FIG. 6: The low-energy spectrum of the cSDW is obtained from exact diagonalization of 4 × 4 × 2 spin-1/2 moments. The following set of Heisenberg exchange coupling constants are used: J ‖ 1 = 0, J ⊥ 1 > 0, J ‖ 2 = 0 . 3 J ⊥ 1 = J ⊥ 2 , and J 0 = − 15 J ⊥ 1 . The dispersion of these spin-1 excited states resembles the spin-wave spectrum predicted for the cSDW in the conventional J$_{1}$ - J$_{2}$ model, where Hund’s rule is obeyed. 13 , 14 Figure 6 also displays that the groundstate is doubly degenerate, which again is consistent with cSDW order over the square lattice. Last, Fig. 5 shows the low-energy spectrum for Hund’s rule coupling J$_{0}$ = − 1 . 35 J ⊥ 1 near a possible quantum phase transition (cf. Fig. 7). The lowest-energy spin-1 excitations at momentum (0 , 0) and ( π/a, 0) are degenerate, and their dispersion resembles the prediction from linear spin-wave theory that is shown by Fig. 2. Notably, the lowest-energy spin-1 excitation does not “dip down” at momentum ( π/a, π/a ). Table I predicts a transition into hidden order at J$_{0}$ = − 0 . 8 J ⊥ 1 in the classical limit at large spin s , however, which suggests that quantum effects renormalize up the critical Hund’s rule coupling in the physically relevant case of spin-1/2 iron 3 d orbitals. Figure 7 shows the evolution of relevant magnetic order parameters with Hund’s rule 15 -253.2 -253 -252.8 -252.6 -252.4 -252.2 -252 -251.8 (1,1) (0,0) (1,0) (2,0) (2,1) (2,2) ENERGY (J 1 ) MOMENTUM ( /2a) COMMENSURATE SDW AT STRONG HUND’S RULE COUPLING S = 0 S = 1 S = 2