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arXiv:cond-mat/0109037 · Physical review. B, Condensed matter

X-boson cumulant approach to the periodic Anderson model

R. Franco, Marcos Sergio Figueira, Mario E. Foglio

Published Jul 26, 2002• 29 citations
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Abstract

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.

DOI
10.1103/physrevb.66.045112

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The periodic Anderson model can be studied in the limit U=\ensuremath∞U=\ensuremath{\infty}U=\ensuremath∞ 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 \ensuremathμ\ensuremath≫Ef\ensuremath{\mu}\ensuremath{\gg}{E}_{f}\ensuremathμ\ensuremath≫Ef​ 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.
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 this paper a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians is applied to the periodic Anderson model with the aim to describe heavy-fermion behavior. In this method high-energetic excitation operators instead of high energetic states are eliminated. We arrive at an effective Hamiltonian for a quasifree system which consists of two noninteracting heavy-quasiparticle bands. The resulting renormalization equations for the parameters of the Hamiltonian are valid for large as well as small degeneracy ${\ensuremath{\nu}}_{f}$ of the angular momentum. An expansion in $1∕{\ensuremath{\nu}}_{f}$ is avoided. Within an additional approximation which adapts the idea of a fixed renormalized fff level ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$, we obtain coupled equations for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$ and the averaged fff occupation ⟨nf⟩⟨{n}_{f}⟩⟨nf​⟩. These equations resemble to a certain extent those of the usual slave boson mean-field (SB) treatment. In particular, for large ${\ensuremath{\nu}}_{f}$ the results for the PRM and the SB approach agree perfectly whereas considerable differences are found for small ${\ensuremath{\nu}}_{f}$.
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.
We investigate Gutzwiller-correlated wave functions as variational ground states for the periodic Anderson model. We extend a recently developed technique for the Hubbard model to the case of two hybridized bands with degeneracy N. We calculate expectation values in the following limits: dilute system (single-impurity case), large degeneracy N, and large dimension d. The mean-field results using the slave-boson approach of Coleman and of Kotliar and Ruckenstein are reproduced in the large-N and large-d limit, respectively. These techniques, therefore, obey variational principles at T=0 in the respective limits and their applicability for T>0 is naturally limited to the Fermi-liquid regime. Evaluations of Gutzwiller-correlated wave functions using the ``Gutzwiller approximation'' are seen to become exact in the limit of large dimensions. A systematic expansion in 1/d can be performed for N=2. In general, corrections are small, of order 1/(dN).
The authors consider a systematic projection method for terminating the Green function equations of motion for Hubbard X-operators. This gives a set of mean fields that depend on the operators chosen. When applied to lowest order in the N-fold-degenerate U to infinity Anderson model an expression for the one-electron Green function is obtained that is exact for N=1 and in the limit N to infinity . This approach avoids the necessity of introducing auxiliary fields, such as constraint fields or slave bosons, to obtain asymptotically exact mean fields in the large-N limit and has general applicability in similar strongly correlated problems.
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.