Reflection and refraction of waves occur at the interface between two different media. These two fundamental interfacial wave phenomena form the basis of fabricating various wave components, such as optical lenses. Classical refraction-now referred to as positive refraction-causes the transmitted wave to appear on the opposite side of the interface normal compared to the incident wave. By contrast, negative refraction results in the transmitted wave emerging on the same side of the interface normal. It has been observed in artificial materials, following its theoretical prediction, and has stimulated many applications including super-resolution imaging. In general, reflection is inevitable during the refraction process, but this is often undesirable in designing wave functional devices. Here we report negative refraction of topological surface waves hosted by a Weyl phononic crystal-an acoustic analogue of the recently discovered Weyl semimetals. The interfaces at which this topological negative refraction occurs are one-dimensional edges separating different facets of the crystal. By tailoring the surface terminations of the Weyl phononic crystal, constant-frequency contours of surface acoustic waves can be designed to produce negative refraction at certain interfaces, while positive refraction is realized at different interfaces within the same sample. In contrast to the more familiar behaviour of waves at interfaces, unwanted reflection can be prevented in our crystal, owing to the open nature of the constant-frequency contours, which is a hallmark of the topologically protected  surface states in Weyl crystals.

Very recently, novel quasiparticles beyond those mimicking the elementary high-energy particles such as Dirac and Weyl fermions have attracted great interest in condensed matter physics and materials science 1-9 . Here we report the first experimental observation of the long-desired quadratic Weyl points 10 by using a three-dimensional chiral metacrystal of sound waves. Markedly different from the newly observed unconventional quasiparticles 5-9 , such as the spin-1 Weyl points and the charge-2 Dirac points that are featured respectively with threefold and fourfold band crossings, the charge-2 Weyl points identified here are simply twofold degenerate, and the dispersions around them are quadratic in two directions and linear in the third one 10 . Besides the essential nonlinear bulk dispersions, we further unveil the exotic double-helicoid surface arcs that emanate from a projected quadratic Weyl point and terminate at two projected conventional Weyl points through Fourier transformation of the scanned surface fields. This unique global surface connectivity provides conclusive evidence for the double topological charges of such unconventional topological nodes.

Dirac semimetals, the materials featured with discrete linearly crossing points (called Dirac points) between four bands, are critical states of topologically distinct phases.Such gapless topological states have been accomplished by a band-inversion mechanism, in which the Dirac points can be annihilated pairwise by perturbations without changing the symmetry of the system. Here, we report an experimental observation of Dirac points that are enforced completely by the crystal symmetry, using a nonsymmorphic three-dimensional phononic crystal. Intriguingly, our Dirac phononic crystal hosts four spiral topological surface states, in which the surface states of opposite helicities intersect gaplessly along certain momentum lines, as confirmed by our further surface measurements. The novel Dirac system may release new opportunities for studying the elusive (pseudo)relativistic physics, and also offer a unique prototype platform for acoustic applications.

Recently, the topological physics in acoustics has been attracting much attention. However, all the studies are aimed to elastic or airborne sound systems.Realizing topological insulators for underwater sound is of great importance, since water is another crucial sound medium in addition to solid and air. Here we report an experimental study on the valley-projected edge states for underwater sound. The edge states are directly observed in our ultrasound scanning experiments, together with a solid evidence for the valley-selective excitation. The experimental data agree well with our numerical results. Prospective applications can be anticipated, such as for underwater sound signal processing and ocean noise control.  Corresponding author. cyqiu@whu.edu.cn 2 / 12

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