Coexistence of All-Order Topological States in a Three-Dimensional Phononic Topological Crystalline Insulator.

Classical-wave topological materials lacking intrinsic half-integer spins are less robust while more tunable. Here, we explore a single 3-dimensional phononic topological crystalline insulator that simultaneously exhibits a whole family of first-order quadratic surface, second-order hinge, and third-order corner states within the same bandgap. Such a topological crystalline insulator hosting all-order phases originates from the different topological nature when hierarchically projected onto different facets and lower dimensions, thus free from trivial cladding crystals. Our work offers an ideal platform for either robust wave propagation or localization in on-demand dimensions and may facilitate dimension division multiplexing technology.

Ideal acoustic quantum spin Hall phase in a multi-topology platform.

Fermionic time-reversal symmetry ([Formula: see text])-protected quantum spin Hall (QSH) materials feature gapless helical edge states when adjacent to arbitrary trivial cladding materials. However, due to symmetry reduction at the boundary, bosonic counterparts usually exhibit gaps and thus require additional cladding crystals to maintain robustness, limiting their applications. In this study, we demonstrate an ideal acoustic QSH with gapless behaviour by constructing a global Tf on both the bulk and the boundary based on bilayer structures. Consequently, a pair of helical edge states robustly winds several times in the first Brillouin zone when coupled to resonators, promising broadband topological slow waves. We further reveal that this ideal QSH phase behaves as a topological phase transition plane that bridges trivial and higher-order phases. Our versatile multi-topology platform sheds light on compact topological slow-wave and lasing devices.

Experimental Demonstration of Bulk-Hinge Correspondence in a Three-Dimensional Topological Dirac Acoustic Crystal.

Three-dimensional topological Dirac semimetal (DSM) is a vital state to explore topological phases and phase transitions. However, its bulk-boundary correspondence is elusive. Here, we experimentally investigate the higher-order hinge states in an acoustic DSM. Not only removable trivial surface states but also robust nontrivial hinge arcs are observed, attributed to the direct correspondence between bulk polarization and hinge charge. We further reveal that a pair of zigzag and bearded hinges possess arcs located in complementary momentum regions. Our work provides solid proof of the bulk-hinge correspondence in DSM and sheds light on the study of topological hierarchy across dimensions.

Acoustic analogues of three-dimensional topological insulators.

Topological insulators (TIs) can host an insulating gapped bulk with conducting gapless boundary states in lower dimensions than the bulk. To date, various kinds of classical wave TIs with gapless symmetry-protected boundary states have been discovered, promising for the efficient confinement and robust guiding of waves. However, for airborne sound, an acoustic analogue of a three-dimensional TI has not been achieved due to its spinless nature. Here, we experimentally demonstrate a three-dimensional topological acoustic crystal with pseudospins using bilayer chiral structures, in which multi-order topological bandgaps are generated step by step via elaborately manipulating the corresponding spatial symmetries. We observe acoustic analogues of 1st-order (two-dimensional gapless surface Dirac cones) and 2nd-order (one-dimensional gapless hinge Dirac dispersion) TIs in three dimensions, supporting robust surface or hinge sound transport. Based solely on spatial symmetry, our work provides a route to engineer the hierarchies of TIs and explore topological devices for three-dimensional spinless systems.