Onset of Quantum Chaos in Random Field Theories

We study the quantum Lyapunov exponent ${\lambda }_L$ in theories with spacetime-independent disorder. We first derive self-consistency equations for the two- and four-point functions for products of $N$ models coupled by disorder at large $N$, generalizing the equations appearing in SYK-like models. We then study families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow ${\lambda }_L$ from weak to strong coupling. We find interesting behaviors, including a discontinuous transition into chaos, mimicking classical KAM theory.