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arXiv:cond-mat/0303456

A quantum-geometrical description of the statistical laws of nature

Wellington da Cruz

Published Mar 21, 2003• 0 citations
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Abstract

We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$ $$ < $$ $$h$$ $$

DOI
10.48550/arxiv.cond-mat/0303456

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We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$ <<< hhh $$
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 hhh (1<h<21 < h < 21<h<2) of the quantum paths of fractons. In this way we have defined, associated to these universality classes, a scaling exponent as κ=1/hκ=1/hκ=1/h, such that when hhh runs into its interval of definition, we obtain $ 1 \gtrsim κ\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.
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