Application of random matrix theory to quasiperiodic systems

We study statistical properties of energy spectra of a tight-binding model on the twodimensional quasiperiodic Ammann-Beenker tiling. Taking into account the symmetries of finite approximants, we find that the underlying universal level-spacing distribution is given by the Gaussian orthogonal random matrix ensemble, and thus differs 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 difference to the Wigner surmise. In a recent paper [1], we investigated energy spectra of quasiperiodic tightbinding 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 reflection symmetry, an “almost symmetry” under rotation by 90 degrees which may influence the level statistics [6], whereas the octagonal patches used in Ref. [5] possess the full D8-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. Preprint submitted to Elsevier Preprint 1 February 2008 Fig. 1. Octagonal cluster of the Ammann-Beenker tiling with 833 vertices and exact D8 symmetry around the central vertex as indicated by the solid and dashed lines. Shadings indicate successive inflation steps of the central octagon. To arrive at this conclusion, we considered in Ref. [1] different patches that approximate the infinite quasiperiodic tiling, both with free and periodic boundary conditions. Exact symmetries were either exploited to block-diagonalize the Hamiltonian, thus splitting the spectrum into its irreducible parts, or avoided altogether by choosing patches without any symmetries. Here, we concentrate on the D8-symmetric octagonal patch shown in Fig. 1. For this case, the Hamiltonian matrix splits into ten blocks according to the −4 −2 0 2 4 E 0.0 0.2 0.4 0.6 0.8