Fig. 4.

Hyperbolic behaviour for the acoustic topological pseudospin-valley states. a The unit cell with increased height. b Numerical results for the projected band structures along the high symmetric directions of the surface BZ. c Zoom-in 3D view of the surface states in an eye shape. d, e Experimental transmission spectra along the $\tilde{\mathrm{\Gamma }}$–$\tilde{\mathrm{X}}$ and $\tilde{\mathrm{\Gamma }}$–$\tilde{\mathrm{Z}}$ directions, respectively. Dashed box represents the frequency region of the eye. The shadow regions indicate the bulk pass bands