Stability of highly-twisted Skyrmions from contact topology

We describe a topological protection mechanism for highly-twisted two-dimensional Skyrmions in systems with Dyloshinskii-Moriya (DM) coupling, where non-zero DM energy density (dubbed “twisting energy density”) acts as a kind of band gap in real space, yielding an N invariant for highly twisted Skyrmions. We prove our result through the application of contact topology by extending our system along a fictitious third dimension, and further establish the structural stability of highly-twisted Skyrmions under arbitrary chirality-preserving distortions. Our results apply for all two-dimensional systems hosting Skyrmion excitations including spin-orbit coupled systems exhibiting quantum Hall ferromagnetism.