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Chin. Phys. B, 2022, Vol. 31(4): 040201    DOI: 10.1088/1674-1056/ac2f2c
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Revealing Chern number from quantum metric

Anwei Zhang(张安伟)1,2,†
1 Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China;
2 Department of Physics, Ajou University, Suwon 16499, Korea
Abstract  Chern number is usually characterized by Berry curvature. Here, by investigating the Dirac model of even-dimensional Chern insulator, we give the general relation between Berry curvature and quantum metric, which indicates that the Chern number can be encoded in quantum metric as well as the surface area of the Brillouin zone on the hypersphere embedded in Euclidean parameter space. We find that there is a corresponding relationship between the quantum metric and the metric on such a hypersphere. We give the geometrical property of quantum metric. Besides, we give a protocol to measure the quantum metric in the degenerate system.
Keywords:  quantum metric      Chern insulator      topological physics  
Received:  01 September 2021      Revised:  04 October 2021      Accepted manuscript online:  13 October 2021
PACS:  02.40.-k (Geometry, differential geometry, and topology)  
  03.65.Vf (Phases: geometric; dynamic or topological)  
  61.82.Ms (Insulators)  
Fund: We would like to thank R. B. Liu for useful discussion and N. Goldman for helpful comment.
Corresponding Authors:  Anwei Zhang     E-mail:  zawcuhk@gmail.com

Cite this article: 

Anwei Zhang(张安伟) Revealing Chern number from quantum metric 2022 Chin. Phys. B 31 040201

[1] Pancharatnam S 1956 Proc. Indian Acad. Sci. A 44 247
[2] Berry M V 1984 Proc. R. Soc. Lond. A 392 45
[3] Simon B 1983 Phys. Rev. Lett. 51 2167
[4] Thouless D J, Kohmoto M, Nightingale M P and den Nijs M 1982 Phys. Rev. Lett. 49 405
[5] Zhang S C and Hu J 2001 Science 294 823
[6] Provost J P and Vallee G 1980 Commun. Math. Phys. 76 289
[7] Wilczek F and Zee A 1984 Phys. Rev. Lett. 52 2111
[8] Ma Y Q, Chen S, Fan H and Liu W M 2010 Phys. Rev. B 81 245129
[9] Zanardi P and Paunkovic N 2006 Phys. Rev. E 74 031123
[10] Carollo A, Valenti D and Spagnolo B 2020 Phys. Rep. 838 1
[11] Zanardi P, Giorda P and Cozzini M 2007 Phys. Rev. Lett. 99 100603
[12] Dey A, Mahapatra S, Roy P and Sarkar T 2012 Phys. Rev. E 86 031137
[13] Yang S, Gu S J, Sun C P and Lin H Q 2008 Phys. Rev. A 78 012304
[14] Garnerone S, Abasto D, Haas S and Zanardi P 2009 Phys. Rev. A 79 032302
[15] Klees R L, Rastelli G, Cuevas J C and Belzig W 2020 Phys. Rev. Lett. 124 197002
[16] Bleu O, Malpuech G, Gao Y and Solnyshkov D D 2018 Phys. Rev. Lett. 121 020401
[17] Palumbo G and Goldman N 2018 Phys. Rev. Lett. 121 170401
[18] Ahn J, Guo G Y and Nagaosa N 2020 Phys. Rev. X 10 041041
[19] Ahn J, Guo G Y, Nagaosa N and Vishwanath A 2021 arXiv:2103.01241[cond-mat.mes-hall]
[20] Zhang Y F, Yang Y Y, Ju Y, Sheng L, Shen R, Sheng D N and Xing D Y 2013 Chin. Phys. B 22 117312
[21] Claassen M, Lee C H, Thomale R, Qi X L and Devereaux T P 2015 Phys. Rev. Lett. 114 236802
[22] Yang L, Ma Y Q and Li X G 2015 Physica B 456 359
[23] Piechon F, Raoux A, Fuchs J N and Montambaux G 2016 Phys. Rev. B 94 134423
[24] Pozo O and de Juan F 2020 Phys. Rev. B 102 115138
[25] Qi X L, Hughes T L and Zhang S. C 2008 Phys. Rev. B 78 195424
[26] Murakami S, Nagaosa N and Zhang S C 2004 Phys. Rev. B 69 235206
[27] Weatherburn C E 1938 An Introduction to Riemannian Geometry and the Tensor Calculus (Cambridge:Cambridge University)
[28] In the hyperspherical coordinates $\hat{d}_1=\cos \theta_1$, $\hat{d}_2=\sin\theta_1\cos \theta_2$, $\hat{d}_3=\sin\theta_1\sin\theta_2\cos \theta_3$, $\hat{d}_4=\sin\theta_1\sin\theta_2\sin\theta_3\cos\theta_4$, $\hat{d}_5=\sin\theta_1\sin\theta_2\sin\theta_3\sin\theta_4$, the non-zero components of the quantum metric tensor are $g_{\theta_1\theta_1}=\frac{1}{2}$, $g_{\theta_2\theta_2}=\frac{1}{2}\sin^2\theta_1$, $g_{\theta_3\theta_3}=\frac{1}{2}\sin^2\theta_1\sin^2\theta_2$, $g_{\theta_4\theta_4}=\frac{1}{2}\sin^2\theta_1\sin^2 \theta_2\sin^2\theta_3$. It can be seen that these diagonal elements are positive.
[29] Eguchi T, Gilkey P B and Hanson A J 1980 Phys. Rep. 66 213
[30] Morgan F 1993 Riemannian Geometry:A Beginner's Guide (Boston:Jones and Bartlett) p. 72
[31] Sugawa S, Salces-Carcoba F, Perry A R, Yue Y and Spielman I B 2018 Science 360 1429
[32] Price H M 2020 Phys. Rev. B 101 205141
[33] Wang Y, Price H M, Zhang B and Chong Y D 2020 Nat. Commun. 11 2356
[34] Yu R, Zhao Y X and Schnyder A P 2020 Natl. Sci. Rev. 7 1288
[35] Ozawa T and Goldman N 2018 Phys. Rev. B 97 201117(R)
[36] Roy R 2014 Phys. Rev. B 90 165139
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