Exact solution for the energy-minimizing magnetization field with positive Hopf index in a helimagnet

Derive an exact analytical form of the magnetization vector field m(r), with fixed length |m| = 1 and positive Hopf index, that minimizes the micromagnetic energy per unit volume for a classical helimagnet whose energy density includes exchange, Dzyaloshinskii–Moriya, Zeeman, and uniaxial anisotropy terms, with the external magnetic field H and anisotropy axis s parallel to the OZ axis.

Background

The paper formulates the minimization problem for the micromagnetic energy of a classical helimagnet with exchange and Dzyaloshinskii–Moriya interactions, Zeeman coupling to an external field, and uniaxial anisotropy, under the constraint that the magnetization field has unit magnitude and a positive Hopf index. Because an exact analytic solution is unavailable, the authors employ a Ritz variational approach based on the Whitehead ansatz to construct trial fields for hopfions.

Establishing an exact solution would provide a rigorous baseline for properties of bulk hopfions and their lattices, including critical fields and size scaling, and could validate or refine the variational and numerical findings presented here.