Inhomogeneous fan-shaped surface state induced by isolated Weyl points in acoustic crystals and the associated multi-frequency sound-wave filters

The nontrivial surface states excited by isolated Weyl points (IWPs) have been scarcely studied to date, primarily due to their circumvention from the Nielsen-Ninomiya no-go theorem. In a groundbreaking study on this topic [Adv. Sci., 10, 2207508 (2023)], we discovered that IWPs can generate a novel nontrivial surface state, namely the multi-fold fan-shaped surface state, which we named. Here, we report another type of fan-shaped surface state generated by an IWP surrounded by a closed Weyl nodal wall (WNW). Unlike previous findings, the fan-shaped surface state discovered here exhibits inhomogeneous in spatial distribution, with significantly varying sizes of the fan blades. Moreover, this surface state can be generated by a charge-four IWP protected by the rotation symmetries {$C_{31}^{+}|$000}, {$C_{2z}|\frac{1}{2}$0$\frac{1}{2}$}, {C_2x|0$\frac{1}{2}\frac{1}{2}$} and the time-reversal symmetry in the space group (SG) No. 198. Importantly, our simulation results of the acoustic crystals in this SG revel that the inhomogeneous fan-shaped surface state can provide multiple channels for acoustic wave transmission without energy dissipation, demonstrating that this kind of nontrivial surface state offers an effective mechanism for designing multi-frequency acoustic wave filters and selectors.