Dynamics of many-body delocalization in the time-dependent Hartree-Fock approximation

We explore dynamics of disordered and quasi-periodic interacting lattice models using a self-consistent time-dependent Hartree-Fock (TDHF) approximation, accessing both large systems (up to L = 400 sites) and very long times (up to t = 10⁵). We find that, in the t → ∞ limit, the many-body localization (MBL) is always destroyed within the TDHF approximation. At the same time, this approximation provides important information on the long-time character of dynamics in the ergodic side of the MBL transition. Specifically, for one-dimensional (1D) disordered chains, we find slow power-law transport up to the longest times, supporting the rare-region (Griffiths) picture. The information on this subdiffusive dynamics is obtained by the analysis of three different observables - temporal decay ∼t^-β of real-space and energy-space imbalances as well as domain wall melting - which all yield consistent results. For two-dimensional (2D) systems, the decay is faster than a power law, in consistency with theoretical predictions that β grows as log t for the decay governed by rare regions. At longest times and moderately strong disorder, β approaches the limiting value β = 1 corresponding to 2D diffusion. In quasi-periodic (Aubry-André) 1D systems, where rare regions are absent, we find considerably faster decay that reaches the ballistic value β = 1 , which provides further support to the Griffiths picture of the slow transport in random systems.