Percolation of the two-dimensional XY model in the flow representation

We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L = 4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K-perc from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K-perc, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K-perc = 1.105 3(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K-BKT = 1.119 3(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.