Fibonacci链上波函数和局域谱测度的分形维数
Fractal dimensions of the wavefunctions and local spectral measures on the Fibonacci chain
摘要：
我们提出了一个理论框架，了解波函数和频谱的广泛研究范式准周期系统，即斐波那契链。我们的分析结果，这是在跳跃振幅的强调制的限制下得到的，与发表的数值数据吻合得很好。在微扰极限下，我们证明了一种新的对称性的波函数置换的网站和能量指数。我们计算的波函数重整化因子，并从他们推导出相应的个别波函数的分形指数的解析表达式，以及它们的全球平均值。波函数的多重分形出现在跳跃振幅比$ρ$的次领先阶。给出了局部谱密度的指数，与数值计算结果非常吻合。有趣的是，我们的分析结果的指数被观察到，以描述系统相当好，即使是值的$ρ$以及微扰理论的适用范围之外。
英文摘要：
We present a theoretical framework for understanding the wavefunctions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in good agreement with published numerical data. In the perturbative limit, we show a new symmetry of wavefunctions under permutation of site and energy indices. We compute the wavefunction renormalization factors and from them deduce analytical expressions for the fractal exponents corresponding to individual wavefunctions, as well as their global averages. The multifractality of wavefunctions is seen to appear at next-to-leading order in the ratio of the hopping amplitudes, $ρ$. Exponents for the local spectral density are given, in extremely good accord with numerical calculations. Interestingly, our analytical results for exponents are observed to describe the system rather well even for values of $ρ$ well outside the domain of applicability of perturbation theory.
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