Abstract
The topological Hall effect (THE) of electrons in skyrmion crystals (SkXs) is strongly related to the quantum Hall effect (QHE) on lattices. This relation suggests to revisit the QHE because its Hall conductivity can be unconventionally quantized. It exhibits a jump and changes sign abruptly if the Fermi level crosses a van Hove singularity. In this Paper, we investigate the unconventional QHE features by discussing band structures, Hall conductivities, and topological edge states for square and triangular lattices; their origin are Chern numbers of bands in the SkX (THE) or of the corresponding Landau levels (QHE). Striking features in the energy dependence of the Hall conductivities are traced back to the band structure without magnetic field whose properties are dictated by the lattice geometry. Based on these findings, we derive an approximation that allows us to determine the energy dependence of the topological Hall conductivity on any two-dimensional lattice. The validity of this approximation is proven for the honeycomb lattice. We conclude that SkXs lend themselves for experiments to validate our findings for the THE and—indirectly—the QHE.
Highlights
With the recent ascent of skyrmions [1,2,3,4,5]—particle-like topologically nontrivial field configurations [6]—to one of the most auspicious research areas in physics, the transport of electrons in a Hall geometry may become of great interest again
In what follows we investigate and explain how lattice effects manifest themselves in the quantum Hall effect (QHE)
After revisiting the topological Hall effect (THE) on a triangular lattice (Section III C) we formulate the approximation for the Hall conductivity of both THE and QHE and check its validity for the honeycomb lattice (Section III D)
Summary
With the recent ascent of skyrmions [1,2,3,4,5]—particle-like topologically nontrivial field configurations [6]—to one of the most auspicious research areas in physics, the transport of electrons in a Hall geometry may become of great interest again. The topological Hall effect (THE) [14,15,16,17,18,19,20,21,22] of electrons in skyrmion crystals—regular arrays of skyrmions—arises from the real-space Berry curvature of the spin texture which produces an emergent magnetic field proportional to nSk(r). We propose a handy approximation for the energy-dependent Hall conductivity that circumvents calculations of the Berry curvature; its validity is checked for the QHE and the THE on a honeycomb lattice.
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