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: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
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.
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