Renormalization method for proving frustration-free local spin chains are gapped

TL;DR

This paper introduces a rigorous renormalization technique leveraging decaying correlations in ground states to prove that certain local spin chains are gapped, advancing understanding in quantum physics and information theory.

Contribution

The paper presents a novel renormalization method for establishing spectral gaps in frustration-free local spin chains, applicable where previous methods failed.

Findings

Successfully proved two models are gapped

Completed proofs where earlier methods were inconclusive

Demonstrated the method's effectiveness in quantum spin systems

Abstract

Key properties of a physical system depend on whether it is gapped, i.e. whether its spectral gap has a positive lower bound that is independent of system size. In quantum information theory, the question of whether a system is gapped has essential computational significance as well. Here, we introduce a rigorous renormalization method to prove that a spin chain is gapped. This approach exploits the fact that ground states of gapped systems exhibit decaying correlations. We apply the method to show that two interesting models are gapped, successfully completing proofs even where the previously established methods are inconclusive.