# How a small quantum bath can thermalize long localized chains

## TL;DR

This paper demonstrates that a small ergodic grain can delocalize a many-body localized chain if the localization length exceeds a critical threshold, supported by numerical and theoretical analysis.

## Contribution

It introduces a simple theory predicting a critical localization length for delocalization, validated by numerical simulations involving a minimal ergodic grain.

## Key findings

- A 3-qubit ergodic grain can delocalize a localized chain beyond a critical localization length.
- The proposed theory accurately predicts the critical localization length and matches numerical results.
- Local operator behavior inside and outside the ergodic grain aligns with theoretical predictions.

## Abstract

We investigate the stability of the many-body localized (MBL) phase for a system in contact with a single ergodic grain, modelling a Griffiths region with low disorder. Our numerical analysis provides evidence that even a small ergodic grain consisting of only 3 qubits can delocalize a localized chain, as soon as the localization length exceeds a critical value separating localized and extended regimes of the whole system. We present a simple theory, consistent with the arguments in [Phys. Rev. B 95, 155129 (2017)], that assumes a system to be locally ergodic unless the local relaxation time, determined by Fermi's Golden Rule, is larger than the inverse level spacing. This theory predicts a critical value for the localization length that is perfectly consistent with our numerical calculations. We analyze in detail the behavior of local operators inside and outside the ergodic grain, and find excellent agreement of numerics and theory.