Quantum operator growth bounds for kicked tops and semiclassical spin chains

We prove bounds on infinite temperature out-of-time-ordered correlation functions in semiclassical spin models, where each site contains a large-S spin degree of freedom. Focusing on the dynamics of a single spin, we prove the finiteness of the Lyapunov exponent in the large-S limit, and numerically find our upper bound on Lyapunov exponents can be within an order of magnitude of numerically computed values in classical and quantum kicked top models. Generalizing our results to coupled large-S spins on lattices, we prove that the butterfly velocity, which characterizes the spatial speed of quantum information scrambling, is finite in the large-S limit. Our work demonstrates how to derive rigorous constraints on quantum dynamics in a large class of models where conventional Lieb-Robinson bounds are not useful. We emphasize qualitative differences between semiclassical large-spin models, and quantum holographic systems including the Sachdev-Ye-Kitaev model.