Phase diagram of the bose Hubbard model

The first reliable analytic calculation of the phase diagram of the bose gas on a d-dimensional lattice with on-site repulsion is presented. In one dimension, the analytic calculation is in excellent agreement with the numerical Monte Carlo results. In higher dimensions, the deviations from the Monte Carlo calculations are larger, but the correct shape of the Mott insulator lobes is still obtained. Explicit expressions for the energy of the Mott and the “defect” phase are given in a strong-coupling expansion. Typeset using REVTEX 1 Strongly interacting quantum systems continue to challenge theoretical physics. Fermionic examples of strong correlations include high temperature superconductors, heavy fermion materials and metal-insulator transitions. The study of these systems has been confined to either numerical simulations which are plagued by finite-size effects (with the notable exception of the infinite-dimensional expansion) or uncontrollable approximations. Similar problems arise in strongly interacting bosonic systems which have attracted a lot of recent interest [1–4]. Physical realizations include short correlation length superconductors, granular superconductors, Josephson arrays, and the dynamics of flux lattices in type II superconductors. The relevant physics of these problems is contained in the bose Hubbard Hamiltonian which describes the competition between kinetic energy and potential energy effects. Various aspects of this model were investigated analytically by mean-field theory [1,5], by renormalization group techniques [1,3] and by projection methods [6]. The bose Hubbard model was also studied with Quantum Monte Carlo methods (QMC) by Batrouni et al. [2] in one dimension (1+1) and by Krauth and Trivedi [7] in two dimensions (2+1). In this contribution, we show that the phase diagram obtained from a strong-coupling expansion has the correct dependence on the dimensionality of the spatial lattice, and agrees with the QMC calculations (to within a few percent). We study the minimal model which contains the key physics of the strongly interacting bose system – the competition between kinetic and potential energy effects: