Signature of chaotic diffusion in band spectra

We investigate the two-point correlations in the band spectra of periodic systems that exhibit chaotic diffusion in the classical limit, in terms of form factors with the winding number as a spatial argument. For times below the Heisenberg time, they contain the full space-time dependence of the classical propagator. They approach constant asymptotes via a regime, reflecting quantal ballistic motion, where they decay by a factor proportional to the number of unit cells. We derive a universal scaling function for the long-time behavior. In the limit of long chains, our results are consistent with expressions obtained by field-theoretical methods. They are substantiated by numerical studies of the kicked rotor and a billiard chain.