Single-Particle Universality of the Many-Body Spectral Form Factor

Abstract

We consider systems of fermions evolved by non-interacting unitary circuits with correlated on-site potentials. When these potentials are drawn from the eigenvalue distribution of a circular random matrix ensemble, the single-particle sector exhibits chaotic dynamics. We study the corresponding many-body spectral statistics and show that the spectral form factor (SFF) can be computed \textit{exactly}. Due to the absence of interactions the SFF grows exponentially in time, a result which we demonstrate through simple arguments, scaling collapses, and closed-form evaluation of the SFF. We study the role of interactions by numerically analyzing a kicked Ising model and find that the SFF crosses over to a linear growth regime consistent with many-body random matrix universality. Our exact results for the SFF provide a baseline for future studies of the crossover between single-particle and many-body random matrix behavior.