Square-root-like higher-order topological states in three-dimensional sonic crystals.

The square-root descendants of higher-order topological insulators were proposed recently, whose topological property is inherited from the squared Hamiltonian. Here we present a three-dimensional (3D) square-root-like sonic crystal by stacking the 2D square-root lattice in the normal () direction. With the nontrivial intralayer couplings, the opened degeneracy at the-direction induces the emergence of multiple acoustic localized modes, i.e., the extended 2D surface states and 1D hinge states, which originate from the square-root nature of the system. The square-root-like higher order topological states can be tunable and designed by optionally removing the cavities at the boundaries. We further propose a third-order topological corner state in the 3D sonic crystal by introducing the staggered interlayer couplings on each square-root layer, which leads to a nontrivial bulk polarization in thedirection. Our work sheds light on the high-dimensional square-root topological materials, and have the potentials in designing advanced functional devices with sound trapping and acoustic sensing.