Hofstadter rules and generalized dimensions of the spectrum of Harper's equation
Hofstadter rules and generalized dimensions of the spectrum of Harper's equation
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霍夫施塔特规则和哈珀方程谱的广义维数
DOI：
10.1088/0305-4470/30/1/009
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
F. Piéchon
中科院分区：
文献类型：
--
作者：
A. Rudinger;F. Piéchon
我们考虑磁场中二维布洛赫电子的哈珀模型。对于通过单胞的无理通量，相应的能谱是一个具有多重分形性质的康托集。为了将哈珀方程的谱的最大和最小分形维数与所涉及的无理数相关联，我们将联合收割机的Hofstadter规则的改进版本与相空间中的半经典分析和隧道效应的结果相结合。对于连分式展开的二次无理数，最大分形维数表现出作为n的函数的振荡行为，这可以用重整化流的结构来解释。最小分形维数的渐近行为由下式给出。由于广义维数与初始局域波包的反常扩散指数有关，我们的结果表明高阶矩的时间演化对n的宇称是敏感的.
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 tunnelling in phase space. For quadratic irrationals with continued fraction expansion the maximal fractal dimension exhibits oscillatory behaviour as a function of n, which can be explained by the structure of the renormalization flow. The asymptotic behaviour of the minimal fractal dimension is given by . 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 is sensible to the parity of n.