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

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 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 disorderinteraction phase diagram such physics may be important.

Disordered two-dimensional superconductors: Roles of temperature and interaction strength

“…In some situations, Bogoliubov-de Gennes-type theories do a good job of capturing spatial charge redistribution driven by disorder; a recent comparison between mean-field and exact approaches in systems with diagonal disorder indicates that although charge redistribution may be well described within mean-field approaches, local spin correlations are not. 28 In the case considered here, however, this discrepancy can be attributed-to some extent-to two intertwined mechanisms. First, there is the fact that a three-component order parameter It is also worth noting that the strong-coupling pure attractive Hubbard model ͑at half filling͒ can be mapped onto an isotropic Heisenberg model.…”

Optical conductivity of a metal-insulator transition for the Anderson-Hubbard model in three dimensions away from half filling

“…We also allow local magnetic moments to develop in the xz-plane, which increases the spin degree of freedom, believed to be important in some circumstances. 30 We also mention the effective-field theory analysis of local moment formation in disordered interacting Fermi liquids by Milovanovic, Sachdev, and Bhatt. 39 The potential importance of such moments in the metalinsulator transition was left as an outstanding question, and will be one of the main aspects addressed in this paper.…”

“…The moments found for the real h ± i HF ground state are strongly noncollineare.g., see our discussion in Ref. 30 . Therefore, the restriction of collinear moments employed in Refs.…”

Real-space variational Gutzwiller wave functions for the Anderson-Hubbard model

“…The only known exact solution for the AH Hamiltonian is in infinite-dimensions, 18 and we thus have to use approximate methods in finite dimensions. The various approximate methods used previously include self-consistent Hartree-Fock theory, 19,20,21,22,23 dynamical mean field theories, 24,25,26,27,28,29,30,31 and exact numerical calculations for small clusters. 20,32,33,34 Of particular relevance to the current work is a recent variational wavefunction approach by Pezzoli et al, 35 which we discuss below.…”

“…The latter comparison is motivated by the fact that UHF has been, until recently, the standard numerical technique for studying disordered systems. Furthermore, a recent work 20 has suggested that UHF actually provides good quantitative results for some local physical quantities in disordered systems (namely that disorder improves the quality of the UHF approximation). We discuss the strengths and limitations of the simple GWF in Sec.…”