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Physical review. B, Condensed matter

Tight-binding treatment of the Hubbard model in infinite dimensions

L. Craco, M. A. Gusmão

Published Jul 15, 1996• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.54.1629

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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.
We develop a perturbation expansion in the intersite hopping around the atomic limit of the Hubbard model. It is valid for arbitrary finite temperatures and interaction strengths. Diagrammatic rules that determine the grand-canonical potential and the Green's functions are derived. They reduce the calculation of any finite-order contribution to simple algebra. This opens the way for series extrapolations from computer-aided high-finite-order evaluations. Discrepancies in earlier expansions around the atomic limit are clarified. The present expansion scheme involves only connected diagrams with unrestricted lattice sums. This allows one to perform a vertex renormalization as for the linked-cluster expansion of the Ising model. The renormalized perturbation expansion can be used to construct self-consistent approximations which are automatically exact in the atomic limit. In the limit of high lattice dimensions, only fully two-particle reducible embeddings of diagrams on the lattice contribute. The single-particle properties of the infinite-dimensional Hubbard model reduce to those of independent tight-binding fermions hopping between dressed sites.
Starting from the exact solution of the Hubbard model in the atomic limit, and treating the hopping term as a perturbation by means of a diagram technique, we discuss the electronic structure of the model in some simple approximations. We show that a metal-insulator transition is obtained above a critical value of the on-site Coulomb interaction in a self-consistent evaluation of the one-particle Green's function to one-loop order. A relationship with results in infinite dimensions is briefly discussed.
We calculate the one-particle density of states for the Mott-Hubbard insulating phase of the Hubbard model on a Bethe lattice in the limit of infinite coordination number. We employ the Kato-Takahashi perturbation theory around the strong-coupling limit to derive the Green function. We show that the Green function for the lower Hubbard band can be expressed in terms of polynomials in the bare hole-hopping operator. We check our technique against the exact solution of the Falicov-Kimball model and give explicit results up to and including second order in the inverse Hubbard interaction. Our results provide a stringent test for analytical and numerical investigations of the Mott-Hubbard insulator and the Mott-Hubbard transition within the dynamical mean-field theory. We find that the Hubbard-III approximation is not satisfactory beyond lowest order, but the local-moment approach provides a very good description of the Mott-Hubbard insulator at strong coupling.
In this work we solve a general Hubbard Hamiltonian for two interacting particles in a periodic and nonperiodic infinite one-dimensional lattice, using a real-space mapping method, the renormalized perturbation expansion (RPE), and the Green-function technique. This Hamiltonian considers a general bond-charge interaction, the on-site interaction, and the general intersite interaction. The real-space method is based on mapping the correlated many-body problem onto an equivalent site- and bond-impurity tight-binding problem in a higher-dimensional space. Analyzing the periodic and the quasiperiodic lattices in this new space, we obtained the analytical solution for the binding condition at the ground state. Our general results for the periodic chain reproduce completely the limit cases of the numerical solution obtained previously and those obtained in reciprocal space
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.
We present an exact mapping of the Hubbard model in infinite dimensions onto a single-impurity Anderson (or Wolff) model supplemented by a self-consistency condition. This provides a mean-field picture of strongly corrrelated systems, which becomes exact as d\ensuremath{\rightarrow}\ensuremath{\infty}. We point out a special integrable case of the mean-field equations, and study the general case using a perturbative renormalization group around the atomic limit. Three distinct Fermi-liquid regimes arise, corresponding to the Kondo, mixed-valence, and empty-orbitals regimes of the single-impurity problem. The Kondo resonance and the satellite peaks of the single-impurity model correspond to the quasiparticle and Hubbard-bands features of the Hubbard model, respectively.