Extended and critical wave functions in a Thue-Morse chain

We study the one-dimensional tight-binding model with site energies arranged in the Thue-Morse sequence. To show the characteristics of eigenstates, we study the trace map, wave functions, and the resistance, and perform a multifractal analysis on the wave functions. Half of all the eigenstates are represented by lattice-like wave functions, the amplitudes of which resemble the Thue-Morse sequences of lower order. These lattice-like wave functions are denoted as chaotic or extended, depending on the lattice size considered. The other half are classified into critical or extended states. We discuss the properties of the electronic spectrum. Our results show that critical states coexist with extended ones; therefore, the spectrum consists of absolutely continuous parts and singular continuous parts.