The Eigenstate Thermalization Hypothesis (ETH) represents a cornerstone in our understanding of the thermalization mechanism in quantum many body systems.
It deals with the matrix elements of physical observables in the energy eigenbasis and relies on ideas borrowed from quantum chaos and random matrix theory.
Inspired by the recent developments in the characterization of chaotic behavior in terms of out of time order correlations we argue that in order to have non trivial multi-point correlation functions one has to consider correlations between matrix elements previously neglected within ETH.
Moreover we show that generic rotationally invariant random matrix models satisfy a simple relation: the probability distribution of off-diagonal elements and the one of half the difference between any two diagonal elements coincide.
In the spirit of ETH we test in different models the hypothesis that the same relation holds in quantum systems that are non-localized, when one considers small energy differences. The relation provides a stringent test of ETH beyond the Gaussian ensemble.