Critical properties of diluted anisotropic magnets near percolation threshold

Abstract

In this paper we study some theoretical aspects of the multicritical behavior of a diluted Heisenberg magnet with Ising-like anisotropy near the percolation threshold. We present three approaches. The first approach is based on the observation that near the percolation threshold the magnetic correlations propagate dominantly along one-dimensional paths. A consequence of this assumption is argued to be that the magnetic correlation function factorizes into two parts, one of which is a purely geometric factor and the other a magnetic correlation function on a chain which appears to be a constrained self-avoiding walk. This simple assumption then helps us understand some salient features of magnetic correlations in diluted systems. Our second approach is based on the scaling theory due to Lubensky and Birgeneau et al. Here we present an analytical, though approximate, calculation of the longitudinal correlation length of an anisotropic Heisenberg chain of classical spins. Our expression is valid in both low- and high-temperature limits and for both low and high anisotropy. This enables us to study the crossover from Heisenberg-like behavior to Ising-like behavior as the percolation point is approached. In the third approach we perform a renormalization-group calculation for a S=1/2-spin Heisenberg model. The fixed points and the flows of the recursion relations are studied. The critical surface of the dominant fixed point in the temperature-concentration-anisotropy space is numerically obtained. From this, we calculate the Curie temperature versus concentration curves as the anisotropy is varied. These are found to be similar to those obtained in the scaling approach.