We argue that, in a basis common to all one-site shift-invariant conserved charges, there is no eigenstate of a noninteracting local spin-$\frac{1}{2}$ 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. Using both analytical and numerical techniques, indeed, we calculate the bipartite Rényi entropies and show 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.

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• Eigenstate thermalization
• Entanglement entropy
• Excited-state quantum phase transitions
• Geometric & topological phases
• Quantum scars
• Spinless fermions
• Quantum spin models
• Integrable systems
• Simulated annealing
• XY model