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 externai fi.elds e that are Grassmann variables. We liave found a simple way to extend the diagrammatic rules to the e f= O 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 --+ ao).

The Periodic Anderson Model (PAM) can be studied in the infinite U limit by employing the Hubbard X operators to project out the unwanted states. We have already studied this problem employing the cumulant expansion with the hybridization as perturbation, but the probability conservation of the local states (completeness) is not usually satisfied when partial expansions like the Chain Approximation (CHA) are employed. Here we treat the problem by a technique inspired in the mean field approximation of Coleman's slave-bosons method, and we obtain a description that avoids the unwanted phase transition that appears in the mean-field slave-boson method both when the chemical potential is greater than the localized level Ef at low temperatures (T) and for all parameters at intermediate T.Comment: Submited to Physical Review B 14 pages, 17 eps figures inserted in the tex

%e present a new approach to the theory of intermediate valence, applicable to a specific kind of such solids. It consists of: (i) diagonalization of all intra-atomic terms in the Hamiltonian, including hybridization; (ii) elimination by means of a projection of all states beyond a lowenergy manifold; (iii) conversion of the remaining states into equivalent fermion states; (iv) expression of the intraand interatomic states in terms of the new fermion operators; and (v) treatment of this new Hamiltonian in a mean-field approximation. This approach, although valid in restricted cases, avoids all the problems of the alternative treatments.

The approximate Green's functions of the localized electrons, obtained by the cumulant expansion of the periodic Anderson model in the limit of infinite Coulomb repulsion, do not satisfy completeness even for the simplest families of diagrams, like the chain approximation. The idea that employing -derivable approximations would solve this difficulty is shown to be false by proving that the chain approximation is -derivable and does not satisfy completeness. After finding a family of diagrams with Green's functions that satisfy completeness, we put forward a conjecture that shows how to select families of diagrams with this property.

In the present work we apply the atomic approach to the single-impurity Anderson model (SIAM). A general formulation of this approach, that can be applied both to the impurity and to the lattice Anderson Hamiltonian, was developed in a previous work (Foglio et al 2009 arxiv: 0903.0139v2 [cond-mat.str-el]). The method starts from the cumulant expansion of the periodic Anderson model, employing the hybridization as a perturbation. The atomic Anderson limit is analytically solved and its sixteen eigenenergies and eigenstates are obtained. This atomic Anderson solution, which we call the AAS, has all the fundamental excitations that generate the Kondo effect, and in the atomic approach is employed as a 'seed' to generate the approximate solutions for finite U. The width of the conduction band is reduced to zero in the AAS, and we choose its position such that the Friedel sum rule is satisfied, close to the chemical potential mu. We perform a complete study of the density of states of the SIAM over the whole relevant range of parameters: the empty dot, intermediate valence, Kondo and magnetic regimes. In the Kondo regime we obtain a density of states that characterizes well the structure of the Kondo peak. To show the usefulness of the method we have calculated the conductance of a quantum dot, side-coupled to a conduction band.