Chaos and operator growth in 2d CFT

TL;DR

The paper shows that in a zero-temperature 2d large‑c CFT, evolution by Hermitian combinations of Virasoro generators drives the OTOC to decay exponentially with a Lyapunov exponent that saturates the proposed bound Λ_L=2α, linking operator growth to Krylov complexity. By identifying H_k as modular Hamiltonians for half-space and orbifold geometries, the authors connect this chaotic behavior to thermal dynamics within a vacuum state and derive the chaos bound using analyticity and KMS-like properties. Across vacuum-block and light-intermediate-state analyses, the bound saturation persists, with generalizations to higher Virasoro modes yielding Λ_L=2αk and Krylov growth K_V^k ∼ e^{2αk t}. The results point to a holographic interpretation with a black hole at inverse temperature β=π/α and motivate extensions to finite temperature and Floquet CFT contexts. Overall, the work provides a cohesive framework tying OTOCs, modular Hamiltonians, and operator growth bounds in 2d CFTs.

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 $2α$, which sets an upper bound on the Lyapunov exponent, $λ_L \leq 2α$. 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.