  The eigenstate entanglement entropy has been recently shown to be a powerful
tool to distinguish integrable from generic quantum-chaotic models. In
integrable models, a unique feature of the average eigenstate entanglement
entropy (over all Hamiltonian eigenstates) is that the volume-law coefficient
depends on the subsystem fraction. Hence, it deviates from the maximal
(subsystem fraction independent) value encountered in quantum-chaotic models.
Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form
expression for the average eigenstate entanglement entropy as a function of the
subsystem fraction. We test its correctness against numerical results for the
quadratic Sachdev-Ye-Kitaev model. We also show that it describes the average
entanglement entropy of eigenstates of the power-law random banded matrix model
(in the delocalized regime), and that it is close but not the same as the
result for quadratic models that exhibit localization in quasimomentum space.
