Higher-order topological insulators (HOTIs) are unique materials hosting topologically protected states, whose dimensionality is at least by 2 lower than that of the bulk. Topological states in such insulators may be strongly confined in their corners which leads to considerable enhancement of nonlinear processes involving such states. However, all nonlinear HOTIs demonstrated so far were built on periodic bulk lattice materials. Here, we demonstrate the first nonlinear photonic HOTI with the fractal origin. Despite their fractional effective dimensionality, the HOTIs constructed here on two different types of the Sierpiński gasket waveguide arrays, may support topological corner states for unexpectedly wide range of coupling strengths, even in parameter regions where conventional HOTIs become trivial. We demonstrate thresholdless spatial solitons bifurcating from corner states in nonlinear fractal HOTIs and show that their localization can be efficiently controlled by the input beam power. We observe sharp differences in nonlinear light localization on outer and multiple inner corners and edges representative for these fractal materials. Our findings not only represent a new paradigm for nonlinear topological insulators, but also open new avenues for potential applications of fractal materials to control the light flow.

高阶拓扑绝缘体 (HOTI) 是具有拓扑保护态的独特材料，其维数至少比本体维数低 2。这种绝缘体中的拓扑状态可能被强烈限制在它们的角落，这导致涉及这些状态的非线性过程的显着增强。然而，迄今为止展示的所有非线性 HOTI 都是建立在周期性体晶格材料上的。在这里，我们展示了第一个具有分形起源的非线性光子HOTI。尽管有效维数为分数，但在两种不同类型的 Sierpiński 垫片波导阵列上构建的 HOTI 可以支持拓扑角态，从而实现意想不到的广泛耦合强度，即使在传统 HOTI 变得微不足道的参数区域也是如此。我们演示了非线性分形 HOTI 中从角态分叉的无阈值空间孤子，并表明它们的定位可以通过输入光束功率有效控制。我们观察到代表这些分形材料的外角和多个内角和边缘上非线性光定位的显着差异。我们的发现不仅代表了非线性拓扑绝缘体的新范例，而且还为分形材料控制光流的潜在应用开辟了新途径。