Memories of initial states and density imbalance in the dynamics of noninteracting and interacting disordered systems

Abstract

We study the dynamics of one- and two-dimensional disordered lattice bosons/fermions initialized to a Fock state with 1 particle on a set of lattice sites ($A$) and 0 particles on the rest of the sites ($\overline{A}$). Such states have been considered in recent ultracold atomic experiments to detect many body localization. For noninteracting systems we establish a universal relation between the long time density imbalance between $A$ and $\overline{A}$ sites, $I(\ensuremath{\infty})$, the localization length ${\ensuremath{\xi}}_{l}$, and the geometry of the initial pattern. For the alternating initial pattern of 1 and 0 particles in one dimension, $I(\ensuremath{\infty})=tanh[a/{\ensuremath{\xi}}_{l}]$, where $a$ is the lattice spacing. For systems with mobility edge, we find analytic relations between $I(\ensuremath{\infty})$, the effective localization length ${\overline{\ensuremath{\xi}}}_{l}$, and the fraction of localized states ${f}_{l}$. The imbalance as a function of disorder shows nonanalytic behavior when the mobility edge passes through a band edge. For interacting bosonic systems, we show that there is a mechanism to retain a finite long-time imbalance in the system even in presence of dissipative and stochastic processes coming from interparticle scattering. The scattering of particles, which lead to a decay of the memory of initial conditions through dissipative processes, also creates excitations in the system. For strong disorder, the excitations act as a local bath, whose noise correlators retain information of the initial pattern. This sustains a finite imbalance at long times in strongly disordered interacting systems.