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

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

2010-10-15

arxiv.org/abs/1010.3075v1

PhysRevE.83.011124 (2011)

Physics

Condensed Matter

Statistical Mechanics

23 pages, 14 figures

Scientific paper

10.1103/PhysRevE.83.011124

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

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