Many-body level statistics of quantum chaos

In classical systems, chaos is usually diagnosed by the strong sensitivity of trajectories to the initial conditions, whereas in quantum systems, such an indicator is served by the energy level statistics instead. While there has been a large number of studies about the level statistics of single-particle chaotic models, much remains unknown for many-body quantum chaos. We first consider a system of noninteracting fermions populating levels of a random matrix ensemble, and find that, apart from an early time slope, its spectral form factor also exhibits an exponential ramp and a plateau which reflect the repulsion between two distant many-body levels. Using a field theoretical approach, we show that this ramp originates from Goldstone modes caused by the spontaneous breakdown of a SU(2) symmetry. In the presence of interactions, these Goldstone modes acquire a mass, leading to the suppression of the exponential ramp.