Correlation Length versus Gap in Frustration-Free Systems

TL;DR

This paper proves a tight bound on the correlation length in frustration-free quantum systems, showing it scales as the inverse square root of the spectral gap, unlike the linear relation in general systems.

Contribution

It establishes a new, tight bound on correlation length in frustration-free systems, revealing a fundamental difference from frustrated systems near criticality.

Findings

Correlation length scales as 1/√ε in frustration-free systems

The bound is tight and improves upon previous results

Highlights a fundamental difference between frustrated and 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 $\epsilon$ 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.