Mott Transition in Lattice Boson Models

Initially introduced in 1989 [9], the Bose-Hubbard model has been the object of much recent work. It represents a simple lattice model of itinerant bosons which interact locally. This model turns out to describe fairly well recent experiments with bosonic atoms in optical lattices [12, 15]. Its low-temperature phase diagram has been uncovered in several studies, both analytical (see e.g. [9, 10, 8]) and numerical [2, 19] ones. When parameters such as the chemical potential or the tunneling amplitude are varied the Bose-Hubbard model exhibits a phase transition from a Mott insulating phase to a Bose-Einstein condensate. Fig. 2, below, depicts its ground state phase diagram. In this paper, we investigate the phase diagram of this model in a mathematically rigorous way. We focus on the situation with a small tunneling amplitude, t, and a small chemical potential, μ. We construct the critical line between Mott and nonMott behavior to lowest order in the ratio t/μ. More precisely, we prove the existence of domains with and without Mott insulator. These domains are separated by a comparatively thin stretch; the domain without Mott insulator is widely believed to be a Bose condensate. Our results establish in particular the occurrence of a “quantum phase transition” in the ground state. Over the years several analytical methods have been developed that are useful for the study of models such as the Bose-Hubbard model. They include a general theory of classical lattice systems with quantum perturbations [3, 5, 6, 16]. These methods can be used to establish the existence of Mott phases for small t; but they only apply to domains of parameters far from the transition lines. The Bose-Hubbard model on the complete graph can be studied rather explicitly and its phase diagram is similar to the one of the finite-dimensional model [4]. Results using reflection positivity are mentioned below and only apply to the hard-core model. A related model with an extra chessboard potential was studied in [1] (see also [17]).