Order by disorder and an emergent Kosterlitz-Thouless phase in a triangular Rydberg array

A programmable quantum simulator using Rydberg-atom array provides a promising route to demystifying quantum many-body physics in strongly correlated systems. Motivated by the recent realization of various quantum magnetic phases on the frustrated Rydberg-atom array, we perform numerically exact quantum Monte Carlo simulations to investigate the exotic states of matter emerging in the model describing the Rydberg atom on a triangular lattice. Our state-of-the-art simulation unveils the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ triangular antiferromagnetic order exists at $1/3$ or $2/3$-Rydberg filling, consistent with the observation in experiments. Remarkably, $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ long-range order arising from order-by-disorder mechanism emerges at $1/2$ filling. At finite temperature, $\mathrm{U}(1)$ symmetry is emergent at $1/2$ filling and a Kosterlitz-Thouless phase transition occurs with increasing temperature. These intriguing phenomena are potentially detected in future Rydberg-atom experiments.