  We describe the zero-temperature phase diagram of a model of a
two-dimensional square-lattice array of neutral atoms, excited into Rydberg
states and interacting via strong van der Waals interactions. Using the
density-matrix renormalization group algorithm, we map out the phase diagram
and obtain a rich variety of phases featuring complex density wave orderings,
upon varying lattice spacing and laser detuning. While some of these phases
result from the classical optimization of the van der Waals energy, we also
find intrinsically quantum-ordered phases stabilized by quantum fluctuations.
These phases are surrounded by novel quantum phase transitions, which we
analyze by finite-size scaling numerics and Landau theories. Our work
highlights Rydberg quantum simulators in higher dimensions as promising
platforms to realize exotic many-body phenomena.
