Metallic collinear antiferromagnets with mirror-symmetric and asymmetric spin splittings

In this paper we theoretically describe a distinct class of two-dimensional N\'eel ordered metallic antiferromagnets on a honeycomblike lattice in which the two magnetic sublattices are connected only by a combination of time-reversal and mirror symmetry operations. As a result of this symmetry, conducting fermions have antiferromagnetic spin splitting consistent with the symmetry, mirror-symmetric spin splitting. It is shown that the anomalous spin Hall effect is expected in such antiferromagnets. We also consider a N\'eel ordered metallic antiferromagnet in which there are no symmetries between the magnetic sublattices and obtain asymmetric spin splitting of conducting fermions. Such systems are ferrimagnets and show the anomalous Hall effect.