Fibonacci准晶：隐维数和多重分形的实例研究
The Fibonacci quasicrystal: case study of hidden dimensions and multifractality
摘要：
准晶独特的电子性质源于它们的长程结构有序性，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度的晶格$D>d$来描述，并且它们的许多性质可以简单地从考虑额外的“隐藏”维度的分析中导出。特别是，最近的理论和实验研究表明，准晶体可以具有从母体晶体继承的拓扑性质。这些性质在这里讨论的最简单的准晶，一维斐波那契链。斐波那契非相互作用紧束缚哈密顿量的特征是谱和态的多重分形，这表现在它的许多物理性质中，特别是在输运中。由于无序和重新进入现象的扰动进行了描述，随着交叉强安德森局域化。由于边界条件的扰动也给出了空间和拓扑电子性质的信息，如超导邻近效应所示。相关的模型，包括声子和混合斐波那契模型进行了讨论，以及推广到其他准周期链和高维扩展。简要讨论了相互作用准周期系统和多体局域化的情况。介绍了一维准晶的一些实验实现及其潜在的应用。
英文摘要：
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 1D Fibonacci chain. The Fibonacci noninteracting tight-binding Hamiltonians are characterized by multifractality of spectrum and states, which is manifested in many of its physical properties, notably in transport. Perturbations due to disorder and re-entrance 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 well as generalizations to other quasiperiodic chains and higher dimensional extensions. Interacting quasiperiodic systems and the case for many body localization are briefly discussed. Some experimental realizations of the 1D quasicrystal and their potential applications are described.
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