  Much has been learned about universal properties of entanglement entropies in
ground states of quantum many-body lattice systems. Here we unveil universal
properties of the average bipartite entanglement entropy of eigenstates of the
paradigmatic quantum Ising model in one dimension. The leading term exhibits a
volume-law scaling that we argue is universal for translationally invariant
quadratic models. The subleading term is constant at the critical field for the
quantum phase transition and vanishes otherwise (in the thermodynamic limit),
i.e., the critical field can be identified from subleading corrections to the
average (over all eigenstates) entanglement entropy.
