Slow dynamics and strong finite-size effects in many-body localization with random and quasi-periodic potential

Abstract

We investigate charge relaxation in disordered and quasi-periodic quantum-wires of spin-less fermions ($t-V$-model) at different inhomogeneity strength $W$ in the localized and nearly-localized regime. Our observable is the time-dependent density correlation function, $\Phi (x,t)$, at infinite temperature. We find that disordered and quasi-periodic models behave qualitatively similar: Although even at longest observation times the width $\Delta x(t)$ of $\Phi (x,t)$ does not exceed significantly the non-interacting localization length, ${\xi }_0$, strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. As a relatively reliable indicator for the boundary towards the many-body localized (MBL) regime even under these conditions, we consider the exponent function $\beta (t)=d\ln \Delta x(t)/d\ln t$. Motivated by our numerical data for $\beta$, we discuss a scenario in which the MBL-phase splits into two subphases: in MBL${}_{\text{A}}$ $\Delta x(t)$ diverges slower than any power, while it converges towards a finite value in MBL${}_{\text{B}}$. Within the scenario the transition between MBL${}_{\text{A}}$ and the ergodic phase is characterized by a length scale, $\xi$, that exhibits an essential singularity $\ln \xi \sim 1/|W-W_{\text{c}}|$. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.

11 Figs, 11 pages