Non-ergodic Extended Regime from the Energy-dependent Localization Length

The nature of the non-ergodic extended (NEE) regime in many-body interacting systems has been studied in the last few years. While Phys. Rev. Lett. 115, 186601 highlights a connection between the NEE regime and the mobility edge, Phys. Rev. B 109, 214309 argues that the NEE regime arises primarily due to the spatial structure of the potential in finite-size systems. In this work, we point out that these different understandings of NEE behavior in finite-size systems can be unified as one--the energy dependence of the localizations length of the eigenstates in the corresponding single-particle model, where the SPME can be viewed as a special case. We consider different models with or without single-particle mobility edges, but their single-particle localizations are all energy-dependent. We observe similar NEE behaviors in these models under finite size by analyzing the statistical properties of entanglement entropy.