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Journal of Physics Condensed Matter

A momentum distribution functional approach to the Hubbard model at half-filling

Simon M. Manning, David M. Edwards

Published Aug 23, 1993• 5 citations
Publisher

Abstract

The authors investigate the half-filled Hubbard model in one and two dimensions using approximations to the energy expressed as functionals of the momentum distribution. The functionals are calculated using the Hartree-Fock approximation and density functional theory in the local-spin-density and self-interaction-corrected (SIC) local-spin-density approximations (LSDA) by expressing the ground-state spin densities as a variational ansatz in which the momentum distribution enters as a variational quantity. The autthors calculate the momentum distribution, energy and local moment, as well as their ansatz orbitals, in all three approximations and find that the SIC always improves upon the LSDA.

DOI
10.1088/0953-8984/5/34/008

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The authors investigate the half-filled Hubbard model in one and two dimensions using approximations to the energy expressed as functionals of the momentum distribution. The functionals are calculated using the Hartree-Fock approximation and density functional theory in the local-spin-density and self-interaction-corrected (SIC) local-spin-density approximations (LSDA) by expressing the ground-state spin densities as a variational ansatz in which the momentum distribution enters as a variational quantity. The autthors calculate the momentum distribution, energy and local moment, as well as their ansatz orbitals, in all three approximations and find that the SIC always improves upon the LSDA.
The density-functional theory for the half-filled extended Hubbard model in one and two dimensions has been studied within the self-interaction-corrected (SIC) local-spin-density (LSD) approximation. The SIC to the total energy is expressed in terms of the Wannier functions which have the most optimal degree of localization. The system at ground state can have spin-density wave (SDW) or charge-density wave (CDW) ordering depending on the relative strength of the on-site repulsion and the nearest-neighbor repulsion. Like the Hartree-Fock approximation, the SIC-LSD scheme predicts a first-order transition between CDW and SDW. However, in the SDW, the nearest-neighbor repulsion V near the transition point tends to increase the band gap for small U and decrease the band gap for large U, with a crossover around $U\ensuremath{\approx}4$ in one dimension, which corresponds to the tricritical point predicted by other theories. The ground-state energies are better described in SIC-LSD than in the Hartree-Fock approximation.
Svane and Gunnarsson's method of the self-interaction corrected local-spin-density approximation (SIC-LSDA) in the density-functional theory was applied to the one-dimensional Hubbard model and the two-dimensional pd Hubbard model in order to study the one particle energy spectrum. The calculated energy band in the SIC-LSDA gives a reasonable band gap in contrast to serious underestimation of it by the LSDA and slight overstimation by the Hartree-Fock approximation. Behavior of SIC localized orbitals is also discussed
Momentum dependent local-ansatz wavefunction approach (MLA) to the correlated electron systems in solids has been further developed to solve best a self-consistent equation for variational parameters at non half-filling. With use of the improved variational scheme we performed the numerical calculations for the non-half-filled band Hubbard model on the hypercubic lattice in infinite dimensions. We verified that the self-consistent scheme significantly improves the correlation energy and the momentum distribution as compared with the original scheme in the MLA. We also demonstrate that the theory improves the standard variational methods such as the Local-Ansatz approach (LA) and the Gutzwiller wavefunction approach (GA); the ground-state energy in the MLA is lower than those of the LA and the GA in the weak and intermediate Coulomb interaction regimes. The double occupation number is shown to be suppressed as compared with the LA. Calculated momentum distribution functions show a distinct momentum dependence, which is qualitatively different from those of the LA and the GA.
Svane and Gunnarsson's method of the self-interaction corrected local-spin-density approximation (SIC-LSDA) in the density-functional theory was applied to the one-dimensional Hubbard model and the two-dimensional pd Hubbard model in order to study the one particle energy spectrum. The calculated energy band in the SIC-LSDA gives a reasonable band gap in contrast to serious underestimation of it by the LSDA and slight overestimation by the Hartree-Fock approximation. Behavior of SIC localized orbitals is also discussed.
The self-interaction correction (SIC) to local-spin-density functional theory is important in antiferromagnetic insulators such as transition metal oxides. To understand better the role of SIC in such systems, Svane and Gunnarsson introduced a discrete local-density functional for the Hubbard model. It is shown that their results, and further results, are obtained rather simply by expressing the energy as a functional of the momentum distribution. New results are given for the energy of the two-dimensional Hubbard model. The method is extended to calculate the transverse wave-vector-dependent susceptibility in a simple case.
We show that the self-interaction-corrected (SIC) local spin-density (LSD) approximation can describe localization in a system where the Coulomb interaction is large compared with the band-width. In contrast with the normal LSD approximation, the SIC approximation can be used for a band with orbital degeneracy, even if the band is not half filled. The method is applied to the one-dimensional Hubbard model for which there is an exact solution. The total energy, the band gap, the local moment, and the momentum distribution are substantially better described than in the LSD approximation. For some of these properties the SIC approach is shown to compare favorably with the Gutzwiller ansatz.