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Physical review. B, Condensed matter

From the atomic limit to a metal-insulator transition in the Hubbard model

L. Craco, M. A. Gusmão

Published Dec 15, 1995• 27 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.52.17135

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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.
We analyze the unanalytical structure of metal-insulator transition (MIT) in infinite dimensions. A multiple-valued structure in Green's function and other thermodynamical quantities with respect to the interaction strength U are observed at low temperatures by introducing a transformation into the dynamical mean-field equation of Hubbard model. A complete description of stable, metastable, and unstable phases is established in the regime Uc1(T)<U<Uc2(T).{U}_{c1}(T)<U<{U}_{c2}(T).Uc1​(T)<U<Uc2​(T). The Maxwell construction is performed to evaluate the MIT line U∗(T).{U}^{*}(T).U∗(T). We show how the first-order MIT at U∗(T){U}^{*}(T)U∗(T) for T>0T>0T>0 evolves into the second-order one at Uc2(0){U}_{c2}(0)Uc2​(0) for T=0.T=0.T=0. The phase diagram near MIT is presented.
We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.
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 single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
A detailed study of electronic phase transitions in the ionic Hubbard model at half filling is presented. Within the dynamical mean field approximation a series of transitions from the band insulator via a metallic state to a Mott--Hubbard insulating phase is found at intermediate values of the one-body potential $\ensuremath{\Delta}$ with increasing the Coulomb interaction UUU. We obtain a critical region in which the metallic phase disappears and a coexistence phase between the band and the Mott insulating state sets in. Our results are consistent with those obtained at low dimensions, thus they provide a concrete description for the charge degrees of freedom of the ionic Hubbard model.