  Starting from the Bardeen-Cooper-Schrieffer (BCS) free energy functional, we
derive the Ginzburg-Landau functional in the presence of a weak homogeneous
magnetic field. We also provide an asymptotic formula for the BCS critical
temperature as a function of the magnetic field. This extends the previous
works arXiv:1102.4001 and arXiv:1410.2352 of Frank, Hainzl, Seiringer and
Solovej to the case of external magnetic fields with non-vanishing magnetic
flux through the unit cell.
