Elliptical stability of hopfions in bulk helimagnets

TL;DR

This paper investigates the elliptical stability of magnetic hopfions in bulk helimagnets, revealing conditions under which they are stable or unstable, and compares their energy to skyrmion lattices, with implications for their nucleation.

Contribution

It provides a variational model analysis of hopfion stability in bulk helimagnets, including stability diagrams and analytical expressions for critical parameters.

Findings

Stable hopfions have energy below that of 2π-skyrmion lattices.

Hopfions can be stable or unstable depending on their internal vortex structure.

A stability diagram on the anisotropy-field phase space is constructed.

Abstract

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.