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

Complex density wave orders and quantum phase transitions in a model of square-lattice Rydberg atom arrays