Cantor spectra and scaling of gap widths in deterministic aperiodic systems.
Cantor spectra and scaling of gap widths in deterministic aperiodic systems.
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DOI：
10.1103/physrevb.39.5834
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发表时间：
1989-03
期刊：
影响因子：
--
通讯作者：
J. Luck
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文献类型：
--
作者：
J. Luck
通过考虑一维紧结合Schrödinger方程的例子，研究了非周期结构的几何和物理性质之间的关系，其中位置势由任意确定性非周期序列给出。在态的积分密度的微扰分析中，能谱中的间隙可以用势序列的傅里叶变换的奇异性来“标记”。这种方法证实了准周期和近周期系统的已知性质，并建议将它们扩展到更一般的序列，例如具有奇异连续傅立叶变换的序列。有强有力的证据表明，对于比准周期模型大得多的一类模型，谱是一个零测度的康托集。还确定了各种间隙宽度对势强度的依赖性：获得了几种不同的行为，例如具有非平凡指数的幂律或本质奇点。这些一般结果与其他各种方法的那些自相似序列的替换，即Thue-Morse序列，周期加倍序列，“圆序列”，和Rudin-Shapiro序列。
The relationship between geometry and physical properties of aperiodic structures is investigated by considering the example of the tight-binding Schrödinger equation in one dimension, where the site potentials are given by an arbitrary deterministic aperiodic sequence. In a perturbative analysis of the integrated density of states, the gaps in the energy spectrum can be ‘‘labeled’’by the singularities of the Fourier transform of the sequence of potentials. This approach confirms known properties of quasiperiodic and almost-periodic systems, and suggests an extension of them to more general sequences, such as those with a singular continuous Fourier transform. There is strong evidence that the spectrum is a Cantor set with zero measure for a much larger class of models than quasiperiodic ones. The dependence of the widths of various gaps on the potential strength is also determined: several different kinds of behavior are obtained, such as a power law with a nontrivial exponent, or an essential singularity. These general results are compared with those of various other approaches for four self-similar sequences generated by substitution, namely the Thue-Morse sequence, the period-doubling sequence, a ‘‘circle sequence,’’and the Rudin-Shapiro sequence.