# Slow propagation in some disordered quantum spin chains

## TL;DR

This paper introduces transmission time to analyze disordered quantum spin chains, linking it to many-body localization, and explores how perturbations affect dynamical localization and transmission times.

## Contribution

It establishes a connection between transmission time, local integrals of motion, and many-body localization, providing new insights into disordered quantum spin chain dynamics.

## Key findings

- Zero-velocity Lieb-Robinson bound implies LIOM representation.
- Sparse perturbations cause transmission times to diverge with distance.
- Exponential dynamical localization leads to diverging transmission times under perturbations.

## Abstract

We introduce the notion of transmission time to study the dynamics of disordered quantum spin chains and prove results relating its behavior to many-body localization properties. We also study two versions of the so-called Local Integrals of Motion (LIOM) representation of spin chain Hamiltonians and their relation to dynamical many-body localization. We prove that uniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies the existence of a LIOM representation of the dynamics as well as a weak converse of this statement. We also prove that for a class of spin chains satisfying a form of exponential dynamical localization, sparse perturbations result in a dynamics in which transmission times diverge at least as a power law of distance, with a power for which we provide lower bound that diverges with increasing sparseness of the perturbation.