Diagnosing Thermalization via Participation Ratio in Disordered Bosonic Chains

TL;DR

This study investigates how disordered bosonic chains thermalize, revealing a strong link between initial state delocalization, measured by Participation Ratio, and the system's tendency to reach thermal equilibrium, with implications for experimental diagnostics.

Contribution

It establishes a direct correlation between initial state delocalization and thermalization in disordered bosonic systems, using Participation Ratio as a predictive tool.

Findings

Eigenstates satisfy ETH within a broad energy window.

High PR states tend to thermalize, low PR states deviate.

Participation Ratio effectively predicts thermalization behavior.

Abstract

We study thermalization in a disordered one-dimensional interacting bosonic system described by the Aubry-Andre model using full exact diagonalization. We find a broad chaotic energy window where the system's eigenstates satisfy the Eigenstate Thermalization Hypothesis (ETH), demonstrated by the smooth energy dependence of observables like entanglement entropy and local particle number, whose fluctuations decrease with system size. Dynamically, we investigate the equilibration of initial Fock states and find that thermalization is not universal. The key finding is a direct and nontrivial correlation between an initial state's delocalization in the energy eigenbasis quantified by the Participation Ratio (PR) and its subsequent equilibration. States with a high PR consistently evolve toward the microcanonical ensemble prediction, whereas those exhibiting a low PR display deviations whose magnitude inversely correlates with the PR value. This connection is quantitatively confirmed by the trace distance, providing a powerful, experimentally relevant diagnostic for predicting which initial states will reach thermal equilibrium.