Correlation Length versus Gap in Frustration-Free Systems

Abstract

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 $\hat{I}\mu$ has correlation length $\hat{I}¾$ upper bounded as $\hat{I}¾=O(1/\hat{I}\mu )$. 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 $\hat{I}¾=O(1/\sqrt{\hat{I}}\mu )$ 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.

v3: published version