Many-body chaos in the antiferromagnetic quantum critical metal

We compute the scrambling rate at the antiferromagnetic (AFM) quantum critical point, using the fixed point theory of Phys. Rev. X $\boldsymbol{7}$, 021010 (2017). At this strongly coupled fixed point, there is an emergent control parameter $w \ll 1$ that is a ratio of natural parameters of the theory. The strong coupling is unequally felt by the two degrees of freedom: the bosonic AFM collective mode is heavily dressed by interactions with the electrons, while the electron is only marginally renormalized. We find that the scrambling rates act as a measure of the "degree of integrability" of each sector of the theory: the Lyapunov exponent for the boson $\lambda_L^{(B)} \sim \mathcal O(\sqrt{w}) \,k_B T/\hbar$ is significantly larger than the fermion one $\lambda_L^{(F)} \sim \mathcal O(w^2) \,k_B T/\hbar$, where $T$ is the temperature. Although the interaction strength in the theory is of order unity, the larger Lyapunov exponent is still parametrically smaller than the universal upper bound of $\lambda_L=2\pi k_B T/\hbar$. We also compute the spatial spread of chaos by the boson operator, whose low-energy propagator is highly non-local. We find that this non-locality leads to a scrambled region that grows exponentially fast, giving an infinite "butterfly velocity" of the chaos front, a result that has also been found in lattice models with long-range interactions.