Phononic Stiefel–Whitney Topology with Hinge Vibrational Modes in 3D Carbon Allotrope 4³T57-CA

As a fragile topological state lacking spin-orbit coupling (SOC) and possessing space-time inversion (<i>PT</i>) symmetry, the Stiefel-Whitney (SW) insulator has received much attention. Up until now, the identification of SW insulators has been extensively suggested for 2D phononic systems but has been rarely considered for 3D phononic systems. 3D carbon allotrope 4³T57-CA has the capability to achieve nontrivial phononic SW topology, which can be distinguished by a nontrivial second SW number. Moreover, 3D 4³T57-CA can host hinge vibrational modes, which are protected by <i>PT</i>.