Edge theories of two-dimensional fermionic symmetry protected topological phases protected by unitary Abelian symmetries

Shang-Qiang Ning Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China Chenjie Wang Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China Qing-Rui Wang Department of Physics, Yale University, New Haven, Connecticut 06511-8499, USA Zheng-Cheng Gu Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

arXiv subject: High Energy Physics - Theory (hep-th) arXiv subject: Strongly Correlated Electrons (cond-mat.str-el) mathscidoc:2206.74021

PHYSICAL REVIEW B, 104, 075151, 2021.8
Abelian Chern-Simons theory, characterized by the so-called K matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry G_f. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry Gf have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if G_f is a nontrivial extension of bosonic symmetry G_b by \mathbb{Z}_f^2, (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge n−1 for type-III bosonic SPT phases protected by (\mathbb{Z}_n)^3 symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry G_f.
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@inproceedings{shang-qiang2021edge,
  title={Edge theories of two-dimensional fermionic symmetry protected topological phases protected by unitary Abelian symmetries},
  author={Shang-Qiang Ning, Chenjie Wang, Qing-Rui Wang, and Zheng-Cheng Gu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616164248313759393},
  booktitle={PHYSICAL REVIEW B},
  volume={104},
  pages={075151},
  year={2021},
}
Shang-Qiang Ning, Chenjie Wang, Qing-Rui Wang, and Zheng-Cheng Gu. Edge theories of two-dimensional fermionic symmetry protected topological phases protected by unitary Abelian symmetries. 2021. Vol. 104. In PHYSICAL REVIEW B. pp.075151. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616164248313759393.
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