Density propagator for many-body localization: finite size effects, transient subdiffusion, and exponential decay

We investigate charge relaxation in quantum-wires of spin-less disordered fermions ($t{-}V$-model). 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 $\Delta x_\varepsilon(t)$ of $\Pi_\varepsilon(x,t)$ exhibits a behavior $d\ln \Delta x_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)$, where the exponent function $\beta_\varepsilon(t){\lesssim}1/2$ is seen to depend strongly on $L$ at all investigated parameter combinations. (i) We confirm the existence of a region in phase space that exhibits subdiffusive dynamics in the sense that $\beta_\varepsilon{<}1/2$ in large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with $\beta_{\varepsilon}{=}1/2$. (ii) We cannot confirm the existence of many-body mobility edges even in regions of the phase-diagram that have been reported to be deep in the delocalized phase. (iii) (Transient) subdiffusion $0<\beta_\varepsilon(t)\lesssim 1/2$, coexists with an enhanced probability for returning to the origin, $\Pi_\varepsilon(0,t)$, decaying much slower than $1/\Delta x_\varepsilon (t)$. Correspondingly, the spatial decay of $\Pi_\varepsilon(x,t)$ is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.