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 are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms generated from functional derivatives of equal-time Green's functions of the type nNn, etc., with n2. It is found that the 's are, in fact, renormalization factors involved in the self-energy and that the structure of the 's resembles that of and contains the same renormalization factors. The renormalization factors are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the 's, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum 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.