Nonlocal order in gapless systems: Entanglement spectrum in spin chains

We show that the entanglement spectrum can be used to define non-local order in gapless spin systems. We find a gap that fully separates a series of generic, high "entanglement energy" levels, from a flat band of levels with specific multiplicities defining the ground state, and remains finite in the thermodynamic limit. We pick the appropriate set of quantum numbers and partition the system in this space, corresponding to a nonlocal real-space cut. Despite the Laughlin state being bulk gapped while the antiferromagnetic spin chain state is bulk gapless, we show that the S=12 Heisenberg antiferromagnet in one dimension has an entanglement spectrum almost identical to that of the Laughlin Fractional Quantum Hall state in two dimensions, revealing the similar field theory of their low-energy bulk and edge excitations, respectively. We also discuss the dimerization transition from entanglement gap scaling.