Lattice supersolidity and simplex phases in a three-level Rydberg system

We investigate quantum phases of matter naturally realizable in a two-dimensional Rydberg array using three different highly-excited Rydberg states as an encoding. The resulting system implements a long-range hopping model of two species of hardcore bosons, which can further be fine-tuned to realize the spin-1 dipolar XY Hamiltonian. We explore the ground-state physics of both models on a range of relevant lattices and numerically characterize the phase diagrams, finding an array of interesting, highly-correlated quantum phases including two distinct lattice supersolids and a trimerized simplex phase fundamentally enabled by the three-state nature of the system.