Instability due to long-range Coulomb interaction in a liquid of Feynman polarons

The interaction Hamiltonian for a dilute classical liquid of polarons à la Feynman in the presence of long-range Coulomb interaction is derived and the dielectric function is computed in mean field. Above a critical concentration the liquid becomes unstable. The onset of the instability is signaled by the softening of a collective optical mode in which all electrons oscillate in phase in their respective self-trapping potential. We associate the instability with a metalization of the system. Optical experiments in slightly doped cuprates and doped nickelates are analyzed within this theory. We discuss why doped cuprates metalize whereas nickelates do not.