Glide-symmetric magnetic topological crystalline insulators with inversion symmetry

It is known that three-dimensional magnetic systems with glide symmetry can be characterized by a $Z_2$ topological invariant together with the Chern number associated with the normal vector of the glide plane, and they are expressed in terms of integrals of the Berry curvature. In the present paper, we study the fate of this topological invariant when inversion symmetry is added while time-reversal symmetry is not enforced. There are two ways to add inversion symmetry, leading to space groups No.~13 and No.~14. In space group No.~13, we find that the glide-$Z_2$ invariant is expressed solely from the irreducible representations at high-symmetry points in $k$-space. It constitutes the $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry-based indicator for this space group, together with another $\mathbb{Z}_2$ representing the Chern number modulo 2. In space group No.~14, we find that the symmetry-based indicator $\mathbb{Z}_2$ is given by a combination of the glide-$Z_2$ invariant and the Chern number. Thus, in space group No.~14, from the irreducible representations at high-symmetry points we can only know possible combinations of the glide-$Z_2$ invariant and the Chern number, but in order to know each value of these topological numbers, we should calculate integrals of the Berry curvature. Finally, we show that in both cases, the symmetry-based indicator $\mathbb{Z}_4$ for inversion symmetric systems leading to the higher-order topological insulators is directly related with the glide-$Z_2$ invariant and the Chern number. As an independent approach to these results, we also construct all invariants from the layer construction for these space groups, and we show complete agreement with the above results for the topological invariants constructed from $k$-space topology.