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

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 $Q=1=1/Q$.