Level-spacing distribution of localized phases induced by quasiperiodic potentials

Level statistics is an important quantity for exploring and understanding localized physics. The level-spacing distribution (LSD) of the disordered localized phase follows Poisson statistics, and many studies naturally apply it to the quasiperiodic localized phase. Here, we analytically obtain the LSD of the quasiperiodic localized phase, and find that it deviates from Poisson statistics. Moreover, based on this level statistics, we derive the ratio of adjacent gaps and find that for a single sample, it is a 8 function, which is in excellent agreement with numerical studies. Additionally, unlike disordered systems, in quasiperiodic systems, there are variations in the LSD across different regions of the spectrum, and the presence of spectral correlations results in nonequivalence between increasing the size and increasing the sample. Our findings carry significant implications for the reevaluation of level statistics in quasiperiodic systems and a profound understanding of the distinct effects of quasiperiodic potential-induced and disorder-induced localization.