Dynamical obstruction to localization in a disordered spin chain

We analyze a one-dimensional XXZ spin chain in a disordered magnetic field. As the main probes of the system's behavior we use the sensitivity of eigenstates to adiabatic transformations, as expressed through the fidelity susceptibility, in conjunction with the low frequency asymptotes of the spectral function. We find that even in the localized regime at strong disorder the eigenstates remain exponentially sensitive in the system size to changes in local magnetic field. We further find strong evidence that the system always remains delocalized in the thermodynamic limit. Instead at moderate values of disorder it enters a universal subdiffusive relaxation regime, where the spectral function of the local longitudinal magnetization is inversely proportional to the frequency, corresponding to logarithmic in time relaxation of its auto-correlation function. This scaling of the spectral function is incompatible with the localization transition in the thermodynamic limit. Our findings are consistent with recent works of J. Suntajs et al.