Tracking locality in the time evolution of disordered systems

Using local density correlation functions for a one-dimensional spin system, we introduce a correlation function difference (CFD) which compares correlations on a given site between a full system of size L and its restriction to pound < L sites. We show that CFD provides useful information on transfer of information in quantum many-body systems by considering the examples of ergodic, Anderson, and many-body localized regimes in a disordered XXZ spin chain. In the ergodic phase, we find that the propagation of CFD is asymptotically faster than the spin transport but slower than the ballistic propagation implied by the Lieb-Robinson bound. In contrast, in the localized cases, we unravel an exponentially slow relaxation of CFD. Connections between CFD and other observables detecting nonlocal correlations in the system are discussed.