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

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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 PB(w) around the origin (in the tail) to have the form of PB(w) wα (PB(w) e⁻βʷγ) (α, β, γ> 0), while in the critical region PB(w) w⁻α' (α' > 0). We also find the level spacing distribution to follow an inverse power law PG(s) s⁻ δ (δ> 0)