MigdalEliashberg

Migdal-Eliashberg theory is a microscopic framework in condensed matter physics that describes superconductivity in conventional metals arising from electron-phonon interactions. It extends the Bardeen-Cooper-Schrieffer (BCS) theory by incorporating retarded, frequency-dependent interactions and strong-coupling effects, allowing quantitative treatment beyond the weak-coupling limit.

A central idea is Migdal’s theorem, which states that in systems where the phonon frequency is small compared with the electronic Fermi energy, vertex corrections are negligibly small. This justifies a self-consistent treatment of electron-phonon coupling without complex vertex corrections. The theory introduces two key functions: the frequency-dependent superconducting gap Δ(ω) and the mass renormalization function Z(ω) (often computed on the imaginary axis as Z(iωn)). The equations, known as the Eliashberg equations, are a set of coupled integral equations that determine Δ and Z from the electron-phonon spectral function α2F(ω) and the Coulomb pseudopotential μ*.

Inputs and outputs: α2F(ω) encodes the strength and spectral distribution of the electron-phonon interaction, while μ* accounts for screened electron-electron repulsion. Solving the equations yields the superconducting transition temperature Tc, the energy gap, and features such as the isotope effect and strong-coupling deviations from BCS.

Applications and limitations: The theory successfully describes many conventional superconductors (for example Pb, Nb, and Nb3Sn) and provides a framework for interpreting tunneling spectra and gap values. It is primarily applicable to phonon-mediated pairing and can be computationally demanding; it is less suitable for unconventional superconductors where other pairing mechanisms are believed to operate.