# Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?

## TL;DR

This paper investigates how the shape of the complex time contour affects the out-of-time-ordered correlation function (OTOC) and its Lyapunov spectrum in various theories, questioning which spectrum reflects true physical chaos.

## Contribution

It demonstrates contour dependence of the Lyapunov spectrum in weakly coupled theories and identifies conditions under which the spectrum is contour-independent, clarifying the physical relevance.

## Key findings

- Lyapunov spectrum depends on contour shape in weakly coupled theories.
- In SYK, the Lyapunov spectrum is contour-independent despite contour dependence of the full OTOC.
- Symmetric contour configuration relates to the physical chaos bound and experimental measures.

## Abstract

We study the contour dependence of the out-of-time-ordered correlation function (OTOC) both in weakly coupled field theory and in the Sachdev-Ye-Kitaev (SYK) model. We show that its value, including its Lyapunov spectrum, depends sensitively on the shape of the complex time contour in generic weakly coupled field theories. For gapless theories with no thermal mass, such as SYK, the Lyapunov spectrum turns out to be an exception; their Lyapunov spectra do not exhibit contour dependence, though the full OTOCs do. Our result puts into question which of the Lyapunov exponents computed from the exponential growth of the OTOC reflects the actual physical dynamics of the system. We argue that, in a weakly coupled $\Phi^4$ theory, a kinetic theory argument indicates that the symmetric configuration of the time contour, namely the one for which the bound on chaos has been proven, has a proper interpretation in terms of dynamical chaos. Finally, we point out that a relation between these OTOCs and a quantity which may be measured experimentally --- the Loschmidt echo --- also suggests a symmetric contour configuration, with the subtlety that the inverse periodicity in Euclidean time is half the physical temperature. In this interpretation the chaos bound reads $\lambda \leq \frac{2\pi}{\beta}= \pi T_{\text{physical}}$.