Higher-Order Topology of Three-Dimensional Strong Stiefel-Whitney Insulators*

The Stiefel-Whitney insulator (SWI) is a two-dimensional topological phase protected by the symmetry under the combination of time reversal $T$ and a two-fold rotation $C_2$. This phase has got attention because it shows new aspects of topological crystalline insulators such as fragile topology and higher-order topology. In this talk, we study the three-dimensional generalization of the Stiefel-Whitney insulator. We show that a $C_2T$-symmetric insulator in 3D can have a stable topological invariant, contrary to its two-dimensional counterpart having fragile band topology. To characterize the bulk band topology further, we develop a new method based on the homotopy class of the symmetry representation for $C_{2z}T$ in a smooth gauge, instead of examining the obstruction to constructing smooth wavefunctions compatible with the reality condition. By using the new method, we show that the 3D topological insulator, dubbed 3D strong SWI, is characterized by the quantized magnetoelectric polarizability, which induces anomalous chiral hinge states along the edges parallel to the $C_2$ rotation axis and massless Dirac fermions on the surfaces normal to the $C_2$ axis. This establishes that a 3D strong SWI is a second-order topological insulator.