Instability of metals with respect to strong electron-phonon interaction

We show that thermal equilibrium between conduction electrons and phonons becomes kinetically unstable when the renormalized electron-phonon coupling exceeds a certain threshold. We prove that negative electronic specific heat, $C_{\mathrm{e}\mathrm{l}}<0$, is sufficient to trigger the instability. Specifically, the instability sets in as soon as the quasiparticle weight becomes negative over a range of energies, even before $C_{\mathrm{e}\mathrm{l}}$ turns negative. This is an inherently nonequilibrium phenomenon, occurring prior to the formation of any equilibrium phase. Depending on the system, it can proceed along different pathways, ultimately resulting in a structural transition to an insulating or metallic state.