Localization of Electronic Wave Functions on Quasiperiodic Lattices

TL;DR

This paper investigates the nature of electronic wave functions on quasiperiodic lattices, revealing different localization behaviors in two- and three-dimensional structures through numerical analysis.

Contribution

It provides a comparative numerical study of eigenstates on 2D Penrose and 3D icosahedral tilings, highlighting their distinct localization properties.

Findings

Power-law localization observed in Penrose tiling eigenstates

No power-law localization in icosahedral tiling eigenstates

Different behaviors of eigenstates in 2D and 3D quasiperiodic systems

Abstract

We study electronic eigenstates on quasiperiodic lattices using a tight-binding Hamiltonian in the vertex model. In particular, the two-dimensional Penrose tiling and the three-dimensional icosahedral Ammann-Kramer tiling are considered. Our main interest concerns the decay form and the self-similarity of the electronic wave functions, which we compute numerically for periodic approximants of the perfect quasiperiodic structure. In order to investigate the suggested power-law localization of states, we calculate their participation numbers and structural entropy. We also perform a multifractal analysis of the eigenstates by standard box-counting methods. Our results indicate a rather different behaviour of the two- and the three-dimensional systems. Whereas the eigenstates on the Penrose tiling typically show power-law localization, this was not observed for the icosahedral tiling.