Knabe's spectral gap method for open boundary conditions

TL;DR

This paper extends Knabe's finite-size spectral gap criterion from periodic to open boundary conditions in frustration-free quantum spin chains, introducing both bulk and edge criteria to analyze spectral gaps.

Contribution

It develops two finite-size criteria for open boundary conditions, including a novel edge criterion to detect edge-localized gapless excitations.

Findings

Established a bulk spectral gap criterion for open chains.

Introduced an edge criterion to exclude edge gapless excitations.

Extended Knabe's method to open boundary conditions.

Abstract

In 1988, Knabe found a "finite-size criterion" to determine whether a frustration-free quantum spin chain with periodic boundary conditions is uniformly gapped in the thermodynamic limit. The criterion provides a threshold for the spectral gap at a finite system size such that, if the threshold is exceeded for a fixed system size, then the chain with periodic boundary conditions is uniformly gapped. We extend Knabe's result to frustration-free spin chains equipped with open boundary conditions. We now obtain two finite-size criteria: The first one is identical to Knabe's criterion and we interpret it as a bulk criterion. The second one controls the spectral gaps at smaller system sizes and can be interpreted as a new edge criterion. Heuristically, it excludes the presence of thermodynamically gapless excitations living near the edge.