Chaos in the butterfly cone

TL;DR

This work establishes a universal bound on the velocity-dependent Lyapunov exponent within the butterfly cone, λ({\bf v}) ≤ 2πT(1 − |{\bf v}|/v_B), extending the MSS chaos bound to local, ballistic spreading of operator growth. It analyzes how the bound is saturated in various large-N systems by a pole–saddle mechanism, deriving a critical velocity v_* < v_B at which maximal chaos sets in, and shows how stringy/gravity corrections and Regge theory determine the VDLE across regimes. The authors connect the VDLE to the leading Regge trajectory in conformal Regge theory, relate chaos to Regge data, and extend the framework to boosted and rotating ensembles, including the OTOC behavior in rotating BTZ black holes. Overall, the paper provides a coherent, model-spanning picture of how chaos propagates in space-time with velocity-dependent growth rates, tying together SYK-like models, holography, Regge theory, and rotating/boosted ensembles with precise bounds and saturation mechanisms.

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, $λ({\bf v})$. We show that this exponent is bounded inside the butterfly cone as $λ({\bf v})\leq 2πT(1-|{\bf v}|/v_B)$, 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 $λ({\bf v})$ 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 $λ({\bf v})$ 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.

Figures (15)

• Figure 1: Illustration of a typical velocity dependent Lyapunov exponent in strongly coupled theories. At small velocities, the function is quadratic, which corresponds to a diffusive spreading of operators. The VDLE saturates the bound \ref{['eq:VDLObound2']} from some finite, order one critical velocity $v_*$, giving a ballistic butterfly front with maximal Lyapunov exponent.
• Figure 2: The bound \ref{['BoundSummary']} for various values of $v_B^\pm$ and for $\beta=2\pi$.
• Figure 3: Definition of ${\bf v}_B^{\pm}({\bf v})$. The dashed half lines connect ${\bf v}$ with the origin, and their intersection with the shapes define ${\bf v}_B^{\pm}({\bf v})$. For the green shape only ${\bf v}_B^+({\bf v})$ is defined. For the orange shape ${\bf v}_B^{+}({\bf v})$ is the farther and ${\bf v}_B^{-}({\bf v})$ is the closer intersection.
• Figure 4: The upper bound \ref{['BoundSummary2']} for $\lambda({\bf v})$ for the two shapes considered in Fig. \ref{['fig:vBpmDef']} (and for $\beta=2\pi$).
• Figure 5: The generic mechanism for saturating the bounds \ref{['eq:VDLObound1']}-\ref{['eq:VDLObound2']} above some critical velocity. The growing contribution to the OTOC is given by an integral, that can be evaluated by saddle point for large $t$. The integration contour is drawn by a blue solid line. As we increase the velocity $v$, the saddle point shifts in the direction of the imaginary axis, and eventually it crosses a pole. After this, the contribution of the pole dominates the integral.