Ab initio theory of magnetic-field-induced odd-frequency two-band superconductivity in MgB2

We develop the anisotropic Eliashberg framework for superconductivity in the presence of an applied magnetic field. Using as input the ab initio calculated electron and phonon band structures and electron-phonon coupling, we solve self-consistently the anisotropic Eliashberg equations for the archetypal superconductor MgB2. We find two self-consistent solutions, time-even two-band superconductivity, as well as unconventional time-odd s-wave spin triplet two-band superconductivity emerging with applied field. We provide the full momentum, frequency, and spin-resolved dependence and magnetic field-temperature phase diagrams of the time-even and time-odd superconducting pair amplitudes and predict fingerprints of this novel odd-frequency state in tunneling experiments.