Functional renormalization group for fermion lattice models in three dimensions: Application to the Hubbard model on the cubic lattice

The channel-decomposed functional renormalization group (FRG) approach, most recently presented in the variant of truncated-unity FRG (TUFRG), has so far been used for various two-dimensional model systems. Yet for many interesting material systems the third spatial dimension is of clear relevance. Therefore, FRG schemes working in three spatial dimensions (3D) are definitely on the wish list. Here we demonstrate that a 3D TUFRG scheme can be set up in a straightforward extension of previous two-dimensional codes and gives physically sensible results with affordable numerical effort, regarding both the qualitative and quantitative descriptions. The computed phase diagram of the three-dimensional Hubbard model at half filling or perfect nesting shows a phase transition to a (pi, pi, pi)-ordered antiferromagnetic ground state for repulsive interactions at an energy scale that compares well with other numerical approaches in the literature. Furthermore, the method allowed us to detect a d-wave pairing and a concurring (pi, pi, 0) antiferromagnetic ground state in the hole-doped Hubbard model.