Ultracold Quantum Gases and the Functional Renormalization Group

Ultracold quantum gases provide an ideal platform to study the transition form micro- to macrophysics both in experiment and theory. In this lecture series we start on the microscopic level and explain the scale separations which underly the precise modeling in atomic ensembles. We then move to larger length scales and address the key many-body quantum condensation phenomena in weakly interacting bosonic and fermionic systems, Bose-Einstein condensation (BEC) and Bardeen-Cooper-Schrieffer (BCS) superfluidity. Going one step beyond, two-body Feshbach resonances allow for a tuning of the microscopic interactions in a controlled way. On the many-body level, this gives rise to the 'BCS-BEC crossover' in fermionic systems, smoothly connecting the two seemingly separate condensation scenarios of BCS-type superfluidity and a (molecular) BEC by passing through a regime of strong interactions. We will then see how functional renormalization group equations can provide a unified picture for the whole phase diagram of the BCS-BEC crossover. In particular, it allows us to resolve physics at very different scales. This ranges from the emergence of an effective microscopic bosonic theory from fundamental fermionic constituents over many-body in-medium effects on intermediate length scales relevant to thermodynamics, to the universal long-wavelength critical behavior close to the finite temperature phase transition. The approach also explains the universality - the insensitivity to microphysical details - of many-body observables close to broad Feshbach resonances. (author)