Functional-derivative study of the Hubbard model. III. Fully renormalized Green's function

The functional-derivative method of calculating the Green's function developed earlier for the Hubbard model is generalized and used to obtain a fully renormalized solution. Higher-order functional derivatives operating on the basic Green's functions, $G$ and $\Gamma$, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms $\Phi$ generated from functional derivatives of equal-time Green's functions of the type $\frac{{\delta }^n〈N〉}{\delta {\epsilon }^n}$, etc., with $n\ge 2$. It is found that the $\Phi \text{'}\mathrm{s}$ are, in fact, renormalization factors involved in the self-energy $\Sigma$ and that the structure of the $\Phi \text{'}\mathrm{s}$ resembles that of $\Sigma$ and contains the same renormalization factors $\Phi$. The renormalization factors $\Phi$ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the $\Phi \text{'}\mathrm{s}$, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum $\omega$ can now be evaluated for all occupations $n$. The Schwinger relation is the only basic relation used in generating this fully self-consistent Green's function, and the Baym-Kadanoff continuity condition is automatically satisfied.