# Signatures of lattice geometry in quantum and topological Hall effect

## TL;DR

This paper explores how lattice geometry influences the quantum and topological Hall effects, revealing unconventional quantization and sign changes linked to band structure features like van Hove singularities.

## Contribution

It introduces an approximation to determine topological Hall conductivity energy dependence on any 2D lattice, validated for the honeycomb lattice, connecting lattice geometry to Hall effect signatures.

## Key findings

- Hall conductivity exhibits abrupt jumps and sign changes at van Hove singularities.
- Band structure properties are dictated by lattice geometry, affecting Hall effects.
- An approximation method for topological Hall conductivity is validated for honeycomb lattices.

## Abstract

The topological Hall effect (THE) of electrons in skyrmion crystals 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 skyrmion crystal (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 twodimensional lattice. The validity of this approximation is proven for the honeycomb lattice. We conclude that skyrmion crystals lend themselves for experiments to validate our findings for the THE and - indirectly - the QHE.