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Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants

Yunqing Ouyang State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China; Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China Qing-Rui Wang Department of Physics, Yale University, New Haven, CT 06511, USA Zheng-Cheng Gu Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China Yang Qi State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China; Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China; Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

Mathematical Physics arXiv subject: Strongly Correlated Electrons (cond-mat.str-el) mathscidoc:2206.22002

Chinese Physics Letters, 38, (12), 2021.11
In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. In this paper, based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.
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@inproceedings{yunqing2021computing,
  title={Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants},
  author={Yunqing Ouyang, Qing-Rui Wang, Zheng-Cheng Gu, and Yang Qi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616162736581201389},
  booktitle={Chinese Physics Letters},
  volume={38},
  number={12},
  year={2021},
}
Yunqing Ouyang, Qing-Rui Wang, Zheng-Cheng Gu, and Yang Qi. Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants. 2021. Vol. 38. In Chinese Physics Letters. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616162736581201389.
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