Classification of vacuum and superselection-sector phases

Determine how to decide whether a given gapped phase represented, for example, by a commuting-projector Hamiltonian is the phase associated with the natural vacuum or the phase associated with a superselection sector.

Background

The paper distinguishes the gapped phase associated with the natural vacuum from phases obtained by treating a superselection-sector state, such as a state containing a nonabelian anyon, as a ground state. These phases can have identical anyon theories while differing in Haag-duality properties.

The authors state that it is not clear how to determine, from a given gapped phase or a representative commuting-projector Hamiltonian, which of these two situations applies. They suggest that approximate Haag duality, and potentially strict Haag duality, may provide a criterion for selecting the phase associated with the natural vacuum, but the decision procedure itself remains unresolved.