Functional renormalization group for multi-orbital Fermi surface instabilities

Technological progress in material synthesis, as well as artificial realization of condensed matter scenarios via ultra-cold atomic gases in optical lattices or epitaxial growth of thin films, is opening the gate to investigate a plethora of unprecedented strongly correlated electron systems. In a large subclass thereof, a metallic state of layered electrons undergoes an ordering transition below some temperature into unconventional states of matter driven by electronic correlations, such as magnetism, superconductivity (SC), or other Fermi surface (FS) instabilities. While these phenomena have been a well-established direction of research in condensed matter for decades, the variety of today's accessible scenarios pose fundamental new challenges to describe them. A core complication is the multi-orbital nature of the low-energy electronic structure of these systems, such as the multi-d orbital nature of electrons in iron-pnictides and transition-metal oxides in general, but also electronic states of matter on lattices with multiple sites per unit cell such as the honeycomb or kagome lattice. In this review, we propagate the functional renormalization group (FRG) as a suited approach to investigate multi-orbital FS instabilities. The primary goal of the review is to describe the FRG in explicit detail and render it accessible to everyone both at a technical and intuitive level. Summarizing recent progress in the field of multi-orbital FS instabilities, we illustrate how the unbiased fashion by which the FRG treats all kinds of ordering tendencies guarantees an adequate description of electronic phase diagrams and often allows to obtain parameter trends of sufficient accuracy to make qualitative predictions for experiments. This review includes detailed and illustrative examples of magnetism and, in particular, SC for the iron-pnictides from the viewpoint of FRG. Furthermore, it discusses candidate scenarios for topological bulk singlet SC and exotic particle–hole condensates on hexagonal lattices such as sodium-doped cobaltates, graphene doped to van-Hove filling, and the kagome Hubbard model. In total, the FRG promises to be one of the most versatile and revealing numerical approaches to address unconventional FS instabilities in future fields of condensed matter research.