ABSTRACT
We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the spin-1/2 isotropic Heisenberg (XXX) chain.  We verify 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.  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.  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).