A model of a 2d non-Fermi liquid with SO(5) symmetry, AF order, and a   d-wave SC gap
  Demanding a consistent quantum field theory description of spin 1/2 particles near a circular Fermi surface in 2d leads to a unique fermionic theory with relevant quartic interactions which has an emergent Lorentz symmetry and automatically has an Sp(4) = SO(5) internal symmetry. The free theory has resistivity linear in temperature. The interacting theory has a low-energy interacting fixed point and is thus a non-Landau/Fermi liquid. Anti-ferromagnetic (AF) and superconducting (SC) order parameters are bilinears in the fields and form the 5-dimensional vector representation of SO(5). An AF phase occurs at low doping which terminates in a first order transition. We incorporate momentum dependent scattering of Cooper pairs near the Fermi surface to 1-loop and derive a new kind of SC gap equation beyond mean field with a d-wave gap solution. Taking into account the renormalization group (RG) scaling properties near the low energy fixed point, we calculate the complete phase diagram as a function of doping, which shows some universal geometric features. The d-wave SC dome terminates on the over-doped side at the fixed point of the RG,which is a quantum critical point. Optimal doping is estimated to occur just below 3/2\pi^2. The critical temperature for SC at optimal doping is set mainly by the universal nodal Fermi velocity and lattice spacing, and is estimated to average around 140K for LSCO. The pseudogap energy scale is identified as the RG scale of the coupling. 
Eliot Kapit and Andr\'e LeClair