<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 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.