Two types of metastable magnetic hopfions

Abstract: Localized magnetic topological solitons with Hopf index of 1 in an unbounded bulk magnet are studied theoretically, starting with the classical micromagnetic Hamiltonian. It is shown analytically that (like Bloch and N\'eel walls in classical magnetism) there are two possible stable configurations of these solitons, which are equivalent topologically. Their stability regions are plotted on a phase diagram in the magnetic field -- uniaxial anisotropy space for different normalized magnetostatic interaction strengths. For some parameters both types of hopfions can be stable.