  Integration between magnetism and topology is an exotic phenomenon in
condensed-matter physics. Here, we propose an exotic phase named topological
crystalline antiferromagnetic state, in which antiferromagnetism intrinsically
integrates with nontrivial topology, and we suggest such a state can be
realized in tetragonal FeS. A combination of first-principles calculations and
symmetry analyses shows that the topological crystalline antiferromagnetic
state arises from band reconstruction induced by pair checker-board
antiferromagnetic order together with band-gap opening induced by intrinsic
spin-orbit coupling in tetragonal FeS. The topological crystalline
antiferromagnetic state is protected by the product of fractional translation
symmetry, mirror symmetry, and time-reversal symmetry, and present some unique
features. In contrast to strong topological insulators, the topological
robustness is surface-dependent. These findings indicate that non-trivial
topological states could emerge in pure antiferromagnetic materials, which
sheds new light on potential applications of topological properties in
fast-developing antiferromagnetic spintronics.
