Justifying a common mantra

When one reads papers about quantum lattice models a key "dogma" that is often implicitly assumed is that a gap in the excitation spectrum (with particular

quantum numbers, e.g. triplets) is equivalent to short-range spatial correlations in some corresponding variable (e.g. spin-spin correlations).

Finding a rigorous justification for these kind a assertions is hard. However, a paper by Matthew Hastings does it under reasonably general conditions.