Electronic structure of the octagonal tiling

An s-band model on the octagonal tiling is studied. The matching rules, the inflation rules, and the standard approximants are reviewed. It is shown that in the approximants the matching rules are only slightly violated. Therefore electronic properties, e.g., the density of states (DOS), converge very fast w.r.t. the approximant size N, e.g., Δ(DOS)∝N−2. In the low kinetic-energy limit the electronic properties of the model are dominated by a one-dimensional aperiodic substitution chain. The Hamiltonian of this chain is given and spectral and transport properties are studied. It is shown that the support of the measure decreases as a power of the approximant size N. The conductance through a macroscopic cell (l∼0.2 mm) is shown to be finite. In the presence of a small random on-site potential (W=0.01) the system becomes insulating.