib and Halperin: $\ensuremath{\nu}=2/a$, and discuss the occurring deviations.
A review is given of Monte Carlo simulations for the phase transition in simple cubic Ising ferro-, ferri-, and antiferromagnets with quenched non-magnetic impurities of concentration x. Special emphasis is given on the description of a fully vectorized multispin coding program and on the problems related to the analysis of the Monte Carlo data. The following three subjects are discussed: 1) For ferro- and antiferromagnets, the critical behaviour of the order parameter is different from the one of the corresponding pure system, in agreement with the prediction of the Harris criterion. The possibly effective exponent β depends on x, whereas for ferrimagnets no such concentration dependence is revealed by the present study for x=0, 0.1, and 0.2.2) For strongly diluted systems, there is a maximum for the Kouvel-Fisher exponent γ(T) of the order-parameter susceptibility outside the critical regime, in accordance with the predictions of the correlated molecular field theory. 3) For the diluted antiferromagnet (x=0.5), a cusp emerges at T c for the uniform susceptibility, in agreement with the theory of Fishman and Aharony.