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(v). We show that this exponent is bounded inside the butterfly cone as λ(v) ≤ 2πT (1 − |v|/vB), where T is the temperature and vB is the butterfly speed. This result generalizes the chaos bound of Maldacena, Shenker and Stanford. We study λ(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*< vB. In this sense, boosting a system enhances chaos. We discuss the connection to conformal Regge theory, where λ(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.
Highlights
Ones carry the sensitivity to chaos, in particular, they decay to zero after a characteristic scrambling time tscr in chaotic systems
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(v)
We show that this exponent is bounded inside the butterfly cone as λ(v) ≤ 2πT (1 − |v|/vB), where T is the temperature and vB is the butterfly speed
Summary
This function satisfies 1 and 2 when V and W are Hermitian operators and the Hamiltonian is bounded from below. In order to have 3, one needs to assume that time ordered correlation functions factorize for times larger than the local thermalization time td ∼ β, and that OTOCs approximately factorize around a time t0 that is larger than the local thermalization time, but much shorter than the scrambling time, td < t0 ts. These assumptions are automatic in a large N theory, or other theories with classical limits, but they are expected to hold more generally for chaotic theories with many local degrees of freedom. Condition 3 follows from these factorization assumptions by applying the Cauchy-Schwarz inequality (which requires unitarity) and the maximum modulus principle inside the strip
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