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.

Figure 1

Fractal spin Chern insulator and spin-polarized edge states. (a) Top view of G(2) fractal lattice. The green, blue, and red cavities denote A, B, and C sites. (b) The structure details of the fractal lattice. (c) Compared with the spin Chern number $C_s$ in Lieb lattice, the topological phase in the fractal lattice, which is characterized by the spin Bott index $B_s$ is compressed. (d) (Left) Numerically calculated energy spectrum of the fractal lattice. The mid-gap presents a hierarchy of edge states including external, middle, and internal edge states. (Inset) Spatial distribution of the bulk (grey), external edge (red), middle edge (blue), and internal edge (green). (Right) Calculated localized density of states. (e) Calculated spin spectrum of the edge states in the gap.

Figure 2

Robustness of topological edge states against defect. (a) Photograph of Sierpinski carpet sample with defect at the outer perimeter. (b) Measured transmission spectra for the sample with and without a defect. (c)–(e) Measured field intensity profiles for the (c) middle, (d) external, and (e) internal edge state at $5868\mathrm{Hz}, 6150\mathrm{Hz}$, and $6050\mathrm{Hz}$, respectively. (f) Measured field intensity profiles for the external edge state with a defect at $6150\mathrm{Hz}$.

Figure 3

Experimental observation of spin-dependent edge transport with Gaussian pulse excitation. (a) The sample photo of phononic fractal lattice. Cyan stars and white arrows represent the acoustic sources and the propagating directions for different pseudospin states, respectively. (b) Measured wave packets at points A, B, C, and D in (a). (c) Comparison of edge state velocities in the fractal lattice with that in crystalline counterpart. The fitted velocities of spin-down/up-polarized edge states in the fractal and crystal lattice are $143.2\mathrm{m}/\mathrm{s}, 146.9\mathrm{m}/\mathrm{s}, 127.7\mathrm{m}/\mathrm{s}$, and $131.6\mathrm{m}/\mathrm{s}$, respectively. (d)–(i) Envelops of propagating pulses in the space-time domain which is measured along the counterclockwise direction, where (d)–(f) and (g)–(i) show the edge states excited by spin down and up source, respectively. The external, middle, and internal edge states are excited by the Gaussian pulses at center frequencies of $6150\mathrm{Hz}, 5868\mathrm{Hz}$, and $6050\mathrm{Hz}$, respectively.