Fig. 1. Acoustic octupole topological insulator.

a Geometry for minimal tight-binding model of octupole topological insulator, ${\gamma }_{\perp }$ and ${\lambda }_{\perp }$ are the intra-cell and inter-cell couplings in the plane perpendicular to $z$ axis, and ${\gamma }_z$ and ${\lambda }_z$ are the intra-cell and inter-cell couplings in the $z$ direction. b Unit cell design of acoustic tetragonal crystal that guarantees the negative coupling at each plaquette of the lattice, geometry parameters are provided in the Methods. c Band structures of the acoustic unit cell in (b), shaded gray region indicates the frequency bandwidth of interested bands (orange colored), inset figures show fourfold degenerate band structures of interest (slight deviation due to meshing in numerical calculation) and high-symmetry lines and points in the tetragonal Brillouin zone, respectively. d Acoustic pressure profile of the acoustic coupled dipolar modes taken at the high-symmetry point A.