Winding Angle Distributions for Random Walks and Flux Lines

TL;DR

This paper analyzes how flux lines wind around defects and impurities, revealing exponential decay in winding angles with boundary-condition-dependent constants, and introduces new distributions for chiral defects with implications for entanglement.

Contribution

It provides the first analytical and numerical characterization of winding angle distributions for flux lines around various defect types, including chiral defects and impurities.

Findings

Winding angle distributions decay exponentially with boundary-condition-dependent constants.

Chiral defects produce asymmetric, broad distributions influenced by chirality.

Impurities tend to narrow winding distributions, reducing entanglements.

Abstract

We study analytically and numerically the winding of a flux line around a columnar defect. Reflecting and absorbing boundary conditions apply to marginal or repulsive defects, respectively. In both cases, the winding angle distribution decays exponentially for large angles, with a decay constant depending only on the boundary condition, but not on microscopic features. New {\it non-universal} distributions are encountered for {\it chiral} defects which preferentially twist the flux line in one direction. The resulting asymmetric distributions have decay constants that depend on the degree of chirality. In particular, strong chirality encourages entanglements and leads to broad distributions. We also examine the windings of flux lines in the presence of point impurities (random bonds). Our results suggest that pinning to impurities reduces entanglements, leading to a narrow (Gaussian) distribution.