Generalized Self-Consistent-Field Theory: Gor'kov Factorization

Abstract

Gor'kov factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)〉〈\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉$, is made the basis of a generalized self-consistent-field (SCF) theory precisely analogous to that previously developed for Hartree factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1)\ensuremath{\psi}(4)〉〈({\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)〉$. 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.