Translation-invariant quasi-free states for fermionic systems and the BCS approximation (Q2920917)

The paper derives a gap equation for the extended theory with direct and exchange terms and investigates the existence of non-trivial solutions for general interaction potentials. The Bardeen-Cooper-Schrieffer (BCS) theory restricts the allowed states of the system of fermionic gases with local pair interactions at low temperature to quasi-free states assuming translation-invariance and \(\mathrm{SU}(2)\) rotation invariance and dismissing the direct and exchange terms in the energy. The resulting BCS functional depends on the temperature, the chemical potential, the interaction potential, the momentum distribution and the Cooper pair wave function \(\alpha\). The characterization of a superfluid phase is motivated by the existence of a minimizer of the BCS functional for which \(\alpha \neq 0\). The paper focuses on the question to what extent it is justifiable to dismiss the direct and exchange terms in the energy. The authors give a rigorous justification for dismissing the two terms for potentials whose range \(\ell\) is short compared to the scattering length and the Fermi wave length. It is shown that for small enough \(\ell\), the system still can be described by the conventional BCS equation if the chemical potential is renormalized appropriately. In the limit \(\ell \to 0\), the spectral gap function converges to a constant function and the authors recover the BCS equation in the usual form. The paper represents the first proof of the existence of pairing in a translation-invariant Bogoliubov-Hartree-Fock (HF) model in the continuum. In the model, the BCS-HF functional is considered, whose infimum over all states describes the negative of the pressure of the system. The system is in a superfluid phase if the minimum of the functional does not attain at a normal state, which is unstable in this case. First, the existence of a superfluid phase for a large class of interaction potentials is characterized. The sufficient conditions on the interaction potentials are found for the BCS-HF functional to have a minimizer. The main result concerns the case of short-range interaction potentials, where the monotonicity in the spectral gap function and the correct definition of the critical temperature is recovered.