Topological invariants for nonsymmorphic symmetries

Topological insulators with time reversal symmetry are known to fall into a Z2 classification. With additional nonsymmorphic symmetry, topological phases can be extended into a more detailed classification, which can be characterized by a Z4 invariant first defined in K theory in Shiozaki's paper [PRB93,195413]. In our work, we reformulate the Z4xZ2 invariant with the non-Abelian Wilson loop for the insulating systems preserving time reversal and a glide symmetry. Furthermore, we also extend the classification to the time-reversal-invariant systems with two glide symmetries. A lot of materials have been proposed to realize the distinct topological phases as well.