期刊
PHYSICAL REVIEW X
卷 11, 期 4, 页码 -出版社
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevX.11.041059
关键词
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资金
- Simons Foundation [651440]
- Harvard Quantum Initiative Postdoctoral Fellowship in Science and Engineering
- Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Center [SFB 1143]
- National Science Foundation [NSF PHY-1748958]
- Heising-Simons Foundation
- Zuckerman Fellowship
- DFG [FOR 1807, PO 1370/2-1, TRR80, 107745057]
- European Research Council under the European Union's Horizon 2020 research and innovation program [771537]
We introduce topological invariants for gapless systems and study the associated boundary phenomena, primarily focusing on the 1 + 1d case where the edge has a topological degeneracy with exponential or algebraic finite-size splitting. Symmetry properties of low-energy conformal field theory (CFT) provide discrete invariants establishing symmetry-enriched quantum criticality, with examples such as the Ising phase and cluster phases. The formalism unifies various examples previously studied in literature, including the classification of symmetry-enriched 1 + 1d Ising CFTs and the construction of topological invariants in higher dimensions.
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/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. Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 + 1d CFT without gapped sectors.
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