Elastic higher-order topological insulator with topologically protected corner states

Abstract: 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 are only demonstrated in discrete mechanical and electromagnetic systems and electrical circuits with the quantized quadrupole polarization. Here, we realize the higher-order topological states in a two-dimensional (2D) continuous elastic system whose energy bands can be well described. 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 but reconfigurable 0D local eigenmodes and provide an excellent platform for the topological transformation of elastic energy among 2D bulk, 1D edge and 0D corner modes.