Projection operator approach to the Bose-Hubbard model

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 z(0) of the lattice and reaches to within 0.5% of quantum Monte Carlo data for lattices with z(0) = 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 z(0)J/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 tau and exponent alpha: J(t) = J(i) + (J(f) - J(i))(t/tau)(alpha). We compute the wave function overlap F, the residual energy Q, the superfluid order parameter Delta(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.