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A local excitation in a quantum many-particle system evolvesdeterministically. A time-reversal procedure, involving the invertion of thesigns of every energy and interaction, should produce an excitation revival:the Loschmidt echo (LE). If somewhat imperfect, only a fraction of theexcitation will refocus. We use such a procedure to show how non-inverted weakdisorder and interactions, when assisted by the natural reversible dynamics,fully degrade the LE. These perturbations enhance diffusion and evenlydistribute the excitation throughout the system. Such a dynamical paradigm,called ergodicity, breaks down when either the disorder or the interactions aretoo strong. These extreme regimes give rise to the well known Andersonlocalization and Mott insulating phases, where quantum diffusion becomesrestricted. Accordingly, regardless of the kinetic energy terms, the excitationremains mainly localized and out-of-equilibrium, and the system behavesnon-ergodically. The LE constitutes a fair dynamical witness for the wholephase diagram since it evidences a surprising topography in which ergodic andnon-ergodic phases interpenetrate each other. Furthermore, we provide anestimation for the critical lines separating the ergodic and non-ergodic phasesaround the Mott and Anderson transitions. The energy uncertainties introducedby disorder and interaction shift these thresholds towards strongerperturbations. Remarkably, the estimations of the critical lines are in goodagreement with the phase diagram derived from the LE dynamics.

Role of energy uncertainties in ergodicity breaking induced by competing interactions and disorder. A dynamical assessment through the Loschmidt echo