Exact solution for the energy-minimizing hopfion configuration in a helimagnet

Derive the exact magnetization field m(r) that minimizes the free energy per unit volume E of a classical helimagnet with exchange stiffness C, Dzyaloshinskii–Moriya interaction parameter D, external magnetic field H, and uniaxial anisotropy constant K (with H and the anisotropy axis s parallel and aligned along the OZ axis), subject to the unit-magnetization constraint |m(r)| = 1 and the requirement that the configuration has a positive Hopf index.

Background

The paper formulates the micromagnetic energy functional for a classical helimagnet including exchange, Dzyaloshinskii–Moriya interaction, Zeeman, and uniaxial anisotropy terms, with the external field and anisotropy axis aligned. The authors seek a magnetization field m(r) of fixed length that minimizes this energy while possessing a positive Hopf index, corresponding to a bulk magnetic hopfion.

Because an exact analytical solution is unavailable, the authors adopt a variational (Ritz) approach using a trial function based on Whitehead’s ansatz and a generalized profile function that may depend on both radial and azimuthal coordinates. While this framework yields numerical approximations and insights (e.g., elliptical stability and lattice behavior), the exact solution to the underlying constrained minimization problem remains explicitly unresolved.