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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?

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Any free fermion Hamiltonian where the Fermi energy is chosen such that it is exactly at the bottom (or top) of the band (in the case of a single band) is of this form: It is obviously gapless, and its ground state is the vacuum, i.e., short-range correlated (or rather uncorrelated).

One such example would be the 1D XX model, $$ H=-\tfrac12\sum (\sigma_x^i\sigma_x^{i+1} + \sigma_y^i\sigma_y^{i+1}) + \sum \sigma_z\ . $$

You can also construct gapless "uncle Hamiltonians" for short-range correlated Matrix Product State (MPS) and Projected Entangled Pair State (PEPS) wavefunctions, see http://arxiv.org/abs/1111.5817, http://arxiv.org/abs/1210.6613.

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