Universality in the BCS - BEC Crossover in Cold Fermion Gases

There are two cornerstones for the description of quantum condensation phenomena: Bose-Einstein condensation (BEC) in bosonic systems and BCSpairing for fermions. The underlying pictures for these phenomena are quite different – BEC is the macroscopic population of a single quantum state, while the BCS mechanism relies on the formation of Cooper pairs. However, both phenomena share a decisive feature in common: they can be described as the spontaneous breaking of the global symmetry of phase rotations, U(1). Due to this similarity, it is plausible that the two different scenarios sketched above are indeed connected by some smooth transition or crossover [1, 2, 3]. In a simple physical picture, the position-space delocalized Cooper pairs characteristic for the BCS regime undergo a localization process throughout the crossover, ending up as effectively pointlike bosonic particles or strongly bound molecules in the BEC regime. At this point ultracold atoms come into play. The presence of Feshbach resonances in fermionic gases such as Li or K offers the unique possibility to tune the interaction strength between the atoms to arbitrary positive and negative values, thereby allowing for an experimental implementation of the crossover [4, 5, 6, 7, 8, 9]. In this contribution , we address the crossover problem based on an atommolecule model which is appropriate for a realistic description of the crossover in cold fermion gases. In sect. 2 we summarize the ingredients of the functional integral formalism developed in refs. [10, 11]. In sect. 3 we discuss universal aspects of the phase diagram encoded in the atom-molecule model, with special emphasis on an additional form of crossover interpolating between universal narrow to broad resonance limits. While the narrow resonance limit describing a situation with large effective range can be solved exactly, broad reso-