Anomaly indicators for topological orders with U(1) and time-reversal symmetry

Matthew F. Lapa, Michael Levin

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


We study anomalies in time-reversal-(Z2T) and U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly (2+1)-dimensional system but can be realized on the surface of a (3+1)-dimensional [(3+1)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., U(1)âŠZ2T and U(1)×Z2T, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of (3+1)D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk (3+1)D SPT phase on a particular manifold and in the presence of a particular background gauge field for the U(1) symmetry.

Original languageEnglish
Article number165129
JournalPhysical Review B
Issue number16
StatePublished - 17 Oct 2019
Externally publishedYes

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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