Fractons and Fractal Statistics

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 $β-$function is defined for a complex conductivity which embodies the classes $h$. The thermodynamics is also considered for a gas of fractons $(h,ν)$ 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 $h$.