Easy plane baby Skyrmions

The baby Skyrme model is studied with a novel choice of potential, $V=\frac{1}{2}{\varphi }_3^2$. 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 $B=1,\dots ,14,18,32$ 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 $B\ge 7$, rectangular or distorted rectangular arrays of $2B$ half lumps are preferred, as close to square as possible.

Figure 1

The minimizer for degree $B=1$.

Figure 2

Plots of ${\varphi }_2$ and ${\varphi }_3$ for $r\in \{2,3,4,5,7.5,10\}$ (the dashed lines, colored blue, green, red, cyan, magenta, and yellow, respectively) and the corresponding best fits (solid, black).

Figure 3

Predicted behavior of two-Skyrmion superpositions. The disks represent half lumps with ${\varphi }_3>0$ (red) and ${\varphi }_3<0$ (blue), while the black arrows represent the scalar dipole moment associated with each Skyrmion.

Figure 4

Sample minimizers for higher degrees. All plots are contour plots of ${\varphi }_3$.

Figure 5

Normalized energy per $4\pi B$ versus $B$ of those configurations which we believe are global minima (crosses), the polygon-shaped solitons (circles), and a least squares fit of the form $a+b{e}^{cB}$ (dashed curve).