Symplectic N and time reversal in frustrated magnetism

Identifying the time-reversal symmetry of spins as a symplectic symmetry, we develop a large N approximation for quantum magnetism that embraces both antiferromagnetism and ferromagnetism. In SU(N), where N>2, not all spins invert under time reversal, so we have introduced a large N treatment that builds interactions exclusively out of the symplectic subgroup [SP(N)] of time-reversing spins, a more stringent condition than the symplectic symmetry of previous SP(N) large N treatments. As a result, we obtain a mean-field theory that incorporates the energy cost of frustrated bonds. When applied to the frustrated square lattice, the ferromagnetic bonds restore the frustration dependence of the critical spin in the Néel phase and recover the correct frustration dependence of the finite temperature Ising transition.