Topological phase transitions in square-octagon lattice with Rashba spin-orbit coupling

Yang Yuan; Chen Shuai; Li Xiao-Bing

doi:10.7498/aps.67.20180624

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:

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).

References

[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

Cited By

[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

[1]	Liu Xiang-Lian, Li Kai-Zhou, Li Xiao-Qiong, Zhang Qiang. Coexistence of quantum spin and valley hall effect in two-dimensional dielectric photonic crystals. Acta Physica Sinica, 2023, 72(7): 074205. doi: 10.7498/aps.72.20221814
[2]	Wang Zhi-Mei, Wang Hong, Xue Nai-Tao, Cheng Gao-Yan. Quantum coherence in spin-orbit coupled quantum dots system. Acta Physica Sinica, 2022, 71(7): 078502. doi: 10.7498/aps.71.20212111
[3]	Jia Liang-Guang, Liu Meng, Chen Yao-Yao, Zhang Yu, Wang Ye-Liang. Research progress of two-dimensional quantum spin Hall insulator in monolayer 1T'-WTe₂. Acta Physica Sinica, 2022, 71(12): 127308. doi: 10.7498/aps.71.20220100
[4]	Li Jia-Rui, Wang Zi-An, Xu Tong-Tong, Zhang Lian-Lian, Gong Wei-Jiang. Topological properties of the one-dimensional ${\cal {PT}}$-symmetric non-Hermitian spin-orbit-coupled Su-Schrieffer-Heeger model. Acta Physica Sinica, 2022, 71(17): 177302. doi: 10.7498/aps.71.20220796
[5]	Chen Xing, Xue Xiao-Bo, Zhang Sheng-Kang, Ma Yu-Quan, Fei Peng, Jiang Yuan, Ge Jun. Ground energy level transition for two-body interacting Fermionic system with spin-orbit coupling and Zeeman interaction. Acta Physica Sinica, 2021, 70(8): 083401. doi: 10.7498/aps.70.20201456
[6]	Zhang Ai-Xia, Jiang Yan-Fang, Xue Ju-Kui. Nonlinear energy band structure of spin-orbit coupled Bose-Einstein condensates in optical lattice. Acta Physica Sinica, 2021, 70(20): 200302. doi: 10.7498/aps.70.20210705
[7]	Xue Hai-Bin, Duan Zhi-Lei, Chen Bin, Chen Jian-Bin, Xing Li-Li. Electron transport through Su-Schrieffer-Heeger chain with spin-orbit coupling. Acta Physica Sinica, 2021, 70(8): 087301. doi: 10.7498/aps.70.20201742
[8]	Shi Ting-Ting, Wang Liu-Jiu, Wang Jing-Kun, Zhang Wei. Some recent progresses on the study of ultracold quantum gases with spin-orbit coupling. Acta Physica Sinica, 2020, 69(1): 016701. doi: 10.7498/aps.69.20191241
[9]	Wang Yan-Lan, Li Yan. Pseudospin states and topological phase transitions in two-dimensional photonic crystals made of dielectric materials. Acta Physica Sinica, 2020, 69(9): 094206. doi: 10.7498/aps.69.20191962
[10]	Li Zhi-Qiang, Wang Yue-Ming. One-dimensional spin-orbit coupling Bose gases with harmonic trapping. Acta Physica Sinica, 2019, 68(17): 173201. doi: 10.7498/aps.68.20190143
[11]	Liang Tao, Li Ming. Integer quantum Hall effect in a spin-orbital coupling system. Acta Physica Sinica, 2019, 68(11): 117101. doi: 10.7498/aps.68.20190037
[12]	Geng Hu, Ji Qing-Shan, Zhang Cun-Xi, Wang Rui. Time-reversal-symmetry broken quantum spin Hall in Lieb lattice. Acta Physica Sinica, 2017, 66(12): 127303. doi: 10.7498/aps.66.127303
[13]	Long Yang, Ren Jie, Jiang Hai-Tao, Sun Yong, Chen Hong. Quantum spin Hall effect in metamaterials. Acta Physica Sinica, 2017, 66(22): 227803. doi: 10.7498/aps.66.227803
[14]	Liu Sheng-Li, Li Jian-Zheng, Cheng Jie, Wang Hai-Yun, Li Yong-Tao, Zhang Hong-Guang, Li Xing-Ao. Doping and Raman scattering of strong spin-orbit-coupling compound Sr2-xLaxIrO4. Acta Physica Sinica, 2015, 64(20): 207103. doi: 10.7498/aps.64.207103
[15]	Chen Dong-Hai, Yang Mou, Duan Hou-Jian, Wang Rui-Qiang. Electronic transport properties of graphene pn junctions with spin-orbit coupling. Acta Physica Sinica, 2015, 64(9): 097201. doi: 10.7498/aps.64.097201
[16]	Chen Guang-Ping. Ground state of a rotating spin-orbit-coupled Bose-Einstein condensate in a harmonic plus quartic potential. Acta Physica Sinica, 2015, 64(3): 030302. doi: 10.7498/aps.64.030302
[17]	Gong Shi-Jing, Duan Chun-Gang. Recent progress in Rashba spin orbit coupling on metal surface. Acta Physica Sinica, 2015, 64(18): 187103. doi: 10.7498/aps.64.187103
[18]	Zhang Lei, Li Hui-Wu, Hu Liang-Bin. Study of stability of persistent spin helix in two-dimensional electron gases with spin-orbit coupling. Acta Physica Sinica, 2012, 61(17): 177203. doi: 10.7498/aps.61.177203
[19]	Dong Quan-Li, Zhang Jie, Yang Jie, Jiang Zhao-Tan. Electronic energy band structures of carbon nanotubeswith spin-orbit coupling interaction. Acta Physica Sinica, 2011, 60(7): 075202. doi: 10.7498/aps.60.075202
[20]	Yu Zhi-Qiang, Xie Quan, Xiao Qing-Quan. Effects of the spin-orbit coupling on X-ray spectrum in special relativity. Acta Physica Sinica, 2010, 59(2): 925-931. doi: 10.7498/aps.59.925

Catalog

Vol. 67, Issue 23 - 2018-12-05

Metrics

Abstract views: 7357
PDF Downloads: 174
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板