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

Abstract

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 $P_B(w) \sim w^α$ ($P_B(w) \sim e^{-βw^γ}$) ($α, β, γ> 0 $), while in the critical region $P_B(w) \sim w^{-α'}$ ($α' > 0$). We also find the level spacing distribution to follow an inverse power law $P_G(s) \sim s^{- δ}$ ($δ> 0$)