Dyson equation approach to many-body Green's functions and self-consistent RPA: Application to the Hubbard model

An approach for particle-hole correlation functions, based on the so-called self-consistent random-phase approximation is developed. This leads to a fully self-consistent RPA-like theory that 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 behavior apart from a general prefactor. The strong-coupling limit of our approach can be described analytically.