Generalized Inverse Participation Numbers in Metallic-Mean Quasiperiodic Systems

Abstract

From the quantum mechanical point of view, the electronic characteristics of quasicrystals are determined by the nature of their eigenstates. A practicable way to obtain information about the properties of these wave functions is studying the scaling behavior of the generalized inverse participation numbers $Z_q\sim N^{-D_q(q-1)}$ with the system size $N$. In particular, we investigate $d$-dimensional quasiperiodic models based on different metallic-mean quasiperiodic sequences. We obtain the eigenstates of the one-dimensional metallic-mean chains by numerical calculations for a tight-binding model. Higher dimensional solutions of the associated generalized labyrinth tiling are then constructed by a product approach from the one-dimensional solutions. Numerical results suggest that the relation $D_q^{d\mathrm{d}}=d{D}_q^{1d}$ holds for these models. Using the product structure of the labyrinth tiling we prove that this relation is always satisfied for the silver-mean model and that the scaling exponents approach this relation for large system sizes also for the other metallic-mean systems.

7 pages, 3 figures

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Cited by in corpus (7)

• Critical eigenstates and their properties in one and two dimensional quasicrystals
• Fraction of delocalized eigenstates in the long-range Aubry-André-Harper model
• Wave Functions, Quantum Diffusion, and Scaling Exponents in Golden-Mean Quasiperiodic Tilings
• Quantum and Spectral Properties of the Labyrinth Model
• Stability of quasicrystalline ultracold fermions to dipolar interactions
• The Fibonacci quasicrystal: case study of hidden dimensions and multifractality
• Missing link between the 2D Quantum Hall problem and 1D quasicrystals