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arXiv math-ph/02090282002-09math-phcond-mathep-thmath.MPmath.NT

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

Wellington da Cruz

备注：: Typos corrected, 8 pages, latex. To appear in Chaos, Solitons and Fractals, 2003
发表信息：: Chaos Solitons Fractals 17 (2003) 975-979
DOI：: 10.1016/S0960-0779(02)00580-5

英文摘要：

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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