  We discuss eigenstate correlations for ergodic, spatially extended many-body
quantum systems, in terms of the statistical properties of matrix elements of
local observables. While the eigenstate thermalization hypothesis (ETH) is
known to give an excellent description of these quantities, the butterfly
effect implies structure beyond ETH. We determine the universal form of this
structure at long distances and small eigenvalue separations for Floquet
systems. We use numerical studies of a Floquet quantum circuit to illustrate
both the accuracy of ETH and the existence of our predicted additional
correlations.
