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

TL;DR

This paper develops a self-consistent RPA-like method based on the Dyson equation for many-body Green's functions, applied to the Hubbard model, yielding promising results for charge and spin susceptibilities.

Contribution

It introduces a fully self-consistent RPA approach satisfying key sum rules and applies it to the Hubbard model for the first time.

Findings

Accurately reproduces charge and spin susceptibilities at half filling

Matches exact results closely, apart from a prefactor

Provides analytical insights in the strong coupling limit

Abstract

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.