Abstract
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 $\sim 1/L^{14}$. 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. Non-trivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched $2+1d$ CFT without gapped sectors.
Highlights
Topological phases of quantum matter are fascinating emergent phenomena, commonly characterized by nonlocal order parameters in the bulk and exotic behavior at the boundary
While we introduce our framework for general symmetry groups G and discuss a variety of examples, for clarity we illustrate these concepts in detail for two particular symmetry groups: the unitary group Z2 × Z2 and the antiunitary Z2 × ZT2
For H00, one can show that μðxÞ ∼ Á Á Á Xn−2Xn−1YnZnþ1 [61]. [Note that in both cases, the nearby symmetry-preserving phase [62] has long-range order limjx−yj→∞hμðxÞμðyÞi ≠ 0—see, e.g., Eq (3)—as expected by Kramers-Wannier duality.] This suggests that the two models are two distinct G-enriched Ising conformal field theory (CFT) distinguished by TμT 1⁄4 Æμ
Summary
Topological phases of quantum matter are fascinating emergent phenomena, commonly characterized by nonlocal order parameters in the bulk and exotic behavior at the boundary. We show that this Ising CFT is topologically enriched by π rotations around the principal axes (a Z2 × Z2 group), explaining the observed edge modes. The concepts developed in this work apply to symmetryenriched CFTs with a general on-site symmetry group G; we refer to them as G-enriched CFTs, or G-CFTs for short This is relevant for condensed matter systems, such as the example described above, and to high-energy physics; in particular, G-CFTs can be related to discrete torsion of orbifold CFTs arising in string theory [42,43,44,45], edge modes have not yet been pointed out in that context.
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