Spectral signatures of chaotic diffusion in systems with and without spatial order

We investigate the two-point correlations in the spectra of extended systems exhibiting chaotic diffusion in the classical limit, in presence and in absence of spatial order. For periodic systems, we express the spectral two-point correlations in terms of form factors with the unit-cell index as a discrete 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 of quantum ballistic motion. In the opposite regime of strong disorder with localized eigenstates, we derive a semiclassical approximation of the form factor that spans the entire transition from metallic to isolating behaviour. The regime of weak breaking of periodicity is accessed from the side of exact order by a perturbation theory for the sets of, without disorder, symmetry-related periodic orbits.