Chess-board Acoustic Crystals with Momentum-space Nonsymmorphic Symmetries

Spatial symmetries appearing in both real and momentum space are of fundamental significance to crystals. However, in the conventional framework, every space group in real space, either symmorphic or nonsymmorphic, corresponds to a symmorphic dual in momentum space. Our experiment breaks the framework by showing that in a 2D acoustic crystal with chess-board pattern of $\pi$ and 0 fluxes, mirror reflections are manifested nonsymmorphically as glide reflections in momentum space. These momentum-space nonsymmorphic symmetries stem from projective, rather than ordinary, representations of the real-space symmetries due to the peculiar flux pattern. Moreover, our experiment demonstrates that the glide reflection can reduce the topological type of the Brillouin zone from the torus to the Klein bottle, resulting in novel topological phases with new topological invariants. Since crystalline topologies are based on momentum-space symmetries, our work paves the way for utilizing engineerable gauge fluxes over artificial crystals to extend the current topological classifications into the broader regime of momentum-space nonsymmorphic symmetries.