No abstract

In the past years classical wave-systems have constituted an excellent platform for emulating complex quantum phenomena. This approach has been especially fruitful in demonstrating topological phenomena in photonics and acoustics: from chiral edge states of Chern insulators and helical edge states of topological insulators to higher-dimensional topological states of quasiperiodic systems and systems with synthetic dimensions. Recently, a new class of topological states localized in more than one dimension of a D-dimensional system, referred to as higher-order topological (HOT) states, has been reported, offering an even more versatile platform to confine and control classical radiation and mechanical motion. However, because experimental research of HOT states has so far been limited to two-dimensional (2D) systems, third and higherorder states have evaded experimental observation. Studying higher-dimensional classical systems therefore opens an opportunity to emulate higher-order topological insulators and explore HOT states beyond second order. In this letter, we design and experimentally study a 3D acoustic metamaterial supporting third-order (0D) topological corner states along with second-order (1D) edge states within the same topological bandgap, thus establishing a full hierarchy of HOT states in three dimensions. The metamaterial is implemented over a versatile additive manufacturing platform, which enables rapid prototyping of metaatoms and metamolecules, which can be snapped together to form 3D metamaterials with complex geometries. The assembled 3D topological metamaterial represents the acoustic analogue of a pyrochlore lattice made of interconnected molecules, and is shown to exhibit topological bulk polarization, leading to the emergence of HOT states localized in all three or in two dimensions.Topological acoustics and mechanics have been explored as a powerful platform for the implementation of a plethora of topological phenomena (1-4). A bias for sound propagation imparted by the angular-momentum carried by a rotating fluid or by rotational motion in mechanical resonators has been used to emulate the effect of magnetic bias and to demonstrate the emergence of quantum Hall effect (QHE)-like states with robust edge transport for acoustics and mechanics (5-11). In parallel, a variety of symmetry-protected topological acoustic states have been reported, including Zak-phase in 1D acoustic lattices (12), quantum spin Hall effect (QSHE) (13-16), nontrivial bulk polarization induced edge and corner states in 2D Kagome lattices (17)(18)(19)(20). More recently, versatile 3D structures fabricated by either direct assembly or by advanced 3D