Green's functions for fermion systems with pairing correlations

Abstract

The Bogoliubov-Valatin canonical transformation is used to transform the Hamiltonian for a system of interacting fermions, to that for a system of quasi particles in which an important part of the pairing interaction has been absorbed into the free quasi particle. Field theoretic Green's functions, which do not in general conserve the number of quasi particles, are introduced to treat the interaction between the quasi particles. These Green's functions are shown to satisfy an infinite set of coupled integral equations, which can be conveniently represented graphically. The infinite set of equations can be terminated by neglecting the interaction Hamiltonian above some arbitrarily chosen equation. The Green's function equations are then solved in the special case where the sum of quasi particles created and annihilated is four, neglecting self-energy terms, and using a pairing plus Coulomb interaction of the type important in superconductivity. A secular equation for the energy eigenvalues is obtained which is examined for the cases of no pairing interaction and no Coulomb interaction. When the Coulomb interaction alone is present the secular equation becomes that obtained by Sawada et al. for plasmons. When the pairing interaction alone is present, the energy spectrum for excitons, bound pairs of quasi particles, is obtained. It is shown that one of the excitons is a phonon.