Two types of metastable hopfions in bulk magnets

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