Physical nature of all the electronic states in the Thue-Morse chain

TL;DR

This paper analytically demonstrates that the Thue-Morse lattice uniquely supports only extended electronic states, lacking localized or critical states, which is unprecedented among aperiodic systems.

Contribution

It provides an exact analytical method to find all eigenfunctions and eigenvalues of the Thue-Morse lattice and reveals its unusual all-extended state spectrum.

Findings

All electronic states in the Thue-Morse chain are extended.

Degenerate eigenstates arise due to lattice symmetry.

Landauer resistivity is zero for degenerate states and scales as L^2 at band edges.

Abstract

We present an analytical method for finding all the electronic eigenfunctions and eigenvalues of the aperiodic Thue-Morse lattice. We prove that this system supports only extended electronic states which is a very unusual behavior for this class of systems, and so far as we know, this is the only example of a quasiperiodic or aperiodic system in which critical or localized states are totally absent in the spectrum. Interestingly we observe that the symmetry of the lattice leads to the existence of degenerate eigenstates and all the eigenvalues excepting the four global band edges are doubly degenerate. We show exactly that the Landauer resistivity is zero for all the degenerate eigenvalues and it scales as $\sim L^2$ ($L=$ system size) at the global band edges. We also find that the localization length $\xi$ is always greater than the system size.