Similarity of eigenstates in generalized labyrinth tilings

TL;DR

This paper investigates the eigenstates of higher-dimensional quasiperiodic tilings constructed from one-dimensional chains, showing that eigenstates become increasingly similar across different tilings as system size grows, enabling analytical derivations.

Contribution

It introduces a method to analyze eigenstates of generalized labyrinth tilings using separable Hamiltonians and inflation rules, revealing eigenstate similarities in large systems.

Findings

Eigenstates of different tilings become similar for large system sizes.

Participation numbers and scaling exponents are derived analytically.

Eigenstates of higher-dimensional tilings can be inferred from one-dimensional results.

Abstract

The eigenstates of d-dimensional quasicrystalline models with a separable Hamiltonian are studied within the tight-binding model. The approach is based on mathematical sequences, constructed by an inflation rule P = {w -> s, s -> sws^(b-1)} describing the weak/strong couplings of atoms in a quasiperiodic chain. Higher-dimensional quasiperiodic tilings are constructed as a direct product of these chains and their eigenstates can be directly calculated by multiplying the energies or wave functions of the chain, respectively. Applying this construction rule, the grid in d dimensions splits into 2^(d-1) different tilings, for which we investigated the characteristics of the wave functions. For the standard two-dimensional labyrinth tiling constructed from the octonacci sequence (b=2) the lattice breaks up into two identical lattices, which consequently yield the same eigenstates. While this is not the case for other b, our numerical results show that the wave functions of the different grids become increasingly similar for large system sizes. This can be explained by the fact that the structure of the 2^(d-1) grids mainly differs at the boundaries and thus for large systems the eigenstates approach each other. This property allows us to analytically derive properties of the higher-dimensional generalized labyrinth tilings from the one-dimensional results. In particular participation numbers and corresponding scaling exponents have been determined.