Generalized self-consistent-field theory: Gor'kov factorization

Gor'kov factorization, ψ†(1)ψ†(2)ψ(3)ψ(4)→ψ†(1) ψ†(2)ψ(3)ψ(4), is made the basis of a generalized self-consistent-field (SCF) theory precisely analogous to that previously developed for Hartree factorization, ψ†(1)ψ†(2)ψ(3)ψ(4)→ψ†(1)ψ(4) (ψ†(2)ψ(3). The generalized SCF method is reviewed in the context of Gor'kov factorization. The corresponding simple SCF theory is developed to illustrate the method and is shown to give a simple Hamiltonian version of the Bardeen-Cooper-Schrieffer theory of super-conductivity and, more generally, of coherent pairing. The notion of an external pairing field is introduced and the corresponding response functions developed via a formalism like that of Kubo. General fluctuation-dissipation theorems are proved for the response functions. An equation of state is then obtained by a formulation analogous to the dielectric formulation. The insertion of the simple SCF approximation to the response function into the equation of state yields a generalization of the usual low-density (or short-range) evaluation of the grand potential to arbitrary temperatures. Screening the bare interaction with this approximate response function converts the former into the t matrix of the independent-pair approximation. The entire analysis gives an underlying unity to the currently disparate treatments of superconductivity, of long-range correlations, and of short-range correlations.