Baby skyrmions on the sphere

TL;DR

This paper investigates two-dimensional skyrmions on a spherical surface, revealing how their properties depend mainly on the product of potential strength and sphere size, with implications for the Quantum Hall Effect.

Contribution

It introduces a model for skyrmions on a sphere, showing the dependence of physical results on L^4 and providing a variational solution to the field equations.

Findings

Order parameter vanishes for L^4  3, leading to uniform density.

For L^4  6, the order parameter approaches unity and density concentrates at poles.

The disoliton remains bound across parameter ranges.

Abstract

We study a model for two-dimensional skyrmions on a sphere of radius L. Such model simulates a skyrmion lattice of density W/(2 \pi L^2), where W is the skyrmion winding number. We show that, to a very good approximation, physical results depend only on the product \alpha L^4, where \alpha is the strength of potential term. In the range \alpha L^4 approx. or less than 3 the order parameter vanishes, there is a uniform distribution of the density over the whole surface and the energy of the W=2 sector lies above twice the energy of the W=1 sector. If \alpha L^4 approx. or greater than 6 the order parameter approaches unity and the density concentrates near one of the poles. Moreover the disoliton is always bound. We also present a variational solution to the field equations for which the pure \alpha L^4-dependence is exact. Finally, some consequences of our results for the Quantum Hall Effect are discussed.