Higher-Order Topological Spin Hall Effect of Sound

We theoretically propose a reconfigurable two-dimensional (2D) hexagonal sonic crystal with higher-order topology protected by the six-fold, C_6, rotation symmetry. The acoustic band gap and band topology can be controlled by rotating the triangular scatterers in each unit cell. In the nontrivial phase, the sonic crystal realizes the topological spin Hall effect in a higher-order fashion: (i) the edge states emerging in the bulk band gap exhibit partial spin-momentum correlation and are gapped due to the reduced spatial symmetry at the edges. (ii) The gapped edge states, on the other hand, stabilize the topological corner states emerging in the edge band gap. The partial spin-momentum correlation is manifested as pseudo-spin-polarization of edge states away from the time-reversal invariant momenta, where the pseudospin is emulated by the acoustic orbital angular momentum. We reveal the underlying topological mechanism using a corner topological index based on the symmetry representation of the acoustic Bloch bands.