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 ($t\text{−}V$ model). Our observable is the time-dependent density propagator ${\mathrm{\Pi }}_{ϵ}(x,t)$, calculated in windows of different energy density $ϵ$ of the many-body Hamiltonian and at different disorder strengths $W$, not exceeding the critical value $W_c$. The width ${\mathrm{\Delta }x}_{ϵ}(t)$ of ${\mathrm{\Pi }}_{ϵ}(x,t)$ exhibits a behavior $d\ln {\mathrm{\Delta }x}_{ϵ}(t)/d\ln t={\beta }_{ϵ}(t)$, where the exponent function ${\beta }_{ϵ}(t)\lesssim 1/2$ 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 ${\beta }_{ϵ}(t)<1/2$ in a large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with ${\beta }_{ϵ}=1/2$. (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 $0<{\beta }_{ϵ}(t)\lesssim 1/2$ coexists with an enhanced probability for returning to the origin ${\mathrm{\Pi }}_{ϵ}(0,t)$, decaying much slower than $1/{\mathrm{\Delta }x}_{ϵ}(t)$. Correspondingly, the spatial decay of ${\mathrm{\Pi }}_{ϵ}(x,t)$ 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.

Figure 1

A qualitative phase diagram of different dynamical regions in the disorder energy-density plane of the $t\text{−}V$ model. At disorder strength $W$ below the many-body localization transition $W_c$, we propose a transient subdiffusive, weakly ergodic dynamical regime with an anomalously slow decay of the return probability.

Figure 2

(a1)–(a3) The time evolution of ${\mathrm{\Delta }x}_{ϵ}(t)$ at $W=2.5$ and $V=1$ near the lower band edge (upper row, $ϵ=0.1$) in the center region (center row, 0.5) and near the upper band edge (lower row, 0.875) for system sizes $L=16$, 20, 24 (dashed traces blue, red, green). Also shown are noninteracting reference traces for $L=16$, 20 ($V=0$, solid lines). (b1)–(b3) Replotting (a1)–(a3) as $d\ln \mathrm{\Delta }{x}_{ϵ}(t)/d\ln t$ over $\mathrm{\Delta }{x}_{ϵ}(t)/\mathrm{\Delta }{x}_{ϵ}^{(0)}$ to highlight finite-size effects. Inset shows the enlargement of the (b3) data for better visibility of trends, including system sizes $L=16$, 18, 20, 22, 24 (bottom to top). (c1)–(c3) Probability to return to the origin. The legends in this column also give the three system sizes in units of the bare localization length. (In all calculations we fix the width of the energy window $\mathrm{\Delta }ϵ=0.1$ [49]).

Figure 3

(a1)–(a3) The density propagator ${\mathrm{\Pi }}_{ϵ}(x,t)$ in the delocalized regime ($ϵ=0.1$, 0.5, 0.875, $W=2.5$, $L=24$) at two times $t=40$, 100. The log-normal plot illustrates non-Gaussian shape. Solid line in (a3) shows a stretched exponential fit with an exponent $\approx 0.7$. (b1)–(d3) The corresponding memory kernel ${\kappa }_{ϵ}(q,t)=q^2/[{\mathrm{\Pi }}_{ϵ}^{-1}(q,t)-1]$, see also Eq. (6), for the case without (b1)–(b3) and with interactions (c1)–(d3). The structure at larger wave numbers illustrates the (nonexponential) short-distance behavior. The absence of effects in time (and system size, not shown) highlights the localized character of the noninteracting kernel ${\kappa }_{ϵ}^{(0)}$. In contrast, the evolution of the interacting kernel is the hallmark of delocalization. (d1)–(d3) The $L$ dependence of ${\kappa }_{ϵ}(t)$.