  Recent numerical work by Bardarson et. al. [Phys. Rev. Lett. 109, 017202
(2012)] revealed a slow, logarithmic in time, growth of entanglement entropy
for initial product states in a putative many-body localized phase. We show
that this surprising phenomenon results from the dephasing due to exponentially
small interaction-induced corrections to the eigenenergies of different states.
For weak interactions, we find that the entanglement entropy grows as \xi ln
(Vt/\hbar), where V is the interaction strength, and \xi is the single-particle
localization length. The saturated value of the entanglement entropy at long
times is determined by the participation ratios of the initial state over the
eigenstates of the subsystem. The proposed mechanism is illustrated with
numerical simulations of small systems. Our work shows that the logarithmic
entanglement growth is a universal phenomenon characteristic of the many-body
localized phase in any number of spatial dimensions, and reveals a broad
hierarchy of dephasing time scales present in such a phase.
