# Out-of-time-order Operators and the Butterfly Effect

## TL;DR

This paper investigates how out-of-time-order operators serve as diagnostics of quantum chaos, connecting classical Lyapunov growth to quantum sensitivity and analyzing quantum corrections near the scrambling time.

## Contribution

It introduces quantum mechanical formulations of OTO operators using phase space and spin expansions, clarifying their relation to classical chaos and quantum corrections.

## Key findings

- Leading term matches classical Lyapunov growth.
- Quantum corrections dominate near the scrambling time.
- OTOs can be expressed via propagators as a quantum divergence measure.

## Abstract

Out-of-time-order (OTO) operators have recently become popular diagnostics of quantum chaos in many-body systems. The usual way they are introduced is via a quantization of classical Lyapunov growth, which measures the divergence of classical trajectories in phase space due to the butterfly effect. However, it is not obvious how exactly they capture the sensitivity of a quantum system to its initial conditions beyond the classical limit. In this paper, we analyze sensitivity to initial conditions in the quantum regime by recasting OTO operators for many-body systems using various formulations of quantum mechanics. Notably, we utilize the Wigner phase space formulation to derive an $\hbar$-expansion of the OTO operator for spatial degrees of freedom, and a large spin $1/s$-expansion for spin degrees of freedom. We find in each case that the leading term is the Lyapunov growth for the classical limit of the system and argue that quantum corrections become dominant at around the scrambling time, which is also when we expect the OTO operator to saturate. We also express the OTO operator in terms of propagators and see from a different point of view how it is a quantum generalization of the divergence of classical trajectories.