  We study the consequences of having translational invariance in space and in
time in many-body quantum chaotic systems. We consider an ensemble of random
quantum circuits, composed of single-site random unitaries and nearest
neighbour couplings, as a minimal model of translational invariant many-body
quantum chaotic systems. We evaluate the spectral form factor (SFF) as a sum
over many-body Feynman diagrams, which simplifies in the limit of large local
Hilbert space dimension $q$. At sufficiently large $t$, diagrams corresponding
to rigid translations dominate, reproducing the chaotic behavior of random
matrix theory (RMT). At finite $t$, we show that translational invariance
introduces additional mechanisms via two novel Feynman diagrams, known as the
crossed and deranged diagrams, which delay the emergence of RMT. Our analytics
suggests the existence of exact scaling forms which describe the approach to
RMT behavior in the scaling limit where both $t$ and $L$ are large while the
ratio between $L$ and $L_\mathrm{Th}(t)$, the many-body Thouless length, is
fixed. We numerically demonstrate, with simulations of two distinct circuit
models, that in such a scaling limit, most microscopic details become
unimportant, and the resulting scaling functions are largely universal,
remarkably being only dependent on a few global properties of the system like
the spatial dimensionality, and the space-time symmetries.
