Two dimensional skyrmions on the sphere

We study a model for two-dimensional skyrmions on a sphere of radius L. Suchmodel simulates a skyrmion lattice of density W/(2 \pi L^2), where W is theskyrmion winding number. We show that, to a very good approximation, physicalresults depend only on the product \alpha L^4, where \alpha is the strength ofpotential term. In the range \alpha L^4 approx. or less than 3 the orderparameter vanishes, there is a uniform distribution of the density over thewhole surface and the energy of the W=2 sector lies above twice the energy ofthe W=1 sector. If \alpha L^4 approx. or greater than 6 the order parameterapproaches unity and the density concentrates near one of the poles. Moreoverthe disoliton is always bound. We also present a variational solution to thefield equations for which the pure \alpha L^4-dependence is exact. Finally,some consequences of our results for the Quantum Hall Effect are discussed.Comment: 9 pages (cover and figs. included), Latex, 2 EPS-figs. include