We consider the entangl ement entropies of energy eigenstates in quantum many-body systems. For th e typical models that allow for a field-theoretical description of the lon g-range physics\, we find that the entanglement entropy of (almost) all ei genstates is described by a single crossover function. The eigenstate ther malization hypothesis (ETH) implies that such crossover functions can be d educed from subsystem entropies of thermal ensembles and that they assume universal scaling forms in quantum-critical regimes. They describe the ful l crossover from the groundstate entanglement scaling for low energies and small subsystem size (area or log-area law) to the extensive volume-law r egime for high energies or large subsystem size. For critical 1d systems\, the scaling function follows from conformal field theory (CFT). We use it to also deduce the scaling function for Fermi liquids in d>\;1 dimensio ns. These analytical results are complemented by numerics for large non-in teracting systems of fermions in d=1\,2\,3 and the harmonic lattice model (free scalar field theory) in d=1\,2. Lastly\, we demonstrate ETH for enta nglement entropies and the validity of the scaling arguments in integrable and non-integrable interacting spin chains.

\n\nReferences: PRL 127
\, 040603 (2021)\; PRA 104\, 022414 (2021)\; arXiv:2010.07265.

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