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Physical review. B, Condensed matter

Localization in the self-interaction-corrected density-functional formalism

A. Svane, O. Gunnarsson

Published Jun 1, 1988• 177 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.37.9919

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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.
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.
An efficient minimization method is presented to find the optimal orthogonal localized orbitals within the self-interaction-corrected (SIC) local spin-density (LSD) approximation. By introducing a unified Hamiltonian, both the canonicalvalence bands and conduction bands are obtained by diagonalizing the same matrix. A general criterion for the existence of the self-consistent localized solutions of the SIC-LSD equations is proposed. As an application, we consider the phase diagram and the band structure of the half-filled two-dimensional extended Hubbard model with the next-nearest-neighbor hopping amplitude ${t}^{\ensuremath{'}}.$ The phase boundary between the charge-density wave (CDW) and the spin-density wave (SDW) is determined by comparing the ground-state energies. For nonzero ${t}^{\ensuremath{'}},$ the CDW and SDW states are unstable over a finite portion of the phase diagram at weak coupling.
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 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.
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
A systematic investigation of two approximate self-interaction corrections (SICs), Perdew-Zunger SIC and Lundin-Eriksson SIC, and the local-density approximation (LDA) is performed for a model Hamiltonian whose exact many-body solution and exact LDA are known. Both SICs as well as LDA are applied in the calculation of ground-state energies, ground-state densities, energy gaps, and impurity densities of one-dimensional Hubbard chains differing in size, particle number, and interaction strength. The orbital-dependent potentials arising from either SIC are treated within the optimized-effective potential method, which we reformulate for the Hubbard model. The delocalization tendency of LDA is confronted with the localization tendency of SIC. A statistical analysis of the resulting data set sheds light on the role of SIC for weakly and strongly interacting particles and allows one to assess the performance of each methodology.