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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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