Exact spectral form factors of non-interacting fermions with Dyson statistics

The spectral form factor (SFF) is a powerful diagnostic of random matrix behavior in quantum many-body systems. We introduce a family of random circuit ensembles whose SFFs can be computed \textit{exactly}. These ensembles describe the evolution of non-interacting fermions in the presence of correlated on-site potentials drawn from the eigenvalue distribution of a circular ensemble. For disorder parameters drawn from the circular unitary ensemble (CUE), we derive an exact closed form for the SFF, valid for any choice of system size $L$ and integer time $t$. When the disorder is drawn from the circular orthogonal or symplectic ensembles (COE and CSE, respectively), we carry out the disorder averages analytically and reduce the computation of the SFF at integer times to a combinatorial problem amenable to transfer matrix methods. In each of these cases the SFF grows exponentially in time, which we argue is a signature of random matrix universality at the single-particle level. Finally, we develop matchgate circuit representations of our circuit ensembles, enabling their experimental realization in quantum simulators.
Comment: This paper provides proofs of results presented in arxiv:2410.07306 in addition to previously unstated results. The reader may benefit from examining both in parallel. Comments welcome!