Phase transitions in a Bose-Hubbard model with cavity-mediated global-range interactions

We study a system with competing short- and global-range interactions in the framework of the Bose-Hubbard model. Using a mean-field approximation we obtain the phase diagram of the system and observe four different phases: a superfluid, a supersolid, a Mott insulator, and a charge-density wave, where the transitions between the various phases can be either of first or second order. We qualitatively support these results using Monte Carlo simulations. An analysis of the low-energy excitations shows that the second-order phase transition from the charge-density wave to the supersolid is associated with the softening of particle-and holelike excitations which give rise to a gapless mode and an amplitude Higgs mode in the supersolid phase. This amplitude Higgs mode is further transformed into a roton mode which softens at the supersolid to superfluid phase transition.