  In this work, we develop a self-consistent Hartree-Fock approach to
theoretically study the far-from-equilibrium quantum dynamics of interacting
fermions, and apply this approach to explore the onset of many-body
localization (MBL) in these systems. We investigate the dynamics of a state
with a nonequilibrium density profile; we find that at weak disorder the
density profile equilibrates rapidly, whereas for strong disorder it remains
frozen on the accessible timescales. We analyze this behavior in terms of the
Hartree-Fock self-energy. At weak disorder the self-energy fluctuates strongly
and can be interpreted as a self-consistent noise process. By contrast, at
strong disorder the self-energy evolves with a few coherent oscillations which
cannot delocalize the system. Accordingly, the non-equilibrium site-resolved
spectral function shows a broad spectrum at weak disorder and sharp spikes at
strong disorder. Our Hartree-Fock theory incorporates spatial fluctuations and
rare-region effects. As a consequence, we find subdiffusive relaxation in
random systems; but, when the system is subjected to weak quasi-periodic
potentials, the subdiffusive response ceases to exist, as rare region effects
are absent in this case. This self-consistent Hartree-Fock approach can be
regarded as a relatively simple theory that captures much of the MBL
phenomenology.
