We consider the entanglement entropies of energy eigenstates in quantum many-body systems. For the typical models that allow for a field-theoretical description of the long-range physics, we find that the entanglement entropy of (almost) all eigenstates is described by a single scaling function. This is predicated on the applicability of the weak or strong eigenstate thermalization hypothesis (ETH), which then implies that the scaling functions can be deduced from subsystem entropies of thermal ensembles. The scaling functions describe the full crossover from the ground state entanglement regime for low energies and small subsystem size (area or log-area law) to the extensive volume-law regime 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 dimensions. These analytical results are complemented by numerics for large non-interacting systems of fermions in d=1,2,3 and the harmonic lattice model (free scalar fieldtheory) in d=1,2. Lastly, we demonstrate ETH for entanglement entropies and the validity of the scaling arguments in integrable and non-integrable interacting spin chains. In particular, we analyze the XXZ and transverse-field Ising models with and without next-nearest-neighbor interactions.

References: arXiv:1905.07760, arXiv:1912.10045, arXiv:2010.07265