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

References in corpus (12)

• Efficient Monte Carlo algorithm and high-precision results for percolation
• Inverted Berezinskii-Kosterlitz-Thouless Singularity and High-Temperature Algebraic Order in an Ising Model on a Scale-Free Hierarchical-Lattice Small-World Network
• The two dimensional XY model at the transition temperature: A high precision Monte Carlo study
• Computing the Roughening Transition of Ising and Solid-On-Solid Models by BCSOS Model Matching
• Phase Transition with the Berezinskii--Kosterlitz--Thouless Singularity in the Ising Model on a Growing Network
• BKT-like transition in the Potts model on an inhomogeneous annealed network
• Complementary algorithms for graphs and percolation
• High-temperature expansion of the magnetic susceptibility and higher moments of the correlation function for the two-dimensional XY model
• The triangular Ising model with nearest- and next-nearest-neighbor couplings in a field
• The Atick-Witten free energy, closed tachyon condensation and deformed Poincare' symmetry
• Percolation and critical O($n$) loop configurations
• Conducting-angle-based percolation in the XY model

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