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International Journal of Modern Physics B

ALLOY ANALOG APPROACH AND THE GUTZWILLER APPROXIMATION AS PARTICULAR CASES OF A MEAN-FIELD THEORY

Armando A. Aligia, M. Avignon

Published Oct 20, 1992• 3 citations
Publisher

Abstract

We develop a new mean field-theory for systems with on-site correlations, like the Hubbard and Anderson models. It consists in a generalization of the saddle-point approximation to the functional integral representation proposed by Kotliar and Ruckenstein to more than one saddle point. It contains also the alloy analog approximation proposed by Hubbard as a particular case. For a half-filled Hubbard model and a model density of states appropriate for three dimensions, we obtain a metal to insulator transition as U increases. The effective Hamiltonian for the insulating phase contains two split bands (rather than only one with an extremely heavy mass) which reproduce correctly the Green’s functions of the atomic limit.

DOI
10.1142/s021797929200147x

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We develop a new mean field-theory for systems with on-site correlations, like the Hubbard and Anderson models. It consists in a generalization of the saddle-point approximation to the functional integral representation proposed by Kotliar and Ruckenstein to more than one saddle point. It contains also the alloy analog approximation proposed by Hubbard as a particular case. For a half-filled Hubbard model and a model density of states appropriate for three dimensions, we obtain a metal to insulator transition as U increases. The effective Hamiltonian for the insulating phase contains two split bands (rather than only one with an extremely heavy mass) which reproduce correctly the Green’s functions of the atomic limit.
We propose a new functional integral representation of the Hubbarda and Anderson models of lattice fermions. The simplest saddle-point approximation leads, at zero temperature, to the results derived from the Gutzwiller variational wave function. This approach uncovers the limitations of the Gutzwiller approximation and clarifies its connection to the "auxiliary-boson" mean-field theory of the Anderson model. This formulation leads to a novel strong-coupling mean-field theory which allows for a unified treatment of antiferromagnetism and ferromagnetism, metal-to-insulator transition, and Kondo compensation effects.
The diagrammatic formulation of the single-site approximation to the Hubbard model immediately yields the exact solution to the Falikov-Kimball model in infinite dimensions. By an extension of the alloy analogy, we then derive a two-fluid model, dividing self-consistently the electron system among itinerating and relatively localized particles. Our theory coincides with the CPA only the at half-filling and u=∞. We start by studying the mean field approximation to our theory and identify a phase boundary in the particle density-interaction strength (n−u) plane at which the Fermi liquid model breaks down. Then the effect of fluctuations in the Fermi liquid regime is considered and, finally, we construct, at half-filling, a model which interpolates between the Fermi liquid and the Mott insulating phases.
The details of the Generalized-Gutzwiller-Approximation in degenerate Hubbard models are presented. It is found that the metal-insulator transition occurs at a finite correlation Uc when the average number of electrons per lattice site is an integer. A derivation is given for the general expression of Uc(x, N) which reproduces all previously known Gutzwiller solutions, including that of the boson Hubbard model. The Uc depends sensitively on both the band degeneracy N and the filling x. Effects of the lattice structure on the metal-insulator transition and the effective mass are discussed.
Degenerate Hubbard models are studied using the Generalized-Gutzwiller-Approximation. It is found that the metal-insulator transition occurs at a finite correlation Uc when the average number of electrons per lattice site is an integer. The critical Uc depends sensitively on both the band degeneracy N and the filling x. A derivation is presented for the general expression of Uc(x,N) which reproduces all previously known Gutzwiller solutions, including that of the Boson Hubbard model. Effects of the lattice structure on the metal-insulator transition and the effective mass are discussed.
The tight-binding diagonally disordered Anderson model with on-site Coulomb interaction is solved in the mean-field hopping approach. Phase diagrams of the metal-insulator transition are obtained from the free energy for the half-filled band and the binary-alloy distribution. For attractive interaction (U<0), the Coulomb term gives additional localizing effect and for some values of the microscopic parameters the temperature induced two transitions (insulator to metal to insulator). For U>0 two different types of behavior are distinguished. For small disorder (W\ensuremath{\le}U/2), the metallic phase can be induced by disorder, while for large W, the disorder always destroys the phase of extended states.
We study the Hubbard model through a functional-integral formulation proposed by Kotliar and Ruckenstein. A careful use of the U(1${)}^{\ensuremath{\bigotimes}3}$ gauge symmetry of the bosonized action allows us to compute the complete one-loop correction to the free energy in the case of half-filled band. We extract the effective mass of the fermion in the metallic phase and show that the noninteracting limit is not recovered for vanishing on-site repulsion. The bosonic Hamiltonian must be adjusted order by order in perturbation theory.