Variational derivative equations for the partition functions of the Hubbard and Anderson models

Abstract

The method of a generating functional of Green’s functions was further developed within the framework of the Hubbard model and single-impurity Anderson model. In contrast to the earlier proposed works, the equations in the variational derivatives for the partition functions are presented here in the closed form, i.e. the role of variables is played by the physical matrix parameters of the systems rather than by the external local fluctuating fields. The solutions to these equations are the generating functionals of different Green’s functions. It is shown that the simplest iterative solutions in terms of the parameters U/W and W/U in the case of the Hubbard model or U/Δ and Δ/U for the Anderson model, where U is the Coulomb repulsion on a site, W is the width of a free electron zone, and Δ is the width of an impurity level, lead to the well-known results of the weak and strong coupling limits.