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 validity 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 groundstate 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 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