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Inspired by recent experiments on many-body localized systems coupled to anenvironment, we apply a Flow Equation method to study the problem of a disorderchain of spinless fermions, coupled via density-density interactions to asecond clean chain of spinless fermions. In particular, we focus on theconditions for the onset of a many-body localized phase in the clean sector ofour model by proximity to the dirty one. We find that a many-body localizationproximity effect in the clean component is established when the density ofdirty fermions exceeds a threshold value. From the flow equation method we findthat, similar to many-body localization in a single chain, the many-bodylocalization proximity effect is also described by an extensive set of localintegrals of motion. Furthermore, by tuning the geometry of the inter-chaincouplings, we show that the dynamics of the model is ruled, on intermediatetime scales, by an emergent set of quasi-conserved charges.

Exploring many-body localization in quantum systems coupled to an environment via Wegner-Wilson flows