# Anyons and Fractional Quantum Hall Effect in Fractal Dimensions

## TL;DR

This paper introduces a new class of fractional quantum Hall systems in fractal dimensions, providing analytical wave functions and demonstrating the presence of anyons across various fractal geometries.

## Contribution

It constructs fractional quantum Hall models in fractal dimensions, offering analytical solutions and numerical evidence of topological order in these unconventional settings.

## Key findings

- Anyons are present in fractal quantum Hall systems.
- Topological phases depend more on local structure than Hausdorff dimension.
- Fractional quantum Hall physics can exist in fractals with non-integer dimensions.

## Abstract

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.