Functional renormalization group at finite density and Bose condensation

We discuss the functional renormalization group and pion condensation in the presence of a finite isospin chemical potential $\mu_I$. We calculate the phase diagram as function of temperature $T$ and $\mu_I$. While the exact effective average action is invariant under certain gauge transformations, the effective action in the local-potential approximation is not. As a consequence, the critical chemical potential $\mu_I^c$ for Bose-Einstein condensation at T=0 is no longer equal to the mass of the condensing mode. We discuss possible solutions to this problem.