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

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.