Quantum chaos in an electron-phonon bad metal

We calculate the scrambling rate $\lambda_L$ and the butterfly velocity $v_B$ associated with the growth of quantum chaos for a solvable large-$N$ electron-phonon system. We study a temperature regime in which the electrical resistivity of this system exceeds the Mott-Ioffe-Regel limit and increases linearly with temperature - a sign that there are no long-lived charged quasiparticles - although the phonons remain well-defined quasiparticles. The long-lived phonons determine $\lambda_L$, rendering it parametrically smaller than the theoretical upper-bound $\lambda_L \ll \lambda_{max}=2\pi T/\hbar$. Significantly, the chaos properties seem to be intrinsic - $\lambda_L$ and $v_B$ are the same for electronic and phononic operators. We consider two models - one in which the phonons are dispersive, and one in which they are dispersionless. In either case, we find that $\lambda_L$ is proportional to the inverse phonon lifetime, and $v_B$ is proportional to the effective phonon velocity. The thermal and chaos diffusion constants, $D_E$ and $D_L\equiv v_B^2/\lambda_L$, are always comparable, $D_E \sim D_L$. In the dispersive phonon case, the charge diffusion constant $D_C$ satisfies $D_L\gg D_C$, while in the dispersionless case $D_L \ll D_C$.