Scaling at the OTOC Wavefront: free versus chaotic models

TL;DR

This paper investigates the wavefront behavior of out-of-time-ordered correlators (OTOCs) in quantum systems, revealing distinct scaling laws for free and chaotic models and proposing a universal connection to chaos through wavefront analysis.

Contribution

It demonstrates that free and chaotic quantum models exhibit different scaling behaviors in OTOC wavefronts, with free models showing power-law and chaotic models exponential decay, and links these to universal wavefront forms.

Findings

Free models exhibit power-law scaling in wavefront coefficients.

Chaotic models show exponential decay in wavefront coefficients.

The wavefront form transitions smoothly between regimes, indicating stability of the Airy form against weak chaos.

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 the leading edge of a propagating Gaussian with the argument $-m(x)\left( x-v_B t \right)^2 +b(x)t$ gives an excellent fit to the region around $x=v_{B}t$ for both the free and chaotic cases. However, the scaling in these two regimes is very different: in the free case the coefficients $m(x)$ and $b(x)$ have an inverse power law dependence on $x$ whereas in the chaotic case they decay exponentially. We conjecture that this result is universal by using catastrophe theory to show that, on the one hand, the wavefront in the free case has to take the form of an Airy function and its local expansion shows that the power law scaling seen in the numerics holds rigorously, and on the other hand an exponential scaling of the OTOC wavefront must be a signature of nonintegrable dynamics. We find that the crossover between the two regimes is smooth and characterized by an S-shaped curve giving the lifting of Airy nodes as a function of a chaos parameter. This shows that the Airy form is qualitatively stable against weak chaos and consistent with the concept of a quantum Kolmogorov-Arnold-Moser theory.