Hausdorff dimension

Sierpinski carpet

Fractal

Physics

Mathematics

Fractal dimension

Mathematical analysis

Geometry

Sierpinski triangle

Combinatorics

Topology (electrical circuits)

Science Bulletin

Topological phases of matter have been extensively investigated in solid-state materials and classical wave systems with integer dimensions. However, topological states in non-integer dimensions remain almost unexplored. Fractals, being self-similar on different scales, are one of the intriguing complex geometries with non-integer dimensions. Here, we demonstrate fractal higher-order topological states with unprecedented emergent phenomena in a Sierpiński acoustic metamaterial. We uncover abundant topological edge and corner states in the acoustic metamaterial due to the fractal geometry. Interestingly, the numbers of the edge and corner states depend exponentially on the system size, and the leading exponent is the Hausdorff fractal dimension of the Sierpiński carpet. Furthermore, the results reveal the unconventional spectrum and rich wave patterns of the corner states with consistent simulations and experiments. This study thus unveils unconventional topological states in fractal geometry and may inspire future studies of topological phenomena in non-Euclidean geometries.

Observation of fractal higher-order topological states in acoustic metamaterials