Anomalous Hall effect in metallic collinear antiferromagnets

Abstract: In this paper we propose and theoretically study minimal models of all N\'{e}el ordered collinear antiferromagnets which show the anomalous Hall effect. For simplicity we consider two-dimensional models of antiferromagnets with two magnetic sublattices on a square lattice. Examples of N\'{e}el ordered ferrimagnet and of Dzyaloshinskii's weak ferromagnet are given. Turov's invariants of existence of the magnetization in N\'{e}el ordered antiferromagnets for proposed systems are analyzed. As a result, we obtain that the anomalous Hall effect is allowed only for certain directions of the N\'{e}el order defined by the crystal lattice symmetries. We derive the mechanism of the anomalous Hall effect in antiferromagnets to be the interplay of momentum-dependent exchange interaction of conducting fermions with the N\'{e}el order, and the spin-orbit coupling consistent with the broken symmetries that allowed for the Turov's invariant in the system.