Topological aspects of solitons in ferromagnets

Two kinds of topological soliton (skymion and magnetic vortex ring) in ferromagnets are studied. They have the common topological origin, a tensor H alpha beta= (n) over right arrow center dot (partial derivative(alpha)(n) over right arrow x partial derivative(beta)(n) over right arrow), which describes the non-trivial distribution of local orientation of magnetization (n) over right arrow at large distances in space. The topological stability of skyrmion is protected by the winding number. Knot-like topological defect as magnetic vortex rings is also studied. On the assumption that magnetic vortex rings are geometric lines, we present their delta-function distribution in ferromagnetic materials. Furthermore, it is briefly shown that Hopf invariant is a proper topological invariant to describe the topology of magnetic vortex rings.