Asymptotic Interactions of Critically Coupled Vortices
N. S. Manton, J. M. Speight
TL;DR
This work addresses the asymptotic interactions of vortices at critical coupling in the Ginzburg-Landau framework by deriving the slow, geodesic dynamics on the vortex moduli space. It develops two complementary methods to obtain the asymptotic metric: a field-theoretic approach via Samols' formula and a physical point-source model in which vortices are composite scalar monopoles with magnetic dipoles, both yielding a metric dominated by a Bessel function $K_0$. The explicit results include the asymptotic two-vortex metric $g = 2\pi dZ d\bar{Z} + 2\pi \left(1 - \frac{q^2}{\pi^2} K_0(2\sigma)\right) (d\sigma^2 + \sigma^2 d\theta^2)$ and its $N$-vortex generalization, together with a Kähler potential and asymptotic Ricci and scalar curvatures. The paper further analyzes two-vortex scattering within this metric, obtaining an analytic leading-order scattering angle $\Theta \approx \frac{q^2}{2\pi} e^{-2a}$ for large impact parameter, and demonstrates consistency with numerical simulations, thereby linking soliton dynamics to monopole-inspired moduli-space geometry.
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.