Structure and electronic properties of Thue-Morse lattices

We study a one-dimensional system which is neither periodic, quasiperiodic, nor random. We find that the structure factor of this system consists of a set of peaks whose heights scale with L, the length of the chain, according to $L^{\alpha \left(k\right)}$. We show that for k<(1/2, α(k)≤α((1/3)=ln3/ln2, so that all of the peaks vanish relative to the peak at the center of the Brillouin zone (which is associated with the periodicity of the underlying lattice) as the system grows. We also prove a number of other properties of these exponents. We discuss the energy spectrum of this system for both weak and strong potentials. We show how the gaps in the two limits are related, and we argue that, despite the expectations of naive perturbation theory, gaps persist in the L→∞ limit.