  We employ the functional renormalization group to investigate the phase
diagram of the $t-t'$ Hubbard model on the square lattice with finite chemical
potential $\mu$ at zero temperature. A unified scheme to derive flow equations
in the symmetric and symmetry broken regimes allows a consistent continuation
of the renormalization flow in the symmetry broken regimes. At the transition
from the symmetric regime to the symmetry broken regimes, our calculation
reveals leading instabilities in the d-wave superconducting and
antiferromagnetic channels. Furthermore, we find a first order transition
between commensurate and incommensurate antiferromagnetism. In the symmetry
broken regimes our flow equations are able to renormalize around a changing
Fermi surface geometry. We find a coexistence of d-wave superconductivity and
antiferromagnetism at intermediate momentum scales k. However, there is a
mutual tendency of superconductivity and antiferromagnetism to repel each other
at even smaller scales k, which leads to the eradication of the coexistence
phase in the limit of macroscopic scales.
