Abstract The properties of a recently discovered unrestricted Hartree‐Fock solution of the Hubbard Hamiltonian is analysed on the square lattice that shows a really two‐dimensional spin polarization of a vortex‐like character. This and other several mean‐field solutions are compared with the exact ground‐state energy and wave function of two and four electrons on a four‐site system (square). Results for two correlated electrons in large 3 × 3 and 4 × 4 clusters are also given and discussed. The vortex mean‐field solution provides the best solution in almost all systems that deviate from half‐filling.
Abstract The properties of a recently discovered unrestricted Hartree‐Fock solution of the Hubbard Hamiltonian is analysed on the square lattice that shows a really two‐dimensional spin polarization of a vortex‐like character. This and other several mean‐field solutions are compared with the exact ground‐state energy and wave function of two and four electrons on a four‐site system (square). Results for two correlated electrons in large 3 × 3 and 4 × 4 clusters are also given and discussed. The vortex mean‐field solution provides the best solution in almost all systems that deviate from half‐filling.
The Hubbard model in a two-dimensional square lattice is studied within the unrestricted Hartree-Fock approximation. A wide variety of solutions are found, as a function of filling and value of the Coulomb repulsion. Their properties range from those of magnetic polarons to a behavior similar to electrons in an external magnetic field (flux phases). The main features of the vortex solution are studied. It is shown that its energy increases logarithmically with system size, and that it becomes more stable for high values of the Coulomb repulsion and doping levels. Also, multivortex configurations are shown to exist. The implications of our results in the qualitative behavior of the Hubbard model as function of doping and strength of the interaction are discussed.
Abstract. A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by 〈Si · Sj〉, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorderinteraction phase diagram such physics may be important. PACS numbers: 71.27.+a,71.23.-k,71.10.-wCritiquing HF for the Anderson-Hubbard Model 2 1.
We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.
A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by < S_i . S_j >, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorder-interaction phase diagram such physics may be important.
The ground-state wave function and energy are calculated for the one-band Hubbard Hamiltonian on a 4\ifmmode\times\else\texttimes\fi{}4 cluster of the square lattice up to first and second order of the on-site Coulomb repulsion U, respectively. Changes induced by a deviation of the number of electrons from half-filling (one electron per lattice site) are closely followed. Charge-charge and spin-spin correlation functions are directly obtained from the ground-state wave function and show the existence of antiferromagnetic correlations that extend over the whole cluster at half-filling. Correlations among sites separated by more than a lattice site diminish very rapidly as holes are introduced into the cluster. The analysis is extended to the negative-U case. Ground-state properties change qualitatively.
The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.