The Phase Diagram of Compact QED Coupled to a Four-Fermi Interaction
  Compact lattice Quantum Electrodynamics (QED) with four species of fermions is simulated with massless quarks by using the $\chi$QED scheme of adding a four-fermi interaction to the action. Simulations directly in the chiral limit of massless quarks are done with high statistics on $8^4$, and $16^4$ lattices, and the phase diagram, parameterized by the gauge and the four-fermi couplings, is mapped out. The line of monopole condensation transitions is separate from the line of chiral symmetry restoration. The simulation results indicate that the monopole condensation transition is first order while the chiral transition is second order. The challenges in determining the Universality class of the chiral transition are discussed. If the scaling region for the chiral transition is sufficiently wide, the $16^4$ simulations predict critical indices far from mean field values. We discuss a speculative scenario in which anti-screening provided by double-helix strands of monopole and anti-monopole loops are the agent that balances the screening of fermion anti-fermion pairs to produce an ultra-violet fixed point in the electric coupling. 
John B. Kogut and Costas G. Strouthos