Electronic properties of the octagonal tiling : a new renormalization-group calculation

We present a new renormalization-group calculation of the octagonal tiling excitation spectrum. Two new self-similar octagonal tilings are introduced, a triangle-kite and a triangle-dart tiling, which are entangled with the standard rhombus-square tiling. The decimation procedure involved in the renormalization group calculation transforms each tiling onto a different one and goes back to the (rescaled) original tiling after three steps. The local and average density of states are calculated. In the case of a Laplacian-like tight-binding Hamiltonian, on the standard rhombussquare octagonal tiling, our results compare reasonably well to numerically derived spectra and improve some previous renormalization group calculations. In the pure hopping case, the renormalization is shown to be inaccurate by comparing to numerical density of states obtained by a continued fraction method