Functional-derivative study of the Hubbard model. II. Self-consistent equation and its complete solution

We develop a self-consistent method to solve the basic equation for the self-energy correction of the Hubbard model obtained in the preceding paper. The term π[Δ] involving second functional derivatives is neglected and the quantities NRσ(t) and CRσ†(t)CR′′σ(t) are initially assumed to be independent of the external fields ε(σ) and ε(σ̄). Under these restrictions, the complete self-energy correction is shown to be expanded in powers of ε(σ) and ε(σ̄) in the form RR′σ(tt′)=ξ0(RR′σ̄t)δtt′+n= 0R′′R′′′ξ(n) (RR′;R′′R′′′) ε(R′′R′′′σt)δtt′, where ξ0 consists of all possible terms linear in ε(σ̄), while ξ(n) is made up of all possible terms of the nth degree in ε(σ̄). Equations for ξ0 and ξ(n) are solved exactly and the resulting series is summed analytically, yielding a compact and complete analytic solution for the restricted equation. The part which is linear in ε is shown to be equal to the perturbation result obtained in the preceding paper, confirming the claim that the perturbation result is exact through terms linear in ε. The method is extended and the effect of δNδε and δC†Cδε is included. The effect is found to eliminate the difficulty that the value of one of the terms in the self-energy correction is abnormally overestimated in the previous result in the split-band, half-filled limit.