Diagnosing nonergodic extended states and many body localization through real-space and Fock-space excitations

We provide real-space and Fock-space (FS) characterizations of nonergodic extended (NEE) phases in two kinds of interacting quasiperiodic chains possessing a mobility edge and critical states in the noninteracting limits of the two models, respectively. We show that the single-particle (SP) excitations in the interacting systems show similar behaviour as that in the noninteracting systems, reflecting the different origins of NEE phases in the two models. In contrast, all SP excitations get delocalized and localized in the ergodic and MBL phases, respectively. We give complementary insights into the distinction of the NEE states from the ergodic and MBL states by computing local FS self-energies and decay length associated, respectively, with the local and the non-local FS propagators. Based on a finite-size scaling analysis of the typical local self-energy, we show that MBL and NEE states exhibit qualitatively similar multifractal character. However, we find that the NEE and MBL states can be distinguished in terms of the decay of the non-local propagator in the FS, whereas the typical local FS self-energy can merely tell them apart.