Magnetic Skyrmions: from lumps to supercompactons

The magnetic Skyrmion is described by one control parameter and one length scale. We study the two extreme limits of the control parameter - infinitely large and vanishing - and find that the magnetic Skyrmion becomes a "restricted" magnetic Skyrmion and an O(3) sigma model lump, respectively. Depending on the potential under consideration, the restricted limit manifests differently. In the case of the Zeeman term, the restricted magnetic Skyrmion becomes a "supercompacton" that develops a discontinuity, whereas for the Zeeman term to the power 3/2 it becomes a normal compacton. In both the lump and the restricted limit the solution is given in exact explicit form. We observe that the case of the Zeeman term squared, which can also be understood as a special combination of the Zeeman term and the easy-plane potential - realizable in the laboratory, the analytically exact solution for all values of the coupling - including the Bogomol'nyi-Prasad-Sommerfield (BPS) case - is also of the lump type. Finally, we notice that certain materials (e.g. Fe$_{1-x}$Co$_x$Si or Mn$_{1-x}$Fe$_x$Ge) have a rather large control parameter $\epsilon$ of order 100, making the restricted limit a suitable rough approximation.