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Physical review. B, Condensed matter

Obtaining localized orbitals and band structure in self-interaction-corrected density-functional theory

Yuannan Xie, Rushan Han, Xinwei Zhang

Published Sep 15, 1999• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.60.8543

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arXiv:cond-mat/0109051•Physical Review Letters•Jan 16, 2002•131 citations
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′.{t}^{\ensuremath{'}}.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′,{t}^{\ensuremath{'}},t\ensuremath′, the CDW and SDW states are unstable over a finite portion of the phase diagram at weak coupling.
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.
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.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
We consider a generalized Hubbard model with on-site interaction U, nearest-neighbor repulsion V, and nearest-neighbor hopping for spin \ensuremath{\sigma}, which depends on the sum of particles ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$ with opposite spin in the two sites involved. The hopping matrix elements are denoted by tAA{\mathrm{t}}_{\mathrm{AA}}tAA​,tAB{\mathrm{t}}_{\mathrm{AB}}tAB​,tBB{\mathrm{t}}_{\mathrm{BB}}tBB​ for ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$=0,1,2, respectively. For 0tAB{\mathrm{t}}_{\mathrm{AB}}tAB​tAA{\mathrm{t}}_{\mathrm{AA}}tAA​=tBB{\mathrm{t}}_{\mathrm{BB}}tBB​, we have determined the regions of parameters for which the ground-state (GS) of the one-dimensional system is a charge-density wave (CDW), a spin-density wave (SDW), or a metal (M), using Hartree-Fock, exact diagonalization of finite chains and quantum Monte Carlo. The results agree qualitatively with the exactly solvable limit of tAB{\mathrm{t}}_{\mathrm{AB}}tAB​=0. For 0tAB{\mathrm{t}}_{\mathrm{AB}}tAB​tAA{\mathrm{t}}_{\mathrm{AA}}tAA​=tBB{\mathrm{t}}_{\mathrm{BB}}tBB​, the GS is a M for sufficiently low values of U and V. In contrast, when tAA{\mathrm{t}}_{\mathrm{AA}}tAA​+tBB{\mathrm{t}}_{\mathrm{BB}}tBB​-2tAB{2\mathrm{t}}_{\mathrm{AB}}2tAB​=0, our results suggest that the GS is either a CDW or a SDW, with the boundary between them lying near the line U=2V.
The square-lattice extended Hubbard model at half filling is studied by a perturbation scheme. By taking into account the ground state of the mean-field theory as a reference state, the scheme treats the collective fluctuations beyond the mean field as the predominant perturbation. The system at ground state can be a spin-density wave (SDW) or a charge-density wave (CDW) depending on the relative strengths of the on-site electron interaction and the nearest-neighbor interaction. The excitation modes above the ground state are analyzed through diagonalizing an effective Hamiltonian. The energy spectra and the wave functions are self-consistently given by the formalism. By comparing the ground-state energies, we investigate the SDW-CDW transition.
We reexamine the ground-state phase diagram of the one-dimensional half-filled Hubbard model with on-site and nearest-neighbor repulsive interactions. We calculate second-order corrections to coupling constants in the weak-coupling renormalization-group approach ( g-ology) to show that the bond-charge-density-wave (BCDW) phase exists for weak couplings in between the charge-density-wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin electrons destabilizes the BCDW state and gives rise to a bicritical point where the CDW-BCDW and SDW-BCDW continuous-transition lines merge into the CDW-SDW first-order transition line.