# Spectral gap and local dynamics imply exponential decay, what about the reverse?

The paper "Spectral Gap and Exponential Decay of Correlations"(https://arxiv.org/abs/math-ph/0507008) proves that if a system has a spectral gap and is local, then ground state expectation values will decay exponentially fast.

My question is: does the converse hold, i.e. given a state in which all correlation functions decay exponentially, does it follow that the state may be chosen as the ground state of a local Hamiltonian with spectral gap?