Bulk-boundary correspondence for intrinsically gapless symmetry-protected topological phases from group cohomology

Intrinsically gapless symmetry-protected topological phases (igSPT) are systems with SPT edge states properties that could not arise in a gapped system the same symmetry and dimensionality. igSPT from which an anomaly low-energy (infrared) group emerges extended anomaly-free microscopic (ultraviolet) symmetry. We construct general framework for constructing lattice models emergent anomalies classified by cohomology, establish direct connection between anomaly, group-extension, gauging extending In many examples, edge-state protection has physically transparent mechanism: UV operations pump lower-dimensional SPTs onto edge, tuning to (multi)critical point different protected IR two- three-dimensional systems, additional possibility is can be satisfied anomalous symmetry-enriched order, we call quotient-symmetry-enriched order (QSET) sharply distinguished nonanomalous SETs phase transition. exactly solvable QSET order.