Quantum Chaos for the Unitary Fermi Gas from the Generalized Boltzmann Equations

In this paper, we study the chaotic behavior of the unitary Fermi gas in both high and low temperature limits by calculating the Quantum Lyapunov exponent defined in terms of the out-of-time-order correlator. We take the method of generalized Boltzmann equations derived from the augmented Keldysh approach \cite{augKeldysh}. At high temperature, the system is described by weakly interacting fermions with two spin components and the Lyapunov exponent is found to be $\lambda_L=21\frac{n}{T^{1/2}}$. Here $n$ is the density of fermions for a single spin component. In the low temperature limit, the system is a superfluid and can be described by phonon modes. Using the effective action derived in \cite{Son}, we find $\lambda_L=9\times 10^3\left(\frac{T}{T_F}\right)^4T$ where $T_F$ is the Fermi energy. By comparing these to existing results of heat conductivity, we find that $D_E\ll v^2 /\lambda_L$ where $D_E$ is the energy diffusion constant and $v$ is some typical velocity. We argue that this is related to the conservation law for such systems with quasi-particles.