Anomalous Hall effect and topological defects in antiferromagnetic Weyl semimetals: Mn${}_3$Sn/Ge

We theoretically study the interplay between bulk Weyl electrons and magnetic topological defects, including magnetic domains, domain walls, and ${\mathbb{Z}}_6$ vortex lines, in the antiferromagnetic Weyl semimetals \mnsn
and Mn${}_3$Ge with negative vector chirality. We argue that these materials possess a hierarchy of energies scales which allows a description of the spin structure and spin dynamics using a XY model with ${\mathbb{Z}}_6$ anisotropy. We propose a dynamical equation of motion for the XY order parameter, which implies the presence of ${\mathbb{Z}}_6$ vortex lines, the double-domain pattern in the presence of magnetic fields, and the ability to control domains with current. We also introduce a minimal electronic model which allows efficient calculation of the electronic structure in the antiferromagnetic configuration, unveiling Fermi arcs at domain walls, and sharp quasi-bound states at ${\mathbb{Z}}_6$ vortices. Moreover, we have shown how these materials may allow electronic-based imaging of antiferromagnetic microstructure, and propose a possible device based on domain-dependent anomalous Hall effect.