Role of Phonons and Band Structure in Metal-Insulator Phase Transition

We present a soluble model of electrons interacting with lattice vibrations which displays metallic, superconducting, and insulating phases. We solve the thermodynamics of this model and discuss its properties. The metal-insulator phase transition is second order for sc and bcc tight-binding bands. With sufficient deviation [(5-10)% for coupling constant $\tilde{g}=0.1\mathrm{}$] from such band structure the transition becomes first order. Above this, the "collective insulator" ceases to exist. Our model might apply to certain transition-series metal oxides, such as V${\mathrm{O}}_2$.