Mean-field phase diagram of the extended Bose-Hubbard model of many-body cavity quantum electrodynamics

We investigate the mean-field phase diagram of the Bose-Hubbard model with infinite-range interactions in two dimensions. This model describes ultracold bosonic atoms confined by a two-dimensional optical lattice and dispersively coupled to a cavity mode with the same wavelength as the lattice. We determine the ground-state phase diagram for a grand-canonical ensemble by means of analytical and numerical methods. Our results mostly agree with the ones reported in Dogra et al. [N. Dogra, F. Brennecke, S. D. Huber, and T. Donner, Phys. Rev. A 94, 023632 (2016)], and have a remarkable qualitative agreement with the quantum Monte Carlo phase diagrams of Flottat et al. [T. Flottat, L. de Forges de Parny, F. Hebert, V. G. Rousseau, and G. G. Batrouni, Phys. Rev. B 95, 144501 (2017)]. The salient differences concern the stability of the supersolid phases, which we discuss in detail. Finally, we discuss differences and analogies between the ground-state properties of all-connected bosons with the ones predicted for repulsively interacting dipolar bosons in two dimensions.