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研究了圆环型波导依照蜂窝结构排列的声子晶体系统中的拓扑相变.利用晶格结构的点群对称性实现赝自旋，并在圆环中引入旋转气流来打破时间反演对称性.通过紧束缚近似模型计算的解析结果表明，没有引入气流时，调节几何参数，系统存在普通绝缘体和量子自旋霍尔效应绝缘体两个相；引入气流后，可以实现新的时间反演对称性破缺的量子自旋霍尔效应相，而增大气流强度，则可以实现量子反常霍尔效应相.这三个拓扑相可以通过自旋陈数来分类.通过有限元软件模拟了多个系统中边界态的传播，发现不同于量子自旋霍尔效应相，量子反常霍尔相系统的表面只支持一种自旋的边界态，并且它无需时间反演对称性保护.We report a new topological phononic crystal in a ring-waveguide acoustic system. In the previous reports on topological phononic crystals, there are two types of topological phases:quantum Hall phase and quantum spin Hall phase. A key point in achieving quantum Hall insulator is to break the time-reversal (TR) symmetry, and for quantum spin Hall insulator, the construction of pseudo-spin is necessary. We build such pseudo-spin states under particular crystalline symmetry (C6v) and then break the degeneracy of the pseudo-spin states by introducing airflow to the ring. We study the topology evolution by changing both the geometric parameters of the unit cell and the strength of the applied airflow. We find that the system exhibits three phases:quantum spin Hall phase, conventional insulator phase and a new quantum anomalous Hall phase.The quantum anomalous Hall phase is first observed in phononics and cannot be simply classified by the Chern number or Z2 index since it results from TR-broken quantum spin Hall phase. We develop a tight-binding model to capture the essential physics of the topological phase transition. The analytical calculation based on the tight-binding model shows that the spin Chern number is a topological invariant to classify the bandgap. The quantum anomalous Hall insulator has a spin Chern number C±=(1,0) indicating the edge state is pseudo-spin orientation dependent and robust against TR-broken impurities.We also perform finite-element numerical simulations to verify the topological differences of the bandgaps. At the interface between a conventional insulator and a quantum anomalous Hall insulator, pseudo-spin dependent one-way propagation interface states are clearly observed, which are strikingly deferent from chiral edge states resulting from quantum Hall insulator and pairs of helical edge states resulting from quantum spin Hall insulator. Moreover, our pseudo-spin dependent edge state is robust against TR-broken impurities, which also sheds lights on spintronic devices.
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Keywords:
- phononic crystal /
- topology /
- quantum anomalous Hall effect
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[1] John S 1987 Phys. Rev. Lett. 58 2486
[2] Yablonovitch E 1987 Phys. Rev. Lett. 58 2059
[3] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[4] Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405
[5] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[6] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[7] König M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766
[8] Qi X L, Wu Y S, Zhang S C 2006 Phys. Rev. B 74 085308
[9] Prodan E 2009 Phys. Rev. B 80 125327
[10] Kitagawa T, Berg E, Rudner M, Demler E 2010 Phys. Rev. B 82 235114
[11] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
[12] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[13] Moore J E 2010 Nature 464 194
[14] Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904
[15] Wang Z, Chong Y D, Joannopoulos J D, Soljačić M 2008 Phys. Rev. Lett. 100 013905
[16] Khanikaev A B, Hossein Mousavi S, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233
[17] Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y,Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196
[18] Lu L, Joannopoulos J D, Soljacic M 2014 Nat. Photon. 8 821
[19] Yang Z, Gao F, Shi X, Lin X, Gao Z, Chong Y, Zhang B 2015 Phys. Rev. Lett. 114 114301
[20] Xiao M, Ma G, Yang Z, Sheng P, Zhang Z Q, Chan C T 2015 Nat. Phys. 11 240
[21] Lu J, Qiu C, Ke M, Liu Z 2016 Phys. Rev. Lett. 116 093901
[22] Fleury R, Sounas D L, Sieck C F, Haberman M R, Alù A 2014 Science 343 516
[23] Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901
[24] He C, Sun X C, Liu X P, Lu M H, Chen Y, Feng L, Chen Y F 2016 Proc. Natl. Acad. Sci. USA 113 4924
[25] Zhang Z, Wei Q, Cheng Y, Zhang T, Wu D, Liu X 2017 Phys. Rev. Lett. 118 084303
[26] Xu L, Wang H X, Xu Y D, Chen H Y, Jiang J H 2016 Opt. Express 24 18059
[27] He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124
[28] Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016
[29] Chen Z G, Wu Y 2016 Phys. Rev. Appl. 5 054021
[30] Haldane F D M 1988 Phys. Rev. Lett. 61 2015
[31] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[32] Li H, Sheng L, Shen R, Shao L B, Wang B, Sheng D N, Xing D Y 2013 Phys. Rev. Lett. 110 266802
[33] Chen Z G, Ni X, Wu Y, He C, Sun X C, Zheng L Y, Lu M H, Chen Y F 2014 Sci. Rep. 4 4613
[34] Alexandradinata A, Fang C, Gilbert M J, Bernevig B A 2014 Phys. Rev. Lett. 113 116403
[35] Liu C X, Zhang R X, van Leeuwen B K 2014 Phys. Rev. B 90 085304
[36] Sakoda K 2012 Opt. Express 20 3898
[37] Liu C X, Qi X L, Zhang H, Dai X, Fang Z, Zhang S C 2010 Phys. Rev. B 82 045122
[38] Chen Z G, Mei J, Sun X C, Zhang X, Zhao J, Wu Y 2017 Phys. Rev. A 95 043827
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