Ensembles of quantum chaotic systems are expected to exhibit random matrix
universality in their energy spectrum. The presence of this universality can be
diagnosed by looking for a linear in time 'ramp' in the spectral form factor,
but for realistic systems this feature is typically only visible after a
sufficiently long time. I will discuss several developments in our understanding
of this intermediate time regime in the spectral form factor. First, I will
describe a hybrid system in which the single particle levels are chaotic while
the many-body levels are those of non-interacting particles. In this case, the
linear ramp is replaced by an exponential ramp. Interacting deformations of the
system can then lead to full many-body quantum chaos, but the approach to a
linear ramp can be slow due to the presence of nearly conserved modes. Motivated
by this problem, I will present a theory of the spectral form factor in systems
with slow modes and will apply the results to a variety of hydrodynamic systems.
Joint work with Mike Winer and Shaokai Jian.