Real-projective-plane hybrid-order topological insulator realized in phononic crystals
Pengtao Lai, Jien Wu, Zhenhang Pu, Qiuyan Zhou, Jiuyang Lu, Hui Liu, Weiyin Deng, Hua Cheng, Shuqi Chen, Zhengyou Liu
Abstract
The manifold of the fundamental domain of the Brillouin zone is always considered to be a torus. However, under the synthetic gauge field, the Brillouin manifold can be modified by the projective symmetries, resulting in unprecedented topological properties. Here, we realize a real-projective-plane hybrid-order topological insulator in a phononic crystal by introducing the ${Z}_{2}$ gauge field. Such an insulator hosts two momentum-space nonsymmorphic reflection symmetries, which change the Brillouin manifold from a torus to a real projective plane. These symmetries can simultaneously lead to a Klein-bottle insulator and quadrupole insulator phases in different bulk gaps. The nonsymmorphic reflection symmetries on Brillouin real-projective-plane, edge states of the Klein-bottle insulator, and corner states of the quadrupole insulator are observed. These results evidence the hybrid-order topology on the Brillouin manifold beyond the torus, and enrich the topological physics.