Slow dynamics in many-body localized system 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, $\\xi_0$, however strong finite-size effects are encountered. We conclude that rare disorder configurations (Griffiths effects) are not a likely cause 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 $\\ln \\xi \\sim 1/|W-W_{\\text{c}_1}|$. Relations to earlier proposals of two-phase scenarios will be discussed.