3D phononic Stiefel–Whitney insulator: A case study of 3D 3,42T32-type carbon allotrope (3,42T32-CA)

The Stiefel–Whitney (SW) insulator, a specific example of a higher-order topological insulator, is supported by spatial and time-reversal (PT) symmetry. This system with a vanishing Chern number exhibits a topology that belongs to the second SW class. Although the identification of SW insulators has been extensively suggested for 2D phononic systems in the past three years, other researchers have not given much consideration to 3D phononic systems with the SW topology. This work shows that a 3D 3,42T32-type carbon allotrope (named 3,42T32-CA) is a candidate material to host the phononic SW topology. Moreover, we present a nontrivial second SW number, represented as w2 = 1, to characterize the phononic SW topology for 3D 3,42T32-CA. Finally, we show that the 3D 3,42T32-CA hosts unconventional bulk-boundary correspondence, i.e., it exhibits 1D hinge vibrational modes in two dimensions lower than the 3D bulk vibrational modes.