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

Abstract

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 ${P}_{B}(w)\ensuremath{\sim}{w}^{\ensuremath{\alpha}}[{P}_{B}(w)\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}{w}^{\ensuremath{\gamma}}}]\phantom{\rule{0.3em}{0ex}}(\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma}>0)$, while in the critical region ${P}_{B}(w)\ensuremath{\sim}{w}^{\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}}\phantom{\rule{0.3em}{0ex}}({\ensuremath{\alpha}}^{\ensuremath{'}}>0)$. We also find the level spacing distribution to follow an inverse power law ${P}_{G}(s)\ensuremath{\sim}{s}^{\ensuremath{-}\ensuremath{\delta}}\phantom{\rule{0.3em}{0ex}}(\ensuremath{\delta}>0)$.