Scrambling versus relaxation in Fermi and non-Fermi liquids

Abstract

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 quasiparticle relaxation rate $1/\ensuremath{\tau}$ and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ characterizing scrambling. In the Fermi liquid states we find that the quasiparticle relaxation rate dictates the Lyapunov exponent. In non-Fermi liquids, where $1/\ensuremath{\tau}\ensuremath{\gg}T$, we find that ${\ensuremath{\lambda}}_{L}$ 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. ${\ensuremath{\lambda}}_{L}$ approaches the general upper bound $2\ensuremath{\pi}T$ 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 nonanalytic function $\ensuremath{\sim}gln(1/g)$ of the coupling constant $g$.