Fractal sets of dual topological quantum numbers

TL;DR

This paper proposes a fractal geometric framework for understanding the fractional quantum Hall effect, linking topological quantum numbers, fractal dimensions, and number theory, potentially offering insights into the Riemann hypothesis.

Contribution

It introduces a novel fractal-based approach to the FQHE that avoids empirical formulas and connects quantum physics with fractal geometry and number theory.

Findings

FQHE exhibits a fractal structure linked to Hausdorff dimensions.

Quantum Hall transitions relate to Farey sequences of rational numbers.

The approach suggests a possible connection to the Riemann hypothesis.

Abstract

The universality classes of the quantum Hall transitions are considered in terms of fractal sets of dual topological quantum numbers filling factors, labelled by a fractal or Hausdorff dimension defined into the interval $1 < h < 2$ and associated with fractal curves. We show that our approach to the fractional quantum Hall effect-FQHE is free of any empirical formula and this characteristic appears as a crucial insight for our understanding of the FQHE. According to our formulation, the FQHE gets a fractal structure from the connection between the filling factors and the Hausdoff dimension of the quantum paths of particles termed fractons which obey a fractal distribution function associated with a fractal von Neumann entropy. This way, the quantum Hall transitions satisfy some properties related to the Farey sequences of rational numbers and so our theoretical description of the FQHE establishes a connection between physics, fractal geometry and number theory. The FQHE as a convenient physical system for a possible prove of the Riemann hypothesis is suggested.