Multifractal Properties of Tribonacci Chains

Abstract

The description of electrons in solids using Bloch’s theorem has allowed for a profound understanding of the electronic band structure of regular crystalline materials. An example of its success is the understanding of topological insulators and superconductors, and their invariants, via the reciprocal space. Aperiodic systems on the other hand, which are systems that break translational symmetry but possess long-range order, cannot be studied using the same tools, rendering their topological nature ambiguous. In this thesis, we review some physical and mathematical properties of aperiodic systems, and discuss some experimental realisations. We introduce two 1D tight-binding models based on the Tribonacci substitution, the hopping and on-site Tribonacci chains, which generalize the Fibonacci chain. For both hopping and on-site models, a perturbative real-space renormalization procedure is developed. We show that the two models are equivalent at the fixed point of the renormalization group flow, and that the renormalization procedure naturally gives the Local Resonator Modes. Additionally, the Rauzy fractal, inherent to the Tribonacci substitution, is shown to serve as the analog of conumbering for the Tribonacci chain. Our work provides new insight into how the internal space of a cut-and-project quasicrystal is used to describe the eigenstates, and can in principle be applied to any cut-and-project quasicrystal. Finally, we construct a Rice-Mele charge pump from the Tribonacci word, which exhibits multilevel topological pumping due to the multifractal nature of the Tribonacci chain’s energy spectrum. This is a first step towards studying the topological properties of the Tribonacci chain.