Identifying two-dimensional $Z_2$ antiferromagnetic topological insulators

We revisit the question of whether a two-dimensional topological insulator may arise in a commensurate N\'eel antiferromagnet, where staggered magnetization breaks both the elementary translation and time reversal, but retains their product as a symmetry. In contrast to the so-called $Z_2$ topological insulators, an exhaustive characterization of antiferromagnetic topological phases with the help of a topological invariant has been missing. We analyze a simple model of an antiferromagnetic topological insulator and chart its phase diagram based on a recently proposed criterion for centrosymmetric systems [Fang et al., Phys. Rev. B 88, 085406 (2013)]. We then adapt two methods, originally designed for paramagnetic systems, and make antiferromagnetic topological phases manifest. The proposed methods apply far beyond the particular example treated in this work, and admit straightforward generalization. We illustrate this by considering a non-centrosymmetric system, where there are no simple criteria to identify topological phases. We also present an explicit construction of edge states in an antiferromagnetic topological insulator.