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

Abstract

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 E2 symmetry is exactly solvable when on-site local potential or disorder is included, while the model without local potential or disorder 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 ρ = 1 with M = 6 sites and that with M = 10 are obtained. The probabilities to detect n particles on a single site, Pn, for n = 0, 1, 2 as functions of the control parameter U/t in these two cases are also calculated. It is shown that the critical point in Pn 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 the model-independent entanglement measure is related with Pn, which, therefore, may be practically useful because Pn is measurable experimentally. [8,9], the density matrix renormalization group (DMRG) method [10][11][12] were used, from which ground state properties of the model have been studied extensively. Extensions of the model to include on-site disorder [1,13,14], longer ranged interactions [5,15], long-range hopping [16,17], infinite-range hopping [1,18], and pair-correlated hopping [19], etc. have also been made, in which, generally, more complicated phase structures emerge. Though many properties of the BH model have been known quite well from the above mentioned approximate calculations, it will be helpful if there is a similar model that can be solved exactly or quasi-exactly, because exactly and quasi-exactly solvable models may offer valuable insight and their solutions may be used as the basis in approximation methods.

The Critical Point Entanglement and Chaos in the Dicke Model

“…However, it is not necessarily true that the system is most entangled at the critical point. For example, the entanglement measure increases with increasing value of the control parameter in the Jaynes-Cummings model [6], which is also the Dicke model with the RWA, the spin chain models [55], the Bose-Hubbard model with on-site repulsion [56], etc., though saturation in the measure will reach beyond the critical point in the strong coupling regime. On the other hand, in general, chaotic systems tend to produce larger entanglement than for regular systems, but there are also exceptions for classically regular systems, as shown in [57,58].…”