Band to Mott transition in the infinite dimensional Holstein model

down at intermediate couplings in the limit of low phonon frequency where it is supposed to be most applicable. By examining the effective hopping and renormalized phonon frequency, I suggest an intuitive explanation for the breakdown of Migdal-Eliashberg theory. For completeness, a description of the breakdown of Migdal's theorem in the limit of high phonon frequency is considered via the Lang-Firsov canonical transformation. The Holstein model can be seen as providing an interpolation scheme between Hubbard-like and Falicov-Kimball like behaviour and it may therefore help elucidate the nature of the transition between band and Mott states. I describe a number of approximations to the self-energy. By comparing these with results from the exact solution in various limits, I find that a second order iterated perturbation scheme may be capable of describing the intermediate behaviour. Application of this method reveals a strongly renormalized hybrid behaviour between band and miniature-Mott states, with unusual consequences for the response functions. In recent years, the study of correlated electron systems has become particularly intense as a result of discoveries such as high-T sub C - superconductivity and colossal magnetoresistance in the manganites. This thesis is a study of the Holstein model, which represents the interaction between electrons and Einstein phonons. It may exhibit metallic, insulating, superconducting and charge ordered phases. It also has very different limiting behaviours. In one limit it can be mapped onto a Hubbard model which describes correlated behaviour and in the other onto a Falicov-Kimball model which describes a band system. This thesis describes the dynamical properties of the Holstein model, including spectral functions, resistivity curves and the quasi-particle effective mass for a wide range of electron-phonon coupling strengths and phonon frequencies. I use dynamical-mean field theory (DMFT), which is exact in the limit of infinite-dimensions and can also be thought of as an approximation to 3D systems. Even with the simplifications which arise in DMFT, further approximations are necessary. I apply Migdal-Eliashberg theory (which has been extremely successful at describing real systems because the phonon frequency is normally much lower than the Fermi-energy) within the DMFT framework to calculate spectral functions of the Holstein model. I also examine the first-order vertex correction, to determine the line of breakdown for ME theory, finding that this approximation may brea