Fractal index, central charge and fractons

TL;DR

This paper introduces a fractal index for particles called fractons, linking their statistics to conformal field theory, central charge, and Hausdorff dimension, revealing new relationships between these concepts.

Contribution

It establishes a novel connection between fractal statistics of fractons, conformal field theory, and mathematical functions like Rogers dilogarithm, introducing a new framework for understanding quasiparticles.

Findings

Derived Fermi velocity in terms of central charge

Linked Hausdorff dimension to conformal anomaly

Connected Rogers dilogarithm with Farey series and Hausdorff dimension

Abstract

We introduce the notion of fractal index associated with the universal class $h$ of particles or quasiparticles, termed fractons, which obey specific fractal statistics. A connection between fractons and conformal field theory(CFT)-quasiparticles is established 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. In this way, we derive the Fermi velocity in terms of the central charge as $v\sim\frac{c[\nu]}{\nu+1}$. The Hausdorff dimension $h$ which labelled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.