Quantum operator growth bounds for kicked tops and semiclassical spin chains

TL;DR

This paper develops bounds on operator growth and information scrambling in large-spin models, showing finite Lyapunov exponents and butterfly velocities, with implications for semiclassical and quantum chaotic systems.

Contribution

It introduces a framework for bounding operator growth in large-$S$ spins, providing tighter bounds than previous Lieb-Robinson bounds and analyzing the classical-quantum differences.

Findings

Finiteness of Lyapunov exponent in large-$S$ limit.

Upper bounds on Lyapunov exponents close to numerical values.

Finite butterfly velocity in coupled large-$S$ spin systems.

Abstract

We present a framework for understanding the dynamics of operator size, and bounding the growth of out-of-time-ordered correlators, in models of large-$S$ spins. Focusing on the dynamics of a single spin, we show the finiteness of the Lyapunov exponent in the large-$S$ limit; our bounds are tighter than the best known Lieb-Robinson-type bounds on these systems. We numerically find our upper bound on Lyapunov exponents is 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 show that the butterfly velocity, which characterizes the spatial speed of quantum information scrambling, is finite as $S\rightarrow\infty$. We emphasize qualitative differences between operator growth in semiclassical large-spin models, and quantum holographic systems including the Sachdev-Ye-Kitaev model.