Two-particle Irreducible Effective Action Approach to Correlated   Electron Systems
  The two-particle irreducible (2PI) effective action theories are employed to study the strongly fluctuating electron systems, under the formalism of the two-dimensional Hubbard model. We obtain the corresponding quantum 2PI effective action after the original classic action of the Hubbard model is bosonized. In our actual calculations, the 2PI effective action is expanded to three loops, in which the leading order (LO) and next-to-leading order (NLO) quantum fluctuations are included. Numerical calculations indicate that the NLO fluctuations should not be neglected when the Coulomb on-site repulsion energy is larger than two times the nearest-neighbor hopping energy. 
Wei-Jie Fu
cond-mat.str-el hep-ph