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Physical Review A

Studies on the six-site Hubbard model: Comparison of various approximations

Mme M. NAUCIEL-BLOCH, Thomas P. Eggarter

Published May 1, 1979• 3 citations
Publisher

Abstract

The authors tested several approximations commonly used in molecular energy calculations by applying them to the six-site half-filled Hubbard model and comparing the results with the exact solution. Three main conclusions follow from this study: (i) for $\frac{\ensuremath{\beta}}{U}$ in the range of interest for molecules, perturbations to the molecular-orbital ground state produce better results than the other more elaborate methods tested; (ii) Nesbet's pair theory is only slightly worse in this range but has the advantage of working well for all values of the parameters, and (iii) perturbative corrections to the unrestricted Hartree-Fock ground state give very poor results when this state has lower symmetry than the Hamiltonian.

DOI
10.1103/physreva.19.1862

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The authors tested several approximations commonly used in molecular energy calculations by applying them to the six-site half-filled Hubbard model and comparing the results with the exact solution. Three main conclusions follow from this study: (i) for \ensuremathβU\frac{\ensuremath{\beta}}{U}U\ensuremathβ​ in the range of interest for molecules, perturbations to the molecular-orbital ground state produce better results than the other more elaborate methods tested; (ii) Nesbet's pair theory is only slightly worse in this range but has the advantage of working well for all values of the parameters, and (iii) perturbative corrections to the unrestricted Hartree-Fock ground state give very poor results when this state has lower symmetry than the Hamiltonian.
We have calculated the ground-state energy and the fluctuation contribution to the free-energy for the Hubbard model at half filling. We have used a gauge that leads to the Hartree-Fock solution in the saddle-point approximation. The mean-field ground-state energy calculated here agrees with the results obtained by Dagotto et al. [Phys. Rev. B 45, 10 741 (1992)] with use of the exact-diagonalization techniques on a 4\ifmmode\times\else\texttimes\fi{}4 site cluster. In the large-U limit, the ground-state energy per site is obtained as Eg{\mathit{E}}_{\mathit{g}}Eg​/N=-U/4-J+O(J2{\mathit{J}}^{2}J2/U). The fluctuation contribution to the free-energy was found to have a stronger temperature and U dependence than the mean-field contribution, and to play a dominant role for strongly correlated systems.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.
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.
We formulate a method for studying the quantum field dynamics of ultracold Bose gases confined within optical lattice potentials, within the lowest Bloch-band Bose-Hubbard model. Our formalism extends the two-sites results of Vardi and Anglin [Phys. Rev. Lett. 86, 568 (2001)] to the general case of MMM lattice sites. The methodology is based on mapping the Bose-Hubbard Hamiltonian to an SU(M)\mathrm{SU}(M)SU(M) pseudospin problem and truncating the resulting hierarchy of dynamical equations for correlation functions, up to pair correlations between SU(M)\mathrm{SU}(M)SU(M) generators. Agreement with few-site exact many-particle calculations is consistently better than the corresponding Hartree-Fock-Bogoliubov approximation. Moreover, our approximation compares favorably with a more elaborate two-particle irreducible effective action formalism, at a fraction of the analytic and numerical effort.
With the purpose of assessing the validity of commonly used approximations, best variational solutions are calculated for a four-site (tetrahedral-cluster) Hubbard model with periodic boundary conditions and compared with exact results. In particular, the optimum number-conserving Bardeen-Cooper-Schrieffer-type variational solution is obtained for two, four, and six electrons, and the optimal resonanting-valence-bond state for four electrons. For an attractive interaction U, the exact ground state of the system is obtained in all four cases. If the interaction U is repulsive, the exact ground state is obtained only when the cluster contains two and six electrons. For four electrons the BCS approximation is only satisfactory for relatively small U. For any positive U the resonating-valence-bond state on the other hand, gives very good (albeit not exact) energies for the ground state.