Curved thin-film limits of chiral Dirichlet energies

TL;DR

This paper studies the limiting behavior of perturbed Dirichlet energies on curved thin films, revealing a novel energy form relevant to magnetic skyrmions and antisymmetric exchange interactions.

Contribution

It establishes the $ ext{Gamma}$-convergence of perturbed Dirichlet energies on curved thin films to a new energy functional, linking geometry with magnetic interactions.

Findings

Convergence to a new energy functional on the submanifold N.

Manifestation of antisymmetric exchange interactions in the limit.

Revealing the influence of curvature on magnetic energy interactions.

Abstract

We investigate the curved thin-film limit of a family of perturbed Dirichlet energies in the space of $H^1$ Sobolev maps defined in a tubular neighborhood of an $(n - 1)$-dimensional submanifold $N$ of $\mathbb{R}^n$ and with values in an $(m - 1)$-dimensional submanifold $M$ of $\mathbb{R}^m$. The perturbation $\mathsf{K}$ that we consider is represented by a matrix-valued function defined on $M$ and with values in $\mathbb{R}^{m \times n}$. Under natural regularity hypotheses on $N$, $M$, and $\mathsf{K}$, we show that the family of these energies converges, in the sense of $\Gamma$-convergence, to an energy functional on $N$ of an unexpected form, which is of particular interest in the theory of magnetic skyrmions. As a byproduct of our results, we get that in the curved thin-film limit, antisymmetric exchange interactions also manifest under an anisotropic term whose specific shape depends both on the curvature of the thin film and the curvature of the target manifold. Various types of antisymmetric exchange interactions in the variational theory of micromagnetism are a source of inspiration and motivation for our work.