Consistent combination of truncated-unity functional renormalization group and mean-field theory

We propose a novel scheme for combining efficiently the truncated-unity functional renormalization group (TUFRG) and the mean-field theory. It follows a method of Wang, Eberlein and Metzner that uses only the two-particle irreducible part of the vertex as an input for the mean-field treatment. In the TUFRG, the neglect of fluctuation effects from other channels in the symmetry-broken regime is represented by applying the random phase approximation (RPA) in each individual channel, below the divergence scale. Then the Bethe-Salpeter equation for the four-point vertex is translated into the RPA matrix equations for the bosonic propagators that relates the singular and irreducible singular modes of the propagators. The universal symmetries for the irreducible singular modes are obtained from the antisymmetry of Grassmann variables. The mean-field equation based on these modes is derived by the saddle-point approximation in the framework of the path-integral formalism. By using our scheme, the power of the TUFRG, as an unbiased tool for identifying the many-body instabilities, could be elevated to a quantitative level, and its application would be extended to a quantitative analysis of the coexisting orders. As an illustration we employ this scheme to study the coexistence phase of the chiral superconductivity and the chiral spin-density wave, predicted near van Hove filling of the honeycomb lattice.