Topological order is a new kind of order in quantum matter, which corresponds to pattern of long-range quantum entanglement. See http://en.wikipedia.org/wiki/Topological_order

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Is there a method for differentiating fractional quantum Hall states aside from finding Chern numbers?

The ground state for a quantum Hall system on a torus with fractional filling factor can be classified by the Chern number, which is why the Hall conductance is quantized. Is there another method or ...