Lossy phononic metamaterials for valley nonreciprocity.

Non-Hermitian physics - characterized by complex band spectra - established a new paradigm in condensed matter systems and metamaterials. Combined to nonreciprocity, non-Hermitian gain can be arbitrarily injected in a system to manipulate its valley dynamics, leading to phenomena such as the amplified topological whispering gallery modes for acoustic lasing. Here, we reveal that the coupling loss can also be used to manipulate valley degrees of freedom in a phononic metamaterial. We theoretically and experimentally show three distinct effects induced by valley nonreciprocity, including unidirectional transport of valley bulk states that functions as a valley filter, valley-dependent skin effects, where bulk states from different valleys localize at opposite boundaries, and valley-projected edge states with boundary-dependent lifetimes that leads to an anomalous beam routing. Owing to the simple control over losses demonstrated here, our findings shed light on the interplay between non-Hermitian and valley physics, providing a route to applications of valley-based devices.