An exact calculation of the free energy in systems with separable interactions. II

Abstract

In this paper we extend the calculation of the free energy in systems with separable interactions given in a previous paper, to a more general class of systems characterized by a hamiltonian which contains a number of separable two-particle operators of the antiferromagnetic type in addition to separable ferromagnetic interactions and one-particle operators. By deriving an upper bound and a lower bound we establish an expression for the free energy which is of the molecular-field type. In the derivation of the lower bound we have used Laplace's method in order to evaluate a multidimensional integral of a function e NG . The proof that the second derivatives at the absolute minimum do not give a contribution in the thermodynamic limit is more complicated than in the ferromagnetic case and is given in detail.