  We study a two-dimensional lattice gas of atoms that are photo-excited to
high-lying Rydberg states in which they interact via the van-der-Waals
interaction. We explore the regime of dominant nearest neighbor interaction
where this system is intimately connected to a quantum version of Baxter's hard
squares model. We show that the strongly correlated ground state of the Rydberg
gas can be analytically described by a projected entangled pair state that
constitutes the ground state of the quantum hard squares model. This
correspondence allows us to identify a first order phase boundary where the
Rydberg gas undergoes a transition from a disordered (liquid) phase to an
ordered (solid) phase.
