Critiquing variational theories of the Anderson-Hubbard model: real-space self-consistent Hartree-Fock solutions

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 wavefunctions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this paper we critique the success of this approximation by making comparisons between such solutions and the exact wavefunctions 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 4 x 4 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 wavefunctions 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.