${\mathbb{Z}}_2$ Vortex Crystals in Tetrahedral Antiferromagnets: Fractional Charges and Topological Magnons

We report the formation of a ${\mathbb{Z}}_2$ vortex crystal in the tetrahedral antiferromagnetic order on a triangular lattice. The noncoplanar tetrahedral state consists of four sublattices with spins oriented along the faces of a tetrahedron in spin space. The long-range order characterized by a ${\mathbb{Z}}_2$ topology arises due to the Dzyaloshinskii-Moriya interaction and appears at zero temperature and without external fields. Each vortex carries a half-integer electric charge relative to the uniform background in itinerant magnets, enabling the emergence of anyonic excitations. Its magnetic excitations include magnetically active gyrotropic and breathing modes, which – under an external magnetic field – carry nontrivial Chern numbers that stabilize chiral magnon edge states.