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arXiv:cond-mat/9708024 · Journal of Physics A Mathematical and General

Cumulant expansion of the periodic Anderson model in infinite dimensions

Mario E. Foglio, Marcos Sergio Figueira

Published Nov 21, 1997• 13 citations
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Abstract

The diagrammatic cumulant expansion for the periodic Anderson model with infinite Coulomb repulsion is considered here for an hypercubic lattice of infinite dimension . The nearest neighbour hopping of the uncorrelated electrons is described exactly by a conduction band, while two different models of hybridization are treated as a perturbation. The same type of simplifications obtained by Metzner for the cumulant expansion of the Hubbard model in the limit of , are also shown to be valid for the periodic Anderson model. The derivation of these properties had to be modified because of the exact treatment of the conduction band.

DOI
10.1088/0305-4470/30/22/024

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The diagrammatic cumulant expansion for the periodic Anderson model with infinite Coulomb repulsion is considered here for an hypercubic lattice of infinite dimension . The nearest neighbour hopping of the uncorrelated electrons is described exactly by a conduction band, while two different models of hybridization are treated as a perturbation. The same type of simplifications obtained by Metzner for the cumulant expansion of the Hubbard model in the limit of , are also shown to be valid for the periodic Anderson model. The derivation of these properties had to be modified because of the exact treatment of the conduction band.
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.
We consider the cumulant expansion of the PAM employing the hybridization as perturbation (Phys. Rev. B 50, 17933 (1994)), and we obtain formally exact one-electron Green's functions (GF). These GF contain effective cumulants that are as difficult to calculate as the original GF, and the Atomic Approximation consists in substituting the effective cumulants by the ones that correspond to the atomic case, namely by taking a conduction band of zeroth width and local hybridization. This approximation has already been used for the case of infinite electronic repulsion U (Phys. Rev. B 62, 7882 (2000)), and here we extend the treatment to the case of finite U. The method can also be applied to the single impurity Anderson model (SIAM), and we give explicit expressions of the approximate GF both for the PAM and the SIAM.
In a large variety of systems, like the high‐temperature superconductors, the dominant parameter of the system is the on‐site Coulomb repulsion that becomes comparable with the kinetic energy in the vicinity of the metal‐insulator transition. This implies that perturbation expansions must include this interaction in the zeroth order Hamiltonian and treat the hopping term as a perturbation. Thus, one gets a situation radically different from the usual one in which the particles are initially free and needs to develop a nonstandard perturbation theory based on a cumulant expansion. In this paper, we revisit the general formalism of the cumulant expansion approach for the single band and two‐band Hubbard model, being interested in introducing a systematic way to perform approximations which take into account cooperative phenomena. As an example we analyze the effect of the charge fluctuations on the single‐particle spectral properties.
A new diagram technique, previously proposed for Hubbard Model, was used for the Periodic Anderson Model (PAM).The Hamiltonian of localized electrons H 0 f and of free conduction electrons H 0 c are considered as a main part of the Hamiltonian of PAM and the hybridization term is treated as a perturbation.H 0 f contains on-site Coulomb interaction and is diagonalized in Hubbard operators.This diagram technique is used for Matsubara Green functions and contains apart from the Wick product of free Green functions new contributions equal to on-site cumulants depending of the Coulomb interaction.If such cumulants are ignored we obtaine the so called Hubbard I approximation for PAM with 3 branches of renormalized energy spectrum of quasiparticles.They are investigated in detail and the chemical potential of the system is determined
The periodic Anderson model can be studied in the limit $U=\ensuremath{\infty}$ by employing the Hubbard X operators to project out the unwanted states. We had already studied this problem by 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. To rectify this situation, we modify the CHA by employing a procedure that was used in the mean-field approximation of Coleman's slave-boson method. Our technique reproduces the features of that method in its region of validity, but avoids the unwanted phase transition that appears in the same method both when $\ensuremath{\mu}\ensuremath{\gg}{E}_{f}$ at low T and for all values of the parameters at intermediate temperatures. Our method also has a dynamic character that is absent from the mean-field slave-boson method.
Exact solutions of the Hubbard model and the periodic Anderson model in the limit of infinite interaction strength are presented. Both models are studied on a D-dimensional decorated hypercubic lattice with periodic boundary conditions for any dimension D\ensuremath{\ge}2 and arbitrary size. The lattice is very similar to the perovskite lattice. In addition to the ground-state energy, a corresponding eigenstate is constructed. This ground state contains at least two particles per unit cell. For the Anderson model, the exact solution is restricted to a surface in the (Ef{\mathit{E}}_{\mathit{f}}Ef​,V) parameter space; however, the resulting relation V(Ef{\mathit{E}}_{\mathit{f}}Ef​) does not lead to unphysical parameters.