Icosahedral Skyrmions

TL;DR

This paper investigates the specific baryon numbers where minimal energy Skyrmions exhibit icosahedral symmetry, proposing a sequence of 'magic' numbers supported by computational and constructive evidence.

Contribution

It introduces a conjecture on magic baryon numbers with icosahedral symmetry, supported by computational methods and new techniques for constructing symmetric rational maps.

Findings

Identification of a sequence of magic baryon numbers: 7, 17, 37, 67, 97.

Development of two new methods for computing symmetric rational maps.

Evidence from simulated annealing and explicit map constructions supports the conjecture.

Abstract

In this paper we aim to determine the baryon numbers at which the minimal energy Skyrmion has icosahedral symmetry. By comparing polyhedra which arise as minimal energy Skyrmions with the dual of polyhedra that minimize the energy of Coulomb charges on a sphere, we are led to conjecture a sequence of magic baryon numbers, B=7,17,37,67,97,... at which the minimal energy Skyrmion has icosahedral symmetry and unusually low energy. We present evidence for this conjecture by applying a simulated annealing algorithm to compute energy minimizing rational maps for all degrees upto 40. Further evidence is provided by the explicit construction of icosahedrally symmetric rational maps of degrees 37, 47, 67 and 97. To calculate these maps we introduce two new methods for computing rational maps with Platonic symmetries.