# Unconventional topological Hall effect in high-topological-number skyrmion crystals

## TL;DR

This paper theoretically investigates the topological Hall effect in high-topological-number skyrmion crystals, revealing quantized Hall conductivity with a fractional relationship to the skyrmion number, differing from conventional skyrmions.

## Contribution

It introduces the study of topological Hall effects in high-$Q$ skyrmion crystals with $Q=2$ and $Q=3$, showing a fractional quantization pattern in Hall conductivity.

## Key findings

- Hall quantization number increases by 1/Q with Fermi energy crossing bands
- Quantized Hall conductivity is linked to emergent magnetic fields and band structures
- Distinct behavior from conventional skyrmions with Q=1

## Abstract

Skyrmions with the topological number $Q$ equal an integer larger than 1 are called high-topological-number skyrmions or high-$Q$ skyrmions. In this work, we theoretically study the topological Hall effect in square-lattice high-$Q$ skyrmion crystals (SkX) with $Q=2$ and $Q=3$. As a result of the emergent magnetic field, Landau-level-like electronic band structure gives rise to quantized Hall conductivity when the Fermi energy is within the gaps between adjacent single band or multiple bands intertwined. We found that different from conventional ($Q=1$) SkX the Hall quantization number increases by $1/Q$ in average when the elevating Fermi energy crosses each band. We attribute the result to the fact that the Berry phase ${\cal{C}}$ is measured in the momentum space and the topological number of a single skyrmion $Q$ is measured in the real space. The reciprocality does not affect the conventional SkX because $Q=1=1/Q$.