We present a renormalization group (RG) analysis of a fermionic hot spot model of interacting electrons on the square lattice. We truncate the Fermi surface excitations to linearly dispersing quasiparticles in the vicinity of eight hot spots on the F
ermi surface, with each hot spot separated from another by the wavevector $(pi, pi)$. This motivated by the importance of these Fermi surface locations to the onset of antiferromagnetic order; however, we allow for all possible quartic interactions between the fermions, and also for all possible ordering instabilities. We compute the RG equations for our model, which depend on whether the hot spots are perfectly nested or not, and relate our results to earlier models. We also compute the RG flow of the relevant order parameters for both Hubbard and $J$, $V$ interactions, and present our results for the dominant instabilities in the nested and non-nested cases. In particular, we find that non-nested hot spots with $J$, $V$ interactions have competing singlet $d_{x^2-y^2}$ superconducting and $d$-form factor incommensurate density wave instabilities. We also investigate the enhancement of incommensurate density waves near experimentally observed wavevectors, and find dominant $d$-form factor enhancement for a range of couplings.
We theoretically study an exactly solvable Gamma matrix generalization of the Kitaev spin model on the ruby lattice, which is a honeycomb lattice with expanded vertices and links. We find this model displays an exceptionally rich phase diagram that i
ncludes: (i) gapless phases with stable spin fermi surfaces, (ii) gapless phases with low-energy Dirac cones and quadratic band touching points, and (iii) gapped phases with finite Chern numbers possessing the values {pm}4,{pm}3,{pm}2 and {pm}1. The model is then generalized to include Ising-like interactions that break the exact solvability of the model in a controlled manner. When these terms are dominant, they lead to a trivial Ising ordered phase which is shown to be adiabatically connected to a large coupling limit of the exactly solvable phase. In the limit when these interactions are weak, we treat them within mean-field theory and present the resulting phase diagrams. We discuss the nature of the transitions between various phases. Our results highlight the richness of possible ground states in closely related magnetic systems.
