We study superconductivity and its competition with antiferromagnetism in the ground state of the two-dimensional Hubbard model at weak coupling using the functional renormalization group within a purely fermionic setting. Two methods for the description of superconductivity are presented: i) The continuation of fermionic flows to the superconducting phase, and ii) a combination of renormalization group flows at high scales with mean-field theory for the description of the formation of order at low scales. Continuing renormalization group flows to the superconducting phase, we determine the d-wave superconducting gap as a function of the interaction, the next-nearest neighbor hopping and the fermionic density. Our results reveal the crucial role of the next-nearest neighbor hopping in the competition between antiferromagnetism and superconductivity and suggest the existence of an optimal value of the next-nearest neighbor hopping for pairing. The combination of renormalization group and mean-field theory allows for a simple treatment of competition and coexistence of different orders, and yields superconducting gaps in good agreement with the renormalization group treatment.

Max Planck Inst. for Solid State Research, Stuttgart, Germany