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 $\gamma$ and defined in the space of ${\mathbb{S}}^2$-valued functions. The perturbation $K$ is represented by a matrix-valued function defined on ${\mathbb{S}}^2$ with values in ${\mathbb{R}}^{3\times 3}$. Under natural regularity conditions on $K$, we show that the family of perturbed Dirichlet energies converges, in the sense of $\mathrm{\Gamma }$-convergence, to a simplified energy functional on $\gamma$. The reduced energy unveils how part of the antisymmetric exchange interactions contribute to an anisotropic term whose specific shape depends on the curvature of $\gamma$. We also discuss the significant implications of our results for studies of ferromagnetic nanowires when Dzyaloshinskii-Moriya interaction (DMI) is present.