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Long-range entanglement (LRE) is the main feature of topological orders. The string-net condensation model was constructed to exhibit LRE.

But the many-body systems of such models do not look like any earthly materials at all but are closer to quantum gravity models. In quantum gravity, nobody sees the detailed structure beyond the Planck scale. But in condensed matter, the microscopic structures are "crystal" clear.

Then how can such strange models be used to explain LRE in laboratory materials?

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The many-body systems that give rise to LRE and string-net condensation are simply quantum spin models and they CAN look like earthly materials, such as the $J_1$-$J_2$ Heisenberg model on square lattice and the Heisenberg model on Kagome lattice for 1/2-spins. The many-body systems that give rise to LRE and string-net condensation DO have a "crystal" clear microscopic structures.

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Thank you! Yes, I remember you discussed about the $J_1\approx J_2$ model at KITPC. Now I realize that it is more of belief than logic. Personally I believe the bottom-up approach is more "helpful" to uncover novel materials (may or may not be "topological"), just like the case of topological insulators. –  ChenChao Dec 29 '12 at 16:33

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