  We study general entanglement properties of the excited states of the one
dimensional translational invariant free fermions and coupled harmonic
oscillators. In particular, using the integrals of motion, we prove that these
Hamiltonians independent of the gap (mass) has infinite excited states that can
be described by conformal field theories with integer or half-integer central
charges. In the case of free fermions, we also show that because of the huge
degeneracy in the spectrum, even a gapless Hamiltonian can have excited states
with an area-law. Finally, we study the universal average entanglement entropy
over eigenstates of the Hamiltonian of the free fermions introduced recently in
L. Vidmar, et al., Phys. Rev. Lett. 119, 020601 (2017)
https://link.aps.org/doi/10.1103/PhysRevLett.119.020601 . In particular, using
a duality relation, we propose a method which can be useful for experimental
measurement of the universal average entanglement entropy. Part of our
conclusions can be extended to the quantum spin chains that are associated with
the free fermions via Jordan-Wigner(JW) transformation.
