Time evolution of non-ergodic extended states

Increasing onsite disorder in an isolated Heisenberg spin-1/2 chain leads to the transition from chaos to many-body localization. We obtain a semi-analytical expression for the entire evolution of the survival probability in the intermediate region between chaos and localization. This expression captures the quantity's initial power-law decay, as well as the correlation hole and the ramp toward saturation. We find very good agreement with our numerical results.