Higher-order topological states in a three-dimensional acoustic metamaterial

Abstract

Topologically gapless edge states, characterized by topological invariants and Berry’s phases of bulk energy bands, provide amazing techniques to robustly control the reflectionless propagation of electrons, photons and phonons. Recently, a new family of topological phases, dictated by the bulk polarization, has been observed, leading to the discovery of the higher-order topological insulators. The higher-order topological states located on the “boundaries of boundaries” can be used to limit and control the transmission of optical waves, acoustic waves and elastic waves. So far, the higher-order topological states have been demonstrated in two-dimensional mechanical, electromagnetic, optical, acoustic and elastic systems. However, the study of higher-order topological states in three-dimensional acoustic systems, which can increase the possibility of topological metamaterial design, is rarely reported. In this letter, we propose a three-dimensional acoustic metamaterial supporting topologically protected second-order hinge states along with topologically protected first-order surface states within the same bandgap. Only the nearest neighbor couplings are considered in this metamaterial, and the coupling strengths can be adjusted by changing the diameters of the connection pipes. The intra-cell coupling strength t 1 is proportional to the diameters of the intra-cell connection pipes d 1, and the inter-cell coupling strength t 2 is proportional to the diameters of the inter-cell connecting pipes d 2. The band structure of the unit cell of three-dimensional acoustic metamaterial with the same intra-cell and inter-cell coupling strengths can form a two-fold nodal line along the K-H direction of its Brillouin zone. When intra-cell and inter-cell coupling strengths are different, the two-fold nodal line is lifted to form a complete bandgap, yielding two different kinds of acoustic metamaterials, that are, trivial and topological non-trivial metamaterials. The numerical results of band structures and eigenfrequencies indicate that: When the intra-cell coupling strength t 1 is less than the inter-cell coupling strength t 2, the acoustic metamaterial has 3 second-order hinge bands, 12 first-order surface bands and 18 second-order hinge eigenfrequencies, 72 first-order surface eigenfrequencies in the complete bandgap range respectively; when the inter-cell coupling strength t 2 is less than the intra-cell coupling strength t 1, there is neither any energy bands nor any eigenfrequencies in the complete bandgap range. The second-order hinge states and first-order surface states have great immunity against defects, favorably evidencing the strong robustness of the topological states. The numerical results of the transmission spectra furtherly verify the above phenomena. When t 1 2, the transmission spectrum of the second-order hinge reaches its peak at 4670 Hz. For the first-order surface transmission spectrum, a high peak, located in the bandgap, is observed around 4555 Hz. When t 1 >t 2, the surface and hinge transmission spectra are very low within the complete bandgap, the acoustic energy is concentrated only at the excitation cylinder, showing that the acoustic wave will not propagate in any part of the acoustic metamaterial in the complete bandgap range and the acoustic metamaterial is trivial. The realization of the higher-order topological states in three-dimensional acoustic metamaterial breaks through the limitation of that in two-dimensional system, realizes the topologically transform of acoustic energy among bulk, surface and hinge states and has the potential application prospect in acoustic energy recovery and high-precision acoustic sensor.