Scrambling versus relaxation in Fermi and non-Fermi liquids

TL;DR

This paper analytically investigates quantum scrambling in generalized Sachdev-Ye-Kitaev models, revealing how the Lyapunov exponent varies across Fermi liquids, non-Fermi liquids, and fast scramblers, and clarifying its relation to quasiparticle relaxation.

Contribution

It provides a detailed analytic study of the Lyapunov exponent in models transitioning between different quantum states, linking scrambling to quasiparticle relaxation rates.

Findings

Lyapunov exponent in Fermi liquids is dictated by quasiparticle relaxation rate.

In non-Fermi liquids, the Lyapunov exponent is linear in temperature, independent of coupling in weak coupling limit.

The Lyapunov exponent approaches the upper bound at the transition to fast scrambling.

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 quasi-particle relaxation rate $1/\tau$ 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 $1/\tau \gg T$, 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 $\sim g \ln (1/g)$ of the coupling constant $g$.