The distinctive electronic properties of quasicrystals stem from their long-range structural order, with invariance under rotations and under discrete scale change, but without translational invariance. $d$-dimensional quasicrystals can be described in terms of lattices of higher dimension ($D>d$), and many of their properties can be simply derived from analyses that take into account the extra “hidden” dimensions. In particular, as recent theoretical and experimental studies have shown, quasicrystals can have topological properties inherited from the parent crystals. These properties are discussed here for the simplest of quasicrystals, the one-dimensional (1D) Fibonacci chain. The Fibonacci noninteracting tight-binding Hamiltonians are characterized by the multifractality of the spectrum and states, which is manifested in many of its physical properties, most notably in transport. Perturbations due to disorder and reentrance phenomena are described, along with the crossover to strong Anderson localization. Perturbations due to boundary conditions also give information on the spatial and topological electronic properties, as is shown for the superconducting proximity effect. Related models including phonon and mixed Fibonacci models are discussed, as are generalizations to other quasiperiodic chains and higher-dimensional extensions. Interacting quasiperiodic systems and the case for many body localization are discussed. Some experimental realizations of the 1D quasicrystal and their potential applications are described.

中文翻译：

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斐波那契准晶体：隐藏维度和多重分形的案例研究

准晶体独特的电子特性源于它们的长程结构有序，在旋转和离散尺度变化下具有不变性，但没有平移不变性。 $d$维准晶可以用更高维的晶格来描述（$D>d$)，并且它们的许多属性可以简单地从考虑额外“隐藏”维度的分析中推导出来。特别是，正如最近的理论和实验研究表明的那样，准晶体可以具有继承自母体晶体的拓扑特性。此处针对最简单的准晶体（一维 (1D) 斐波那契链）讨论了这些属性。斐波那契非相互作用紧束缚哈密顿量的特点是光谱和状态的多重分形，这体现在它的许多物理性质上，最显着的是在传输中。描述了由于无序和重入现象引起的扰动，以及与强安德森定位的交叉。由于边界条件引起的扰动也提供了关于空间和拓扑电子特性的信息，如超导邻近效应所示。讨论了包括声子和混合斐波那契模型在内的相关模型，以及对其他准周期链和高维扩展的推广。讨论了相互作用的准周期系统和许多身体定位的情况。描述了 1D 准晶的一些实验实现及其潜在应用。