Critiquing variational theories of the Anderson–Hubbard model: real-space self-consistentHartree–Fock solutions

Abstract

A simple and commonly employed approximate technique with which one can examinespatially disordered systems when strong electronic correlations are present is based on theuse of real-space unrestricted self-consistent Hartree–Fock wavefunctions. In such anapproach the disorder is treated exactly while the correlations are treated approximately. Inthis paper we critique the success of this approximation by making comparisonsbetween such solutions and the exact wavefunctions for the Anderson–Hubbardmodel. Due to the sizes of the complete Hilbert spaces for these problems, thecomparisons are restricted to small one-dimensional chains, up to ten sites, and a4 × 4 two-dimensionalcluster, 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 from1/2-filling. This approximation is based on a variational approach which minimizes theHartree–Fock energy, and we have completed comparisons of the exact and Hartree–Fockenergies. However, in order to assess the success of this approximation in reproducingground-state correlations we have completed comparisons of the local charge and spincorrelations, including the calculation of the overlap of the Hartree–Fock wavefunctionswith those of the exact solutions. We find that this approximation reproduces the localcharge densities to quite a high accuracy, but that the local spin correlations,as represented by , are not as well represented. In addition to these comparisons, we discuss the properties ofthe spin degrees of freedom in the HF approximation, and where in the disorder–interactionphase diagram such physics may be important.