Fig. 2: Topological edge states and non-Hermitian skin modes.

a Calculated eigen-spectrum for a supercell with PBC along the \(x\) direction and OBC along the \(y\) direction in the Hermitian regime. Acoustic pressure field profiles for the two marked states (gray diamond and yellow star) are presented, showing a typical extended bulk state and a topological edge state, whose mode patterns are standing waves due to the spin-degeneracy. b The same as a, only with acoustic loss. Due to the non-Hermitian skin effect, the original bulk states and topological edge states are no longer distinguishable and become edge-like skin modes, as shown by the acoustic pressure field profiles for the two marked states (orange diamond and star). Interestingly, the non-Hermitian skin effect is spin-polarized, with spin-up states accumulating toward the lower boundary while the spin-down states accumulating toward the upper boundary. This feature is in sharp contrast to the topological edge states in a. c, d Experimental measurements on the extended bulk states and the non-Hermitian skin modes, respectively. The transmission spectra are measured for the bulk (gray) and edge (blue or orange) pump-probes. The corresponding acoustic field profiles for two excited states (marked by the gray and orange arrows) are also presented, with a piece of SC sample sketched to illustrate the scanning domain (see Supplementary Information for more details), where the black thick arrow indicates the input plane-wave-like signal.