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We demonstrate that geometric frustration and long-range interactions bothpromote order-by-disorder in the antiferromagnetic transverse-field Ising modelon the ruby lattice. To this end we investigate the quantum phase diagram fortruncated $J_1$-$J_2$-$J_3$ Ising interactions. In the low-field limit wederive an effective quantum dimer model, analyzing how the extensiveground-state degeneracy at zero field is lifted by two distinctorder-by-disorder scenarios. We support our analysis by studying thegap-closing of the high-field phase using series expansions. For $J_2>J_3$, wefind an emergent clock-ordered phase at low fields, stabilized by resonatingplaquettes, and a 3d-XY quantum phase transition to the polarized high-fieldphase. For $J_3>J_2$, an order-by-disorder mechanism stabilizes a distinct$k=(0,0)$ order and a quantum phase transition in the 3d-Ising universalityclass is observed. In contrast to the triangular lattice, on the ruby latticealgebraically decaying long-range interactions favor the clock-orderedlow-field phase and therefore allow a robust implementation in existing Rydbergatom quantum simulators.

Order-by-disorder in the antiferromagnetic long-range transverse-field Ising model on the ruby lattice