Higher-order topological insulators, featured with multidimensional topologically protected boundary states, have attracted much research interest. However, in the existing higher-order topological insulators, topological corner states mostly emerge in a single band gap, which restricts their potential applications in photonics and acoustics. Here, a higher-order topological insulator hosting tri-band corner and edge states is proposed theoretically and realized experimentally based on a generalized honeycomb lattice where each sublattice site is generalized to a trimer. The triple topological band gaps are induced by the multiple degrees of freedom at each trimer, exhibiting enriched valley degrees of freedom and rich topological in-gap corner states when tuning the on-site energies. Both the numerical and experimental results demonstrate the existence of tri-band second-order corner states. The results may pave the way for multiband integrated acoustic devices.

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• Edge states
• Electronic structure
• Symmetry protected topological states
• Topological effects in acoustic systems
• Topological phase transition
• Valley degrees of freedom
• Acoustic metamaterials
• Topological insulators