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

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) approximate to 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 approximate to 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.