We consider the many-body quantum Gibbs state for the Bose–Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schrödinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.
We study the dissipative dynamics of a one-dimensional bosonic system described in terms of the bipartite Bose-Hubbard model with alternating gain and loss. This model exhibits the
PT symmetry under some specific conditions and features a
PT-symmetry phase transition. It is characterized by an order parameter corresponding to the population imbalance between even and odd sites,similar to the continuous phase transitions in the Hermitian realm. In the noninteracting limit, we solve the problem exactly and compute the parameter dependence of the order parameter. The interacting limit is addressed at the mean-field level, which allows us to construct the phase diagram for the model. We find that both the interaction and dissipation rates induce
PT-symmetry breaking. On the other hand, periodic modulation of the dissipative coupling in time stabilizes the
PT-symmetric regime. Our findings are corroborated numerically on a tight-binding chain with gain and loss.
We propose an extended BCS–Hubbard model and investigate its ground state phase diagram in an external magnetic field. By mapping the model onto a model of spinless fermions coupled with conserving Z 2 variables which are mimicked by pseudospins, the model is shown to be exactly solvable along the symmetric lines for an arbitrary on-site Hubbard interaction on the bipartite lattice. In the zero field limit, the ground states exhibit an antiferromagnetic order of pseudospins. In the large field limit, on the other hand, the pseudospins are fully polarized ordered. With the increase of the applied field, a first-order phase transition occurs between these kinds of phases when the on-site Coulomb interaction is less than a critical value U c. Above this critical U c, a novel intermediate phase emerges between the fully polarized and antiferromagnetic phases. The ground states in this phase are macroscopically degenerate, like in a spin ice, and the corresponding entropy scales linearly with the lattice size at zero temperature.