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Gutzwiller Approximation in Degenerate Hubbard Models

Jian Ping Lu

Published Jan 1, 1996• 5 citations
OpenAlex

Abstract

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. PACS numbers: 71.10.+x,71.30.+h,74.70.W Typeset using REVTEX 1 The simple Hubbard model, where each lattice site is occupied by at most two electrons, has been extensively studied in recent years as a model of strongly correlated electron systems. 1,2 But in many systems, such as molecular solids C60 and AxC60, 3 the conduction band is degenerate which allows the occupation of more than two electrons per lattice site. For these systems, studying the degenerate Hubbard model on a lattice is the first step toward the understanding of electron-electronic correlations. Yet, there has been little progress since the pioneering works of Chao and Gutzwiller in early 1970s. 4 Recently, we succeeded in obtaining the Gutzwiller-like analytical solutions to degenerate Hubbard models using a scheme similar to the original Gutzwiller-Approximation. 5 A brief report of our results has been published. 6 The solution was used successfully to interpret the unusual metal-insulator transitions observed in fullerides AxC60. 6,7 In this paper we present details of our derivation and the approximation. In addition, effects of the crystal lattice structure on the metal-insulator transition and the effective mass are also investigated. We consider the N-fold degenerate Hubbard model defined on a lattice

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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. PACS numbers: 71.10.+x,71.30.+h,74.70.W Typeset using REVTEX 1 The simple Hubbard model, where each lattice site is occupied by at most two electrons, has been extensively studied in recent years as a model of strongly correlated electron systems. 1,2 But in many systems, such as molecular solids C60 and AxC60, 3 the conduction band is degenerate which allows the occupation of more than two electrons per lattice site. For these systems, studying the degenerate Hubbard model on a lattice is the first step toward the understanding of electron-electronic correlations. Yet, there has been little progress since the pioneering works of Chao and Gutzwiller in early 1970s. 4 Recently, we succeeded in obtaining the Gutzwiller-like analytical solutions to degenerate Hubbard models using a scheme similar to the original Gutzwiller-Approximation. 5 A brief report of our results has been published. 6 The solution was used successfully to interpret the unusual metal-insulator transitions observed in fullerides AxC60. 6,7 In this paper we present details of our derivation and the approximation. In addition, effects of the crystal lattice structure on the metal-insulator transition and the effective mass are also investigated. We consider the N-fold degenerate Hubbard model defined on a lattice
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 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.
An approximate solution scheme, similar to the Gutzwiller approximation, is presented for the Baeriswyl and the Baeriswyl-Gutzwiller variational wave functions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wave function a metal-insulator transition is found at half filling, where the metallic phase (U<Uc)(U<{U}_{c})(U<Uc​) corresponds to the Hartree-Fock solution, the insulating phase is one with finite double occupations arising from bound excitons. This transition can be viewed as the ``inverse'' of the Brinkman-Rice transition. Close to but away from half filling, the U>UcU>{U}_{c}U>Uc​ phase displays a finite Fermi step, as well as double occupations originating from bound excitons. As the filling is changed away from half-filling bound excitons are suppressed. For the Baeriswyl-Gutzwiller wave function at half filling a metal-insulator transition between the correlated metallic and excitonic insulating state is found. Away from half-filling bound excitons are suppressed quicker than for the Baeriswyl wave function.
We review the recently proposed extension of the Gutzwiller approximation, M. Schiro' and M. Fabrizio, Phys. Rev. Lett. 105, 076401 (2010), designed to describe the out-of-equilibrium time-evolution of a Gutzwiller-type variational wave function for correlated electrons. The method, which is strictly variational in the limit of infinite lattice-coordination, is quite general and flexible, and it is applicable to generic non-equilibrium conditions, even far beyond the linear response regime. As an application, we discuss the quench dynamics of a single-band Hubbard model at half-filling, where the method predicts a dynamical phase transition above a critical quench that resembles the sharp crossover observed by time-dependent dynamical mean field theory. We next show that one can actually define in some cases a multi-configurational wave function combination of a whole set of mutually orthogonal Gutzwiller wave functions. The Hamiltonian projected in that subspace can be exactly evaluated and is equivalent to a model of auxiliary spins coupled to non-interacting electrons, closely related to the slave-spin theories for correlated electron models. The Gutzwiller approximation turns out to be nothing but the mean-field approximation applied to that spin-fermion model, which displays, for any number of bands and integer fillings, a spontaneous Z2Z_2Z2​ symmetry breaking that can be identified as the Mott insulator-to-metal transition.
Mott metal-insulator transitions in an M-fold orbitally degenerate Hubbard model are studied by means of a generalization of the linearized dynamical mean-field theory. The method allows for an efficient and reliable determination of the critical interaction Uc for any integer filling n and different M at zero temperature. For half-filling a linear dependence of Uc on M is found. Inclusion of the (full) Hund's rule exchange J results in a strong reduction of Uc. The transition turns out to change qualitatively from continuous for J=0 to discontinuous for any finite J. This result is confirmed using the exact-diagonalization method for M=2.