We show the boundedness of entanglement entropy for (bipartite) pure states of quantum spin chains implies split property of subsystems. As a corollary the infinite volume ground states for 1-dim spin chains with the spectral gap between the ground state energy and the rest of spectrum have the split property. We see gapless excitation exists for the spinless Fermion on $\bfZ$ if the ground state is non-trivial and translationally invariant and the U(1) gauge symmetry is unbroken. Here we do not assume uniqueness of ground states for all finite volume Hamiltonians.
$β$-Boundedness, Semipassivity, and the KMS-Condition
The complete set of infinite volume ground states for Kitaev's abelian quantum double models
Entanglement entropy for a particle coupled with its surrounding
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