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Physical review. B, Condensed matter

Critical behavior of the simple cubic Ising model with quenched site impurities

David Paul Landau

Published Sep 1, 1980• 79 citations
Publisher

Abstract

The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.

DOI
10.1103/physrevb.22.2450

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We have used a Monte Carlo renormalization-group method to study the Baxter-Wu model with random, quenched, nonmagnetic site impurities. The results clearly show that the impurities change the critical exponents from the zero-impurity exact value of \ensuremath{\nu}=(2/3) to \ensuremath{\nu}=0.97\ifmmode\pm\else\textpm\fi{}0.04. The magnetic and crossover exponents do not change measurably with the addition of the impurities. We also find that the fixed point for the model with quenched impurities is further from the starting Hamiltonian than it was for the pure model.

Physical review. B, Condensed matter•Sep 1, 1985•19 citations
The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.
We have used an importance-sampling Monte Carlo technique to study the Baxter-Wu model with various fractions of the lattice sites occupied by random, quenched, nonmagnetic, site impurities. We found the system had long-time fluctuations which were caused by the creation and motion of domain boundaries. For this reason our study required the use of a large number of Monte Carlo steps per spin. The data were analyzed using finite-size scaling to extract the infinite-lattice critical exponents and critical amplitudes from the Monte Carlo simulation of finite lattices. This analysis of the data for the pure Baxter-Wu model yielded results which agreed well with exact results and series-expansion predictions. The addition of quenched site impurities caused a dramatic change in the critical behavior. The pure-lattice critical exponents ($\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=\frac{2}{3}$, ${\ensuremath{\gamma}}_{m}\ensuremath{\simeq}1.17$) change upon the addition of only a few percent impurities to $\ensuremath{\nu}=1.00\ifmmode\pm\else\textpm\fi{}0.07$, $\ensuremath{\alpha}\ensuremath{\lesssim}0.0$, and ${\ensuremath{\gamma}}_{m}=1.95\ifmmode\pm\else\textpm\fi{}0.08$. The phase diagram as a function of impurity concentration and an estimate for the infinite-lattice percolation limit are also given.
Using the Monte Carlo technique the authors have studied a two-dimensional Ising model with quenched non-magnetic impurities. In particular they calculated the temperature dependence of the order parameter, of the zero-field susceptibility and of the specific heat. These results indicate that a second-order phase transition is preserved for p>or=0.9 where 1-p denotes the concentration of randomly distributed non-magnetic sites. Moreover, they yield beta approximately=1/8 and gamma approximately=7/4, indicating that these exponents equal the corresponding ones for the pure Ising model.
A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.
A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.
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.
We have used a Monte Carlo renormalization-group method to study the Baxter-Wu model with random, quenched, nonmagnetic site impurities. The results clearly show that the impurities change the critical exponents from the zero-impurity exact value of \ensuremath{\nu}=(2/3) to \ensuremath{\nu}=0.97\ifmmode\pm\else\textpm\fi{}0.04. The magnetic and crossover exponents do not change measurably with the addition of the impurities. We also find that the fixed point for the model with quenched impurities is further from the starting Hamiltonian than it was for the pure model.