  We study a model of interacting fermions in one dimension subject to random,
uncorrelated onsite disorder. The model realizes an interaction-driven quantum
phase transition between an ergodic and a many-body localized phase (MBL). We
propose a single-particle framework to characterize these phases by the
eigenstates (the natural orbitals) and the eigenvalues (the occupation
spectrum) of the one-particle density matrix (OPDM) in many-body eigenstates.
As a main result, we find that the natural orbitals are localized in the MBL
phase, but delocalized in the ergodic phase. This qualitative change in these
single-particle states is a many-body effect, since without interactions the
single-particle energy eigenstates are all localized. The occupation spectrum
in the ergodic phase is thermal in agreement with the eigenstate thermalization
hypothesis, while in the MBL phase the occupations preserve a discontinuity at
an emergent Fermi edge. This suggests that the MBL eigenstates are weakly
dressed Slater determinants, with the eigenstates of the underlying Anderson
problem as reference states. We discuss the statistical properties of the
natural orbitals and of the occupation spectrum in the two phases and as the
transition is approached. Our results are consistent with the existing picture
of emergent integrability and localized integrals of motion, or quasiparticles,
in the MBL phase. We emphasize the close analogy of the MBL phase to a
zero-temperature Fermi liquid: in the studied model, the MBL phase is
adiabatically connected to the Anderson insulator and the occupation-spectrum
discontinuity directly indicates the presence of quasiparticles localized in
real space. Finally, we show that the same picture emerges for interacting
fermions in the presence of an experimentally-relevant bichromatic lattice and
thereby demonstrate that our findings are not limited to a specific model.
