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The channel-decomposed functional renormalization group (FRG) approach, mostrecently in the variant of truncated-unity-(TU-)FRG, has so far been used forvarious two-dimensional model systems. Yet, for many interesting materialsystems the third spatial dimension is of clear relevance. Therefore FRGschemes working in three spatial dimensions (3D) are definitely on thewishlist. Here we demonstrate that a 3D TUFRG scheme can be set up instraightforward extension of previous 2D codes and gives physically sensibleresults with affordable numerical effort, both regarding the qualitative aswell as the quantitative description. The computed phase diagram of thethree-dimensional Hubbard model at half filling or perfect nesting shows aphase transition to a \((\pi,\pi,\pi)\)-ordered antiferromagnetic ground statefor repulsive interactions at an energy scale that compares well with othernumerical approaches in the literature. Furthermore, the method allowed us todetect a \(d\)-wave pairing and a concurring \((\pi,\pi,0)\) antiferromagneticground state in the hole doped Hubbard model.

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