Phase-wide propagation of Haag duality

Determine whether Haag duality at a single point in a gapped phase of matter implies Haag duality throughout the entire phase.

Background

Haag duality is an operator-algebraic condition asserting that the observable algebra associated with a region coincides with the commutant of the algebra associated with its complement. The paper notes that Haag duality has recently been proved for all Levin–Wen models, but emphasizes that it is not known whether validity at one representative of a gapped phase forces validity at every representative.

This question is distinct from the paper’s main result: the paper establishes that nonabelian anyon sectors violate Haag duality and approximate Haag duality, but it does not resolve whether Haag duality at a single point can generally propagate throughout a gapped phase. The issue is important because approximate Haag duality is known to be stable under perturbations, whereas strict Haag duality need not have been shown to share that phase-wide stability.