Many-body chaos in the antiferromagnetic quantum critical metal

Recently, the scrambling rate as defined from the exponential growth of regularized out-of-time-ordered correlators has been used as a measure of integrability in quantum many-body systems. We compute the scrambling rate at the antiferromagnetic (AFM) quantum critical point, using the fixed point theory of Phys. Rev. X 7, 021010 (2017). At this strongly coupled fixed point, there is an emergent small control parameter w 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. The scrambling rates of both degrees of freedom are linear in temperature (up to logarithms), but come with very different powers of w, indicating the different "degrees of integrability" of the two sectors of the theory. Although the interaction strength is of order unity, the larger Lyapunov exponent is still parametrically smaller than the universal upper bound. We also show that due to the non-local nature of the boson propagator, its effective "butterfly velocity" of the chaos front is infinite.