Eliashberg equations derived from the functional renormalization group
  We describe how the fermionic functional renormalization group (fRG) flow of a Cooper+forward scattering problem can be continued into the superconducting state. This allows us to reproduce from the fRG flow the fundamental equations of the Eliashberg theory for superconductivity at all temperatures including the symmetry-broken phase. We discuss possible extensions of this approach like the inclusion of vertex corrections. 
Carsten Honerkamp, Manfred Salmhofer