Non-Abelian Topological Phases and Their Quotient Relations in Acoustic Systems

Non-Abelian topological phases (NATPs) exhibit enigmatic intrinsic physics distinct from wellestablished Abelian topological phases, while lacking straightforward configuration and manipulation, especially for classical waves. In this Letter, we exploit novel braiding-type couplings among a pair of triple-component acoustic dipoles, which act as functional elements with effective imaginary couplings. Sequencing them in one dimension allows us to generate acoustic NATPs in a compact yet time-reversal invariant Hermitian system. We further provide the whole phase diagram that encompasses all i, j, and k non-Abelian phases, and directly demonstrate their unique quotient relations via different end point states. Our NATPs based on real-space braiding may inspire the exploration of acoustic devices with noncommutative characters.