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Physical review. B, Solid state

Moment-Generated Solution to the Hubbard Narrow-Energy-Band Model

D. M. Esterling, H. C. Dubin

Published Dec 1, 1972• 10 citations
Publisher

Abstract

A solution to the Hubbard model is presented which invokes the single assumption that the low-energy peak in the spectral weight function can be well approximated by a function which duplicates its zeroth, first, and second moments to a given order in the bandwidth. The solution takes advantage of the rigorous moment relations of Harris and Lange. A simple, exact, and unambiguous technique to evaluate the correlation functions, required for the moment relations, in the atomic limit is developed. These functions are used to calculate the moments which in turn generate a spectral weight function with no restriction on the electron density. The ground-state energy for the paramagnetic and ferromagnetic configurations are calculated and compared. The model is found to be ferromagnetic for moderate densities.

DOI
10.1103/physrevb.6.4276

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A solution to the Hubbard model is presented which invokes the single assumption that the low-energy peak in the spectral weight function can be well approximated by a function which duplicates its zeroth, first, and second moments to a given order in the bandwidth. The solution takes advantage of the rigorous moment relations of Harris and Lange. A simple, exact, and unambiguous technique to evaluate the correlation functions, required for the moment relations, in the atomic limit is developed. These functions are used to calculate the moments which in turn generate a spectral weight function with no restriction on the electron density. The ground-state energy for the paramagnetic and ferromagnetic configurations are calculated and compared. The model is found to be ferromagnetic for moderate densities.
We must understand the physics of strongly interacting electrons if we are to have a fundamental understanding of magnetism. This paper treats the simplest model with this property — the Hubbard model. The moments of the spectral weight function (SWF), derived by Harris and Lange, are calculated in the atomic limit. These moments are used to generate an approximate SWF (including lifetime effects). The paramagnetic and ferromagnetic ground state energies are calculated and compared. There is no restriction on the electron density.
A companion paper derived exact relations for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension and any lattice structure. These relations involve moments of the spectral weight function but differ from earlier work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle equal time correlation function in the infinite U limit. Here the relations offer rich insights into the model behavior and offer exacting tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. Specific tests are applied to mean field theory solutions to the Hubbard model as well as some one dimensional results. In each case, substantial problems are exposed. The relations are also used to determine a best(moment-preserving) solution subject only to the one assumption of a dominant single peak in the lower Hubbard band spectral weight function. As shown in the companion paper, the second central moment of the lower Hubbard band spectral weight function is not zero, as might be expected, as the electron density approaches one electron per site (paramagnetic case). That moment is large, offering a significant test for any approximate solution.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension,any lattice structure and general hopping matrix. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.
Hubbard's model for studying correlation effects in systems with narrow energy bands is analyzed by means of a technique which allows the calculation of moments of the individual peaks in the spectral weight function for single-particle excitations. The analysis of the zeroth moments of the peaks shows that the total weight in the bands depends on the strength of the kinetic-energy term in the Hamiltonian even though the bands may be narrow and widely separated. This conclusion is illustrated and verified by an exact calculation for the case when there are only two lattice sites. Analysis of first and higher moments yields results for nonmagnetic or paramagnetic phases which are in qualitative agreement with Hubbard's improved solution. However, we find that (a) there occurs a spin-dependent shift in the band energies which has not been obtained by other treatments of the model and which energetically favors ferromagnetism, and (b) single-particle excitations are more heavily damped in antiferromagnetic than in isomorphic paramagnetic phases.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension and any lattice structure. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.
We discuss the approximate calculation of a Green's function and its spectral density from its low-order moments by the use of ratio-extrapolation techniques to estimate and factor off the dominant shape of the band edges. The techniques are illustrated, following Brinkman and Rice, on the problem of a single hole propagating in ferromagnetic, antiferromagnetic, and random electron-spin configurations in the atomic limit of the Hubbard model; the methods are, however, of wider applicability.