Non-perturbative treatment of the attractive Hubbard model

Abstract

We have used a Hubbard–Stratonovich transformation to convert the quartic Hubbard problem of interacting electrons into more tractable quadratic problem of non-interacting electrons coupled to a Bose field. The Dyson and the Bethe-Salpeter (BS) equations for the corresponding Green functions are derived by applying a functional derivative technique combined with the method of Legendre transform. This approach allows us to obtain exact relations between single- and two-particle quantities. We have compared the self-consistent relationship between mass operator and two-particle Green function with the similar relationship in the two-particle self-consistent (TPSC) approximation, and came up with the conclusion that the accuracy of the TPSC approximation is questionable because: (i) if equal-time pair-correlation function is set to one, the mass operator in TPSC approximation differs significantly from the exact mass operator; and (ii) the TPSC approximation does not take into account the Hartree part of the mass operator. The latter results in missing exchange interaction in the BS equation. A generalized random-phase approximation is applied to demonstrate the important role of the exchange interaction when BS equation is used to study collective excitation spectrum of the attractive Hubbard model.