We present Floquet fractal topological insulators: photonic topological insulators in a fractal-dimensional lattice consisting of helical waveguides. The helical modulation induces an artificial gauge field and leads to a trivial-to-topological phase transition. The quasi-energy spectrum shows the existence of topological edge states corresponding to real-space Chern number 1. We study the propagation of light along the outer edges of the fractal lattice and find that wavepackets move along the edges without penetrating into the bulk or backscattering even in the presence of disorder. In a similar vein, we find that the inner edges of the fractal lattice also exhibit robust transport when the fractal is of sufficiently high generation. Finally, we find topological edge states that span the circumference of a hybrid half-fractal, half-honeycomb lattice, passing from the edge of the honeycomb lattice to the edge of the fractal structure virtually without scattering, despite the transition from two dimensions to a fractal dimension. Our system offers a realizable experimental platform to study topological fractals and provides new directions for exploring topological physics.

我们提出了 Floquet 分形拓扑绝缘体：由螺旋波导组成的分形维晶格中的光子拓扑绝缘体。螺旋调制引起人工规范场并导致平凡到拓扑的相变。准能谱显示存在与实空间陈数1相对应的拓扑边缘态。我们研究了光沿分形晶格外边缘的传播，发现波包沿边缘移动而没有穿透到体中或发生后向散射即使存在混乱。同样，我们发现，当分形的生成足够高时，分形晶格的内边缘也表现出稳健的传输。最后，我们发现拓扑边缘态跨越混合半分形、半蜂窝晶格的圆周，从蜂窝晶格的边缘传递到分形结构的边缘，几乎没有散射，尽管从二维过渡到二维分形维数。我们的系统为研究拓扑分形提供了一个可实现的实验平台，为探索拓扑物理提供了新的方向。