On the critical finite-size gap scaling for frustration-free Hamiltonians

We prove that the critical finite-size gap scaling for frustration-free Hamiltonians is of inverse-square type. The novelty of this note is that the result is proved on general graphs and for general finite-range interactions. Therefore, the inverse-square critical gap scaling is a robust, universal property of finite-range frustration-free Hamiltonians. This places further limits on their ability to produce conformal field theories in the continuum limit. Our proof refines the divide-and-conquer strategy of Kastoryano and the second author through the refined Detectability Lemma of Gosset--Huang.