One self-consistent closure for chains of equations for Green's functions. An anisotropic heisenberg model

This article present a self-consistent approach to computation of the correlation function in the method of Green's functions. The basis for the approach is representation of the desired Green's function in the form of a chain of fractions that is subsequently closed. The closure is based on the use of concrete relations imposed on the higher-order correlation functions. General expressions for the correlation functions and conditions for self-consistency of the computations are presented. The method has been tested by computing the magnetization and critical temperature in a Heisenberg model with arbitrary anisotropy parameter. We obtain general expressions for these quantities. The critical temperature obtained is less than the corresponding value given by the molecular-field approximation. The latter approximation also leads to an overestimate of the magnetization values. It is shown that no critical transition is possible for any value of the anisotropy parameter. The corresponding inequalities are obtained. The method is compared with a method for self-consistent computation of correlation functions that was proposed earlier by the author.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 1, pp. 46–52, January, 1996.