Octupole topological insulating phase protected by a three-dimensional momentum-space nonsymmorphic group

Recent advancements in quantum polarization theory have propelled the exploration of topological insulators (TIs) into the realm of higher-order systems, leading to the study of the celebrated two-dimensional (2D) quadrupole and 3D octupole TIs. Traditionally, these topological phases have been associated with the toroidal topology of the conventional Brillouin zone. This paper reports the discovery of a novel octupole topological insulating phase protected by a 3D momentum-space nonsymmorphic group emerging within the framework of the Brillouin 3D real projective space ([Formula: see text]). We theoretically propose the model and its corresponding topological invariant, experimentally construct this insulator within a topological circuit framework and capture the octupole insulating phase as a localized impedance peak at the circuit's corner. Furthermore, our [Formula: see text] circuit stands out as a pioneering 3D model to simultaneously exhibit both intrinsic, termination-independent symmetry-protected topological phases and extrinsic, termination-dependent surface-obstructed topological phases within the symmetry-protected topological phases. Our results broaden the topological landscape and provide insights into the band theory within the manifold of the Brillouin [Formula: see text] space.