Glide-symmetric topological crystalline insulator phase in a nonprimitive lattice

TL;DR

This paper derives formulas for glide-symmetry protected Z2 topological invariants in nonprimitive lattices, extending previous results and confirming them through layer-construction methods, advancing understanding of topological crystalline insulators.

Contribution

It provides a new formula for glide-Z2 invariants in nonprimitive lattices and relates them to known primitive lattice formulas, including cases with added inversion symmetry.

Findings

Derived a formula for glide-Z2 invariant in nonprimitive lattices.

Connected the glide-Z2 invariant to Fu-Kane-like formulas with inversion symmetry.

Validated invariants through layer-construction approach.

Abstract

We study the topological crystalline insulator phase protected by the glide symmetry, which is characterized by the Z2 topological number. In the present paper, we derive a formula for the Z2 topological invariant protected by glide symmetry in a nonprimitive lattice, from that in a primitive lattice. We establish a formula for the glide-Z2 invariant for the space group No. 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-Z2 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-Z2 invariantwhen the inversion symmetry is added, and the space group becomes No. 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.