Rational map approximation for Skyrmions

The rational map approximation writes the angular dependence of a Skyrmion using a rational map $R:S^2\to S^2$ and determines a radial profile variationally. The degree of $R$ equals the baryon number, while zeros of its Wronskian mark directions of vanishing angular baryon density.

This degree-four rational map obeys $R(iz)=1/R(z)$ and $R((iz+1)/(1-iz))=e^{2\pi i/3}R(z)$. These pair the generators of the rotational symmetry group of a cube with target rotations, giving combined spatial-isospin symmetry in the Skyrme model. Its spatial half-turn kernel is a Klein four-group. The map gives a cubic four-Skyrmion approximation; preserving only its Wronskian zero set does not guarantee the same rotational symmetry.

For a degree-$N$ rational map, the angular Jacobian of a rational map has average $N$, so the Cauchy-Schwarz inequality gives $\mathcal{I}\ge N^2$. In dimensionless Skyrme model units the radial energy within the rational-map ansatz isFirst minimize $\mathcal{I}$ over admissible maps, then minimize over profiles with $f(0)=\pi$, $f(\infty )=0$. This gives a restricted variational approximation and an upper bound on the unrestricted minimum in that topological sector, not an exact multi-Skyrmion solution. Houghton, Manton and Sutcliffe developed this construction in arxiv.org/abs/hep-th/9705151.