The family of topological Hall effects for electrons in skyrmion crystals

Hall effects of electrons can be produced by an external magnetic field, spin–orbit coupling or a topologically non-trivial spin texture. The topological Hall effect (THE) – caused by the latter – is commonly observed in magnetic skyrmion crystals. Here, we show analogies of the THE to the conventional Hall effect (HE), the anomalous Hall effect (AHE), and the spin Hall effect (SHE). In the limit of strong coupling between conduction electron spins and the local magnetic texture the THE can be described by means of a fictitious, “emergent” magnetic field. In this sense the THE can be mapped onto the HE caused by an external magnetic field. Due to complete alignment of electron spin and magnetic texture, the transverse charge conductivity is linked to a transverse spin conductivity. They are disconnected for weak coupling of electron spin and magnetic texture; the THE is then related to the AHE. The topological equivalent to the SHE can be found in antiferromagnetic skyrmion crystals. We substantiate our claims by calculations of the edge states for a finite sample. These states reveal in which situation the topological analogue to a quantized HE, quantized AHE, and quantized SHE can be found.