Observation of topological phases without crystalline counterparts

Topological phases have been extensively studied primarily in crystalline systems with translational symmetry. Recent theoretical studies, however, have demonstrated the existence of topological phases in quasicrystals that are absent in crystals. Despite numerous experimental observations of topological phases in various crystalline systems, observing these phases without crystalline counterparts remains challenging due to very complex models. Here, we design a practically realizable tight-binding model with nearest-neighbor hopping on the Ammann-Beenker quasicrystalline lattice. This model respects eight-fold rotational and chiral symmetries, resulting in a higher-order topological phase with eight zero-energy corner modes that have no crystalline counterparts. We experimentally explore the topological phase in an acoustic quasicrystal. Surprisingly, we also discover symmetry-protected zero-energy modes near the center of the quasicrystal in a topologically trivial phase, a phenomenon not seen in crystals. We further experimentally observe these modes in a topologically trivial acoustic quasicrystal. Our work represents the first experimental observation of topological phases in quasicrystals without crystalline counterparts, paving the way for the study of exotic topological physics in quasicrystals.