Twisted bilayers of quasicrystals which lend to moir\'e patterns can be studied using two dimensional Penrose lattices. These lattices are widely used to model quasiperiodic systems that exhibit the same diffraction pattern as the decagonal quasicrystal or real quasicrystal. A decagonal quasicrystal consists of a periodic piling of quasicrystal layers. A poor layer stacking in a decagonal quasicrystal can form a moir\'e pattern, visualized by families of lines of destructive interference which intersect each other. This moir\'e pattern can be seen as €˜regions with phason flip defects, consisting of discreet reordering of atoms that finish in a rearrangement of the cells inside the structure. The phason flip can diffuse along linear quasiperiodic arrangements called Conway worms, which make up the moir\'e pattern. In this work, we analyzed the electronic structure of two coupled Conway worms with the effects of phason flips. The phason flips generate localized zero-mode states near the Fermi level of the energy spectrum, which could play a central role to explain superconductivity in twisted bilayers of quasicrystals.
Huipe-Domratcheva et al. (Tue,) studied this question.