Abstract
We study general 2D fermionic topological orders enriched by the mirror symmetry with $\mathcal{M}^2=1$. It is known that certain mirror symmetry enriched fermionic topological orders (mirror SETs) are anomalous, in the sense that they cannot be realized in strict two dimensions but have to live on the surface of 3D topological crystalline superconductors. Mirror anomaly, or equivalently 3D topological crystalline superconductor, has a $\mathbb{Z}_{16}$ classification. In this work, we derive an explicit expression, namely an \emph{anomaly indicator}, for the $\mathbb{Z}_{16}$ mirror anomaly for general fermionic mirror SETs. This derivation is based on the recently developed folding approach, originally proposed for bosonic topological orders. We generalize it to fermion systems. Through this approach, we establish a direct bulk-boundary correspondence between surface fermionic topological orders and 3D bulk topological crystalline superconductors. In addition, during the derivation, we obtain some general properties of fermionic topological orders as well as a few constraints on properties of fermionic mirror SETs.
Highlights
The discovery of three-dimensional (3D) time-reversal symmetric topological insulators and topological superconductors has attracted tremendous attention in recent years, both experimentally and theoretically [1,2]
We study the bulk-boundary correspondence for 3D fermionic symmetry-protected topological (SPT) phases with mirror symmetry only, namely topological crystalline superconductors (TCSCs), under the assumption that the surface is a mirrorsymmetric topological order
It is obvious that (1) and (2) are very similar. This similarity is expected from the topological crystalline equivalence principle [49], which states that classifications of SPT/symmetry-enriched topological (SET) phases are equivalent for internal and crystalline symmetries
Summary
The discovery of three-dimensional (3D) time-reversal symmetric topological insulators and topological superconductors has attracted tremendous attention in recent years, both experimentally and theoretically [1,2]. The ’t Hooft anomalies in d dimensional field theories with a symmetry group G have a one-to-one correspondence to the d + 1 dimensional SPT phases of the same group. We study the bulk-boundary correspondence for 3D fermionic SPT phases with mirror symmetry only, namely topological crystalline superconductors (TCSCs), under the assumption that the surface is a mirrorsymmetric topological order. It is obvious that (1) and (2) are very similar This similarity is expected from the topological crystalline equivalence principle [49], which states that classifications of SPT/SET phases are equivalent for internal (such as time-reversal) and crystalline (such as mirror) symmetries. In Appendix, we discuss some general properties of fermionic topological orders after the fermion parity is gauged
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