  The fractional quantum Hall effect is a paradigm of topological order and has
been studied thoroughly in two dimensions. Here, we construct a new 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(4)/ln(5). 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.
