Glide-symmetric topological crystalline insulator phase in a nonprimitive lattice

We study the topological crystalline insulator phase protected by the glide symmetry, which is characterized by the ${Z}₂$ topological number. In the present paper, we derive a formula for the ${Z}₂$ topological invariant protected by glide symmetry in a nonprimitive lattice from that in a primitive lattice. We establish a formula for the glide-${Z}₂$ invariant for the space group No. $\mathbit{9}$ with glide symmetry in the base-centered lattice by folding the Brillouin zone into that of the primitive lattice where the formula for the glide-${Z}₂$ invariant is known. The formula is written in terms of integrals of the Berry curvatures and Berry phases in the $k$ space. We also derive a formula of the glide-${Z}₂$ invariant when the inversion symmetry is added, and the space group becomes $\mathbit{15}$. This reduces the formula into the Fu-Kane-like formula, expressed in terms of the irreducible representations at high-symmetry points in $k$ space. We also construct these topological invariants by the layer-construction approach, and the results completely agree with those from the $k$-space approach.