The reduced Hartree-Fock model with self-generated magnetic fields

We study the well-posedness of the reduced Hartree-Fock model for molecules and perfect crystals when taking into account a self-generated magnetic field. We exhibit a critical value α_c > 0 such that if the fine structure constant α is smaller than α_c, then the corresponding system is stable, whereas if α is greater than α_c, it is unstable. We give an explicit characterization of α_c as a minimization problem over the set of zero-modes, and we prove that the critical values for the molecular case and the periodic case coincide. Finally, we prove the existence of minimizers when the system is neutral or positively charged.