In two-dimensional lattices, the Lieb lattice is known for its unique wave properties that emerge from the interplay between flat bands and Dirac bands. Unlike the Kagome lattice, where the flat band is positioned differently, the flat band in the Lieb lattice is situated precisely in the middle of the Dirac band. Leveraging this distinctive feature, we have designed a two-dimensional Lieb lattice based on acoustic resonators. Our findings reveal that this lattice exhibits first-order edge states and zero-order corner states, with the intriguing observation of topologically protected flat bands. While topological wave regulation in classical wave systems of integer-dimensional lattices, such as the Lieb and Kagome lattices, has been extensively explored, the topological wave regulation in non-integer-dimensional fractal lattices remains largely uncharted territory. To address this gap, we have designed a Sierpinski carpet lattice with non-integer dimensions, also based on acoustic resonators. Our research demonstrates that this fractal lattice possesses topological states with fractal characteristics, including a rich array of edge states and inner-outer corner states. Moreover, we have discovered that the numbers of bulk states, edge states, and inner corner states in this lattice exhibit an exponential relationship with the self-similar iteration times of the acoustic system. Notably, the high-order corner states display novel spectral characteristics and diverse modal vibration patterns. These results provide compelling evidence for the existence of high-order topological states in low-dimensional lattices, potentially offering new insights and avenues for the development of low-order topological wave regulation.

Dr. Xianfeng Man received his Ph.D. degree in mechanical engineering from Hunan University in 2019. From 2017 to 2019, he served as a visiting scholar at the University of Technology Sydney. In 2020, he joined the School of Mechanical and Electrical Engineering of Changsha University. In 2023, he joined the Postdoctoral Research Station in Mechanics at Hunan University. His main research focuses on acoustic metamaterials for wave modulation. He is currently leading projects including the Excellent Young Scholars Project of the Education Department of Hunan Province, the Hunan Provincial Natural Science Foundation for Young Scholars and the General Program, and one Military-Civil Integration Project. He has published 16 academic papers in journals such as Science Bulletin, JASA, JSV, et al.