Do correlations in local quantum spin systems always decay exponentially or algebraically?

Question

Consider translation-invariant quantum spin systems, that is qu-d-its on a lattice with a geometrically local Hamiltonian. Usually, such models are either gapped (in an ordered/disordered phase) or gapless (at a critical point).

Now, consider 2-point correlation functions, i.e. connected ground state correlators $C_{O_1,O_2}(i,j)$ between two fixed local observables $O_1,O_2$ located around sites $i$ and $j$. I'm interested in the asymptotic behavior of such correlations in dependence of the distance $r=\|i-j\|$.

For gapped models we would expect those correlations to decay exponentially, that is $$ |C(r)|<C_e e^{-\beta r}$$ for some correlation length $\beta>0$. If we're in a symmetry-breaking phase and choose a ground state which doesn't fully break the symmetry, the correlations won't go to $0$ but will still exponentially quickly converge to a constant: $$ |C(r)-C_0|<C_e e^{-\beta r}$$ For gapless systems on the other hand, one expects CFT-like algebraic decays. However, in a lattice model I'd assume this only holds up to exponentially small corrections, $$ |C(r)-(C_0 r^{\alpha_0}+C_1 r^{\alpha_1}+\ldots)| < C_e e^{-\beta r}$$

My question: Is it true that in any local quantum spin systems the correlations converge to an algebraic decay exponentially quickly? What is the evidence for this? Are there any proofs for special cases? Or, are there counter-examples of Hamiltonians which have sub-exponential but super-polynomial decay of correlations (e.g., $C(r)<e^{-\sqrt{r}}$)?

One can ask the same question for thermal correlation functions of local classical statistical systems such as the Ising model, and I'd expect that the behavior is the same.

This post imported from StackExchange Physics at 2023-08-31 12:46 (UTC), posted by SE-user Andi Bauer