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arXiv · 2609.17037

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

Zhiqiang Huang
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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.

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Zhiqiang Huang. 2026-09-15. Eigenstate thermalization beyond the envelope: exact two-point overlap statistics in random free fermions. https://arxiv.org/abs/2609.17037

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