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I have seen from literature that the $Z_2$ lattice gauge theory in 2d could be mapped into a quantum Ising model with gauge constraints on the Hilbert space by dual transformation. The deconfined phase in $Z_2$ gauge theory corresponds to paramagnetic state in quantum Ising model. Does that mean there is topological entanglement entropy (EE) in the quantum Ising model in paramagnetic state? If yes, does that EE corresponds to topological order in quantum Ising model?

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    $\begingroup$ No, there is no TEE in transverse Ising model. The mapping is not a local unitary transformation. In fact it is quite a non-local one that messes up the topological sectors. $\endgroup$ – Meng Cheng May 5 '15 at 3:21
  • $\begingroup$ @MengCheng A bit more elaboration, and this would be a perfect answer! $\endgroup$ – Danu May 5 '15 at 5:32
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Duality transformation does not preserve topological order, and hence not preserving the topological entanglement entropy. The quantum Ising model has no topological entanglement entropy. See this related question for more discussions.

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  • $\begingroup$ Thanks for the link! I have not noticed that before I asked $\endgroup$ – qc2014 May 5 '15 at 13:52

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