Wave Functions, Quantum Diffusion, and Scaling Exponents in Golden-Mean Quasiperiodic Tilings

TL;DR

This paper investigates wave functions and quantum diffusion in high-dimensional quasiperiodic tilings derived from Fibonacci chains, revealing multifractal properties and structure-dependent transport behaviors through renormalization group analysis.

Contribution

It introduces a numerical approach to large quasiperiodic systems and relates wave function properties to the underlying structure using a renormalization group framework.

Findings

Wave functions are multifractal in quasiperiodic tilings.

Wave-packet dynamics show structure-dependent scaling behavior.

Proposes bounds for wave packet spreading in different regimes.

Abstract

We study the properties of wave functions and the wave-packet dynamics in quasiperiodic tight-binding models in one, two, and three dimensions. The atoms in the one-dimensional quasiperiodic chains are coupled by weak and strong bonds aligned according to the Fibonacci sequence. The associated d-dimensional quasiperiodic tilings are constructed from the direct product of d such chains, which yields either the hypercubic tiling or the labyrinth tiling. This approach allows us to consider rather large systems numerically. We show that the wave functions of the system are multifractal and that their properties can be related to the structure of the system in the regime of strong quasiperiodic modulation by a renormalization group (RG) approach. We also study the dynamics of wave packets to get information about the electronic transport properties. In particular, we investigate the scaling behaviour of the return probability of the wave packet with time. Applying again the RG approach we show that in the regime of strong quasiperiodic modulation the return probability is governed by the underlying quasiperiodic structure. Further, we also discuss lower bounds for the scaling exponent of the width of the wave packet and propose a modified lower bound for the absolute continuous regime.