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

We study the three-dimensional generalization of the two-dimensional Stiefel-Whitney insulator protected by the combination of a two-fold rotation $C_2$ and time-reversal $T$ symmetries. We show that a $C_2T$-symmetric three-dimensional insulator 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 three-dimensional topological insulator, dubbed a three-dimensional strong Stiefel-Whitney insulator is characterized by the quantized magnetoelectric polarizability, which induces anomalous chiral hinge states along the edges parallel to the $C_2$ rotation axis and two-dimensional massless Dirac fermions on the surfaces normal to the $C_2$ axis. This establishes that a three-dimensional strong Stiefel-Whitney insulator is a second-order topological insulator.