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 $\pi \left[\Delta \right]$ involving second functional derivatives is neglected and the quantities $〈N_{R\sigma }\mathrm{}\left(t\right)\mathrm{}〉$ and $〈C_{R\sigma }^{†}\mathrm{}\left(t\right)\mathrm{}{C}_{R^{\prime \prime }\sigma }\mathrm{}\left(t\right)\mathrm{}〉$ are initially assumed to be independent of the external fields $\epsilon \left(\sigma \right)$ and $\epsilon \left(\overline{\sigma }\right)$. Under these restrictions, the complete self-energy correction is shown to be expanded in powers of $\epsilon \left(\sigma \right)$ and $\epsilon \left(\overline{\sigma }\right)$ in the form $\Sigma {R{R}^{\prime }\sigma }^{}\left(t{t}^{\prime }\right)={\xi }_0\left(R{R}^{\prime }\overline{\sigma }t\right){\delta }_{t{t}^{\prime }}+\Sigma {n=0\mathrm{}}^{}\Sigma {R^{\prime \prime }}^{}\Sigma {R^{\prime \prime \prime }}^{}{\xi }^{\mathrm{}\left(n\right)\mathrm{}}(R{R}^{\prime };R^{\prime \prime }{R}^{\prime \prime \prime })\epsilon \left(R^{\prime \prime }{R}^{\prime \prime \prime }\sigma t\right){\delta }_{t{t}^{\prime }}$, where ${\xi }_0$ consists of all possible terms linear in $\epsilon \left(\overline{\sigma }\right)$, while ${\xi }^{\mathrm{}\left(n\right)\mathrm{}}$ is made up of all possible terms of the $n$th degree in $\epsilon \left(\overline{\sigma }\right)$. Equations for ${\xi }_0$ and ${\xi }^{\mathrm{}\left(n\right)\mathrm{}}$ 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 $\epsilon$ 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 $\epsilon$. The method is extended and the effect of $\frac{\delta 〈N〉}{\delta \epsilon }$ and $\frac{\delta 〈C^{†}C〉}{\delta \epsilon }$ 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.