Non-Abelian Three-Loop Braiding Statistics for 3D Fermionic Topological Phases

Jing-Ren Zhou Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong Qing-Rui Wang Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong Chenjie Wang Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China Zheng-Cheng Gu Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Mathematical Physics arXiv subject: High Energy Physics - Theory (hep-th) arXiv subject: Strongly Correlated Electrons (cond-mat.str-el) mathscidoc:2206.22003

Nature Communications, 12, 3191, 2021.5
Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing universal topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for loop-like excitations for 3D fermionic topological phases. Most surprisingly, we discovered new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with fermionic particles). The simplest example of such non-Abelian braiding statistics can be realized in interacting fermionic systems with a gauge group Z2×Z8 or Z4×Z4, and the physical origin of non-Abelian statistics can be viewed as attaching an open Majorana chain onto a pair of linked loops, which will naturally reduce to the well known Ising non-Abelian statistics via the standard dimension reduction scheme. Moreover, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also give rise to an alternative way to classify FSPT phases with unitary symmetries. We further compare the classification results for FSPT phases with arbitrary Abelian total symmetry G^f and find systematical agreement with previous studies using other methods. We believe that the proposed framework of understanding three-loop braiding statistics (including both Abelian and non-Abelian cases) in interacting fermion systems applies for generic fermonic topological phases in 3D.
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@inproceedings{jing-ren2021non-abelian,
  title={Non-Abelian Three-Loop Braiding Statistics for 3D Fermionic Topological Phases},
  author={Jing-Ren Zhou, Qing-Rui Wang, Chenjie Wang, and Zheng-Cheng Gu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616163125035157390},
  booktitle={Nature Communications},
  volume={12},
  pages={3191},
  year={2021},
}
Jing-Ren Zhou, Qing-Rui Wang, Chenjie Wang, and Zheng-Cheng Gu. Non-Abelian Three-Loop Braiding Statistics for 3D Fermionic Topological Phases. 2021. Vol. 12. In Nature Communications. pp.3191. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616163125035157390.
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