Eigenstate thermalization hypothesis (ETH) and integrability in quantum spin chains

TL;DR

This paper examines the eigenstate thermalization hypothesis in the integrable XXX spin chain, showing that fluctuations decay as L^{-1/2} and that typical eigenstates have volume-law entanglement, contrasting with non-integrable systems.

Contribution

It provides numerical evidence that ETH holds for typical eigenstates in an integrable model and characterizes their entanglement and fluctuation scaling behaviors.

Findings

Eigenstate-to-eigenstate fluctuations decay as L^{-1/2}

Typical eigenstates exhibit volume-law entanglement entropy

Fluctuations are normally distributed in the thermodynamic limit

Abstract

We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the spin-1/2 isotropic Heisenberg (XXX) chain. We provide numerical evidence that ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as L^{-1/2}, with L the size of the chain. This is in contrast with the exponential decay that is found in generic non-integrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the XXX chain. We numerically verify that typical mid-spectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).