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For the typical quantum many-body systems that obey the eigenstatethermalization hypothesis (ETH), we argue that the entanglement entropy of(almost) all energy eigenstates is described by a single crossover function.The ETH implies that the crossover functions can be deduced from subsystementropies of thermal ensembles and have universal properties. These functionscapture the full crossover from the groundstate entanglement regime at lowenergies and small subsystem size (area or log-area law) to the extensivevolume-law regime at high energies or large subsystem size. For criticalone-dimensional systems, a universal scaling function follows from conformalfield theory (CFT) and can be adapted for nonlinear dispersions. We use it toalso deduce the crossover scaling function for Fermi liquids in $d>1$dimensions. The analytical results are complemented by numerics for largenon-interacting systems of fermions in $d=1$, $2$, and $3$ dimensions and havealso been confirmed for bosonic systems and non-integrable spin chains.Furthermore, one can deduce the distribution function for eigenstateentanglement.

Eigenstate entanglement: Crossover from the ground state to volume laws