The BCS - BEC Crossover in Ultracold Fermion Gases
The continuous crossover between a Bardeen-Cooper-Schrieffer (BCS)-type superfluid of fermion pairs and a Bose-Einstein condensate (BEC) of tightly bound bosonic molecules can be attributed to the spontaneous breaking of global $U(1)$ gauge symmetry which underlies both quantum condensation phenomena. Recently much attention has been paid to this problem, since Feshbach resonances allow for an experimental implementation of crossover physics in cold fermion gases. The strong interactions close to resonance call for an analysis beyond Mean Field Theory. We develop a systematic functional integral approach for the description of this phenomenon. Starting from a Yukawa-type atom-molecule model, a symmetry analysis allows to both construct the equation of state and to classify the thermodynamic phases in a unified way. The onset of superfluidity is signalled by the emergence of a massless Goldstone mode associated with the broken continuous U(1) symmetry. Beyond Mean Field, we include fluctuations of the molecule field self-consistently via the solution of suitable Schwinger-Dyson equations. The phase diagram is computed, and a variety of universal features are established. A new form of crossover from an exactly solvable narrow resonance limit to broad resonances or pointlike interactions is found. At low temperature our results agree well with quantum Monte Carlo simulations and recent experiments. Our approach is further developed in the frame of functional renormalization group equations. While the effective bosonic theory in the BEC regime shows the characteristics of a Bogoliubov theory for small temperatures, the phase transition is of second order.
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