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In this paper: https://doi.org/10.1088/1367-2630/14/3/033044 it is show that for Kitaev toric code looses topological entanglement entropy over long times if it is thermally opened.

What is an example of a system which does not loose topological entanglement entropy over time at finite temperatures?

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The Toric Code in four space dimensions is thermally stable. (The excitations are loop-like, rather than point-like, and possess a tension, and thus tend to stay small.)

Some references:

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  • $\begingroup$ in addition, Haah's code in 3D is topologically-ordered and also stable at finite temperature $\endgroup$ – 4xion Jun 30 at 0:45
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    $\begingroup$ @4xion Not in the same way as the 4D toric code (the energy barrier is much lower). Also, since the question is about topological entropies, I'm not sure about the situation in that regard in Haah's model. $\endgroup$ – Norbert Schuch Jun 30 at 9:59
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    $\begingroup$ (Of course, it should be noted that also the 4D Toric Code loses topological order over sufficiently long times, so it really boils down to what scaling of size vs. time is considered acceptable.) $\endgroup$ – Norbert Schuch Jun 30 at 9:59

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