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Many-body localization is a fascinating theoretical concept describing theintricate interplay of quantum interference, i.e. localization, with many-bodyinteraction induced dephasing. Numerous computational tests and also severalexperiments have been put forward to support the basic concept. Typically,averages of time-dependent global observables have been considered, such as thecharge imbalance. We here investigate within the disordered spin-less Hubbard($t-V$) model how dephasing manifests in time dependent variances ofobservables. We find that after quenching a N\'eel state the local chargedensity exhibits strong temporal fluctuations with a damping that is sensitiveto disorder $W$: variances decay in a power law manner, $t^{-\zeta}$, with anexponent $\zeta(W)$ strongly varying with $W$. A heuristic argument suggeststhe form, $\zeta\approx\alpha(W)\xi_\text{sp}$, where $\xi_\text{sp}(W)$denotes the noninteracting localization length and $\alpha(W)$ characterizesthe multifractal structure of the dynamically active volume fraction of themany-body Hilbert space. In order to elucidate correlations underlying thedamping mechanism, exact computations are compared with results from thetime-dependent Hartree-Fock approximation. Implications for experimentallyrelevant observables, such as the imbalance, will be discussed.

Dephasing in strongly disordered interacting quantum wires