Symmetry representation approach to topological invariants in C2zT -symmetric systems

Abstract

We study the homotopy classification of symmetry representations to describe the bulk topological invariants protected by the combined operation of a twofold rotation ${C}_{2z}$ 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 ${C}_{2z}T$, is the homotopy invariant that characterizes the second homotopy class of the matrix representation of ${C}_{2z}T$. As an application of our result, we show that the three-dimensional (3D) bulk topological invariant for the ${C}_{2z}T$-protected topological crystalline insulator proposed by C. Fang and L. Fu in Phys. Rev. B 91, 161105 (2015), 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.