2019/05/31 by Ruben Verresen, Ryan Thorngren, Nick G. Jones +1 · 1 citation
Physics and Astronomy · #cond-mat.str-el #cond-mat.stat-mech #hep-th #quant-ph
paper · pdf · doi:10.1103/physrevx.11.041059
published as Phys. Rev. X 11, 041059 (2021) · we generalized our framework to 2+1d, and added more examples
arxiv created 2021/12/16 · arxiv updated 2022/01/06
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 former is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the latter arises between the gapped Ising and cluster phases: this symmetry-enriched Ising CFT has an edge mode with finite-size splitting ∼ 1/L14. 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. Non-trivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2+1d CFT without gapped sectors.