Cumulant expansion of the periodic Anderson model: General derivation.

Abstract

We extend to the periodic Anderson model (PAM) the diagrammatic expansion in cumulants that was employed by Hubbard to study his model of a narrow band of strongly correlated electrons. The PAM is a lattice of localized and strongly correlated electrons with spin one-half and without orbital degeneracy, hybridized with a wide band of uncorrelated conduction electrons. We have extended the model by considering localized electronic states with an arbitrary scheme of energy levels: this extension would be useful to study intermediate valence compounds of Eu or Tm with the present formalism. We give the rules for the diagrammatic calculation of the grand canonical potential and of the Green's functions for the general model: only connected diagrams appear in those calculations and the lattice sums are unrestricted. To generate the cumulant averages it was necessary to employ external fields \ensuremath{\xi} that are Grassman variables. We have found a simple way to extend the diagrammatic rules to the \ensuremath{\xi}\ensuremath{\ne}0 case. The absence of excluded site restrictions, that leads to complicated excluded volume problems in other treatments, and the existence of linked cluster expansions, are features of the cumulant expansion. As an application of the present method, we have calculated the occupation numbers of localized and conduction electrons for the PAM in the limit of infinite Coulomb repulsion (U\ensuremath{\rightarrow}\ensuremath{\infty}).