Topological refraction in dual-band valley sonic crystals

Valley pseudospins, as quantum states of energy extrema in momentum space, have been introduced from condensed-matter systems into classical sound systems, and several valley sonic crystals (VSCs) have been realized experimentally. However, in the existing VSCs, topological kink states generally appear in a single band gap, which apparently has become an obstacle for multiband topological sound devices. To overcome this challenge, we here experimentally demonstrate dual-band VSCs, in which robust valley kink states exist in two separated bulk band gaps. More interestingly, two opposite valleys separately located in two band gaps are locked to a single propagation direction, which arises from the fact that the bands below two band gaps show opposite valley Chern numbers at the K/K\ensuremath{'} valley. This double valley-locking phenomenon has been demonstrated via measuring the topological refraction of the kink states into the ambient space at a zigzag termination. We observe positive refraction at the lower band gap, whereas the coexistence of positive and negative refraction at the higher band gap. Additionally, we observe the robust valley transport through the sharp corners at two band gaps. The designed VSCs with the dual-band topological refraction and robust valley transport could find potential applications in multiband and multidirectional devices.