Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1999-05-31
Int.J.Mod.Phys.A15:3805-3828,2000
Physics
High Energy Physics
High Energy Physics - Theory
25pages, latex, additional references. To appear in International Journal of Modern Physics A (2000)
Scientific paper
10.1016/S0217-751X(00)00231-5
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 $h$.
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