Quantum Chaos of the Bose-Fermi Kondo model at the intermediate temperature

TL;DR

This paper investigates quantum chaos in the Bose-Fermi Kondo model at intermediate temperatures, revealing how the Lyapunov exponent varies with coupling parameters and challenging previous assumptions about chaos at quantum critical points.

Contribution

It provides the first detailed calculation of the Lyapunov exponent in the Bose-Fermi Kondo model at intermediate temperatures, showing its dependence on coupling strengths.

Findings

Lyapunov exponent increases with Kondo coupling J_K.

Lyapunov exponent decreases with impurity-bosonic bath coupling g.

Chaos is not maximized at the quantum critical point.

Abstract

We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large $N$ limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent $\lambda_L$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling $J_K$ increases, and it can reach an order of $\lambda_L\sim T$ as $J_K$ approaches the $MCK$ point. Furthermore, we also demonstrate that $\lambda_L$ decreases monotonously as the impurity and bosonic bath coupling $g$ increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi liquid nature.