Chin. Phys. Lett.  2021, Vol. 38 Issue (12): 127101    DOI: 10.1088/0256-307X/38/12/127101
Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants
Yunqing Ouyang1,2, Qing-Rui Wang3†, Zheng-Cheng Gu4*, and Yang Qi1,2,5*
1State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
2Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China
3Department of Physics, Yale University, New Haven, CT 06511, USA
4Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
5Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
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Yunqing Ouyang, Qing-Rui Wang, Zheng-Cheng Gu et al  2021 Chin. Phys. Lett. 38 127101
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Abstract 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. 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.
Received: 12 September 2021      Editors' Suggestion Published: 27 November 2021
PACS:  71.27.+a (Strongly correlated electron systems; heavy fermions)  
  02.40.Re (Algebraic topology)  
  03.65.Vf (Phases: geometric; dynamic or topological)  
Fund: Supported by the National Natural Science Foundation of China (Grant No. 11874115), the Hong Kong Research Grants Council (Grant No. GRF 14306918), and ANR/RGC Joint Research Scheme (Grant No. A-CUHK402/18).
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Articles by authors
Yunqing Ouyang
Qing-Rui Wang
Zheng-Cheng Gu
and Yang Qi
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