Searching for anomalous spin transport in the disordered XXZ Heisenberg chain

We study the spin transport in the ergodic phase of the disordered XXZ Heisenberg chain, an archetypal model for many-body localization proposed to show subdiffusive transport. We analyze the statistics of the spin autocorrelation function by applying the dissipation-assisted operator evolution method, granting access to large system sizes. We find that our results lie on the boundary between diffusion and weak subdiffusion. We gain more insight into the quantum data by considering a classical single particle random walk, displaying a tunable subdiffusive behavior. Our analytical and numerical results for this classical model reveal three distinct transport regimes, characterized by subiffusive behavior, by diffusion with anomalous power law fluctuations in the statistics of the return probability, and by standard diffusion, respectively. By comparing to the behavior of the disordered Heisenberg chain, we find that the distribution of the spin autocorrelation function shows a power law tail, consistent with weak subdiffusion or diffusion with anomalous fluctuations. Our results highlight the challenges in detecting anomalous transport from finite size and finite time quantum simulations.