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Recent experimental and theoretical efforts have focused on the effect ofdissipation on quantum many-body systems in their many-body localized (MBL)phase. While in the presence of dephasing noise such systems reach a uniqueergodic state, their dynamics is characterized by slow relaxation manifested innon-exponential decay of self-correlations. Here we shed light on a currentlymuch debated issue, namely the role of interactions for this relaxationdynamics. We focus on the experimentally relevant situation of the evolutionfrom an initial charge density wave in the presence of strong dephasing noise.We find a crossover from a regime dominated by disorder to a regime dominatedby interactions, with a concomitant change of time correlators from stretchedexponential to compressed exponential form. The strongly interacting regime canbe explained in terms of nucleation and growth dynamics of relaxing regions -reminiscent of the kinetics of crystallization in soft matter systems - andshould be observable experimentally. This interaction-driven crossover suggeststhat the competition between interactions and noise give rise to a much richerstructure of the MBL phase than anticipated so far.

Role of interactions in a dissipative many-body localized system