Microscopic derivation of Ginzburg–Landau theory and the BCS critical temperature shift in general external fields

We consider the Bardeen–Cooper–Schrieffer (BCS) free energy functional with weak and macroscopic external electric and magnetic fields and derive the Ginzburg–Landau functional. We also provide an asymptotic formula for the BCS critical temperature as a function of the external fields. This extends our previous results in Deuchert et al. (Microscopic derivation of Ginzburg-Landau theory and the BCS critical temperature shift in a weak homogeneous magnetic field, PMP 4(1), 1–89 (2023)) for the constant magnetic field to general magnetic fields with a nonzero magnetic flux through the unit cell.