Fig. 2.

Topological pseudospin-valley states with saddle surfaces. a One-half of the surface BZ projected onto the k_xz plane. b The schematic of the interface. c Numerical results for the projected band structures along the high symmetric directions of the surface BZ. The shadow regions indicate the bulk pass bands. d Zoom-in surface states for the whole surface BZ. e Acoustic pseudospins (marked in c) are hybridized by symmetric (S) and anti-symmetric (A) states with ±π/2 phases delay, forming S ± iA. f Acoustic pseudospin sphere on the base of S (noted as S_y), iA_x and iA_z states, where red and green circles represent opposite chirality, and the corresponding arrows represent pseudospins