We investigate the eigenstate thermalization hypothesis (ETH) in integrablemodels, focusing on the spin-1/2 isotropic Heisenberg (XXX) chain. We providenumerical evidence that ETH holds for typical eigenstates (weak ETH scenario).Specifically, using a numerical implementation of state-of-the-art Bethe ansatzresults, we study the finite-size scaling of the eigenstate-to-eigenstatefluctuations of the reduced density matrix. We find that fluctuations arenormally distributed, and their standard deviation decays in the thermodynamiclimit as L^{-1/2}, with L the size of the chain. This is in contrast with theexponential decay that is found in generic non-integrable systems. Based on ourresults, it is natural to expect that this scenario holds in other integrablespin models and for typical local observables. Finally, we investigate theentanglement properties of the excited states of the XXX chain. We numericallyverify that typical mid-spectrum eigenstates exhibit extensive entanglemententropy (i.e., volume-law scaling).
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