Ground-state properties of the attractive one-dimensional Bose-Hubbard model

We study the ground state of the attractive one-dimensional Bose-Hubbard model, and in particular the nature of the crossover between the weak interaction and strong interaction regimes for finite system sizes. Indicator properties such as the gap between the ground and first excited energy levels, and the incremental ground-state wave function overlaps are used to locate different regimes. Using mean-field theory we predict that there are two distinct crossovers connected to spontaneous symmetry breaking of the ground state. The first crossover arises in an analysis valid for large L with finite N, where L is the number of lattice sites and N is the total particle number. An alternative approach valid for large N with finite L yields a second crossover. For small system sizes we numerically investigate the model and observe that there are signatures of both crossovers. We compare with exact results from Bethe ansatz methods in several limiting cases to explore the validity for these numerical and mean-field schemes. The results indicate that for finite attractive systems there are generically three ground-state phases of the model.