Dynamical detection of extended non-ergodic states in many-body quantum systems

We show that the exponent of the power-law decay observed in the time-evolution of certain observables of one-dimensional interacting quantum systems, in the vicinity of the many-body localization transition, is connected to the multifractal dimension $D_2$. This dimension serves as an indicator of correlations among components of the initial state from which the dynamics originate and its a hallmark of the presence of extended but non-ergodic states. To illustrate this, we present results for the XXZ model under a quasiperiodic Harper's potential and in the presence of on-site quenched disorder.