  The hallmark of a skyrmion crystal (SkX) is the topological Hall effect
(THE). In this Article, we predict and explain an unconventional behavior of
the topological Hall conductivity in SkXs. In simple terms, the spin texture of
the skyrmions causes an inhomogeneous emergent magnetic field whose associated
Lorentz force acts on the electrons. By making the emergent field homogeneous,
the THE is mapped onto the quantum Hall effect (QHE). Consequently, each
electronic band of the SkX is assigned to a Landau level. This correspondence
of THE and QHE allows to explain the unconventional behavior of the THE of
electrons in SkXs. For example, a skyrmion crystal on a triangular lattice
exhibits a quantized topological Hall conductivity with steps of $2 \cdot
e^2/h$ below and with steps of $1 \cdot e^2/h$ above the van Hove singularity.
On top of this, the conductivity shows a prominent sign change at the van Hove
singularity. These unconventional features are deeply connected to the topology
of the structural lattice.
