Correlation Length versus Gap in Frustration-Free Systems

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 ɛ has correlation length ξ upper bounded as ξ =O (1 /ɛ ). 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 ξ =O (1 /√{ɛ }) 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.