Classification of intrinsically gapless topological phases
In plain words
Some gapless (critical) states carry protected edge modes that no gapped phase with the same symmetry can have, and a general list of such gapless topological phases is missing.
Precise statement
Classify symmetry-enriched gapless phases of 1D and 2D lattice models, including gapless SPT and intrinsically gapless SPT phases defined by an emergent anomaly in the low-energy symmetry action, and determine which carry protected edge modes. An answer is a classification in terms of symmetry extensions and anomalies with a lattice example in each class.
What would settle it
A classification scheme in terms of symmetry extensions and anomalies, verified by explicit lattice realizations of every class in 1D and 2D.
Status in the literature
Unverified note
Gapless SPT phases were introduced in 1D (Scaffidi, Parker, Vasseur 2017), extended to intrinsically gapless SPT (Thorngren, Vishwanath, Verresen 2021) and classified in 1+1D by topological holography (Wen and Potter 2023); a general 2D classification is open as of 2026.