Acoustic mirror Hofstadter insulator with projective parity-time symmetry

In condensed matter physics, symmetry profoundly governs the fundamentals of topological matter. The emergence of new topological phase is typically linked to the enrichment of symmetries. Different parity-time symmetry relations ( PT ) 2 = ± 1 distinguish between spinless and spinful physical systems. In spinless systems, creating pseudo-spins can realize fragile topological phase but not break the time-reversal symmetry. Therefore, growing attentions were recently focused on the topological phase in spinless systems. Here we follow the previous theory and break the framework of crystallographic symmetry groups by utilizing the projective symmetry ( P M T ) 2 = −1 in a Z 2 gauge field to experimentally realize the Mirror Hofstadter Insulator (MHI) in a bilayer twisted Hofstadter model. We experimentally demonstrate that the MHI is actually a novel topological phase in classical wave systems, contradicting the popular belief that a “static” spinless system is unlikely to support Chern Insulators. In experiments, the edge modes were unambiguously observed with odd-shaped boundaries, confirming the topological features. The clockwise and anti-clockwise edge states with opposite group velocities were completely separated via an energy drain. In addition, we demonstrate that MHI has robust topological whispering gallery modes. Our work establishes a foundation for investigating exotic topological effects arising from the interplays between artificial gauge fields and wave systems.