Asymptotic Interactions of Critically Coupled Vortices

TL;DR

This paper derives the asymptotic metric for multiple vortices in the Ginzburg-Landau model at critical coupling, linking geometric, physical, and scattering properties of vortices.

Contribution

It provides a new explicit asymptotic form of the moduli space metric for multiple vortices and connects it to a physical particle model.

Findings

Asymptotic metric expressed via Bessel functions

Extension to N vortices with explicit formulas

Analysis of vortex scattering and curvature properties

Abstract

At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. The asymptotic form of the metric for two well separated vortices is shown here to be expressible in terms of a Bessel function. A straightforward extension gives the metric for N vortices. The asymptotic metric is also shown to follow from a physical model, where each vortex is treated as a point-like particle carrying a scalar charge and a magnetic dipole moment of the same magnitude. The geodesic motion of two well separated vortices is investigated, and the asymptotic dependence of the scattering angle on the impact parameter is determined. Formulae for the asymptotic Ricci and scalar curvatures of the N-vortex moduli space are also obtained.