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

TL;DR

This study uses a worm algorithm to simulate the 2D XY model in flow representation, revealing a percolation transition that occurs before the BKT transition, with implications for understanding phase transitions in related systems.

Contribution

First to analyze the geometric percolation transition in the flow representation of the 2D XY model using large-scale simulations.

Findings

Percolation transition occurs at $K_{perc} = 1.1053(4)$

Percolation threshold is below the BKT transition point

Finite-size scaling describes the percolation transition as second-order

Abstract

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 $K_{\rm perc}$ 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 qusi-long-range order associated with spin properties. Near $K_{\rm perc}$, 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 $K_{\rm perc}=1.105 \, 3(4)$ is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point $K_{\rm BKT} = 1.119 \, 3(10)$, which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed lights on a variety of classical and quantum systems of BKT phase transitions.