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.