Crystalline Spectral Form Factors

We investigate a crystalline-like behavior of the spectral form factor (SFF) in unitary quantum systems with extremely strong energy level repulsion. Using low-temperature Coulomb gas as a qualitative model of repulsive eigenvalues, we derive the Debye-Waller factor suppressing periodic oscillations of the SFF and estimate the order of its discontinuities near multiples of the Heisenberg time. We also reproduce this crystalline-like behavior using local shallow circuits. Our results lay the foundation for future studies of the spectral statistics in perturbed quantum cellular automata and quantum permutation circuits, where crystalline SFF naturally emerges.