Topological magnetic crystalline insulators and co-representation theory

Gapless surface states of time reversal invariant topological insulators are protected by the anti-unitary nature of the time reversal operation. Very recently, this idea was generalized to magnetic structures, in which time reversal symmetry is explicitly broken, but there is still an anti-unitary symmetry operation combining time reversal symmetry and crystalline symmetry. These topological phases in magnetic structures are dubbed "topological magnetic crystalline insulators". In this work, we present a general theory of topological magnetic crystalline insulators in different types of magnetic crystals based on the co-representation theory of magnetic crystalline symmetry groups. We construct two concrete tight-binding models of topological magnetic crystalline insulators, the $\hat{C}_4\Theta$ model and the $\hat{\bf \tau}\Theta$ model, in which topological surface states and topological invariants are calculated explicitly. Moreover, we check different types of anti-unitary operators in magnetic systems and find that the systems with $\hat{C}_4\Theta$, $\hat{C}_6\Theta$ and $\hat{\bf \tau}\Theta$ symmetry are able to protect gapless surface states. Our work will pave the way to search for topological magnetic crystalline insulators in realistic magnetic materials.