One of the key goals of quantum chaos is to establish a clean relationship between the observed universal spectral fluctuations of simple quantum systems and random matrix theory. For single particle systems with fully chaotic classical counterparts, the problem has been essentially solved by M. Berry within the so-called diagonal approximation of semiclassical periodic-orbit sums.
In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront also for simple many-body quantum systems, such as locally interacting spin chains. Such systems seem to display two universal types of of behavior which are nowadays usually termed as `many-body localized phase' and `ergodic phase’. In the ergodic phase, the spectral fluctuations are typically excellently described by random matrix theory, despite simplicity of interactions and lack of any external source of disorder.
After giving a broad overview of the problem, I will outline a heuristic derivation of random matrix spectral form factor for clean non-integrable spin chains, an example of which is the Ising chain in a tilted (transverse + longitudinal) periodically kicking magnetic field. I will also present a specific model of the same type with nearest-neighbor interactions where random matrix spectral form factor can be rigorously proven.