Fractality-induced topological phase squeezing and devil's staircase

We propose and experimentally demonstrate the first example of Chern fractal insulators in an acoustic Sierpinski lattice. Through introducing fractality into the well-known Haldane model, we find that the topological phase diagram is significantly squeezed by about 0.5 times, compared with that of the original Haldane model. The energy spectra for different generations of a fractal lattice exhibits a hierarchy of self-similar energy bands in the form of a devil's staircase. With local acoustic measurements, we fabricate the sonic sample based on the theoretical model and experimentally confirm the fractality-induced squeezing and observe the one-way edge states that are protected by a robust mobility gap. Our work demonstrates the fundamental interplay between the fractality and topological Haldane insulator, and may provide new directions for the advanced control of sound waves.