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The transverse field Ising model in 1+1d has two phases: a symmetric "disordered" phase and a symmetry-breaking "ordered" phase. Both of these phases have a finite excitation gap. In the symmetry-breaking phase, there is long-range order, that is $\langle Z_i Z_j\rangle \neq 0$ for $|i - j| \rightarrow \infty$. ($Z_i$ is the Pauli $z$ operator on site $i$.)

However, I thought that gapped states should have short-range, i.e., exponentially decaying, correlations. In this answer, Dominic Else says

A ground state of gapped Hamiltonian must have correlations which decay exponentially with distance (this has been proved rigorously by Hastings and Koma).

My question is: why are these facts (long-range correlations in gapped, symmetry-breaking phase and short-range correlations in gapped phase) not in contradiction with each other? I think I must be missing something simple, since both sides of this are well-understood.

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    $\begingroup$ I want to say that it's really the fluctuations $\tilde{Z}_i = Z_i - \left < Z_i \right >$ which have exponentially decaying correlations in the broken phase. $\endgroup$ Oct 20 '21 at 20:37
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If you look at the paper by Hastings and Koma, https://arxiv.org/abs/math-ph/0507008, they claim to prove the following:

When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption.

The key word here is "connected." For the $Z_i$'s (which trivially commute with each other so the above sentence applies), this implies that it is $$ C(i,j) = \langle Z_i Z_j \rangle - \langle Z_i \rangle \langle Z_j \rangle $$ which decays exponentially at large |i - j|, not the correlator $\langle Z_i Z_j \rangle$. It is surely these connected correlations which Dominic meant in his answer on characterizing gapped ground states.

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