First order metal-insulator transision in band overlap mechanism

We present a method to analyze the metal-insulator transition (MIT) due to the band overlap mechanism. It is based on a model with the knowledge of the homogeneous electron gas, combined with results based on the quasiparticle self-consistent $GW$ method. Because of the long-range nature of the Coulomb interaction, the MIT occurs as the first-order phase transition, that is, the band gap becomes negative (band overlap) suddenly at some critical lattice constant.