Topological edge states in concave hexagonal gyroscope phononic crystals

The study of gyroscopic phononic crystals (GPCs) opens new directions for topological acoustics. Based on the propagation of torsional waves in the structure, a GPC with lattice of concave hexagonal is proposed by introducing the gyroscope element into the concave hexagonal infinite periodic structure. The bandgap characteristics of GPC are analyzed, and the mechanisms of opening the Dirac cone and generating topological edge states due to changes in gyroscope torque are discussed. It is discovered that through the breaking of structural symmetry and time-reversal symmetry, two band gaps can be opened where two topological edge states can be found with a same topological GPC arrangement. Subsequently, the influence of gyroscope rotation velocity on the bandgap is meticulously examined, uncovering phenomena such as band inversion and the manifestation of valley Hall edge states. The investigation is extended to analyzing the supercell of the topological GPC. The wave propagation characteristics at topological interfaces in the two new band gaps with different arrangements are discussed, and the directivity difference between the topological edge states of the upper and lower band gaps is revealed. Moreover, the robustness of the topological edge states of GPC to defects is also proven.