The fractional quantum Hall effect is a paradigm of topological order and has been studied thoroughly in two dimensions. Here, we construct a different type of fractional quantum Hall system, which has the special property that it lives in fractal dimensions. We provide analytical wave functions and exact few-body parent Hamiltonians, and we show numerically for several different Hausdorff dimensions between 1 and 2 that the systems host anyons. We also find examples of fractional quantum Hall physics in fractals with Hausdorff dimension 1 and $ln\left(4\right)/ln\left(5\right)$. Our results suggest that the local structure of the investigated fractals is more important than the Hausdorff dimension to determine whether the systems are in the desired topological phase. The study paves the way for further investigations of strongly correlated topological systems in fractal dimensions.

中文翻译：

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分形维数中的Anyon和分数量子霍尔效应

分数量子霍尔效应是拓扑顺序的范例，已在二维上进行了深入研究。在这里，我们构造了另一种类型的分数量子霍尔系统，该系统具有分形维数的特殊性质。我们提供了解析波函数和确切的小体母哈密顿量，并通过数字显示了系统承载任意正则的1到2之间的几个不同Hausdorff维数。我们还发现了Hausdorff维数为1和$ln（4）/ln（5）$。我们的结果表明，要确定系统是否处于所需的拓扑阶段，所研究的分形的局部结构比Hausdorff维数更为重要。该研究为进一步研究分形维数高度相关的拓扑系统铺平了道路。