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

doi:10.1088/0256-307X/38/12/127101

Chin. Phys. Lett.

2021, Vol. 38

Issue (12): 127101 DOI: 10.1088/0256-307X/38/12/127101

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES

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

Yunqing Ouyang^1,2, Qing-Rui Wang^3†, Zheng-Cheng Gu^4*, and Yang Qi^1,2,5*

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

Cite this article:

Yunqing Ouyang, Qing-Rui Wang, Zheng-Cheng Gu et al 2021 Chin. Phys. Lett. 38 127101

Download: PDF(612KB) PDF(mobile)(1178KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

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).

TRENDMD:

URL:

https://cpl.iphy.ac.cn/10.1088/0256-307X/38/12/127101 OR https://cpl.iphy.ac.cn/Y2021/V38/I12/127101

Service
	E-mail this article
	E-mail Alert
	RSS
Articles by authors
	Yunqing Ouyang
	Qing-Rui Wang
	Zheng-Cheng Gu
	and Yang Qi

[1]	Wen X G 2017 Rev. Mod. Phys. 89 041004
[2]	Wen X G 2019 Science 363 eaal3099
[3]	Gu Z C and Wen X G 2009 Phys. Rev. B 80 155131
[4]	Chen X, Liu Z X, and Wen X G 2011 Phys. Rev. B 84 235141
[5]	Chen X, Gu Z C, Liu Z X, and Wen X G 2012 Science 338 1604
[6]	Chen X, Gu Z C, Liu Z X, and Wen X G 2013 Phys. Rev. B 87 155114
[7]	Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[8]	Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[9]	Fu L 2011 Phys. Rev. Lett. 106 106802
[10]	Affleck I, Kennedy T, Lieb E H, and Tasaki H 1987 Phys. Rev. Lett. 59 799
[11]	Gu Z C and Wen X G 2014 Phys. Rev. B 90 115141
[12]	Kapustin A, Thorngren R, Turzillo A, and Wang Z 2015 J. High Energy Phys. 2015(12) 052
[13]	Gaiotto D and Kapustin A 2016 Int. J. Mod. Phys. A 31 1645044
[14]	Kapustin A and Thorngren R 2017 J. High Energy Phys. 2017(10) 080
[15]	Gaiotto D and Johnson-Freyd T 2019 J. High Energy Phys. 2019(05) 007
[16]	Wang Q R and Gu Z C 2018 Phys. Rev. X 8 011055
[17]	Wang J, Ohmori K, Putrov P, Zheng Y, Wan Z, Guo M, Lin H, Gao P, and Yau S T 2018 Prog. Theor. Exp. Phys. 2018 053A01
[18]	Cheng M, Bi Z, You Y Z, and Gu Z C 2018 Phys. Rev. B 97 205109
[19]	Lan T, Zhu C, and Wen X G 2019 Phys. Rev. B 100 235141
[20]	Wang Q R and Gu Z C 2020 Phys. Rev. X 10 031055
[21]	Brumfiel G and Morgan J 2016 arXiv:1612.02860 [math.AT]
[22]	Brumfiel G and Morgan J 2018 arXiv:1803.08147 [math.GT]
[23]	Tantivasadakarn N 2017 Phys. Rev. B 96 195101
[24]	Levin M and Gu Z C 2012 Phys. Rev. B 86 115109
[25]	Wang C and Levin M 2014 Phys. Rev. Lett. 113 080403
[26]	Wang C and Levin M 2015 Phys. Rev. B 91 165119
[27]	Cheng M, Tantivasadakarn N, and Wang C 2018 Phys. Rev. X 8 011054
[28]	Wang J, Wen X G, and Yau S T 2019 Ann. Phys. 409 167904
[29]	Chen X, Burnell F J, Vishwanath A, and Fidkowski L 2015 Phys. Rev. X 5 041013
[30]	Barkeshli M, Bonderson P, Cheng M, and Wang Z 2019 Phys. Rev. B 100 115147
[31]	Barkeshli M and Cheng M 2020 SciPost Phys. 8 28
[32]	Bulmash D and Barkeshli M 2020 Phys. Rev. Res. 2 043033
[33]	Ning S Q, Zou L, and Cheng M 2020 Phys. Rev. Res. 2 043043
[34]	Qi Y Sptset Package https://github.com/yangqi137/SptSet
[35]	GAP 2019 GAP-Groups, Algorithms, and Programming Version 4.10.2 https://www.gap-system.org
[36]	Ryu S, Schnyder A P, Furusaki A, and Ludwig A W W 2010 New J. Phys. 12 065010
[37]	Wang Q R, Qi Y, and Gu Z C 2019 Phys. Rev. Lett. 123 207003
[38]	Joyner D 2007 arXiv:0706.0549 [math.GR]
[39]	Brown K S 2012 Cohomology of Groups (Berlin: Springer) vol 87
[40]	Ellis G and Luyen L V 2012 J. Symb. Comput. 47 1309
[41]	Ellis G 2004 J. Symb. Comput. 38 1077
[42]	Ellis G 2019 HAP, Homological Algebra Programming Version 1.19 https://gap-packages.github.io/hap
[43]	Wall C T C 1961 Math. Proc. Cambridge Philos. Soc. 57 251
[44]	Wallpaper group, https://en.wikipedia.org/wiki/Wallpaper/_group
[45]	Thorngren R and Else D V 2018 Phys. Rev. X 8 011040
[46]	Cheng M and Wang C 2018 arXiv:1810.12308 [cond-mat.str-el]
[47]	Zhang J H, Yang S, Qi Y, and Gu Z C 2020 arXiv:2012.15657 [cond-mat.str-el]

Viewed

Full text

Abstract