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

Non-Abelian topological phases (NATPs) are highly sought-after candidate states for quantum computing and communication while lacking straightforward configuration and manipulation, especially for classical waves. In this work, we exploit novel braid-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 reciprocal 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 endpoint states. Our NATPs based on real-space braiding may inspire the exploration of acoustic devices with non-commutative characters.