For the typical quantum many-body systems that obey the eigenstate thermalization 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 subsystem entropies of thermal ensembles and have universal properties. These functions capture the full crossover from the ground-state entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. For critical one-dimensional systems, a universal scaling function follows from conformal field theory and can be adapted for nonlinear dispersions. We use it to also deduce the crossover scaling function for Fermi liquids in $d>1$ dimensions. The analytical results are complemented by numerics for large noninteracting systems of fermions in $d\le 3$ dimensions and have also been confirmed for bosonic systems and nonintegrable spin chains.

• Received 30 May 2019
• Revised 27 May 2021
• Accepted 14 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.040603