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The Hubbard Model: Exact Results in the Strong Correlation Limit.

D. M. Esterling

Published Jul 29, 2018• 0 citations
OpenAlex

Abstract

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.

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The Hubbard Model: Exact Results in the Strong Coupling Limit

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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.

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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.

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

D. M. Esterling

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.

arXiv:1807.09223•Jul 24, 2018•0 citations
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.
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.
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 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.
Using strong-coupling perturbation theory for the Hubbard model, explicit expressions are obtained for the integrated weights, energy positions, and widths of the upper and lower Hubbard bands separately, both for the optical and photoemission spectrum. In one dimension all expressions can be explicitly evaluated using the large-U Bethe-ansatz wave function. The k\ensuremath{\rightarrow}-dependent moments for the one-particle spectrum are compared with the closely related two-pole ansatz. The strong momentum dependence of the spectra demonstrates the importance of the k\ensuremath{\rightarrow}-dependent correlation functions in the spectral moments, which are often neglected. In order to estimate corrections due to the neglect of higher-order terms (in t/U), all results are compared with numerical-diagonalization data for intermediate-U values. The sum-rule results show a rapid spectral-weight transfer from the high- to the low-energy regime upon doping, similar to what is observed experimentally in one-particle and optical spectra of Cu (high-Tc{\mathit{T}}_{\mathit{c}}Tc​) and Ni compounds. Such fast weight redistributions as a function of the carrier density are a natural consequence of the strong correlations in these materials. The redistribution of intensity in the optical conductivity is connected with the dilution of the spin system by the added holes. For the one-particle spectrum the decrease of weight in the upper Hubbard band away from half filling can be understood by state counting at U=\ensuremath{\infty}, and the effect is strongly enhanced for small doping by both the first- and second-order terms in t/U. Altogether, the one-particle and optical spectra show the importance of the three-site hopping term. Therefore the t-J model does not represent well the spectral properties of the Hubbard model at large U. The local spin order is essential and determines both the k\ensuremath{\rightarrow} dependence of the one-particle spectrum and the intensities in the optical spectrum.
The asymptotic expression for the one-particle Green's function in the strong coupling limit of the double degenerate Hubbard model containing multiplet terms is obtained for arbitrary band filling within the irreducible Green's function approach. An explicit analytic expression is obtained for the mean-field Greens's function, conserving the sum rules for the first few spectral moments. Stability of homogeneous phases is analised in terms of a single flip spin and orbital excitations. It is shown that degeneracy has an additional enhancing effect on correlation driven interband transfer of spectral weight. Discovery of superconductivity in fullerides and of materials with colossal magnetoresistance during recent years have increased the interest for electron systems with physical properties being essentially dependent on orbital degeneracy and strong correlation effects. In many cases the orbitally degenerate Hubbard model (DHM) incorporates the key features of such materials [1], although there are even fewer exact results established for DHM than for the nondegenerate Hubbard model (NHM). In view of complexity of correlated systems, corrections to the atomic limit of one-particle excitation spectrum and spectral weights of correlated bands found by Harris and Lange for NHM [2] by means of perturbation theory based on canonical transformations provide an important exact result which serves as a test for different approximations. A similar derivation for DHM has not been carried out yet. In the present paper the Green's function for the strong coupling limit of the DHM is obtained by means of irreducible Green's function approach [3] - [6] which is closely related to canonical perturbation theory because of its spectral moment conserving construction. As was explicitly proved in [7] for the NHM, the first couples of spectral moments of correlated bands obtained within mean-field decoupling are identical to those of canonical perturbation theory to first order in / tU ( t is the n.n. hopping amplitude and U the Coulomb repulsion).
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.