  In a recent Letter [Phys. Rev. Lett. {\bf{125}}, 180604 (2020)], we
introduced a closed-form analytic expression for the average bipartite von
Neumann entanglement entropy of many-body eigenstates of random quadratic
Hamiltonians. Namely, of Hamiltonians whose single-particle eigenstates have
random coefficients in the position basis. A paradigmatic Hamiltonian for which
the expression is valid is the quadratic Sachdev-Ye-Kitaev (SYK2) model in its
Dirac fermion formulation. Here we show that the applicability of our result is
much broader. Most prominently, it is also relevant for local Hamiltonians such
as the three-dimensional (3D) Anderson model at weak disorder. Moreover, it
describes the average entanglement entropy in Hamiltonians without
particle-number conservation, such as the SYK2 model in the Majorana fermion
formulation and the 3D Anderson model with additional terms that break
particle-number conservation. We extend our analysis to the average bipartite
second R{\'e}nyi entanglement entropy of eigenstates of the same quadratic
Hamiltonians, which is derived analytically and tested numerically. We
conjecture that our results for the entanglement entropies of many-body
eigenstates apply to quadratic Hamiltonians whose single-particle eigenstates
exhibit quantum chaos, to which we refer as quantum-chaotic quadratic
Hamiltonians.
