Density Propagator for Many-Body Localization: Finite-Size Effects, Transient Subdiffusion, and Exponential Decay

We investigate charge relaxation in quantum wires of spinless disordered fermions (t-V model). Our observable is the time-dependent density propagator Pi(epsilon)(x.t), calculated in windows of different energy density epsilon of the many-body Hamiltonian and at different disorder strengths W, not exceeding the critical value W-c. The width Delta x(epsilon) (t) of Pi(epsilon)(x,t) exhibits a behavior d ln Delta x(epsilon) (t) / d ln t = beta(epsilon)(t) where the exponent function beta(epsilon)(t) less than or similar to 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(epsilon)(t) < 1/2 in a large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with beta(epsilon) = 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(epsilon)(t) less than or similar to 1/2 coexists with an enhanced probability for returning to the origin. Pi(epsilon),(0,t) decaying much slower than 1/Delta x(epsilon) (t) Correspondingly, the spatial decay of Pi(epsilon)(x,t) is far from Gaussian, being exponential or even slower. On a phenomenological level, our findings are broadly consistent with the effects of strong disorder and (fractal) Griffiths regions.