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 ${\lambda }_L\approx 3.2T/N$. 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.

• Received 21 March 2017

DOI:https://doi.org/10.1103/PhysRevD.96.065005