The phases with spontaneously broken symmetries corresponding to antiferromagnetic and $d$-wave superconducting order in the two-dimensional $t-t^{\text{'}}$-Hubbard model are investigated by means of the functional renormalization group. The introduction of composite boson fields in the magnetic, charge density, and superconducting channels allows an efficient parametrization of the four-fermion vertex and the study of regimes where either the antiferromagnetic or superconducting order parameter, or both, are nonzero. We compute the phase diagram and the temperature dependence of the order parameter below the critical temperature, where antiferromagnetic and superconducting order show a tendency of mutual exclusion.

• Received 12 January 2011

DOI:https://doi.org/10.1103/PhysRevB.83.155125