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Physical Review Letters

Incompleteness of the Critical Exponent Description for Ferromagnetic Systems Containing Random Impurities

Barry M. McCoy

Published Aug 18, 1969• 203 citations
Publisher

Abstract

We consider the magnetic properties and spin correlation functions of a two-dimensional Ising model with random impurities near the critical temperature and find that the usual parametrization provided by the exponents $\ensuremath{\gamma}$, $\ensuremath{\delta}$, and $\ensuremath{\eta}$ and the correlation length $\ensuremath{\xi}$ for $T\ensuremath{\gtrsim}{T}_{c}$ does not describe our results. Experimental implications of this breakdown of the critical exponent forms are discussed.

DOI
10.1103/physrevlett.23.383

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We consider the magnetic properties and spin correlation functions of a two-dimensional Ising model with random impurities near the critical temperature and find that the usual parametrization provided by the exponents \ensuremathγ\ensuremath{\gamma}\ensuremathγ, \ensuremathδ\ensuremath{\delta}\ensuremathδ, and \ensuremathη\ensuremath{\eta}\ensuremathη and the correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ for T\ensuremath≳TcT\ensuremath{\gtrsim}{T}_{c}T\ensuremath≳Tc​ does not describe our results. Experimental implications of this breakdown of the critical exponent forms are discussed.
The effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.
The critical properties of random ferromagnets with pure specific-heat exponent ${\ensuremath{\alpha}}_{P}=0$ are analyzed using renormalization-group methods. The origin and form of the impurity-induced logarithmic corrections for the random two-dimensional Ising model are discussed. Corrections develop only for the specific heat, which presents a new singularity of the form ${|\mathrm{ln}|T\ensuremath{-}{T}_{c}||}^{1\ensuremath{-}\ensuremath{\mu}}$ as well as a new critical exponent $\ensuremath{\mu}$.
We consider two-dimensional Ising models with randomly distributed ferromagnetic bonds and study the local critical behavior at defect lines by extensive Monte Carlo simulations. Both for ladder and chain type defects, non-universal critical behavior is observed: the critical exponent of the defect magnetization is found to be a continuous function of the strength of the defect coupling. Analyzing corresponding stability conditions, we obtain new evidence that the critical exponent ννν of the bulk correlation length of the random Ising model does not depend on dilution, i.e. ν=1ν=1ν=1.
In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special empha-size is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually avoiding lines) the function is governed by the power law of 1/ra with an universal exponent a≈2. This result sup-ports our findings about the numerical values of the critical exponents governing magnetic phase transition in the 3D Ising model with long-range-correlated disorder.