Gauss chains: Quadratic to quintic

Abstract

Gauss chains are one-dimensional nonperiodic lattices in which lattice sites are located at zj=jnd, with j∈W, n∈N, and d being the underlying lattice constant. They may potentially be realized in semiconductor superlattices or tailored chain molecules. In recent work, we have characterized the electronic delocalized states for the quadratic Gauss chain n=2 obtained using a one-dimensional transfer-matrix approach applied to the Kronig–Penney model. We here extend those results to cases of larger n, illustrating the approach for n=3, 4, and 5. Beginning with the structure factor, we find that the case n=2 has a visually evident structure lacking for n>2, and this structure (or lack thereof) is reflected in the statistics of the structure factor. Turning to the electronic structure, the delocalized-state spectrum for each n is singular-continuous with delocalized states at wavevectors k for all rational multiples rs of πd with r,s∈N coprime in the limit of weak onsite potential parameter λ. Most states, however, become localized at a state-dependent threshold value of |λ|. The case n=2 appears to exhibit delocalized-state spectra that most clearly reveal the hidden symmetry of these systems. Gauss chains with various n, therefore, provide a way (in principle) to realize a class of functions of number-theoretic importance.