Functional renormalization group for fermions in three dimensions

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The goal of this theory project is to devise and apply a functional renormalization group (fRG) scheme for interacting electrons in three-dimensional (3D) lattices. Over the last two decades, fRG has been used to explore interaction effects in a multitude of zero-, one- and two-dimensional model systems, but 3D systems were essentially out of range because of numerical limitations. Particular advantages of the fRG are its unbiasedness as to what the dominant interaction tendencies at low scales are, and that the functional nature of the flow equations allows one to obtain a rather detailed picture of the effective low-energy physics, e.g. for the effective interactions. These strengths would of course also be helpful in the analysis of 3D interacting electron systems. Recent advances in the fRG, most importantly a technique called vertex decomposition, have made the description of the effective interaction much more efficient, both regarding numerical aspects as well as for the physical understanding of its structure. Hence, new extensions can be addressed. Here, we plan to develop vertex decomposition techniques for general three-dimensional interacting electron systems and to apply them to important model cases. As a first application of the 3D fRG approach, we envisage the study of linear and quadratic band crossing point models that have recently been considered as hosting non-Fermi liquid behavior, unconventional charge screening and interaction-induced topological states.