The transition between a metallic and a pseudogap phase in high-temperature cuprate superconductors is the subject of experimental investigations but has not been settled theoretically, even within the context of the Hubbard model. We apply the Cluster Dynamical Impurity Approximation (CDIA)  to the anisotropic Hubbard model on the square lattice at zero temperature and finite doping. This approach can detect a first-order transition between two metallic states: a pseudogap state at low doping, with a depleted density of states at the Fermi level, and a correlated metal at higher doping.
The purpose of this talk will be to argue, with numerical evidence, for the existence of a first-order phase transition in the two-dimensional, anisotropic Hubbard model, between a metallic phase and a pseudogap phase as doping is reduced towards half-filling. The nature of the pseudogap state is an important and still unresolved issue in the physics of high-temperature superconductors.
 David S\'ene\'echal,“Bath optimization in the cellular dynamical mean-field theory,” Phys. Rev. B, 81, 235125 (2010).