Real-projective-plane hybrid-order topological insulator realized in phononic crystals

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 Z₂ 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.