  To gain insight into the physics of the metal insulator transition and the
effectiveness of cluster dynamical mean field theory (DMFT) we have used one,
two and four site dynamical mean field theory to solve a polaron model of
electrons coupled to a classical phonon field. The cluster size dependence of
the metal to polaronic insulator phase boundary is determined along with
electron spectral functions and cluster correlation functions. Pronounced
cluster size effects start to occur in the intermediate coupling region in
which the cluster calculation leads to a gap and the single-site approximation
does not. Differences (in particular a sharper band edge) persist in the strong
coupling regime. A partial density of states is defined encoding a generalized
nesting property of the band structure; variations in this density of states
account for differences between the dynamical cluster approximation and the
cellular-DMFT implementations of cluster DMFT, and for differences in behavior
between the single band models appropriate for cuprates and the multiband
models appropriate for manganites. A pole or strong resonance in the self
energy is associated with insulating states; the momentum dependence of the
pole is found to distinguish between Slater-like and Mott-like mechanisms for
metal insulator transition. Implications for the theoretical treatment of doped
manganites are discussed.
