Self-consistent $T$-matrix theory of superconductivity

Using principles of the Fadeev-Lovelace-Watson multiple scattering expansion, a $T$-matrix approximation is derived which coincides with the Galitskii-Feynman $T$ matrix in the normal state and yields the gap in the superconducting state. Unlike other $T$-matrix approaches, the theory satisfies not only the self-consistent Thouless criterion but also the Baym-Kadanoff conditions for a conserving theory in equilibrium. In single-mode approximation it simplifies to the Eliashberg theory.

Figure 1

Condensate-mediated self-interaction. Arrows represent bare fermionic Green’s functions $G_0$ and wavy lines are boson-mediated interactions. (a) Schematic picture of four interacting particles of initial momenta $k$, $p$, $q$, and $m$. Due to the Pauli principle all particles are in different states, in particular $m\ne k$ and $q\ne k$, and $p\ne m$. (b),(c) Corresponding Feynman diagrams for the Green’s function of the particle $k$ and for the thermodynamical potential. For lines closed into loops momenta are summed over with no restriction.

Figure 2

$T$-matrix approximations in diagrams. Both approximations have a self-energy constructed from the many-body $T$ matrix. The interaction carried by boson propagators, shown by wavy lines, is included in the ladder approximation. Thick arrows are self-energy dressed Green’s functions, while thin arrows are bare Green’s functions.