Long persistence of localization in a disordered anharmonic chain beyond the atomic limit

TL;DR

This paper rigorously analyzes how localization persists in a disordered anharmonic chain, showing that transport is extremely slow as anharmonicity diminishes, and introduces a novel perturbation approach beyond the atomic limit.

Contribution

It develops a new perturbation theory around the harmonic system with fixed interactions, extending analysis beyond the atomic limit and addressing ongoing debates in the field.

Findings

Transport occurs only on extremely long time scales as anharmonicity decreases.

The developed perturbation method compares favorably with numerical results.

The work extends theoretical understanding of localization in disordered chains.

Abstract

We establish rigorous bounds on the decorrelation time and thermal transport in the disordered Klein-Gordon chain with a quartic on-site potential, governed by a parameter $\lambda$. At $\lambda = 0$, the chain is harmonic, and any form of transport is fully suppressed by Anderson localization. For the anharmonic system, at $\lambda > 0$, our results show that decorrelation and transport can occur only on time scales that grow faster than any polynomial in $1/\lambda$ as $\lambda \to 0$. From a technical perspective, the main novelty of our work is that we don't restrict ourselves to the atomic limit. Instead, we develop perturbation theory around the harmonic system with a fixed harmonic interaction between nearby oscillators. This allows us to compare our mathematical results with previous numerical work and contribute to resolving an ongoing debate, as detailed in a companion paper arXiv:2308.10572.