A quantum-geometrical description of fracton statistics

TL;DR

This paper introduces a quantum-geometrical framework for describing fracton statistics, linking fractal paths, fractional quantum Hall effect, and number theory through a novel distribution function and entropy.

Contribution

It develops a fractal distribution function and entropy for fractons, connecting quantum geometry with fractional statistics in two-dimensional systems.

Findings

Derived a fractal distribution function for fractons

Established a connection between fracton statistics and the fractional quantum Hall effect

Linked fractal geometry with quantum statistical mechanics

Abstract

We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$, a fractal distribution function associated with a fractal von Neumann entropy. Fractons are charge-flux systems defined in two-dimensional multiply connected space and they carry rational or irrational values of spin. This formulation can be considered in the context of the fractional quantum Hall effect-FQHE and number theory.