Low-temperature studies of random Ising models

Abstract

A brief discussion is given of the application of the methods of low-temperature or excitation expansions to random Ising models. A specific investigation is then undertaken of the low-temperature behaviour of the random-field model A relationship is established between the δ-function distribution with equal probabilities of positive and negative fixed fields H ₀, and a site percolation process on the lattice with p = 1/2. At T = 0 successive percolation clusters give rise to first-order transitions at different values of H ₀. There is a significant difference in behaviour between lattices for which pc < 1/2 which have an infinite cluster and pc > 1/2 which do not. For standard lattices, no matter how small is H ₀ there are overturned clusters, but for the Bethe lattice there is a range of H ₀ for which the ground state is one of ferromagnetic order. By allowing the coordination number to become large in the latter system, the results of the mean-field approximation are reproduced. The above considerations do not apply to a Gaussian distribution of fields, and the absence of a first-order transition can then be understood. Since large clusters overturn for small fields, there are clear indications of metastable behaviour.