The symplectic-N t-J model and s$_\pm$ superconductors

The possible discovery of $s_\pm$ superconducting gaps in the moderately correlated iron-based superconductors has raised the question of how to properly treat $s_\pm$ gaps in strongly correlated superconductors. Unlike the d-wave cuprates, the Coulomb repulsion does not vanish by symmetry, and a careful treatment is essential. Thus far, only the weak correlation approaches have included this Coulomb pseudopotential, so here we introduce a symplectic N treatment of the t-J model that incorporates the strong Coulomb repulsion through the complete elimination of on-site pairing. Through a proper extension of time-reversal symmetry to the large N limit, symplectic-N is the first superconducting large N solution of the t-J model. For d-wave superconductors, the previous uncontrolled mean field solutions are reproduced, while for $s_\pm$ superconductors, the SU(2) constraint enforcing single occupancy acts as a pair chemical potential adjusting the location of the gap nodes. This adjustment can capture the wide variety of gaps proposed for the iron based superconductors: line and point nodes, as well as two different, but related full gaps on different Fermi surfaces.