[Many papers][1] said that AHE and SHE share the same mechanism. A simple description of this theory is that the charge carriers have different spin directions can have opposite transverse movements by some mechanism. Therefore, one can get SHE. In general, this happens to non-ferromagnetic materials and there is no applied magnetic field in SHE. Although there are charge carriers with opposite spin at two sides of the sample, the amount of these two kinds of carriers are equal. That's why one can not get a voltage difference between two sides of sample.

  However, if the material is ferro-magnetic, then the amount of spin-up and spin-down are usually not equal, no matter the applied field is zero or not. In this case, if a SHE happens, an AHE also happens. Since it is much easier to measure charge instead of spin, we focus on its AHE and SHE is ignored. Also, personal opinion, people tend to connect SHE with no magnetic field situation(also non-magnetic materials). As for AHE, it is related to magnetic field and magnetic materials. Even when the applied field is zero, ferro-magnetic materials have an inner field. So maybe that's why the misunderstanding comes from that AHE and SHE are exclusive. 

  Finally, back to the paper you mentioned. It talks about a special situation, which is anti-ferromagnetism. Those antiferromagnetic materials have special spin textures. To be simple, you can just imagine that the amount of spin-up and spin-down are equal. So, when there is no applied field, a normal SHE happens. When there is an applied field, spin-up are more than spin-down, so you get AHE. BTW, actually this Mn3X has a complex spin texture, not just spin-up and spin-down, but it is easier for you to understand.

  [1]: https://doi.org/10.1103/RevModPhys.87.1213