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.

我们使用自洽时间相关的Hartree-Fock（TDHF）近似方法，探索无序和准周期相互作用晶格模型的动力学，同时访问两个大型系统（直至 $\mathrm{大号}=400$ 网站）和很长的时间（最多 $Ť=\mathrm{1个}{0}^5$）。我们发现，在$Ť\to \infty$在极限情况下，多体定位（MBL）始终在TDHF逼近范围内被破坏。同时，这种近似提供了有关MBL过渡的遍历一侧动力学的长期特征的重要信息。具体来说，对于一维（1D）无序链，我们发现直到最长时间的幂律传输速度都很慢，这支持了稀有区域（Griffiths）的情况。通过分析三个不同的可观测量（时间衰减）获得有关此亚扩散动力学的信息$〜{Ť}^{-\beta }$真实空间和能量空间的不平衡以及畴壁融化，这些都会产生一致的结果。对于二维（2D）系统，其衰减比幂定律快，这与理论预测一致：$\beta$ 成长为 $日志Ť$对于由稀有地区控制的衰变。在最长的时间和中等强度的疾病中，$\beta$ 接近极限值 $\beta =\mathrm{1个}$对应于2D扩散。在缺少稀有区域的准周期（Aubry-André）1D系统中，我们发现衰减更快，达到弹道值$\beta =\mathrm{1个}$，这为格里菲斯关于随机系统中缓慢传输的图景提供了进一步的支持。