Magnetic Hopfions as Local Rotations of the Uniform Magnetization Background

Topologically nontrivial 3D magnetization textures with the nonzero Hopf index are considered. The calculation approach is based on the theory of toroidal hopfions developed within the classical field theory for infinite media. The Hopf mapping of 3D coordinate space R³(r) to the unit sphere m² = 1 in the magnetization space m(r) can be parametrized by some angles. It is shown that these angles can be used to define a local rotation SO(3) matrix to restore the hopfion magnetization configuration from the uniformly magnetized state for any values of the Hopf indices. The rotation angles are calculated explicitly using the toroidal coordinates for the 3D radius vector r. The role of the hopfion field vorticities in the formation of the hopfion magnetization patterns is considered. It is shown that the azimuthal vorticity plays the main role, whereas the poloidal vorticity is of the second importance. Anti‐hopfions are introduced for the negative hopfion field vorticities.