Many-body chaos in the antiferromagnetic quantum critical metal

Abstract

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 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 ${\lambda }_L^{(B)}\sim \mathcal{O}(\sqrt{w})k_{B}T/\hslash$ is significantly larger than the fermion one ${\lambda }_L^{(F)}\sim \mathcal{O}(w^2)k_{B}T/\hslash$, 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 ${\lambda }_L=2\pi k_{B}T/\hslash$. 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.

arXiv comment10+6 pages, 11 figures

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