# Long-range correlations in transverse field Ising model

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.

• 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. Oct 20, 2021 at 20:37

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.