Electronic wave functions of quasiperiodic systems in momentum space

In quasicrystalline tilings often multifractal electronic wave functions can be found. In order to obtain a better insight into their localization properties, we study the wave functions of quasiperiodic tilings in momentum space. The models are based on one-dimensional quasiperiodic chains, in which the atoms are coupled by weak and strong bonds aligned according to the metallic-mean sequences. The associated hypercubic tilings and labyrinth tilings in d dimensions are then constructed from the direct product of d such chains. The results show that each wave function is described by a hierarchy of wave vectors and is always dominated by a single wave vector which is directly related to the energy eigenvalue of the wave function. The corresponding spectral function of the systems shows a hierarchy of branches with different intensities. Each branch is a copy of the main branch containing the dominant wave vectors for each wave function. Using perturbation theory and a renormalization group approach, we determine the shape of the branches for the limit of weak and strong coupling. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013