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

Abstract

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 $\ensuremath{\Gamma}$, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms $\ensuremath{\Phi}$ generated from functional derivatives of equal-time Green's functions of the type $\frac{{\ensuremath{\delta}}^{n}〈N〉}{\ensuremath{\delta}{\ensuremath{\epsilon}}^{n}}$, etc., with $n\ensuremath{\ge}2$. It is found that the $\ensuremath{\Phi}'\mathrm{s}$ are, in fact, renormalization factors involved in the self-energy $\ensuremath{\Sigma}$ and that the structure of the $\ensuremath{\Phi}'\mathrm{s}$ resembles that of $\ensuremath{\Sigma}$ and contains the same renormalization factors $\ensuremath{\Phi}$. The renormalization factors $\ensuremath{\Phi}$ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the $\ensuremath{\Phi}'\mathrm{s}$, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum $\ensuremath{\omega}$ 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.