Theory of Spin Glass by the Method of the Distribution Function of an Effective Field

Abstract

The theory of the random Ising model formulated in terms of the distribution function of the effective field in the pair and in the cluster approximations is reviewed. An integral equation for the distribution function is derived. The integral equation has a solution for the paramagnetic state, a solution for the ferromagnetic state, a solution for the antiferromagnetic state, and solutions for the spin glass state. The phase diagram, and the ground state energy and entropy are calculated. The ground state entropy of the model is shown to be a small positive quantity contrary to the Sherrington­ Kirkpatrick infinitely long-ranged model. The phase diagram derived from the cluster approximation well explains the experimental phase diagrams, in particular, of fcc spin glass and the spin glass of EupSr1-pS. The distribution function for the spin glass in the pair approximation at T=O with a continu­ ous distribution is obtained analytically. They are composed of a-functions or of a-functions and a quadratic continuous function. In this paper the treatment of random spin systems, especially of the spin glass problem by the method of the distribution function of the effective field developed by our group is reviewed. A cluster (pair, triangle, square, tetrahedron, etc.) is taken in a given lattice (square, triangle, simple cubic, hexagonal, face-centered cubic, etc.). An effective field with a distribution is assumed to be applied to each vertex of the cluster. The partition function of the cluster is calculated exactly in terms of these effective fields. Self-consistent relation leads to an integral equation for the distribution function of the effective fields. The solution of the integral equation gives the physical quantities, the phase diagrams and the ground state properties. We consider random Ising models on a given crystal lattice. The hamiltonian &, the density matrix p, and the free energy Fare given by