Hybrid skyrmions in magnetic multilayer thin films are half-integer hopfions

The magnetic skyrmions found in multilayer thin films have a three-dimensional hybrid structure with depth-dependent helicity. The authors develop here a theoretical framework for these hybrid skyrmions based on the Hopf fibration, which captures the knotted structure of their field lines. They show that hybrid skyrmions can be considered half-integer Hopfions, and use their framework to relate the topological charge of partially twisted hybrid skyrmions to their surface domain wall helicity, providing a connection between experimental measurements and underlying topology.