Onsager reaction terms for quantum many-body systems: Application to antiferromagnetic and superconducting order in the Hubbard model

We demonstrate that an expansion in powers of the strength of the interaction, of the free energy at fixed order parameter, can be used to generate and correct mean-field theories for interacting quantum many-body systems. The first two terms in the expansion generally yield ordinary Hartree-Fock mean-field theory and the next term gives an ‘‘Onsager reaction field’’ correction to Hartree-Fock theory. This method can be used to directly generate expansions for inverse susceptibilities. We illustrate the method for the one- and two-dimensional Hubbard model, for which we consider corrections to mean-field theories of antiferromagnetism for the repulsive-U half-filled case and superconductivity in the attractive-U case. These corrections give a quantitative account superior to that of the random-phase approximation (RPA) for the correlation energy at small and intermediate values of U. For susceptibilities, we recover from the first two terms in the expansion the usual RPA results, while higher-order terms give systematic corrections to the RPA susceptibilities. For the case of superconductivity in the repulsive-U Hubbard model, we show that the higher-order terms in the expansion must be considered to determine whether or not an instability exists. We find that there is no superconducting instability in the repulsive-U case, at least towards ordinary singlet or triplet pairing. We also find no evidence for a superconducting instability driven by a coexisting antiferromagnetic order.