We study statistical properties of matrix elements entering the eigenstate thermalization hypoth-esis by studying the observables written in the energy eigenbasis and truncated to small micro-canonical windows. We put forward a picture, that below certain energy scale collective statisticalproperties of matrix elements exhibit emergent unitary symmetry. In particular, below this scale thespectrum of the microcanonically truncated operator exhibits universal behavior for which we intro-duce readily testable criteria. We support this picture by numerical simulations and demonstrateexistence of emergent unitary symmetry scale for all considered operators in chaotic many-bodyquantum systems. We discuss operator and system-size dependence of this energy scale and put ourfindings into context of previous works exploring emergence of random-matrix behavior in narrowenergy windows.

This work has been funded by theDeutsche Forschungsgemeinschaft (DFG), under GrantNo. 397107022, No. 397067869, and No. 397082825,within the DFG Research Unit FOR 2692, under GrantNo. 355031190. A. D. is supported by the NationalScience Foundation under Grant No. PHY 2310426.This work was performed in part at Aspen Center forPhysics, which is supported by National Science Foun-dation grant PHY-2210452. J. R. acknowledges fundingfrom the European Union’s Horizon Europe research andinnovation programme, Marie Sklodowska-Curie grantno. 101060162, and the Packard Foundation through aPackard Fellowship in Science and Engineering.