The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality

Abstract

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.

Dephasing-enhanced performance in quasiperiodic thermal machines

“…The Fibonacci model has a fractal spectrum, with critical single-particle eigenfunctions [21,[38][39][40]. This fractality of We notice that they do not depend on the thermodynamic configurations.…”

“…In this work we focus on a specific example from the family of quasiperiodic systems, the Fibonacci model [21,39]. We take a one-dimensional (1D) tight-binding chain of noninteracting fermions, described by the following Hamiltonian…”

“…These systems, often called quasicrystals, are known to possess highly non-trivial singular continuous spectra with fractal structure [18], which leads to the appearance of critical states [19,20] which are neither extended nor localized. These unique spectral features can in fact induce localization and anomalous transport without the presence of interactions [21][22][23][24][25][26][27]. Perhaps the most celebrated example is the Aubry-André-Harper (AAH) model [28,29] which displays a transition from a completely delocalized to a completely localized phase at a finite potential strength.…”

“…is the Fibonacci model where the lattice energies are generated by a substitution rule. The Fibonacci quasicrystal has unusual properties such as a critical energy spectrum across all energy scales [21,[38][39][40], without a localization transition. This spectral criticality gives rise to anomalous transport exponents varying continuously with the potential strength, so that it is possible to tune the transport regime from superdiffusive to subdiffusive [25,39,[41][42][43].…”

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“…This is partly because the lack of periodicity severely limits applicable theoretical methods. An exception would be one dimension, where the effect of a quasiperiodic potential [10][11][12] on interacting fermions has been studied since early days [13][14][15][16] , and has attracted a resurgent attention in recent years [17][18][19] , particularly in connection with the ultracold-atom experiment 20 .…”