Approximation des phases aleatoires self-consistante. Applications a des systemes de fermions fortement correles

Abstract : In the present thesis we have applied the self consistent RPA (SCRPA) to the Hubbard model with a small number of sites (a chain of 2, 4, 6, ... sites). Earlier SCRPA had produced very good results in other models like the pairing model of Richardson. It was therefore interesting to see what kind of results the method is able to produce in the case of a more complex model like the Hubbard model. To our great satisfaction the case of two sites with two electrons (half-filling) is solved exactly by the SCRPA. This may seem a little trivial but the fact is that other respectable approximations like "GW" or the approach with the Gutzwiller wave function yield results still far from exact. With this promising starting point, the case of 6 sites at half willing was considered next. For that case, evidently, SCRPA does not any longer give exact results. However, they are still excellent for a wide range of values of the coupling constant U, covering for instance the phase transition region towards a state with non zero magnetisation. We consider this as a good success of the theory. Non the less the case of 4 sites (a plaquette), as indeed all cases with 4n sites at half filling, turned out to have a problem because of degeneracies at the Hartree-Fock level. A generalisation of the present method, including in addition to the pairs, quadruples of Fermions operators (called second RPA) is proposed to also include exactly the plaquette case in our approach. This is therfore a very interesting perspective of the present work.
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Mohsen Jemai. Approximation des phases aleatoires self-consistante. Applications a des systemes de fermions fortement correles. Physique mathématique [math-ph]. Université Paris Sud - Paris XI, 2004. Français. ⟨tel-00006530⟩

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