  The apparent randomness of chaotic eigenstates in interacting quantum systems
hides subtle correlations dynamically imposed by their finite energy per
particle. These correlations are revealed when Berrys approach for chaotic
eigenfunctions in single-particle systems is lifted into many-body space. We
achieve this by a many-body semiclassics analysis, appropriate for the
mesoscopic regime of large but finite number of particles. We then identify the
universality of both the cross-correlations and the Gaussian distribution of
expansion coefficients as the signatures of chaotic eigenstates. Combined,
these two aspects imprint a distinctive backbone to the morphology of
eigenstates that we check against extensive quantum simulations. The
universality of eigenstate correlations for fixed energy density is then a
further signature of many-body quantum chaos that, while consistent with the
eigenstate thermalization hypothesis, lies beyond random matrix theory.
