A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree $\pm1$ harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$

TL;DR

This paper characterizes skyrmions in ultrathin ferromagnetic films as energy minimizers and links their profiles to harmonic maps, providing existence results and a rigidity theorem for degree ±1 harmonic maps.

Contribution

It introduces a new variational framework for skyrmions in 2D ferromagnetic films and establishes a quantitative rigidity result for harmonic maps of degree ±1.

Findings

Existence of skyrmion minimizers under certain parameters.

Asymptotic profiles of skyrmions as harmonic maps.

Rigidity theorem for degree ±1 harmonic maps.

Abstract

We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree $\pm 1$ harmonic maps from the two-dimensional sphere to itself.