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Condensed Matter Physics

Asymmetric Hubbard model within generating functional approach in dynamical mean field theory

Ihor V. Stasyuk, O. B. Hera

Published Jan 1, 2006• 3 citations
Publisher

Abstract

In the paper a new analytic approach to the solution of the effective single-site problem in the dynamical mean field theory is developed.The approach is based on the method of the Kadanoff-Baym generating functional in the form developed by Izyumov et al.It makes it possible to obtain a closed equation in functional derivatives for the irreducible part of the single-site particle Green's function; the solution is constructed iteratively.As an application of the proposed approach the asymmetric Hubbard model (AHM) is considered.The inverse irreducible part Ξ -1 σ of the single-site Green's function is constructed in the linear approximation with respect to the coherent potential Jσ.Basing on the obtained result, the Green's function of itinerant particles in the Falicov-Kimball limit of AHM is considered, and the decoupling schemes in the equations of motion approach (GH3 approximation, decoupling by Jeschke and Kotliar) are analysed.

DOI
10.5488/cmp.9.3.587

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In the paper a new analytic approach to the solution of the effective single-site problem in the dynamical mean field theory is developed.The approach is based on the method of the Kadanoff-Baym generating functional in the form developed by Izyumov et al.It makes it possible to obtain a closed equation in functional derivatives for the irreducible part of the single-site particle Green's function; the solution is constructed iteratively.As an application of the proposed approach the asymmetric Hubbard model (AHM) is considered.The inverse irreducible part Ξ -1 σ of the single-site Green's function is constructed in the linear approximation with respect to the coherent potential Jσ.Basing on the obtained result, the Green's function of itinerant particles in the Falicov-Kimball limit of AHM is considered, and the decoupling schemes in the equations of motion approach (GH3 approximation, decoupling by Jeschke and Kotliar) are analysed.
The asymmetric Hubbard model is used in investigating the lattice gas of the moving particles of two types.The model is considered within the dynamical mean-field method.The effective single-site problem is formulated in terms of the auxiliary Fermi-field.To solve the problem an approximate analytical method based on the irreducible Green's function technique is used.This approach is tested on the Falicov-Kimball limit (when the mobility of ions of either type is infinitesimally small) of the infinite-U case of the model considered.The dependence of chemical potentials on concentration is calculated using the one-particle Green's functions, and different approximations are compared with the exact results obtained thermodynamically.The densities of states of localized particles are obtained for different temperatures and particle concentrations.The phase transitions are investigated for the case of the Falicov-Kimball limit in different thermodynamic regimes.
An approximate analytical scheme of the dynamical mean-field theory (DMFT), that is used for electron systems with Hubbard correlations and is exact in the limit of the infinite dimensionality of a space, is developed.The effective single-site problem arising in the framework of this method is formulated in terms of the auxiliary Fermi-field.The irreducible Green's function technique with the projecting on the Hubbard basis of Fermi-operators is used for its solution.A system of DMFT equations is obtained in the approximation which is a generalization of the Hubbard-III approximation and combines it with a self-consistent renormalization of the local electron levels.It is shown that the proposed approach includes as simple specific cases a number of known approximations (Hubbard-III, AA, MAA, . . . ) based on the assumption of the single-site structure of the electron selfenergy.
We discuss the infinite dimension limit of the Hubbard model by means of a perturbative expansion of the one-particle Green's function around the atomic limit. The diagrammatic structure is simplified in this limit, allowing a formal resummation that reproduces a previously proposed mapping to a single-site mean-field problem. The method provides a natural way of addressing this effective problem by means of a perturbative expansion in the local mean field. This gives the correct exact result for the Falicov-Kimball model, which is used as a starting point to study the Hubbard case. \textcopyright{} 1996 The American Physical Society.
The asymptotic expression for the one-particle Green's function in the strong coupling limit of the double degenerate Hubbard model containing multiplet terms is obtained for arbitrary band filling within the irreducible Green's function approach. An explicit analytic expression is obtained for the mean-field Greens's function, conserving the sum rules for the first few spectral moments. Stability of homogeneous phases is analised in terms of a single flip spin and orbital excitations. It is shown that degeneracy has an additional enhancing effect on correlation driven interband transfer of spectral weight. Discovery of superconductivity in fullerides and of materials with colossal magnetoresistance during recent years have increased the interest for electron systems with physical properties being essentially dependent on orbital degeneracy and strong correlation effects. In many cases the orbitally degenerate Hubbard model (DHM) incorporates the key features of such materials [1], although there are even fewer exact results established for DHM than for the nondegenerate Hubbard model (NHM). In view of complexity of correlated systems, corrections to the atomic limit of one-particle excitation spectrum and spectral weights of correlated bands found by Harris and Lange for NHM [2] by means of perturbation theory based on canonical transformations provide an important exact result which serves as a test for different approximations. A similar derivation for DHM has not been carried out yet. In the present paper the Green's function for the strong coupling limit of the DHM is obtained by means of irreducible Green's function approach [3] - [6] which is closely related to canonical perturbation theory because of its spectral moment conserving construction. As was explicitly proved in [7] for the NHM, the first couples of spectral moments of correlated bands obtained within mean-field decoupling are identical to those of canonical perturbation theory to first order in / tU ( t is the n.n. hopping amplitude and U the Coulomb repulsion).
An approximate analytical scheme of the dynamical mean field theory (DMFT) is developed for the description of the electron (ion) lattice systems with Hubbard correlations within the asymmetric Hubbard model where the chemical potentials and electron transfer parameters depend on an electron spin (a sort of ions). Considering a complexity of the problem we test the approximation in the limiting case of the infinite-UUU spinless Falicov--Kimball model. Despite the fact that the Falicov--Kimball model can be solved exactly within DMFT, the densities of states of localized particles have not been completely investigated off half filling. We use the approximation to obtain the spectra of localized particles for various particle concentrations (chemical potentials) and temperatures. The effect of a phase separation phenomenon on the spectral function is considered.
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.