PHYSICAL REVIEW B VOLUME 43, NUMBER 4 1 FEBRUARY 1991 Qnsager reaction terms for quantum many-body systems: Application to antiferromagnetic and superconducting order in the Hubbard model Antoine Georges' and Jonathan S. Yedidia Department of Physics, Joseph Henry Laboratories, Jadwin Hall, Princeton University, Princeton, New Jersey 08544 (Received 20 August 1990) 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 suscepti- bilities. 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 supe- rior to that of the random-phase approximation (RPA) for the correlation energy at small and inter- mediate 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. I. INTRODUCTION In this paper we present a new and very simple method to derive mean-field theories and corrections to those theories for interacting quantum many-body systems which exhibit some kind of long-range order. The method is an expansion in powers of the strength of the interaction of the free energy 3 (I ) considered at a fixed order parameter m. The first two terms in the expansion will generally reproduce the ordinary Hartree-Fock ap- proximation, and higher-order terms can be used to derive corrections in a simple and systematic way. The technique can actually be applied to any Hamiltonian which can be decomposed into a part which can be diago- nalized plus a remainder. In this paper, we consider as an example the popular Hubbard model, ' and although the method is easily generalized to finite temperature T, we will restrict ourselves to T =0. Our method of deriving a mean-field theory and its corrections is very useful for at least two kinds of situa- tions. (i) For low-dimensional systems where fiuctuations are large. For example, the ordinary Hartree-Fock mean- field theory of antiferromagnetism in the Hubbard model predicts that half-filled one-dimensional chains will ex- hibit antiferromagnetic long-range order, in contradic- tion with known results. When we calculate the correc- tions to Hartree-Fock mean-field theory using our tech- nique, we find that, indeed, fluctuations destroy the long- range order in one dimension, while in two or more di- mensions, the corrections merely weaken the long-range order. (ii) When there is no sensible Hartree-Fock mean-field theory because (in our language) one of the first two terms of the expansion of the free energy does not depend on the order parameter (usually for some symmetry reason). Such a situation actually arises for the case of superconductivity in the repulsive- U Hubbard model. For these cases, it is crucial to consider the higher-order terms in the expansion in order to assess the possible ex- istence of the relevant order parameter. Our technique can also be used to calculate any desired susceptibilities. The fact that we work at a fixed order parameter means that we effectively resum infinitely many ordinary Feynman diagrams, as is demonstrated by the fact that the susceptibilities that we obtain from just the first two terms of our expansion are identical for the Hubbard model to the ordinary random-phase approxi- mation (RPA) susceptibilities. However, our technique has the advantage that it will work even for models for which an RPA resummation is impossible. In fact, it ac- tually provides a direct systematic expansion for inUerse susceptibilities. The outline of this paper is as follows. In Sec. II, we il- lustrate our method for the case of the half-filled, repulsive-U Hubbard model where antiferromagnetic or- der is expected in two or more dimensions. We use our corrections to the Hartree-Fock mean-field theory to cal- culate the "correlation energy" in this case. In Sec. III, we apply our technique to generate and correct mean- field theories of superconductivity in the attractive-U Hubbard model, while in Secs. IV, and V, we consider the 43 3475 1991 The American Physical Society