Symplectic-N, the t-J1-J2 model, and the iron-based superconductors

Symplectic-$N$ has proven to be a useful large $N$ technique for spin Hamiltonians, and here we extend it to treat charge fluctuations by introducing symplectic Hubbard operators. Requiring that the anti-commutator of two symplectic Hubbard operators generate a symplectic spin flip yields a large $N$ generalization of Wen and Lee's $SU(2)$ slave boson Hubbard operators[1]. As a test application to the iron-based superconductors, we solve the symplectic $t-J_1-J_2$ model, motivated by the observation that a 45 degree rotation of the extended s-wave order parameter gives an $s_{\pm}$ order parameter. [1] X.G. Wen and P. Lee, {\it Phys. Rev. Lett.} {\bf 76}, 503-506 (1996).