Topological insulators are electronic systems with an insulating bulk and topologically protected boundary states. Conventional 2D topological insulators induce 1D edge states. Recent studies indicate that lower-dimensional topological states are also possible in electronic systems, which, however, has been confirmed only in Bismuth in experiments . In this talk, I will show that lower-dimensional topological wave trapping can be achieved in photonic and acoustic systems through several different mechanisms. First, with concurrent real-space and wavevector-space topology, we show topological 0D light-trapping on a dislocation . Second, with two different schemes of higher-order topological insulators, we demonstrate topological 0D corner states in photonic crystals and sonic crystals [3,4]. Finally, we show that a novel topological state, named as hierarchical topological multipole insulators, which exhibits both topological dipole and quadrupole orders in a single system, can be realized in macroscopic sonic crystals . The underlying physics indicates the possibility of achieving frequency-stable topological cavity states in two-dimensional photonic/phononic chips that may enable scalable quantum photonic interface and other unprecedented functions in integrated photonics/acoustics.
 F. Schindler et al. Higher-order topology in bismuth. Nature Physics 14 , 918-924 (2018).
 Fei-Fei Li, Hai-Xiao Wang, Zhan Xiong, Qun Lou, Ping Chen, Rui-Xin Wu, Yin Poo, Jian-Hua Jiang, and Sajeev John. Topological light-trapping on a dislocation. Nature Communications 9 , 2462 (2018)
 Bi Ye Xie, Hong Fei Wang, Hai-Xiao Wang, Xue Yi Zhu, Jian-Hua Jiang, Ming Hui Lu, and Yan Feng Chen. Second-order photonic topological insulator with corner states. arXiv:1805.07555. Physical Review Letters under review.
 Xiujuan Zhang, Hai-Xiao Wang, Zhi-Kang Lin, Yuan Tian, Biye Xie, Ming-Hui Lu, Yan-Feng Chen, and Jian-Hua Jiang. Observation of second-order topological insulators in sonic crystals. arXiv:1806.10028. Nature Physics under review.
 Zhi-Kang Lin, Xiujuan Zhang, Hai-Xiao Wang, Yuan Tian, Ming-Hui Lu, Yan-Feng Chen, and Jian-Hua Jiang. Discovery of hierarchical topological multipole insulators. Submitted to Science .