Reduced theory of symmetric and antisymmetric exchange interactions in nanowires

We investigate the behavior of minimizers of perturbed Dirichlet energies supported on a wire generated by a regular simple curve γ and defined in the space of S²-valued functions. The perturbation K is represented by a matrix-valued function defined on S² with values in R3X3. Under natural regularity conditions on K, we show that the family of perturbed Dirichlet energies converges, in the sense of Γ-convergence, to a simplified energy functional on γ. The reduced energy unveils how part of the antisymmetric exchange interactions contribute to an anisotropic term whose specific shape depends on the curvature of γ. We also discuss the significant implications of our results for studies of ferromagnetic nanowires when Dzyaloshinskii-Moriya interaction (DMI) is present.