Glide-symmetric topological crystalline insulators with inversion symmetry

It is known that three-dimensional systems with glide symmetry can be characterized by a $Z_2$ topological invariant, and it is expressed in terms of integrals of the Berry curvature. In the present paper, we study the fate of this topological invariant when the inversion symmetry is added. There are two ways to add the inversion symmetry, leading to the space groups No.~13 and No.~14. In the space group 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 $Z_2\times Z_2$ symmetry-based indicator for this space group, together with another $Z_2$ representing the Chern number modulo 2. In the space group 14, we find that the symmetry-based indicator $Z_2$ is given by a combination of the glide-$Z_2$ invariant and the Chern number. Thus, in the space group 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 $Z_4$ for inversion symmetric systems leading to the higher-order topological insulators is directly related with the glide-$Z_2$ invariant and the Chern number.