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四极子拓扑绝缘体是人们提出的第一类高阶拓扑绝缘体, 它具有量子化的四极矩而偶极矩为零. 四极子拓扑绝缘体拓宽了传统的体-边对应关系, 从而观察到了更低维度的拓扑边界态. 最近, 由局域在位错附近的拓扑缺陷态主导的体-位错对应关系引起了许多研究者的关注, 其将晶格倒易空间的拓扑结构与位错态的出现联系起来. 本文研究了声学四极子拓扑绝缘体中的位错态. 在具有非平庸相的声学四极子拓扑绝缘体中嵌入部分具有平庸相的晶格, 此时在由两种具有不同拓扑相晶格形成边界的角落处就会产生可以用1/2量化分数电荷表征的位错态. 通过在系统内部引入缺陷, 验证了此拓扑位错态的鲁棒性. 此外, 还证明了通过运用不同嵌入晶格的方式可以随意设计位错态的位置. 本工作中研究的拓扑位错态拓宽了人工结构中高阶拓扑物态的种类, 并为高阶拓扑绝缘体在声学中的应用(如声传感和高性能能量收集)提供了新的思路.Quadrupole topological insulator (QTI) is the first proposed higher-order topological phase of matter with quantized quadrupole moment but zero dipole moment. The QTI has expanded widely the traditional bulk-boundary correspondence, thereby the lower-dimensional topological boundary state can be observed. The recent interest has turned to the bulk-dislocation correspondence, which dominates the topological states localized to disclinations, and links the reciprocal-space topology of lattices with the appearance of dislocation states. Recently, many research groups have turned the studies of dislocation defects to classical wave systems. In these researches, the method of inducing dislocation defects is to remove a portion of the lattices of topological insulator and then rearrange the remaining lattices of the topological insulator. Through such a method, the micro structure of the lattices is changed, but it is difficult to realize in the actual operation. In this work, we study the dislocation defect states in acoustic QTIs. The acoustic QTI is designed by reversing the magnitude of the intracellular and extracellular coupling in the system, and the bulk energy bands and topological corner states are studied. Subsequently, by introducing partial trivial lattices into acoustic QTI structure, the dislocation bound states are generated in the corner formed by two different topological phases, which can be characterized by a 1/2 quantized fractional charge. The robustness of the topological dislocation states is verified by introducing the imperfection inside the system. Further, it is demonstrated that the dislocation positions can be designed at will. Without changing the microstructure of the lattice, we successfully modulate the line dislocation states and bulk dislocation states. The topological dislocation states studied in this work broaden the types of higher-order topological states in artificial structures, and provide new insights into the acoustic applications of higher-order topological insulators, such as sensing and high-performance energy harvesting.
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图 1 四极子拓扑绝缘体的紧束缚模型和对应的声学模型 (a)二维紧束缚模型结构示意图, 每个黄色的圆圈代表一个无自旋的电子轨道, 蓝色方框内表示一个原胞, 蓝线表示正耦合, 红线表示负耦合, 细线表示胞内耦合, 粗线表示胞间耦合; (b)声学四极子拓扑绝缘体的原胞结构, 黄色矩形柱为声学谐振腔, 代表无自旋的电子轨道, 两根耦合管的直接连接(红色标记)表示负耦合, 交叉连接(蓝色标记)表示正耦合; (c)声子晶体的能带结构(黑色曲线), 红色散点是根据紧束缚模型计算的, 可以看出, 由声子晶体与由紧束缚模型计算得到的Pz模平坦带吻合得很好; (d)由$ 4 \times 4 $个有限单元计算得到的声学四极子拓扑绝缘体的本征频谱, 在Pz模打开的带隙范围内观察到角态(红点)是四极子拓扑绝缘体为高阶拓扑绝缘体的特征之一
Fig. 1. Tight-binding (TB) model and the corresponding acoustic model for QTI. (a) Schematic of the two-dimensional TB model. Each yellow circle represents a spinless electronic orbital, and the light shaded square sketches a four-site unit cell. The blue lines indicate the positive hoppings and red lines indicate negative hoppings. The thin lines represent intracellular hoppings and the thick lines represent the intercellular hoppings. (b) The unit cell structure of the corresponding acoustic model. The yellow rectangular pillars are the resonant cavities, representing spinless electronic orbitals. The straight link of two tubes (marked with red) indicates the negative coupling, and the crossing link (marked with blue) indicates the positive coupling. (c) The band structures for the acoustic crystal (black curves). The red scatter are calculated from TB model. It can be seen that the coupled bands from Pz modes in acoustic crystal agree well with these from TB model. (d) The band structures for a finite-size acoustic structure with $ 4 \times 4 $unit cells. The corner states (marked red) are observed within the gap of Pz bands, which is one of the characteristics of higher-order QTI.
图 2 线位错态 (a)调制线位错态的模型结构, 橙色矩形表示嵌入四极子拓扑绝缘体的平庸晶格, 黑色细线表示将原始的强耦合反转为弱耦合, 红色圆圈表示位错点; (b)线位错态的本征频谱, 可以观察到两个线位错态(实空间中的拓扑态)位于Pz模平坦带打开的带隙范围内(红点); (c), (d)本征频率分别为8298.2 Hz和8298.4 Hz的两个线位错态的本征场分布
Fig. 2. Line-dislocation states. (a) The model structure to introduce the line-dislocation states. The orange rectangle represents the trival lattice, embedded into the QTI. Black thin lines indicate the weak couplings which are reversed from original strong couplings. The red circles are marked as the dislocation sites. (b) The band structures for the line-dislocation model. It can be observed that two line-dislocation states (topological states in real space) are located within the gap of Pz bands (marked as red). (c), (d) The eigen fields distribution for the two line-dislocation states with frequency of 8298.2 Hz and 8298.4 Hz, respectively.
图 3 体位错态 (a)调制体位错态的模型结构, 橙色矩形表示嵌入四极子拓扑绝缘体中的平庸晶格, 黑色实线表示弱耦合, 红色圆圈表示位错点; (b)调制体位错态的声子晶体拓扑绝缘体计算得到的本征频谱, 两个体位错态位于Pz模打开的带隙范围内(红点标记); (c), (d)频率分别为8145.4 Hz和8146.1 Hz的两个体位错态的本征场分布
Fig. 3. The bulk-dislocation states. (a) The model structure to introduce the bulk-dislocation states. The orange rectangle represents the trival lattice, embedded into the QTI. Black thin lines indicate the weak couplings. The red circles are marked as the dislocation sites. (b) The band structures for the bulk-dislocation model. Two bulk-dislocation states are observed within the gap of Pz bands (marked as red). (c), (d) The eigen fields distribution for the two bulk-dislocation states with frequency of 8145.4 Hz and 8146.1 Hz, resepectively.
图 4 线位错态和体位错态的鲁棒性 (a)验证线位错态鲁棒性的声学模型, 黑色圆圈表示长和宽都为7.5 mm (原始腔的1.5倍)的更大谐振腔, 表明在声学四极子拓扑绝缘体内部引入了缺陷; (b)对应于(a)中模型的本征频谱, 插图展示了线位错态的本征场分布情况; (c)验证体位错态鲁棒性的声学模型, 在声学四极子拓扑绝缘体内引入长和宽都为7.5 mm (原始腔的1.5倍)的更大谐振腔(黑色圆圈标记); (d)对应于(c)的本征频谱, 插图为体位错态的本征场分布, 可以看出, 两种情况下的特征频率和本征场分布几乎保持不变
Fig. 4. Robustness of line-dislocation and bulk-dislocation states. (a) The acoustic model to verify the robustness of line-dislocation states. The bold circle denotes a bigger resonant cavity with length of 7.5 mm (1.5 times of that of original cavity), indicating the introduction of imperfection inside acoustic QTI. (b) The corresponding the band structure for the model shown in (a), insets display the eigen fields for the dislocation states. (c) The acoustic model to verify the robustness of bulk-dislocation states. A bigger resonant cavity with length of 7.5 mm (1.5 times of that of original cavity) are introduced inside acoustic QTI (marked as bold circle). (d) The corresponding the band structure for the model shown in (c), insets display the eigen fields for the dislocation states. It can be seen that both the eigen frequency and eigen field keeps almost unchanged for the two cases.
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[1] Benalcazar W A, Bernevig B A, Hughes T L 2017 Phys. Rev. B 96 245115
Google Scholar
[2] Benalcazar W A, Bernevig B A, Hughes T L 2017 Science 357 61
Google Scholar
[3] Serra-Garcia M, Peri V, Susstrunk R, Bilal O R, Larsen T, Villanueva L G, Huber S D 2018 Nature 555 342
Google Scholar
[4] Huang X Q, Lu J Y, Yan Z B, Yan M, Deng W Y, Chen G, Liu Z Y 2022 Sci. Bull. 67 488
Google Scholar
[5] 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
Google Scholar
[6] Biesenthal T, Maczewsky L J, Yang Z, Kremer M, Segev M, Szameit A, Heinrich M 2022 Science 376 1114
Google Scholar
[7] Song Z D, Fang Z, Fang C 2017 Phys. Rev. Lett. 119 246402
Google Scholar
[8] Xue H R, Yang Y H, Zhang B L 2022 Nat. Rev. Mater. 7 974
Google Scholar
[9] Ezawa M 2018 Phys. Rev. Lett. 120 026801
Google Scholar
[10] Xie B, Wang H X, Zhang X, Zhan P, Jiang J H, Lu M, Chen Y 2021 Nat. Rev. Phys. 3 520
Google Scholar
[11] Pu Z H, He H L, Luo L C, Ma Q Y, Ye L P, Ke M Z, Liu Z Y 2023 Phys. Rev. Lett. 130 116103
Google Scholar
[12] Ma Q Y, Pu Z H, Ye L P, Lu J Y, Huang X Q, Ke M Z, He H L, Deng W Y, Liu Z Y 2024 Phys. Rev. Lett. 132 066601
Google Scholar
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Google Scholar
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Google Scholar
[15] 胡军容, 孔鹏, 毕仁贵, 邓科, 赵鹤平 2022 71 054301
Google Scholar
Hu J R, Kong P, Bi R G, Deng K, Zhao H P 2022 Acta Phys. Sin. 71 054301
Google Scholar
[16] Mittal S, Orre V V, Zhu G, Gorlach M A, Poddubny A, Hafezi M 2019 Nat. Photonics 13 692
Google Scholar
[17] Qi Y J, Qiu C Y, Xiao M, He H L, Ke M Z, Liu Z Y 2020 Phys. Rev. Lett. 124 206601
Google Scholar
[18] Li C A, Wu S S 2020 Phys. Rev. B 101 195309
Google Scholar
[19] Li C A, Fu B, Hu Z A, Li J, Shen S Q 2020 Phys. Rev. Lett. 125 166801
Google Scholar
[20] Qi Y J, He H L, Xiao M 2022 Appl. Phys. Lett. 120 212202
Google Scholar
[21] Ye L P, Qiu C Y, Xiao M, Li T Z, Du J, Ke M Z, Liu Z Y 2022 Nat. Commun. 13 508
Google Scholar
[22] Yamada S S, Li T, Lin M, Peterson C W, Hughes T L, Bahl G 2022 Nat. Commun. 13 2035
Google Scholar
[23] Ran Y, Zhang Y, Vishwanath A 2009 Nat. Phys. 5 298
Google Scholar
[24] Grinberg I H, Lin M, Benalcazar W A, Hughes T L, Bahl G 2020 Phys. Rev. Appl. 14 064042
Google Scholar
[25] Xue H R, Jia D, Ge Y, Guan Y J, Wang Q, Yuan S Q, Sun H X, Chong Y D, Zhang B L 2021 Phys. Rev. Lett. 127 214301
Google Scholar
[26] Teo J C, Hughes T L 2017 Annu. Rev. Condens. Matter Phys. 8 211
Google Scholar
[27] Peterson C W, Li T, Jiang W, Hughes T L, Bahl G 2021 Nature 589 376
Google Scholar
[28] Liu Y, Leung S W, Li F F, Lin Z K, Tao X F, Poo Y, Jiang J H 2021 Nature 589 381
Google Scholar
[29] Wei H, Tsui D, Paalanen M, Pruisken A 1988 Phys. Rev. Lett. 61 1294
Google Scholar
[30] Roberts E, Behrends J, Béri B 2020 Phys. Rev. B 101 155133
Google Scholar
[31] Liu F, Wakabayashi K 2017 Phys. Rev. Lett. 118 076803
Google Scholar
[32] Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F 2019 Phys. Rev. Lett. 122 233903
Google Scholar
[33] Yang Y B, Li K, Duan L M, Xu Y 2021 Phys. Rev. B 103 085408
Google Scholar
[34] Wang H X, Lin Z K, Jiang B, Guo G Y, Jiang J H 2020 Phys. Rev. Lett. 125 146401
Google Scholar
[35] He L, Addison Z, Mele E J, Zhen B 2020 Nat. Commun. 11 3119
Google Scholar
[36] Benalcazar W A, Li T, Hughes T L 2019 Phys. Rev. B 99 245151
Google Scholar
[37] Weiner M, Ni X, Li M, Alù A, Khanikaev A B 2020 Sci. Adv. 6 eaay4166
Google Scholar
[38] Wheeler W A, Wagner L K, Hughes T L 2019 Phys. Rev. B 100 245135
Google Scholar
[39] Zhang X, Xie B Y, Wang H F, Xu X, Tian Y, Jiang J H, Lu M H, Chen Y F 2019 Nat. Commun. 10 5331
Google Scholar
[40] Kang B, Shiozaki K, Cho G Y 2019 Phys. Rev. B 100 245134
Google Scholar
[41] Ni X, Li M, Weiner M, Alù A, Khanikaev A B 2020 Nat. Commun. 11 2108
Google Scholar
[42] Zheng S, Man X, Kong Z L, Lin Z K, Duan G, Chen N, Yu D, Jiang J H, Xia B 2022 Sci. Bull. 67 2069
Google Scholar
[43] Kim H R, Hwang M S, Smirnova D, Jeong K Y, Kivshar Y, Park H G 2020 Nat. Commun. 11 5758
Google Scholar
[44] Xie B Y, Wang H F, Wang H X, Zhu X Y, Jiang J H, Lu M H, Chen Y F 2018 Phys. Rev. B 98 205147
Google Scholar
[45] 李荫铭, 孔鹏, 毕仁贵, 何兆剑, 邓科 2022 71 244302
Google Scholar
Li Y M, Kong P, Bi R G, He Z J, Deng K 2022 Acta Phys. Sin. 71 244302
Google Scholar
[46] Chen Y Y, Liu D Y, Wu Y, Yu P, Liu Y J 2023 Int. J. Mech. Sci. 239 107884
Google Scholar
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