Energy Levels of Quasiperiodic Hamiltonians, Spectral Unfolding, and Random Matrix Theory

We consider a tight-binding Hamiltonian defined on the quasiperiodic Ammann-Beenker tiling. Although the density of states (DOS) is rather spiky, the integrated DOS (IDOS) is quite smooth and can be used to perform spectral unfolding. The effect of unfolding on the integrated level-spacing distribution is investigated for various parts of the spectrum which show different behaviour of the DOS. For energy intervals with approximately constant DOS, we find good agreement with the distribution of the Gaussian orthogonal random matrix ensemble (GOE) even without unfolding. For energy ranges with fluctuating DOS, we observe deviations from the GOE result. After unfolding, we always recover the GOE distribution. In two recent papers [1, 2], we investigated the energy-level statistics of two-dimensional quasiperiodic Hamiltonians, concentrating on the case of the eight-fold Ammann-Beenker tiling [3] shown in Fig. 1. The Hamiltonian contains solely constant hopping elements along the edges of the tiles in Fig. 1. Numerical results suggest that typical eigenstates of the model are multifractal [4]. In [1, 2], we numerically calculated the level-spacing distribution P (s) and the ∆3 and Σ2 statistics [5], and found perfect agreement with the results for the GOE. One ingredient of the calculation was the so-called unfolding procedure, explained below, which corrects for the fluctuations in the DOS of the model Hamiltonian. It is well known that, for