Eigenstate thermalization beyond the envelope: exact two-point overlap statistics in random free fermions

Abstract

Eigenstate thermalization constrains the smooth dependence of observables on energy, but it does not by itself fix the microscopic statistics of the overlaps between many-body eigenstates and a chosen basis: the smooth envelope is a one-point statement, and the fluctuation field it leaves undetermined carries a structured two-point covariance. We establish this distinction, and solve that fluctuation field exactly, in random free fermions---an ensemble of Slater-determinant eigenstates built from a Gaussian orthogonal single-particle Hamiltonian, whose eigenstate thermalization was established by Magán. In this ensemble every channel overlap is a minor of a Haar-distributed orthogonal matrix, so its statistics follow from classical random-matrix theory. The one-point law is exactly flat for every channel, with an exact moment hierarchy that is not Porter--Thomas: small intensities are enhanced algebraically rather than by the exponential Porter--Thomas form, and the mean sector carries a negative, order-one correlation correction of purely normalization origin. The two-point covariance closes exactly at the Gaussian fixed point: it is organized by the number of one-body modes shared by two eigenstates and by the number of modes shared by two channels, and its energy-resolved form factorizes into this geometry times the classical convolution of the single-particle semicircle, from which the two-resolvent covariance follows by an integral transform. Exact finite-size computations confirm all closed forms, and the structural identities of the dictionary hold at machine precision. The results provide an exactly solvable microscopic realization of the two-point fluctuation sector underlying multi-resolvent descriptions of eigenstate thermalization: a smooth envelope does not determine fine-grained fluctuations.