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The Hausdorff dimension of fractal sets and fractional quantum Hall effect

Wellington da Cruz
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Abstract

We consider Farey series of rational numbers in terms of {\it fractal sets} labeled by the Hausdorff dimension with values defined in the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$ and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of {\it dual topological quantum numbers}, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that {\it the universality classes of the quantum Hall transitions are classified in terms of $h$}. The connection between Number Theory and Physics appears naturally in this context.

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Wellington da Cruz. 2002-09-15. The Hausdorff dimension of fractal sets and fractional quantum Hall effect. https://doi.org/10.1016/s0960-0779(02)00580-5

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