# Chaos in the butterfly cone

## TL;DR

This paper extends the chaos bound in quantum many-body systems to include velocity-dependent Lyapunov exponents within the butterfly cone, revealing how chaos propagates and saturates in different models and regimes.

## Contribution

It introduces a velocity-dependent chaos bound, generalizes the Maldacena-Shenker-Stanford bound, and explores its implications in SYK models, holography, and conformal Regge theory.

## Key findings

- The velocity-dependent Lyapunov exponent is bounded by a linear function of velocity.
- In some models, the bound is saturated at a critical velocity below the butterfly speed.
- Boosting a system can enhance chaos, leading to saturation of the bound.

## Abstract

A simple probe of chaos and operator growth in many-body quantum systems is the out of time ordered four point function. In a large class of local systems, the effects of chaos in this correlator build up exponentially fast inside the so called butterfly cone. It has been previously observed that the growth of these effects is organized along rays and can be characterized by a velocity dependent Lyapunov exponent, $\lambda({\bf v})$. We show that this exponent is bounded inside the butterfly cone as $\lambda({\bf v})\leq 2\pi T(1-|{\bf v}|/v_B)$, where $T$ is the temperature and $v_B$ is the butterfly speed. This result generalizes the chaos bound of Maldacena, Shenker and Stanford. We study $\lambda({\bf v})$ in some examples such as two dimensional SYK models and holographic gauge theories, and observe that in these systems the bound gets saturated at some critical velocity $v_*<v_B$. In this sense, boosting a system enhances chaos. We discuss the connection to conformal Regge theory, where $\lambda({\bf v})$ is related to the spin of the leading large $N$ Regge trajectory, and controls the four point function in an interpolating regime between the Regge and the light cone limit. Finally, we comment on the generalization of the chaos bound to boosted and rotating ensembles and clarify some recent results on this in the literature.