  We analyze the zero-temperature phases of an array of neutral atoms on the
kagome lattice, interacting via laser excitation to atomic Rydberg states.
Density-matrix renormalization group calculations reveal the presence of a wide
variety of complex solid phases with broken lattice symmetries. In addition, we
identify a novel regime with dense Rydberg excitations that has a large
entanglement entropy and no local order parameter associated with lattice
symmetries. From a mapping to the triangular lattice quantum dimer model, and
theories of quantum phase transitions out of the proximate solid phases, we
argue that this regime could contain one or more phases with topological order.
Our results provide the foundation for theoretical and experimental
explorations of crystalline and liquid states using programmable quantum
simulators based on Rydberg atom arrays.
