We study minimizers $\boldsymbol{ m}\colon \mathbb R^2\to\mathbb S^2$ of the energy functional \begin{ align*} E_\sigma(\boldsymbol{ m}) = \int_{ \mathbb R^2} \bigg(\frac 12 |\nabla\boldsymbol{ m}|^2 +\sigma^2 \boldsymbol{ m} \cdot \nabla \times\boldsymbol{ m} +\sigma^2 m_3^2 \bigg)\, dx\,, \end{ align*} for $0<\sigma\ll 1$, with prescribed topological degree \begin{ align*} Q(\boldsymbol{ m})=\frac{ 1}{ 4\pi} \int_{ \mathbb R^2}\boldsymbol{ m} \cdot \partial_1 \boldsymbol{ m}\times\partial_2\boldsymbol{ m}\, dx =\pm 1\,. \end{ align*} This model arises in thin ferromagnetic films with Dzyaloshinskii-Moriya interaction and easy-plane anisotropy, where these minimizers represent bimeron configurations. We prove their existence, and describe them precisely as perturbations of specific M\"obius maps: we establish in particular that they are localized at scale of order $1/|\ln(\sigma^2)|$. The proof follows a strategy introduced by Bernand-Mantel, Muratov and Simon (Arch. Ration. Mech. Anal., 2021) for a similar model with easy-axis anisotropy, but requires several adaptations to deal with the less coercive easy-plane anisotropy and different symmetry properties.
Categories: math.AP