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We employ the functional renormalization group to investigate the phasediagram of the $t-t'$ Hubbard model on the square lattice with finite chemicalpotential $\mu$ at zero temperature. A unified scheme to derive flow equationsin the symmetric and symmetry broken regimes allows a consistent continuationof the renormalization flow in the symmetry broken regimes. At the transitionfrom the symmetric regime to the symmetry broken regimes, our calculationreveals leading instabilities in the d-wave superconducting andantiferromagnetic channels. Furthermore, we find a first order transitionbetween commensurate and incommensurate antiferromagnetism. In the symmetrybroken regimes our flow equations are able to renormalize around a changingFermi surface geometry. We find a coexistence of d-wave superconductivity andantiferromagnetism at intermediate momentum scales k. However, there is amutual tendency of superconductivity and antiferromagnetism to repel each otherat even smaller scales k, which leads to the eradication of the coexistencephase in the limit of macroscopic scales.

Exploring the Hubbard Model on the Square Lattice at Zero Temperature with a Bosonized Functional Renormalization Approach