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 PB(w) around the origin (in the tail) to have the form of ${P}B(w){~}{w}^{{α}}[{P}B(w){~}{e}^{{-}{β}{w}^{{γ}}}]{{0.3em}{0ex}}({α},{β},{γ}>0)$, while in the critical region ${P}B(w){~}{w}^{{-}{{α}}^{{'}}}{{0.3em}{0ex}}({{α}}^{{'}}>0)$. We also find the level spacing distribution to follow an inverse power law ${P}G(s){~}{s}^{{-}{δ}}{{0.3em}{0ex}}({δ}>0)$.