We present a detailed derivation of a multisite mean-field theory (MSMFT) used to describe the Mott-insulator to superfluid transition of bosonic atoms in optical lattices. The approach is based on partitioning the lattice into small clusters which are decoupled by means of a mean-field approximation. This approximation invokes local superfluid order parameters defined for each of the boundary sites of the cluster. The resulting MSMFT grand potential has a nontrivial topology as a function of the various order parameters. An understanding of this topology provides two different criteria for the determination of the Mott insulator superfluid phase boundaries. We apply this formalism to d-dimensional hypercubic lattices in one, two, and three dimensions and demonstrate the improvement in the estimation of the phase boundaries when MSMFT is utilized for increasingly larger clusters, with the best quantitative agreement found for d = 3. The MSMFT is then used to examine a linear dimer chain in which the onsite energies within the dimer have an energy separation of . This system has a complicated phase diagram within the parameter space of the model, with many distinct Mott phases separated by superfluid regions.MCINTOSH, PISARSKI, GOODING, AND ZAREMBA PHYSICAL REVIEW A 86, 013623 (2012) II. FORMALISM: MULTISITE MEAN-FIELD THEORY A. Derivation in one dimensionOur work is based on the BH Hamiltonian [2] in the grand canonical ensemble. With the assumption of a single orbital per site, this Hamiltonian is given bywhere the index i labels the sites of the optical lattice andĉ † i andĉ i are site creation and annihilation operators (henceforth referred to as site operators), respectively; the number operator for site i is given byn i =ĉ † iĉ i . The system parameters include the onsite energies ε i at each lattice site, the tunneling energy J ij between sites i and j , and the intrasite interaction energy U i . To a good approximation it is sufficient to ignore interactions between bosons on different sites and hopping between sites further apart than the nearest-neighbor distance [2,37]. Furthermore, we will restrict our considerations to the case where the interaction parameter has a common value U for all sites and a hopping parameter J for all nearest-neighbor pairs. Generalizations to more complex situations such as superlattices [15][16][17] can be readily accommodated in the MSMFT that we develop. The final parameter in the BH Hamiltonian is the chemical potential μ which controls the 013623-2 013623-3 MCINTOSH, PISARSKI, GOODING, AND ZAREMBA PHYSICAL REVIEW A 86, 013623 (2012)

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.