Elliptical stability of hopfions in bulk helimagnets

Magnetic hopfions are three-dimensional topological solitons with non-zero Hopf index ${\cal H}$ in the vector field of material's local magnetization. In this Letter elliptical stability of hopfions with ${\cal H}=1$ in a classical helimagnet is studied on the basis of a variational model. It is shown that, depending on their internal structure (vortex and antivortex tubes ordering), the hopfions can either be stable in a bulk magnet or unstable with respect to elongation along their central axis. It is found that the energy of stable hopfions is always below the energy of the $2\pi$-skyrmion lattice in the same material, suggesting the possibility to use $2\pi$-skyrmions as a precursor for hopfion nucleation. Stability diagram for hopfions on the magnetic anisotropy-field phase diagram is computed numerically. Explicit analytical expressions for some of its critical lines are derived.