Fractal statistics, fractal index and fractons

TL;DR

This paper introduces the concept of fractal index and explores the properties of fractons, particles obeying fractal statistics, linking them to conformal field theory, Hausdorff dimension, and number theory.

Contribution

It establishes a novel connection between fractal statistics, conformal field theory, and number theory through the fractal index and Hausdorff dimension.

Findings

Relation between fractons and CFT-quasiparticles

Connection between Hausdorff dimension and conformal anomaly

Link between Rogers dilogarithm, Farey series, and fractal index

Abstract

The concept of fractal index is introduced in connection with the idea of universal class $h$ of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field theory(CFT)-quasiparticles taking into account the central charge $c[\nu]$ and the particle-hole duality $\nu\longleftrightarrow\frac{1}{\nu}$, for integer-value $\nu$ of the statistical parameter. The Hausdorff dimension $h$ which labelled the universal classes of particles and the conformal anomaly are therefore related. We also establish a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.