Order-by-disorder in the antiferromagnetic J₁-J₂-J₃ transverse-field Ising model on the ruby lattice

We investigate the quantum phase diagram of the J₁-J₂-J₃ antiferromagnetic transverse-field Ising model on the ruby lattice. In the low-field limit we derive an effective quantum dimer model, analyzing how the extensive ground-state degeneracy at zero field is lifted by an order-by-disorder scenario. We support our analysis by studying the gap-closing of the high-field phase using series expansions. For J₂>J₃, we find a columnar phase at low fields, followed by a clock-ordered phase stabilized by resonating plaquettes at intermediate field values, and an emergent 3d-XY quantum phase transition to the polarized high-field phase. For J₃>J₂, an order-by-disorder mechanism stabilizes a distinct k=(0,0) order and a quantum phase transition in the 3d-Ising universality class is observed. Further, we discuss the possible implementation of the columnar- and clock-ordered phase in existing Rydberg atom quantum simulators. When taking into account the full algebraically decaying long-range interactions on the ruby lattice, we find that long-range interactions favor the same ground state as the quantum fluctuations induced by a transverse field, which could make the ruby lattice a promising candidate for the realization of a clock-ordered phase.