Nonabelian Topological Hydrodynamics

The transport and dynamics of abelian topological charges (e.g. winding, vortices, skyrmions, hedgehogs and hopfions) which are classified according to an abelian homotopy group has been well-studied by the magnetic community, with many works discussing how they may be generated and controlled. Here, we discuss a transport theory for nonabelian topological charges, and investigate the potential to realize this exotic topological excitation in a 1-dimensional system. Using formalism inspired by braid theory, we introduce the braid density as a nonabelian charge rooted in the higher-order Vassiliev invariant. We find that the braid density exhibits striking differences from the abelian winding density, and propose a model in which a build-up of the braiding realizes a domain wall.