A fractal-like structure for the fractional quantum Hall effect

TL;DR

This paper proposes a fractal geometric framework for understanding the fractional quantum Hall effect, linking Hausdorff dimensions, particle spin, and fractal entropy to explain experimental data without empirical formulas.

Contribution

It introduces a novel fractal-based model connecting quantum paths, spin, and entropy to explain FQHE phenomena, providing a foundational perspective beyond empirical approaches.

Findings

The model accounts for all known FQHE filling factors.

It relates Hausdorff dimension to particle spin via a fractal formula.

Provides a new geometric perspective on charge-flux systems.

Abstract

We have pursued in the literature a fractal-like structure for the fractional quantum Halll effect-FQHE which consider the Hausdorff dimension associated with the quantum mechanics paths and the spin of the particles or quasiparticles termed fractons. These objects carry rational or irrational values of spin and satisfy a fractal distribution function associated with a fractal von Neumann entropy. We show that our approach offers {\it a rationale} for all FQHE data including possible filling factors suggested by some authors. Our formulation is free of any empirical formula and this characteristic appears as a foundational insight for this FQHE-phenomenon. The connection between a geometrical parameter, the Hausdorff dimension $h$, associated with the quantum paths and the spin $s$ of particles, $h=2-2s$, $0 < s < {1/2}$, is a physical analogous to the fractal dimension formula, $\Delta(\Gamma)=2-H$, of the graph of the functions in the context of the fractal geometry, where $H$ is knwon as H\"older exponent, with $ 0 < H < 1$. We emphasize that our fractal approach to the fractional spin particles gives us a new perspective for charge-flux systems defined in two-dimensional multiply connected space.