Fig. 1: The acoustic HOTI and the spin-polarized higher-order non-Hermitian skin effect.

a Unit-cell of the SC, consisting of four site CRAW ring resonators, coupled to each other by smaller link ring resonators. Each site ring comprises 200 alternating layers of rigid material (gray color) and air (the white background) and each link ring contains 160 alternating layers. The geometric parameters are taken as \({t}_{s}=0.4\) mm, \({t}_{a}=1\) mm and \(w=5.7\) mm, with the lattice constant \(a=22\) cm (in the simulations, the SC structure is scaled up by 9.5% to model the geometric errors in the experimental samples). The gap between the site rings and the link rings is quantified by \(\delta\), which is tuned to control the intra-cell and inter-cell couplings. Non-Hermiticity is introduced by adding acoustic dissipation in the green regions (realized by porous sponge). b Acoustic band structures calculated for the Hermitian SC (i.e., \(\gamma =0\)) with \(\delta =-0.7\) mm, \(0\) and \(0.7\) mm (inset shows the Brillouin zone). A parity inversion at the X point is observed (the lowest band gap is considered), as indicated by the flip of the “+” and “\(-\)” signs (respectively representing the \(s\)-/\(d\)-like states and the \(p\)-like states). This is associated with the topological phase transition. c Break-down of the conventional BBC in the non-Hermitian cases (\(\gamma \ne 0\)), signified by the inconsistence of the critical band gap-closing points under the PBC (labeled by the colored shadings) and OBC (labeled by the dotted lines). This indicates the emergence of non-Hermitian skin effect. d, e Physical pictures of the spin-polarized higher-order non-Hermitian skin effect. The left panels illustrate the Hermitian cases while the right panels depict the cases with acoustic loss.