Quantum phase transitions of fractons (a fractal scaling theory for the FQHE)

We consider the quantum phase transitions of fractons in correspondence with the quantum phase transitions of the fractional quantum Hall effect-FQHE. We have that the Hall states can be modelled by fractons, known as charge-flux systems which satisfy a fractal distribution function associated with a fractal von Neumann entropy. In our formulation, the universality classes of the fractional quantum Hall transitions, are considered as fractal sets of dual topological quantum numbers filling factors labelled by the Hausdorff dimension $h$ ($1 < h < 2$) of the quantum paths of fractons. In this way we have defined, associated to these universality classes, a scaling exponent as $\kappa=1/h$, such that when $h$ runs into its interval of definition, we obtain $ 1 \gtrsim \kappa \gtrsim 0.5 $. The behavior of this scaling exponent, topological in character, is in agreement with some experimental values claimed in the literature. Thus, according to our approach we have a fractal scaling theory for the FQHE which distinguishes diverse universality classes for the fractional quantum Hall transitions.