Symmetry and uniqueness of degree $-1$ minimizers of a chiral magnetic energy

Abstract

We consider the two-dimensional energy of a chiral magnet, consisting of the exchange energy, a bulk Dzyaloshinskii--Moriya interaction with constant $κ\ne0$, and the Zeeman energy in an applied field $h>0$. We prove that the minimizers of this energy among maps of degree $-1$ are exactly the translates of one axially symmetric Bloch skyrmion, whose helicity is determined by the sign of $κ$. This holds for $h \geq c κ^2$, where $c \in(π^2 / 16,1)$ is determined by the slope of the radial profile at the origin; in particular, it holds in the whole range $h\geκ^2$, which answers a question raised by Melcher and by Döring and Melcher. The proof is a calibration argument. Using the profile of the skyrmion, we write the energy density of an arbitrary map as a weighted topological density, plus a divergence, plus a nonnegative remainder, and we show that this identity can be integrated for every map of finite energy. The method also gives the uniqueness of the monotone solution of the profile equation for $h\geκ^2/2$.