Towards an effective theory of skyrmion crystals

We consider multiskyrmion configurations in $2\mathrm{D}$ ferromagnets with Dzyaloshinskii–Moriya interaction and the magnetic field, using the stereographic projection method. In the absence of Dzyaloshinskii–Moriya interaction, $D$, and the field, $B$, the skyrmions do not interact and the exact multiskyrmion solution is a sum of individual projections. In certain range of $B, D\ne 0$, skyrmions become stable and form a hexagonal lattice. The shape of one skyrmion on the plane is fully determined by $D$ and $B$. We describe multiskyrmion configurations by simple sums of individual skyrmion projections, of the same shape and adjusted scale. This procedure reveals pairwise and triple interactions between skyrmions, and the energy of proposed hexagonal structure is found in a good agreement with previous studies. It allows an effective theory of skyrmion structures in terms of variables, referring to individual skyrmions, i.e., their position, size and phase, elliptic distortions, etc.