The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality

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.