# Onset of many-body chaos in the $O(N)$ model

## TL;DR

This paper calculates how chaos emerges in the 2+1 dimensional $O(N)$ model at finite temperature, showing exponential growth of operator commutators with a temperature-dependent rate and ballistic operator spreading.

## Contribution

It provides the first leading-order calculation of chaos growth rate and butterfly velocity in the $O(N)$ model using $1/N$ expansion at finite temperature.

## Key findings

- Chaos growth rate $oxed{ ext{~} rac{3.2 T}{N}}$
- Butterfly velocity $v_B/c oxed{ ext{~} 1}$
- Growth of chaos is slow at large $N$ due to weak interactions.

## Abstract

The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with $N$ components in the $(2+1)$-dimensional $O(N)$ nonlinear sigma model to leading order in $1/N$. The system is taken to be in thermal equilibrium at a temperature $T$ above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted $\lambda_L$. At large $N$ the growth of chaos as measured by $\lambda_L$ is slow because the model is weakly interacting, and we find $\lambda_L \approx 3.2 T/N$. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a "butterfly velocity" given by $v_B/c \approx 1$ where $c$ is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of $\lambda_L$ and $v_B$ in the neighboring symmetry broken and unbroken phases.