The electronic orders in Hubbard models on a Kagome lattice at van Hove filling are of intense current interest and debate. We study this issue using the singular-mode functional renormalization group theory. We discover a rich variety of electronic instabilities under short range interactions. With increasing on-site repulsion U , the system develops successively ferromagnetism, intra unitcell antiferromagnetism, and charge bond order. With nearest-neighbor Coulomb interaction V alone (U = 0), the system develops intra-unit-cell charge density wave order for small V , s−wave superconductivity for moderate V , and the charge density wave order appears again for even larger V . With both U and V , we also find spin bond order and chiral d x 2 −y 2 + idxy superconductivity in some particular regimes of the phase diagram. We find that the s-wave superconductivity is a result of charge density wave fluctuations and the squared logarithmic divergence in the pairing susceptibility. On the other hand, the d-wave superconductivity follows from bond order fluctuations that avoid the matrix element effect. The phase diagram is vastly different from that in honeycomb lattices because of the geometrical frustration in the Kagome lattice.

We study the electronic instabilities of near 1/4 electron doped graphene using the singular-mode functional renormalization group, with a self-adaptive k-mesh to improve the treatment of the van Hove singularities, and variational Monte-Carlo method. At 1/4 doping the system is a chiral spin density wave state exhibiting the anomalous quantized Hall effect. When the doping drops below 1/4, the d x 2 −y 2 + idxy Cooper pairing becomes the leading instability. Our results suggest that near 1/4 electron-or hole-doping (away from the neutral point) the graphene is either a Chern insulator or a topoligical superconductor.

In several recent experiments the superconducting gap of a single-unit-cell-thick FeSe film on SrTiO 3 substrate has been observed by scanning tunneling spectroscopy and angle-resolved photoemission spectroscopy. The value of the superconducting gap is about nine times larger than that of the bulk FeSe under ambient pressure, suggesting a much higher pairing energy scale and T c than all other iron-based superconductors and thus calling for a better understanding of its superconducting mechanism. In this paper we study the effects of screening due to the SrTiO 3 ferroelectric phonons on Cooper pairing in FeSe. We conclude that it can significantly enhance the energy scale of Cooper pairing and even change the pairing symmetry. Our results also raise some concerns on whether phonons can be completely ignored for bulk iron-based superconductors.