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We study the many-body localization aspects of single-particle mobility edgesin fermionic systems. We investigate incommensurate lattices and randomdisorder Anderson models. Many-body localization and quantum nonergodicproperties are studied by comparing entanglement and thermal entropy, and bycalculating the scaling of subsystem particle number fluctuations,respectively. We establish a nonergodic extended phase as a genericintermediate phase (between purely ergodic extended and nonergodic localizedphases) for the many-body localization transition of non-interacting fermionswhere the entanglement entropy manifests a volume law (`extended'), but thereare large fluctuations in the subsystem particle numbers (`nonergodic'). Weargue such an intermediate phase scenario may continue holding even for themany-body localization in the presence of interactions as well. We find formany-body states in non-interacting 1d Aubry-Andre and 3d Anderson models thatthe entanglement entropy density and the normalized particle-number fluctuationhave discontinuous jumps at the localization transition where the entanglemententropy is sub-thermal but obeys the "volume law". In the vicinity of thelocalization transition we find that both the entanglement entropy and theparticle number fluctuations obey a single parameter scaling. We argue usingnumerical and theoretical results that such a critical scaling behavior shouldpersist for the interacting many-body localization problem with importantconsequences. Our work provides persuasive evidence in favor of there being twotransitions in many-body systems with single-particle mobility edges, the firstone indicating a transition from the purely localized nonergodic many-bodylocalized phase to a nonergodic extended many-body metallic phase, and thesecond one being a transition eventually to the usual ergodic many-bodyextended phase.

Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge