The introduction of the concept of valley pseudospin to phononic crystals has made a remarkable topologically protected interface transport of sound, which opens a novel research area referred to as valley Hall topological insulators. Here, we demonstrate the simultaneous multi-band edge states of shear vertical waves in two-dimensional phononic crystals with veins. The multi-band edge states are topologically valley-protected and are obtained by simultaneously gapping multiple Dirac points at K (or K′) under the inversion symmetry breaking. As the relative radius of the two adjacent steel columns varies, the band diagram undergoes a topological transition which can be characterized by topological charge distributions and opposite valley Chern numbers. Subsequently, the vortex chirality of the bulk valley modes is unveiled. With numerical simulations, simultaneous multi-band valley dependent edge states and the associated valley-protected backscattering suppression around the curved waveguide are further demonstrated. Our work could become a promising platform for applications of multi-functional topological acoustic devices.

The study for exotic topological effects of sound has attracted uprising interests in fundamental physics and practical applications. Based on the concept of valley pseudospin, we demonstrate the topological valley transport of plate-mode waves in a homogenous thin plate with periodic stubbed surface, where a deterministic two-fold Dirac degeneracy is form by two plate modes. We show that the topological property can be controlled by the height of stubs deposited on the plate. By adjusting the relative heights of adjacent stubs, the valley vortex chirality and band inversion are induced, giving rise to a phononic analog of valley Hall phase transition. We further numerically demonstrate the valley states of plate-mode waves with robust topological protection. Our results provide a new route to design unconventional elastic topological insulators and will significantly broaden its practical application in the engineering field.

Topological defects with symmetry-breaking phase transitions have captured much attention.Vortex generated by topological defects exhibits exotic properties and its flow direction can be switched by altering the spin configurations. Contrary to electromagnetic and acoustic domains, the topological transport of elastic waves in periodic structures with topological defects is not well explored due to the mode conversion between the longitudinal and transverse modes. Here, we propose an elastic topological insulator with spontaneously broken symmetry based on the topological theory of defects and homotopy theory. Multiple topological transitions for elastic waves are achieved by topologically modifying the ellipse orientation in a triangular lattice of elliptical cylinders. The solid system, independent of the number of molecules in order parameter space, breaks through the limit of the point-group symmetry to emulate elastic pseudospin-orbit coupling. The transport robustness of the edge states is experimentally demonstrated. Our approach provides new possibilities for controlling and transporting elastic waves.Topological defects in ordered media [1] with spontaneously broken symmetry have attracted an enormous interest due to their nontrivial topology, which can play a central role in physical processes such as phase transitions [2][3][4]. Their topological origin and fundamental behavior was first described by the Kibble-Zurek mechanism as a continuous system is quenched across a phase transition into an ordered state [5,6]. In recent years, such topological defects have been extensively studied in various branches of physics. It has been shown that the topological defects with +1/2 or -1/2 topological charge can govern cell motion [7][8][9], and even arise in fatigue of materials [10]. Antivortices and vortices can be generated in three-dimensional nonporous ferroelectric structures with topological defects [11]. These exotic physical phenomena and unprecedented material properties imply that topological defects can be leveraged to explore quantum behavior of classical waves and new forms of topological orders in

Zero-angle refraction of elastic waves in metamaterials has attracted attention for its extraordinary wave collimation properties. However, earlier implementations relied on the specific flat equifrequency curve of the phononic crystals suffer from a narrow range of incident angles or operating bandwidths, which severely hinders the exploration and design of functional devices. Here, we propose an elastic near-zero refractive index metamaterial of a triangular lattice to realize topological zero refraction with arbitrary angles of incidence and wide working frequency range. Topological robustness of the zero-angle refraction of pseudospin-Hall edge state against defects is experimentally demonstrated. Furthermore, tunable wave mode conversion associated with the zero-angle refraction is revealed and discussed. These results provide a paradigm for the simultaneous control of the refraction properties of longitudinal and transverse waves that can be employed for designing the topological elastic antennas and elastic wave collimator.

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