Observation of topological phase with critical localization in a quasi-periodic lattice

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.