Random-Phase-Approximation Excitation Spectra for Bose-Hubbard Models

We obtain the excitation spectra of the following three generalized Bose-Hubbard (BH) models: (1) a two-species generalization of the spinless BH model, (2) a single-species, spin-1 BH model, and (3) the extended Bose-Hubbard model (EBH) for spinless interacting bosons of one species. In all the phases of these models we provide a unified treatment of random-phase-approximation (RPA) excitation spectra. These spectra have gaps in all the MI phases and gaps in the DW phases in the EBH model; they are gapless in all the SF phases in these models and in the SS phases in the EBH model. We obtain the dependence of (a) gaps $\Delta$ and (b) the sound velocity $u_s$ on the parameters of these models and examine $\Delta$ and $u_s$ as these systems go through phase transitions. At the SF-MI transitions in the spin-1 BH model, $u_s$ goes to zero continuously (discontinuously) for MI phases with an odd (even) number of bosons per site; this is consistent with the natures of these transition in mean-field theory. In the SF phases of these models, our excitation spectra agree qualitatively, at weak couplings, with those that can be obtained from Gross-Pitaevskii-type models. We compare the results of our work with earlier studies of related models and discuss implications for experiments.