Abstract : The so-called Self-Consistent RPA (SCRPA) is applied to rticle-hole
correlation functions in the Hubbard model.
For a general many-body Green's function, this method is obtained by establishing a Dyson Equation, and by replacing the full mass operator by its instantaneous contributions. The Green's function is then given in terms of a system of non-linear integral equations which can be solved self-consistently. The solution satisfies the energy-weighted sum rule and several other theorems. For single- and two-particle Green's functions, the approach is shown to be connected to a variational principle.
The charge and longitudinal spin correlation functions in the Hubbard model are treated in SCRPA. Neglecting the connected two-body densities in its mass operator yields another, simpler self-consistent theory, the renormalized RPA. Both approaches are discussed and compared to standard RPA.
In the one-dimensional Hubbard model, the renormalized RPA is
established and solved numerically for charge-density correlation
functions. The charge and the longitudinal spin susceptibility, the
momentum distribution function and several ground state properties are calculated and compared to the exact results. At half band filling and for strong interactions, the renormalized RPA has an analytic solution which agrees, apart from a prefactor, with the corresponding series expansion of the Bethe ansatz solution. As expected, specific one-dimensional features, such as Luttinger liquid behaviour, could not be reproduced. Our approach provides a rather generic description, which could be quite realistic in higher dimensions.
Parts of this work can be found in
"Dyson Equation Approach to Many-Body Greens Functions and
Self-Consistent RPA, First Application to the Hubbard Model"
Steffen Schäfer, Peter Schuck, Phys. Rev. B 59, 1712-1733 (1999).