Finite-size scaling of eigenstate thermalization

According to the eigenstate thermalization hypothesis (ETH), even isolated quantum systems can thermalize because the eigenstate-to-eigenstate fluctuations of typical observables vanish in the limit of large systems. Since isolated systems are by nature finite, the finite-size scaling of such fluctuations is a central aspect of the ETH. We propose that for generic non-integrable systems these fluctuations scale with a universal power law in the dimension of the Hilbert space. We present extensive multiple-system numerical evidence for this scaling law and provide supporting arguments. We also show how the scaling changes when approaching integrability.