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, $Φ(x,t)$, at infinite temperature. We find that disordered and quasi-periodic models behave qualitatively similar: Although even at longest observation times the width $Δx(t)$ of $Φ(x,t)$ does not exceed significantly the non-interacting localization length, $ξ_0$, 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 $β(t) {=} d\ln Δx(t) / d\ln t$. Motivated by our numerical data for $β$, we discuss a scenario in which the MBL-phase splits into two subphases: in MBL$_\text{A}$ $Δx(t)$ 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, $ξ$, that exhibits an essential singularity $\ln ξ\sim 1/|W-W_\text{c}|$. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.