Elastic Higher-Order Topological Insulator with Topologically Protected Corner States.

Topologically gapless edge states, characterized by topological invariants and Berry's phases of bulk energy bands, provide amazing techniques to robustly control the reflectionless propagation of electrons, photons, and phonons. Recently, a new family of topological phases, dictated by the bulk polarization, has been observed, leading to the discovery of the higher-order topological insulators (HOTIs). So far, the HOTIs have been demonstrated in mechanical and electromagnetic systems and electrical circuits with quantized quadrupole polarization and, more recently, have been experimentally realized in optical and acoustic systems. Here, we realize the higher-order topological states in a two-dimensional (2D) continuous elastic system. We experimentally observe the gapped one-dimensional (1D) edge states, the trivially gapped zero-dimensional (0D) corner states, and the topologically protected 0D corner states. Compared with the trivial corner modes, the topological ones, immunizing against defects, are robustly localized at the obtuse-angled but not the acute-angled corners. The topological shape-dependent corner states open a new route for the design of the topologically protected and reconfigurable 0D localized resonances and provide an excellent platform for the topological transformation of the elastic energy among 2D bulk, 1D edge, and 0D corner modes.