Glide-symmetric topological crystalline insulators with inversion symmetry

It is known that three-dimensional systems with glide symmetry can be characterized by a Z₂ topological invariant in class A, and it is expressed in terms of integrals of the Berry curvature. In this presentation, we consider what happens to the glide-Z₂ invariant, when the inversion symmetry is added. There are two ways to add the inversion symmetry, leading to the space group No. 13 or No. 14. In the space group 13, we find that the glide-Z₂ invariant is expressed in terms of the irreducible representations at high-symmetric points in the moment space, which constitutes the Z₂×Z₂ symmetry-based indicators for this space group, together with the Chern number modulo 2. In the space group 14, we find that the symmetry-based indicator Z₂ is given by the combination of the glide-Z₂ invariant and the Chern number. From the irreducible representations at high-symmetry points in this case, we can only know possible combinations of the glide-Z₂ invariant and the Chern number. Moreover, we show that in both cases the Z₄ strong topological invariant for inversion symmetric systems is directly related to the glide-Z₂ invariant and the Chern number. Finally, we also investigate topological invariants for glide-symmetric systems with nonprimitive lattice with and without inversion symmetry.