Matrix mean-field theory for the paramagnetic susceptibility of disordered magnets

A matrix-based mean-field theory for the paramagnetic susceptibility of disordered Heisenberg and Ising magnets with conventional long-range order is outlined. The wave-vector-dependent static susceptibility is expressed in terms of the eigenvalues and eigenvectors of a symmetrized exchange matrix. The resulting expression is subsequently written as an integral involving a weighted eigenvalue distribution of the matrix. The conditions under which the distribution can be approximated by a single delta function are investigated, and a criterion is established for the validity of the delta-function approximation for the critical temperature. The criterion and the position of the delta function are written in terms of the moments of the distribution which are expressed as products of the elements of the exchange matrix. An alternative approach in which the susceptibility is expressed as the Laplace transform of a linear combination of functions obtained by integrating a set of coupled first-order differential equations is outlined. The theory is applied to a dilute ferromagnet with exponential exchange interactions. The limitations of the theory for strongly disordered systems are discussed, and it is pointed out that when all of the eigenvectors are localized, there is no phase transition in the matrix mean-field approximation.