Many-Body Quantum Chaos and Space-time Translational Invariance

Abstract: 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.