Fractons and Fractal Statistics

TL;DR

This paper explores fractons, a class of anyons with fractal statistics, and their role in the Fractional Quantum Hall Effect, revealing new dualities, connections to modular groups, and thermodynamic properties.

Contribution

It introduces a novel framework linking fracton classes to quantum phase transitions, modular groups, and Farey sequences, advancing understanding of fractal statistics in quantum systems.

Findings

Filling factors in FQHE correspond to fracton classes

Duality between classes h and 3-h is established

Farey sequences relate to equivalence classes h

Abstract

Fractons are anyons classified into equivalence classes and they obey a specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension $h$. We consider this approach in the context of the Fractional Quantum Hall Effect (FQHE) and the concept of duality between such classes, defined by $\tilde{h}=3-h$ shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes $h$ and the modular group for the quantum phase transitions of the FQHE is also obtained. A $\beta-$function is defined for a complex conductivity which embodies the classes $h$. The thermodynamics is also considered for a gas of fractons $(h,\nu)$ with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes $h$.