The family of topological Hall effects for electrons in skyrmion crystals

TL;DR

This paper explores the various topological Hall effects in skyrmion crystals, establishing analogies with conventional Hall, anomalous Hall, and spin Hall effects, and analyzing their conditions and quantization through theoretical calculations.

Contribution

It introduces a unified framework describing the topological Hall effect and its analogies to other Hall effects in skyrmion crystals, including conditions for quantization.

Findings

The topological Hall effect can be mapped onto the conventional Hall effect via an emergent magnetic field.

The transverse charge and spin conductivities are linked or disconnected depending on electron spin coupling.

Quantized topological Hall effects can be identified through edge state calculations.

Abstract

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.