Symmetry representation approach to topological invariants in C2zT -symmetric systems

Abstract

© 2019 American Physical Society.We study the homotopy classification of symmetry representations to describe the bulk topological invariants protected by the combined operation of a twofold rotation C2z and time-reversal T symmetries. We define topological invariants as obstructions to having smooth Bloch wave functions compatible with a momentum-independent symmetry representation. When the Bloch wave functions are required to be smooth, the information on the band topology is contained in the symmetry representation. This implies that the d-dimensional homotopy class of the unitary matrix representation of the symmetry operator corresponds to the d-dimensional topological invariants. Here, we prove that the second Stiefel-Whitney number, a two-dimensional (2D) topological invariant protected by C2zT, is the homotopy invariant that characterizes the second homotopy class of the matrix representation of C2zT. As an application of our result, we show that the three-dimensional (3D) bulk topological invariant for the C2zT-protected topological crystalline insulator proposed by C. Fang and L. Fu in Phys. Rev. B 91, 161105 (2015)PRBMDO1098-012110.1103/PhysRevB.91.161105, which we call the 3D strong Stiefel-Whitney insulator, is identical to the quantized magnetoelectric polarizability. The bulk-boundary correspondence associated with the quantized magnetoelectric polarizability shows that the 3D strong Stiefel-Whitney insulator has chiral hinge states as well as 2D massless surface Dirac fermions. This shows that the 3D strong Stiefel-Whitney insulator has the characteristics of both the first- and the second-order topological insulators, simultaneously, which is in consistence with the recent classification of higher-order topological insulators protected by an order-two symmetry