Statistics of spectra for critical quantum chaos in one-dimensional quasiperiodic systems

We study spectral statistics of one-dimensional quasiperiodic 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)similar tow(alpha)[P-B(w)similar toe(-betawgamma)] (alpha,beta,gamma>0), while in the critical region P-B(w)similar tow(-alpha') (alpha(')>0). We also find the level spacing distribution to follow an inverse power law P-G(s)similar tos(-delta) (delta>0).