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 onedimensional (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 acuteangled 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.
Topologically protected gapless edge states are phases of quantum matter which behave as massless Dirac fermions, immunizing against disorders and continuous perturbations. Recently, a new class of topological insulators (TIs) with topological corner states have been theoretically predicted in electric systems, and experimentally realized in two-dimensional (2D) mechanical and electromagnetic systems, electrical circuits, optical and sonic crystals, and elastic phononic plates. Here, we demonstrate a pseudospin-valley-coupled phononic TI, which simultaneously exhibits gapped edge states and topological corner states. Pseudospin-orbit coupling edge states and valley-polarized edge state are respectively induced by the lattice deformation and the symmetry breaking. When both of them coexist, these topological edge states will be greatly gapped and the topological corner state emerges. Under direct field measurements, the robust edge propagation behaving as an elastic waveguide and the topological corner mode working as a robust localized resonance are experimentally confirmed. The pseudospin-valley coupling in our phononic TIs can be well-controlled which provides a reconfigurable platform for the multiple edge and corner states, and exhibits well applications in the topological elastic energy recovery and the highly sensitive sensing.
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