Gor'kov-Hedin-Baym Equations for Quantum Many-Body Systems with Spin-Dependent Interactions

Abstract

Driven by the need to understand and determine the presence of non-trivial 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.

Figures (8)

• Figure 1: (color online) Schematic of self-consistent cycle for (a) the full set of Gor'kov-Hedin equations, (b) the GW approximation, and where the vertex is restricted to (c) phonon or (d) electron degrees of freedom.
• Figure 2: (color online) Diagrammatic representation of the various contributions to the Hartree potential in the harmonic approximation. (a) electronic Hartree self-energy, (b) clamped-ion potential, and (c) Debye-Waller self-energy.
• Figure 3: (color online) Diagrammatic representation of the (a) electron-hole and (b) Cooper pair breaking excitations within the RPA that contribute to the electronic polarizability. A schematic of the electronic band diagrams of a (a) semiconductor and a (b) superconductor are shown above to illustrate each associated excitation process.
• Figure 4: (color online) Diagrammatic representation of the GW self-energy: (a) electronic, and (b) clamped-ion and (c) Fan-Migdal phonon contributions to the self-energ. (d)-(f) Illustrate the effect of spin-dependent interactions on a spin up electron for both (d) conventional and (e)-(f) anomalous components of the self-energy (see text for details). The dashed teal lines represent either the electronic $(w_{e})$ or phonon $(w_{ph})$ screened interactions.
• Figure 5: (color online) Diagrammatic representation of the exact self-energy with contributions from the (a) Hartree, (b) Fock/GW, and (c) effective interactions $\Gamma$ explicitly shown.