From Lyapunov Growth to Weak Quantum Chaos

Abstract

Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator $C_r(t)$, its spatial sum $\bar C(t) = \sum_r C_r(t)$, and the symmetrically regularised connected dOTOC $D(t)$ of two extensive observables. We show that Lieb--Robinson locality gives $\bar C(t) \lesssim t^d$. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, $D(t) \lesssim t^d$, improving our earlier suboptimal $t^{3d}$ bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates $D(t)$ to $\bar C(t)$ and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.