Easy plane baby skyrmions

TL;DR

This paper investigates baby Skyrme models with an 'easy plane' potential, revealing discrete symmetries in skyrmions, developing a long-range interaction model, and numerically finding minimal energy configurations for various charges.

Contribution

It introduces a novel potential in the baby Skyrme model, analyzes the resulting discrete symmetries, and develops a scalar dipole interaction model validated by numerical simulations.

Findings

Skyrmions have only discrete symmetry with the new potential.

A scalar dipole model accurately predicts two-skyrmion interactions.

Numerical minimizers form regular polygons or rectangular arrays depending on charge.

Abstract

The baby Skyrme model is studied with a novel choice of potential, $V=1/2 \phi_3^2$. This "easy plane" potential vanishes at the equator of the target two-sphere. Hence, in contrast to previously studied cases, the boundary value of the field breaks the residual SO(2) internal symmetry of the model. Consequently, even the unit charge skyrmion has only discrete symmetry and consists of a bound state of two half lumps. A model of long-range inter-skyrmion forces is developed wherein a unit skyrmion is pictured as a single scalar dipole inducing a massless scalar field tangential to the vacuum manifold. This model has the interesting feature that the two-skyrmion interaction energy depends only on the average orientation of the dipoles relative to the line joining them. Its qualitative predictions are confirmed by numerical simulations. Global energy minimizers of charges B=1,...,14,18,32 are found numerically. Up to charge B=6, the minimizers have 2B half lumps positioned at the vertices of a regular 2B-gon. For charges B >= 7, rectangular or distorted rectangular arrays of 2B half lumps are preferred, as close to square as possible.