Acoustic higher-order topological corner states in a low-symmetry Lieb lattice

Among the various two-dimensional Bravais lattices, Lieb lattice systems exhibit unique physics regarding the coexistence of flat bands and Dirac bands, with the flat bands positioned in the middle of them, and have been extensively studied. As a low-symmetry lattice, the formation of the flat band in the Lieb lattice is attributed to the destructive interference of Bloch wave functions caused by lattice symmetry. However, the higher-order topological properties of the Lieb lattice have been insufficiently investigated in the field of acoustics. In this work, we design a two-dimensional topological acoustic system by rationally arranging resonators of acoustic metamaterials in a Lieb lattice. The presence of topologically protected flat bands, first-order edge states, and zero-dimensional corner states in the Lieb lattice is observed on simulations. Subsequently, numerical and experimental observations reveal a flat band capable of forming localized bulk states, with corner states distinctly separated from them. Notably, corner states are localized not only in the corner but also in the independent resonator with "pointy" edges. Furthermore, the topological states of Lieb lattice and their number are related to their three types of polymers from a structural perspective. The proposed work presents an experimentally viable framework for investigating topologically protected flat bands and the high-order topological insulator within the context of the Lieb lattice.