Driven by the need to understand and determine the presence of nontrivial superconductivity in real candidate materials, we present a generalized set of self-consistent Gor'kov-Hedin-Baym equations with spin-dependent electron-electron and electron-phonon interactions. This extends Hedin's original equations to treat quantum many-body systems where electronic and lattice correlations along with relativistic effects coexist on the same footing with superconductivity. The leading order self-energies yields a generalization of the Migdal-Eliashberg theory and by iterating this set of equations generalized ladder vertex corrections naturally emerge.

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• Spin-orbit coupling
• Topological superconductors
• Green's function methods
• Methods in superconductivity