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

## TL;DR

This paper investigates how initial state memories and density imbalances evolve in disordered lattice systems, revealing universal relations in non-interacting cases and persistent imbalances due to bath-like excitations in interacting systems.

## Contribution

It establishes universal relations between long-time density imbalance, localization length, and initial pattern geometry in non-interacting disordered systems, and explores memory retention mechanisms in interacting systems.

## Key findings

- Universal relation between imbalance and localization length in non-interacting systems.
- Non-analytic behavior of imbalance near mobility edge transitions.
- Persistent imbalance in interacting systems due to bath-like excitations.

## Abstract

We study the dynamics of one and two dimensional disordered lattice bosons/fermions initialized to a Fock state with a pattern of $1$ and $0$ particles on $A$ and ${\bar A}$ sites. For non-interacting systems we establish a universal relation between the long time density imbalance between $A$ and ${\bar A}$ site, $I(\infty)$, the localization length $\xi_l$, and the geometry of the initial pattern. For alternating initial pattern of $1$ and $0$ particles in 1 dimension, $I(\infty)=\tanh[a/\xi_l]$, where $a$ is the lattice spacing. For systems with mobility edge, we find analytic relations between $I(\infty)$, the effective localization length $\tilde{\xi}_l$ and the fraction of localized states $f_l$. The imbalance as a function of disorder shows non-analytic behaviour when the mobility edge passes through a band edge. For interacting bosonic systems, we show that dissipative processes lead to a decay of the memory of initial conditions. However, the excitations created in the process act as a bath, whose noise correlators retain information of the initial pattern. This sustains a finite imbalance at long times in strongly disordered interacting systems.