Scrambling versus relaxation in Fermi and non-Fermi liquids

We compute the Lyapunov exponent characterizing quantum scrambling in a family of generalized Sachdev-Ye-Kitaev models, which can be tuned between different low temperature states from Fermi liquids, through non-Fermi liquids to fast scramblers. The analytic calculation, controlled by a small coupling constant and large $N$, allows us to clarify the relations between the quasi-particle relaxation rate $1\mathrm{/}\tau$ and the Lyapunov exponent ${\lambda }_L$ characterizing scrambling. In the Fermi liquid states we find that the quasi-particle relaxation rate dictates the Lyapunov exponent. In non-Fermi liquids, where $1\mathrm{/}\tau \gg T$, we find that ${\lambda }_L$ is always $T$-linear with a prefactor that is independent of the coupling constant in the limit of weak coupling. Instead it is determined by a scaling exponent that characterizes the relaxation rate. ${\lambda }_L$ approaches the general upper bound $2\pi T$ at the transition to a fast scrambling state. Finally in a marginal Fermi liquid state the exponent is linear in temperature with a prefactor that vanishes as a non analytic function $\sim g\ln (1\mathrm{/}g)$ of the coupling constant $g$.