Scaling at the OTOC Wavefront: Integrable versus chaotic models

Abstract

Out of time ordered correlators (OTOCs) are useful tools for investigating foundational questions such as thermalization in closed quantum systems because they can potentially distinguish between integrable and nonintegrable dynamics. Here we discuss the properties of wavefronts of OTOCs by focusing on the region around the main wavefront at $x=v_{B}t$, where $v_{B}$ is the butterfly velocity. Using a Heisenberg spin model as an example, we find that a propagating Gaussian with the argument $-m(x)\left( x-v_B t \right)^2 +b(x)t$ gives an excellent fit for both the integrable case and the chaotic case. However, the scaling in these two regimes is very different: in the integrable case the coefficients $m(x)$ and $b(x)$ have an inverse power law dependence on $x$ whereas in the chaotic case they decay exponentially. In fact, the wavefront in the integrable case is a rainbow caustic and catastrophe theory can be invoked to assert that power law scaling holds rigorously in that case. Thus, we conjecture that exponential scaling of the OTOC wavefront is a robust signature of a nonintegrable dynamics.