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

Highlights: • TDHF dynamics for 1D random, 1D quasi-periodic, and 2D random interacting systems. • Characterization by running power-law exponent up to very long times (10⁵). • Slow subdiffusive dynamics of many-body delocalization in random systems. • Importance of rare spots (Griffiths physics) for subdiffusive dynamics demonstrated. • Absence of strict many-body localization within TDHF 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 $t=10^5$). We find that, in the $t\to \infty$ 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 $\sim t^{-\beta }$ 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 $\beta$ grows as $logt$ for the decay governed by rare regions. At longest times and moderately strong disorder, $\beta$ approaches the limiting value $\beta =1$ 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 $\beta =1$, which provides further support to the Griffiths picture of the slow transport in random systems.