Projection operator approach to the Bose-Hubbard model

Abstract

We develop a projection operator formalism for studying both the zero-temperature equilibrium phase diagram and the nonequilibrium dynamics of the Bose-Hubbard model. Our work, which constitutes an extension of that of Trefzger and Sengupta [ Phys. Rev. Lett. 106 095702 (2011)], shows that the method provides an accurate description of the equilibrium zero-temperature phase diagram of the Bose-Hubbard model for several lattices in two and three dimensions. We show that the accuracy of this method increases with the coordination number z₀ of the lattice and reaches to within 0.5% of quantum Monte Carlo data for lattices with z₀=6. We compute the excitation spectra of the bosons using this method in the Mott and the superfluid phases and compare our results with mean-field theory. We also show that the same method may be used to analyze the nonequilibrium dynamics of the model both in the Mott phase and near the superfluid-insulator quantum critical point where the hopping amplitude J and the on-site interaction U satisfy z₀J/U«1. In particular, we study the nonequilibrium dynamics of the model both subsequent to a sudden quench of the hopping amplitude J and during a ramp from J_i to J_f characterized by a ramp time τ and exponent α: J(t)=J_i+(J_f−J_i)(t/τ)α. We compute the wave function overlap F, the residual energy Q, the superfluid order parameter Δ(t), the equal-time order parameter correlation function C(t), and the defect formation probability P for the above-mentioned protocols and provide a comparison of our results to their mean-field counterparts. We find that Q, F, and P do not exhibit the expected universal scaling. We explain this absence of universality and show that our results for linear ramps compare well with the recent experimental observations.