Dyson Equation Approach to Many-Body Greens Functions and Self-Consistent RPA, First Application to the Hubbard Model

An approach for particle-hole correlation functions, based on the so-called SCRPA, is developed. This leads to a fully self-consistent RPA-like theory which satisfies the $f$-sum rule and several other theorems. As a first step, a simpler self-consistent approach, the renormalized RPA, is solved numerically in the one-dimensional Hubbard model. The charge and the longitudinal spin susceptibility, the momentum distribution and several ground state properties are calculated and compared with the exact results. Especially at half filling, our approach provides quite promising results and matches the exact behaviour apart from a general prefactor. The strong coupling limit of our approach can be described analytically.