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Both examples you ask about are for gapless systems, but in your edit you then bring up some notions related to the gapped systems, so let me address both. Moreover it will turn out that understanding the gapped case well (which is simpler) almost naturally implies the gapless case. Topological invariants for band insulators Suppose we have our Brillouin ...


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The IQH state cannot be changed to a product state via local unitary transformation (ie cannot be smoothly deformed into a product state without phase transition), thus, according to the original definition of long-range-entanglement in http://arxiv.org/abs/1004.3835, IQH state is a long-range-entangled state (ie has topological order). Later Kitaev ...



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