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

TL;DR

This paper proves that frustration-free Hamiltonians exhibit a universal inverse-square scaling of their finite-size energy gap at criticality, regardless of graph structure or ground state correlations.

Contribution

It establishes the inverse-square gap scaling as a robust, universal property of finite-range frustration-free Hamiltonians, refining existing proof techniques.

Findings

Inverse-square critical gap scaling proven for frustration-free Hamiltonians

Applicable to general graphs embedded in R^D and finite-range interactions

Limits the ability of such Hamiltonians to produce conformal field theories

Abstract

We prove that the critical finite-size gap scaling for frustration-free Hamiltonians is of inverse-square type. The result covers general graphs embedded in $\mathbb R^D$ and general finite-range interactions without requiring assumptions about the ground state correlations. 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.