Ab initio theory of superconductivity in a magnetic field. I. Spin density functional theory for superconductors and Eliashberg equations

We present a first-principles approach to describe magnetic and superconducting systems and the phenomena of competition between these electronic effects. We develop a density functional theory SpinSCDFT by extending the Hohenberg-Kohn theorem and constructing the noninteracting Kohn-Sham system. An exchange-correlation functional for SpinSCDFT is derived from the Sham-Schlüter connection between the SpinSCDFT Kohn-Sham and a self-energy in Eliashberg approximation. The reference Eliashberg equations for superconductors in the presence of magnetism are also derived and discussed.