  Long-range magnetic textures, such as magnetic skyrmion, give rise to rich
transport properties in magnetic metals, such as the anomalous Hall effect
related to spin chirality, a.k.a. topological Hall effect. In addition to the
topological Hall effect, recent studies on non-centrosymmetric magnets find
that the spin-orbit interaction of itinerant electrons gives rise to novel
contributions related to spin chirality, i.e., the chiral Hall effect. In this
work, we discuss that the spin-orbit interaction has a distinct, yet
significant, effect on the anomalous Hall effect related to spin chirality in
centrosymmetric magnets. Using a scattering theory method, we find that the
anomalous Hall effect related to scalar spin chirality in a two-dimensional
Luttinger model is suppressed by more than one order of magnitude compared to
the quadratic dispersion, and the contributions similar to the chiral Hall
effect in Rashba model vanishes. At the same time, a novel term that gives
different Hall conductivity for the Bloch and Neel skyrmions occurs, thereby
enabling the detection of the skyrmion helicity. The striking differences
demonstrate the rich effect of crystal symmetry on the chirality-related
anomalous Hall effect in materials with strong spin-orbit interaction.
