The Hausdorff dimension of fractal sets and fractional quantum Hall effect

TL;DR

This paper links the Hausdorff dimension of fractal sets to the fractional quantum Hall effect, classifying quantum Hall transitions through number theory and fractal geometry.

Contribution

It introduces a novel classification of quantum Hall transitions using Hausdorff dimensions of fractal sets and Farey series properties.

Findings

Quantum Hall transitions classified by fractal set dimensions.

Farey series properties relate to quantum Hall filling factors.

Number theory provides a framework for understanding quantum Hall universality classes.

Abstract

We consider Farey series of rational numbers in terms of {\it fractal sets} labeled by the Hausdorff dimension with values defined in the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$ and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of {\it dual topological quantum numbers}, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that {\it the universality classes of the quantum Hall transitions are classified in terms of $h$}. The connection between Number Theory and Physics appears naturally in this context.