Stability and Relaxation of Electron-Phonon Systems at Strong Coupling

Abstract

We study the kinetic properties of Migdal-Eliashberg theory (MET) for electron-phonon interactions at strong coupling. Previously, the system was found to be dynamically unstable at large couplings within a Markovian quantum Boltzmann approximation. Using the quasiclassical Kadanoff-Baym equation, we show that this instability is absent when non-Markovian effects are taken into account. We also analyze the long-time relaxation dynamics at strong-couplin, and find a clear imprint of the non-Markovian nature of the dynamics: in an intermediate temperature range between the Debye frequency and the coupling strength, the relaxation rate becomes linear in temperature with Planckian slope, while the leading relaxation mode acquires a large imaginary component, producing strongly oscillatory decay. Our results imply that any instability which preempts superconductivity must arise from effects beyond MET, such as phonon softening or polaronic effects at finite electronic bandwidth. This shows that the characteristic phonon frequency is not a fundamental bound on $T_c$, and leaves open a route to higher $T_c$ in multi-orbital electron-phonon systems.