Fig. 2: Acoustic realizations.

a Acoustic unit cell of phase I. b \({\theta }_{z}\) evolution of the projected dispersion for a supercell that has finite size of \(14.5a* 14.5a\) in \(x\)-\(y\) plane, where gapless corner states (red and magenta dots) appear in the common band gap of the bulk (olive dots) and edge states (green dots). The edge truncations of the 2D acoustic supercell are the same as those used in TBM (Fig. 1j). c The same as b, but for the acoustic supercell with interlayer couplers (purple in f), where supercell has the size of \(3.5a* 3.5a\) in \(x\)-\(y\) plane. During the calculation, the periodic Bloch boundary condition is applied in the \(z\) direction. The topological pump of corner states is represented by blue surfaces, whereas the bulk bands are labeled by olive surfaces. d The pumping curve of states at \(f=4.16\,{{{{{\rm{kHz}}}}}}\), corresponding to the intersection illustrated by the blue horizontal sheet in c. e Pressure field distributions of the topological states from right bottom corner to left bottom corner localization with the change of the pumping parameter \({\theta }_{z}\) from \(0.25{{{{{\rm{\pi }}}}}}\) to \(0.75{{{{{\rm{\pi }}}}}}\) along the \(z\) direction. f Numerical demonstration of topological corner-edge-corner transport at \(f=4.16\,{{{{{\rm{kHz}}}}}}\). Inset shows the details of the cavity with interlayer couplers (purple). g The same as f, but for the defect system. Inset presents the details of the defects. The cyan dots and black arrows in f, g represent the point sources and the propagations of sound, respectively.