Gapless Topological Phases and Symmetry-Enriched Quantum Criticality

We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the 1 1d case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors. An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as 1=L 14 . In addition to such new cases, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer-including a complete characterization of symmetry-enriched 1 1d Ising CFTs. Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 1d CFT without gapped sectors.