Non-existence of Skyrmion-Skyrmion and Skyrmion-anti-Skyrmion static equilibria

TL;DR

This paper proves that classical Skyrmions and anti-Skyrmions cannot form static equilibrium configurations due to their inherent attractive or repulsive interactions, regardless of symmetry assumptions.

Contribution

It provides a rigorous proof that static Skyrmion-Skyrmion and Skyrmion-anti-Skyrmion equilibria are impossible under broad symmetry conditions, extending previous numerical and analytical insights.

Findings

Skyrmions always repel Skyrmions

Skyrmions always attract anti-Skyrmions

No static equilibrium configurations exist

Abstract

We consider classical static Skyrmion-anti-Skyrmion and Skyrmion-Skyrmion configurations, symmetric with respect to a reflection plane, or symmetric up to a $G$-parity transformation respectively. We show that the stress tensor component completely normal to the reflection plane, and hence its integral over the plane, is negative definite or positive definite respectively. Classical Skyrmions always repel classical Skyrmions and classical Skyrmions always attract classical anti-Skyrmions and thus no static equilibrium, whether stable or unstable, is possible in either case. No other symmetry assumption is made and so our results also apply to multi-Skyrmion configurations. Our results are consistent with existing analyses of Skyrmion forces at large separation, and with numerical results on Skymion-anti-Skyrmion configurations in the literature which admit a different reflection symmetry. They also hold for the massive Skyrme model. We also point out that reflection symmetric self-gravitating Skyrmions or black holes with Skyrmion hair cannot rest in symmetric equilibrium with self-gravitating anti-Skyrmions.