Spin Chern insulator in a phononic fractal lattice

The spin Chern topological phases are more natural in solid-state systems and are thought to exist in two or three dimensions. To date, there is no evidence for the existence of spin Chern topological phase in non-integer dimension. Fractal offers a platform for exploring novel topological phases and phenomena in noninteger dimension. Here, based on a phononic fractal lattice, we experimentally demonstrate the presence of the spin Chern phase in noninteger dimension. We find that the spin Chern phase is compressed in the fractal lattice compared to the crystal lattice. We also highlight the robustness and unidirectionality of spin-polarized topologically protected edge states even the momentum space is ill defined. Interestingly, sound travels faster at the boundaries of the fractal lattice than in crystal lattice. Abundant spin-polarized edge states and increased velocities not only may inspire further study in other noninteger dimensional systems, but also provide an opportunity for the design of multichannel on-chip communication devices.