  We argue that, in a basis common to all one-site shift invariant conserved
charges, there is no eigenstate of a noninteracting local spin-1/2 chain
Hamiltonian that satisfies the area law if the ground state has half-integer
central charge. That is to say, in those models all (quasi)local one-site shift
invariant conserved operators are gapless. From the standpoint of bipartite
entanglement properties, we show indeed that there are three distinct one-site
shift invariant noninteracting models, two of which are equivalent to the XX
model (for one of them the transformation breaks one-site shift invariance) and
the other to the critical Ising model. The former class has two locally
distinct one-site shift invariant excited states satisfying the area law; the
latter two classes have none.
