Probing eigenstate thermalization in the weakly interacting Holstein polaron model

The origin of chaos in quantum systems remains a central challenge in many-body physics. Here, we use the adiabatic gauge potential (AGP) as a tool to investigate the transition from integrability to chaos in the one-dimensional Holstein polaron model. By combining exact diagonalization with an analysis of eigenstate expectation values (EEVs) and the statistical properties of off-diagonal matrix elements, we provide a detailed characterization of thermalization indicators in this model. In the integrable regime, we observe distributions of EEVs and power-law decays of their outliers with system size, consistent with the breakdown of the eigenstate thermalization hypothesis (ETH). Notably, in the noninteracting limit, standard indicators of quantum chaos, such as the mean level spacing, fail to capture the absence of integrability breaking. As integrability is broken, the off-diagonal matrix elements exhibit Gaussian statistics with variances matching random-matrix predictions, while EEV fluctuations vanish rapidly, signaling the onset of quantum chaos. Our results establish the AGP, with matrix element statistics, as a framework to probe the crossover from integrability to chaos in electron-phonon coupled systems.