缀饰格子中时间反演对称破缺的量子自旋霍尔效应

耿虎; 计青山; 张存喜; 王瑞

doi:10.7498/aps.66.127303

摘要

研究了缀饰格子中的量子自旋霍尔效应，模型中同时考虑了Rashba自旋轨道耦合和交换场的作用.缀饰格子具有简立方对称性，以零能平带和单狄拉克锥结构为主要特点.在缀饰格子中，不论是实现量子自旋霍尔效应还是量子反常霍尔效应，都需要一个不为零的内禀自旋轨道耦合作用来打开一个完全的体能隙，这与石墨烯等六角格子模型有着很大的不同.在交换场破坏了时间反演对称性的情况下，以自旋陈数为标志的量子自旋霍尔效应仍然能够存在，边缘态和极化率的相关结果也证明了这一结论.结果表明自旋陈数比z2拓扑数在表征量子自旋霍尔效应方面有着更广泛的适用范围，相应的结论为利用磁场控制量子自旋霍尔效应提出了一个理论模型和依据.

关键词:

Abstract

In this paper, the time-reversal (TR) symmetry broken quantum spin Hall (QSH) in Lieb lattice is investigated in the presence of both Rashba spin-orbit coupling (SOC) and uniform exchange field. The Lieb lattice has a simple cubic symmetry, and it has three different sites in each unit cell. The most distinctive feature of this model is that it contains only one Dirac-cone in the first Brillouin zone, where the upper dispersive band and the lower dispersive band touch the middle zero-energy band at M point and form a cone-like dispersion. The intrinsic SOC is essentially needed to open the full energy gap in the bulk. When the intrinsic SOC is nonzero, all the band structures are separated everywhere in the Brillouin zone and can be characterized by some topological invariants. The exact QSH first put forward by Kane and Mele in 2005 is characterized by the z2 number. The protection from the TR symmetry ensures the gapless crossing in the surface state in the bulk gap. In our model, the presence of the exchange field breaks the TR symmetry, which results in opening a small gap in the crossing point and the z2 topological order is not suitable for the system. This kind of state is a TR symmetry broken QSH, which is characterized by the spin Chern numbers. The spin Chern numbers have a much wider scope of application than z2 index. It is suitable for both TR symmetry system and the TR symmetry broken system. For Lieb lattice ribbons, the spin polarization and the wave-function distributions are obtained numerically. There exists a weak scattering between the counter-propagating states in the TR symmetry broken QSH, and the spin transport along the boundary with a low dissipation replaces the dissipationless spin current in a TR symmetry system. In experiment, such a system can be realized by the two-dimensional Fermi gases in optical lattice with Lieb symmetry. The above conclusions are expected to give theoretical guidance in the spin device and the quantum information.

Keywords:

作者及机构信息

1.
浙江海洋大学电子信息科学与工程系, 舟山 316022;

2.
浙江海洋大学东海科学与技术学院, 舟山 316000

通信作者: 王瑞, wangrui@zjou.edu.cn

基金项目: 国家自然科学基金（批准号：11304281，10547001）和浙江省自然科学基金（批准号：LY13D060002）资助的课题.

Authors and contacts

1.
Department of Electronic Information Science and Engineering, Zhejiang Ocean University, Zhoushan 316022, China;

2.
Department of Donghai Science and Engineering, Zhejiang Ocean University, Zhoushan 316000, China

Corresponding author: Wang Rui, wangrui@zjou.edu.cn

Funds: Project supported by National Natural Science Foundation of China (Grant Nos. 11304281, 10547001) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LY13D060002).

参考文献

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[8]	Qi X L, Zhang S C 2010 Physics Today 63 33
[9]	Li Z J, Li Q, Chen Z G, Li H B, Fang Y 2014 Chin. Phys. B 23 028102
[10]	Thouless D J, Kohmoto M, Nightingale M P, Den Nijs M 1982 Phys. Rev. Lett. 49 405
[11]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[12]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[13]	Konig M, Wiedmann S, Brune C, Roth A, Buthmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766
[14]	Sheng D N, Weng Z Y, Sheng L, Haldane F D M 2006 Phys. Rev. Lett. 97 036808
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[17]	Qiao Z H, Yang S A, Feng W X, Tse W K, Ding J, Yao Y G, Wang J, Niu Q 2010 Phys. Rev. B 82 161414
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[20]	Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[21]	Raghu S, Chung S B, Qi X L, Zhang S C 2010 Phys. Rev. Lett. 104 116401
[22]	Yu R, Zhang W, Zhang H J, Zhang S C, Dai X, Fang Z 2010 Science 329 61
[23]	Wu C 2008 Phys. Rev. Lett. 101 186807
[24]	Guo H M, Franz M 2009 Phys. Rev. B 80 113102
[25]	Zhang Z Y 2011 J. Phys. Condens. Matter 23 365801
[26]	Ishizuka H, Motome Y 2013 Phys. Rev. B 87 081105
[27]	Kargarian M, Fiete G A 2010 Phys. Rev. B 82 085106
[28]	Chen W C, Liu R, Wang Y F, Gong C D 2012 Phys. Rev. B 86 085311
[29]	Ohgushi K, Murakami S, Nagaosa N 2000 Phys. Rev. B 62 R6065
[30]	Wang Z, Zhang P 2008 Phys. Rev. B 77 125119
[31]	Shen R, Shao L B, Wang B, Xing D Y 2010 Phys. Rev. B 81 041410
[32]	Beugeling W, Everts J C, Morais S C 2012 Phys. Rev. B 86 195129
[33]	Zhao A, Shen S Q 2012 Phys. Rev. B 85 085209
[34]	Weeks C, Franz M 2010 Phys. Rev. B 82 085310
[35]	Sun K, Fradkin E 2008 Phys. Rev. B 78 245122
[36]	He Y, Moore J, Varma C M 2012 Phys. Rev. B 85 155106
[37]	Stanescu T D, Galitski V, Vaishnav J Y, Clark C W, Das Sarma S 2009 Phys. Rev. A 79 053639
[38]	Zhu S L, Fu H, Wu C J, Zhang S C, Duan L M 2006 Phys. Rev. Lett. 97 240401
[39]	Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885
[40]	Goldman N, Urban D F, Bercioux D 2011 Phys. Rev. A 83 063601
[41]	Gibertini M, Singha A, Pellegrini V, Polini M, Vignale G, Pinczuk A, Pfeiffer L N, West K W 2009 Phys. Rev. B 79 241406
[42]	Zhang C, Tewari S, Lutchyn R M, Das Sarma S 2008 Phys. Rev. Lett. 101 160401
[43]	Chosh P, Sau J D, Tewari S, Das Sarma S 2010 Phys. Rev. B 82 184525
[44]	Temari S, Sau J D 2012 Phys. Rev. Lett. 109 150408

施引文献

[1]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[2]	Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559
[3]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[4]	Zhang H J, Xu Y, Wang J, Chang K, Zhang S C 2014 Phys. Rev. Lett. 112 216803
[5]	Miao M S, Yan Q, van de Wall C G, Lou W K, Li L L, Chang K 2012 Phys. Rev. Lett. 109 186803
[6]	Zhang D, Lou W K, Miao M S, Zhang S C, Chang K 2013 Phys. Rev. Lett. 111 156402
[7]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
[8]	Qi X L, Zhang S C 2010 Physics Today 63 33
[9]	Li Z J, Li Q, Chen Z G, Li H B, Fang Y 2014 Chin. Phys. B 23 028102
[10]	Thouless D J, Kohmoto M, Nightingale M P, Den Nijs M 1982 Phys. Rev. Lett. 49 405
[11]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[12]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[13]	Konig M, Wiedmann S, Brune C, Roth A, Buthmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766
[14]	Sheng D N, Weng Z Y, Sheng L, Haldane F D M 2006 Phys. Rev. Lett. 97 036808
[15]	Yang Y Y, Xu Z, Sheng L, Wang B G, Xing D Y, Sheng D N 2011 Phys. Rev. Lett. 107 066602
[16]	Pradan E 2009 Phys. Rev. B 80 125327
[17]	Qiao Z H, Yang S A, Feng W X, Tse W K, Ding J, Yao Y G, Wang J, Niu Q 2010 Phys. Rev. B 82 161414
[18]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015
[19]	Onoda M, Nagaosa N 2003 Phys. Rev. Lett. 90 206601
[20]	Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[21]	Raghu S, Chung S B, Qi X L, Zhang S C 2010 Phys. Rev. Lett. 104 116401
[22]	Yu R, Zhang W, Zhang H J, Zhang S C, Dai X, Fang Z 2010 Science 329 61
[23]	Wu C 2008 Phys. Rev. Lett. 101 186807
[24]	Guo H M, Franz M 2009 Phys. Rev. B 80 113102
[25]	Zhang Z Y 2011 J. Phys. Condens. Matter 23 365801
[26]	Ishizuka H, Motome Y 2013 Phys. Rev. B 87 081105
[27]	Kargarian M, Fiete G A 2010 Phys. Rev. B 82 085106
[28]	Chen W C, Liu R, Wang Y F, Gong C D 2012 Phys. Rev. B 86 085311
[29]	Ohgushi K, Murakami S, Nagaosa N 2000 Phys. Rev. B 62 R6065
[30]	Wang Z, Zhang P 2008 Phys. Rev. B 77 125119
[31]	Shen R, Shao L B, Wang B, Xing D Y 2010 Phys. Rev. B 81 041410
[32]	Beugeling W, Everts J C, Morais S C 2012 Phys. Rev. B 86 195129
[33]	Zhao A, Shen S Q 2012 Phys. Rev. B 85 085209
[34]	Weeks C, Franz M 2010 Phys. Rev. B 82 085310
[35]	Sun K, Fradkin E 2008 Phys. Rev. B 78 245122
[36]	He Y, Moore J, Varma C M 2012 Phys. Rev. B 85 155106
[37]	Stanescu T D, Galitski V, Vaishnav J Y, Clark C W, Das Sarma S 2009 Phys. Rev. A 79 053639
[38]	Zhu S L, Fu H, Wu C J, Zhang S C, Duan L M 2006 Phys. Rev. Lett. 97 240401
[39]	Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885
[40]	Goldman N, Urban D F, Bercioux D 2011 Phys. Rev. A 83 063601
[41]	Gibertini M, Singha A, Pellegrini V, Polini M, Vignale G, Pinczuk A, Pfeiffer L N, West K W 2009 Phys. Rev. B 79 241406
[42]	Zhang C, Tewari S, Lutchyn R M, Das Sarma S 2008 Phys. Rev. Lett. 101 160401
[43]	Chosh P, Sau J D, Tewari S, Das Sarma S 2010 Phys. Rev. B 82 184525
[44]	Temari S, Sau J D 2012 Phys. Rev. Lett. 109 150408

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