“…Moreover, the original Dirac cones at the K and K’ points are opened to form a complete photonic bandgap, resulting in the edge states at the interfaces of different valley Hall phases [ 23 , 24 , 25 ], which are immune to backscattering [ 28 , 29 ] caused by defects and acute light channels. In recent years, valley topological insulators (TIs) have been widely studied in optical, acoustic, and electronic systems, such as valley topological robust transport [ 18 , 19 , 20 ], topological photon routing [ 24 , 30 ], unidirectional light transport [ 24 ], valley topological edge state frequency tuning [ 13 ], valley topological acoustic wave group velocity modulation based on phononic crystals [ 13 ], and topological spin–valley filtering effects [ 31 , 32 ]. These studies have opened up unprecedented application opportunities for valley TIs in the fields of tunable acoustics, topological photonics, and the emerging field of nontrivial states.…”