  Hastings established exponential decay of correlations for ground states of
gapped quantum many-body systems. A ground state of a (geometrically) local
Hamiltonian with spectral gap $\epsilon$ has correlation length $\xi$ upper
bounded as $\xi=O(1/\epsilon)$. In general this bound cannot be improved. Here
we study the scaling of the correlation length as a function of the spectral
gap in frustration-free local Hamiltonians, and we prove a tight bound
$\xi=O(1/\sqrt\epsilon)$ in this setting. This highlights a fundamental
difference between frustration-free and frustrated systems near criticality.
The result is obtained using an improved version of the combinatorial proof of
correlation decay due to Aharonov, Arad, Vazirani, and Landau.
