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 $P_G(s) sim s^{- delta}$ ($delta > 0$)