Quantum Chaos in an Electron-Phonon Bad Metal

We calculate the scrambling rate and the butterfly velocity 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 the scrambling rate, rendering it parametrically smaller than the theoretical upper-bound. Significantly, the chaos properties seem to be intrinsic - and are the same for electronic and phononic operators. We consider both dispersive and non-dispersive phonons. In either case, we find that the scrmabling rate is proportional to the inverse phonon lifetime, and the butterfly velocity is proportional to the effective phonon velocity. The thermal and chaos diffusion constants are always comparable, while the charge diffusion constant may be either larger or smaller than the chaos diffusion constant.