$\mathrm{\Gamma }$-limit for two-dimensional charged magnetic zigzag domain walls

Charged domain walls are a type of domain walls in thin ferromagnetic films which appear due to global topological constraints. The non-dimensionalized micromagnetic energy for a uniaxial thin ferromagnetic film with in-plane magnetization $m\in {\mathbb{S}}^1$ is given by \begin{align*} E_\epsilon[m] \ =
\epsilon|\nabla m|{L^2}^2 + \frac {1}{\epsilon} |m \cdot e_2|{L^2}^2

• \frac{\pi\lambda}{2|\ln\epsilon|} |\nabla \cdot (m-M)|_{\dot H^{-\frac{1}{2}}}^2, \end{align*} where magnetization in $e_1$-direction is globally preferred and where $M$ is an arbitrary fixed background field to ensure global neutrality of magnetic charges. We consider a material in the form a thin strip and enforce a charged domain wall by suitable boundary conditions on $m$. In the limit $ϵ\to 0$ and for fixed $\lambda >0$, corresponding to the macroscopic limit, we show that the energy $\mathrm{\Gamma }$-converges to a limit energy where jump discontinuities of the magnetization are penalized anisotropically. In particular, in the subcritical regime $\lambda \le 1$ one-dimensional charged domain walls are favorable, in the supercritical regime $\lambda >1$ the limit model allows for zigzaging two-dimensional domain walls.