Rashba自旋轨道耦合下square-octagon晶格的拓扑相变

杨圆; 陈帅; 李小兵

doi:10.7498/aps.67.20180624

摘要

本文研究了各向同性square-octagon晶格在内禀自旋轨道耦合、Rashba自旋轨道耦合和交换场作用下的拓扑相变，同时引入陈数和自旋陈数对系统进行拓扑分类.系统在自旋轨道耦合和交换场的影响下会出现许多拓扑非平庸态，包括时间反演对称破缺的量子自旋霍尔态和量子反常霍尔态.特别的是，在时间反演对称破缺的量子自旋霍尔效应中，无能隙螺旋边缘态依然能够完好存在.调节交换场或者填充因子的大小会导致系统发生从时间反演对称破缺的量子自旋霍尔态到自旋过滤的量子反常霍尔态的拓扑相变.边缘态能谱和自旋谱的性质与陈数和自旋陈数的拓扑刻画完全一致.这些研究成果为自旋量子操控提供了一个有趣的途径.

关键词:

Abstract

Motivated by the square-octagon lattice which supports topological phases over a wide range of parameters and a number of interesting quantum phase transitions in the phase diagram when considering the intrinsic spin-orbit coupling, we investigate the topological phase transitions in the isotropic square-octagon lattice combining the effects of both spin-orbit couplings and exchange field. The inversion symmetry and time-reversal symmetry are broken when both Rashba spin-orbit coupling and exchange field are present. The Z₂ index is not applicable for quantum spin Hall systems without time-reversal symmetry, but the spin Chern number remains valid even in the absence of time-reversal symmetry. Therefore, we use the Chern number and spin Chern number to describe the topological properties of the system. We explore that a variety of topologically nontrivial states appear with changing the exchange field, including time-reversal-symmetry-broken quantum spin Hall states and quantum anomalous Hall states. The phase transition between these topological phases is accompanied by the closing of band gaps. Interestingly, the quantum spin Hall effect described by nonzero spin Chern number is found to remain intact when the time-reversal symmetry is broken. Furthermore, the variation of the amplitude of the exchange field and filling factor drive interesting topological phase transitions from the time-reversal-symmetry-broken quantum spin Hall phase to spin-filtered quantum anomalous Hall phase. A spin-filtered quantum anomalous Hall phase is characterized by the presence of edge states with only one spin component, which provides an interesting route towards quantum spin manipulation. We also present the band structures, edge state wave functions, and spin polarizations of the different topological phases in the system. It is demonstrated that the energy spectra of edge states are in good agreement with the topological characterization based on the Chern number and spin Chern number. In particular, we observe that gapless edge states can appear in a time-reversal-symmetry-broken quantum spin Hall system, but the corresponding spin spectrum gap remains open on the edges. Recently, an important functional material ZnO with quasi square-octagon lattice has been found experimentally. Consequently, the results found in our work are helpful for understanding the property of square-octagon lattice and studying the real materials with square-octagon structure.

Keywords:

作者及机构信息

1. 江苏科技大学张家港校区, 张家港 215600;

2. 南京大学, 固体微结构物理国家重点实验室, 南京 210093

基金项目: 国家自然科学基金（批准号：11647145）、南京大学固体微结构物理国家重点实验室开放课题（批准号：M31024）和江苏省高校自然科学研究项目（批准号：16KJB430012）资助的课题.

Authors and contacts

1. Zhangjiagang Campus, Jiangsu University of Science and Technology, Zhangjiagang 215600, China;

2. National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11647145), the National Laboratory of Solid State Microstructures (Grant No. M31024), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 16KJB430012).

参考文献

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[45]	Lin Y J, Jiménez-García K, Spielman I B 2011 Nature 471 83

施引文献

[1]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[2]	Bernevig B A, Zhang S C 2005 Phys. Rev. Lett. 96 106802
[3]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
[4]	Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[5]	Ren Y F, Qiao Z H, Niu Q 2016 Rep. Prog. Phys. 79 066501
[6]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[7]	Moore J E, Balents L 2007 Phys. Rev. B 75 121306
[8]	Prodan E 2009 Phys. Rev. B 80 125327
[9]	Prodan E 2010 New J. Phys. 12 065003
[10]	Sheng D N, Weng Z Y, Sheng L, Haldane F D M 2006 Phys. Rev. Lett. 97 036808
[11]	Yang Y Y, Xu Z, Sheng L, Wang B G, Xing D Y, Sheng D N 2011 Phys. Rev. Lett. 107 066602
[12]	Du L J, Knez I, Sullivan G, Du R R 2015 Phys. Rev. Lett. 114 096802
[13]	Yao Y G, Ye F, Qi X L, Zhang S C, Fang Z 2007 Phys. Rev. B 75 041401
[14]	Min H, Hill J E, Sinitsyn N A, Sahu B R, Kleinman L, MacDonald A H 2006 Phys. Rev. B 74 165310
[15]	Liu C C, Jiang H, Yao Y 2011 Phys. Rev. B 84 195430
[16]	Qiao Z, Yang S A, Feng W, Tse W K, Ding J, Yao Y, Wang J, Niu Q 2010 Phys. Rev. B 82 161414
[17]	Qiao Z, Jiang H, Li X, Yao Y, Niu Q 2012 Phys. Rev. B 85 115439
[18]	Zhang Z Y 2011 J. Phys. Condens. Matter 23 365801
[19]	Chen M S, Wan S L 2012 J. Phys. Condens. Matter 24 325502
[20]	Guo H M, Franz M 2009 Phys. Rev. B 80 113102
[21]	Rüegg A, Wen J, Fiete G A 2010 Phys. Rev. B 81 205115
[22]	Zhou T, Zhang J, Xue Y, Zhao B, Zhang H, Jiang H, Yang Z 2016 Phys. Rev. B 94 235449
[23]	Kargarian M, Fiete G A 2010 Phys. Rev. B 82 085106
[24]	Liu X P, Chen W C, Wang Y F, Gong C D 2013 J. Phys. Condens. Matter 25 305602
[25]	Bao A, Tao H S, Liu H D, Zhang X Z, Liu W M 2015 Sci. Rep. 4 6918
[26]	Bao A, Zhang X F, Zhang X Z 2015 Chin. Phys. B 24 050310
[27]	Zhang L, Wang F 2017 Phys. Rev. Lett. 118 087201
[28]	Kang Y T, Yang F, Yao D X 2017 arXiv: 1801.00220. https://arxiv.org/abs/1801.00220
[29]	Yang Y, Yang J, Li X, Zhao Y 2018 Phys. Lett. A 382 723
[30]	Panahi P S, Struck J, Hauke P, Bick A, Plenkers W, Meineke G, Becker C, Windpassinger P, Lewenstein M, Sengstock K 2011 Nat. Phys. 7 434
[31]	Jo G B, Guzman J, Thomas C K, Hosur P, Vishwanath A, StamperKurn D M 2012 Phys. Rev. Lett. 108 045305
[32]	He M R, Yu R, Zhu J 2012 Angew. Chem. 124 7864
[33]	Fukui T, Hatsugai Y, Suzuki H 2005 J. Phys. Soc. Jpn. 74 1674
[34]	Taillefumier M, Dugaev V K, Canals B, Lacroix C, Bruno P 2008 Phys. Rev. B 78 155330
[35]	Hatsugai Y 1993 Phys. Rev. B 48 11851
[36]	Hatsugai Y 1993 Phys. Rev. Lett. 71 3697
[37]	Sun K, Fradkin E 2008 Phys. Rev. B 78 245122
[38]	Goldman N, Beugeling W, Smith C M 2012 Europhys. Lett. 97 23003
[39]	Beugeling W, Goldman N, Smith C M 2012 Phys. Rev. Lett. 86 075118
[40]	Li H C, Sheng L, Shen R, Shao L B, Wang B G, Sheng D N, Xing D Y 2013 Phys. Rev. Lett. 110 266802
[41]	Miao M, Yan Q, van de Walle C, Lou W, Li L, Chang K 2012 Phys. Rev. Lett. 109 186803
[42]	Zhang D, Lou W, Miao M, Zhang S, Chang K 2013 Phys. Rev. Lett. 111 156402
[43]	Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T 2014 Nature 515 237
[44]	Lin Y J, Compton R L, Jiménez-García K, Porto J V, Spielman I B 2009 Nature 462 628
[45]	Lin Y J, Jiménez-García K, Spielman I B 2011 Nature 471 83

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