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. Of
course, isolated systems are by nature finite, and the main way of computing
such quantities is through numerical evaluation for finite-size systems.
Therefore, the finite-size scaling of the fluctuations of eigenstate
expectation values is a central aspect of the ETH. In this work, we present
numerical evidence that for generic non-integrable systems these fluctuations
scale with a universal power law D−1/2 with the dimension D of the
Hilbert space. We provide heuristic arguments, in the same spirit as the ETH,
to explain this universal result. Our results are based on the analysis of
three families of models, and several observables for each model. Each family
includes integrable members, and we show how the system size where the
universal power law becomes visible is affected by the proximity to
integrability.Comment: 9 pages, 8 figures; accepted for publication in Phys. Rev.
