Bose–Einstein condensation in the excited band and the energy spectrum of the Bose–Hubbard model

Based on the mean field approximation, we investigate the transition into the Bose–Einstein condensate phase in the Bose–Hubbard model with two local states and boson hopping in only the excited band. In the hard-core boson limit, we study the instability associated with this transition, which appears at excitation energies $\delta<|t_0'|$, where $|t_0'|$ is the particle hopping parameter. We discuss the conditions under which the phase transition changes from second to first order and present the corresponding phase diagrams $(\Theta,\mu)$ and $(|t_0'|,\mu)$, where $\Theta$ is the temperature and $\mu$ is the chemical potential. Separation into the normal and Bose–Einstein condensate phases is possible at a fixed average concentration of bosons. We calculate the boson Green's function and one-particle spectral density using the random phase approximation and analyze changes in the spectrum of excitations of the “particle” or “hole” type in the region of transition from the normal to the Bose–Einstein condensate phase.