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Motivated by experiments performed by Landig et al. [Nature 532, 476-479], weconsider a two dimensional Bose gas in an optical lattice, trapped inside asingle mode superradiant Fabry Perot cavity. The cavity mediates infinite rangecheckerboard interactions between the atoms, which produces competition betweenMott insulator, charge density wave, superfluid and supersolid phases. Wecalculate the phase diagram of this Bose gas in a homogeneous system and in thepresence of a harmonic trap.

Lattice bosons with infinite range checkerboard interactions