We study the ground-state phase diagram of Rydberg atoms in a triangular-prism optical tweezer array using the density matrix renormalization group. Unlike two-leg Rydberg ladders with ${\mathbb{Z}}_2$ leg-exchange symmetry, the triangular prism possesses ${\mathbb{D}}_3$ symmetry, giving rise to a richer set of ordered phases and transitions. At relatively small blockade radius, a ${\mathbb{Z}}_2\times {\mathbb{Z}}_3$ density-wave phase appears at large detuning and breaks ${\mathbb{Z}}_2$ translational and ${\mathbb{Z}}_3$ rotational symmetry while preserving the reflection symmetry of the triangular rung. Upon decreasing detuning, it melts via a Berezinskii-Kosterlitz-Thouless (BKT) transition into a critical phase and subsequently enters a disordered phase through a second BKT transition, forming a commensurate regime described at low energies by a ${\mathbb{Z}}_6$ clock model. At larger blockade radius, a ${\mathbb{D}}_3$ phase with one Rydberg excitation per triangle emerges through a first-order transition. When double excitations on neighboring triangles are suppressed, rung-trimerized density waves develop as detuning increases from the disordered phase. These phases break translational symmetry while preserving ${\mathbb{D}}_3$ symmetry, and their melting transitions exhibit critical behaviors similar to those in Rydberg chains and two-leg ladders. In addition, we identify a geometry-dependent crossover within the ${\mathbb{Z}}_2$ rung-trimerized phase, followed by a first-order transition into a ${\mathbb{Z}}_2\times {\mathbb{D}}_3$ phase with complete ${\mathbb{D}}_3$ symmetry breaking. We also find floating phases with incommensurate quasi-long-range order between rung-trimerized states.

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• BKT transition
• Critical exponents
• Dipolar Rydberg atoms
• First order phase transitions
• Phase diagrams
• Quantum phase transitions
• Quantum simulation
• Density matrix renormalization group
• Finite-size scaling