Generalized Inverse Participation Numbers in Metallic-Mean Quasiperiodic Systems

TL;DR

This paper investigates the scaling behavior of generalized inverse participation numbers in metallic-mean quasiperiodic systems, revealing a proportional relation between dimensions and eigenstate localization properties.

Contribution

It provides a numerical and analytical study of eigenstates in metallic-mean quasicrystals, establishing a relation between their fractal dimensions across dimensions.

Findings

The relation D_q^{dD} = d D_q^{1D} holds for these models.

Numerical results support the proportionality of fractal dimensions across dimensions.

Analytical proof confirms the relation for the silver-mean model.

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 $D_q^{d\mathrm{d}} = d D_q^\mathrm{1d}$ 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.