The electron-phonon problem reconsidered

The phase diagram of the electron-phonon (e-p) problem, as a function of e-p coupling and temperature, is presented. We work with the Holstein model and present determinant quantum Monte Carlo results for both large and infinite ion mass. In weak-coupling the system is a Fermi-liquid (and ultimately a “low temperature” superconductor), while beyond a critical coupling strength there is a transition to a (π,π) charge-density wave state. At elevated temperatures we find a cross-over to a pseudo-gap regime, corresponding to the formation of bound pairs of electrons with large effective mass (bipolarons). We compare MC results both with predictions from Migdal theory and strong-coupling expansions. For sufficiently weak coupling the Migdal approximation works extremely well but breaks down catastrophically upon entering the pseudo-gap regime, where the physics is accurately described by an Ising lattice gas model, as expected from the strong-coupling expansion.