Statistics of Spectra for One-dimensional Quasi-Periodic Systems at the Metal-Insulator Transition

We study spectral statistics of one-dimensional quasi-periodic systems at the metal-insulator transition. Several types of spectral statistics are observed at the critical points, lines, and region. On the critical lines, we find the bandwidth distribution $P_B(w)$ around the origin (in the tail) to have the form of $P_B(w)\sim w^{\alpha }$ ($P_B(w)\sim e^{-\beta w^{\gamma }}$) ($\alpha ,\beta ,\gamma >0$), while in the critical region $P_B(w) \sim w^{-\alpha’}$($\alpha’ > 0$). We also find the level spacing distribution to follow an inverse power law $P_G(s)\sim s^{-\delta }$ ($\delta >0$)