Phononic Stiefel–Whitney Topology with Hinge\nVibrational Modes in 3D Carbon Allotrope 4<sup>3</sup>T57-CA

As a fragile topological\nstate lacking spin–orbit coupling\n(SOC) and possessing space–time inversion (<i>PT</i>) symmetry, the Stiefel–Whitney (SW) insulator has received\nmuch attention. Up until now, the identification of SW insulators\nhas been extensively suggested for 2D phononic systems but has been\nrarely considered for 3D phononic systems. 3D carbon allotrope 4<sup>3</sup>T57-CA has the capability to achieve nontrivial phononic SW\ntopology, which can be distinguished by a nontrivial second SW number.\nMoreover, 3D 4<sup>3</sup>T57-CA can host hinge vibrational modes,\nwhich are protected by <i>PT</i>.