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Physical review. B, Condensed matter

Variational results as saddle-point approximations: The Anderson impurity model

Kurt Schönhammer

Published Aug 1, 1990• 17 citations
Publisher

Abstract

The slave-boson functional-integral approach to the Hubbard model by Kotliar and Ruckenstein is applied to the spin-degenerate Anderson impurity model. At T=0 a simple saddle-point approximation to the functional integral is shown to be equivalent to a special case of a variational ansatz presented by the author in the 1970's. The question of how to choose the freedom in the form of the hopping Hamiltonian in the enlarged Hilbert space including the slave bosons is discussed.

DOI
10.1103/physrevb.42.2591

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The slave-boson functional-integral approach to the Hubbard model by Kotliar and Ruckenstein is applied to the spin-degenerate Anderson impurity model. At T=0 a simple saddle-point approximation to the functional integral is shown to be equivalent to a special case of a variational ansatz presented by the author in the 1970's. The question of how to choose the freedom in the form of the hopping Hamiltonian in the enlarged Hilbert space including the slave bosons is discussed.
We consider slave boson formulations of the Hubbard model, the t–J model and the Anderson model, in which spin and charge fermion degrees of freedom are represented by Bose fields. The approach has been applied to the Hubbard model generalizing the formulation of Kotliar and Ruckenstein. There the Gutzwiller solution is recovered as a saddle point. The method allows to find other types of mean field solutions in a systematic way. Dynamical correlation functions in mean field approximation as well as fluctuation contributions are also calculated.
We revisit the Kotliar-Ruckenstein (KR) slave boson saddle point evaluation for a two-site correlated electron model. As the model can be solved analytically, it is possible to compare the KR saddle point results with the exact many-particle levels. The considered two-site cluster mimics an infinite-UUU single-impurity Anderson model with a nearest-neighbor Coulomb interaction: one site is strongly correlated with an infinite local Coulomb repulsion, which hybridizes with the second site, on which the local Coulomb repulsion vanishes. Making use of the flexibility of the representation, we introduce appropriate weight factors in the KR saddle point scheme. Ground-state and all excitation levels agree with the exact diagonalization results. Thermodynamics and correlation functions may be recovered in a suitably renormalized saddle point evaluation.
Based on the Kotliar--Ruckenstein slave-boson scheme, we develop a configuration-interaction (CI) approach that is suitable to evaluate energy corrections for symmetry-broken saddle-point solutions. The theory is applied to spin-polaron states in the Hubbard model and compared with analogous results obtained within the Hartree--Fock approximation. In addition, we show that within the infinite D\mathcal{D}D prescription of the Gutzwiller method a CI approach does not correct the variational result since in the thermodynamic limit matrix elements between different inhomogeneous states vanish due to an ``orthogonality catastrophe.''
A slave-boson functional-integral method has been developed for the Hubbard model with arbitrary, orbital degeneracy DDD. Its saddle-point mean-field theory is equivalent to the Gutzwiller approximation, as in the case of single-band Hubbard model. Our theory is applied to the doubly degenerate (D=2D = 2D=2) model, and numerical calculations have been performed for this model in the paramagnetic states. The effect of the exchange interaction on the metal-insulator (MI) transition is discussed. The critical interaction for the MI transition is analytically calculated as functions of orbital degeneracy and electron occupancy.
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 Kotliar-Ruckenstein slave-boson scheme is used to allow for an unrestricted variation of the bosonic and fermionic fields on the saddle-point level. Various inhomogeneous solutions, such as spin polarons and domain walls, are discussed within the two-dimensional Hubbard model and compared with results of unrestricted Hartree-Fock (HF) calculations. We find that the present approach drastically reduces the polarization of these states and leads to increased delocalized wave functions as compared to the HF model. The interaction between two spin polarons turns out to be attractive over a wide range of the on-site repulsion UUU. In addition we obtain the crossover from vertical to diagonal domain walls at a higher value of UUU than predicted by HF.