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29 March 2024
 
  » arxiv » cond-mat/9610177

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Hofstadter Rules and Generalized Dimensions of the Spectrum of Harper's Equation
Andreas Rudinger ; Frederic Piechon ;
Date 24 Oct 1996
Subject Mesoscopic Systems and Quantum Hall Effect | cond-mat.mes-hall
AbstractWe consider the Harper model which describes two dimensional Bloch electrons in a magnetic field. For irrational flux through the unit-cell the corresponding energy spectrum is known to be a Cantor set with multifractal properties. In order to relate the maximal and minimal fractal dimension of the spectrum of Harper’s equation to the irrational number involved, we combine a refined version of the Hofstadter rules with results from semiclassical analysis and tunneling in phase space. For quadratic irrationals $omega$ with continued fraction expansion $omega = [0;overline{n}]$ the maximal fractal dimension exhibits oscillatory behavior as a function of $n$, which can be explained by the structure of the renormalization flow. The asymptotic behavior of the minimal fractal dimension is given by $amin sim { m const.} ln n / n$. As the generalized dimensions can be related to the anomalous diffusion exponents of an initially localized wavepacket, our results imply that the time evolution of high order moments $< r^{q} >, q o infty$ is sensible to the parity of $n$.
Source arXiv, cond-mat/9610177
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