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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,它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 insulators (QTIs) are the first proposed higher-order topological phase of matter with quantized quadrupole moment but zero dipole moment. The QTIs have expanded widely the traditional bulk-boundary correspondence, giving rise to the observation of lower-dimensional topological boundary states. The recent interest turns to bulk-dislocation correspondence, which dominates the topological states localized to disclinations, and links the reciprocal-space topology of a lattice to the emergence of dislocation states. Recently, many research groups have turned the studies of dislocation defects to classical waves systems. The methods to induce the dislocation defects in these researches are to remove part of the lattices of topological insulator and then rearrange the remaining of the topological insulator. However, through such methods, the micro structures of the lattices were changed, it is difficult to realize in the actual operation. In this paper, we study the dislocation defect states in acoustic QTIs. The acoustic QTIs 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 structures, the dislocation bound states are generated at the corner formed by two different topological phases, which can be characterized by a ½ 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 have successfully achieved the modulation of line dislocation states and bulk dislocation states. The topological dislocation states studied in this work broaden the classification of higher-order topological phases in artificial structures, and provide new insights into the acoustic applications of higher-order topology, such as sensing and high-performance energy harvesting. Figure. (a) The tight-binding (TB) model and the corresponding acoustic model for QTI. (b) The corresponding band structures for the acoustic QTI. (c) The model structure to create the bulk-dislocation states. (d) The corresponding band structures for the bulk-dislocation model. In the following, we briefly present the key importance in this work. The tight-binding (TB) model is employed to characterize the quadrupole topological insulator as shown in Fig. (a), the single cell is marked by the light shaded square sketches in the middle of Fig (a). The corresponding band structure for the acoustic QTI model is displayed in Fig. (b). It can be observed that the coupled Pz modes in resonant cavities are splitted into two pairs of flat bands in QTI, which are separated by a gap with frequency ranging from 7558.7 Hz to 8359.8 Hz. Each pair of flat bands are almost double degenerated. In Fig. (c), we schematically exhibit the bulk-dislocation defect of TB model by inserting one row of trivial lattices (marked with orange rectangle) in the QTI. Along the defect, we choose the site at will to form the bulk dislocation (marked with red circles). The black thin lines indicate the weak couplings. The corresponding band structure for the bulk-dislocation defect is displayed in Fig. (d). It can be clearly seen that two bulk-dislocation states (red dots) are created within the gap of coupled Pz modes.
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