An effective field theory for non-maximal quantum chaos

In non-maximally quantum chaotic systems, the exponential behavior of out-of-time-ordered correlators (OTOCs) results from summing over exchanges of an infinite tower of higher "spin" operators. We construct an effective field theory (EFT) to capture these exchanges in $(0+1)$ dimensions. The EFT generalizes the one for maximally chaotic systems, and reduces to it in the limit of maximal chaos. The theory predicts the general structure of OTOCs both at leading order in the $1/N$ expansion ($N$ is the number of degrees of freedom), and after resuming over an infinite number of higher order $1/N$ corrections. These general results agree with those previously explicitly obtained in specific models. We also show that the general structure of the EFT can be extracted from the large $q$ SYK model.