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.

Topological phononic states, which facilitate unique acoustic transport around defects and disorders, have significantly revolutionized our scientific cognition of acoustic systems. Here, by introducing a zone folding mechanism, we realize the topological phase transition in a double Dirac cone of the rotatable triangular phononic crystal with $C_{3v}$ symmetry. We then investigate the distinct topological edge states on two types of interfaces of our phononic insulators. The first one is a zigzag interface which simultaneously possesses a symmetric mode and an antisymmetric mode. Hybridization of the two modes leads to a robust pseudospin-dependent one-way propagation. The second one is a linear interface with a symmetric mode or an antisymmetric mode. The type of mode is dependent on the topological phase transition of the phononic insulators. Based on the rotatability of triangular phononic crystals, we consider several complicated contours defined by the topological zigzag interfaces. Along these contours, the acoustic waves can unimpededly transmit without backscattering. Our research develops a route for the exploration of the topological phenomena in experiments and provides an excellent framework for freely steering the acoustic backscattering-immune propagation within topological phononic structures.

In condensed matter physics, topologically protected edge transportation has drawn extensive attention over recent years. Thus far, the topological valley edge states have been produced near the Dirac cones fixed at the high-symmetry points of the Brillouin zone. In this paper, we demonstrate a unique valleylike phononic crystal (PnC) with the position-varying Dirac cones at the high-symmetry lines of the Brillouin zone boundary. The emergence of such Dirac cones, characterized by the vortex structure in a momentum space, is attributed to the unavoidable band crossing protected by the mirror symmetry. The Dirac cones can be unbuckled and a complete band gap can be induced through breaking the mirror symmetry. Interestingly, by simply rotating the square columns, we realize the valleylike vortex states and the band inversion effect which leads to the valley Hall phase transition. Along the valleylike PnC interfaces separating two distinct acoustic valley Hall phases, the valleylike protected edge transport of sound in domain walls is observed in both the simulations and the experiments. These results are promising for the exploration of alternative topological phenomena in the valleylike PnCs beyond the graphenelike lattice.