Low density limit of BCS theory and Bose-Einstein condensation of fermion pairs
This paper considers the low-density limit in the case where the interparticle interaction does allow for two-particle bound states assuming that the Bardeen-Cooper-Schrieffer (BCS) theory approximation is correct. The authors consider a macroscopic sample of a system of spin \(1/2\) fermions at zero temperature. The treatment is restricted to three spatial dimensions, but the analysis is applied to any dimension. The interaction among the fermions is described by a local two-body potential and the particles are subjected to external electric and/or magnetic fields. It is shown that the macroscopic variations in the pair density are, to leading order, correctly described by the Gross-Pitaevskii (GP) functional describing a Bose-Einstein condensate of fermion pairs with an effective repulsive interaction. By allowing for a large class of possible interactions between the particles, a sole assumption concerns the existence of two-particle bound states. The existence of these states is very important because, in their absence, one obtains an ideal Fermi gas in the low-density limit, up to exponentially small corrections. The authors consider the effect of weak and slowly varying external fields which due to external magnetic and electric potentials are written via a small parameter \(h\). First, the rescaled BCS functional is written and the BCS state of the system is a minimizer of this functional over all admissible states. Then, the GP functional is defined and the relation between the BCS energy and the GP energy is stated for small parameter \(h\) in the frame of the main theorem of the paper. Semiclassical estimates needed in the proof of the main theorem are stated; they are very important to obtain the bounds in the main theorem. It is shown that they are an extension to zero temperature of the similar expressions at positive temperature obtained by the authors before. An upper bound on the BCS energy is obtained by using the variational principle, and then the lower bound is stated. Moreover, the structure of approximate minimizers is investigated which leads to a definition of the order parameter. The proof, presented in the paper, follows closely the proof of the theorem under similar considerations at positive temperature given by the authors before.
scientific article; zbMATH DE number 6084514
scientific article