Fig. 2. Higher-order topology and corresponding boundary states of the acoustic tetragonal crystal.

a–c nth-order Wannier bands obtained based on the first-principle FEM simulation. a First-order Wannier bands v_z of the selective energy bands below the bandgap shown in Fig. 1c. b Second-order Wannier bands ${\nu }_x^{-z}$ of the Wannier-sector −v_z. c Third-order Wannier bands ${\nu }_y^{-x,-z}$ of the Wannier sector $-{\nu }_x^{-z}$. d–f Energy bands show hierarchy topological states in which (d), energy bands of the supercell lattice (110) supporting the surface states (yellow-colored bands). e Energy bands of the supercell lattice (010) supporting the hinge states (blue-colored bands). f Energy diagram of a finite lattice composed of 3 × 3 × 3 unit cells supporting mid-gap edge polarization-induced corner states. g–i Surface, hinge and corner field distributions over the finite lattice integrated from the respective boundary eigenstates inside the bulk bandgap in (f). Surface and hinge states in the vertical boundaries are mixed and represented by gray-colored bands in (f).