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We design a two-dimensional acoustic crystal (AC) to obtain topologically protected edge states for sound waves. The AC is composed of a triangular array of a complex unit cell consisting of two identical triangle-shaped steel rods arranged in air. The steel rods are placed on the vertices of the hexagonal unit cell so that the whole lattice possesses the C6v symmetry. We show that by simply rotating all triangular rods around their respective centers by 180 degrees, a topological phase transition can be achieved, and more importantly, such a transition is accomplished with no need of changing the fill ratios or changing the positions of the rods. Interestingly, the achieved topologically nontrivial band gap has a very large frequency width, which is really beneficial to future applications. The topological properties of the AC are rooted in the spatial symmetries of the eigenstates. It is well known that there are two doubly-degenerate eigenstates at the point for a C6v point group, and they are usually called the p and d states in electronic system. By utilizing the spatial symmetries of the p and d states in the AC, we can construct the pseudo-time reversal symmetry which renders the Kramers doubling in this classical system. We find pseudospin states in the interface between topologically trivial and nontrivial ACs, where anticlockwise (clockwise) rotational behaviors of time-averaged Poynting vectors correspond to the pseudospin-up (pseudospin-down) orientations of the edge states, respectively. These phenomena are very similar to the real spin states of quantum spin Hall effect in electronic systems. We also develop an effective Hamiltonian for the associated bands to characterize the topological properties of the AC around the Brillouin zone center by the kp perturbation method. We calculate the spin Chern numbers of the ACs, and reveal the inherent link between the band inversion and the topological phase transition. With full-wave simulations, we demonstrate the one-way propagation of sound waves along the interface between topologically distinct ACs, and demonstrate the robustness of the edge states against different types of defects including bends, cavity and disorder. Our design provides a new way to realize acoustic topological effects in a wide frequency range spanning from infrasound to ultrasound. Potential applications and acoustic devices based on our design are expected, so that people can manipulate and transport sound waves in a more efficient way.
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Keywords:
- acoustic crystals /
- topological phase transition /
- band inversion /
- pseudospin
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[1] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[2] Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405
[3] Laughlin R B 1983 Phys. Rev. Lett. 50 1395
[4] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757
[5] Bernevig B A, Zhang S C 2006 Phys. Rev. Lett. 96 106802
[6] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801
[7] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[8] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
[9] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
[10] Wang Z, Chong Y D, Joannopoulos J D, Soljačić M 2008 Phys. Rev. Lett. 100 013905
[11] Wang Z, Chong Y D, Joannopoulos J D, Soljačić M 2009 Nature 461 772
[12] Fang Y T, He H Q, Hu J X, Chen L K, Wen Z 2015 Phys. Rev. A 91 033827
[13] Fleury R, Sounas D L, Sieck C F, Haberman M R, Al A 2014 Science 343 516
[14] Yang Z J, Gao F, Shi X H, Lin X, Gao Z, Chong Y D, Zhang B L 2015 Phys. Rev. Lett. 114 114301
[15] Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016
[16] Chen Z G, Wu Y 2016 Phys. Rev. Appl. 5 054021
[17] Khanikaev A B, Fleury R, Mousavi S H, Al A 2015 Nat. Commun. 6 8260
[18] Fleury R, Sounas D L, Al A 2015 Phys. Rev. B 91 174306
[19] Fleury R, Khanikaev A B, Alu A 2016 Nat. Commun. 7 11744
[20] Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368
[21] He C, Li Z, Ni X, Sun X C, Yu S Y, Lu M H, Liu X P, Chen Y F 2016 Appl. Phys. Lett. 108 031904
[22] Wei Q, Tian Y, Zuo S Y, Cheng Y, Liu X J 2017 Phys. Rev. B 95 094305
[23] Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901
[24] Wang H X, Xu L, Chen H Y, Jiang J H 2016 Phys. Rev. B 93 235155
[25] Mei J, Chen Z G, Wu Y 2016 Sci. Rep. 6 32752
[26] 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
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[28] 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
[29] Li Y, Mei J 2015 Opt. Express 23 12089
[30] Li Y, Wu Y, Mei J 2014 Appl. Phys. Lett. 105 014107
[31] Dai H Q, Liu T T, Jiao J R, Xia B Z, Yu D J 2017 J. Appl. Phys. 121 135105
[32] Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901
[33] Ma T, Shvets G 2016 New J. Phys. 18 025012
[34] Lu J Y, Qiu C Y, Ye L P, Fan X Y, Ke M Z, Zhang F,Liu Z Y 2017 Nat. Phys. 13 369
[35] Mei J, Wu Y, Chan C T, Zhang Z Q 2012 Phys. Rev. B 86 035141
[36] Wu Y 2014 Opt. Express 22 1906
[37] Lu J Y, Qiu C Y, Xu S J, Ye Y T, Ke M Z, Liu Z Y 2014 Phys. Rev. B 89 134302
[38] Shen S Q, Shan W Y, Lu H Z 2011 Spin 1 33
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