Application of random matrix theory toquasiperiodic

We study statistical properties of energy spectra of a tight-binding model on the two-dimensional quasiperiodic Ammann-Beenker tiling. Taking into account the symmetries of nite approximants, we nd that the underlying universal level-spacing distribution is given by the Gaussian orthogonal random matrix ensemble, and thus diiers from the critical level-spacing distribution observed at the metal-insulator transition in the three-dimensional Anderson model of disorder. Our data allow us to see the diierence to the Wigner surmise. In a recent paper 1], we investigated energy spectra of quasiperiodic tight-binding models, concentrating on the case of the octagonal Ammann-Beenker tiling 2] shown in Fig. 1. The Hamiltonian is restricted to constant hopping matrix elements along the edges of the tiles in Fig. 1. Previous studies of the same model had led to diverging results on the level statistics: For periodic approximants, level repulsion was observed 3,4], and the level-spacing distribution P (s) was argued to follow a log-normal distribution 4]. On the other hand, for octagonal patches with an exact eightfold symmetry and free boundary conditions, level clustering was found 5]. On the basis of our numerical results for P (s) and the spectral rigidity 3 6], compiled in Ref. 1], we concluded that the underlying universal level-spacing distribution of this system is given by the Gaussion orthogonal random matrix ensemble (GOE) 6,7]. Concerning the contradictory results of previous investigations, we attribute these to the non-trivial symmetry properties of the octagonal tiling. The periodic approximants studied in Refs. 3,4] show, besides an exact re-ection symmetry, an \almost symmetry" under rotation by 90 degrees which may innuence the level statistics 6], whereas the octagonal patches used in Ref. 5] possess the full D 8-symmetry of the regular octagon. Hence the level statistics observed in this case is that of a superposition of seven completely independent subspectra, and therefore rather close to a Poisson law.