Super-Poissonian Behavior of the Energy Specrum in the Non-Ergodic Extended Regime

Non-ergodic extended phase between the ergodic extended metallic phase and the localized phase has drawn much recent interest in different fields such as Such a the Sachdev-Ye-Kitaev model in high-energy physics and quantum gravity, to the interacting many-body localization in condensed matter physics and quantum computing. This phase is characterized by fractal behavior of the wavefunctions, and a postulated correlated mini-band structure of the energy spectrum. The signature in the spectra is expected on intermediate energy scales. Nevertheless, due to the Thouless energy and ambiguities in the unfolding procedure, the usual methods for characterizing these scales, e.g., rigidity or number variance, are inconclusive. We show that by using the singular value decomposition method, clear evidence for a super-Poissonian behavior in this regime emerges, for the Rosenzweig-Porter model as well as for anisotopic spin chains in the many-particle delocalized regime. This behavior is consistent with a picture of correlated mini-bands.