Mean-field theory of nearly many-body localized metals

We develop a mean-field theory of the metallic phase near the many-body localization (MBL) transition, using the observation that a system near the MBL transition should become an increasingly slow heat bath for its constituent parts. As a first step, we consider the properties of a many-body localized system coupled to a generic ergodic bath whose characteristic dynamical time scales are much slower than those of the system. As we discuss, a wide range of experimentally relevant systems fall into this class; we argue that relaxation in these systems is dominated by collective many-particle rearrangements, and compute the associated time scales and spectral broadening. We then use the observation that the self-consistent environment of any region in a nearly localized metal can itself be modeled as a slowly fluctuating bath to outline a self-consistent mean-field description of the nearly localized metal and the localization transition. In the nearly localized regime, the spectra of local operators are highly inhomogeneous and the typical local spectral linewidth is narrow. The local spectral linewidth is proportional to the dc conductivity, which is small in the nearly localized regime. This typical linewidth and the dc conductivity go to zero as the localized phase is approached, with a scaling that we calculate, and which appears to be in good agreement with recent experimental results.