We investigate the competing Fermi surface instabilities in the kagome tight-binding model. Specifically, we consider on-site and short-range Hubbard interactions in the vicinity of van Hove filling of the dispersive kagome bands where the fermiology promotes the joint effect of enlarged density of states and nesting. The sublattice interference mechanism devised by Kiesel and Thomale [Phys. Rev. B 86, 121105 (2012)] allows us to explain the intricate interplay between ferromagnetic fluctuations and other ordering tendencies. On the basis of the functional renormalization group used to obtain an adequate low-energy theory description, we discover finite angular momentum spin and charge density wave order, a twofold degenerate d-wave Pomeranchuk instability, and f-wave superconductivity away from van Hove filling. Together, this makes the kagome Hubbard model the prototypical scenario for several unconventional Fermi surface instabilities.

The band structure of graphene exhibits van Hove singularities (VHS) at doping x = ±1/8 away from the Dirac point. Near the VHS, interactions effects, enhanced due to the large density of states, can give rise to various many-body phases at experimentally accessible temperatures. We study the competition between different many-body instabilities in graphene using functional renormalization group (FRG). We predict a rich phase diagram, which, depending on long range hopping as well as screening strength and absolute scale of the Coulomb interaction, contains a d + id-wave superconducting (SC) phase, or a spin density wave phase at the VHS. The d + id state is expected to exhibit quantized charge and spin Hall response, as well as Majorana modes bound to vortices. In the vicinity of the VHS, we find singlet d + id-wave as well as triplet f -wave SC phases.

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, or other Fermi surface instabilities. While this type of phenomena has 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 Fermi surface 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 Fermi surface 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 illustrations of magnetism and, in particular, superconductivity for the iron pnictides from the viewpoint of FRG. Furthermore, it discusses candidate scenarios for topological bulk singlet superconductivity 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 Fermi surface instabilities in future fields of condensed matter research.

We investigate the superconducting phase in the KxBa1−xFe2As2 122 compounds from moderate to strong hole-doping regimes. Using functional renormalization group, we show that while the system develops a nodeless anisotropic s ± order parameter in the moderately doped regime, gapping out the electron pockets at strong hole doping drives the system into a nodal (cos kx+cos ky)(cos kx− cos ky) d-wave superconducting state. This is in accordance with recent experimental evidence from measurements on KFe2As2 which observe a nodal order parameter in the extreme doping regime. The magnetic instability is strongly suppressed.PACS numbers: 74.20.Mn, 74.20.Rp, 74.25.Jb, 74.72.Jb The most elementary questions in the field of ironbased superconductors, such as the symmetry of the order parameter in the superconducting (SC) state, are still under vivid debate. The complexities involve an intricate band structure, a diversity of different material compounds which exhibit sometimes contradictory behavior, and the proximity of various symmetry-broken phases. Due to best single-crystal quality, the most studied pnictide compounds belong to the 122 family such as BaFe 2 As 2 . Their crystal structure is tetragonal I4/mmm, where the Fe and As atoms arrange into layers; the intralayer hybridization is dominant but, unlike other pnictide compounds such as the 1111 family, the inter-layer hybridization is also important. Soon after their discovery [1], the 122 compounds have been synthesized not only with Ba as a substituent between the FeAs layers, but also with K, Cs, and Sr. The SC transition temperatures achieved were up to 37 K [2].The current theoretical opinion on the SC order parameter has converged on a nodeless s ± order parameter that changes sign between the electron (e) pockets and hole (h) pockets. This order parameter comes out of both the strong and the weak-coupling pictures of the iron-based superconductors [3][4][5][6][7], and owes its origin to the pnictide Fermi surface (FS) topology of h pockets at the Γ and e pockets at the X (π, 0)/(0, π) point of the unfolded Brillouin zone. The dominant scattering contributions originate from h pocket scattering at Γ to e pockets at X, yielding the s ± SC order parameter for the doped case and the collinear antiferromagnetic phase in the undoped case. Detailed nesting properties of the pockets, the multi-orbital character of the FS, and the presence or absence of a third h pocket at M (π, π) in the unfolded Brillouin zone complicate this picture. For the 1111 compounds, it was shown that the absence of the M h pocket (whose Fermi level can be significantly tuned by the pnictogen height through replacing As by P) can modify the SC order parameter anisotropy from a nodeless to a nodal s ± phase, which gives the correct material trend for As-P substitution in other pnictide families [7][8][9]. With small exceptions, the anisotropic extended s-wave scenario (and its extension to the nodal s ± ) was consistent with experimental findings for most of the pnictide compounds [10][1...

We put forward a scenario that explains the difference between the order-parameter character in arsenide (As) and phosphorous (P) iron-based superconductors. Using functional renormalization group to analyze it in detail, we find that nodal superconductivity on the electron pockets (hole pocket gaps are always nodeless) can naturally appear when the hole pocket at (π,π) in the unfolded Brillouin zone is absent, as is the case in LaOFeP. There, electron-electron interactions render the gap on the electron pockets softly nodal (of s(±) form). When the pocket of d(xy) orbital character is present, intraorbital interactions with the d(xy) part of the electron Fermi surface drives the superconductivity nodeless.