Density propagator for many-body localization: finite size effects, transient subdiffusion, (stretched-) exponentials

We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain. Our observable is the time-dependent density propagator, $\Pi_{\varepsilon}(x,t)$, calculated in windows of different energy density, $\varepsilon$, of the many-body Hamiltonian and at different disorder strengths, $W$, not exceeding the critical value $W_\text{c}$. The width $\dex_\varepsilon(t)$ of $\Pie(x,t)$ exhibits a behavior $d\ln \dex_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)$, where $\beta_\varepsilon(t){\lesssim}1/2$ is seen to depend strongly on $L$ at all investigated parameter combinations. (i) We do not find a region in phase space that exhibits subdiffusive dynamics in the sense that $\beta{<}1/2$ in the thermodynamic limit. Instead, subdiffusion may be transient, giving way eventually to conventional diffusive behavior, $\beta{=}1/2$. (ii) (Transient) subdiffusion $0{<}\beta_\varepsilon(t)\lesssim 1/2$, coexists with an enhanced probability for returning to the origin, $\Pie(0,t)$, decaying much slower than $1/\dex_\varepsilon (t)$. Correspondingly, the spatial decay of $\Pie(x,t)$ is far from Gaussian, i.e. exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and Griffiths regions.