Functional renormalization for the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation crossover.

We review the functional renormalization group (RG) approach to the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCS-BEC) crossover for an ultracold gas of fermionic atoms. Formulated in terms of a scale-dependent effective action, the functional RG interpolates continuously between the atomic or molecular microphysics and the macroscopic physics on large length scales. We concentrate on the discussion of the phase diagram as a function of the scattering length and the temperature, which is a paradigm example for the non-perturbative power of the functional RG. A systematic derivative expansion provides for both a description of the many-body physics and its expected universal features as well as an accurate account of the few-body physics and the associated BEC and BCS limits.