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

Abstract

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 $\ensuremath{\pi}[\ensuremath{\Delta}]$ involving second functional derivatives is neglected and the quantities $〈{N}_{R\ensuremath{\sigma}}(t)〉$ and $〈{C}_{R\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}(t){C}_{{R}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\sigma}}(t)〉$ are initially assumed to be independent of the external fields $\ensuremath{\epsilon}(\ensuremath{\sigma})$ and $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$. Under these restrictions, the complete self-energy correction is shown to be expanded in powers of $\ensuremath{\epsilon}(\ensuremath{\sigma})$ and $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$ in the form $\ensuremath{\Sigma}{R{R}^{\ensuremath{'}}\ensuremath{\sigma}}^{}(t{t}^{\ensuremath{'}})={\ensuremath{\xi}}_{0}(R{R}^{\ensuremath{'}}\overline{\ensuremath{\sigma}}t){\ensuremath{\delta}}_{t{t}^{\ensuremath{'}}}+\ensuremath{\Sigma}{n=0}^{}\ensuremath{\Sigma}{{R}^{\ensuremath{'}\ensuremath{'}}}^{}\ensuremath{\Sigma}{{R}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}}^{}{\ensuremath{\xi}}^{(n)}(R{R}^{\ensuremath{'}};{R}^{\ensuremath{'}\ensuremath{'}}{R}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})\ensuremath{\epsilon}({R}^{\ensuremath{'}\ensuremath{'}}{R}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}\ensuremath{\sigma}t){\ensuremath{\delta}}_{t{t}^{\ensuremath{'}}}$, where ${\ensuremath{\xi}}_{0}$ consists of all possible terms linear in $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$, while ${\ensuremath{\xi}}^{(n)}$ is made up of all possible terms of the $n$th degree in $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$. Equations for ${\ensuremath{\xi}}_{0}$ and ${\ensuremath{\xi}}^{(n)}$ 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 $\ensuremath{\epsilon}$ 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 $\ensuremath{\epsilon}$. The method is extended and the effect of $\frac{\ensuremath{\delta}〈N〉}{\ensuremath{\delta}\ensuremath{\epsilon}}$ and $\frac{\ensuremath{\delta}〈{C}^{\ifmmode\dagger\else\textdagger\fi{}}C〉}{\ensuremath{\delta}\ensuremath{\epsilon}}$ 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.