Elastic three-dimensional phononic topological insulators with Dirac hierarchy

Three-dimensional (3D) phononic topological insulators (TIs) featuring two-dimensional (2D) surface states and one-dimensional (1D) hinge states have opened up a new route for multi-dimensional robust wave transport, providing unprecedented methods for integrated acoustic sensors and energy harvesting devices. However, aiming at the elastic 3D phononic TI with gapless surface states and hinge states, the realization of elastic 3D phononic TIs with gapless surface states and hinge states is a significant challenge due to the complicated multi-mode polarization of elastic waves in 3D structures. In this study, we demonstrate an elastic 3D phononic TI with a Dirac hierarchy by elaborately operating the corresponding spatial symmetries of the chiral honeycomb lattice. First, a 3D double Dirac cone of elastic wave can be achieved by doubling the lattice along the out-of-plane direction to fold two iso-frequency Weyl points. The topological phase transitions and 2D gapless two-fold Dirac surface states of elastic wave are realized by breaking the half-lattice spatial translation symmetry. Subsequently, based on the Brillouin zone folding along the in-plane direction, the 2D gapless two-fold surface Dirac cones are folded into four-fold surface Dirac cones. Finally, by inducing the relative radius of adjacent holes to break the in-plane spatial inversion symmetry, the fourfold surface Dirac cones are gapped and associated with a surface state inversion, in which the gapless 1D hinge Dirac dispersion is achieved. This research offers a route for engineering the hierarchies of TIs in 3D elastic wave systems and provides new possibilities for designing 3D ultrasonic devices with unconventional functions.