Legendre transformation in Hubbard and Anderson models

Abstract

The previously derived variational-derivative equations for the Hubbard model and for the single-impurity Anderson model after the Legendre transformation are represented in the form of a set of two nonlinear integral equations. The one-particle propagators of the number of particles and of the momentum that are defined by these equations exhibit a regular behavior in three limiting cases. First, for both models in the limit of the zero width of the conduction band (W/U = 0) there is obtained a result known as the atomic limit. Second, it has been shown that in the limit of U/W < 1 the Pauli principle in the form of an additional coupling equation excludes from the perturbation-theory series some class of diagrams that are present in the standard expansion. Finally, for the case of U = ∞ and N e = N at - 1 in the framework of the Hubbard model there has been obtained an equation that agrees with the exact Nagaoka statement on the saturated ferromagnetism. A calculation has been performed of the density of impurity electron states in the symmetrical Anderson model in the paramagnetic phase for various values of the parameters of the Coulomb interaction U/πΓ and temperature T/Γ, where Γ is the width of the localized impurity level. The calculation results are in good agreement with the results obtained by other methods. © 2011 Pleiades Publishing, Ltd.