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arXiv:1807.09223

The Hubbard Model: A Computer Lab in the Strong Coupling Region

D. M. Esterling

Published Jul 24, 2018• 0 citations
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Abstract

A challenge - and opportunity - is offered to the Hubbard Model community of solutions extant for the strong coupling region. A rigorous and quantitatively demanding test - a Computer Lab - is presented based on certain exact results for the strong coupling limit derived in a companion paper. The test offers rich insights into the essential physics of the model, specifically the quasiparticle energy and width of the spectral weight function in the lower Hubbard band. The width of the spectral weight function, related to electron scattering, is large and has a strong momentum dependence, even for infinite coupling. The test is applied to mean field solutions as well as some one dimensional results. In each case, substantial problems are exposed. The method is also used to determine a "best" (i.e. moment preserving) solution, subject to only one assumption of a single dominant peak for the lower Hubbard band spectral weight function.

DOI
10.48550/arxiv.1807.09223

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A challenge - and opportunity - is offered to the Hubbard Model community of solutions extant for the strong coupling region. A rigorous and quantitatively demanding test - a Computer Lab - is presented based on certain exact results for the strong coupling limit derived in a companion paper. The test offers rich insights into the essential physics of the model, specifically the quasiparticle energy and width of the spectral weight function in the lower Hubbard band. The width of the spectral weight function, related to electron scattering, is large and has a strong momentum dependence, even for infinite coupling. The test is applied to mean field solutions as well as some one dimensional results. In each case, substantial problems are exposed. The method is also used to determine a "best" (i.e. moment preserving) solution, subject to only one assumption of a single dominant peak for the lower Hubbard band spectral weight function.
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.
Exact spectral moment relations are derived and presented for the low temperature, strong coupling limit of the Hubbard model, for any dimension, lattice structure and electron density. These results generate an exact, rigorous and quantitative test for proposed solutions to the Hubbard model that claim to be valid in the strong coupling, low temperature region. The test complements analyses for weak coupling and high temperatures, offering insights into the essential physics of the model, specifically the quasiparticle energy (centroid) and lifetime (width) of the spectral weight function in the infinite coupling, finite hopping limit. Numerical results are given for nearest neighbour hopping in 1, 2 and 3D.
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.
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.
The spectral function of a single hole in the two-dimensional Hubbard model is studied numerically through exact diagonalization for a nine-site cluster over a wide range of interaction strengths U. In the weak-interaction limit, the spectral function consists of sharp quasiparticle peaks corresponding to the allowed single-particle excitations of the noninteracting system. As the interaction strength increases, the spectral weight begins to spread over a large range of energies, and satellite structures appear at higher excitation energy. The band-narrowing effect and the quasiparticle weight are investigated. The results are discussed in connection with other analytical and numerical studies on the spectral function.