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Physical review. B, Condensed matter

Self-consistent spin-spin and density-density correlations in the Hubbard model

Chris Gearhart, J. Charles Hicks

Published Sep 15, 1995• 1 citation
Publisher

Abstract

A linear combination of Hubbard projection operators is used to linearize the equations of motion for the Hubbard model. The Green's funtions in this linearization scheme depend explicitly on the value of the nearest-neighbor density-density correlation function. For the half-filled case we calculate this correlation function by taking variations of the equations of motion with respect to a density field. Three-body vertex corrections are ignored. The expressions we get for the correlation functions along with the equations of motion for the Green's functions form a system of equations that can be solved self-consistently. We solve these equations, both with and without effective vertex corrections, for a half-filled 12-site chain. These results are compared to numerically exact Lanczos solutions and to standard mean-field calculations. Calculations of on-site charge corelation, ground-state energy, and long-range spin-spin correlations for intermediate values of U show a significant improvement over standard mean-field techniques.

DOI
10.1103/physrevb.52.7787

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A linear combination of Hubbard projection operators is used to linearize the equations of motion for the Hubbard model. The Green's funtions in this linearization scheme depend explicitly on the value of the nearest-neighbor density-density correlation function. For the half-filled case we calculate this correlation function by taking variations of the equations of motion with respect to a density field. Three-body vertex corrections are ignored. The expressions we get for the correlation functions along with the equations of motion for the Green's functions form a system of equations that can be solved self-consistently. We solve these equations, both with and without effective vertex corrections, for a half-filled 12-site chain. These results are compared to numerically exact Lanczos solutions and to standard mean-field calculations. Calculations of on-site charge corelation, ground-state energy, and long-range spin-spin correlations for intermediate values of U show a significant improvement over standard mean-field techniques.
Ground-state energies and on-site density-density correlations are calculated for the one-dimensional Hubbard model using a linear combination of the Hubbard projection operators. The mean-field coefficients in the resulting linearized equations of motion (EOM) depend on both one-particle static expectation values as well as static two-particle correlations. To test the model, the one-particle expectation values are determined self-consistently while using Lanczos-determined values for the two-particle correlation terms. Ground-state energies and on-site density-density correlations are then compared as a function of U to the corresponding Lanczos values on a 12-site Hubbard chain for 1/2 and 5/12 fillings. To further demonstrate the validity of the technique, the static correlation functions are also calculated using a similar EOM approach, which ignores the effective vertex corrections for this problem, and compares those results as well for a 1/2-filled chain. These results show marked improvement over standard mean-field techniques.
The ground-state properties of the half-filled Hubbard model are investigated in the framework of lattice density functional theory. The single-particle density matrix ${\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}$ is regarded as the central variable of the many-body problem, where iii and jjj refer to the lattice sites and $\ensuremath{\sigma}$ to the spin. The interaction-energy functional $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ is calculated exactly for representative finite periodic systems by performing exact Lanczos diagonalizations. The relationship between $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ and the entropy $S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}]$ of independent fermions with natural-orbital occupations ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$ is analyzed. A simple approximation to the interaction energy of the half-filled Hubbard model is proposed, which takes the form $W=W(S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}])$. Using this functional we derive the ground-state energy, kinetic energy, average number of double occupations, charge distribution, magnetic susceptibility, and field-induced spin polarization in one-, two-, and three-dimensional periodic lattices. The limit of infinite dimensions is also explored. The accuracy of the method is assessed by comparison with available exact numerical or analytical results. Goals, limitations, and possible extensions of the domain of applicability of the functional are discussed.
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 study the effects of a bond-charge interaction (or correlated hopping) on the properties of the extended [i.e., with both on-site (U) and nearest-neighbor (V) repulsions] Hubbard model in one dimension at half-filling. Energy gaps and correlation functions are calculated by Lanczos diagonalization on finite systems. We find that, irrespective of the sign of the bond-charge interaction, X, the charge-density-wave (CDW) state is more robust than the spin-density-wave (SDW) state. A small bond-charge interaction term is enough to make the differences between the CDW and SDW correlation functions much less dramatic than when X=0. For X=t and fixed V2t (t is the uncorrelated hopping integral), there is an intermediate phase between a charge ordered phase and a phase corresponding to singly occupied sites, the nature of which we clarify: it is characterized by a succession of critical points, each of which corresponds to a different density of doubly occupied sites. We also find an unusual slowly decaying staggered spin-density correlation function, which is suggestive of some degree of ordering. No enhancement of pairing correlations was found for any X in the range examined. \textcopyright{} 1996 The American Physical Society.
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.
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.