Hexagonal structure of baby Skyrmion lattices

We study the zero-temperature crystalline structure of baby Skyrmions by applying a full-field numerical minimization algorithm to baby Skyrmions placed inside different parallelogramic unit cells and imposing periodic boundary conditions. We find that within this setup, the minimal energy is obtained for the hexagonal lattice, and that in the resulting configuration the Skyrmion splits into quarter Skyrmions. In particular, we find that the energy in the hexagonal case is lower than the one obtained on the well-studied rectangular lattice, in which splitting into half Skyrmions is observed.

Figure 1

The parameterization of the fundamental unit-cell parallelogram $\Lambda$ (in black) and the two-torus ${\mathbb{T}}_2$ into which it is mapped (in gray).

Figure 2

Charge-two Skyrmions in the pure $O(3)$ case: Contour plots of the charge densities ranging from low density (violet) to high density (red) for various parallelogram settings, all of which saturate the energy bound $E=4\pi B=8\pi$.

Figure 3

Charge-two Skyrmions in the Skyrme case with ${\kappa }^2=0.03$ and $\rho =2$: Contour plots of the charge densities for the hexagonal, square, and other fundamental cells ranging from low density (violet) to high density (red). As the captions of the individual subfigures indicate, the hexagonal configuration has the lowest energy.

Figure 4

Total energy $E$ (divided by $8\pi$) of the charge-two Skyrmion in the hexagonal lattice ($\square$—Skyrme case and $◊$—general case) and in the square lattice ($\blacksquare$—Skyrme case and $⧫$—general case) as function of the Skyrmion density (in the Skyrme case, ${\kappa }^2=0.03$ and in the general case ${\kappa }^2=0.03$ and ${\mu }^2=0.1$). Note the existence of an optimal density (at $\rho \approx 0.14$) in the general case, for which the energy attains a global minimum. Figure (b) is an enlargement of the lower left corner of figure (a).

Figure 5

Energy difference (in %) between the hexagonal setting and the square setting, in the Skyrme case ($\square$) and in the general case ($⧫$), as function of the Skyrmion density. As the density decreases, the energy difference between the two lattice types is reduced.

Figure 6

Charge-two Skyrmions in the general case with ${\mu }^2=0.1$ and ${\kappa }^2=0.03$: Contour plots of the charge densities of the minimal energy configurations in the hexagonal setting for different densities. Here, the energetically most favorable density is $\rho =0.14$. The plot colors range from low density (violet) to high density (red).