Two-dimensional skyrmions on the sphere

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 less than or similar to 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 greater than or similar to 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.