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

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