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

TL;DR

This work develops a rigorous perturbative framework around a locally Anderson-insulating harmonic chain to bound decorrelation and energy transport in a disordered Klein-Gordon chain with weak anharmonicity. By treating the anharmonic term as a perturbation of the harmonic dynamics and introducing a probabilistic $Z$-collection formalism to control small denominators, the authors prove decorrelation bounds and Green-Kubo-type limits that persist beyond the atomic limit. The approach yields time-scale bounds that grow faster than any polynomial in $1/\lambda$ as $\lambda\to 0$, supporting the notion of asymptotic localization even with fixed harmonic coupling. The results provide a rigorous link between long-time transport properties and the localized harmonic spectrum, clarifying the limitations of numerical studies and guiding future explorations of many-body localization in classical disordered lattices.

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 $λ$. At $λ= 0$, the chain is harmonic, and any form of transport is fully suppressed by Anderson localization. For the anharmonic system, at $λ> 0$, our results show that decorrelation and transport can occur only on time scales that grow faster than any polynomial in $1/λ$ as $λ\to 0$. 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.

Theorems & Definitions (53)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1
• proof : Proof of Theorem \ref{['th: decorrelation']}
• Proposition 2
• proof : Proof of Theorem \ref{['th: current']}
• Proposition 3
• proof : Proof of Theorem \ref{['th: Green Kubo']}