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Journal of Physics Condensed Matter

The cumulant expansion of the periodic Anderson model; completeness and the -derivable approximation

Marcos Sergio Figueira, Mario E. Foglio

Published Jul 1, 1996• 10 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/8/27/012

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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.
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 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.
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
It is sometimes convenient to emphasize the local aspects of a part of a crystalline system, and use the corresponding localized states to build a basis of the states of the whole system. In many cases it is only a subspace of these local states that is relevant, and the Hubbard operators provide a fairly simple way to write the corresponding projected Hamiltonian. Two examples of this type of treatment are presented int this work. The first is a Co2+ impurity in a MgO crystal interacting through a Jahn-Teller term with the crystal phonons, and it is shown how this interaction affects the electronic Raman scattering. The second is the Anderson lattice when the local electrons have an infinite Coulomb repulsion, and a diagrammatic expansion with cumulants is discussed. We propose a method to obtain approximate Green's functions for the Anderson lattice that employs the exact solutions of an atomic problem, and the corresponding spectral density of the local electrons is calculated.
This work is mainly devoted to the description of processes that involve the interaction between an atom and a surface, in which a strong Coulomb repulsion on the atomic site (U)(U)(U) limits the charge exchange to one electron (infinite-UUU limit). In this limit, the Anderson Hamiltonian for a many-fold (N)(N)(N) of states localized on the atomic site can be represented in terms of auxiliary bosons and physical operators in the mixed boson-electron space can be defined. In this work the Hamiltonian is solved by defining appropriate Green's functions for physical operators. Then we solved the equations of motion of these Green's functions, up to a second order in the atom-surface coupling, either for the stationary case or for a real time-dependent problem. We show that our approach reproduces the known exact results in the nondegenerate (N=1)(N=1)(N=1) case, and for $N\ensuremath{\succ}1$ gives excellent agreement with exact calculations and approximations valid for large NNN (the 1∕N1∕N1∕N expansion). Finally, the accurate description of dynamical processes is shown by the comparison with the exact results available for a small four-level system. In this case we also compare with results obtained by using the noncrossing approximation and with the usual spinless model calculation.