Hofstadter rules and generalized dimensions of the spectrum of Harper's equation

Abstract

We 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 $\amin \sim {\rm const.} \ln n / n$. 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 $< r^{q} >, q \to \infty$ is sensible to the parity of $n$.