Skyrmions, Fullerenes and Rational Maps

TL;DR

This paper compares two methods for calculating Skyrmions with baryon numbers less than 23, finding that most solutions resemble fullerene structures and that different approaches yield consistent results.

Contribution

It introduces a combined use of rational map ansatz and full nonlinear simulations to analyze Skyrmions, highlighting structural similarities and energy approximations.

Findings

Most Skyrmions for B>6 are fullerene-like structures.

Good agreement between rational map and full nonlinear solutions.

Multiple Skyrmion configurations with similar energies exist for the same B.

Abstract

We apply two very different approaches to calculate Skyrmions with baryon number B less than 23. The first employs the rational map ansatz, where approximate charge B Skyrmions are constructed from a degree B rational map between Riemann spheres. We use a simulated annealing algorithm to search for the minimal energy rational map of a given degree B. The second involves the numerical solution of the full non-linear time dependent equations of motion, with initial conditions consisting of a number of well separated Skyrmion clusters. In general, we find a good agreement between the two approaches. For B greater than 6 almost all the solutions are of fullerene type, that is, the baryon density isosurface consists of twelve pentagons and 2B-14 hexagons arranged in a trivalent polyhedron. There are exceptional cases where this structure is modified, which we discuss in detail. We find that for a given value of B there are often many Skyrmions, with different symmetries, whose energies are very close to the minimal value, some of which we discuss. We present rational maps which are good approximations to these Skyrmions and accurately compute their energy by relaxation using the full non-linear dynamics.