Energy spectra, wavefunctions and quantum diffusion for quasiperiodic systems

TL;DR

This paper investigates the spectral properties, eigenstates, and quantum diffusion in one- and two-dimensional quasiperiodic systems, revealing multifractal eigenstates and anomalous diffusion behaviors through numerical analysis.

Contribution

It provides a detailed numerical study of energy spectra, eigenstates, and quantum diffusion in quasiperiodic models, including the silver mean chain and labyrinth tiling, highlighting their multifractal and critical properties.

Findings

Energy spectra are singular continuous or fractal-like depending on modulation strength.

Eigenstates are multifractal and critical in nature.

Quantum diffusion exhibits power-law behavior with exponents related to spectral properties.

Abstract

We study energy spectra, eigenstates and quantum diffusion for one- and two-dimensional quasiperiodic tight-binding models. As our one-dimensional model system we choose the silver mean or `octonacci' chain. The two-dimensional labyrinth tiling, which is related to the octagonal tiling, is derived from a product of two octonacci chains. This makes it possible to treat rather large systems numerically. For the octonacci chain, one finds singular continuous energy spectra and critical eigenstates which is the typical behaviour for one-dimensional Schr"odinger operators based on substitution sequences. The energy spectra for the labyrinth tiling can, depending on the strength of the quasiperiodic modulation, be either band-like or fractal-like. However, the eigenstates are multifractal. The temporal spreading of a wavepacket is described in terms of the autocorrelation function C(t) and the mean square displacement d(t). In all cases, we observe power laws for C(t) and d(t) with exponents -delta and beta, respectively. For the octonacci chain, 0<delta<1, whereas for the labyrinth tiling a crossover is observed from delta=1 to 0<delta<1 with increasing modulation strength. Corresponding to the multifractal eigenstates, we obtain anomalous diffusion with 0<beta<1 for both systems. Moreover, we find that the behaviour of C(t) and d(t) is independent of the shape and the location of the initial wavepacket. We use our results to check several relations between the diffusion exponent beta and the fractal dimensions of energy spectra and eigenstates that were proposed in the literature.