  We study the localization problem of one-dimensional interacting spinless
fermions in an incommensurate optical lattice, which changes from an extended
phase to a nonergoic many-body localized phase by increasing the strength of
the incommensurate potential. We identify that there exists an intermediate
regime before the system enters the many-body localized phase, in which both
the localized and extended many-body states coexist, thus the system is divided
into three different phases, which can be characterized by normalized
participation ratios of the many-body eigenstates and distributions of natural
orbitals of the corresponding one-particle density matrix. This is very
different from its noninterating limit, in which all eigenstaes undergo a
delocaliztion-localization transtion when the strength of the incommensurate
potential exceeds a critical value.
