The Hausdorff dimension of fractal sets and fractional quantum Hall effect

Details The Hausdorff dimension of fractal sets and fractional quantum Hall effect The Hausdorff dimension of fractal sets and fractional quantum Hall effect

2002-09-15

arxiv.org/abs/math-ph/0209028v4

Chaos Solitons Fractals 17 (2003) 975-979

Physics

Mathematical Physics

Typos corrected, 8 pages, latex. To appear in Chaos, Solitons and Fractals, 2003

Scientific paper

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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