Lattice bosons with infinite range checkerboard interactions

Motivated by experiments performed by Landig et al. [Nature 532, 476-479], we consider a two dimensional Bose gas in an optical lattice, trapped inside a single mode superradiant Fabry Perot cavity. The cavity mediates infinite range checkerboard interactions between the atoms, which produces competition between Mott insulator, charge density wave, superfluid and supersolid phases. We calculate the phase diagram of this Bose gas in a homogeneous system and in the presence of a harmonic trap.