The t-t'-U model and the cuprate materials

We have studied the two-dimensional model, by means of the Composite Operator Method to analyze the possibility to handle the complexity of the experimental situation for the cuprates. Using relations containing the Pauli principle, we have been able to fix the dynamics in a fully self-consistent way Furthermore, the recovery of the Pauli principle has assured us to satisfy the relations coming from the particle-hole transformation. To check our solution we have compared our results for the local quantities with the ones coming from numerical schemes with a very good agreement. We have computed the structure of the energy bands, the shape of the Fermi surface and the relative position of the van Hove singularity. We have shown that the parameter is essential in modifying features of the Fermi surface and the density of states. The occurrence of such behaviors might give some insight in the comprehension of the mechanism that differentiates between various cuprates. The comparison with experimental data has shown that the Hubbard model, by varying the value of the parameter, is capable to describe both ( , ) and ( , ), that share the property to be 1-layer cuprates. Actually, by the same choice of parameters, it is possible to reproduce both the shape of the Fermi surface and the relative position of the van Hove singularity. The same is not true for , where we found that, in the context of the model, there is not an unique choice of parameters which could reproduce both the shape of the Fermi surface and the band structure. This result might suggest weakness of the model in describing two-layer compounds. This can be read as a clear signal that two-dimensional Hubbard-like model can play an important role in describing the physics of the 1-layer cuprates superconductors, but that the multi-layer one need some more complex models. We have also shown that the model presents an incommensurate phase at finite doping, that disappears by increasing temperature, and that the magnetic scattering become isotropic as is increased [31]. The occurrence of such an evolution of the magnetic fluctuations can be related to the spreading of the nesting vector in the momentum space. That is, the evolution in the dynamical response of a spatially uniform electron liquid from pseudo-nested to a roughly circular hole-like Fermi surface. Indeed, a scenario emerges where the shape of the Fermi surface and the incommensurate spin fluctuations are connected showing that ARPES and neutron scattering experiments are intimately related. We also provided a fully self-consistent treatment of the paramagnetic and the antiferromagnetic phases. Near half filling the critical value of the Coulomb repulsion as function of and the temperature dependence of the magnetization and internal energy have been studied and analyzed in relation with the ones obtained by the renormalization group [36, 37]