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

Abstract

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 $p_{ij}=\max (0,1-e^{-2K{s}_i^x{s}_j^x})$, where $K>0$ governs the percolation process. A line of percolation thresholds $K_{\mathrm{c}}(J)$ is found in the low-temperature range $J\ge J_{\mathrm{c}}$, 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 $K\ge K_{\mathrm{c}}(J)$, and the associated critical exponent depends on $J$ and $K$. Along the threshold line $K_{\mathrm{c}}(J)$, 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 $K_{\mathrm{c}}(J)$ line is of the Berezinskii-Kosterlitz-Thouless type.

23 pages, 14 figures