Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensional XY model

We study a percolation problem on a substrate formed by two-dimensional XY spin configurations using Monte Carlo methods. For a given spin configuration, we construct percolation clusters by randomly choosing a direction x in the spin vector space, and then placing a percolation bond between nearest-neighbor sites i and j with probability p(ij)=max(0,1-e(-2Ks(i)(x)s(j)(x))), where K>0 governs the percolation process. A line of percolation thresholds K(c)(J) is found in the low-temperature range J≥J(c), where J>0 is the XY coupling strength. Analysis of the correlation function g(p)(r), defined as the probability that two sites separated by a distance r belong to the same percolation cluster, yields algebraic decay for K≥K(c)(J), and the associated critical exponent depends on J and K. Along the threshold line K(c)(J), the scaling dimension for g(p) is, within numerical uncertainties, equal to 1/8. On this basis, we conjecture that the percolation transition along the K(c)(J) line is of the Berezinskii-Kosterlitz-Thouless type.