Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems

TL;DR

This paper introduces a homotopy classification framework for symmetry representations in $C_{2z}T$-symmetric systems, linking topological invariants to matrix homotopy classes and revealing new insights into higher-order topological insulators.

Contribution

It establishes a homotopy-based approach to classify topological invariants in $C_{2z}T$-symmetric systems, connecting the second Stiefel-Whitney number to matrix homotopy classes and identifying the 3D strong Stiefel Whitney insulator with quantized magnetoelectric polarizability.

Findings

The second Stiefel-Whitney number is the homotopy invariant for $C_{2z}T$ symmetry.

The 3D strong Stiefel Whitney insulator's invariant equals the quantized magnetoelectric polarizability.

The 3D strong Stiefel Whitney insulator exhibits both first and second order topological features.

Abstract

We study the homotopy classification of symmetry representations to describe the bulk topological invariants protected by the combined operation of a two-fold 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 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 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(R) (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 hinges 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 order and the second order topological insulators, simultaneously, which is consistent with the recent classification of higher-order topological insulators protected by an order-two symmetry.