Many-body quantum chaos and space-time translational invariance

Getting a grip on the chaotic properties of quantum systems is difficult. Now, the effect of translational invariance in space in time in an ensemble of random quantum circuits is shown to lead to largely universal scaling laws describing the system without the need of knowing microscopic details. We study the consequences of having translational invariance in space and time in many-body quantum chaotic systems. We consider ensembles of random quantum circuits as minimal models of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor as a sum over many-body Feynman diagrams in the limit of large local Hilbert space dimension q. At sufficiently large t, diagrams corresponding to rigid translations dominate, reproducing the random matrix theory (RMT) behaviour. At finite t, we show that translational invariance introduces additional mechanisms via two novel Feynman 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-Th(t), the many-body Thouless length, is fixed. We numerically demonstrate, with simulations of two distinct circuit models, that the resulting scaling functions are universal in the scaling limit.

Automated summary

The effect of translational invariance in space and time on many-body quantum chaotic systems is studied using random quantum circuits. It is found that there are universal scaling laws describing the system, even without knowing the microscopic details. The emergence of random matrix theory behavior is delayed by translational invariance, which introduces additional mechanisms via novel Feynman diagrams.