  The quantum Hall effect is generally understood for free electron gases, in
which topologically protected edge states between Landau levels (LLs) form
conducting channels at the edge of the sample. In periodic crystals, the LLs
are imprinted with lattice properties; plateaus in the transverse Hall
conductivity are not equidistant in energy anymore. Herein, crystals with a
polyatomic basis are considered. For a stripe arrangement of different atoms,
the band structure resorts nontrivially and exhibits strong oscillations that
form a salient pattern with very small bandgaps. The Hall conductivity strongly
decreases for energies within these bands, and only sharp peaks remain for
energies in the gap. These effects are traced back to open orbits in the
initial band structure; the corresponding LLs are formed from states with
positive and negative effective mass. The partial cancellation of transverse
charge conductivity also holds for different polyatomic stripe lattices and
even when the magnetic field is replaced by a topologically nontrivial spin
texture. The topological Hall effect is suppressed in the presence of magnetic
skyrmions. The discussion is complemented by calculations of Hofstadter
butterflies and orbital magnetization.
