Exact Solutions of an Extended Bose-Hubbard Model with E 2 Symmetry

An extended Bose-Hubbard (BH) model with number-dependent multi-site and infinite-range hopping is proposed, which, similar to the original BH model, describes a phase transition between the delocalized superfluid (SF) phase and localized Mott insulator (MI) phase. It is shown that this extended model with local Euclidean E (2) symmetry is exactly solvable when on-site local potentials are included, while the model without local potentials is quasi-exactly solvable, which means only a part of the excited states including the ground state being exactly solvable. As applications of the exact solution for the ground state, phase diagram of the model in 1D without local potential and on-site disorder for filling factor rho = 1 with M = 6, M = 10, and M = 14 sites are obtained. The ground state probabilities to detect n particles on a single site, P (n) , for n = 0, 1, 2 as functions of the control parameter U/t in these cases are also calculated. It is shown that the critical point in P (n) and in the entanglement measure is away from that of the SF-MI transition determined in the phase analysis. It is also shown that the model-independent entanglement measure is related with P (n) , which, therefore, may be practically useful because P (n) is measurable experimentally. The ground state expectation value of local particle numbers, the ground state local particle number fluctuations, the ground state probabilities to detect n particles on every site, and the entanglement measure have also been studied in the model for N = M = 4 with the two-body onsite repulsion and a local confining harmonic potential. The connection between these quantities and the entanglement observed previously is verified.