Anomaly indicators and bulk-boundary correspondences for three-dimensional interacting topological crystalline phases with mirror and continuous symmetries

2021/05/06 by Shang-Qiang Ning, Bin-Bin Mao, Zhengqiao Li +1
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Anomaly (physics) #Anyon #Boundary (topology) #Combinatorics #Geometry #Homogeneous space #Mathematics #Physics #Quantum #Quantum many-body systems #Quantum mechanics #Surface (topology) #Symmetry (geometry) #Symmetry protected topological order #Theoretical physics #Topological Materials and Phenomena #Topological order #Topological quantum computer #Topology (electrical circuits) #cond-mat.str-el #hep-th #math-ph #math.MP

paper · pdf · doi:10.1103/physrevb.104.075111

published as Phys. Rev. B 104, 075111 (2021) · 44 pages, 3 figures, 6 tables

arxiv created 2021/05/06 · openalex created_date 2021/05/10 · openalex publication_date 2021/08/09 · arxiv updated 2021/08/18 · openalex updated_date 2026/08/05

Abstract

We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric, and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of U(1), SU(2), or SO(3). In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous tenfold-way classification, with the time-reversal symmetry replaced by the mirror symmetry and with strong interaction taken into account. For surface SETs, the most general interplay between symmetries and anyon excitations is considered. Based on the previously proposed dimensional reduction and folding approaches, we rederive the classification of bulk SPT phases and define a complete set of bulk topological invariants for every symmetry group under consideration and then derive explicit expressions of the bulk invariants in terms of surface topological properties (such as topological spin, quantum dimension) and symmetry properties (such as mirror fractionalization, fractional charge or spin). These expressions are our quantitative bulk-boundary correspondences. Meanwhile, the bulk topological invariants can be interpreted as anomaly indicators for the surface SETs which carry 't Hooft anomalies of the associated symmetries whenever the bulk is topologically nontrivial. Hence, the quantitative bulk-boundary correspondences provide an easy way to compute the 't Hooft anomalies of the surface SETs. Moreover, our anomaly indicators are complete. Our derivations of the bulk-boundary correspondences and anomaly indicators are explicit and physically transparent. The anomaly indicators obtained in this work can be straightforwardly translated to their time-reversal counterparts that apply to the usual topological insulators and topological superconductors, due to a known correspondence between mirror and time-reversal topological phases.

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