Disorder and localization have dramatic influence on the topological properties of a quantum system. While strong disorder can close the band gap thus depriving topological materials of topological features, disorder may also induce topology from trivial band structures, wherein topological invariants are shared by completely localized states. Here we experimentally investigate a fundamentally distinct scenario where topology is identified in a critically localized regime, with eigenstates neither fully extended nor completely localized. Adopting the technique of momentum-lattice engineering for ultracold atoms, we implement a one-dimensional, generalized Aubry-André model with both diagonal and off-diagonal quasi-periodic disorder in momentum space, and characterize its localization and topological properties through dynamic observables. We then demonstrate the impact of interactions on the critically localized topological state, as a first experimental endeavor toward the clarification of many-body critical phase, the critical analogue of the many-body localized state.

无序和局域化对量子系统的拓扑性质有显着影响。虽然强无序可以关闭带隙从而剥夺拓扑材料的拓扑特征，但无序也可能从微不足道的带结构诱导拓扑，其中拓扑不变量由完全局部化的状态共享。在这里，我们通过实验研究了一个根本不同的场景，在该场景中，拓扑结构在临界局部区域中被识别，本征态既不完全扩展也不完全局部化。采用超冷原子的动量晶格工程技术，我们在动量空间中实现了一个具有对角线和非对角线准周期无序的一维广义 Aubry-André 模型，并通过动态观测表征其局域化和拓扑性质。