# Eigenstate thermalization and quantum chaos in the Holstein polaron model

## TL;DR

This paper investigates whether the Holstein polaron model, a minimal electron-phonon system, exhibits quantum chaos and thermalization properties consistent with the eigenstate thermalization hypothesis (ETH) using exact diagonalization.

## Contribution

It demonstrates that despite its simplicity, the Holstein polaron model shows spectral and eigenstate properties indicative of quantum chaos and obeys ETH, extending understanding to solid-state systems.

## Key findings

- Spectral statistics match random-matrix ensemble predictions
- Eigenstates exhibit properties consistent with quantum chaos
- ETH holds for both diagonal and off-diagonal matrix elements

## Abstract

The eigenstate thermalization hypothesis (ETH) is a successful theory that provides sufficient criteria for ergodicity in quantum many-body systems. Most studies were carried out for Hamiltonians relevant for ultracold quantum gases and single-component systems of spins, fermions, or bosons. The paradigmatic example for thermalization in solid-state physics are phonons serving as a bath for electrons. This situation is often viewed from an open-quantum system perspective. Here, we ask whether a minimal microscopic model for electron-phonon coupling is quantum chaotic and whether it obeys ETH, if viewed as a closed quantum system. Using exact diagonalization, we address this question in the framework of the Holstein polaron model. Even though the model describes only a single itinerant electron, whose coupling to dispersionless phonons is the only integrability-breaking term, we find that the spectral statistics and the structure of Hamiltonian eigenstates exhibit essential properties of the corresponding random-matrix ensemble. Moreover, we verify the ETH ansatz both for diagonal and offdiagonal matrix elements of typical phonon and electron observables, and show that the ratio of their variances equals the value predicted from random-matrix theory.