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Perhaps in order to understand the previous answer you should follow the bellow first: There are three different behaviours when $N\rightarrow \infty$ of $<\sigma_{0,0},\sigma,_{N,N}>$ for fixed temperature T. whether $T>T_c$, $T=T_c$ or $T<T_c$ the correlation function $<\sigma_{0,0},\sigma,_{N,N}>$ in each of this cases has a different ...

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What you are observing is the critical slowdown - getting to the equilibrium near to the critical temperature is very hard because of huge fluctuations in the system. Therefore, as you are not averaging over enough runs, it is no wonder that you are getting very large deviations from the (expected) average values. This problem is less severe for more ...

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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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