Acoustic higher-order topological states in kagome lattice with split-ring resonators

The next-nearest-neighbour (NNN) hoppings can induce many unique topological phenomena but are rarely considered in acoustics. This work investigates the higher-order topological states induced by the NNN hoppings in acoustic kagome lattice with split-ring resonators (SRRs). We build an extended tight-binding model (TBM) considering NNN hoppings to predict the topological corner modes (CMs) of the acoustic system and obtain the coupling strengths between the SRRs in different rotational angles. During rotating the SRRs, two nontrivial and one trivial systems characterized by bulk polarization can be obtained. In these two nontrivial systems, four novel CMs induced by the NNN hoppings can be observed in the first bandgap. Furthermore, the interactions between resonant and Bragg scattering effect of the SRRs are researched and the defect modes (DMs) are realized in the second bandgap. Finally, the CMs predicted by the extended TBM and DMs match well with the numerical and experimental ones. These discoveries unveil the intricate physics driven by the NNN hoppings, paving the way for innovative approaches to the design of tunable sound energy localization and multi-band metamaterials.