Chaos and operator growth in 2d CFT

TL;DR

This paper investigates the exponential decay of out-of-time-ordered correlators in a zero temperature 2D CFT under Virasoro evolution, demonstrating saturation of a bound on chaos and linking modular Hamiltonian dynamics to thermal behavior.

Contribution

It shows that the Virasoro generators form the modular Hamiltonian, leading to thermal dynamics at zero temperature and saturating the chaos bound in 2D CFTs.

Findings

OTOC decays exponentially with Lyapunov exponent saturating the bound

Virasoro generators form the modular Hamiltonian of the CFT

Thermal dynamics emerge from modular Hamiltonian evolution at zero temperature

Abstract

We study the out-of-time-ordered correlator (OTOC) in a zero temperature two dimensional conformal field theory (CFT) under evolution by a Liouvillian composed of the Virasoro generators. A bound was conjectured in arXiv:1812.08657 on the growth of the OTOC set by the Krylov complexity which is a measure of operator growth. The latter grows as an exponential of time with exponent $2\alpha$, which sets an upper bound on the Lyapunov exponent, $\lambda_L \leq 2\alpha$. We find that for a two dimensional zero temperature CFT, the OTOC decays exponentially with a Lyapunov exponent which saturates this bound. We show that these Virasoro generators form the modular Hamiltonian of the CFT with half space traced out. Therefore, evolution by this modular Hamiltonian gives rise to thermal dynamics in a zero temperature CFT. Leveraging the thermal dynamics of the system, we derive this bound in a zero temperature CFT using the analyticity and boundedness properties of the OTOC.