Microscopic derivation of Ginzburg-Landau equations for coexistent states of superconductivity and magnetism

Ginzburg-Landau (GL) equations for the coexistent states of superconductivity and magnetism are derived microscopically from the extended Hubbard model with on-site repulsive and nearest-neighbor attractive interactions. In the derived GL free energy a cubic term that couples the spin-singlet and spin-triplet components of superconducting order parameters (SCOP) with magnetization exists. This term gives rise to a spin-triplet SCOP near the interface between a spin-singlet superconductor and a ferromagnet, consistent with previous theoretical studies based on the Bogoliubov de Gennes method and the quasiclassical Green's function theory. In coexistent states of singlet superconductivity and antiferromagnetism it leads to the occurrence of pi-triplet SCOPs.