Rigorous and simple results on very slow thermalization, or quasi-localization, of the disordered quantum chain

TL;DR

This paper presents a rigorous result demonstrating quasi-localization in strongly disordered quantum chains, showing that certain local quantities remain nearly static for extremely long times, highlighting ergodicity-breaking due to disorder.

Contribution

It provides a simple, rigorous proof of quasi-localization in disordered quantum chains, establishing a set of quasi-local quantities with negligible dynamics over exponentially long times.

Findings

Quasi-local quantities remain nearly static up to exponential times in disorder strength.

The proof of quasi-localization is short and transparent for disordered quantum chains.

Demonstrates ergodicity-breaking induced by strong disorder.

Abstract

This paper originates from lectures delivered at the summer school "Fundamental Problems in Statistical Physics XV" in Bruneck, Italy, in 2021. We give a brief and limited introduction into ergodicity-breaking induced by disorder. As the title suggests, we include a simple yet rigorous and original result: For a strongly disordered quantum chain, we exhibit a full set of quasi-local quantities whose dynamics is negligible up to times of order $\exp\{ c (\log W)^{2-\epsilon}\}$, with $\epsilon < 1$, $c$ a numerical constant, and $W$ the disorder strength. Such a result, that is often referred to as "quasi-localization", can in principle be obtained in other systems as well, but for a disordered quantum chain, its proof is relatively short and transparent.