I thought this must have been asked before, but couldn't find it through search.
It was proved by Hastings and Koma in arXiv:math-ph/0507008, given a Hamiltonian satisfying certain locality conditions, that the existence of an energy gap implies that all correlation functions with respect to the ground state sector are short-range. This applies to fermionic systems as well as bosonic systems. The contrapositive is, of course, that long-range correlation implies gaplessness.
However, this leaves the possibility that a system is simultaneously short-range correlated and gapless. Can someone provide a few examples?