Entanglement dynamics with diffusive transport

We investigate how the incoherent (diffusive) transport of conserved quantities affects the growth of entanglement in generic non-integrable systems. We develop a general picture by considering random unitary circuit dynamics with a U(1) symmetry which allows for an efficient numerical treatment and, in a certain coarse-grained limit, leads to a simple equation of motion for the entanglement entropy, which takes the form of a random, space-time dependent surface growth model. Based on this we argue that an initial state with large-scale inomogeneities in the conserved densities gives rise to an uneven entanglement profile across the system, with features whose "height" and "width" both grow as t^1/2 . We confirm this prediction by numerically investigating a variety of different initial states, both in the random circuit model away from the coarse-grained limit, and in a deterministic spin chain. Moreover, we investigate the effects of charge-fluctuation and find that these can lead to a slow-down in entanglement growth even for spatially homogenous states.