Ideal acoustic quantum spin Hall phase in a multi-topology platform

Fermionic time-reversal symmetry ( $${T}_{f}$$ )-protected quantum spin Hall (QSH) materials feature gapless helical edge states when adjacent to arbitrary trivial cladding materials. However, due to symmetry reduction at the boundary, bosonic counterparts usually exhibit gaps and thus require additional cladding crystals to maintain robustness, limiting their applications. In this study, we demonstrate an ideal acoustic QSH with gapless behaviour by constructing a global Tf on both the bulk and the boundary based on bilayer structures. Consequently, a pair of helical edge states robustly winds several times in the first Brillouin zone when coupled to resonators, promising broadband topological slow waves. We further reveal that this ideal QSH phase behaves as a topological phase transition plane that bridges trivial and higher-order phases. Our versatile multi-topology platform sheds light on compact topological slow-wave and lasing devices. Here the authors investigate a comprehensive topological phase diagram of bilayer hexagonal acoustic lattice, including ideal quantum spin Hall phase with gapless helical edge states. They realize a broadband topological slow wave.