Dephasing enhanced spin transport in the ergodic phase of a many-body localizable system

We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime when bulk dephasing is present. We find that while dephasing always renders the transport diffusive, there is nonetheless a remnant of the diffusive to sub-diffusive transition found in a system without dephasing manifested in the behaviour of the diffusion constant with the dephasing strength. By studying finite-size effects we show numerically and theoretically that this feature is caused by the competition between large crossover length scales associated to disorder and dephasing that control the dynamics observed in the thermodynamic limit. We demonstrate that this competition may lead to a dephasing enhanced transport in this model.