Dirac Hierarchy in Acoustic Topological Insulators

Dirac cones are essential features of the electronic band structure of materials like graphene and topological insulators (TIs). Lately, this avenue has found a growing interest in classical wave physics by using engineered artificial lattices. Here, we demonstrate an acoustic 3D honeycomb lattice that features a Dirac hierarchy comprising an eightfold bulk Dirac cone, a 2D fourfold surface state Dirac cone, and a 1D twofold hinge state Dirac cone. The lifting of the Dirac degeneracy in each hierarchy authorizes the 3D lattice to appear as a first-order TI with 2D topological surface states, a second-order TI exhibiting 1D hinge states, and a third-order TI of 0D midgap corner states. Analytically we discuss the topological origin of the surface, hinge, and corner states, which are all characterized by out-of-plane and in-plane winding numbers. Our study offers new routes to control sound and vibration for acoustic steering and guiding, on-chip ultrasonic energy concentration, and filtering to name a few.