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Generic rotationally invariant random matrix models satisfy a simplerelation: the probability distribution of off-diagonal elements and the one ofhalf the difference between any two diagonal elements coincide. In the spiritof the Eigenstate Thermalization Hypothesis (ETH), we test the hypothesis thatthe same relation holds in quantum systems that are non-localized, when oneconsiders small energy differences. The relation provides a stringent test ofETH beyond the Gaussian ensemble. We apply it to a disordered spin chain, theSYK model and a Floquet system.

Eigenstate thermalization and rotational invariance in ergodic quantum systems